text: 2 escape times (which are rigorously connected to two-time probability densities) within the novel approach (Sec. II D). We summarize and conclude in Sec. IV. The paper contains four appendices with additional theoretical results (e.g., the second member of the FPE hierarchy), and technical aspects of the calculations. I. THEORETICAL FRAMEWORK A. Delayed Langevin and Fokker-Planck equation We consider an overdamped Brownian particle at the time-dependent stochastic position χ(t) which represents the ∈ dynamical quantity of interest. The particle moves in a static external potential V (x), with spatial variable x Ω, s where Ω denotes the spatial domain of the system. The particle is further subject to a “delay force” F , which depends d− on the instantaneous and on the delayed particles position, i.e., at a given time t on χ(t) and χ(t τ), with τ > 0 being the delay time. The total deterministic force is hence F(x, x )=F (x) + F (x, x ), (1) τ s d τ ∈ with x Ω being a second spatial variable needed due to the two involved particle positions in F , and F (x) = − τ d s ∂xVs(x) being the negative of the spatial derivative of Vs. The delaye(cid:112)d Langevin equation (LE) reads − − dχ(t)/dt = γ 1F [χ(t), χ(t τ)] + 2D Γ(t), (2) 0 T where γ is the friction coeﬃcient of the surrounding medium at temperature and Γ(t) denotes the Langevin force. (cid:104) (cid:105) ∀ (cid:104) (cid:48) (cid:105) − (cid:48) ∀ (cid:48) The latter introduces additive Gaussian white noise to the system, i.e., Γ(t) =0 t and Γ(t)Γ(t ) =δ(t t ) t, t , (cid:104) (cid:105) with ... denoting the ensemble average and δ(t) the Delta distribution. Further, D is the short-time diﬀusion T 0 coeﬃcient satisfying D = k /γ (with k being the Boltzmann constant) [44]. We measure the time in units of 0 B B the “Brownian” time scale τ = σ2/D , where σ is the particle diameter. This is the time in which a free Brownian B 0 particle (no external forces) travels over a distance equal to its own size. As usual for delay equations, the evolution of the dynamical variable (here χ) depends on a given history function ∀ ∈ − φ, which serves as an initial condition χ(t)=φ(t), t [ τ, 0]. We emphasize that the theoretical framework discussed in this paper does not just apply to Brownian particles, but to other natural and artiﬁcial systems as well. Having this in mind, the friction and diﬀusion coeﬃcient, the Brownian time scale and the particle diameter introduced before just provide energy, time and spatial length scales, which one can adjust to the system under consideration. As ﬁrstly shown in [22], it is possible to derive from the delayed LE a Fokker-Planck equation (FPE) for the (one- (cid:104) − (cid:105) time) probability density ρ (x, t) = δ(x χ(t)) in a similar manner to the Markovian case. We particularly refer 1 to Ref. [23], which presents a derivation based upon Novikov’s theorem [51]. Novikov’s theorem generally links the variational derivative of an arbitrary functional of the Langevin force, Λ[Γ], to the correlations between that functional and the Langevin force. For Gaussian white noise it reads [2(cid:42)3, 51]: (cid:43) (cid:10) (cid:11) δΛ[Γ] Λ[Γ]Γ(t) = . (3) δΓ(t) (cid:10) (cid:11) { − − − } A central step in the derivation in [23] is the usage of Novikov’s theorem to express the correlations δ(x χ(t))δ(x χ(t τ)) Γ(t) τ by the related variational derivatives. For the case of additiv(cid:104)e noise, the del(cid:105)ayed FPE for ρ is given by [22, 23, 32] 1 (cid:98) − − ∂ ρ (x, t) = γ 1∂ F(x, t)ρ (x, t) t 1 x 1 + D ∂ ρ (x, t), (4a) 0 xx 1 where ##LTLine## ##LTLine##

text: escape

text: times

text: (which

text: are

text: rigorously

text: connected

text: to

text: two-time

text: probability

text: densities)

text: within

text: the

text: novel

text: approach

text: (Sec.

text: II

text: D).

text: We

text: summarize

text: and

text: conclude

text: in

text: Sec.

text: IV.

text: The

text: paper

text: contains

text: four

text: appendices

text: with

text: additional

text: theoretical

text: results

text: (e.g.,

text: the

text: second

text: member

text: of

text: the

text: FPE

text: hierarchy),

text: and

text: technical

text: aspects

text: of

text: the

text: calculations.

text: I.

label: section

text: THEORETICAL

text: FRAMEWORK

text: A.

text: Delayed

text: Langevin

text: and

text: Fokker-Planck

text: equation

text: We

text: consider

text: an

text: overdamped

text: Brownian

text: particle

text: at

text: the

text: time-dependent

text: stochastic

text: position

text: χ(t)

text: which

text: represents

text: the

text: dynamical

text: quantity

text: of

text: interest.

text: The

text: particle

text: moves

text: in

text: a

text: static

text: external

text: potential

text: V

text: (x),

text: with

text: spatial

text: variable

text: x

text: Ω,

text: s

text: where

text: Ω

text: denotes

text: the

text: spatial

text: domain

text: of

text: the

text: system.

text: The

text: particle

text: is

text: further

text: subject

text: to

text: a

text: “delay

text: force”

text: F

text: which

text: depends

text: d−

text: on

text: the

text: instantaneous

text: and

text: on

text: the

text: delayed

text: particles

text: position,

text: i.e.,

text: at

text: a

text: given

text: time

text: t

text: on

text: χ(t)

text: and

text: χ(t

text: τ),

text: with

text: τ

text: being

text: the

text: delay

text: time.

text: The

text: total

text: deterministic

text: force

text: is

text: hence

text: F(x,

text: x

text: )=F

text: (x)

text: F

text: (x,

text: x

text: τ

text: s

text: d

text: τ

text: with

text: x

text: Ω

text: being

text: a

text: second

text: spatial

text: variable

text: needed

text: due

text: to

text: the

text: two

text: involved

text: particle

text: positions

text: in

text: F

text: and

text: F

text: (x)

text: τ

text: d

text: s

text: ∂xVs(x)

text: being

text: the

text: negative

text: of

text: the

text: spatial

text: derivative

text: of

text: Vs.

text: The

text: Langevin

text: equation

text: (LE)

text: reads

text: dχ(t)/dt

text: γ

text: [χ(t),

text: χ(t

text: τ)]

text: Γ(t),

text: T

text: where

text: γ

text: is

text: the

text: friction

text: coeﬃcient

text: of

text: the

text: surrounding

text: medium

text: at

text: temperature

text: and

text: Γ(t)

text: denotes

text: the

text: Langevin

text: force.

text: The

text: latter

text: introduces

text: additive

text: Gaussian

text: white

text: noise

text: to

text: the

text: system,

text: i.e.,

text: Γ(t)

text: t

text: and

text: Γ(t)Γ(t

text: =δ(t

text: t

text: t,

text: t

text: with

text: denoting

text: the

text: ensemble

text: average

text: and

text: δ(t)

text: the

text: Delta

text: distribution.

text: Further,

text: D

text: is

text: the

text: short-time

text: diﬀusion

text: T

text: coeﬃcient

text: satisfying

text: D

text: k

text: /γ

text: (with

text: k

text: being

text: the

text: Boltzmann

text: constant)

text: We

text: measure

text: the

text: time

text: in

text: units

text: of

text: B

text: B

text: the

text: “Brownian”

text: time

text: scale

text: τ

text: where

text: σ

text: is

text: the

text: particle

text: diameter.

text: This

text: is

text: the

text: time

text: in

text: which

text: a

text: free

text: Brownian

text: B

text: particle

text: (no

text: external

text: forces)

text: travels

text: over

text: a

text: distance

text: equal

text: to

text: its

text: own

text: size.

text: As

text: usual

text: for

text: delay

text: equations,

text: the

text: evolution

text: of

text: the

text: dynamical

text: variable

text: (here

text: χ)

text: depends

text: on

text: a

text: given

text: history

text: function

text: φ,

text: which

text: serves

text: as

text: an

text: initial

text: condition

text: χ(t)=φ(t),

text: t

text: τ,

text: We

text: emphasize

text: that

text: the

text: theoretical

text: framework

text: discussed

text: in

text: this

text: paper

text: does

text: not

text: just

text: apply

text: to

text: Brownian

text: particles,

text: but

text: to

text: other

text: natural

text: and

text: artiﬁcial

text: systems

text: as

text: well.

text: Having

text: this

text: in

text: mind,

text: the

text: friction

text: and

text: diﬀusion

text: coeﬃcient,

text: the

text: Brownian

text: time

text: scale

text: and

text: the

text: particle

text: diameter

text: introduced

text: before

text: just

text: provide

text: energy,

text: time

text: and

text: spatial

text: length

text: scales,

text: which

text: one

text: can

text: adjust

text: to

text: the

text: system

text: under

text: consideration.

text: As

text: ﬁrstly

text: shown

text: in

text: it

text: is

text: possible

text: to

text: derive

text: from

text: the

text: delayed

text: LE

text: a

text: Fokker-Planck

text: equation

text: (FPE)

text: for

text: the

text: (one-

text: time)

text: probability

text: density

text: ρ

text: (x,

text: t)

text: δ(x

text: χ(t))

text: in

text: a

text: similar

text: manner

text: to

text: the

text: Markovian

text: case.

text: We

text: particularly

text: refer

text: to

text: Ref.

text: which

text: presents

text: a

text: derivation

text: based

text: upon

text: Novikov’s

text: theorem

text: Novikov’s

text: theorem

text: generally

text: links

text: the

text: variational

text: derivative

text: of

text: an

text: arbitrary

text: functional

text: of

text: the

text: Langevin

text: force,

text: Λ[Γ],

text: to

text: the

text: correlations

text: between

text: that

text: functional

text: and

text: the

text: Langevin

text: force.

text: For

text: Gaussian

text: white

text: noise

text: it

text: reads

text: δΛ[Γ]

text: Λ[Γ]Γ(t)

text: δΓ(t)

text: A

text: central

text: step

text: in

text: the

text: derivation

text: in

text: is

text: the

text: usage

text: of

text: Novikov’s

text: theorem

text: to

text: express

text: the

text: correlations

text: δ(x

text: χ(t))δ(x

text: χ(t

text: τ))

text: Γ(t)

text: τ

text: by

text: the

text: related

text: variational

text: derivatives.

text: For

text: the

text: case

text: of

text: noise,

text: the

text: FPE

text: for

text: ρ

text: is

text: given

text: by

text: ρ

text: (x,

text: t)

text: γ

text: 1∂

text: F(x,

text: t)ρ

text: (x,

text: t)

text: t

text: x

text: D

text: ρ

text: (x,

text: t),

text: xx

text: where

text: ##LTLine##

text: ##LTLine##


text: 5 With Eq. (7), the ﬁrst member of the approximate FPE hierarchy (where the force linearization is applied to the second and all higher members) accordingly reads (cid:98) γD ∂ ρFLC(x) = ∂ [FFLC(x)ρFLC(x)], (9a) 0 xx 1,ss x 1,ss where (cid:90) (cid:98) | ∀ ∈ FFLC(x) = F(x, x )ρlin,i(x x; τ)dx , x Ωi. (9b) τ c,ss τ τ Ω In Eq. (9a), ρFLC is the steady-state one-time probability density obtained within the FLC approach. Notice that 1,ss the deterministic force F appearing in the integral in Eqs. (9b) is not linearized. Equa(cid:98)tions (7-9b) [together with the linearization rule (6)] form a closed set of equations. The here presented closure of the FPE hierarchy is the cornerstone of our approach. Since we have a general expression for ρlin,i [Eq. (7)] with c,ss which FFLC can be calculated separately, the FLC formally converts the delayed FPE for ρ itself into a closed, 1,ss quasi-Markovian one [Eq. (9a)]. The fact that the delay is still present in our approximate FPE, becomes apparent by considering the special case of a linear force F(cid:98). Then, Eqs. (9) coincide with the corresponding exact delayed FPE. Furthermore, one can easily see from Eqs. (9) that the usual FPE is recovered when the delay time vanishes, or when F becomes independent of x : in both cases FFLC =F. τ 3. Vanishing steady state probability current The key equations of the FLC [Eqs. (7-9b)] are valid for steady s(cid:98)tates of, in principle, arbitrary systems which can be meaningfully linearized according to the procedure described in Sec. I C 1. A particularly simple situation arises, − if additionally the steady-state probability current given by J = [FFLC(x) γD ∂ ]ρFLC(x) vanishes. If J = 0, the 0 x 1,ss formal solution of Eq. (9a) takes the simple Boltzmann-like(cid:2) form (cid:3) − − T ρFLC(x) = Z 1 exp V FLC(x)/(k ) (10) 1,ss eﬀ B with the eﬀective static potential (cid:90) (cid:98) − x (cid:48) (cid:48) V FLC(x) = dx FFLC(x ), (11) (cid:98) eﬀ xˆ ∈ where FFLC is given in Eq. (9b), and xˆ Ω is arbitrary but ﬁxed. Here and in the following, we denote the normalization constant by Z. We stress that the assumption J = 0 simpliﬁes the analysis, but it is not a necessary condition. Also for non-zero currents, Eq. (9a) can be treated using standard techniques for (Markovian) Fokker-Planck equations [44, 45]. II. APPLICATIONS We now apply the FLC to two generic examples, involving a multistable and a bistable static potential combined with a linear delay force. In the subsequent section , we ﬁrst deﬁne this force and provide some results which apply to both model systems. In Secs. and we then present results from the FLC and compare its performance to the two main approaches known from the literature, i.e., the perturbation theory (PT) and the small delay expansion. As a test of the diﬀerent approximations, we provide numerical results from Brownian dynamics (BD) simulations. Details about the numerical methods are given in the Appendix D. A. Linear delay force, general results ≥ We consider a linear delay force, with amplitude k 0, − − F (x, x )= k(x x ), (12) d τ τ

label: paragraph

text: With

text: Eq.

text: the

text: ﬁrst

text: member

text: of

text: the

text: approximate

text: FPE

text: hierarchy

text: (where

text: the

text: force

text: linearization

text: is

text: applied

text: to

text: the

text: second

text: and

text: all

text: higher

text: members)

text: accordingly

text: reads

label: equation

text: γD

text: ρFLC(x)

text: [FFLC(x)ρFLC(x)],

text: xx

text: x

text: where

text: FFLC(x)

text: F(x,

text: x

text: )ρlin,i(x

text: x;

text: τ)dx

text: x

text: Ωi.

text: τ

text: c,ss

text: τ

text: τ

text: Ω

text: In

text: Eq.

text: ρFLC

text: is

text: the

text: steady-state

text: one-time

text: probability

text: density

text: obtained

text: within

text: the

text: FLC

text: approach.

text: Notice

text: that

text: the

text: deterministic

text: force

text: F

text: appearing

text: in

text: the

text: integral

text: in

text: Eqs.

text: is

text: not

text: linearized.

text: [together

text: with

text: the

text: linearization

text: rule

text: form

text: a

text: closed

text: set

text: of

text: equations.

text: The

text: here

text: presented

text: closure

text: of

text: the

text: FPE

text: hierarchy

text: is

text: the

text: cornerstone

text: of

text: our

text: approach.

text: Since

text: we

text: have

text: a

text: general

text: expression

text: for

text: ρlin,i

text: [Eq.

text: with

text: c,ss

text: which

text: FFLC

text: can

text: be

text: calculated

text: separately,

text: the

text: FLC

text: formally

text: converts

text: the

text: delayed

text: FPE

text: for

text: ρ

text: itself

text: into

text: a

text: closed,

text: quasi-Markovian

text: one

text: [Eq.

text: The

text: fact

text: that

text: the

text: delay

text: is

text: still

text: present

text: in

text: our

text: approximate

text: FPE,

text: becomes

text: apparent

text: by

text: considering

text: the

text: special

text: case

text: of

text: a

text: linear

text: force

text: Then,

text: Eqs.

text: coincide

text: with

text: the

text: corresponding

text: exact

text: delayed

text: FPE.

text: Furthermore,

text: one

text: can

text: easily

text: see

text: from

text: Eqs.

text: that

text: the

text: usual

text: FPE

text: is

text: recovered

text: when

text: the

text: delay

text: time

text: vanishes,

text: or

text: when

text: F

text: becomes

text: independent

text: of

text: x

text: in

text: both

text: cases

text: FFLC

text: =F.

text: τ

text: Vanishing

text: steady

text: state

text: probability

text: current

text: The

text: key

text: equations

text: of

text: the

text: FLC

text: [Eqs.

text: are

text: valid

text: for

text: steady

text: of,

text: in

text: principle,

text: arbitrary

text: systems

text: which

text: can

text: be

text: meaningfully

text: linearized

text: according

text: to

text: the

text: procedure

text: described

text: in

text: Sec.

text: I

text: C

text: A

text: particularly

text: simple

text: situation

text: arises,

text: if

text: additionally

text: the

text: steady-state

text: probability

text: current

text: given

text: by

text: J

text: [FFLC(x)

text: γD

text: ]ρFLC(x)

text: vanishes.

text: If

text: J

text: the

text: x

text: formal

text: solution

text: of

text: Eq.

text: takes

text: the

text: simple

text: form

text: T

text: ρFLC(x)

text: Z

text: exp

text: V

text: FLC(x)/(k

text: eﬀ

text: B

text: with

text: the

text: eﬀective

text: static

text: potential

text: x

text: V

text: FLC(x)

text: dx

text: FFLC(x

text: eﬀ

text: xˆ

text: where

text: FFLC

text: is

text: given

text: in

text: Eq.

text: and

text: xˆ

text: Ω

text: is

text: arbitrary

text: but

text: ﬁxed.

text: Here

text: and

text: in

text: the

text: following,

text: we

text: denote

text: the

text: normalization

text: constant

text: by

text: Z.

text: We

text: stress

text: that

text: the

text: assumption

text: J

text: simpliﬁes

text: the

text: analysis,

text: but

text: it

text: is

text: not

text: a

text: necessary

text: condition.

text: Also

text: for

text: non-zero

text: currents,

text: Eq.

text: can

text: be

text: treated

text: using

text: standard

text: techniques

text: for

text: (Markovian)

text: Fokker-Planck

text: equations

text: II.

text: APPLICATIONS

text: We

text: now

text: apply

text: the

text: FLC

text: to

text: two

text: generic

text: examples,

text: involving

text: a

text: multistable

text: and

text: a

text: bistable

text: static

text: potential

text: combined

text: with

text: a

text: linear

text: delay

text: force.

text: In

text: the

text: subsequent

text: section

text: we

text: ﬁrst

text: deﬁne

text: this

text: force

text: and

text: provide

text: some

text: results

text: which

text: apply

text: to

text: both

text: model

text: systems.

text: In

text: Secs.

text: and

text: we

text: then

text: present

text: results

text: from

text: the

text: FLC

text: and

text: compare

text: its

text: performance

text: to

text: the

text: two

text: main

text: approaches

text: known

text: from

text: the

text: literature,

text: i.e.,

text: the

text: perturbation

text: theory

text: (PT)

text: and

text: the

text: small

text: delay

text: expansion.

text: As

text: a

text: test

text: of

text: the

text: diﬀerent

text: approximations,

text: we

text: provide

text: numerical

text: results

text: from

text: Brownian

text: dynamics

text: (BD)

text: simulations.

text: Details

text: about

text: the

text: numerical

text: methods

text: are

text: given

text: in

text: the

text: Appendix

text: D.

text: A.

text: Linear

text: delay

text: force,

text: general

text: results

text: We

text: consider

text: a

text: linear

text: delay

text: force,

text: with

text: amplitude

text: k

text: F

text: (x,

text: x

text: k(x

text: x

text: d

text: τ

text: τ


text: 8 FLC 2 PT OU PT s.t. small τ F ≡0 1 d BD 0 1 0 1 2 x/σ FIG. 1. (Color online) One peak of the periodic steady state probability density ρ in the “washboard” with delay force. 1,ss Red symbols: numerical data (BD), thick red line: force linearization closure (FLC), blue dashed line: perturbation theory with Ornstein–Uhlenbeck approximation (PT OU), blue thin line: PT with short time propagator (PT s.t.), dashed-dotted ∝ {− T } ≡ line: small delay expansion (small τ), and solid black line: density without delay force ρ (x) exp V (x)/(k ) (F 0). T 1,ss s B d The delay force amplitude and delay time are set to k=8 k /σ2 and τ =τ , respectively. Here and in the following plots the T B B barrier height is set to ∆V =8 k . B ∈ − − T ∈ all other considered values of τ and k in the tested range τ/τ [10 2, 10] and k/(σ 2k ) [0, 32]. To compare B B approximation and simulation results more systematically, we calculate the moments µ of the distributions within − ≤ ≤ n one potential well (for x values between two maxima of V ), e.g., of ρ ( πσ x πσ). Due to the spatial inversion s 1,ss symmetry of the total potential around the enclosed minimum, all odd moments vanish. We hence consider the second (cid:104) − (cid:105) (cid:104) (cid:105) central moment µ = (χ µ )2 (with µ = χ ) of one peak of the distribution. Figure 2 shows µ vs. the delay 2 1 ss 1 ss 2 time τ for two exemplary values of delay force amplitude. We ﬁnd that in the presence of the delay force (12), the variance of the distribution is reduced. This is not surprising, since the delay force arises from a quadratic potential which “traps” the particle and hence enhances the conﬁning eﬀect of V . Within the considered parameter regime, s ≈ the variance is seen to decrease with increasing τ, until a saturation value is approached at τW 0.5 τ (due to the sat B strongly increasing simulation times, we did not(cid:98)simulate much larger τ). On the level of the probability densities and the FPE, the saturation of µ at large τ indicates that in the considered | 2 range of delay times, the delay-averaged drift F, and hence ρ (x x; τ), are essentially constant for τ > τW . In c,ss τ sat other words, the conditional probability for the time diﬀerence τ remains essentially unchanged, when τ is further increased. This suggests some kind of relaxation mechanism within the valley, where the history of the stochastic process becomes “washed out” on the level of ensemble averaged quantities, like the probability densities. In this context it is interesting to note that, for the case without delay force (k =0), the relaxation time within a potential ≈ (cid:48)(cid:48) T T well is of the order τW γ/V (x ) = 2 τ k /∆V (see Ref. [45], p. 348). In the present case, ∆V = 8 k , that is ≈ ir s min B B B τW 0.25 τ . The saturation of µ thus sets in after the density relaxation within a potential well, consistent with ir B 2 our expectation. A diﬀerent situation occurs for much larger τ values than the ones considered in Fig. 2, in particular, when τ gets )x( ρ ss,1 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##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##

text: FLC

font: default

text: PT

text: OU

text: PT

text: s.t.

text: small

text: τ

text: F

text: ≡0

text: d

text: BD

text: x/σ

text: FIG.

text: (Color

label: caption

text: online)

text: One

text: peak

text: of

text: the

text: periodic

text: steady

text: state

text: probability

text: density

text: ρ

text: in

text: the

text: “washboard”

text: with

text: delay

text: force.

text: Red

text: symbols:

text: numerical

text: data

text: (BD),

text: thick

text: red

text: line:

text: force

text: linearization

text: closure

text: (FLC),

text: blue

text: dashed

text: line:

text: perturbation

text: theory

text: with

text: Ornstein–Uhlenbeck

text: approximation

text: (PT

text: OU),

text: blue

text: thin

text: line:

text: PT

text: with

text: short

text: time

text: propagator

text: (PT

text: s.t.),

text: dashed-dotted

text: T

text: line:

text: small

text: delay

text: expansion

text: (small

text: τ),

text: and

text: solid

text: black

text: line:

text: density

text: without

text: delay

text: force

text: ρ

text: (x)

text: exp

text: V

text: (x)/(k

text: (F

text: T

text: s

text: B

text: d

text: The

text: delay

text: force

text: amplitude

text: and

text: delay

text: time

text: are

text: set

text: to

text: k

text: and

text: τ

text: =τ

text: respectively.

text: Here

text: and

text: in

text: the

text: following

text: plots

text: the

text: T

text: B

text: B

text: barrier

text: height

text: is

text: set

text: to

text: ∆V

text: k

text: B

text: T

text: all

text: other

text: considered

text: values

text: of

text: τ

text: and

text: k

text: in

text: the

text: tested

text: range

text: τ/τ

text: and

text: k/(σ

text: To

text: compare

text: B

text: B

text: approximation

text: and

text: simulation

text: results

text: more

text: systematically,

text: we

text: calculate

text: the

text: moments

text: µ

text: of

text: the

text: distributions

text: within

text: n

text: one

text: potential

text: well

text: (for

text: x

text: values

text: between

text: two

text: maxima

text: of

text: V

text: e.g.,

text: of

text: ρ

text: πσ

text: x

text: πσ).

text: Due

text: to

text: the

text: spatial

text: inversion

text: s

text: symmetry

text: of

text: the

text: total

text: potential

text: around

text: the

text: enclosed

text: minimum,

text: all

text: odd

text: moments

text: vanish.

text: We

text: hence

text: consider

text: the

text: second

text: central

text: moment

text: µ

text: (χ

text: µ

text: (with

text: µ

text: χ

text: of

text: one

text: peak

text: of

text: the

text: distribution.

text: Figure

text: shows

text: µ

text: vs.

text: the

text: delay

text: ss

text: ss

text: time

text: τ

text: for

text: two

text: exemplary

text: values

text: of

text: delay

text: force

text: amplitude.

text: We

text: ﬁnd

text: that

text: in

text: the

text: presence

text: of

text: the

text: delay

text: force

text: the

text: variance

text: of

text: the

text: distribution

text: is

text: reduced.

text: This

text: is

text: not

text: surprising,

text: since

text: the

text: delay

text: force

text: arises

text: from

text: a

text: quadratic

text: potential

text: which

text: “traps”

text: the

text: particle

text: and

text: hence

text: enhances

text: the

text: conﬁning

text: eﬀect

text: of

text: V

text: Within

text: the

text: considered

text: parameter

text: regime,

text: s

text: the

text: variance

text: is

text: seen

text: to

text: decrease

text: with

text: increasing

text: τ,

text: until

text: a

text: saturation

text: value

text: is

text: approached

text: at

text: τW

text: τ

text: (due

text: to

text: the

text: sat

text: B

text: strongly

text: increasing

text: simulation

text: times,

text: we

text: did

text: much

text: larger

text: τ).

text: On

text: the

text: level

text: of

text: the

text: probability

text: densities

text: and

text: the

text: FPE,

text: the

text: saturation

text: of

text: µ

text: at

text: large

text: τ

text: indicates

text: that

text: in

text: the

text: considered

text: range

text: of

text: delay

text: times,

text: the

text: delay-averaged

text: drift

text: F,

text: and

text: hence

text: ρ

text: (x

text: x;

text: τ),

text: are

text: essentially

text: constant

text: for

text: τ

text: τW

text: In

text: c,ss

text: τ

text: sat

text: other

text: words,

text: the

text: conditional

text: probability

text: for

text: the

text: time

text: diﬀerence

text: τ

text: remains

text: essentially

text: unchanged,

text: when

text: τ

text: is

text: further

text: increased.

text: This

text: suggests

text: some

text: kind

text: of

text: relaxation

text: mechanism

text: within

text: the

text: valley,

text: where

text: the

text: history

text: of

text: the

text: stochastic

text: process

text: becomes

text: “washed

text: out”

text: on

text: the

text: level

text: of

text: ensemble

text: averaged

text: quantities,

text: like

text: the

text: probability

text: densities.

text: In

text: this

text: context

text: it

text: is

text: interesting

text: to

text: note

text: that,

text: for

text: the

text: case

text: without

text: delay

text: force

text: (k

text: the

text: relaxation

text: time

text: within

text: a

text: potential

text: T

text: T

text: well

text: is

text: of

text: the

text: order

text: τW

text: γ/V

text: (x

text: τ

text: k

text: /∆V

text: (see

text: Ref.

text: p.

text: In

text: the

text: present

text: case,

text: ∆V

text: k

text: that

text: is

text: ir

text: s

text: min

text: B

text: B

text: B

text: τW

text: τ

text: The

text: saturation

text: of

text: µ

text: thus

text: sets

text: in

text: after

text: the

text: density

text: relaxation

text: within

text: a

text: potential

text: well,

text: consistent

text: with

text: ir

text: B

text: our

text: expectation.

text: A

text: diﬀerent

text: situation

text: occurs

text: for

text: much

text: larger

text: τ

text: values

text: than

text: the

text: ones

text: considered

text: in

text: Fig.

text: in

text: particular,

text: when

text: τ

text: gets

text: )x(

text: ρ

text: ##LTFigure##

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text: 9 0.3 −2 k =2 k Tσ B −2 k =4 k Tσ B 0.2 FLC PT OU PT s.t. 0.1 small τ BD -2 -1 0 1 log (τ/τ ) 10 B FIG. 2. (Color online) Second moment µ of the steady state probability density within one well of the “washboard” potential, − ≤ ≤ 2 e.g., of ρ ( πσ x πσ). Symbols show results from Brownian dynamics (BD), and thick solid lines from force linearization 1,ss T T closure (FLC). Cyan color is k = 2 k /σ2. Black is k = 4 k /σ2, with thin solid line: perturbation theory with short time B B ≡ propagator (PT s.t.), dashed line: perturbation theory with Ornstein–Uhlenbeck ρFd 0 (PT OU), and dot-dashed line: small c,ss τ Taylor expansion (small τ). into the range of the mean escape times (numerical results for the escape times are provided in Sec. II D). Then we expect again a τ-dependency of ρ and therewith of µ , since the transport between the potential valleys becomes c,ss 2 important. We now compare the density from the FLC approach with corresponding data from the small delay (Taylor) expansion [Eq. (16) and (17)] and from the PT approach. Within the latter, we either use the short time propagator [see Eq. (19)], or, the corresponding Ornstein–Uhlenbeck approximation [see Eq. (20) and above]. As visible in Fig. 1 and 2, the PT generally overestimates the height of the density peak. Approximating the conditional probability with the short time propagator yields worse results than using ρ of the corresponding Ornstein–Uhlenbeck process. This c,ss observation matches our expectations, see Sec. . At least when the Ornstein–Uhlenbeck approximation is used, the PT reproduces the quantitative behavior of the function µ (τ), see Fig. 2. On the contrary, the small delay expansion 2 fails completely for larger delay times. We conclude that the FLC generates the best approximation of the steady-state density distribution in the periodic | | potential, especially in the regime of large τ or large k. This is the regime where the “perturbation”, i.e. F , is not d small compared to the force applied by the static potential, such that the basic assumption of the perturbation theory is not fulﬁlled. The small delay expansion, on the other hand, involves a truncated Taylor expansion in τ, making it plausible that also this approach fails for large delay times. C. Bistable static potential As a second example, we consider the double-well potential tnemom dnoces µ 2 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##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##

text: −2

text: k

text: k

text: Tσ

text: B

text: −2

text: k

text: k

text: Tσ

text: B

text: FLC

text: PT

text: OU

text: PT

text: s.t.

text: small

text: τ

text: BD

text: log

text: (τ/τ

text: B

text: FIG.

text: (Color

text: online)

text: Second

text: moment

text: µ

text: of

text: the

text: steady

text: state

text: probability

text: density

text: within

text: one

text: well

text: of

text: the

text: “washboard”

text: potential,

text: e.g.,

text: of

text: ρ

text: πσ

text: x

text: πσ).

text: Symbols

text: show

text: results

text: from

text: Brownian

text: dynamics

text: (BD),

text: and

text: thick

text: solid

text: lines

text: from

text: force

text: linearization

text: T

text: T

text: closure

text: (FLC).

text: Cyan

text: color

text: is

text: k

text: k

text: Black

text: is

text: k

text: k

text: with

text: thin

text: solid

text: line:

text: perturbation

text: theory

text: with

text: short

text: time

text: B

text: B

text: propagator

text: (PT

text: s.t.),

text: dashed

text: line:

text: perturbation

text: theory

text: with

text: Ornstein–Uhlenbeck

text: ρFd

text: (PT

text: OU),

text: and

text: dot-dashed

text: line:

text: small

text: c,ss

text: τ

text: Taylor

text: expansion

text: (small

text: τ).

text: into

text: the

text: range

text: of

text: the

text: mean

text: escape

text: times

text: (numerical

text: results

text: for

text: the

text: escape

text: times

text: are

text: provided

text: in

text: Sec.

text: II

text: D).

text: Then

text: we

text: expect

text: again

text: a

text: τ-dependency

text: of

text: ρ

text: and

text: therewith

text: of

text: µ

text: since

text: the

text: transport

text: between

text: the

text: potential

text: valleys

text: becomes

text: c,ss

text: important.

text: We

text: now

text: compare

text: the

text: density

text: from

text: the

text: FLC

text: approach

text: with

text: corresponding

text: data

text: from

text: the

text: small

text: delay

text: (Taylor)

text: expansion

text: [Eq.

text: and

text: and

text: from

text: the

text: PT

text: approach.

text: Within

text: the

text: latter,

text: we

text: either

text: use

text: the

text: short

text: time

text: propagator

text: [see

text: Eq.

text: or,

text: the

text: corresponding

text: Ornstein–Uhlenbeck

text: approximation

text: [see

text: Eq.

text: and

text: above].

text: As

text: visible

text: in

text: Fig.

text: and

text: the

text: PT

text: generally

text: overestimates

text: the

text: height

text: of

text: the

text: density

text: peak.

text: Approximating

text: the

text: conditional

text: probability

text: with

text: the

text: short

text: time

text: propagator

text: yields

text: worse

text: results

text: than

text: using

text: ρ

text: of

text: the

text: corresponding

text: Ornstein–Uhlenbeck

text: process.

text: This

text: c,ss

text: observation

text: matches

text: our

text: expectations,

text: see

text: Sec.

text: At

text: least

text: when

text: the

text: Ornstein–Uhlenbeck

text: approximation

text: is

text: used,

text: the

text: PT

text: reproduces

text: the

text: quantitative

text: behavior

text: of

text: the

text: function

text: µ

text: (τ),

text: see

text: Fig.

text: On

text: the

text: contrary,

text: the

text: small

text: delay

text: expansion

text: fails

text: completely

text: for

text: larger

text: delay

text: times.

text: We

text: conclude

text: that

text: the

text: FLC

text: generates

text: the

text: best

text: approximation

text: of

text: the

text: steady-state

text: density

text: distribution

text: in

text: the

text: periodic

text: potential,

text: especially

text: in

text: the

text: regime

text: of

text: large

text: τ

text: or

text: large

text: k.

text: This

text: is

text: the

text: regime

text: where

text: the

text: “perturbation”,

text: i.e.

text: F

text: is

text: not

text: d

text: small

text: compared

text: to

text: the

text: force

text: applied

text: by

text: the

text: static

text: potential,

text: such

text: that

text: the

text: basic

text: assumption

text: of

text: the

text: perturbation

text: theory

text: is

text: not

text: fulﬁlled.

text: The

text: small

text: delay

text: expansion,

text: on

text: the

text: other

text: hand,

text: involves

text: a

text: truncated

text: Taylor

text: expansion

text: in

text: τ,

text: making

text: it

text: plausible

text: that

text: also

text: this

text: approach

text: fails

text: for

text: large

text: delay

text: times.

text: C.

text: Bistable

text: static

text: potential

text: As

text: a

text: second

text: example,

text: we

text: consider

text: the

text: double-well

text: potential

text: tnemom

text: dnoces

text: µ

text: ##LTFigure##

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text: The space of stable weak equivalence classes of measure-preserving actions ∗ † Lewis Bowen and Robin Tucker-Drob May 30, 2020 Abstract The concept of (stable) weak containment for measure-preserving actions of a count- able group Γ is analogous to the classical notion of (stable) weak containment of unitary representations. If Γ is amenable then the Rokhlin lemma shows that all essentially free actions are weakly equivalent. However if Γ is non-amenable then there can be many diﬀerent weak and stable weak equivalence classes. Our main result is that the set of stable weak equivalence classes naturally admits the structure of a Choquet sim- Z plex. For example, when Γ = this simplex has only a countable set of extreme points but when Γ is a nonamenable free group, this simplex is the Poulsen simplex. We also show that when Γ contains a nonabelian free group, this simplex has uncountably many strongly ergodic essentially free extreme points. Keywords: weak containment, pmp actions MSC:37A35 Contents 1 Introduction 2 1.1 Related literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 ∗ supported in part by NSF grant DMS-1500389, NSF CAREER Award DMS-0954606 † supported in part by NSF grant DMS-1600904 1 ##LTLine##

text: The

label: title

text: space

text: of

text: stable

text: weak

text: equivalence

text: classes

text: of

text: measure-preserving

text: actions

label: author

text: Lewis

text: Bowen

text: and

text: Robin

text: Tucker-Drob

text: May

text: Abstract

text: The

label: abstract

text: concept

text: of

text: (stable)

text: weak

text: containment

text: for

text: measure-preserving

text: actions

text: of

text: a

text: count-

text: able

text: group

text: Γ

text: is

text: analogous

text: to

text: the

text: classical

text: notion

text: of

text: (stable)

text: weak

text: containment

text: of

text: unitary

text: representations.

text: If

text: Γ

text: is

text: amenable

text: then

text: the

text: Rokhlin

text: lemma

text: shows

text: that

text: all

text: essentially

text: free

text: actions

text: are

text: weakly

text: equivalent.

text: However

text: if

text: Γ

text: is

text: non-amenable

text: then

text: there

text: can

text: be

text: many

text: diﬀerent

text: weak

text: and

text: stable

text: weak

text: equivalence

text: classes.

text: Our

text: main

text: result

text: is

text: that

text: the

text: set

text: of

text: stable

text: weak

text: equivalence

text: classes

text: naturally

text: admits

text: the

text: structure

text: of

text: a

text: Choquet

text: sim-

text: Z

text: plex.

text: For

text: example,

text: when

text: Γ

text: this

text: simplex

text: has

text: only

text: a

text: countable

text: set

text: of

text: extreme

text: points

text: but

text: when

text: Γ

text: is

text: a

text: nonamenable

text: free

text: group,

text: this

text: simplex

text: is

text: the

text: Poulsen

text: simplex.

text: We

text: also

text: show

text: that

text: when

text: Γ

text: contains

text: a

text: nonabelian

text: free

text: group,

text: this

text: simplex

text: has

text: uncountably

text: many

text: strongly

text: ergodic

text: essentially

text: free

text: extreme

text: points.

text: Keywords:

text: weak

text: containment,

text: pmp

text: actions

text: Contents

text: Introduction

text: Related

text: literature

text: supported

text: in

text: part

text: by

text: NSF

text: grant

text: NSF

text: CAREER

text: Award

text: supported

text: in

text: part

text: by

text: NSF

text: grant

text: ##LTLine##


text: { } Corollary 6.2. [SW is lower semi-continuous] If µ is a sequence in P and lim µ = µ∞ i i i i then ⊆ SW (µ∞) lim inf SW (µ ). i i Z Remark 1. SW is not continuous in general. For example, consider the case when Γ = . It ∈ (cid:121) is possible to ﬁnd a sequence of measures µ P such that Γ (CantorΓ, µ ) is essentially i i ∈ free for all i but lim µ = δ is the Dirac measure on a ﬁxed point x CantorΓ. By the i i x (cid:54) Rokhlin Lemma, SW (µ ) = P for all i and SW (µ ) = SW (δ ) since SW (δ ) is the subspace i i x x of measures supported on ﬁxed points. Proof. Since SW is compact, after passing to a subsequence, we may assume that lim SW (µ ) = Γ i i ∈ ∈ ≺ SW (ν) for some ν P. Since µ∞ = lim µ it follows that µ∞ SW (ν). Thus µ∞ ν i i s ⊆ and therefore SW (µ∞) SW (ν). 7 Convex integrals and couplings Let Perg denote the extreme points of P = Prob (CantorΓ). Let Prob(Perg) denote Γ → the space of Borel probability measures on Perg. Let π : CantorΓ Perg be an ergodic decomposition map. By deﬁnition this means that π is a Γ-invariant Borel map satisfying • ∈ {(cid:82) ∈ } For each e Perg, e( x CantorΓ : π(x) = e ) = 1. • ∈ For each µ P, µ = ∈ e dπ∗(µ). e Perg (cid:48) Furthermore, π is unique in the following sense: if π is another such map then the set { (cid:54) (cid:48) } ∈ x : π(x) = π (x) is µ-null for all µ P [GS00]. → ∈ Let π∗ : P Prob(Perg) be the associated aﬃne map which takes a measure µ P to ∈ its ergodic decomposition π∗(µ) Prob(Perg). In what follows we will abuse notation and ∈ ∈ write π(µ) for π∗(µ). If κ Prob(P) then w(cid:90)e let β(κ) P denote the Barycenter of κ. By deﬁnition, β(κ) = µ dκ(µ). P ∈ So β(π(µ)) = µ, and if κ Prob(Perg) then π(β(κ)) = κ. 16 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Corollary

text: [SW

text: is

text: lower

text: semi-continuous]

text: If

text: µ

text: is

text: a

text: sequence

text: in

text: P

text: and

text: lim

text: µ

text: µ∞

text: i

text: i

text: i

text: i

text: then

text: SW

text: (µ∞)

text: lim

text: inf

text: SW

text: (µ

text: i

text: i

text: Z

text: Remark

text: SW

text: is

text: not

text: continuous

text: in

text: general.

text: For

text: example,

text: consider

text: the

text: case

text: when

text: Γ

text: It

text: is

text: possible

text: to

text: ﬁnd

text: a

text: sequence

text: of

text: measures

text: µ

text: P

text: such

text: that

text: Γ

text: (CantorΓ,

text: µ

text: is

text: essentially

text: i

text: i

text: free

text: for

text: all

text: i

text: but

text: lim

text: µ

text: δ

text: is

text: the

text: Dirac

text: measure

text: on

text: a

text: ﬁxed

text: point

text: x

text: CantorΓ.

text: By

text: the

text: i

text: i

text: x

text: Rokhlin

text: Lemma,

text: SW

text: (µ

text: P

text: for

text: all

text: i

text: and

text: SW

text: (µ

text: SW

text: (δ

text: since

text: SW

text: (δ

text: is

text: the

text: subspace

text: i

text: i

text: x

text: x

text: of

text: measures

text: supported

text: on

text: ﬁxed

text: points.

text: Proof.

text: Since

text: SW

text: is

text: compact,

text: after

text: passing

text: to

text: a

text: subsequence,

text: we

text: may

text: assume

text: that

text: lim

text: SW

text: (µ

text: Γ

text: i

text: i

text: SW

text: (ν)

text: for

text: some

text: ν

text: P.

text: Since

text: µ∞

text: lim

text: µ

text: it

text: follows

text: that

text: µ∞

text: SW

text: (ν).

text: Thus

text: µ∞

text: ν

text: i

text: i

text: s

text: and

text: therefore

text: SW

text: (µ∞)

text: SW

text: (ν).

text: Convex

text: integrals

text: and

text: couplings

text: Let

text: Perg

text: denote

text: the

text: extreme

text: points

text: of

text: P

text: Prob

text: (CantorΓ).

text: Let

text: Prob(Perg)

text: denote

text: Γ

text: the

text: space

text: of

text: Borel

text: probability

text: measures

text: on

text: Perg.

text: Let

text: π

text: CantorΓ

text: Perg

text: be

text: an

text: ergodic

text: decomposition

text: map.

text: By

text: deﬁnition

text: this

text: means

text: that

text: π

text: is

text: a

text: Γ-invariant

text: Borel

text: map

text: satisfying

text: For

text: each

text: e

text: Perg,

text: e(

text: x

text: CantorΓ

text: π(x)

text: e

text: For

text: each

text: µ

text: P,

text: µ

text: e

text: dπ∗(µ).

text: e

text: Perg

text: Furthermore,

text: π

text: is

text: unique

text: in

text: the

text: following

text: sense:

text: if

text: π

text: is

text: another

text: such

text: map

text: then

text: the

text: set

text: x

text: π(x)

text: π

text: (x)

text: is

text: µ-null

text: for

text: all

text: µ

text: P

text: Let

text: π∗

text: P

text: Prob(Perg)

text: be

text: the

text: associated

text: aﬃne

text: map

text: which

text: takes

text: a

text: measure

text: µ

text: P

text: to

text: its

text: ergodic

text: decomposition

text: π∗(µ)

text: Prob(Perg).

text: In

text: what

text: follows

text: we

text: will

text: abuse

text: notation

text: and

text: write

text: π(µ)

text: for

text: π∗(µ).

text: If

text: κ

text: Prob(P)

text: then

text: let

text: β(κ)

text: P

text: denote

text: the

text: Barycenter

text: of

text: κ.

text: By

text: deﬁnition,

text: β(κ)

text: µ

text: dκ(µ).

text: P

text: So

text: β(π(µ))

text: µ,

text: and

text: if

text: κ

text: Prob(Perg)

text: then

text: π(β(κ))

text: κ.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: { } partition Q n of Y satisfying i i=1 | ∩ − ∩ | µ(γaP P ) ν(γbQ Q ) < (cid:15) i j i j ∈ ≤ ≤ (cid:121) ∈ ∈ for all γ S and 1 i, j n (where the action of Γ aX is denoted γax for γ Γ, x X ∼ ≺ for example). We say a is weakly equivalent to b, denoted a b, if both a b and ≺ b a. Z The Rokhlin Lemma is essentially equivalent to the statement that for the group Γ = all essentially free1 pmp actions are weakly equivalent. Indeed, as remarked in [Kec12], this statement holds for all countable amenable groups. However it fails for nonamenable groups because strong ergodicity is an invariant of weak equivalence [Kec10, Prop. 10.6]. This motivates the problem of providing a description of the set of all weak equivalence classes, denoted by W , for a given group Γ. Γ We start with an equivalent deﬁnition of weak containment. Let Cantor denote any − space homeomorphic to a Cantor set. Let Γ act on CantorΓ by (γx)(f) = x(γ 1f). Let Prob (CantorΓ) denote the space of all Γ-invariant Borel probability measures on CantorΓ Γ equipped with the weak* topology. It is well-known that Prob (CantorΓ) is a Choquet Γ simplex: this means it is a compact convex subset of a locally convex topological vector ∈ space with the property that every element µ Prob (CantorΓ) can be uniquely written Γ as a convex integral of extreme points of Prob (CantorΓ). Γ (cid:121) ⊆ Given an action a := Γ a(X, µ), let Factor(a) Prob (CantorΓ) denote the set of Γ → measures of the form Φ∗µ where Φ : X CantorΓ is a Γ-equivariant measurable map ◦ − and Φ∗µ = µ Φ 1. The weak* closure of Factor(a) is denoted W (a). It follows from ≺ ⊆ [AW13] that a b if and only if W (a) W (b) (see also [TD15, Prop. 3.6]). So the map (cid:55)→ a W (a) induces an injective map from the set of weak equivalence classes into the set of closed subsets of Prob (CantorΓ). We equip the latter with the Vietoris topology, and Γ W with the subspace topology. This topology, considered in [TD15], is a reformulation of Γ a construction due to Abert-Elek. The main result of [AE11] is that W is compact (an Γ alternative proof is given in [TD15]). This motivates the question: what sort of subsets of Prob (CantorΓ) can have the form Γ W (a)? This is addressed in [AW13]: if a is strongly ergodic then W (a) is contained in the 1An action is essentially free if almost every point has trivial stabilizer. 3 ##LTLine##

text: partition

text: Q

text: n

text: of

text: Y

text: satisfying

text: i

text: µ(γaP

text: P

text: ν(γbQ

text: Q

text: i

text: j

text: i

text: j

text: for

text: all

text: γ

text: S

text: and

text: i,

text: j

text: n

text: (where

text: the

text: action

text: of

text: Γ

text: aX

text: is

text: denoted

text: γax

text: for

text: γ

text: Γ,

text: x

text: X

text: for

text: example).

text: We

text: say

text: a

text: is

text: weakly

text: equivalent

text: to

text: b,

text: denoted

text: a

text: b,

text: if

text: both

text: a

text: b

text: and

text: b

text: a.

text: Z

text: The

text: Rokhlin

text: Lemma

text: is

text: essentially

text: equivalent

text: to

text: the

text: statement

text: that

text: for

text: the

text: group

text: Γ

text: all

text: essentially

text: pmp

text: actions

text: are

text: weakly

text: equivalent.

text: Indeed,

text: as

text: remarked

text: in

text: this

text: statement

text: holds

text: for

text: all

text: countable

text: amenable

text: groups.

text: However

text: it

text: fails

text: for

text: nonamenable

text: groups

text: because

text: strong

text: ergodicity

text: is

text: an

text: invariant

text: of

text: weak

text: equivalence

text: Prop.

text: This

text: motivates

text: the

text: problem

text: of

text: providing

text: a

text: description

text: of

text: the

text: set

text: of

text: all

text: weak

text: equivalence

text: classes,

text: denoted

text: by

text: W

text: for

text: a

text: given

text: group

text: Γ.

text: Γ

text: We

text: start

text: with

text: an

text: equivalent

text: deﬁnition

text: of

text: weak

text: containment.

text: Let

text: Cantor

text: denote

text: any

text: space

text: homeomorphic

text: to

text: a

text: Cantor

text: set.

text: Let

text: Γ

text: act

text: on

text: CantorΓ

text: by

text: (γx)(f)

text: x(γ

text: Let

text: Prob

text: (CantorΓ)

text: denote

text: the

text: space

text: of

text: all

text: Γ-invariant

text: Borel

text: probability

text: measures

text: on

text: CantorΓ

text: Γ

text: equipped

text: with

text: the

text: weak*

text: topology.

text: It

text: is

text: well-known

text: that

text: Prob

text: (CantorΓ)

text: is

text: a

text: Choquet

text: Γ

text: simplex:

text: this

text: means

text: it

text: is

text: a

text: compact

text: convex

text: subset

text: of

text: a

text: locally

text: convex

text: topological

text: vector

text: space

text: with

text: the

text: property

text: that

text: every

text: element

text: µ

text: Prob

text: (CantorΓ)

text: can

text: be

text: uniquely

text: written

text: Γ

text: as

text: a

text: convex

text: integral

text: of

text: extreme

text: points

text: of

text: Prob

text: (CantorΓ).

text: Γ

text: Given

text: an

text: action

text: a

text: Γ

text: a(X,

text: µ),

text: let

text: Factor(a)

text: Prob

text: (CantorΓ)

text: denote

text: the

text: set

text: of

text: Γ

text: measures

text: of

text: the

text: form

text: Φ∗µ

text: where

text: Φ

text: X

text: CantorΓ

text: is

text: a

text: Γ-equivariant

text: measurable

text: map

text: and

text: Φ∗µ

text: µ

text: Φ

text: The

text: weak*

text: closure

text: of

text: Factor(a)

text: is

text: denoted

text: W

text: (a).

text: It

text: follows

text: from

text: that

text: a

text: b

text: if

text: and

text: only

text: if

text: W

text: (a)

text: W

text: (b)

text: (see

text: also

text: Prop.

text: So

text: the

text: map

text: a

text: W

text: (a)

text: induces

text: an

text: injective

text: map

text: from

text: the

text: set

text: of

text: weak

text: equivalence

text: classes

text: into

text: the

text: set

text: of

text: closed

text: subsets

text: of

text: Prob

text: (CantorΓ).

text: We

text: equip

text: the

text: latter

text: with

text: the

text: Vietoris

text: topology,

text: and

text: Γ

text: W

text: with

text: the

text: subspace

text: topology.

text: This

text: topology,

text: considered

text: in

text: is

text: a

text: reformulation

text: of

text: Γ

text: a

text: construction

text: due

text: to

text: Abert-Elek.

text: The

text: main

text: result

text: of

text: is

text: that

text: W

text: is

text: compact

text: (an

text: Γ

text: alternative

text: proof

text: is

text: given

text: in

text: This

text: motivates

text: the

text: question:

text: what

text: sort

text: of

text: subsets

text: of

text: Prob

text: (CantorΓ)

text: can

text: have

text: the

text: form

text: Γ

text: W

text: (a)?

text: This

text: is

text: addressed

text: in

text: if

text: a

text: is

text: strongly

text: ergodic

text: then

text: W

text: (a)

text: is

text: contained

text: in

text: the

text: action

text: is

text: essentially

text: free

text: if

text: almost

text: every

text: point

text: has

text: trivial

text: stabilizer.

text: ##LTLine##


text: (cid:48) Therefore, after replacing ν by ν if necessary, we may assume without loss of generality that ≺ SW (ν) = W (ν) so that in fact µ ν. N Fix a non-principal ultraﬁlter U on . Let (CantorΓ, ν ) denote the ultrapower of U U (CantorΓ, ν). As π(ν) is a measure on P which concentrates on Perg, the ultrapower π(ν) U is a measure on the space P which concentrates on the set [Perg] which we identify with U (cid:81) Perg. By Proposition 8.4, we have that π(ν) = lim ∗π(ν) . In particular, for π(ν) -almost U (cid:81) U U U ∈ ∈ ∈ every [e ] Perg, we have lim → e Perg. For each [e ] Perg we let [e ] denote n U n U n n U U n ⊆ the measure on CantorΓ determined by [e ]([A ]) = lim → e (A ) for A CantorΓ U U n n n U n n n Borel. ≺ → → Since µ ν there exist Borel factor maps Φ : CantorΓ CantorΓ with (Φ )∗ν µ. n (cid:81) n → Let Φ : (CantorΓ) CantorΓ be the ultralimit function given by Φ([x ]) = lim → Φ (x ). U (cid:81) n n U n n By [TD15, Proposition 3.11] we have Φ∗(ν ) = lim → (Φ )∗ν = µ and Φ∗ [e ] = lim → (Φ )∗e U n U n U n n U n n ∈ (cid:55)→ for every [e ] Perg. The map [e ] Φ∗ [e ] = lim → (Φ )∗e , from Perg to P, is (cid:82) n U (cid:82) (cid:81) n U n n U n n U therefore measurable by Proposition 8.4. By [TD15, Proposition A.1], the decomposition (cid:90) ν = e dπ(ν)(e) yields ν = [e ] dπ(ν) ([e ]), and hence U U n U n µ = Φ∗(ν ) = lim (Φ )∗e dπ(ν) ([e ]). (2) U → n n U n n U × Let ρ be the measure on P P deﬁned by (cid:90) (cid:16) (cid:17) × ρ = π lim (Φ )∗e δ dπ(ν) ([e ]). (3) n→U n n limn→U en U n × Then ρ concentrates on Perg Perg, and (2) and Proposition 8.4 show that ρ is a coupling of π(µ) and π(ν). ⊆ × \ Claim 2. Let V P be open. Then ρ(B (Perg B )) = 0. V V (cid:110) (cid:111) Proof of Claim. Suppose not. Then the expression (3) implies that π(ν) (D ) > 0, where U 0 ∈ ∈ (cid:54)∈ D = [e ] Perg : lim (Φ )∗e C and lim e B . 0 n U → n n V → n V n U n U 23

text: Therefore,

text: after

text: replacing

text: ν

text: by

text: ν

text: if

text: necessary,

text: we

text: may

text: assume

text: without

text: loss

text: of

text: generality

text: that

text: SW

text: (ν)

text: W

text: (ν)

text: so

text: that

text: in

text: fact

text: µ

text: ν.

text: N

text: Fix

text: a

text: non-principal

text: ultraﬁlter

text: U

text: on

text: Let

text: (CantorΓ,

text: ν

text: denote

text: the

text: ultrapower

text: of

text: U

text: U

text: (CantorΓ,

text: ν).

text: As

text: π(ν)

text: is

text: a

text: measure

text: on

text: P

text: which

text: concentrates

text: on

text: Perg,

text: the

text: ultrapower

text: π(ν)

text: U

text: is

text: a

text: measure

text: on

text: the

text: space

text: P

text: which

text: concentrates

text: on

text: the

text: set

text: [Perg]

text: which

text: we

text: identify

text: with

text: U

text: Perg.

text: By

text: Proposition

text: we

text: have

text: that

text: π(ν)

text: lim

text: ∗π(ν)

text: In

text: particular,

text: for

text: π(ν)

text: -almost

text: U

text: U

text: U

text: U

text: every

text: [e

text: Perg,

text: we

text: have

text: lim

text: e

text: Perg.

text: For

text: each

text: [e

text: Perg

text: we

text: let

text: [e

text: denote

text: n

text: U

text: n

text: U

text: n

text: n

text: U

text: U

text: n

text: the

text: measure

text: on

text: CantorΓ

text: determined

text: by

text: [e

text: ]([A

text: lim

text: e

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text: for

text: A

text: CantorΓ

text: U

text: U

text: n

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text: µ

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text: Φ

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text: Low noise sensitivity analysis of -minimization in oversampled (cid:96) q ∗ systems Haolei Weng and Arian Maleki Abstract ∈ R The class of (cid:96) -regularized least squares (LQLS) are considered for estimating β p from its n noisy q linear observations y = Xβ +w. The performance of these schemes are studied under the high-dimensional asymptotic setting in which the dimension of the signal grows linearly with the number of measurements. In this asymptotic setting, phase transition diagrams (PT) are often used for comparing the performance of different estimators. PT speciﬁes the minimum number of observations required by a certain estimator to recover a structured signal, e.g. a sparse one, from its noiseless linear observations. Although phase transition analysis is shown to provide useful information for compressed sensing, the fact that it ignores the measurement noise not only limits its applicability in many application areas, but also may lead to misunderstandings. For instance, consider a linear regression problem in which n > p and the signal is not exactly sparse. If the measurement noise is ignored in such systems, regularization techniques, such as LQLS, seem to be irrelevant since even the ordinary least squares (OLS) returns the exact solution. However, it is well-known that if n is not much larger than p then the regularization techniques improve the performance of OLS. In response to this limitation of PT analysis, we consider the low-noise sensitivity analysis. We show that this analysis framework (i) reveals the advantage of LQLS over OLS, (ii) captures the difference between different LQLS estimators even when n > p, and (iii) provides a fair comparison among different estimators in high signal-to-noise ratios. As an application of this framework, we will show that under mild conditions LASSO outperforms other LQLS even when the signal is dense. Finally, by a simple transformation we → ∞ connect our low-noise sensitivity framework to the classical asymptotic regime in which n/p and characterize how and when regularization techniques offer improvements over ordinary least squares, and which regularizer gives the most improvement when the sample size is large. Key Words: High-dimensional linear model, (cid:96) -regularized least squares, ordinary least squares, q LASSO, phase transition, asymptotic mean square error, second-order expansion, classical asymp- totics. 1 Introduction 1.1 Problem Statement In modern data analysis, one of the fundamental models that has been extensively studied is the high- ∈ R ∈ R × dimensional linear model y = Xβ + w with y n, X n p, where the length of the signal p is at the same order or in some cases larger than the number of observations n. Since the ordinary least squares ∗ 1 ##LTLine##

text: Low

text: noise

text: sensitivity

text: analysis

text: of

text: -minimization

text: in

text: oversampled

text: q

text: systems

text: Haolei

text: Weng

text: and

text: Arian

text: Maleki

text: Abstract

text: R

text: The

text: class

text: of

text: -regularized

text: least

text: squares

text: (LQLS)

text: are

text: considered

text: for

text: estimating

text: β

text: p

text: from

text: its

text: n

text: noisy

text: q

text: linear

text: observations

text: y

text: Xβ

text: +w.

text: The

text: performance

text: of

text: these

text: schemes

text: are

text: studied

text: under

text: the

text: high-dimensional

text: asymptotic

text: setting

text: in

text: which

text: the

text: dimension

text: of

text: the

text: signal

text: grows

text: linearly

text: with

text: the

text: number

text: of

text: measurements.

text: In

text: this

text: asymptotic

text: setting,

text: phase

text: transition

text: diagrams

text: (PT)

text: are

text: often

text: used

text: for

text: comparing

text: the

text: performance

text: of

text: different

text: estimators.

text: PT

text: speciﬁes

text: the

text: minimum

text: number

text: of

text: observations

text: required

text: by

text: a

text: certain

text: estimator

text: to

text: recover

text: a

text: structured

text: signal,

text: e.g.

text: a

text: sparse

text: one,

text: from

text: its

text: noiseless

text: linear

text: observations.

text: Although

text: phase

text: transition

text: analysis

text: is

text: shown

text: to

text: provide

text: useful

text: information

text: for

text: compressed

text: sensing,

text: the

text: fact

text: that

text: it

text: ignores

text: the

text: measurement

text: noise

text: not

text: only

text: limits

text: its

text: applicability

text: in

text: many

text: application

text: areas,

text: but

text: also

text: may

text: lead

text: to

text: misunderstandings.

text: For

text: instance,

text: consider

text: a

text: linear

text: regression

text: problem

text: in

text: which

text: n

text: p

text: and

text: the

text: signal

text: is

text: not

text: exactly

text: sparse.

text: If

text: the

text: measurement

text: noise

text: is

text: ignored

text: in

text: such

text: systems,

text: regularization

text: techniques,

text: such

text: as

text: LQLS,

text: seem

text: to

text: be

text: irrelevant

text: since

text: even

text: the

text: ordinary

text: least

text: squares

text: (OLS)

text: returns

text: the

text: exact

text: solution.

text: However,

text: it

text: is

text: well-known

text: that

text: if

text: n

text: is

text: not

text: much

text: larger

text: than

text: p

text: then

text: the

text: regularization

text: techniques

text: improve

text: the

text: performance

text: of

text: OLS.

text: In

text: response

text: to

text: this

text: limitation

text: of

text: PT

text: analysis,

text: we

text: consider

text: the

text: low-noise

text: sensitivity

text: analysis.

text: We

text: show

text: that

text: this

text: analysis

text: framework

text: (i)

text: reveals

text: the

text: advantage

text: of

text: LQLS

text: over

text: OLS,

text: (ii)

text: captures

text: the

text: difference

text: between

text: different

text: LQLS

text: estimators

text: even

text: when

text: n

text: p,

text: and

text: (iii)

text: provides

text: a

text: fair

text: comparison

text: among

text: different

text: estimators

text: in

text: high

text: signal-to-noise

text: ratios.

text: As

text: an

text: application

text: of

text: this

text: framework,

text: we

text: will

text: show

text: that

text: under

text: mild

text: conditions

text: LASSO

text: outperforms

text: other

text: LQLS

text: even

text: when

text: the

text: signal

text: is

text: dense.

text: Finally,

text: by

text: a

text: simple

text: transformation

text: we

text: connect

text: our

text: low-noise

text: sensitivity

text: framework

text: to

text: the

text: classical

text: asymptotic

text: regime

text: in

text: which

text: n/p

text: and

text: characterize

text: how

text: and

text: when

text: regularization

text: techniques

text: offer

text: improvements

text: over

text: ordinary

text: least

text: squares,

text: and

text: which

text: regularizer

text: gives

text: the

text: most

text: improvement

text: when

text: the

text: sample

text: size

text: is

text: large.

text: Key

text: Words:

text: High-dimensional

text: linear

text: model,

text: -regularized

text: least

text: squares,

text: ordinary

text: least

text: squares,

text: q

text: LASSO,

text: phase

text: transition,

text: asymptotic

text: mean

text: square

text: error,

text: second-order

text: expansion,

text: classical

text: asymp-

text: totics.

text: Introduction

text: Problem

text: Statement

text: In

text: modern

text: data

text: analysis,

text: one

text: of

text: the

text: fundamental

text: models

text: that

text: has

text: been

text: extensively

text: studied

text: is

text: the

text: high-

text: R

text: R

text: dimensional

text: linear

text: model

text: y

text: Xβ

text: w

text: with

text: y

text: n,

text: X

text: n

text: p,

text: where

text: the

text: length

text: of

text: the

text: signal

text: p

text: is

text: at

text: the

text: same

text: order

text: or

text: in

text: some

text: cases

text: larger

text: than

text: the

text: number

text: of

text: observations

text: n.

text: Since

text: the

text: ordinary

text: least

text: squares

text: ##LTLine##


text: (OLS) estimate is not accurate in the high-dimensional regime, researchers have proposed a wide range of regularization techniques and recovery algorithms to go beyond OLS. The existence of a variety of algorithms and regularizers has in turn called for platforms that can provide fair comparisons among them. One of the most popular platforms is the phase transitions analysis (PT). Intuitively speaking, phase transition diagram measures the minimum number of observations an algorithm or an estimator requires to recover β. To understand the limitations of PT we study the performance of the following (cid:96) -regularized least squares (LQLS), a.k.a. bridge q estimators Frank and Friedman (1993), ∈ 1(cid:107) − (cid:107) (cid:107) (cid:107) ≤ ≤ βˆ(λ, q) arg min y Xβ 2 + λ β q, 1 q 2, (1) 2 2 q β (cid:107) · (cid:107) ≥ where is the usual (cid:96) norm and λ 0 is a tuning parameter. This family covers LASSO (Tibshirani, q q 1996) and Ridge (Hoerl and Kennard, 1970), two well known estimates in high-dimensional statistics and compressed sensing. (cid:107) − (cid:107) Phase transition analysis studies the asymptotic mean square error (AMSE) βˆ(λ, q) β 2/p under the → ∞ → 2 asymptotic setting p and n/p δ. Then, it considers w = 0 and calculates the smallest δ for which (cid:107) − (cid:107) inf lim →∞ βˆ(λ, q) β 2/p = 0. In this paper, we consider situations in which β is not exactly sparse. As is λ p 2 intuitively expected and will be discussed later in the paper, the phase transition analysis implies that for every ≤ ≤ (cid:107) − (cid:107) (cid:107) − 1 q 2, if δ > 1, then inf lim →∞ βˆ(λ, q) β 2/p = 0 and if δ < 1, then inf lim →∞ βˆ(λ, q) (cid:107) (cid:54) λ p 2 λ p β 2/p = 0. This simple application of PT reveals some of the limitations of the phase transition analysis: 2 1. Phase transition analysis is concerned with w = 0, and when β is not sparse, LQLS with different values of q have the same phase transition at δ = 1. Hence, it is not clear whether regularization can improve the performance of ordinary least squares (OLS) and if it does, which regularizer is the best. We expect the choice of regularizer to matter when we add some noise to the measurements. 2. Phase transition diagram is not sensitive to the magnitudes of the elements of β. Again, intuitively speaking, this seems to have a major impact on the performance of different estimators when the noise is present in the system. This paper aims to present a generalization of the phase transition, called low-noise sensitivity analysis, in which it is assumed that a small amount of noise is present in the measurements. Then it calculates AMSE of each estimate. This framework has the following two main advantages over the phase transition analysis: 1. It reveals certain phenomena that are important in applications and are not captured by PT analysis. For instance, one immediately sees the impact of the regularizer and the magnitudes of the elements of β on the AMSE. Furthermore, these relations are expressed explicitly and can be interpreted easily. 2. It provides a bridge between the phase transition analysis proposed in compressed sensing, and the classi- → ∞ cal large sample-size asymptotic (n/p ). We will discuss some of the implications of this connection for the classical asymptotics in Section . As a consequence, low noise sensitivity analysis enables us to present a fair comparison among different LQLS, and reveal different factors that affect their performance. Here is the organization of the rest of this paper: In Section , we discuss some related works and highlight the differences from ours. In section , we formally introduce the asymptotic framework adopted in our analyses. In Section , we present and discuss our main contributions in details. In Section , we prove all the main results. 2 ##LTLine##

text: (OLS)

text: estimate

text: is

text: not

text: accurate

text: in

text: the

text: high-dimensional

text: regime,

text: researchers

text: have

text: proposed

text: a

text: wide

text: range

text: of

text: regularization

text: techniques

text: and

text: recovery

text: algorithms

text: to

text: go

text: beyond

text: OLS.

text: The

text: existence

text: of

text: a

text: variety

text: of

text: algorithms

text: and

text: regularizers

text: has

text: in

text: turn

text: called

text: for

text: platforms

text: that

text: can

text: provide

text: fair

text: comparisons

text: among

text: them.

text: One

text: of

text: the

text: most

text: popular

text: platforms

text: is

text: the

text: phase

text: transitions

text: analysis

text: (PT).

text: Intuitively

text: speaking,

text: phase

text: transition

text: diagram

text: measures

text: the

text: minimum

text: number

text: of

text: observations

text: an

text: algorithm

text: or

text: an

text: estimator

text: requires

text: to

text: recover

text: β.

text: To

text: understand

text: the

text: limitations

text: of

text: PT

text: we

text: study

text: the

text: performance

text: of

text: the

text: following

text: -regularized

text: least

text: squares

text: (LQLS),

text: a.k.a.

text: bridge

text: q

text: estimators

text: Frank

text: and

text: Friedman

text: βˆ(λ,

text: q)

text: arg

text: min

text: y

text: Xβ

text: λ

text: β

text: q,

text: q

text: q

text: β

text: where

text: is

text: the

text: usual

text: norm

text: and

text: λ

text: is

text: a

text: tuning

text: parameter.

text: This

text: family

text: covers

text: LASSO

text: (Tibshirani,

text: q

text: q

text: and

text: Ridge

text: (Hoerl

text: and

text: Kennard,

text: two

text: well

text: known

text: estimates

text: in

text: high-dimensional

text: statistics

text: and

text: compressed

text: sensing.

text: Phase

text: transition

text: analysis

text: studies

text: the

text: asymptotic

text: mean

text: square

text: error

text: (AMSE)

text: βˆ(λ,

text: q)

text: β

text: under

text: the

text: asymptotic

text: setting

text: p

text: and

text: n/p

text: δ.

text: Then,

text: it

text: considers

text: w

text: and

text: calculates

text: the

text: smallest

text: δ

text: for

text: which

text: inf

text: lim

text: βˆ(λ,

text: q)

text: β

text: In

text: this

text: paper,

text: we

text: consider

text: situations

text: in

text: which

text: β

text: is

text: not

text: exactly

text: sparse.

text: As

text: is

text: λ

text: p

text: intuitively

text: expected

text: and

text: will

text: be

text: discussed

text: later

text: in

text: the

text: paper,

text: the

text: phase

text: transition

text: analysis

text: implies

text: that

text: for

text: every

text: q

text: if

text: δ

text: then

text: inf

text: lim

text: βˆ(λ,

text: q)

text: β

text: and

text: if

text: δ

text: then

text: inf

text: lim

text: βˆ(λ,

text: q)

text: λ

text: p

text: λ

text: p

text: β

text: This

text: simple

text: application

text: of

text: PT

text: reveals

text: some

text: of

text: the

text: limitations

text: of

text: the

text: phase

text: transition

text: analysis:

text: Phase

text: transition

text: analysis

text: is

text: concerned

text: with

text: w

text: and

text: when

text: β

text: is

text: not

text: sparse,

text: LQLS

text: with

text: different

text: values

text: of

text: q

text: have

text: the

text: same

text: phase

text: transition

text: at

text: δ

text: Hence,

text: it

text: is

text: not

text: clear

text: whether

text: regularization

text: can

text: improve

text: the

text: performance

text: of

text: ordinary

text: least

text: squares

text: (OLS)

text: and

text: if

text: it

text: does,

text: which

text: regularizer

text: is

text: the

text: best.

text: We

text: expect

text: the

text: choice

text: of

text: regularizer

text: to

text: matter

text: when

text: we

text: add

text: some

text: noise

text: to

text: the

text: measurements.

text: Phase

text: transition

text: diagram

text: is

text: not

text: sensitive

text: to

text: the

text: magnitudes

text: of

text: the

text: elements

text: of

text: β.

text: Again,

text: intuitively

text: speaking,

text: this

text: seems

text: to

text: have

text: a

text: major

text: impact

text: on

text: the

text: performance

text: of

text: different

text: estimators

text: when

text: the

text: noise

text: is

text: present

text: in

text: the

text: system.

text: This

text: paper

text: aims

text: to

text: present

text: a

text: generalization

text: of

text: the

text: phase

text: transition,

text: called

text: low-noise

text: sensitivity

text: analysis,

text: in

text: which

text: it

text: is

text: assumed

text: that

text: a

text: small

text: amount

text: of

text: noise

text: is

text: present

text: in

text: the

text: measurements.

text: Then

text: it

text: calculates

text: AMSE

text: of

text: each

text: estimate.

text: This

text: framework

text: has

text: the

text: following

text: two

text: main

text: advantages

text: over

text: the

text: phase

text: transition

text: analysis:

text: It

text: reveals

text: certain

text: phenomena

text: that

text: are

text: important

text: in

text: applications

text: and

text: are

text: not

text: captured

text: by

text: PT

text: analysis.

text: For

text: instance,

text: one

text: immediately

text: sees

text: the

text: impact

text: of

text: the

text: regularizer

text: and

text: the

text: magnitudes

text: of

text: the

text: elements

text: of

text: β

text: on

text: the

text: AMSE.

text: Furthermore,

text: these

text: relations

text: are

text: expressed

text: explicitly

text: and

text: can

text: be

text: interpreted

text: easily.

text: It

text: provides

text: a

text: bridge

text: between

text: the

text: phase

text: transition

text: analysis

text: proposed

text: in

text: compressed

text: sensing,

text: and

text: the

text: classi-

text: cal

text: large

text: sample-size

text: asymptotic

text: (n/p

text: We

text: will

text: discuss

text: some

text: of

text: the

text: implications

text: of

text: this

text: connection

text: for

text: the

text: classical

text: asymptotics

text: in

text: Section

text: As

text: a

text: consequence,

text: low

text: noise

text: sensitivity

text: analysis

text: enables

text: us

text: to

text: present

text: a

text: fair

text: comparison

text: among

text: different

text: LQLS,

text: and

text: reveal

text: different

text: factors

text: that

text: affect

text: their

text: performance.

text: Here

text: is

text: the

text: organization

text: of

text: the

text: rest

text: of

text: this

text: paper:

text: In

text: Section

text: we

text: discuss

text: some

text: related

text: works

text: and

text: highlight

text: the

text: differences

text: from

text: ours.

text: In

text: section

text: we

text: formally

text: introduce

text: the

text: asymptotic

text: framework

text: adopted

text: in

text: our

text: analyses.

text: In

text: Section

text: we

text: present

text: and

text: discuss

text: our

text: main

text: contributions

text: in

text: details.

text: In

text: Section

text: we

text: prove

text: all

text: the

text: main

text: results.

text: ##LTLine##


text: ∗ We now turn to analyzing R (χ , σ): q (cid:90) ∗ − E ∗ − − E ∗ − R (χ , σ) 1 = (η (B/σ + Z; χ ) B/σ Z)2 + 2 (∂ η (B/σ + Z; χ ) 1) q 1 1 1 − ∗ ≥ − E | | ≤ ∗ ≥ − µ/σ+χ ≥ − ∗ − ∗ 2 1( B/σ + Z χ ) 2 φ(z)dz 4χ φ(µ/σ χ ) − − ∗ µ/σ χ − ∗ − (≥d) E |8φ2(χ |µ≥/σ)∗ (∼e) −8φ2(µ/σ), 1( Z + B/σ χ ) E | | ≥ ∗ → ∗ where (d) is due to (25) and (e) holds because 1( Z + B/σ χ ) 1 and χ = o(σ). This result ∗ − combined with R (χ , σ) 1 < 0 from (24) ﬁnishes the proof. q ∈ We are in position to derive the expansion of AMSE(λ∗ , 1, σ ). Similarly as in the proof for q (1, 2], ,1 w we can use Lemma 10 to derive (23) for q = 1. Then we apply Lemma 10 again to obtain − δ ∗ − δ − ∗ AMSE(λ∗ , 1, σ ) − σ2 = σ¯2R (χ (σ¯), σ¯) − (σ¯2 σ¯2R (σ (σ¯), σ¯)/δ) ,1 w δ 1 w q q δ 1 q q ∗ − δσ¯2(R (χ (σ¯), σ¯) 1) − − − = q q− = o(exp( µ¯2/σ¯2)) = o(exp( µ˜2(δ 1)/(δσ2 ))), δ 1 w where 0 < µ˜ < µ¯ < µ. This closes the proof. 4.4 Proof of Theorem 4 ≤ Similar to the proof of Theorem 3, we consider two cases, i.e. 1 < q 2 and q = 1, and prove them separately. We will follow closely the roadmap illustrated in Section . 4.4.1 Proof for the case 1 < q < 2 Again all the results in this section can be proved easily for q = 2. We will only consider 1 < q < 2. Before we start the proof of our main result, we mention a simple lemma that will be used multiple times in our proof. − → Lemma 11. Let T (σ) and χ(σ) be two nonnegative sequences with the property: χ(σ)T q 2(σ) 0, as → σ 0. Then, η (T (σ), χ(σ)) q lim = 1. → σ 0 T (σ) Proof. The proof is a simple application of scale invariance property of η , i.e, Lemma 6 part (iii). We have q η (T (σ), χ(σ)) − lim q = lim η (1; χ(σ)T q 2(σ)) = 1, → → q σ 0 T (σ) σ 0 where the last step is the result of Lemma 6 part (ii). − Our ﬁrst goal is to show that when χ = Cσq, then lim → R(χ,σ) 1 is a negative constant by choosing an σ 0 σ2 appropriate C. However, since this proof is long, we break it to several steps. These steps are summarized in Lemmas 12, 13, and 14. Then in Lemma 15 we employ these three results to show that if χ = Cσq, then 21 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: We

text: now

text: turn

text: to

text: analyzing

text: R

text: (χ

text: σ):

text: q

text: E

text: E

text: R

text: (χ

text: σ)

text: (η

text: (B/σ

text: Z;

text: χ

text: B/σ

text: η

text: (B/σ

text: Z;

text: χ

text: q

text: E

text: µ/σ+χ

font: unknown

text: B/σ

text: Z

text: χ

text: φ(z)dz

text: 4χ

text: φ(µ/σ

text: χ

text: µ/σ

text: χ

text: (≥d)

text: E

text: |µ≥/σ)∗

text: (∼e)

text: Z

text: B/σ

text: χ

text: E

text: where

text: (d)

text: is

text: due

text: to

text: and

text: (e)

text: holds

text: because

text: Z

text: B/σ

text: χ

text: and

text: χ

text: o(σ).

text: This

text: result

text: combined

text: with

text: R

text: (χ

text: σ)

text: from

text: ﬁnishes

text: the

text: proof.

text: q

text: We

text: are

text: in

text: position

text: to

text: derive

text: the

text: expansion

text: of

text: AMSE(λ∗

text: σ

text: Similarly

text: as

text: in

text: the

text: proof

text: for

text: q

text: w

text: we

text: can

text: use

text: Lemma

text: to

text: derive

text: for

text: q

text: Then

text: we

text: apply

text: Lemma

text: again

text: to

text: obtain

text: δ

text: δ

text: AMSE(λ∗

text: σ

text: σ2

text: σ¯2R

text: (χ

text: (σ¯),

text: σ¯)

text: (σ¯2

text: σ¯2R

text: (σ

text: (σ¯),

text: σ¯)/δ)

text: w

text: δ

text: w

text: q

text: q

text: δ

text: q

text: q

text: δσ¯2(R

text: (χ

text: (σ¯),

text: σ¯)

text: q

text: q−

text: o(exp(

text: o(exp(

text: µ˜2(δ

text: δ

text: w

text: where

text: µ˜

text: µ¯

text: µ.

text: This

text: closes

text: the

text: proof.

text: Proof

text: of

text: Theorem

text: Similar

text: to

text: the

text: proof

text: of

text: Theorem

text: we

text: consider

text: two

text: cases,

text: i.e.

text: q

text: and

text: q

text: and

text: prove

text: them

text: separately.

text: We

text: will

text: follow

text: closely

text: the

text: roadmap

text: illustrated

text: in

text: Section

text: Proof

text: for

text: the

text: case

text: q

text: Again

text: all

text: the

text: results

text: in

text: this

text: section

text: can

text: be

text: proved

text: easily

text: for

text: q

text: We

text: will

text: only

text: consider

text: q

text: Before

text: we

text: start

text: the

text: proof

text: of

text: our

text: main

text: result,

text: we

text: mention

text: a

text: simple

text: lemma

text: that

text: will

text: be

text: used

text: multiple

text: times

text: in

text: our

text: proof.

text: Lemma

text: Let

text: T

text: (σ)

text: and

text: χ(σ)

text: be

text: two

text: nonnegative

text: sequences

text: with

text: the

text: property:

text: χ(σ)T

text: q

text: as

text: σ

text: Then,

text: η

text: (T

text: (σ),

text: χ(σ))

text: q

text: lim

text: σ

text: T

text: (σ)

text: Proof.

text: The

text: proof

text: is

text: a

text: simple

text: application

text: of

text: scale

text: invariance

text: property

text: of

text: η

text: i.e,

text: Lemma

text: part

text: (iii).

text: We

text: have

text: q

text: η

text: (T

text: (σ),

text: χ(σ))

text: lim

text: q

text: lim

text: η

text: χ(σ)T

text: q

text: q

text: σ

text: T

text: (σ)

text: σ

text: where

text: the

text: last

text: step

text: is

text: the

text: result

text: of

text: Lemma

text: part

text: (ii).

text: Our

text: ﬁrst

text: goal

text: is

text: to

text: show

text: that

text: when

text: χ

text: Cσq,

text: then

text: lim

text: R(χ,σ)

text: is

text: a

text: negative

text: constant

text: by

text: choosing

text: an

text: σ

text: σ2

text: appropriate

text: C.

text: However,

text: since

text: this

text: proof

text: is

text: long,

text: we

text: break

text: it

text: to

text: several

text: steps.

text: These

text: steps

text: are

text: summarized

text: in

text: Lemmas

text: and

text: Then

text: in

text: Lemma

text: we

text: employ

text: these

text: three

text: results

text: to

text: show

text: that

text: if

text: χ

text: Cσq,

text: then

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text: → → Our goal is to show that P 0 as σ 0. Since these intervals have different forms, we consider ﬁve − i ≤ ≤ ∗ ∗ ∗ different cases (i) i = 1, (ii) i = 0, (iii) 1 i m , (iv) i = m + 1, and (v) i = m + 2 and for each case → |I| I we show that P 0. Let denote the Lebesgue measure of an interval . For the ﬁrst term, we have for a i positive constant C˜− , (cid:90) 1 (cid:90) (cid:90) (cid:90) √ ∞ ≤ − (cid:90) 1(cid:90) ≤ − C˜−1σ log(1/σ) 1 P− σq 2 − φ(z)dzdF (b) σq 2 − φ(z)dzdF (b) 1 I χq(q 1) I χq(q 1) 0 ∞ −1 0 −1 +σq−2 √ (cid:112) 1− φ(z)dzdF (b) (cid:112) C˜−1σ log(1/σ) I−1 χq(q 1) ≤ σq−2φ(0)|I−1|P(|B| ≤ C˜−1(cid:112)σ log(1/σ)) σq−2φ(C˜−1 log(cid:112)(1/σ) − loσg(q1−/(cid:15)σ))|I−1| − + − χq(q 1) χq(q 1) ≤ σq−(cid:15)−2P(|B| ≤ C˜− σ log(1/σ)) σq−(cid:15)−2φ(C˜−1 log(1/σ)) O(1) 1 + O(1) 2 log(1/σ) log(1/σ) − − ≤ O(1)(log(1/σ))−q2+(cid:15) + O(1)σq (cid:15) 2+C˜−2 1/8 → 0, (28) log(1/σ) P | | − where we have used the condition ( B < t) = O(t2 q+(cid:15)) to obtain the last inequality and the last statement holds by choosin(cid:90)g C˜−(cid:90)1 large enough. We next analyze the term P0. For a constant C˜0 > 0 we have ∞ ≤ − 1 P σq 2 (cid:90) | (cid:90) | − φ(z)dzdF (b) 0 0 √I0\I−1 ηq(b/σ + z; χ) 2 q − (cid:90)C˜0σ log(1/σ) (cid:90) 1 = σq 2 | | − φ(z)dzdF (b) I \I η (b/σ + z; χ) 2 q 0 ∞ 0 −1 q − √ 1 (cid:16) (cid:112) (cid:17) +σq 2 (cid:112) | | − φ(z)dzdF (b) C˜(cid:0)0σ log(1/σ)(cid:1) I0\I−1 ηq(b/σ + z; χ) 2 q (cid:0) (cid:1) − − |I | ≤ σq−2φ(0)|I0|−P(|B| <− C˜0σ log(1/σ)) + σq 2φ C˜0 −log(1/σ−) log(σ1/σ) 0 . (29) η2 q σq (cid:15) ; χ η2 q σq (cid:15) ; χ q log(1/σ) q log(1/σ) | | ≥ − ∈ I We have used the fact that b/σ + z σq (cid:15) for z / − in the last step. Note that according to Lemma log(1/σ) 1 − − ∝ − − → 11, since ( σq (cid:15) )q 2χ σq2 (1+(cid:15))q+2(cid:15)(log(1/σ))2 q 0, we obtain log(1/σ) − σq (cid:15) log(1/σ) li→m − = 1. σ 0 η ( σq (cid:15) ; χ) q log(1/σ) With the above result, it is clear that the second term of the upper bound in (29) vanishes if choosing sufﬁciently large C˜0. Regarding the(cid:112)ﬁrst term we know (cid:0) (cid:1) − |I |P | | σq 2 ( B < C˜ σ log(1/σ)) ∝ − − 0 − −0 σq2 ((cid:15)+2)q+3(cid:15)+1(log(1/σ))(cid:15)+42 3q = o(1). η2 q σq (cid:15) ; χ q log(1/σ) 23 ##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##

text: Our

text: goal

text: is

text: to

text: show

text: that

text: P

text: as

text: σ

text: Since

text: these

text: intervals

text: have

text: different

text: forms,

text: we

text: consider

text: ﬁve

text: i

text: different

text: cases

text: (i)

text: i

text: (ii)

text: i

text: (iii)

text: i

text: m

text: (iv)

text: i

text: m

text: and

text: (v)

text: i

text: m

text: and

text: for

text: each

text: case

text: |I|

text: I

text: we

text: show

text: that

text: P

text: Let

text: denote

text: the

text: Lebesgue

text: measure

text: of

text: an

text: interval

text: For

text: the

text: ﬁrst

text: term,

text: we

text: have

text: for

text: a

text: i

text: positive

text: constant

text: C˜−

text: C˜−1σ

text: P−

text: σq

text: φ(z)dzdF

text: (b)

text: σq

text: φ(z)dzdF

text: (b)

text: I

text: χq(q

text: I

text: χq(q

text: −1

text: −1

text: 1−

text: φ(z)dzdF

text: (b)

text: C˜−1σ

text: χq(q

text: σq−2φ(C˜−1

text: χq(q

text: χq(q

text: C˜−

text: σ

text: P

text: where

text: we

text: have

text: used

text: the

text: condition

text: B

text: t)

text: to

text: obtain

text: the

text: last

text: inequality

text: and

text: the

text: last

text: statement

text: holds

text: by

text: large

text: enough.

text: We

text: next

text: analyze

text: the

text: term

text: For

text: a

text: constant

text: we

text: have

text: P

text: σq

text: φ(z)dzdF

text: (b)

text: ηq(b/σ

text: z;

text: χ)

text: q

text: σq

text: φ(z)dzdF

text: (b)

text: I

text: \I

text: η

text: (b/σ

text: z;

text: χ)

text: q

text: −1

text: q

text: +σq

text: φ(z)dzdF

text: (b)

text: ηq(b/σ

text: z;

text: χ)

text: q

text: |I

text: C˜0σ

text: σq

text: 2φ

text: η2

text: q

text: σq

text: χ

text: η2

text: q

text: σq

text: χ

text: q

text: q

text: I

text: We

text: have

text: used

text: the

text: fact

text: that

text: b/σ

text: z

text: σq

text: for

text: z

text: in

text: the

text: last

text: step.

text: Note

text: that

text: according

text: to

text: Lemma

text: since

text: σq

text: )q

text: 2χ

text: q

text: we

text: obtain

text: σq

text: li→m

text: σ

text: η

text: σq

text: χ)

text: q

text: With

text: the

text: above

text: result,

text: it

text: is

text: clear

text: that

text: the

text: second

text: term

text: of

text: the

text: upper

text: bound

text: in

text: vanishes

text: if

text: choosing

text: sufﬁciently

text: large

text: Regarding

text: term

text: we

text: know

text: |I

text: |P

text: σq

text: B

text: C˜

text: σ

text: −0

text: η2

text: q

text: σq

text: χ

text: q

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