text: 18 ∇ where φ˜ = (∂φ˜ /∂y , . . . , ∂φ˜ /∂y ) and J (θ) is the Jacobian matrix of G for y = 0. λi λi 1 λi n1 G (Note that φ˜ (0, θ) = 0, according to Property 2.) Injecting these two approximations into (7), λi we obtain for the ﬁrst order terms ∇ ∇ d φ˜ ∇ JT (θ) φ˜ (0, θ) + ω λi (0, θ) = λ φ˜ (0, θ) . G λi dθ i λi ∇ where the derivative of φ˜ (0, θ) is well-deﬁned according to the assumption. The solution is λi given by ∇ ∇ φ˜ (0, θ) = Ψ(θ) φ˜ (0, 0) eλiθ/ω (16) λi λi where the fundamental matrix Ψ(θ) satisﬁes Ψ(0) = I and dΨ − 1 = JT Ψ . dθ ω G − We remark that the differentiation of Ψ 1Ψ = I yields − dΨ 1 − − dΨ − 1 − = Ψ 1 Ψ 1 = Ψ 1JT dθ dθ ω G or equivalently dΦ 1 = J Φ (17) G dθ ω − with Φ = Ψ T . We can rewrite (16) as ∇ ∇ ΦT (θ) φ˜ (0, θ) = φ˜ (0, 0) eλiθ/ω λi λi and for θ = 2π, we obtain ∇ ∇ ΦT (2π) φ˜ (0, 0) = φ˜ (0, 0) eλiT . (18) λi λi ∇ · (cid:54) Since the gradient satisﬁes φ (xγ) v = 0, it has a component tangent to the isochron, in the λi i ∇ (cid:54) transverse direction related to y, so that φ˜ (0, 0) = 0. The relationship (18) then implies that λi exp(λ T ) is an eigenvalue of the monodromy matrix Φ(2π). It follows from (17) that exp(λ T ) i i is a Floquet multiplier of the limit cycle, or equivalently λ is a Floquet exponent. In addition, i ∇ − φ˜ (0, 0) is a left eigenvector of Φ(2π), so that it is perpendicular to N 2 Floquet vectors λi May 26, 2020 DRAFT ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: φ˜

text: (∂φ˜

text: /∂y

text: ∂φ˜

text: /∂y

text: and

text: J

text: (θ)

text: is

text: the

text: Jacobian

text: matrix

text: of

text: G

text: for

text: y

text: λi

text: λi

text: λi

text: G

text: (Note

text: that

text: φ˜

text: θ)

text: according

text: to

text: Property

text: Injecting

text: these

text: two

text: approximations

text: into

text: λi

text: we

text: obtain

text: for

text: the

text: ﬁrst

text: order

text: terms

text: d

text: φ˜

text: JT

text: (θ)

text: φ˜

text: θ)

text: ω

text: λi

text: θ)

text: λ

text: φ˜

text: θ)

text: G

text: λi

text: dθ

text: i

text: λi

text: where

text: the

text: derivative

text: of

text: φ˜

text: θ)

text: is

text: well-deﬁned

text: according

text: to

text: the

text: assumption.

text: The

text: solution

text: is

text: λi

text: given

text: by

text: φ˜

text: θ)

text: Ψ(θ)

text: φ˜

text: eλiθ/ω

text: λi

text: λi

text: where

text: the

text: fundamental

text: matrix

text: Ψ(θ)

text: satisﬁes

text: I

text: and

text: dΨ

text: JT

text: Ψ

text: dθ

text: ω

text: G

text: We

text: remark

text: that

text: the

text: differentiation

text: of

text: Ψ

text: 1Ψ

text: I

text: yields

text: dΨ

text: dΨ

text: Ψ

text: Ψ

text: Ψ

text: dθ

text: dθ

text: ω

text: G

text: or

text: equivalently

text: dΦ

text: J

text: Φ

text: G

text: dθ

text: ω

text: with

text: Φ

text: Ψ

text: T

text: We

text: can

text: rewrite

text: as

text: ΦT

text: (θ)

text: φ˜

text: θ)

text: φ˜

text: eλiθ/ω

text: λi

text: λi

text: and

text: for

text: θ

text: we

text: obtain

text: ΦT

text: φ˜

text: φ˜

text: eλiT

text: λi

text: λi

text: Since

text: the

text: gradient

text: satisﬁes

text: φ

text: (xγ)

text: v

text: it

text: has

text: a

text: component

text: tangent

text: to

text: the

text: isochron,

text: in

text: the

text: λi

text: i

text: transverse

text: direction

text: related

text: to

text: y,

text: so

text: that

text: φ˜

text: The

text: relationship

text: then

text: implies

text: that

text: λi

text: exp(λ

text: T

text: is

text: an

text: eigenvalue

text: of

text: the

text: monodromy

text: matrix

text: It

text: follows

text: from

text: that

text: exp(λ

text: T

text: i

text: i

text: is

text: a

text: Floquet

text: multiplier

text: of

text: the

text: limit

text: cycle,

text: or

text: equivalently

text: λ

text: is

text: a

text: Floquet

text: exponent.

text: In

text: addition,

text: i

text: φ˜

text: is

text: a

text: left

text: eigenvector

text: of

text: so

text: that

text: it

text: is

text: perpendicular

text: to

text: N

text: Floquet

text: vectors

text: λi

text: May

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text: 23 Estimation of the basin of attraction: According to the result of Proposition 2, we can investigate the global stability of a ﬁxed point by computing the Koopman eigenfunctions with the above numerical method. In addition, the method can be used to estimate the basin of attraction of the equilibrium. This is performed as follows: (i) build a candidate Lyapunov function of the form (14), (ii) consider the region where this function is decreasing along the trajectories, and (iii) the largest closed level set of the Lyapunov function which is included in that region provides an inner approximation of the basin of attraction. This procedure can also be used to verify the results obtained with Taylor polynomials of small degree and to yield an approximate (conservative) region of stability. Example 1. The dynamics − x˙ = x 1 2 − x˙ = x x + x2x 2 1 2 1 2 of the well-known Van der Pol oscillator, here in backward-time, admit an unstable limit cycle which corresponds to the boundary of the basin of attraction of the stable origin. As shown in Figure 1(a), the method provides an accurate estimation of the basin of attraction. Example 2. The dynamics x˙ = x (26) 1 2 − 1 − x˙ = 2x + x3 x (27) 2 1 3 1 2 is characterized by a locally stable ﬁxed point at the origin and two unstable saddle points √ ± at ( 6, 0). Figure 1(b) shows that the numerical method provides a good estimation of the basin of attraction (delimited by the stable manifold of the saddle points). However, the largest approximation is obtained with a 14th-order Taylor expansion and the method cannot capture the complete geometry of the basin. As explained below, this is due to the fact that the eigenfunctions admit a singularity at the saddle points and are analytic only on a subset of the basin of attraction. May 26, 2020 DRAFT ##LTLine## ##LTLine##

label: paragraph

text: Estimation

text: of

text: the

text: basin

text: of

text: attraction:

text: According

text: to

text: the

text: result

text: of

text: Proposition

text: we

text: can

text: investigate

text: the

text: global

text: stability

text: of

text: a

text: ﬁxed

text: point

text: by

text: computing

text: the

text: Koopman

text: eigenfunctions

text: with

text: the

text: above

text: numerical

text: method.

text: In

text: addition,

text: the

text: method

text: can

text: be

text: used

text: to

text: estimate

text: the

text: basin

text: of

text: attraction

text: of

text: the

text: equilibrium.

text: This

text: is

text: performed

text: as

text: follows:

text: (i)

text: build

text: a

text: candidate

text: Lyapunov

text: function

text: of

text: the

text: form

text: (ii)

text: consider

text: the

text: region

text: where

text: this

text: function

text: is

text: decreasing

text: along

text: the

text: trajectories,

text: and

text: (iii)

text: the

text: largest

text: closed

text: level

text: set

text: of

text: the

text: Lyapunov

text: function

text: which

text: is

text: included

text: in

text: that

text: region

text: provides

text: an

text: inner

text: approximation

text: of

text: the

text: basin

text: of

text: attraction.

text: This

text: procedure

text: can

text: also

text: be

text: used

text: to

text: verify

text: the

text: results

text: obtained

text: with

text: Taylor

text: polynomials

text: of

text: small

text: degree

text: and

text: to

text: yield

text: an

text: approximate

text: (conservative)

text: region

text: of

text: stability.

text: Example

text: The

text: dynamics

text: x˙

text: x

text: x˙

text: x

text: x

text: of

text: the

text: well-known

text: Van

text: der

text: Pol

text: oscillator,

text: here

text: in

text: backward-time,

text: admit

text: an

text: unstable

text: limit

text: cycle

text: which

text: corresponds

text: to

text: the

text: boundary

text: of

text: the

text: basin

text: of

text: attraction

text: of

text: the

text: stable

text: origin.

text: As

text: shown

text: in

text: Figure

text: the

text: method

text: provides

text: an

text: accurate

text: estimation

text: of

text: the

text: basin

text: of

text: attraction.

text: Example

text: The

text: dynamics

text: x˙

text: x

text: x˙

text: x

text: is

text: characterized

text: by

text: a

text: locally

text: stable

text: ﬁxed

text: point

text: at

text: the

text: origin

text: and

text: two

text: unstable

text: saddle

text: points

text: at

text: Figure

text: shows

text: that

text: the

text: numerical

text: method

text: provides

text: a

text: good

text: estimation

text: of

text: the

text: basin

text: of

text: attraction

text: (delimited

text: by

text: the

text: stable

text: manifold

text: of

text: the

text: saddle

text: points).

text: However,

text: the

text: largest

text: approximation

text: is

text: obtained

text: with

text: a

text: Taylor

text: expansion

text: and

text: the

text: method

text: cannot

text: capture

text: the

text: complete

text: geometry

text: of

text: the

text: basin.

text: As

text: explained

text: below,

text: this

text: is

text: due

text: to

text: the

text: fact

text: that

text: the

text: eigenfunctions

text: admit

text: a

text: singularity

text: at

text: the

text: saddle

text: points

text: and

text: are

text: analytic

text: only

text: on

text: a

text: subset

text: of

text: the

text: basin

text: of

text: attraction.

text: May

text: DRAFT

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text: 24 5 5th order 2 10th order 12th order 14th order x2 x2 0 0 5th order 10th order −2 20th order 40th order −5 −2 x1 0 2 −5 0 x1 5 (a) Example 1 (b) Example 2 Figure 1. The Taylor expansion-based method can be used to estimate the basin of attraction (black curve) of the stable ﬁxed point (black dot). (a) Better approximations are obtained as the degree of the Taylor polynomials is increased. (b) The best result is obtained with a 14th-order Taylor expansion. (In the two examples, λ is complex and the Lyapunov function is 1 V | | | | = φ = φ .) λ1 λ2 Example 3 (Non-analytic eigenfunctions). For the dynamics − − − − x˙ = 3/4x 1/8x + 1/4x x 1/4x2 1/2x3 , 1 1 2 1 2 2 1 − − x˙ = 1/8x x , 2 1 2 − × − the origin is globally stable in X = [ 2, 2] [ 2, 2]. However, the Taylor expansion of the eigenfunctions diverges in X, so that the method cannot prove global stability in this region. This is explained by the fact that the eigenfunctions are not analytic on X. Poincare´ linearization theorem implies that the Koopman eigenfunctions are analytic on the largest complex ball ∗ ∗ (cid:54) ∗ centered at the stable ﬁxed point x that does not contain a zero z = x of F (z) (see e.g. ∗ ≈ − ± [5]). Here, the vector ﬁeld admits two complex zeros z 0.035 1.211i (acting as ﬁctitious C ﬁxed points for the vector ﬁeld in ), so that the disk of analyticity of the eigenfunctions has a (cid:107) ∗(cid:107) ≈ radius z 1.212. B. Bernstein polynomial-based method for the ﬁxed point Since the eigenfunctions are always continuous (but maybe not analytic) on the region of attraction, we can approximate them with polynomials, according to Weierstrass theorem. Also, May 26, 2020 DRAFT ##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## 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text: 28 of the equilibrium, so that global stability can always be proven in some region close to the equilibrium (i.e. inside a closed level set of a candidate Lyapunov function of the form (14)). The method is therefore useful to obtain successive approximations of the region of stability: Taylor polynomials of higher degree are gradually added to the truncated basis and the computation of higher order Taylor coefﬁcients relies on the results obtained for lower orders in previous computations. In contrast, for the Bernstein polynomial-based method, inaccurate results cannot be used even in the neighborhood of the equilibrium. The degree of the polynomials must be increased and a different basis of Bernstein polynomials must be considered. The two methods require only a short analytic pre-processing (i.e. expansion of the vector ﬁeld in the chosen polynomial basis) and compute the Koopman eigenfunctions of planar systems in a few minutes. C. Bernstein polynomial-based method for the limit cycle When the attractor is a limit cycle, the eigenfunctions can be computed on a polynomial basis in the direction transverse to the attractor and on a Fourier basis in the direction tangential to the attractor. (Recall that, according to [17], an appropriate space of functions is the space of polynomials with coefﬁcients corresponding to periodic functions on the attractor.) Since the presence of an unstable ﬁxed point implies that the eigenfunctions are analytic only in a (usually close) neighborhood of the limit cycle, it is necessary to use Bernstein polynomials rather than monomials. We assume that the dynamics are two-dimensional (N = 2) and that the origin is contained inside the limit cycle (possibly after a translation of the dynamics). We consider polar-type coordinates (θ, y) such that x = xγ(θ) + (y + ∆)e (θ), where xγ(θ) is a point of the limit r cycle, e (θ) is a (not necessarily unit) radial vector aligned with xγ(θ), and ∆ is a constant. We r { | − } suppose here that the limit cycle—i.e. the set (θ, y) θ = [0, 2π), y = ∆ —can be parametrized by the variable θ. In the new coordinates, we have the dynamics θ˙ = F (θ, y), y˙ = F (θ, y) (see θ y (46)-(47) in Appendix A) and the eigenvalue equation (7) is given by ∂φ ∂φ F λ + F λ = λφ . (32) y θ λ ∂y ∂θ May 26, 2020 DRAFT ##LTLine## ##LTLine##

text: of

text: the

text: equilibrium,

text: so

text: that

text: global

text: stability

text: can

text: always

text: be

text: proven

text: in

text: some

text: region

text: close

text: to

text: the

text: equilibrium

text: (i.e.

text: inside

text: a

text: closed

text: level

text: set

text: of

text: a

text: candidate

text: Lyapunov

text: function

text: of

text: the

text: form

text: The

text: method

text: is

text: therefore

text: useful

text: to

text: obtain

text: successive

text: approximations

text: of

text: the

text: region

text: of

text: stability:

text: Taylor

text: polynomials

text: of

text: higher

text: degree

text: are

text: gradually

text: added

text: to

text: the

text: truncated

text: basis

text: and

text: the

text: computation

text: of

text: higher

text: order

text: Taylor

text: coefﬁcients

text: relies

text: on

text: the

text: results

text: obtained

text: for

text: lower

text: orders

text: in

text: previous

text: computations.

text: In

text: contrast,

text: for

text: the

text: Bernstein

text: polynomial-based

text: method,

text: inaccurate

text: results

text: cannot

text: be

text: used

text: even

text: in

text: the

text: neighborhood

text: of

text: the

text: equilibrium.

text: The

text: degree

text: of

text: the

text: polynomials

text: must

text: be

text: increased

text: and

text: a

text: different

text: basis

text: of

text: Bernstein

text: polynomials

text: must

text: be

text: considered.

text: The

text: two

text: methods

text: require

text: only

text: a

text: short

text: analytic

text: pre-processing

text: (i.e.

text: expansion

text: of

text: the

text: vector

text: ﬁeld

text: in

text: the

text: chosen

text: polynomial

text: basis)

text: and

text: compute

text: the

text: Koopman

text: eigenfunctions

text: of

text: planar

text: systems

text: in

text: a

text: few

text: minutes.

text: C.

text: Bernstein

text: polynomial-based

text: method

text: for

text: the

text: limit

text: cycle

text: When

text: the

text: attractor

text: is

text: a

text: limit

text: cycle,

text: the

text: eigenfunctions

text: can

text: be

text: computed

text: on

text: a

text: polynomial

text: basis

text: in

text: the

text: direction

text: transverse

text: to

text: the

text: attractor

text: and

text: on

text: a

text: Fourier

text: basis

text: in

text: the

text: direction

text: tangential

text: to

text: the

text: attractor.

text: (Recall

text: that,

text: according

text: to

text: an

text: appropriate

text: space

text: of

text: functions

text: is

text: the

text: space

text: of

text: polynomials

text: with

text: coefﬁcients

text: corresponding

text: to

text: periodic

text: functions

text: on

text: the

text: attractor.)

text: Since

text: the

text: presence

text: of

text: an

text: unstable

text: ﬁxed

text: point

text: implies

text: that

text: the

text: eigenfunctions

text: are

text: analytic

text: only

text: in

text: a

text: (usually

text: close)

text: neighborhood

text: of

text: the

text: limit

text: cycle,

text: it

text: is

text: necessary

text: to

text: use

text: Bernstein

text: polynomials

text: rather

text: than

text: monomials.

text: We

text: assume

text: that

text: the

text: dynamics

text: are

text: two-dimensional

text: (N

text: and

text: that

text: the

text: origin

text: is

text: contained

text: inside

text: the

text: limit

text: cycle

text: (possibly

text: after

text: a

text: translation

text: of

text: the

text: dynamics).

text: We

text: consider

text: polar-type

text: coordinates

text: (θ,

text: y)

text: such

text: that

text: x

text: xγ(θ)

text: (y

text: ∆)e

text: (θ),

text: where

text: xγ(θ)

text: is

text: a

text: point

text: of

text: the

text: limit

text: r

text: cycle,

text: e

text: (θ)

text: is

text: a

text: (not

text: necessarily

text: unit)

text: radial

text: vector

text: aligned

text: with

text: xγ(θ),

text: and

text: is

text: a

text: constant.

text: We

text: r

text: suppose

text: here

text: that

text: the

text: limit

text: cycle—i.e.

text: the

text: set

text: (θ,

text: y)

text: θ

text: y

text: —can

text: be

text: parametrized

text: by

text: the

text: variable

text: θ.

text: In

text: the

text: new

text: coordinates,

text: we

text: have

text: the

text: dynamics

text: θ˙

text: F

text: (θ,

text: y),

text: y˙

text: F

text: (θ,

text: y)

text: (see

text: θ

text: y

text: in

text: Appendix

text: A)

text: and

text: the

text: eigenvalue

text: equation

text: is

text: given

text: by

text: ∂φ

text: ∂φ

text: F

text: λ

text: F

text: λ

text: λφ

text: y

text: θ

text: λ

text: ∂y

text: ∂θ

text: May

text: DRAFT

text: ##LTLine##

text: ##LTLine##


text: 6 ⊂ ∈ 2) There exists a neighborhood V(cid:92) X of A such that the limit set ω(x) of every x V is in A, i.e. (cid:44) { } ⊆ ∀ ∈ ω(x) ϕt(x), t > T A x V , ∈R T where denotes the closure of the set; 3) There exists no strictly smaller closed subset satisfying the above properties. Deﬁnition 3 (Global stability). An attractor (or a set) A is • stable if, for every (cid:15) > 0, there exists δ > 0 such that (cid:107) − (cid:107) ∈ ⇒ (cid:107) − (cid:107) ∀ ≥ min x y < δ , x X min ϕt(x) y < (cid:15) t 0 ; ∈ ∈ y A y A ⊆ ∈ • globally attractive in X if ω(x) A for all x X; • globally asymptotically stable if it is stable and globally attractive. Decomposition of the Koopman operator: As a preliminary to the main results of this F ⊆ F \ section, we introduce the subspace of functions with support on A = X A, i.e. Ac c F { ∈ F| ∀ ∈ } = f f(x) = 0 x A . Ac ∈ ∈ ∈ R ◦ ∈ F ∈ If x A, one has ϕt(x) A for all t , so that Utf(x) = f ϕt(x) = 0 for f and x A. Ac F F It follows that is invariant under Ut, and we denote by Ut the restriction of Ut to . Sim- Ac Ac Ac F| F| { | → C| ∈ F} ilarly, we consider the set of observables restricted to A, i.e. = f : A f . A A A ∈ F| Note that the extension f¯ of f to X, such that f¯ = f on A and f¯ = 0 on A , is generally A c F ⊆ F| → F| not in C0(X). The Koopman operator Ut : is associated with the ﬂow on A, A A A | R × → | | ◦ | | i.e. ϕ : A A, and is rigorously deﬁned by Ut (f ) = f ϕt = (Utf) . A A A A A A General stability results: Stability properties of the attractor are captured by the restriction Ut of the Koopman operator. The result is summarized in the following proposition (see also Ac [13]). Proposition 1. The attractor A of (2) is globally attractive in X if and only if ∀ ∈ F lim Ut f = 0 f , (5) t→∞ Ac Ac May 26, 2020 DRAFT ##LTLine## ##LTLine##

label: list

text: There

text: exists

text: a

text: neighborhood

text: X

text: of

text: A

text: such

text: that

text: the

text: limit

text: set

text: ω(x)

text: of

text: every

text: x

text: V

text: is

text: in

text: A,

text: i.e.

text: ω(x)

text: ϕt(x),

text: t

text: T

text: A

text: x

text: V

text: ∈R

text: T

text: where

text: denotes

text: the

text: closure

text: of

text: the

text: set;

text: There

text: exists

text: no

text: strictly

text: smaller

text: closed

text: subset

text: satisfying

text: the

text: above

text: properties.

text: Deﬁnition

text: (Global

text: stability).

text: An

text: attractor

text: (or

text: a

text: set)

text: A

text: is

text: stable

text: if,

text: for

text: every

text: there

text: exists

text: δ

text: such

text: that

text: min

text: x

text: y

text: δ

text: x

text: X

text: min

text: ϕt(x)

text: y

text: t

text: y

text: A

text: y

text: A

text: globally

text: attractive

text: in

text: X

text: if

text: ω(x)

text: A

text: for

text: all

text: x

text: X;

text: globally

text: asymptotically

text: stable

text: if

text: it

text: is

text: stable

text: and

text: globally

text: attractive.

text: Decomposition

text: of

text: the

text: Koopman

text: operator:

text: As

text: a

text: preliminary

text: to

text: the

text: main

text: results

text: of

text: this

text: F

text: F

text: section,

text: we

text: introduce

text: the

text: subspace

text: of

text: functions

text: with

text: support

text: on

text: A

text: X

text: A,

text: i.e.

text: Ac

text: c

text: F

text: F|

text: f

text: f(x)

text: x

text: A

text: Ac

text: R

text: F

text: If

text: x

text: A,

text: one

text: has

text: ϕt(x)

text: A

text: for

text: all

text: t

text: so

text: that

text: Utf(x)

text: f

text: ϕt(x)

text: for

text: f

text: and

text: x

text: A.

text: Ac

text: F

text: F

text: It

text: follows

text: that

text: is

text: invariant

text: under

text: Ut,

text: and

text: we

text: denote

text: by

text: Ut

text: the

text: restriction

text: of

text: Ut

text: to

text: Sim-

text: Ac

text: Ac

text: Ac

text: F|

text: F|

text: C|

text: F}

text: ilarly,

text: we

text: consider

text: the

text: set

text: of

text: observables

text: restricted

text: to

text: A,

text: i.e.

text: f

text: A

text: f

text: A

text: A

text: A

text: F|

text: Note

text: that

text: the

text: extension

text: f¯

text: of

text: f

text: to

text: X,

text: such

text: that

text: f¯

text: f

text: on

text: A

text: and

text: f¯

text: on

text: A

text: is

text: generally

text: A

text: c

text: F

text: F|

text: F|

text: not

text: in

text: The

text: Koopman

text: operator

text: Ut

text: is

text: associated

text: with

text: the

text: ﬂow

text: on

text: A,

text: A

text: A

text: A

text: R

text: i.e.

text: ϕ

text: A

text: A,

text: and

text: is

text: rigorously

text: deﬁned

text: by

text: Ut

text: (f

text: f

text: ϕt

text: (Utf)

text: A

text: A

text: A

text: A

text: A

text: A

text: General

text: stability

text: results:

text: Stability

text: properties

text: of

text: the

text: attractor

text: are

text: captured

text: by

text: the

text: restriction

text: Ut

text: of

text: the

text: Koopman

text: operator.

text: The

text: result

text: is

text: summarized

text: in

text: the

text: following

text: proposition

text: (see

text: also

text: Ac

text: Proposition

text: The

text: attractor

text: A

text: of

text: is

text: globally

text: attractive

text: in

text: X

text: if

text: and

text: only

text: if

text: F

text: lim

text: Ut

text: f

text: f

text: t→∞

text: Ac

text: Ac

text: May

text: DRAFT

text: ##LTLine##

text: ##LTLine##


text: (cid:89) (cid:80) √ | − | − − Φ˜((cid:126)λ1, . . . ,(cid:126)λN) = (cid:126)λi (cid:126)λj 1 ξ e N2 i (cid:126)λ2i , (3.17) i>j satisﬁes equation (3.16) to leading order in the large N expansion and for the “sta- (cid:112) bilised” energy E we obtain: 1 − E = 1 ξ + O(1/N) . (3.18) 2 To verify the consistency of this approximation we treat the two body interaction term as a perturbation δV to the potential of relative order 1/N, therefore the corresponding correction to the energy δE would also be of order 1/N and we conclude that as long as ∼ E 1 we can trust the wave function (3.17). Note that this suggests that the parameter ξ is restricted to ξ < 1, which is consistent with the general arguments (presented in the ≥ previous section) that the model is unstable for ξ 1. Let us comment on the properties of the eigenvalue distribution for ξ < 1.   The probability density corresp(cid:88)onding to the wa(cid:88)ve function (3.17) can be written as:   √ − | | − 1 ξ − Φ˜ 2 = exp N (cid:126)λ2 + log((cid:126)λ (cid:126)λ )2 . (3.19) i 2 (cid:54) i j i i=j (cid:90) The corresponding eigenvalue dis(cid:89)tribution: N | | ρ((cid:126)λ) = dpλ Φ˜((cid:126)λ,(cid:126)λ , . . . ,(cid:126)λ ) 2 (3.20) i 2 N i=2 is then equivalent to the eigenvalue distribution of the commuting matrix model de- (cid:88) (cid:88) scribed by the action: √ − − 1 ξ − S = N (cid:126)λ2 log((cid:126)λ (cid:126)λ )2 , (3.21) i 2 (cid:54) i j i i=j which modulo normalisation constants was analysed in refs. [15]-[18]. The correspond- ing saddle point equation is given by: (cid:88) − √ 1 1 (cid:126)λ (cid:126)λ 1 − ξ (cid:126)λi = N ((cid:126)λ i− (cid:126)λ j)2 , (3.22) j i j (cid:54) j=i comparing this to equation (2.6) in ref. [18] we conclude that all of the results of [18] for gaussian potential are valid in here provided we multiply the radius of the distribution by − ≤ ≤ a factor of (1 ξ )1/4. Therefore, we obtain that for 2 p 4 the radius of the distribution 4 is given by [18]: – 13 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: ξ

text: e

text: i

text: i>j

text: satisﬁes

text: equation

text: to

text: leading

text: order

text: in

text: the

text: large

text: N

text: expansion

text: and

text: for

text: the

text: “sta-

text: bilised”

text: energy

text: E

text: we

text: obtain:

text: E

text: ξ

text: To

text: verify

text: the

text: consistency

text: of

text: this

text: approximation

text: we

text: treat

text: the

text: two

text: body

text: interaction

text: term

text: as

text: a

text: perturbation

text: δV

text: to

text: the

text: potential

text: of

text: relative

text: order

text: therefore

text: the

text: corresponding

text: correction

text: to

text: the

text: energy

text: δE

text: would

text: also

text: be

text: of

text: order

text: and

text: we

text: conclude

text: that

text: as

text: long

text: as

text: E

text: we

text: can

text: trust

text: the

text: wave

text: function

text: Note

text: that

text: this

text: suggests

text: that

text: the

text: parameter

text: ξ

text: is

text: restricted

text: to

text: ξ

text: which

text: is

text: consistent

text: with

text: the

text: general

text: arguments

text: (presented

text: in

text: the

text: previous

text: section)

text: that

text: the

text: model

text: is

text: unstable

text: for

text: ξ

text: Let

text: us

text: comment

text: on

text: the

text: properties

text: of

text: the

text: eigenvalue

text: distribution

text: for

text: ξ

text: The

text: probability

text: density

text: to

text: the

text: function

text: can

text: be

text: written

text: as:

text: ξ

text: Φ˜

text: exp

text: N

text: i

text: i

text: j

text: i

text: i=j

text: The

text: corresponding

text: eigenvalue

text: N

text: dpλ

text: i

text: N

text: is

text: then

text: equivalent

text: to

text: the

text: eigenvalue

text: distribution

text: of

text: the

text: commuting

text: matrix

text: model

text: de-

text: scribed

text: by

text: the

text: action:

text: ξ

text: S

text: N

text: i

text: i

text: j

text: i

text: i=j

text: which

text: modulo

text: normalisation

text: constants

text: was

text: analysed

text: in

text: refs.

text: The

text: correspond-

text: ing

text: saddle

text: point

text: equation

text: is

text: given

text: by:

text: ξ

text: N

text: i−

text: j

text: i

text: j

text: j=i

text: comparing

text: this

text: to

text: equation

text: in

text: ref.

text: we

text: conclude

text: that

text: all

text: of

text: the

text: results

text: of

text: for

text: gaussian

text: potential

text: are

text: valid

text: in

text: here

text: provided

text: we

text: multiply

text: the

text: radius

text: of

text: the

text: distribution

text: by

text: a

text: factor

text: of

text: ξ

text: Therefore,

text: we

text: obtain

text: that

text: for

text: p

text: the

text: radius

text: of

text: the

text: distribution

text: is

text: given

text: by

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##


text: R result in equation (A.10) can be split into contributions from the ﬂag directions and K x contribution from the mixed terms , the two terms read: (cid:88) λ (cid:88) R 1 − 1 ∆2 jk x = N (cid:54) ∆2 (cid:88)2 (cid:54) (cid:54) ∆2(cid:88)∆2 , (A.12) i=j ij i=j=k ij ik K − − 1 − ∆(cid:126) . ∆(cid:126) ij ik = (4p 2) 4 . (A.13) λ (cid:54) ∆2 (cid:54) (cid:54) ∆2 ∆2 i=j ij i=j=k ij ik (cid:113) | | Equation (A.12) is our expression for the curvature of a ﬂag manifold with radii ∆(cid:126) = ij ∆2 . ij References [1] E. Brezin, C. Itzykson, G. Parisi and J. B. Zuber, “Planar Diagrams,” Commun. Math. Phys. 59, 35 (1978). [2] Y. Shimamune, “On the Phase Structure of Large N Matrix Models and Gauge Models,” Phys. Lett. B 108 (1982) 407. [3] T. Banks, W. Fischler, S. H. Shenker and L. Susskind, “M theory as a matrix model: A Conjecture”, Phys. Rev. D55 (1997) 5112 [hep-th/9610043]. [4] P. K. Townsend, “The eleven-dimensional supermembrane revisited”, Phys. Lett. B350 (1995) 184 [hep-th/9501068]. [5] D. E. Berenstein, J. M. Maldacena and H. S. Nastase, “Strings in ﬂat space and pp waves from N=4 Super Yang Mills,” AIP Conf. Proc. 646 (2003) 3. [6] O. Aharony, O. Bergman and D. L. Jaﬀeris, “Fractional M2-branes,” JHEP 0811 (2008) 043 [arXiv:0807.4924 [hep-th]]. [7] S. Kovacs, Y. Sato and H. Shimada, “Membranes from monopole operators in ABJM theory: large angular momentum and M-theoretic AdS /CFT ,” arXiv:1310.0016 [hep-th]. 4 3 [8] J. Hoppe “Quantum Theory Of A Massless Relativistic Surface And A Two Dimensional Bound State Problem”, Ph.D. Thesis, Massachusetts Institute of Technology, (1982). [9] B. de Wit, J. Hoppe and H. Nicolai, “On the Quantum Mechanics of Supermembranes,” Nucl. Phys. B 305 (1988) 545. [10] D. Berenstein, JHEP 0601, 125 (2006) [hep-th/0507203]. [11] B. S. DeWitt, “Dynamical theory in curved spaces. 1. A Review of the classical and quantum action principles,” Rev. Mod. Phys. 29 (1957) 377. [12] K. S. Cheng, “Quantization of a general dynamical system by feynman’s path integration formulation,” J. Math. Phys. 13 (1972) 1723. [13] H. Kleinert, “Quantum Mechanics and path integrals in Spaces With Curvature and Torsion,” Mod. Phys. Lett. A 4 (1989) 2329. [14] L. D. Landau, E. M. Lifshitz, “Quantum Mechanics” Volume 3 of A Course of Theoretical Physics, Pergamon Press, 1965 [15] D. Berenstein, D. H. Correa and S. E. Vazquez, “All loop BMN state energies from matrices,” JHEP 0602, 048 (2006) [hep-th/0509015]. – 18 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: R

text: result

text: in

text: equation

text: can

text: be

text: split

text: into

text: contributions

text: from

text: the

text: ﬂag

text: directions

text: and

text: K

text: x

text: contribution

text: from

text: the

text: mixed

text: terms

text: the

text: two

text: terms

text: read:

text: λ

text: R

text: ∆2

text: jk

text: x

text: N

text: ∆2

text: i=j

text: ij

text: i=j=k

text: ij

text: ik

text: K

text: ij

text: ik

text: λ

text: ∆2

text: ∆2

text: ∆2

text: i=j

text: ij

text: i=j=k

text: ij

text: ik

text: Equation

text: is

text: our

text: expression

text: for

text: the

text: curvature

text: of

text: a

text: ﬂag

text: manifold

text: with

text: radii

text: ij

text: ∆2

text: ij

text: References

text: E.

text: Brezin,

text: C.

text: Itzykson,

text: G.

text: Parisi

text: and

text: J.

text: B.

text: Zuber,

text: “Planar

text: Diagrams,”

text: Commun.

text: Math.

text: Phys.

text: Y.

text: Shimamune,

text: “On

text: the

text: Phase

text: Structure

text: of

text: Large

text: N

text: Matrix

text: Models

text: and

text: Gauge

text: Models,”

text: Phys.

text: Lett.

text: B

text: T.

text: Banks,

text: W.

text: Fischler,

text: S.

text: H.

text: Shenker

text: and

text: L.

text: Susskind,

text: “M

text: theory

text: as

text: a

text: matrix

text: model:

text: A

text: Conjecture”,

text: Phys.

text: Rev.

text: P.

text: K.

text: Townsend,

text: “The

text: eleven-dimensional

text: supermembrane

text: revisited”,

text: Phys.

text: Lett.

text: D.

text: E.

text: Berenstein,

text: J.

text: M.

text: Maldacena

text: and

text: H.

text: S.

text: Nastase,

text: “Strings

text: in

text: ﬂat

text: space

text: and

text: pp

text: waves

text: from

text: Super

text: Yang

text: Mills,”

text: AIP

text: Conf.

text: Proc.

text: O.

text: Aharony,

text: O.

text: Bergman

text: and

text: D.

text: L.

text: Jaﬀeris,

text: “Fractional

text: JHEP

text: [hep-th]].

text: S.

text: Kovacs,

text: Y.

text: Sato

text: and

text: H.

text: Shimada,

text: “Membranes

text: from

text: monopole

text: operators

text: in

text: ABJM

text: theory:

text: large

text: angular

text: momentum

text: and

text: M-theoretic

text: AdS

text: /CFT

text: [hep-th].

text: J.

text: Hoppe

text: “Quantum

text: Theory

text: Of

text: A

text: Massless

text: Relativistic

text: Surface

text: And

text: A

text: Two

text: Dimensional

text: Bound

text: State

text: Problem”,

text: Ph.D.

text: Thesis,

text: Massachusetts

text: Institute

text: of

text: Technology,

text: B.

text: de

text: Wit,

text: J.

text: Hoppe

text: and

text: H.

text: Nicolai,

text: “On

text: the

text: Quantum

text: Mechanics

text: of

text: Supermembranes,”

text: Nucl.

text: Phys.

text: B

text: D.

text: Berenstein,

text: JHEP

text: B.

text: S.

text: DeWitt,

text: “Dynamical

text: theory

text: in

text: curved

text: spaces.

text: A

text: Review

text: of

text: the

text: classical

text: and

text: quantum

text: action

text: principles,”

text: Rev.

text: Mod.

text: Phys.

text: K.

text: S.

text: Cheng,

text: “Quantization

text: of

text: a

text: general

text: dynamical

text: system

text: by

text: feynman’s

text: path

text: integration

text: formulation,”

text: J.

text: Math.

text: Phys.

text: H.

text: Kleinert,

text: “Quantum

text: Mechanics

text: and

text: path

text: integrals

text: in

text: Spaces

text: With

text: Curvature

text: and

text: Torsion,”

text: Mod.

text: Phys.

text: Lett.

text: A

text: L.

text: D.

text: Landau,

text: E.

text: M.

text: Lifshitz,

text: “Quantum

text: Mechanics”

text: Volume

text: of

text: A

text: Course

text: of

text: Theoretical

text: Physics,

text: Pergamon

text: Press,

text: D.

text: Berenstein,

text: D.

text: H.

text: Correa

text: and

text: S.

text: E.

text: Vazquez,

text: “All

text: loop

text: BMN

text: state

text: energies

text: from

text: matrices,”

text: JHEP

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: which is our ﬁnal expression for the distribution ρ˜ and agrees with the result in ref. [1], as it should, since we have only made the semiclassical analysis more explicit. 2.1 Gaussian potential Let us now focus on the speciﬁc case of a gaussian potential V (X) = X2/2. In this case the rescaled individual wave functions f˜ satisfy the Schr¨odinger equation for the one (cid:20) n(cid:21) dimensional harmonic oscillator: − 1 ∂2 1 + λ2 f˜ (λ) = ε f˜ (λ) , (2.28) 2N2 ∂λ2 2 n n n where: − H√(λ)e N λ2/2 (1/2 + n) f˜ (λ) = n ; ε = ; (2.29) n n π1/22nn! N and H (λ) are the Hermite polynomials. The corresponding multi-particle wave func- n tion is given by: (cid:88) (cid:80) − φ˜(λ1 . . . λN) = CN H(cid:89)i1(λ1) . . . HiN (λN(cid:80)) e N i λ2i/2 i1,...,iN − − − = 2N(N2 1) CN (λi λj) e N i λ2i/2 , (2.30) (cid:81) i>j − − N 1 where C 2 = N! (π1/22nn!) and we have used the properties of the Hermite poly- N n=0 nomials to arrive at the last equality. Equation (2.30) suggests that the bosonic wave (cid:89) function ψ˜ before the change of variables (2.12) is given by: − − Ψ˜ (λ1 . . . λN) = 2N(N2 1) CN e N λ2i/2 . (2.31) i Therefore, the bosonic wave function of the ground state of the system is a product of individual wave functions. We could have easily guessed this result even directly in the bosonic description without introducing the new wave function (2.12). Note that this is true only for the case of a gaussian potential. In fact, in the next section we will show that this holds even for the case of more than one commuting matrices with a gaussian potential [10]. 3 Commuting matrix model In this section we focus on the quantisation of the commuting matrix model. While we will set up the problem for general interaction potential the main ﬁndings that we present are for the special case of gaussian potential, ﬁrst discussed in ref. [10]. Our starting point is the lagrangian: – 8 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: which

text: is

text: our

text: ﬁnal

text: expression

text: for

text: the

text: distribution

text: ρ˜

text: and

text: agrees

text: with

text: the

text: result

text: in

text: ref.

text: as

text: it

text: should,

text: since

text: we

text: have

text: only

text: made

text: the

text: semiclassical

text: analysis

text: more

text: explicit.

label: section

text: Gaussian

text: potential

text: Let

text: us

text: now

text: focus

text: on

text: the

text: speciﬁc

text: case

text: of

text: a

text: gaussian

text: potential

text: V

text: (X)

text: In

text: this

text: case

text: the

text: rescaled

text: individual

text: wave

text: functions

text: f˜

text: satisfy

text: the

text: Schr¨odinger

text: equation

text: for

text: the

text: one

text: dimensional

text: harmonic

text: oscillator:

text: ∂2

text: λ2

text: f˜

text: (λ)

text: ε

text: f˜

text: (λ)

text: ∂λ2

text: n

text: n

text: n

text: where:

text: H√(λ)e

text: N

text: n)

text: f˜

text: (λ)

text: n

text: ε

text: n

text: n

text: N

text: and

text: H

text: (λ)

text: are

text: the

text: Hermite

text: polynomials.

text: The

text: corresponding

text: multi-particle

text: wave

text: func-

text: n

text: tion

text: is

text: given

text: by:

text: φ˜(λ1

text: λN)

text: CN

text: HiN

text: e

text: N

text: i

text: CN

text: (λi

text: λj)

text: e

text: N

text: i

text: i>j

text: N

text: where

text: C

text: N!

text: and

text: we

text: have

text: used

text: the

text: properties

text: of

text: the

text: Hermite

text: poly-

text: N

text: nomials

text: to

text: arrive

text: at

text: the

text: last

text: equality.

text: Equation

text: suggests

text: that

text: the

text: bosonic

text: wave

text: function

text: ψ˜

text: before

text: the

text: change

text: of

text: variables

text: is

text: given

text: by:

text: Ψ˜

text: λN)

text: CN

text: e

text: N

text: i

text: Therefore,

text: the

text: bosonic

text: wave

text: function

text: of

text: the

text: ground

text: state

text: of

text: the

text: system

text: is

text: a

text: product

text: of

text: individual

text: wave

text: functions.

text: We

text: could

text: have

text: easily

text: guessed

text: this

text: result

text: even

text: directly

text: in

text: the

text: bosonic

text: description

text: without

text: introducing

text: the

text: new

text: wave

text: function

text: Note

text: that

text: this

text: is

text: true

text: only

text: for

text: the

text: case

text: of

text: a

text: gaussian

text: potential.

text: In

text: fact,

text: in

text: the

text: next

text: section

text: we

text: will

text: show

text: that

text: this

text: holds

text: even

text: for

text: the

text: case

text: of

text: more

text: than

text: one

text: commuting

text: matrices

text: with

text: a

text: gaussian

text: potential

text: Commuting

text: matrix

text: model

text: In

text: this

text: section

text: we

text: focus

text: on

text: the

text: quantisation

text: of

text: the

text: commuting

text: matrix

text: model.

text: While

text: we

text: will

text: set

text: up

text: the

text: problem

text: for

text: general

text: interaction

text: potential

text: the

text: main

text: ﬁndings

text: that

text: we

text: present

text: are

text: for

text: the

text: special

text: case

text: of

text: gaussian

text: potential,

text: ﬁrst

text: discussed

text: in

text: ref.

text: Our

text: starting

text: point

text: is

text: the

text: lagrangian:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Front. Comput. Sci. 2014, X(X): XXXXX 11 (a) (b) Figure 1: Illustration of the domains and operators used in the paper as well as the requirements of CS: sparsity in the transform domain, incoherence of the undersampling artifacts and the need for nonFliinge.a5r rec(oan)sItrlluucsttiroantitohnatoefntfhorecedsosmpaarisnistyand operators used in [57] as the requirements of CS: sparsity in the transform domain, incoherence of the under- sampling artifacts and the need for nonlinear reconstruction that enforces sparsity. (b) Diagram of the single pixel camera. The image x is reﬂected oﬀ a digital micro-mirror device (DMD) array whose mirror orientations are modulated in the pseudorandom pattern supplied by the random number generator (RNG) [58]. 3. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inform. Theory, Inform. Theory, vol. 52, no. 1, pp. 6–18, Jan. 2006. 32 vol. 52, pp. 1289–1306, Jul. 2006. 12. J. Tropp and A. Gilbert, “Signal recovery from random measurements 4. E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: via orthogonal matching pursuit,” IEEE Trans. Inform. Theory, vol. 53, Exact signal reconstruction from highly incomplete frequency infor- no. 12, pp. 4655–4666, Dec. 2007. mation,” IEEE Trans. Inform. Theory, vol. 52, pp. 489–509, Feb. 2006. 13. R. Calderbank, S. Howard, and S. Jafarpour, “Construction of a large 5. E. Candès and T. Tao, “Near optimal signal recovery from random pro- class of deterministic sensing matrices that satisfy a statistical isometry jections: Universal encoding strategies,” IEEE Trans. Inform. Theory, property,” IEEE J. Selected Topics in Signal Processing, vol. 4, no. 2, vol. 52, pp. 5406–5425, 2006. pp. 358–374, Apr. 2010. 6. E. Candes and M. Wakin, “An introduction to compressive sampling,” 14. D. L. Donoho and M. Elad, “Optimally sparse representation in general IEEE Signal Process. Mag., vol. 25, no. 2, pp. 21–30, Mar. 2008. (nonorthogonal) dictionaries via l minimization,” Proc. Nat. Acad. 1 7. R. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag., Sci., vol. 100, no. 5, pp. 2197–2202, Mar. 2003. vol. 24, no. 4, pp. 118–121, Jul. 2007. 15. J. Tropp, “Greed is good: algorithmic results for sparse approxima- 8. E. Candes and J. Romberg, “Practical signal recovery tion,” IEEE Trans. Inform. Theory, vol. 50, no. 10, pp. 2231–2242, from random projections,” 2005, preprint. [Online]. Available: Oct. 2004. www.acm.caltech.edu/ emmanuel/papers/PracticalRecovery.pdf 16. E. J. Candès and Y. Plan, “Near-ideal model selection by l1 minimiza- 9. J. Haupt and R. Nowak, “Signal reconstruction from noisy random pro- tion,” Ann. Stat., vol. 37, no. 5, pp. 2145-–2177, Oct. 2009. jections,” IEEE Trans. Inform. Theory, vol. 52, no. 9, pp. 4036–4048, 17. M. Rudelson and R. Vershynin, “On sparse reconstruction from Fourier Sep. 2006. and Gaussian measurements,” Commun. Pure and Applied Math., 10. E. Candes and J. Romberg, “Quantitative robust uncertainty principles vol. 61, pp. 1025-–1045, Dec. 2008. and optimally sparse decompositions,” Found. Comput. Math.s, vol. 6, 18. M. Duarte and Y. Eldar, “Structured compressed sensing: From theory no. 8, pp. 227–254, 2006. to applications,” IEEE Trans. Signal Processing, vol. 59, no. 9, pp. 11. D. Donoho, M. Elad, and V. Temlyakov, “Stable recovery of sparse 4053–4085, Sep. 2011. overcomplete representations in the presence of noise,” IEEE Trans. 19. E. Candès and Y. Plan, “A probabilistic and RIPless theory of com- ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: Front.

text: Comput.

text: Sci.

text: X(X):

text: XXXXX

text: (a)

text: (b)

text: Figure

text: Illustration

text: of

text: the

text: domains

text: and

text: operators

text: used

text: in

text: the

text: paper

text: as

text: well

text: as

text: the

text: requirements

text: of

text: CS:

text: sparsity

text: in

text: the

text: transform

text: domain,

text: incoherence

text: of

text: the

text: undersampling

text: artifacts

text: and

text: the

text: need

text: for

text: operators

text: used

text: in

text: as

text: the

text: requirements

text: of

text: CS:

text: sparsity

text: in

text: the

text: transform

text: domain,

text: incoherence

text: of

text: the

text: under-

text: sampling

text: artifacts

text: and

text: the

text: need

text: for

text: nonlinear

text: reconstruction

text: that

text: enforces

text: sparsity.

text: (b)

text: Diagram

text: of

text: the

text: single

text: pixel

text: camera.

text: The

text: image

text: x

text: is

text: reﬂected

text: oﬀ

text: a

text: digital

text: micro-mirror

text: device

text: (DMD)

text: array

text: whose

text: mirror

text: orientations

text: are

text: modulated

text: in

text: the

text: pseudorandom

text: pattern

text: supplied

text: by

text: the

text: random

text: number

text: generator

text: (RNG)

text: D.

text: L.

text: Donoho,

text: “Compressed

text: sensing,”

text: IEEE

text: Trans.

text: Inform.

text: Theory,

text: Inform.

text: Theory,

text: vol.

text: no.

text: pp.

text: Jan.

text: vol.

text: pp.

text: Jul.

text: J.

text: Tropp

text: and

text: A.

text: Gilbert,

text: “Signal

text: recovery

text: from

text: random

text: measurements

text: E.

text: Candès,

text: J.

text: Romberg,

text: and

text: T.

text: Tao,

text: “Robust

text: uncertainty

text: principles:

text: via

text: orthogonal

text: matching

text: pursuit,”

text: IEEE

text: Trans.

text: Inform.

text: Theory,

text: vol.

text: Exact

text: signal

text: reconstruction

text: from

text: highly

text: incomplete

text: frequency

text: infor-

text: no.

text: pp.

text: Dec.

text: mation,”

text: IEEE

text: Trans.

text: Inform.

text: Theory,

text: vol.

text: pp.

text: Feb.

text: R.

text: Calderbank,

text: S.

text: Howard,

text: and

text: S.

text: Jafarpour,

text: “Construction

text: of

text: a

text: large

text: E.

text: Candès

text: and

text: T.

text: Tao,

text: “Near

text: optimal

text: signal

text: recovery

text: from

text: random

text: pro-

text: class

text: of

text: deterministic

text: sensing

text: matrices

text: that

text: satisfy

text: a

text: statistical

text: isometry

text: jections:

text: Universal

text: encoding

text: strategies,”

text: IEEE

text: Trans.

text: Inform.

text: Theory,

text: property,”

text: IEEE

text: J.

text: Selected

text: Topics

text: in

text: Signal

text: Processing,

text: vol.

text: no.

text: vol.

text: pp.

text: pp.

text: Apr.

text: E.

text: Candes

text: and

text: M.

text: Wakin,

text: “An

text: introduction

text: to

text: compressive

text: sampling,”

text: D.

text: L.

text: Donoho

text: and

text: M.

text: Elad,

text: “Optimally

text: sparse

text: representation

text: in

text: general

text: IEEE

text: Signal

text: Process.

text: Mag.,

text: vol.

text: no.

text: pp.

text: Mar.

text: (nonorthogonal)

text: dictionaries

text: via

text: l

text: minimization,”

text: Proc.

text: Nat.

text: Acad.

text: R.

text: Baraniuk,

text: “Compressive

text: sensing,”

text: IEEE

text: Signal

text: Process.

text: Mag.,

text: Sci.,

text: vol.

text: no.

text: pp.

text: Mar.

text: vol.

text: no.

text: pp.

text: Jul.

text: J.

text: Tropp,

text: “Greed

text: is

text: good:

text: algorithmic

text: results

text: for

text: sparse

text: approxima-

text: E.

text: Candes

text: and

text: J.

text: Romberg,

text: “Practical

text: signal

text: recovery

text: tion,”

text: IEEE

text: Trans.

text: Inform.

text: Theory,

text: vol.

text: no.

text: pp.

text: from

text: random

text: projections,”

text: preprint.

text: [Online].

text: Available:

text: Oct.

text: E.

text: J.

text: Candès

text: and

text: Y.

text: Plan,

text: “Near-ideal

text: model

text: selection

text: by

text: minimiza-

text: J.

text: Haupt

text: and

text: R.

text: Nowak,

text: “Signal

text: reconstruction

text: from

text: noisy

text: random

text: pro-

text: tion,”

text: Ann.

text: Stat.,

text: vol.

text: no.

text: pp.

text: Oct.

text: jections,”

text: IEEE

text: Trans.

text: Inform.

text: Theory,

text: vol.

text: no.

text: pp.

text: M.

text: Rudelson

text: and

text: R.

text: Vershynin,

text: “On

text: sparse

text: reconstruction

text: from

text: Fourier

text: Sep.

text: and

text: Gaussian

text: measurements,”

text: Commun.

text: Pure

text: and

text: Applied

text: Math.,

text: E.

text: Candes

text: and

text: J.

text: Romberg,

text: “Quantitative

text: robust

text: uncertainty

text: principles

text: vol.

text: pp.

text: Dec.

text: and

text: optimally

text: sparse

text: decompositions,”

text: Found.

text: Comput.

text: Math.s,

text: vol.

text: M.

text: Duarte

text: and

text: Y.

text: Eldar,

text: “Structured

text: compressed

text: sensing:

text: From

text: theory

text: no.

text: pp.

text: to

text: applications,”

text: IEEE

text: Trans.

text: Signal

text: Processing,

text: vol.

text: no.

text: pp.

text: D.

text: Donoho,

text: M.

text: Elad,

text: and

text: V.

text: Temlyakov,

text: “Stable

text: recovery

text: of

text: sparse

text: Sep.

text: overcomplete

text: representations

text: in

text: the

text: presence

text: of

text: noise,”

text: IEEE

text: Trans.

text: E.

text: Candès

text: and

text: Y.

text: Plan,

text: “A

text: probabilistic

text: and

text: RIPless

text: theory

text: of

text: com-

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text: ##LTFigure##

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text: ##LTFigure##