text: ∈ S R ∈ S R ∈ S R Proposition 14. Let f, g ( d), σ ( 2d), a ( 2d). Then (cid:101) (cid:101) (cid:101) (cid:101) ∗ ∗ ∗ a) T a = W a σ, where W a σ is the Weyl operator with symbol a σ σ − ∈ R (and σ(z) = σ( z), z 2d). ∗ b) (T a) = T a, i.e. the adjoint of a Cohen operator is the Cohen σ σ operator corresponding to conjugated sesq(cid:101)uilinear form and symbol. (cid:0) (cid:1) ⊗ F− ∗ c) (T af, g) = (k, f g), i.e. k = A 1[a σ] is the Schwartz kernel σ 2 (cid:55)→ − F of the operator T a, where A : φ(x, t) φ x+t, x t , and is the σ 2 2 Fourier transform with respect to the second half of variables. We omit the proof which is a straightforward computation. The previous proposition leads to an interesting general view on the behavior of the diﬀerent rules of association symbol-operator, improperly called “quantizations”. By (a) all quantizations can be seen as “deforma- tion” of the Weyl quantization where the symbol at ﬁrst undergoes a ﬁltering by the convolution with a ﬁxed kernel σ. In absence of ﬁltering, i.e. when σ = δ, we have the “pure” Weyl quantization. Many characteristics of the quantization can be then read from the Co- hen kernel σ. For example it is natural to ask whether the correspondence → a T a is a quantization in the “classical” sense, i.e. it assigns self-adjoint σ operators to real symbols. We see from (b) that this happens if and only if the kernel σ is itself real. The fact that the Weyl correspondence is a quantization in the classical sense corresponds to the fact that the Dirac δ ∈ S(cid:48) R is real (a distribution u ( 2d) is “real” if its value on real-valued test functions is real). Furthermore, we see from (c) that the correspondence between Schwartz kernels and operators is nothing else than formula (3) where, as sesquilinear ⊗ form, is taken the skew-tensor product f g, which is not in the Cohen class. An explicit computation shows that the expression of the operator T a σ acting on a(cid:90)f(cid:90)unction f is (cid:18) (cid:19) − − t + u − T af(t) = e2πiξ(t u)a(x, ω)σ x , ω ξ f(u) dx dω du dξ, σ R ×R 2 2d 2d (20) in particular we remark that it is a classical pseudo-diﬀerential operator 16 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S

text: R

text: S

text: R

text: S

text: R

text: Proposition

text: Let

text: f,

text: g

text: d),

text: σ

text: a

text: Then

text: a)

text: T

text: a

text: W

text: a

text: σ,

text: where

text: W

text: a

text: σ

text: is

text: the

text: Weyl

text: operator

text: with

text: symbol

text: a

text: σ

text: σ

text: R

text: (and

text: σ(z)

text: σ(

text: z),

text: z

text: b)

text: (T

text: a)

text: T

text: a,

text: i.e.

text: the

text: adjoint

text: of

text: a

text: Cohen

text: operator

text: is

text: the

text: Cohen

text: σ

text: σ

text: operator

text: corresponding

text: to

text: conjugated

text: form

text: and

text: symbol.

text: F−

text: c)

text: (T

text: af,

text: g)

text: (k,

text: f

text: g),

text: i.e.

text: k

text: A

text: σ]

text: is

text: the

text: Schwartz

text: kernel

text: σ

text: F

text: of

text: the

text: operator

text: T

text: a,

text: where

text: A

text: φ(x,

text: t)

text: φ

text: x+t,

text: x

text: t

text: and

text: is

text: the

text: σ

text: Fourier

text: transform

text: with

text: respect

text: to

text: the

text: second

text: half

text: of

text: variables.

text: We

text: omit

text: the

text: proof

text: which

text: is

text: a

text: straightforward

text: computation.

text: The

text: previous

text: proposition

text: leads

text: to

text: an

text: interesting

text: general

text: view

text: on

text: the

text: behavior

text: of

text: the

text: diﬀerent

text: rules

text: of

text: association

text: symbol-operator,

text: improperly

text: called

text: “quantizations”.

text: By

text: (a)

text: all

text: quantizations

text: can

text: be

text: seen

text: as

text: “deforma-

text: tion”

text: of

text: the

text: Weyl

text: quantization

text: where

text: the

text: symbol

text: at

text: ﬁrst

text: undergoes

text: a

text: ﬁltering

text: by

text: the

text: convolution

text: with

text: a

text: ﬁxed

text: kernel

text: σ.

text: In

text: absence

text: of

text: ﬁltering,

text: i.e.

text: when

text: σ

text: δ,

text: we

text: have

text: the

text: “pure”

text: Weyl

text: quantization.

text: Many

text: characteristics

text: of

text: the

text: quantization

text: can

text: be

text: then

text: read

text: from

text: the

text: Co-

text: hen

text: kernel

text: σ.

text: For

text: example

text: it

text: is

text: natural

text: to

text: ask

text: whether

text: the

text: correspondence

text: a

text: T

text: a

text: is

text: a

text: quantization

text: in

text: the

text: “classical”

text: sense,

text: i.e.

text: it

text: assigns

text: self-adjoint

text: σ

text: operators

text: to

text: real

text: symbols.

text: We

text: see

text: from

text: (b)

text: that

text: this

text: happens

text: if

text: and

text: only

text: if

text: the

text: kernel

text: σ

text: is

text: itself

text: real.

text: The

text: fact

text: that

text: the

text: Weyl

text: correspondence

text: is

text: a

text: quantization

text: in

text: the

text: classical

text: sense

text: corresponds

text: to

text: the

text: fact

text: that

text: the

text: Dirac

text: δ

text: R

text: is

text: real

text: (a

text: distribution

text: u

text: is

text: “real”

text: if

text: its

text: value

text: on

text: real-valued

text: test

text: functions

text: is

text: real).

text: Furthermore,

text: we

text: see

text: from

text: (c)

text: that

text: the

text: correspondence

text: between

text: Schwartz

text: kernels

text: and

text: operators

text: is

text: nothing

text: else

text: than

text: formula

text: where,

text: as

text: sesquilinear

text: form,

text: is

text: taken

text: the

text: skew-tensor

text: product

text: f

text: g,

text: which

text: is

text: not

text: in

text: the

text: Cohen

text: class.

text: An

text: explicit

text: computation

text: shows

text: that

text: the

text: expression

text: of

text: the

text: operator

text: T

text: a

text: σ

text: acting

text: on

text: f

text: is

text: t

text: u

text: T

text: af(t)

text: u)a(x,

text: ω)σ

text: x

text: ω

text: ξ

text: f(u)

text: dx

text: dω

text: du

text: dξ,

text: σ

text: R

text: ×R

text: in

text: particular

text: we

text: remark

text: that

text: it

text: is

text: a

text: classical

text: pseudo-diﬀerential

text: operator

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: We also remark that similar types of operators are considered in [1] where boundedness and Schatten-Von Neuman results are obtained. However the estimates in [1] are not given in terms of the Cohen kernel and the type of operators considered there can not cover the totality of the Cohen class operators. Actually the sesquilinear forms associated with the operators in [1] enjoy the following uncertainty principle: if their support has ﬁnite measure then at least one of the two entries is null, ([1], Thm. 1.4.3) and this is not shared by all the sesquilinear forms in the Cohen class, see Example 5 in [8]. 5 Donoho-Stark UP within the Cohen Class In this section we give a new formulation of the classical Donoho-Stark UP in the context of the Cohen class, by using the boundedness results founded previously. Consider the operator T a associated with a generic time-frequency rep- σ resentation in the Cohen class, cf. (3). We have the following result, that is the analogous for Cohen operators of Theorem 12. R Theorem 17. Let T, Ω be measurable sets in d. Let χ be the function (cid:82) T χ ×R (x, ω) = χ (x) and χ be the function χR × (x, ω) = χ (ω), with cor- T d (cid:82) T Ω d Ω Ω responding operators T χT and T χΩ associated with a generic representation σ σ − Qσ. We suppose that the ke−rnel σ is such that F1(t) = T×Rd σ(x t, ω)dxdω and F2((cid:107)ω) (cid:107)= R≤d×Ω σ(x, η ω) dx dη are (real) non negative function sat- isfying F ∞ 1, j = 1, 2. Assume that T χT and T χΩ satisfy the ε- j σ σ concentration condition, i.e., (cid:107) (cid:107) ≥ − (cid:107) (cid:107) (cid:107) (cid:107) ≥ − (cid:107) (cid:107) T χT f 2 (1 ε2 ) f 2 and T χΩf 2 (1 ε2 ) f 2; σ 2 T 2 σ 2 Ω 2 moreover, suppose that {(cid:107) (cid:107) (cid:107) (cid:107) } ≤ (cid:82)min T χT , T χΩ (cid:82) 1. (23) σ B(L2) σ B(L2) − − Deﬁne G1(t) = ∈Rd σ(∞t, ω) dω and G2(ξ) = Rd σ(x, ξ) dx. Then for every s , s , p , p , r [1, ] satisfying 1/s + 1/p = 1 + 1/2r, j = 1, 2, we 1 2 1 2 j j have (cid:18) (cid:19) | | | | ≥ − − 1 T 1/s1 Ω 1/s2 (1 ε ε ) . T Ω d (cid:107) (cid:107) (cid:107) (cid:107) A A A A A A2 G G s1 s2 p1 p2 r2+r1 2r2−r1 1 p1 2 p2 18 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: We

label: paragraph

text: also

text: remark

text: that

text: similar

text: types

text: of

text: operators

text: are

text: considered

text: in

text: where

text: boundedness

text: and

text: Schatten-Von

text: Neuman

text: results

text: are

text: obtained.

text: However

text: the

text: estimates

text: in

text: are

text: not

text: given

text: in

text: terms

text: of

text: the

text: Cohen

text: kernel

text: and

text: the

text: type

text: of

text: operators

text: considered

text: there

text: can

text: not

text: cover

text: the

text: totality

text: of

text: the

text: Cohen

text: class

text: operators.

text: Actually

text: the

text: sesquilinear

text: forms

text: associated

text: with

text: the

text: operators

text: in

text: enjoy

text: the

text: following

text: uncertainty

text: principle:

text: if

text: their

text: support

text: has

text: ﬁnite

text: measure

text: then

text: at

text: least

text: one

text: of

text: the

text: two

text: entries

text: is

text: null,

text: Thm.

text: and

text: this

text: is

text: not

text: shared

text: by

text: all

text: the

text: sesquilinear

text: forms

text: in

text: the

text: Cohen

text: class,

text: see

text: Example

text: in

text: Donoho-Stark

text: UP

text: within

text: the

text: Cohen

text: Class

text: In

text: this

text: section

text: we

text: give

text: a

text: new

text: formulation

text: of

text: the

text: classical

text: Donoho-Stark

text: UP

text: in

text: the

text: context

text: of

text: the

text: Cohen

text: class,

text: by

text: using

text: the

text: boundedness

text: results

text: founded

text: previously.

text: Consider

text: the

text: operator

text: T

text: a

text: associated

text: with

text: a

text: generic

text: time-frequency

text: rep-

text: σ

text: resentation

text: in

text: the

text: Cohen

text: class,

text: cf.

text: We

text: have

text: the

text: following

text: result,

text: that

text: is

text: the

text: analogous

text: for

text: Cohen

text: operators

text: of

text: Theorem

text: R

text: Theorem

text: Let

text: T,

text: Ω

text: be

text: measurable

text: sets

text: in

text: d.

text: Let

text: χ

text: be

text: the

text: function

text: T

text: χ

text: ×R

text: (x,

text: ω)

text: χ

text: (x)

text: and

text: χ

text: be

text: the

text: function

text: χR

text: (x,

text: ω)

text: χ

text: (ω),

text: with

text: cor-

text: T

text: d

text: T

text: Ω

text: d

text: Ω

text: Ω

text: responding

text: operators

text: T

text: χT

text: and

text: T

text: χΩ

text: associated

text: with

text: a

text: generic

text: representation

text: σ

text: σ

text: Qσ.

text: We

text: suppose

text: that

text: the

text: ke−rnel

text: σ

text: is

text: such

text: that

text: T×Rd

text: σ(x

text: t,

text: ω)dxdω

text: and

text: R≤d×Ω

text: σ(x,

text: η

text: ω)

text: dx

text: dη

text: are

text: (real)

text: non

text: negative

text: function

text: sat-

text: isfying

text: F

text: j

text: Assume

text: that

text: T

text: χT

text: and

text: T

text: χΩ

text: satisfy

text: the

text: ε-

text: j

text: σ

text: σ

text: concentration

text: condition,

text: i.e.,

text: T

text: χT

text: f

text: ε2

text: f

text: and

text: T

text: χΩf

text: ε2

text: f

text: σ

text: T

text: σ

text: Ω

text: moreover,

text: suppose

text: that

text: T

text: χT

text: T

text: χΩ

text: σ

text: σ

text: Deﬁne

text: ∈Rd

text: σ(∞t,

text: ω)

text: dω

text: and

text: Rd

text: σ(x,

text: ξ)

text: dx.

text: Then

text: for

text: every

text: s

text: s

text: p

text: p

text: r

text: satisfying

text: j

text: we

text: j

text: j

text: have

text: T

text: Ω

text: ε

text: ε

text: T

text: Ω

text: d

text: A

text: A

text: A

text: A

text: A

text: G

text: G

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Reasoning in the same way for T χΩ, we get σ (cid:107) − (cid:107) ≤ (cid:107) (cid:107) f T χΩf 2 ε2 f 2. σ 2 Ω 2 (cid:107) (cid:107) ≤ We assume for simplicity that T χΩ 1, cf. (23) (in the other case σ B(L2) the proof is similar). We then have (cid:107) − (cid:107) ≤ (cid:107) − (cid:107) (cid:107) − (cid:107) f T χΩT χT f f T χΩf + T χΩf T χΩT χT f σ σ 2 σ 2 σ σ σ 2 ≤ (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) − (cid:107) ε f + T χΩ f T χT f Ω 2 σ B(L2) σ 2 ≤ (cid:107) (cid:107) · (cid:107) (cid:107) ε f + 1 ε f Ω 2 T 2 (cid:107) (cid:107) = (ε + ε ) f . Ω T 2 Then (cid:107) (cid:107) ≥ (cid:107) (cid:107) − (cid:107) − (cid:107) T χΩT χT f f f T χΩT χT f σ σ 2 2 σ σ 2 ≥ (cid:107) (cid:107) − (cid:107) (cid:107) f (ε + ε ) f 2 Ω T 2 − − (cid:107) (cid:107) = (1 ε ε ) f , T Ω 2 and, from this, it follows that − − ≤ (cid:107) (cid:107) 1 ε ε T χΩT χT . (27) Ω T σ σ B(L2) ∈ ∞ From (24), and using H¨older i(cid:100)nequality we have, for every r, k [1, ] (cid:107) (cid:107) (cid:107) (cid:107) T χΩT χT f = F F f σ σ 2 2 1 2 ≤ (cid:107) (cid:107) (cid:107) (cid:107) Ad (cid:48) (cid:48) F F f (cid:48) (cid:48) (2r ) 2 2r 1 (2r ) ≤ (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) Ad (cid:48) (cid:48) F F (cid:48) (cid:48) f (cid:48) (cid:48) (cid:48). (2r ) 2 2r 1 (2r ) k (2r ) k (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) We choose k such that (2r ) k = 2, that implies (2r ) k = 2r. Observe now ∗ ∗ that F = χ G and F = χ G . Then, for every s , s , p , p such 1 T (cid:16) 1 2 Ω 2 (cid:17) 1 2 1 2 that 1/s + 1/p = 1 + 1/2r, j = 1, 2, by Young inequality we get j j (cid:107) (cid:107) ≤ d (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) T χΩT χT f A (cid:48) (cid:48)A A A A A2 (cid:48) χ χ G G f , σ σ 2 (2r ) s1 s2 p1 p2 (2r) T s1 Ω s2 1 p1 2 p2 2 that, together with (27), implies the thesis. 20 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Reasoning

text: in

text: the

text: same

text: way

text: for

text: T

text: χΩ,

text: we

text: get

text: σ

text: f

text: T

text: χΩf

text: ε2

text: f

text: σ

text: Ω

text: We

text: assume

text: for

text: simplicity

text: that

text: T

text: χΩ

text: cf.

text: (in

text: the

text: other

text: case

text: σ

text: the

text: proof

text: is

text: similar).

text: We

text: then

text: have

label: equation

text: f

text: T

text: χΩT

text: χT

text: f

text: f

text: T

text: χΩf

text: T

text: χΩf

text: T

text: χΩT

text: χT

text: f

text: σ

text: σ

text: σ

text: σ

text: σ

text: σ

text: ε

text: f

text: T

text: χΩ

text: f

text: T

text: χT

text: f

text: Ω

text: σ

text: σ

text: ε

text: f

text: ε

text: f

text: Ω

text: T

text: (ε

text: ε

text: f

text: Ω

text: T

text: Then

text: T

text: χΩT

text: χT

text: f

text: f

text: f

text: T

text: χΩT

text: χT

text: f

text: σ

text: σ

text: σ

text: σ

text: f

text: (ε

text: ε

text: f

text: Ω

text: T

text: ε

text: ε

text: f

text: T

text: Ω

text: and,

text: from

text: this,

text: it

text: follows

text: that

text: ε

text: ε

text: T

text: χΩT

text: χT

text: Ω

text: T

text: σ

text: σ

text: From

text: and

text: using

text: H¨older

text: we

text: have,

text: for

text: every

text: r,

text: k

text: T

text: χΩT

text: χT

text: f

text: F

text: F

text: f

text: σ

text: σ

text: Ad

text: F

text: F

text: f

text: Ad

text: F

text: F

text: f

text: k

text: k

text: We

text: choose

text: k

text: such

text: that

text: k

text: that

text: implies

text: k

text: Observe

text: now

text: that

text: F

text: χ

text: G

text: and

text: F

text: χ

text: G

text: Then,

text: for

text: every

text: s

text: s

text: p

text: p

text: such

text: T

text: Ω

text: that

text: j

text: by

text: Young

text: inequality

text: we

text: get

text: j

text: j

text: d

text: T

text: χΩT

text: χT

text: f

text: A

text: A

text: A

text: A

text: χ

text: χ

text: G

text: G

text: f

text: σ

text: σ

text: T

text: Ω

text: that,

text: together

text: with

text: implies

text: the

text: thesis.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: References [1] D. Bayer. Bilinear Time-Frequency Distributions and Pseudo- diﬀerential Operators, PhD Thesis, Univ. Wien, 2011. [2] M. Benedicks. On Fourier transforms of functions supported on sets of ﬁnite Lebesgue measure, J. Math. Anal. Appl. 106 (1), 180–183, 1985. [3] P. Boggiatto, E. Carypis, and A. Oliaro. Windowed-Wigner represen- tations in the Cohen class and uncertainty principles, J. Geom. Anal., 23, no. 4, 1753–1779, 2013. [4] P. Boggiatto, E. Carypis, and A. Oliaro. Local uncertainty principles for the Cohen class, J. Math. Anal. Appl, 419, no. 2, 1004–1022, 2014. [5] P. Boggiatto, E. Carypis, and A. Oliaro. Two aspects of the Donoho– Stark uncertainty principle, J. Math. Anal. Appl, 434, no. 2, 1489– 1503, 2016. [6] P. Boggiatto, G. De Donno and A. Oliaro, Weyl quantization of Lebesgue spaces, Mathematische Nachrichten 282, no. 12, 1656–1663, 2009. [7] P. Boggiatto, G. De Donno and A. Oliaro, Time-Frequency representa- tions of Wigner type and Pseudo-diﬀerential Operators, Trans. Amer. Math. Soc. 362, no. 9, 4955–4981, 2010. [8] P. Boggiatto, C. Fern´andez, A. Galbis. Supports of representations in the Cohen class, J. Fourier Anal. Appl. 17 (6) (2011), 1180-1197. [9] P. Boggiatto, A. Oliaro, M. W. Wong, Lp-boundedness and compact- ness of localization operators, J. Math. Anal. Appl., 322 (2005), 193– 206. [10] L. Cohen. Time-Frequency Analysis. Prentice Hall Signal Proc. Series, New Jersey, 1995. [11] L. Cohen. The uncertainty principle for the short-time Fourier trans- form Proc. Int. Soc. Opt. Eng., 22563, 80-90, 1995. [12] E. Cordero, M. de Gosson, F. Nicola. Born-Jordan Pseudo-Diﬀerential Operators with Symbols in the Shubin Classes. arXiv:1603.03403. [13] E. Cordero, F. Nicola. Sharp integral bounds for Wigner distributions. arXiv:1605.00481. 21

text: References

label: reference

text: D.

text: Bayer.

text: Bilinear

text: Time-Frequency

text: Distributions

text: and

text: Pseudo-

text: diﬀerential

text: Operators,

text: PhD

text: Thesis,

text: Univ.

text: Wien,

text: M.

text: Benedicks.

text: On

text: Fourier

text: transforms

text: of

text: functions

text: supported

text: on

text: sets

text: of

text: ﬁnite

text: Lebesgue

text: measure,

text: J.

text: Math.

text: Anal.

text: Appl.

text: P.

text: Boggiatto,

text: E.

text: Carypis,

text: and

text: A.

text: Oliaro.

text: Windowed-Wigner

text: represen-

text: tations

text: in

text: the

text: Cohen

text: class

text: and

text: uncertainty

text: principles,

text: J.

text: Geom.

text: Anal.,

text: no.

text: P.

text: Boggiatto,

text: E.

text: Carypis,

text: and

text: A.

text: Oliaro.

text: Local

text: uncertainty

text: principles

text: for

text: the

text: Cohen

text: class,

text: J.

text: Math.

text: Anal.

text: Appl,

text: no.

text: P.

text: Boggiatto,

text: E.

text: Carypis,

text: and

text: A.

text: Oliaro.

text: Two

text: aspects

text: of

text: the

text: Donoho–

text: Stark

text: uncertainty

text: principle,

text: J.

text: Math.

text: Anal.

text: Appl,

text: no.

text: P.

text: Boggiatto,

text: G.

text: De

text: Donno

text: and

text: A.

text: Oliaro,

text: Weyl

text: quantization

text: of

text: Lebesgue

text: spaces,

text: Mathematische

text: Nachrichten

text: no.

text: P.

text: Boggiatto,

text: G.

text: De

text: Donno

text: and

text: A.

text: Oliaro,

text: Time-Frequency

text: representa-

text: tions

text: of

text: Wigner

text: type

text: and

text: Pseudo-diﬀerential

text: Operators,

text: Trans.

text: Amer.

text: Math.

text: Soc.

text: no.

text: P.

text: Boggiatto,

text: C.

text: Fern´andez,

text: A.

text: Galbis.

text: Supports

text: of

text: representations

text: in

text: the

text: Cohen

text: class,

text: J.

text: Fourier

text: Anal.

text: Appl.

text: P.

text: Boggiatto,

text: A.

text: Oliaro,

text: M.

text: W.

text: Wong,

text: Lp-boundedness

text: and

text: compact-

text: ness

text: of

text: localization

text: operators,

text: J.

text: Math.

text: Anal.

text: Appl.,

text: L.

text: Cohen.

text: Time-Frequency

text: Analysis.

text: Prentice

text: Hall

text: Signal

text: Proc.

text: Series,

text: New

text: Jersey,

text: L.

text: Cohen.

text: The

text: uncertainty

text: principle

text: for

text: the

text: short-time

text: Fourier

text: trans-

text: form

text: Proc.

text: Int.

text: Soc.

text: Opt.

text: Eng.,

text: E.

text: Cordero,

text: M.

text: de

text: Gosson,

text: F.

text: Nicola.

text: Born-Jordan

text: Pseudo-Diﬀerential

text: Operators

text: with

text: Symbols

text: in

text: the

text: Shubin

text: Classes.

text: E.

text: Cordero,

text: F.

text: Nicola.

text: Sharp

text: integral

text: bounds

text: for

text: Wigner

text: distributions.


text: ∈ ∞ As it is easily veriﬁed, for every q [1, + ), we have lim → ∞ H(p, q) = ∞p + 1, therefore it is convenient to extend (5) by setting H(q, + ) = 1, and also ∞ ∞ H(+ , + ) = 1. With this agreement we have the following result. ≤ ≤ ∞ (cid:48) ≤ ≤ (cid:48) ≤ (cid:48) ≤ Theorem 3. Let 2 p and p q p (so that also p q p). Then the Gabor transform deﬁnes a bounded sesquilinear map ∈ R × (cid:48) R → ∈ R V : (f, g) Lq( d) Lq ( d) V f Lp( 2d), g and (cid:107) (cid:107) ≤ (cid:107) (cid:107) (cid:107) (cid:107) V f H(p, q) f g (cid:48). (6) g p q q ∞ Proof. The case p = + is a trivial application of Young’s inequality which (cid:107) (cid:107) ≤ (cid:107) (cid:107) (cid:107) (cid:107) immediately yields the estimate V f ∞ f g (cid:48). Suppose now that p < ∞ g q q (cid:98) + . We recall that the Gabor transform can be written as Fourier transform · of a product (cf. [18], Thm. 3.3.2), namely V f(x, ω) = (f τ g)(ω), where g − x the translation operator τ is deﬁned as τ h(t) := h(t x). We have the x x following estimation: (cid:32) (cid:33) (cid:90) (cid:18)(cid:90) (cid:19) (cid:98) (cid:107)V f(cid:107) = (cid:18)(cid:90) |(f · τ g)(ω)(cid:19)|pdω p1·p dx 1/p g p x R R d d (cid:98) (cid:18)(cid:90) (cid:107) · (cid:107) (cid:19) 1/p = (f τ g)(ω) pdx R x p d ≤ (cid:32) (cid:107) · (cid:107) 1/p (cid:33) (cid:90)Apd(cid:48) (cid:18)f(cid:90) τ g p(cid:48)dx (cid:19) Rd p x p (cid:101) (cid:48) 1/p | | (cid:48)| − | (cid:48) p/p = Adp(cid:48) (cid:18)(cid:90)R (cid:16) R f(y) p g((cid:17)x y) p(cid:19)dy dx d d (cid:101) (cid:48) (cid:48) 1 p(cid:48) = Ad(cid:48) |f(cid:101)|p(cid:48) ∗ |g|p(cid:48)(x) p/p dx p p p R d (cid:107)| | (cid:48) ∗ | | (cid:48)(cid:107) (cid:48) = Ad(cid:48) f p g p 1/p(cid:48), (cid:101) p p/p (cid:101) − | | (cid:48) where g(x) = g( x). Let us now apply the Young’s inequality to f p (cid:48) and |g|p(cid:48) which are in Lpq(cid:48) and Lpq(cid:48) respectively, and denote s = q(cid:48) = q(p−1), (cid:48) − − p p t = q(cid:48) = q(p−1), and r = p(cid:48) = p 1. Then p p(q 1) p 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: As

text: it

text: is

text: easily

text: veriﬁed,

text: for

text: every

text: q

text: we

text: have

text: lim

text: H(p,

text: q)

text: ∞p

text: therefore

text: it

text: is

text: convenient

text: to

text: extend

text: by

text: setting

text: H(q,

text: and

text: also

text: H(+

text: With

text: this

text: agreement

text: we

text: have

text: the

text: following

text: result.

text: Theorem

text: Let

text: p

text: and

text: p

text: q

text: p

text: (so

text: that

text: also

text: p

text: q

text: p).

text: Then

text: the

text: Gabor

text: transform

text: deﬁnes

text: a

text: bounded

text: sesquilinear

text: map

text: R

text: R

text: R

text: V

text: (f,

text: g)

text: Lq(

text: d)

text: Lq

text: d)

text: V

text: f

text: Lp(

text: g

text: and

text: V

text: f

text: H(p,

text: q)

text: f

text: g

text: g

text: p

text: q

text: q

text: Proof.

text: The

text: case

text: p

text: is

text: a

text: trivial

text: application

text: of

text: Young’s

text: inequality

text: which

text: immediately

text: yields

text: the

text: estimate

text: V

text: f

text: f

text: g

text: Suppose

text: now

text: that

text: p

text: g

text: q

text: q

text: We

text: recall

text: that

text: the

text: Gabor

text: transform

text: can

text: be

text: written

text: as

text: Fourier

text: transform

text: of

text: a

text: product

text: (cf.

text: Thm.

text: namely

text: V

text: f(x,

text: ω)

text: (f

text: τ

text: g)(ω),

text: where

text: g

text: x

text: the

text: translation

text: operator

text: τ

text: is

text: deﬁned

text: as

text: τ

text: h(t)

text: h(t

text: x).

text: We

text: have

text: the

text: x

text: x

text: following

text: estimation:

text: |(f

text: τ

text: dx

text: g

text: p

text: x

text: R

text: R

text: d

text: d

text: (f

text: τ

text: g)(ω)

text: pdx

text: R

text: x

text: p

text: d

text: τ

text: g

text: Rd

text: p

text: x

text: p

text: p/p

text: R

text: f(y)

text: p

text: y)

text: dx

text: d

text: d

text: p/p

text: dx

text: p

text: p

text: p

text: R

text: d

text: f

text: p

text: g

text: p

text: p

text: p/p

text: where

text: g(x)

text: g(

text: x).

text: Let

text: us

text: now

text: apply

text: the

text: Young’s

text: inequality

text: to

text: f

text: p

text: and

text: which

text: are

text: in

text: and

text: respectively,

text: and

text: denote

text: s

text: p

text: p

text: t

text: and

text: r

text: p

text: Then

text: p

text: p(q

text: p

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##


text: (cid:107) ≤ Remark 6. In the particular case q = p, Proposition 5 reads Wig (f, g) − (cid:107) (cid:107) (cid:107) (cid:107) p 2 d4d/pAd f g (cid:48), a result which appears in Wong [29]. p p p We also recall that the boundedness result of Proposition 5 without estimation of the boundedness constant appears in [6], Prop 3.1. From Prop. 3.2 of the same paper [6] one can further deduce that in the remaining cases for q and p we do not have boundedness. We can therefore summarize the situation as follows. ∈ ∞ Proposition 7. Let p, q [1, ], then R × (cid:48) R → R Wig : Lq( d) Lq ( d) Lp( 2d) (cid:48) ≤ ≤ is bounded if and only if p q p. (Remark that this means no boundedness for p < 2). Motivated by Proposition 7, it is natural to investigate the cases R × R → R Wig : Lr( d) Ls( d) Lp( 2d) where not necessarily r, s are conjugate indices and, along these lines, we present next a discussion that will yield to a complete characterization of the cases of boundedness of the Wigner transform on Lebesgue spaces, which, although not strictly needed in the following sections, in our opinion has an interest in itself. The same results, with suitably adapted constants, hold for the Gabor transform. We start by the case of non-boundedness on the diagonal of the indices space. ∈ ∞ R × R → Proposition 8. Let p [1, ], then the map Wig : Lq( d) Lq( d) R (cid:54) Lp( 2d) is not continuous if q = 2. ∈ S R Proof. Consider a function f ( d) and let f (x) = f(λx) be the corre- λ f (x) sponding dilation by λ > 0. Let ϕ (x) = (cid:107)λ (cid:107) be the normalization of f , then an easy computation gives λ (cid:16) (cid:17) fλ 2 λ − 1 1 d (cid:107) (cid:107) λ 2 q (cid:107) (cid:107) ϕ = (cid:107) (cid:107) f . λ q q f 2 (cid:16) (cid:17) Given f as above, 1 ω Wig (ϕ )(x, ω) = (cid:107) (cid:107) Wig (f) λx, , λ f 2 λ 2 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Remark

text: In

text: the

text: particular

text: case

text: q

text: p,

text: Proposition

text: reads

text: Wig

text: (f,

text: g)

text: p

text: f

text: g

text: a

text: result

text: which

text: appears

text: in

text: Wong

text: p

text: p

text: p

text: We

text: also

text: recall

text: that

text: the

text: boundedness

text: result

text: of

text: Proposition

text: without

text: estimation

text: of

text: the

text: boundedness

text: constant

text: appears

text: in

text: Prop

text: From

text: Prop.

text: of

text: the

text: same

text: paper

text: one

text: can

text: further

text: deduce

text: that

text: in

text: the

text: remaining

text: cases

text: for

text: q

text: and

text: p

text: we

text: do

text: not

text: have

text: boundedness.

text: We

text: can

text: therefore

text: summarize

text: the

text: situation

text: as

text: follows.

text: Proposition

text: Let

text: p,

text: q

text: then

text: R

text: R

text: R

text: Wig

text: Lq(

text: d)

text: Lq

text: d)

text: Lp(

text: is

text: bounded

text: if

text: and

text: only

text: if

text: p

text: q

text: p.

text: (Remark

text: that

text: this

text: means

text: no

text: boundedness

text: for

text: p

text: Motivated

text: by

text: Proposition

text: it

text: is

text: natural

text: to

text: investigate

text: the

text: cases

text: R

text: R

text: R

text: Wig

text: Lr(

text: d)

text: Ls(

text: d)

text: Lp(

text: where

text: not

text: necessarily

text: r,

text: s

text: are

text: conjugate

text: indices

text: and,

text: along

text: these

text: lines,

text: we

text: present

text: next

text: a

text: discussion

text: that

text: will

text: yield

text: to

text: a

text: complete

text: characterization

text: of

text: the

text: cases

text: of

text: boundedness

text: of

text: the

text: Wigner

text: transform

text: on

text: Lebesgue

text: spaces,

text: which,

text: although

text: not

text: strictly

text: needed

text: in

text: the

text: following

text: sections,

text: in

text: our

text: opinion

text: has

text: an

text: interest

text: in

text: itself.

text: The

text: same

text: results,

text: with

text: suitably

text: adapted

text: constants,

text: hold

text: for

text: the

text: Gabor

text: transform.

text: We

text: start

text: by

text: the

text: case

text: of

text: non-boundedness

text: on

text: the

text: diagonal

text: of

text: the

text: indices

text: space.

text: R

text: R

text: Proposition

text: Let

text: p

text: then

text: the

text: map

text: Wig

text: Lq(

text: d)

text: Lq(

text: d)

text: R

text: Lp(

text: is

text: not

text: continuous

text: if

text: q

text: S

text: R

text: Proof.

text: Consider

text: a

text: function

text: f

text: d)

text: and

text: let

text: f

text: (x)

text: f(λx)

text: be

text: the

text: corre-

text: λ

text: f

text: (x)

text: sponding

text: dilation

text: by

text: λ

text: Let

text: ϕ

text: (x)

text: be

text: the

text: normalization

text: of

text: f

text: then

text: an

text: easy

text: computation

text: gives

text: λ

text: fλ

text: λ

text: d

text: λ

text: q

text: ϕ

text: f

text: λ

text: q

text: q

text: f

text: Given

text: f

text: as

text: above,

text: ω

text: Wig

text: (ϕ

text: )(x,

text: ω)

text: Wig

text: (f)

text: λx,

text: λ

text: f

text: λ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3 A Donoho-Stark UP for localization operators We revise in this section the content of Sections 2 and 3 of [5] correcting a ﬂaw in the proof of Theorem 6, which expresses a Donoho-Stark uncertainty principle in terms of localization operators and leads to an improvement of the classical Donoho-Stark es(cid:90)timate, which now we specify correctly. Let us ﬁrst recall that a localization operator is a map of the type: −→ f La f = a(x, ω)V f(x, ω) µ τ ψ dxdω (8) φ,ψ R φ ω x 2d R ∈ R ∈ ∞ acting on L2( d), with symbol a Lq( 2d), for q [1, ], and “window” ∈ R − functions φ, ψ L2( d). Here µ τ ψ(t) = e2πiωtψ(t x) are time-frequency ω x shifts of ψ(t). We refer to the available vast literature (e.g. [9, 10, 15, 16, 18, 28]) for the motivations and the meaning of these operators in time- frequency and harmonic analysis, as well as for extensions to more general functional settings. Their use in the Donoho-Stark UP relies on the following boundedness estimate, which appears in [5] (Lemma 4). ∈ R ∈ ∞ Lemma 11. Let φ, ψ L2( d), q [1, ] and consider the quantization (see (8)): ∈ R → ∈ R L : a Lq( 2d) La B(L2( d)). φ,ψ φ,ψ (cid:18) (cid:19) Then the following estimation holds (cid:48) (cid:107) (cid:107) ≤ 1 d/q (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) La (cid:48) φ ψ a , φ,ψ B(L2) q 2 2 q with 1 + 1(cid:48) = 1, and setting ( 1(cid:48) )q1(cid:48) = 1 for q = 1. q q q 9 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: section

text: A

text: Donoho-Stark

text: UP

text: for

text: localization

text: operators

text: We

text: revise

text: in

text: this

text: section

text: the

text: content

text: of

text: Sections

text: and

text: of

text: correcting

text: a

text: ﬂaw

text: in

text: the

text: proof

text: of

text: Theorem

text: which

text: expresses

text: a

text: Donoho-Stark

text: uncertainty

text: principle

text: in

text: terms

text: of

text: localization

text: operators

text: and

text: leads

text: to

text: an

text: improvement

text: of

text: the

text: classical

text: Donoho-Stark

text: which

text: now

text: we

text: specify

text: correctly.

text: Let

text: us

text: ﬁrst

text: recall

text: that

text: a

text: localization

text: operator

text: is

text: a

text: map

text: of

text: the

text: type:

text: f

text: La

text: f

text: a(x,

text: ω)V

text: f(x,

text: ω)

text: µ

text: τ

text: ψ

text: dxdω

text: φ,ψ

text: R

text: φ

text: ω

text: x

text: R

text: R

text: acting

text: on

text: d),

text: with

text: symbol

text: a

text: Lq(

text: for

text: q

text: and

text: “window”

text: R

text: functions

text: φ,

text: ψ

text: d).

text: Here

text: µ

text: τ

text: ψ(t)

text: x)

text: are

text: time-frequency

text: ω

text: x

text: shifts

text: of

text: ψ(t).

text: We

text: refer

text: to

text: the

text: available

text: vast

text: literature

text: (e.g.

text: for

text: the

text: motivations

text: and

text: the

text: meaning

text: of

text: these

text: operators

text: in

text: time-

text: frequency

text: and

text: harmonic

text: analysis,

text: as

text: well

text: as

text: for

text: extensions

text: to

text: more

text: general

text: functional

text: settings.

text: Their

text: use

text: in

text: the

text: Donoho-Stark

text: UP

text: relies

text: on

text: the

text: following

text: boundedness

text: estimate,

text: which

text: appears

text: in

text: (Lemma

text: R

text: Lemma

text: Let

text: φ,

text: ψ

text: d),

text: q

text: and

text: consider

text: the

text: quantization

text: (see

text: R

text: R

text: L

text: a

text: Lq(

text: La

text: d)).

text: φ,ψ

text: φ,ψ

text: Then

text: the

text: following

text: estimation

text: holds

text: d/q

text: La

text: φ

text: ψ

text: a

text: φ,ψ

text: q

text: q

text: with

text: and

text: setting

text: for

text: q

text: q

text: q

text: q

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 FIG. 7. (Color online) The numerically computed spatial cor- FIG. 6. (Color online) The spatial correlation functions of the relation functions of the ﬁrst order, i.e. p = 1, for diamond ﬁrst order, i.e. p = 1, for diamond (top) and sawtooth (bot- (top) and sawtooth (bottom) chains at frustration f = 0.8. tom) chains numerically computed for diﬀerent temperatures The temperature k T = 0.01E was chosen. At variance k T = 10E (red), 0.99E (blue) and 0.01E (brown). The B J B J J J with the low frustration, ferromagnetic regime, the correla- frustration f = 0 was used. tion function vanishes beyond few unit cell distances. The observed ﬂuctuations on larger distances are due to ﬁnite tem- perature and ﬁnite size eﬀects. (see (14) and (18)). As expected ξ grows with decreasing 1 temperature for low frustration. However as f > fD(ST) c , the length ξ essentially drops to zero for all tempera- an increase of ξ . The growth is diminishing as the crit- 1 1 tures and remains of order one, i.e. of few unit cells in ical frustration fD(ST) value is approached and spatial ≥ c the highly frustrated phase f fcD(ST). correlations vanish. Upon entering the highly frustrated ≥ However there are unexpected spatial correlations hid- regime f fD(ST) and approaching f = 1 the lengths c den in the higher order correlation functions Cp(n) with again start to show growth with decreasing temperature p > 1. Indeed, the Monte-Carlo simulation for the di- and a long-range ordering establishes at f = 1, T = 0. amond chain with the largest frustration f = 1, shows Similar eﬀect is observed in higher order correlation func- that the long-range correlations are restored in CD(n) tions for several other values of f > f for both models. 2 c for low temperatures. This result is displayed in Fig. 9 VI. CONCLUSIONS (similar behavior is observed for the sawtooth chain (not shown)). While CD(n) decays to zero beyond 2 unit cells, 1 We have studied analytically and using Monte-Carlo the C (n) shows long-range oscillatory behavior with sign 2 simulations the static and dynamic properties of frus- alternation. This is in good agreement with the analyti- trated arrays of Josephson junction classical nonlinear cal results (see Fig. 4). oscillators. Such periodic arrays are characterized by Similarly to the p = 1 case (see Fig. 8) the long-range Josephson coupling strengths in every cell that can have correlations developing at f = 1 in the diamond and saw- diﬀerent signs. We have considered two models of such tooth chains can b∝e quant−iﬁed by the correlation lengths frustrated arrays, namely, the diamond and sawtooth ξ2 and ξ3 (Cp(n) exp( n/ξp)) respectively as shown chains (see Figs. 1). in Fig. 10. In the low frustration phase f < fD(ST) both In the linear regime the dynamics of these arrays is c lengths grow with decreasing temperature reﬂecting the determined by a multi-band dispersion relation ω(k) (see development of the ferromagnetic ordering, similarly to Figs. 2 and 3). Such spectrum can be experimentally ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: FIG.

text: (Color

label: caption

text: online)

text: The

text: numerically

text: computed

text: spatial

text: cor-

text: FIG.

text: (Color

text: online)

text: The

text: spatial

text: correlation

text: functions

text: of

text: the

text: relation

text: functions

text: of

text: the

text: ﬁrst

text: order,

text: i.e.

text: p

text: for

text: diamond

text: ﬁrst

text: order,

text: i.e.

text: p

text: for

text: diamond

text: (top)

text: and

text: sawtooth

text: (bot-

text: (top)

text: and

text: sawtooth

text: (bottom)

text: chains

text: at

text: frustration

text: f

text: tom)

text: chains

text: numerically

text: computed

text: for

text: diﬀerent

text: temperatures

text: The

text: temperature

text: k

text: T

text: was

text: chosen.

text: At

text: variance

text: k

text: T

text: (red),

text: (blue)

text: and

text: (brown).

text: The

text: B

text: J

text: B

text: J

text: J

text: J

text: with

text: the

text: low

text: frustration,

text: ferromagnetic

text: regime,

text: the

text: correla-

text: frustration

text: f

text: was

text: used.

text: tion

text: function

text: vanishes

text: beyond

text: few

text: unit

text: cell

text: distances.

text: The

text: observed

text: ﬂuctuations

text: on

text: larger

text: distances

text: are

text: due

text: to

text: ﬁnite

text: tem-

text: perature

text: and

text: ﬁnite

text: size

text: eﬀects.

text: (see

text: and

text: As

text: expected

text: ξ

text: grows

text: with

text: decreasing

text: temperature

text: for

text: low

text: frustration.

text: However

text: as

text: f

text: fD(ST)

text: c

text: the

text: length

text: ξ

text: essentially

text: drops

text: to

text: zero

text: for

text: all

text: tempera-

text: an

text: increase

text: of

text: ξ

text: The

text: growth

text: is

text: diminishing

text: as

text: the

text: crit-

text: tures

text: and

text: remains

text: of

text: order

text: one,

text: i.e.

text: of

text: few

text: unit

text: cells

text: in

text: ical

text: frustration

text: fD(ST)

text: value

text: is

text: approached

text: and

text: spatial

text: c

text: the

text: highly

text: frustrated

text: phase

text: f

text: fcD(ST).

text: correlations

text: vanish.

text: Upon

text: entering

text: the

text: highly

text: frustrated

text: However

text: there

text: are

text: unexpected

text: spatial

text: correlations

text: hid-

text: regime

text: f

text: fD(ST)

text: and

text: approaching

text: f

text: the

text: lengths

text: c

text: den

text: in

text: the

text: higher

text: order

text: correlation

text: functions

text: Cp(n)

text: with

text: again

text: start

text: to

text: show

text: growth

text: with

text: decreasing

text: temperature

text: p

text: Indeed,

text: the

text: Monte-Carlo

text: simulation

text: for

text: the

text: di-

text: and

text: a

text: long-range

text: ordering

text: establishes

text: at

text: f

text: T

text: amond

text: chain

text: with

text: the

text: largest

text: frustration

text: f

text: shows

text: Similar

text: eﬀect

text: is

text: observed

text: in

text: higher

text: order

text: correlation

text: func-

text: that

text: the

text: long-range

text: correlations

text: are

text: restored

text: in

text: CD(n)

text: tions

text: for

text: several

text: other

text: values

text: of

text: f

text: f

text: for

text: both

text: models.

text: c

text: for

text: low

text: temperatures.

text: This

text: result

text: is

text: displayed

text: in

text: Fig.

text: VI.

text: CONCLUSIONS

text: (similar

text: behavior

text: is

text: observed

text: for

text: the

text: sawtooth

text: chain

text: (not

text: shown)).

text: While

text: CD(n)

text: decays

text: to

text: zero

text: beyond

text: unit

text: cells,

text: We

text: have

text: studied

text: analytically

text: and

text: using

text: Monte-Carlo

text: the

text: C

text: (n)

text: shows

text: long-range

text: oscillatory

text: behavior

text: with

text: sign

text: simulations

text: the

text: static

text: and

text: dynamic

text: properties

text: of

text: frus-

text: alternation.

text: This

text: is

text: in

text: good

text: agreement

text: with

text: the

text: analyti-

text: trated

text: arrays

text: of

text: Josephson

text: junction

text: classical

text: nonlinear

text: cal

text: results

text: (see

text: Fig.

text: oscillators.

text: Such

text: periodic

text: arrays

text: are

text: characterized

text: by

text: Similarly

text: to

text: the

text: p

text: case

text: (see

text: Fig.

text: the

text: long-range

text: Josephson

text: coupling

text: strengths

text: in

text: every

text: cell

text: that

text: can

text: have

text: correlations

text: developing

text: at

text: f

text: in

text: the

text: diamond

text: and

text: saw-

text: diﬀerent

text: signs.

text: We

text: have

text: considered

text: two

text: models

text: of

text: such

text: tooth

text: chains

text: can

text: b∝e

text: quant−iﬁed

text: by

text: the

text: correlation

text: lengths

text: frustrated

text: arrays,

text: namely,

text: the

text: diamond

text: and

text: sawtooth

text: ξ2

text: and

text: ξ3

text: (Cp(n)

text: exp(

text: n/ξp))

text: respectively

text: as

text: shown

text: chains

text: (see

text: Figs.

text: in

text: Fig.

text: In

text: the

text: low

text: frustration

text: phase

text: f

text: fD(ST)

text: both

text: In

text: the

text: linear

text: regime

text: the

text: dynamics

text: of

text: these

text: arrays

text: is

text: c

text: lengths

text: grow

text: with

text: decreasing

text: temperature

text: reﬂecting

text: the

text: determined

text: by

text: a

text: multi-band

text: dispersion

text: relation

text: ω(k)

text: (see

text: development

text: of

text: the

text: ferromagnetic

text: ordering,

text: similarly

text: to

text: Figs.

text: and

text: Such

text: spectrum

text: can

text: be

text: experimentally

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: 8 noting, that the frustration strength f = 1 for which C , p = 2 and p = 3 correlators show long-range or- p der for the diamond and sawtooth chains respectively, is not unique. There are other values of frustration f where higher order correlators C (n) display long-range p behavior at low temperatures. A few examples for the ≈ diamond chain are f = 0.75 with p = 3, f 0.71 with ≈ p = 4, f = 0.81 with p = 5, f 0.85 with p = 7 ≈ and f 0.78 with p = 8. Interestingly there are no oscillations in C (n) for f = 0.81. The geometry of the 5 diamond and sawtooth chains studied strongly suggests that similar results hold for other corner sharing chains. The ﬂatband construction of Ref. 15 relying on repetition of mini-arrays appears particularly promising. Finally, we note that quantum tunneling between two equivalent states in every cell can result in a unique ground state and a corresponding crossover between clas- sical highly degenerated ground state and macroscopic quantum state at low temperatures. ACKNOWLEDGMENTS FIG. 10. (Color online) The density plot of the correlation length ξ (top) and ξ (bottom) vs. frustration f and tem- 2 3 perature T; brighter color corresponds to larger value of the correlation lengths. The data are extracted from the Monte- The authors would like to thank S. Flach for useful Carlo simulated C (n) and C (n) for the diamond and saw- discussions. M. V. F. acknowledges the ﬁnancial support 2 3 tooth chains respectively. For f < fD(ST) the T-dependence of the Ministry of Education and Science of the Russian c of ξ2, ξ3 follows that of ξ1, reﬂecting ferromagnetic order- Federation in the framework of−Increa−se Competitiveness ing. For f > fcDST the long-range order is developing in the Program of NUST ”MISiS” K2 2017 067 and the State higher order spatial→correlation functions C2 (diamond) and Program 3.3360.2017. This work was also supported by C (sawtooth) as f 1. 3 the Institute for Basic Science in Korea (IBS-R024-D1). ∗ Rigorous versus numerical results,” Phys. Rev. B 76, 220402 (2007). 1 Arkady Pikovsky, Michael Rosenblum, and Ju¨rgen Kurths, S9ynJcuhlrieonniVzaidtiaoln,:BeanouˆıntivDeorusac¸lotc,oRnc´eempyt iMnonssoenrlii,neaanrdsPciaetnrciceks,Butaud, “Interaction induced delocalization for two particles in a Vol. 12 (Cambridge university press, 2003). periodic potential,” Phys. Rev. Lett. 85, 3906–3909 (2000). 2 Giovanni Filatrella, Niels Falsig Pedersen, and Kurt Wiesenfeld1,0“HBiegnho-iqtcDavoiuty¸co-itndanudceJduslyiennchVroidnaizl,at“iPonaiirninogscoifllacotoorpaerrrapyasir,”s in a fully frustrated josephson-junction chain,” Phys. Rev. Lett. Phys. Rev. E 61, 2513–2518 (2000). 88, 227005 (2002). 3 P. Jung, S. Butz, M. Marthaler, M. V. Fistul, J. Leppa¨kanga1s1, VJu.liPe.nKVoisdhaell,etPsa,traicnkdBAu.taVu.dU, sBtiennoovit, “DMouuc¸ltoits,tabainlidtyRa´enmdy Mosseri, “Disorder and interactions in aharonov-bohm cages,” switching in a superconducting metamaterial,” Nat. Comm. 5, 37P3h0yEs.PR–ev(2. 0B146)4., 155306 (2001). 4 A.V. Ustinov, “Solitons in josephson junctions,” Phys. D: Non1l2inD. PanhieenloLmeyeknaam1,2S3e,rg3e1j5F–la3c2h9, O(1m99ri8)B,aahnantu-TarleIindteelr,naantidonAanlton S. Desyatnikov, “Flat band states: Disorder and nonlinearity,” Conference of the Center for Nonlinear Studies. Phys. Rev. B 88, 224203 (2013). 5 P. Binder, D. Abraimov, A. V. Ustinov, S. Flach, and Y. Z1o3loStaerryguejk,Fl“aOchb,seDrvaantiieolnLeoyfkbarmea,tJhoesrhsuianDjo.sBepohdsyofneltl,adPdeetersr,”Matthies, and Anton S. Desyatnikov, “Detangling ﬂat bands into Phys. Rev. Lett. 84, 745–748 (2000). fano lattices,” EPL (Europhysics Letters) 105, 30001 (2014). 6 C. Macklin, K. O’Brien, D. Hover, M. E. Schwartz, V. Bolkhovsk14y,RX..GZh. aDniga,sWan.dDG. Ooulivveeira,Ja.nDd.,I.“SOidridgiaqmi,i“rAulneesafro–rqtuhaenctounms-truction of localized eigenstates of the hubbard model in decorated limited josephson traveling-wave parametric ampliﬁer,” Science 3l5a0tt,i3ce0s7,–”3S1c0i.(2R0e1p5.).5, 16852 (2015). 7 Elliott H. Lieb, “Two theorems on the hubbard model,” Phys.1R5 eLvu.iLs eMtto.r6a2le,s-1I2n0o1s–t1ro2z0a4a(n1d98R9)o.drigo A. Vicencio, “Simple method to construct ﬂat-band lattices,” Phys. Rev. A 94, 043831 8 Oleg Derzhko, Andreas Honecker, and Johannes Richter, “Low-t(e2m01p6e)r.ature thermodynamics for a ﬂat-band ferromagnet: ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: noting,

text: that

text: the

text: frustration

text: strength

text: f

text: for

text: which

text: C

text: p

text: and

text: p

text: correlators

text: show

text: long-range

text: or-

text: p

text: der

text: for

text: the

text: diamond

text: and

text: sawtooth

text: chains

text: respectively,

text: is

text: not

text: unique.

text: There

text: are

text: other

text: values

text: of

text: frustration

text: f

text: where

text: higher

text: order

text: correlators

text: C

text: (n)

text: display

text: long-range

text: p

text: behavior

text: at

text: low

text: temperatures.

text: A

text: few

text: examples

text: for

text: the

text: diamond

text: chain

text: are

text: f

text: with

text: p

text: f

text: with

text: p

text: f

text: with

text: p

text: f

text: with

text: p

text: and

text: f

text: with

text: p

text: Interestingly

text: there

text: are

text: no

text: oscillations

text: in

text: C

text: (n)

text: for

text: f

text: The

text: geometry

text: of

text: the

text: diamond

text: and

text: sawtooth

text: chains

text: studied

text: strongly

text: suggests

text: that

text: similar

text: results

text: hold

text: for

text: other

text: corner

text: sharing

text: chains.

text: The

text: ﬂatband

text: construction

text: of

text: Ref.

text: relying

text: on

text: repetition

text: of

text: mini-arrays

text: appears

text: particularly

text: promising.

text: Finally,

text: we

text: note

text: that

text: quantum

text: tunneling

text: between

text: two

text: equivalent

text: states

text: in

text: every

text: cell

text: can

text: result

text: in

text: a

text: unique

text: ground

text: state

text: and

text: a

text: corresponding

text: crossover

text: between

text: clas-

text: sical

text: highly

text: degenerated

text: ground

text: state

text: and

text: macroscopic

text: quantum

text: state

text: at

text: low

text: temperatures.

text: ACKNOWLEDGMENTS

text: FIG.

text: (Color

text: online)

text: The

text: density

text: plot

text: of

text: the

text: correlation

text: length

text: ξ

text: (top)

text: and

text: ξ

text: (bottom)

text: vs.

text: frustration

text: f

text: and

text: tem-

text: perature

text: T;

text: brighter

text: color

text: corresponds

text: to

text: larger

text: value

text: of

text: the

text: correlation

text: lengths.

text: The

text: data

text: are

text: extracted

text: from

text: the

text: Monte-

text: The

text: authors

text: would

text: like

text: to

text: thank

text: S.

text: Flach

text: for

text: useful

text: Carlo

text: simulated

text: C

text: (n)

text: and

text: C

text: (n)

text: for

text: the

text: diamond

text: and

text: saw-

text: discussions.

text: M.

text: V.

text: F.

text: acknowledges

text: the

text: ﬁnancial

text: support

text: tooth

text: chains

text: respectively.

text: For

text: f

text: fD(ST)

text: the

text: T-dependence

text: of

text: the

text: Ministry

text: of

text: Education

text: and

text: Science

text: of

text: the

text: Russian

text: c

text: of

text: ξ3

text: follows

text: that

text: of

text: reﬂecting

text: ferromagnetic

text: order-

text: Federation

text: in

text: the

text: framework

text: of−Increa−se

text: Competitiveness

text: ing.

text: For

text: f

text: fcDST

text: the

text: long-range

text: order

text: is

text: developing

text: in

text: the

text: Program

text: of

text: NUST

text: ”MISiS”

text: and

text: the

text: State

text: higher

text: order

text: spatial→correlation

text: functions

text: (diamond)

text: and

text: Program

text: This

text: work

text: was

text: also

text: supported

text: by

text: C

text: (sawtooth)

text: as

text: f

text: the

text: Institute

text: for

text: Basic

text: Science

text: in

text: Korea

text: Rigorous

text: versus

text: numerical

text: results,”

text: Phys.

text: Rev.

text: B

text: Arkady

text: Pikovsky,

text: Michael

text: Rosenblum,

text: and

text: Ju¨rgen

text: Kurths,

text: “Interaction

text: induced

text: delocalization

text: for

text: two

text: particles

text: in

text: a

text: Vol.

text: (Cambridge

text: university

text: press,

text: periodic

text: potential,”

text: Phys.

text: Rev.

text: Lett.

text: Giovanni

text: Filatrella,

text: Niels

text: Falsig

text: Pedersen,

text: and

text: Kurt

text: in

text: a

text: fully

text: frustrated

text: josephson-junction

text: chain,”

text: Phys.

text: Rev.

text: Lett.

text: Phys.

text: Rev.

text: E

text: P.

text: Jung,

text: S.

text: Butz,

text: M.

text: Marthaler,

text: M.

text: V.

text: Fistul,

text: J.

text: sBtiennoovit,

text: “DMouuc¸ltoits,tabainlidtyRa´enmdy

text: Mosseri,

text: “Disorder

text: and

text: interactions

text: in

text: aharonov-bohm

text: cages,”

text: switching

text: in

text: a

text: superconducting

text: metamaterial,”

text: Nat.

text: Comm.

text: A.V.

text: Ustinov,

text: “Solitons

text: in

text: josephson

text: junctions,”

text: Phys.

text: D:

text: S.

text: Desyatnikov,

text: “Flat

text: band

text: states:

text: Disorder

text: and

text: nonlinearity,”

text: Conference

text: of

text: the

text: Center

text: for

text: Nonlinear

text: Studies.

text: Phys.

text: Rev.

text: B

text: P.

text: Binder,

text: D.

text: Abraimov,

text: A.

text: V.

text: Ustinov,

text: S.

text: Flach,

text: and

text: Y.

text: and

text: Anton

text: S.

text: Desyatnikov,

text: “Detangling

text: ﬂat

text: bands

text: into

text: Phys.

text: Rev.

text: Lett.

text: fano

text: lattices,”

text: EPL

text: (Europhysics

text: Letters)

text: C.

text: Macklin,

text: K.

text: O’Brien,

text: D.

text: Hover,

text: M.

text: E.

text: Schwartz,

text: V.

text: aDniga,sWan.dDG.

text: of

text: localized

text: eigenstates

text: of

text: the

text: hubbard

text: model

text: in

text: decorated

text: limited

text: josephson

text: traveling-wave

text: parametric

text: ampliﬁer,”

text: Science

text: Elliott

text: H.

text: Lieb,

text: “Two

text: theorems

text: on

text: the

text: hubbard

text: model,”

text: eLvu.iLs

text: A.

text: Vicencio,

text: “Simple

text: method

text: to

text: construct

text: ﬂat-band

text: lattices,”

text: Phys.

text: Rev.

text: A

text: Oleg

text: Derzhko,

text: Andreas

text: Honecker,

text: and

text: Johannes

text: Richter,

text: thermodynamics

text: for

text: a

text: ﬂat-band

text: ferromagnet:

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: If the x data does not contain missing values, it is straightforward to build a “recognition network” that predicts µ | and Σ | from x using only matrix-vector multiplies, as per equations 1 and 2 without z x z x the v subscripts, as the matrix inverse can be computed once and stored. However, if there is missing − data then there is a diﬀerent Σ | for each of the 2D 1 non-trivial patterns of missingness, and thus z xv exact inference cannot be carried out eﬃciently in a single feedforward inference network. (cid:124) Equation 3 can be rewritten in an interesting way. Let v = (w , . . . , w ) denote the jth row of j j1 jK W . Then (cid:88) D (cid:124) − − (cid:124) I + W MΨ 1MW = I + m ψ 1v v , (6) K K j jj j j (cid:81) j=1 | ∝ i.e. where the second term is a sum of rank-1 matrices. This arises from the fact that p(z x) | p(z, x) = p(z) D p(x z), and can be related to the product of Gaussian experts construction j=1 j (Williams and Agakov, 2002). Note also that an information-theoretic argument that conditioning on additional variables never increases entropy shows that Σ | will have a determinant no larger than Σ | 2. z x z xv Vincent et al. (2008) introduced the denoising autoencoder (DAE), where the goal is to train an autoencoder to take a corrupted data point as input, and predict the original, uncorrupted data point as its output. In Vincent et al. (2008) the stochastic corruption process sets a randomly-chosen set of the inputs to zero; the goal is then to reconstruct the corrupted inputs from the non-corrupted ones. As we have seen above, if the data is centered, then setting input values to zero is equivalent to mean imputation. Despite this, the motivation for denoising autoencoders was not so much to handle missing data, but to more robustly learn about the structure of the underlying data manifold. Also, as seen from our results above, a simple feedforward net that ignores the pattern of missingness (as in the DAE) cannot perform exact inference even for the factor analysis case. However, we can develop ideas relating to the DAE to create a denoising encoder (DE). In our situation we have complete training data (i.e. without missingness). Thus for each x we can obtain the corresponding posterior for z (as per eqs. 1 and 2 without any missing data). We then create a pattern of missingness and apply it to x to obtain xmi. One can then train a regression model from xmi to the corresponding posterior mean of z. Note that this averages over the patterns of missingness, rather than | ∼ handling each one separately. The decoder for the DE is the exact solution p(x z) N(W z + µ, Ψ). A limitation of the DE is that to train it we need examples of the patterns of missingness that occur in the data, which is not the case for the exact or approximate methods described above. 3 Experiments We trained a PPCA model on the Frey faces dataset3, which consists of 1965 frames of a greyscale × video sequence with resolution 20 28 pixels. Pixel intensities were rescaled to lie between -1 and 1. We used 43 latent components for the model, explaining 90% of the data variability. 80% of the data frames were randomly selected as a training set, and the remainder were held out as a test set. We estimate the parameters W, µ, σ2 of the model using maximum likelihood, and obtain the analytic solutions as in Tipping and Bishop (1999). In the case of the denoising encoder, we create one missingness-corrupted pattern xmi for each training example when estimating the regression model. Given a trained PPCA model we investigate various approaches for handling missing data with a data imputation task. Given a partially observed data example, the task is to predict the values of the missing data variables using the trained model. We consider two patterns of missing data. In the ﬁrst setting each data variable is independently dropped out with probability 0.5. In the second setting one of the four image quarters is dropped out in each example (see Fig. 1). 2The entropy argument strictly applies when taking expectations over x or x , but for Gaussian distributions the v predictive uncertainty is independent of the value of x or x , so it does imply the desired conclusion. v 3Available from https://cs.nyu.edu/~roweis/data/frey_rawface.mat. 3 ##LTLine##

text: If

text: the

text: x

text: data

text: does

text: not

text: contain

text: missing

text: values,

text: it

text: is

text: straightforward

text: to

text: build

text: a

text: “recognition

text: network”

text: that

text: predicts

text: µ

text: and

text: Σ

text: from

text: x

text: using

text: only

text: matrix-vector

text: multiplies,

text: as

text: per

text: equations

text: and

text: without

text: z

text: x

text: z

text: x

text: the

text: v

text: subscripts,

text: as

text: the

text: matrix

text: inverse

text: can

text: be

text: computed

text: once

text: and

text: stored.

text: However,

text: if

text: there

text: is

text: missing

text: data

text: then

text: there

text: is

text: a

text: diﬀerent

text: Σ

text: for

text: each

text: of

text: the

text: non-trivial

text: patterns

text: of

text: missingness,

text: and

text: thus

text: z

text: xv

text: exact

text: inference

text: cannot

text: be

text: carried

text: out

text: eﬃciently

text: in

text: a

text: single

text: feedforward

text: inference

text: network.

text: Equation

text: can

text: be

text: rewritten

text: in

text: an

text: interesting

text: way.

text: Let

text: v

text: (w

text: w

text: denote

text: the

text: jth

text: row

text: of

text: j

text: jK

text: W

text: Then

text: D

text: I

text: W

text: MΨ

text: I

text: m

text: ψ

text: v

text: K

text: K

text: j

text: jj

text: j

text: j

text: i.e.

text: where

text: the

text: second

text: term

text: is

text: a

text: sum

text: of

text: matrices.

text: This

text: arises

text: from

text: the

text: fact

text: that

text: p(z

text: x)

text: p(z,

text: x)

text: p(z)

text: D

text: p(x

text: z),

text: and

text: can

text: be

text: related

text: to

text: the

text: product

text: of

text: Gaussian

text: experts

text: construction

text: j

text: (Williams

text: and

text: Agakov,

text: Note

text: also

text: that

text: an

text: information-theoretic

text: argument

text: that

text: conditioning

text: on

text: additional

text: variables

text: never

text: increases

text: entropy

text: shows

text: that

text: Σ

text: will

text: have

text: a

text: determinant

text: no

text: larger

text: than

text: Σ

text: z

text: x

text: z

text: xv

text: Vincent

text: et

text: al.

text: introduced

text: the

text: denoising

text: autoencoder

text: (DAE),

text: where

text: the

text: goal

text: is

text: to

text: train

text: an

text: autoencoder

text: to

text: take

text: a

text: corrupted

text: data

text: point

text: as

text: input,

text: and

text: predict

text: the

text: original,

text: uncorrupted

text: data

text: point

text: as

text: its

text: output.

text: In

text: Vincent

text: et

text: al.

text: the

text: stochastic

text: corruption

text: process

text: sets

text: a

text: randomly-chosen

text: set

text: of

text: the

text: inputs

text: to

text: zero;

text: the

text: goal

text: is

text: then

text: to

text: reconstruct

text: the

text: corrupted

text: inputs

text: from

text: the

text: non-corrupted

text: ones.

text: As

text: we

text: have

text: seen

text: above,

text: if

text: the

text: data

text: is

text: centered,

text: then

text: setting

text: input

text: values

text: to

text: zero

text: is

text: equivalent

text: to

text: mean

text: imputation.

text: Despite

text: this,

text: the

text: motivation

text: for

text: denoising

text: autoencoders

text: was

text: not

text: so

text: much

text: to

text: handle

text: missing

text: data,

text: but

text: to

text: more

text: robustly

text: learn

text: about

text: the

text: structure

text: of

text: the

text: underlying

text: data

text: manifold.

text: Also,

text: as

text: seen

text: from

text: our

text: results

text: above,

text: a

text: simple

text: feedforward

text: net

text: that

text: ignores

text: the

text: pattern

text: of

text: missingness

text: (as

text: in

text: the

text: DAE)

text: cannot

text: perform

text: exact

text: inference

text: even

text: for

text: the

text: factor

text: analysis

text: case.

text: However,

text: we

text: can

text: develop

text: ideas

text: relating

text: to

text: the

text: DAE

text: to

text: create

text: a

text: denoising

text: encoder

text: (DE).

text: In

text: our

text: situation

text: we

text: have

text: complete

text: training

text: data

text: (i.e.

text: without

text: missingness).

text: Thus

text: for

text: each

text: x

text: we

text: can

text: obtain

text: the

text: corresponding

text: posterior

text: for

text: z

text: (as

text: per

text: eqs.

text: and

text: without

text: any

text: missing

text: data).

text: We

text: then

text: create

text: a

text: pattern

text: of

text: missingness

text: and

text: apply

text: it

text: to

text: x

text: to

text: obtain

text: xmi.

text: One

text: can

text: then

text: train

text: a

text: regression

text: model

text: from

text: xmi

text: to

text: the

text: corresponding

text: posterior

text: mean

text: of

text: z.

text: Note

text: that

text: this

text: averages

text: over

text: the

text: patterns

text: of

text: missingness,

text: rather

text: than

text: handling

text: each

text: one

text: separately.

text: The

text: decoder

text: for

text: the

text: DE

text: is

text: the

text: exact

text: solution

text: p(x

text: z)

text: N(W

text: z

text: µ,

text: Ψ).

text: A

text: limitation

text: of

text: the

text: DE

text: is

text: that

text: to

text: train

text: it

text: we

text: need

text: examples

text: of

text: the

text: patterns

text: of

text: missingness

text: that

text: occur

text: in

text: the

text: data,

text: which

text: is

text: not

text: the

text: case

text: for

text: the

text: exact

text: or

text: approximate

text: methods

text: described

text: above.

text: Experiments

text: We

text: trained

text: a

text: PPCA

text: model

text: on

text: the

text: Frey

text: faces

text: which

text: consists

text: of

text: frames

text: of

text: a

text: greyscale

text: video

text: sequence

text: with

text: resolution

text: pixels.

text: Pixel

text: intensities

text: were

text: rescaled

text: to

text: lie

text: between

text: and

text: We

text: used

text: latent

text: components

text: for

text: the

text: model,

text: explaining

text: of

text: the

text: data

text: variability.

text: of

text: the

text: data

text: frames

text: were

text: randomly

text: selected

text: as

text: a

text: training

text: set,

text: and

text: the

text: remainder

text: were

text: held

text: out

text: as

text: a

text: test

text: set.

text: We

text: estimate

text: the

text: parameters

text: W,

text: µ,

text: σ2

text: of

text: the

text: model

text: using

text: maximum

text: likelihood,

text: and

text: obtain

text: the

text: analytic

text: solutions

text: as

text: in

text: Tipping

text: and

text: Bishop

text: In

text: the

text: case

text: of

text: the

text: denoising

text: encoder,

text: we

text: create

text: one

text: missingness-corrupted

text: pattern

text: xmi

text: for

text: each

text: training

text: example

text: when

text: estimating

text: the

text: regression

text: model.

text: Given

text: a

text: trained

text: PPCA

text: model

text: we

text: investigate

text: various

text: approaches

text: for

text: handling

text: missing

text: data

text: with

text: a

text: data

text: imputation

text: task.

text: Given

text: a

text: partially

text: observed

text: data

text: example,

text: the

text: task

text: is

text: to

text: predict

text: the

text: values

text: of

text: the

text: missing

text: data

text: variables

text: using

text: the

text: trained

text: model.

text: We

text: consider

text: two

text: patterns

text: of

text: missing

text: data.

text: In

text: the

text: ﬁrst

text: setting

text: each

text: data

text: variable

text: is

text: independently

text: dropped

text: out

text: with

text: probability

text: In

text: the

text: second

text: setting

text: one

text: of

text: the

text: four

text: image

text: quarters

text: is

text: dropped

text: out

text: in

text: each

text: example

text: (see

text: Fig.

text: entropy

text: argument

text: strictly

text: applies

text: when

text: taking

text: expectations

text: over

text: x

text: or

text: x

text: but

text: for

text: Gaussian

text: distributions

text: the

text: v

text: predictive

text: uncertainty

text: is

text: independent

text: of

text: the

text: value

text: of

text: x

text: or

text: x

text: so

text: it

text: does

text: imply

text: the

text: desired

text: conclusion.

text: v

text: from

text: ##LTLine##


text: Global means Mean Denoising Denoising Exact inference imputation encoder encoder* ± ± ± ± ± Random 4.5323 0.0047 1.8675 0.0017 1.0330 0.0006 (1.0330 0.0006) 0.5913 0.0001 ± ± ± ± ± Quarters 4.4642 0.0065 2.5732 0.0045 1.0043 0.0011 1.6863 0.0021 0.6998 0.0005 × Table 1: Mean squared imputation error ( 102) with standard errors for various imputation methods on PPCA generated data. For a particular data example the squared imputation error is the average ∗ squared error over imputed values for that example. Denoising encoder is trained on random patterns of missingness, whereas Denoising encoder is trained on the same type of missingness patterns as it is tested on. We consider four approaches to handling missing data. The ﬁrst method is simply to inpaint the missing variables with their training set means, without using the latent variable model at all. This is denoted by “global means”. The second approach is to use mean imputation for x , and to then m use the approximate inference method as in eq. 5—this is denoted as “mean imputation”. As a third method we train a denoising encoder as described above, using a linear regression model. The ﬁnal method is to use exact inference as per eqs. 1 and 2. 3.1 Results We present two sets of results, ﬁrstly for data generated from the PPCA model ﬁtted to the Frey data (denoted PPCA-Frey data), and secondly for the Frey data itself. For the PPCA-Frey data, the mean squared imputation error results are shown in Table 1. In this case 400 test cases are generated from the PPCA model. For the DE case, 1600 training cases are also generated to train the DE model. For the denoising autoencoder on the quarters data, we train it either on data with missing quarters, or on data with pixels with a random pattern of missingness—the ∗ latter is denoted as Denoising encoder in the tables. In Table 1 we see that mean imputation (eq. 5) improves over global means, but that training the denoising autoencoder (which uses the same input xmi as mean imputation) gives further improvements. ∗ Note that on the quarters data, the Denoising encoder results are quite a lot worse than the DE which had been trained on missing-quarters training data. For both the random and quarters data, the exact inference method is the best, as expected. For the Frey data the mean-squared imputation error results are shown in Table 2, and a comparison on a speciﬁc image is shown in Figure 1. Again mean imputation improves noticeably over global means, but it still gives considerably more error than using the denoising encoder or exact inference. For data with random missingness exact inference signiﬁcantly outperforms the denoising encoder on the test set. For the quarters dataset, we observe slightly better performance of the denoising encoder ∗ than exact inference on the test set. Notice also that when using the Denoising encoder (training using random patterns of missingness) the performance is signiﬁcantly worse. The slight performance gain of the DE on the quarters data over exact inference may be due to the fact that exact inference applies to the PPCA model, but as the Frey data was not generated from this model there can be some room for improvement over the “exact” method. Also note that for the quarters data only four patterns of missingness are used, so the DE network is likely to be able to learn about this more easily than random missingness. There are noticeable diﬀerences between the training and test set errors for the DE model. This can be at least partially explained by noting that on the training set the DE model has access to the posterior for z based on the complete data, so it is not surprising that it does better here. 4 Conclusions Above we have discussed various approaches to handling missing data with variational autoencoders. We have shown how to calculate exactly the latent posterior distribution for the factor analysis (FA) model in the presence of missing data. This solution exhibits a non-trivial dependence on the pattern of missingness, requiring a diﬀerent matrix inversion for each pattern. We have also outlined an alternative method, the denoising encoder, which predicts the latent z from the corrupted xmi via linear 4 ##LTLine## ##LTLine## ##LTLine##

text: Global

label: table

text: means

text: Mean

text: Denoising

text: Denoising

text: Exact

text: inference

text: imputation

text: encoder

text: encoder*

text: Random

text: Quarters

text: Table

text: Mean

text: squared

text: imputation

text: error

text: with

text: standard

text: errors

text: for

text: various

text: imputation

text: methods

text: on

text: PPCA

text: generated

text: data.

text: For

text: a

text: particular

text: data

text: example

text: the

text: squared

text: imputation

text: error

text: is

text: the

text: average

text: squared

text: error

text: over

text: imputed

text: values

text: for

text: that

text: example.

text: Denoising

text: encoder

text: is

text: trained

text: on

text: random

text: patterns

text: of

text: missingness,

text: whereas

text: Denoising

text: encoder

text: is

text: trained

text: on

text: the

text: same

text: type

text: of

text: missingness

text: patterns

text: as

text: it

text: is

text: tested

text: on.

text: We

text: consider

text: four

text: approaches

text: to

text: handling

text: missing

text: data.

text: The

text: ﬁrst

text: method

text: is

text: simply

text: to

text: inpaint

text: the

text: missing

text: variables

text: with

text: their

text: training

text: set

text: means,

text: without

text: using

text: the

text: latent

text: variable

text: model

text: at

text: all.

text: This

text: is

text: denoted

text: by

text: “global

text: means”.

text: The

text: second

text: approach

text: is

text: to

text: use

text: mean

text: imputation

text: for

text: x

text: and

text: to

text: then

text: m

text: use

text: the

text: approximate

text: inference

text: method

text: as

text: in

text: eq.

text: is

text: denoted

text: as

text: “mean

text: imputation”.

text: As

text: a

text: third

text: method

text: we

text: train

text: a

text: denoising

text: encoder

text: as

text: described

text: above,

text: using

text: a

text: linear

text: regression

text: model.

text: The

text: ﬁnal

text: method

text: is

text: to

text: use

text: exact

text: inference

text: as

text: per

text: eqs.

text: and

text: Results

text: We

text: present

text: two

text: sets

text: of

text: results,

text: ﬁrstly

text: for

text: data

text: generated

text: from

text: the

text: PPCA

text: model

text: ﬁtted

text: to

text: the

text: Frey

text: data

text: (denoted

text: PPCA-Frey

text: data),

text: and

text: secondly

text: for

text: the

text: Frey

text: data

text: itself.

text: For

text: the

text: PPCA-Frey

text: data,

text: the

text: mean

text: squared

text: imputation

text: error

text: results

text: are

text: shown

text: in

text: Table

text: In

text: this

text: case

text: test

text: cases

text: are

text: generated

text: from

text: the

text: PPCA

text: model.

text: For

text: the

text: DE

text: case,

text: training

text: cases

text: are

text: also

text: generated

text: to

text: train

text: the

text: DE

text: model.

text: For

text: the

text: denoising

text: autoencoder

text: on

text: the

text: quarters

text: data,

text: we

text: train

text: it

text: either

text: on

text: data

text: with

text: missing

text: quarters,

text: or

text: on

text: data

text: with

text: pixels

text: with

text: a

text: random

text: pattern

text: of

text: missingness—the

text: latter

text: is

text: denoted

text: as

text: Denoising

text: encoder

text: in

text: the

text: tables.

text: In

text: Table

text: we

text: see

text: that

text: mean

text: imputation

text: (eq.

text: improves

text: over

text: global

text: means,

text: but

text: that

text: training

text: the

text: denoising

text: autoencoder

text: (which

text: uses

text: the

text: same

text: input

text: xmi

text: as

text: mean

text: imputation)

text: gives

text: further

text: improvements.

text: Note

text: that

text: on

text: the

text: quarters

text: data,

text: the

text: Denoising

text: encoder

text: results

text: are

text: quite

text: a

text: lot

text: worse

text: than

text: the

text: DE

text: which

text: had

text: been

text: trained

text: on

text: missing-quarters

text: training

text: data.

text: For

text: both

text: the

text: random

text: and

text: quarters

text: data,

text: the

text: exact

text: inference

text: method

text: is

text: the

text: best,

text: as

text: expected.

text: For

text: the

text: Frey

text: data

text: the

text: mean-squared

text: imputation

text: error

text: results

text: are

text: shown

text: in

text: Table

text: and

text: a

text: comparison

text: on

text: a

text: speciﬁc

text: image

text: is

text: shown

text: in

text: Figure

text: Again

text: mean

text: imputation

text: improves

text: noticeably

text: over

text: global

text: means,

text: but

text: it

text: still

text: gives

text: considerably

text: more

text: error

text: than

text: using

text: the

text: denoising

text: encoder

text: or

text: exact

text: inference.

text: For

text: data

text: with

text: random

text: missingness

text: exact

text: inference

text: signiﬁcantly

text: outperforms

text: the

text: denoising

text: encoder

text: on

text: the

text: test

text: set.

text: For

text: the

text: quarters

text: dataset,

text: we

text: observe

text: slightly

text: better

text: performance

text: of

text: the

text: denoising

text: encoder

text: than

text: exact

text: inference

text: on

text: the

text: test

text: set.

text: Notice

text: also

text: that

text: when

text: using

text: the

text: Denoising

text: encoder

text: (training

text: using

text: random

text: patterns

text: of

text: missingness)

text: the

text: performance

text: is

text: signiﬁcantly

text: worse.

text: The

text: slight

text: performance

text: gain

text: of

text: the

text: DE

text: on

text: the

text: quarters

text: data

text: over

text: exact

text: inference

text: may

text: be

text: due

text: to

text: the

text: fact

text: that

text: exact

text: inference

text: applies

text: to

text: the

text: PPCA

text: model,

text: but

text: as

text: the

text: Frey

text: data

text: was

text: not

text: generated

text: from

text: this

text: model

text: there

text: can

text: be

text: some

text: room

text: for

text: improvement

text: over

text: the

text: “exact”

text: method.

text: Also

text: note

text: that

text: for

text: the

text: quarters

text: data

text: only

text: four

text: patterns

text: of

text: missingness

text: are

text: used,

text: so

text: the

text: DE

text: network

text: is

text: likely

text: to

text: be

text: able

text: to

text: learn

text: about

text: this

text: more

text: easily

text: than

text: random

text: missingness.

text: There

text: are

text: noticeable

text: diﬀerences

text: between

text: the

text: training

text: and

text: test

text: set

text: errors

text: for

text: the

text: DE

text: model.

text: This

text: can

text: be

text: at

text: least

text: partially

text: explained

text: by

text: noting

text: that

text: on

text: the

text: training

text: set

text: the

text: DE

text: model

text: has

text: access

text: to

text: the

text: posterior

text: for

text: z

text: based

text: on

text: the

text: complete

text: data,

text: so

text: it

text: is

text: not

text: surprising

text: that

text: it

text: does

text: better

text: here.

text: Conclusions

text: Above

text: we

text: have

text: discussed

text: various

text: approaches

text: to

text: handling

text: missing

text: data

text: with

text: variational

text: autoencoders.

text: We

text: have

text: shown

text: how

text: to

text: calculate

text: exactly

text: the

text: latent

text: posterior

text: distribution

text: for

text: the

text: factor

text: analysis

text: (FA)

text: model

text: in

text: the

text: presence

text: of

text: missing

text: data.

text: This

text: solution

text: exhibits

text: a

text: non-trivial

text: dependence

text: on

text: the

text: pattern

text: of

text: missingness,

text: requiring

text: a

text: diﬀerent

text: matrix

text: inversion

text: for

text: each

text: pattern.

text: We

text: have

text: also

text: outlined

text: an

text: alternative

text: method,

text: the

text: denoising

text: encoder,

text: which

text: predicts

text: the

text: latent

text: z

text: from

text: the

text: corrupted

text: xmi

text: via

text: linear

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: regression. Our experimental results show the eﬀectiveness of these two methods on the Frey faces dataset with random or structured missingness. A limitation of the DE method is that the structure of the missingness needs to be known at training time—our results demonstrate that performance deteriorates markedly when the missingness assumptions at train and test time diﬀer. Possible future directions include the use of nonlinear DE models, and the development of methods to handle missing data in nonlinear autoencoders. Acknowledgements CW thanks Alfredo Nazabal for helpful discussions on eq. 6. The work of CW is supported in part by EPSRC grant EP/N510129/1 to the Alan Turing Institute. CN is supported by a PhD studentship from the EPSRC CDT in Data Science EP/L016427/1. References Bishop, C. M. (2006). Pattern Recognition and Machine Learning. Springer. Dray, S. and Josse, J. (2015). Principal components analysis with missing values: a comparitive survey of methods. Plant Ecology, 216:657–667. Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., and Bengio, Y. (2014). Generative Adversarial Nets. In Advances in Neural Information Processing Systems 27, pages 2672–2680. Kingma, D. P. and Welling, M. (2014). Auto-Encoding Variational Bayes. In ICLR. Little, R. J. A. and Rubin, D. B. (1987). Statistical Analysis with Missing Data. Wiley. Rezende, D. J., Mohamed, S., and Wierstra, D. (2014). Stochastic Backpropagation and Approximate Inference in Deep Generative Models. In ICML. Tipping, M. E. and Bishop, C. M. (1999). Probabilistic principal components analysis. J. Roy. Statistical Society B, 61(3):611–622. Vincent, P., Larochelle, H., Bengio, Y., and Manzagol, P. (2008). Extracting and Composing Robust Features with Denoising Autoencoders. In Proceedings of the Twenty-ﬁfth International Conference on Machine Learning (ICML‘08), pages 1096–1103. Williams, C. K. I. and Agakov, F. V. (2002). Products of Gaussians and Probabilistic Minor Component Analysis. Neural Computation, 14(5):1169–1182. 6

text: regression.

text: Our

text: experimental

text: results

text: show

text: the

text: eﬀectiveness

text: of

text: these

text: two

text: methods

text: on

text: the

text: Frey

text: faces

text: dataset

text: with

text: random

text: or

text: structured

text: missingness.

text: A

text: limitation

text: of

text: the

text: DE

text: method

text: is

text: that

text: the

text: structure

text: of

text: the

text: missingness

text: needs

text: to

text: be

text: known

text: at

text: training

text: time—our

text: results

text: demonstrate

text: that

text: performance

text: deteriorates

text: markedly

text: when

text: the

text: missingness

text: assumptions

text: at

text: train

text: and

text: test

text: time

text: diﬀer.

text: Possible

text: future

text: directions

text: include

text: the

text: use

text: of

text: nonlinear

text: DE

text: models,

text: and

text: the

text: development

text: of

text: methods

text: to

text: handle

text: missing

text: data

text: in

text: nonlinear

text: autoencoders.

text: Acknowledgements

text: CW

text: thanks

text: Alfredo

text: Nazabal

text: for

text: helpful

text: discussions

text: on

text: eq.

text: The

text: work

text: of

text: CW

text: is

text: supported

text: in

text: part

text: by

text: EPSRC

text: grant

text: to

text: the

text: Alan

text: Turing

text: Institute.

text: CN

text: is

text: supported

text: by

text: a

text: PhD

text: studentship

text: from

text: the

text: EPSRC

text: CDT

text: in

text: Data

text: Science

text: References

text: Bishop,

text: C.

text: M.

text: Pattern

text: Recognition

text: and

text: Machine

text: Learning.

text: Springer.

text: Dray,

text: S.

text: and

text: Josse,

text: J.

text: Principal

text: components

text: analysis

text: with

text: missing

text: values:

text: a

text: comparitive

text: survey

text: of

text: methods.

text: Plant

text: Ecology,

text: Goodfellow,

text: I.,

text: Pouget-Abadie,

text: J.,

text: Mirza,

text: M.,

text: Xu,

text: B.,

text: Warde-Farley,

text: D.,

text: Ozair,

text: S.,

text: Courville,

text: A.,

text: and

text: Bengio,

text: Y.

text: Generative

text: Adversarial

text: Nets.

text: In

text: Advances

text: in

text: Neural

text: Information

text: Processing

text: Systems

text: pages

text: Kingma,

text: D.

text: P.

text: and

text: Welling,

text: M.

text: Auto-Encoding

text: Variational

text: Bayes.

text: In

text: ICLR.

text: Little,

text: R.

text: J.

text: A.

text: and

text: Rubin,

text: D.

text: B.

text: Statistical

text: Analysis

text: with

text: Missing

text: Data.

text: Wiley.

text: Rezende,

text: D.

text: J.,

text: Mohamed,

text: S.,

text: and

text: Wierstra,

text: D.

text: Stochastic

text: Backpropagation

text: and

text: Approximate

text: Inference

text: in

text: Deep

text: Generative

text: Models.

text: In

text: ICML.

text: Tipping,

text: M.

text: E.

text: and

text: Bishop,

text: C.

text: M.

text: Probabilistic

text: principal

text: components

text: analysis.

text: J.

text: Roy.

text: Statistical

text: Society

text: B,

text: Vincent,

text: P.,

text: Larochelle,

text: H.,

text: Bengio,

text: Y.,

text: and

text: Manzagol,

text: P.

text: Extracting

text: and

text: Composing

text: Robust

text: Features

text: with

text: Denoising

text: Autoencoders.

text: In

text: Proceedings

text: of

text: the

text: Twenty-ﬁfth

text: International

text: Conference

text: on

text: Machine

text: Learning

text: pages

text: Williams,

text: C.

text: K.

text: I.

text: and

text: Agakov,

text: F.

text: V.

text: Products

text: of

text: Gaussians

text: and

text: Probabilistic

text: Minor

text: Component

text: Analysis.

text: Neural

text: Computation,


text: 0.2 0.2 DOS DOS DOS by SMW DOS by SMW 0.15 0.15 0.1 0.1 0.05 0.05 0 0 1000 2000 3000 4000 20 40 60 80 100 eV eV Figure 3.2: Left: DOS for Ethanol molecule vs. its recovering by using the trace of matrix resolvents; Right: zoom on the small energy interval. 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 500 1000 1500 2000 2500 3000 3500 4000 matrix size Figure 3.3: Algorithm in Theorem 3.2: the rescaled CPU time T/R2 versus n. complexity of these calculations (reduced model) by using a certain smoothness in t in the × parametric matrix resolvent by introducing the low rank approximation of the large n2 M matrix E − − K − − ∈ R × (t) = [E(t ) 1, . . . , E(t ) 1] and (t) = [K(t ) 1, . . . , K(t ) 1] R2 M 1 M 1 M − − obtained by concatenation of vectorized matrices E(t ) 1 and K(t ) 1, m = 1, . . . , M, m m respectively. The idea is that − − − E(t) 1 = blockdiag[P (t) 1, D(t) 1] ∈ deﬁnes an analytic matrix family on the spectral interval t [0, a], and so is the family of { − } core matrices K 1(t) . This favorable property allows the model reduction via low rank E (cid:88)K ∈ approximation of the matrix families (t) and (t), t τ. Suppose that the representations − RK K(t) 1 = c (t)K k k k=1 (cid:88) and − − − RE E(t) 1 = blockdiag[P (t) 1, D(t) 1] = p (t)E m m m=1 14 2 R/T emit delacser ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: DOS

text: DOS

text: DOS

text: by

text: SMW

text: DOS

text: by

text: SMW

text: eV

text: eV

text: Figure

text: Left:

text: DOS

text: for

text: Ethanol

text: molecule

text: vs.

text: its

text: recovering

text: by

text: using

text: the

text: trace

text: of

text: matrix

text: resolvents;

text: Right:

text: zoom

text: on

text: the

text: small

text: energy

text: interval.

text: matrix

text: size

text: Figure

text: Algorithm

text: in

text: Theorem

text: the

text: rescaled

text: CPU

text: time

text: versus

text: n.

text: complexity

text: of

text: these

text: calculations

text: (reduced

text: model)

text: by

text: using

text: a

text: certain

text: smoothness

text: in

text: t

text: in

text: the

text: parametric

text: matrix

text: resolvent

text: by

text: introducing

text: the

text: low

text: rank

text: approximation

text: of

text: the

text: large

text: M

text: matrix

text: E

text: K

text: R

text: (t)

text: [E(t

text: E(t

text: and

text: (t)

text: [K(t

text: K(t

text: M

text: M

text: M

text: obtained

text: by

text: concatenation

text: of

text: vectorized

text: matrices

text: E(t

text: and

text: K(t

text: m

text: M,

text: m

text: m

text: respectively.

text: The

text: idea

text: is

text: that

text: E(t)

text: blockdiag[P

text: (t)

text: D(t)

text: deﬁnes

text: an

text: analytic

text: matrix

text: family

text: on

text: the

text: spectral

text: interval

text: t

text: a],

text: and

text: so

text: is

text: the

text: family

text: of

text: core

text: matrices

text: K

text: This

text: favorable

text: property

text: allows

text: the

text: model

text: reduction

text: via

text: low

text: rank

text: E

text: approximation

text: of

text: the

text: matrix

text: families

text: (t)

text: and

text: (t),

text: t

text: τ.

text: Suppose

text: that

text: the

text: representations

text: RK

text: K(t)

text: c

text: (t)K

text: k

text: k

text: and

text: RE

text: E(t)

text: blockdiag[P

text: (t)

text: D(t)

text: p

text: (t)E

text: m

text: m

text: R/T

text: emit

text: delacser

text: ##LTFigure##

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