text: (cid:88) (cid:88) (cid:2) (cid:3) ∞ n E (r, ω) = k2E a (ω)N(3) (r, ω) + b (ω)M(3) (r, ω) . (8) sca nm nm nm nm nm − n=1 m= n For the incident ﬁeld that has to be ﬁnite in the center of the coordinate systems the expansion reads as (cid:88) (cid:88) (cid:2) (cid:3) ∞ − n E (r, ω) = k2E p (ω)N(1) (r, ω) + q (ω)M(1) (r, ω) . (9) inc nm nm nm nm nm − n=1 m= n k2 = ε(ω)µ(ω) (ω2/c2) is the dispersion relation in the surrounding medium characterized by the permittivity ε(ω) and the permeability µ(ω) and − | | (n m)! E = E in(2n + 1) (10) nm inc (n + m)! | | is a scaling factor with E being the magnitude of the incident ﬁeld. The expansion inc coeﬃcients a (ω) and b (ω) are called scattering coeﬃcients. The expansion coeﬃcients nm nm p (ω) and q (ω) describe the illumination. They are analytically known for a plane nm nm wave or a Gaussian beam, but they can be also kept entirely free to expand an arbitrary illumination close to the scattering object. The link between the scattering coeﬃcients and the incident coeﬃcients is given by the T-matrix of the object. This matrix contains the entire information on how a speciﬁc objects responds to an external electromagnetic ﬁeld and reads as         a · p = T , (11) b q a and b are concatenated vectors containing the scattering coeﬃcients of the outgoing wave and p and q are concatenated vectors containing the coeﬃcients of the incident wave. The T-matrix links them. 11 ##LTLine##

label: equation

text: n

text: E

text: (r,

text: ω)

text: a

text: (r,

text: ω)

text: b

text: (r,

text: ω)

text: sca

text: nm

text: nm

text: nm

text: nm

text: nm

text: m=

text: n

text: For

text: the

text: incident

text: ﬁeld

text: that

text: has

text: to

text: be

text: ﬁnite

text: in

text: the

text: center

text: of

text: the

text: coordinate

text: systems

text: the

text: expansion

text: reads

text: as

text: n

text: E

text: (r,

text: ω)

text: p

text: (r,

text: ω)

text: q

text: (r,

text: ω)

text: inc

text: nm

text: nm

text: nm

text: nm

text: nm

text: m=

text: n

text: ε(ω)µ(ω)

text: is

text: the

text: dispersion

text: relation

text: in

text: the

text: surrounding

text: medium

text: characterized

text: by

text: the

text: permittivity

text: ε(ω)

text: and

text: the

text: permeability

text: µ(ω)

text: and

text: (n

text: m)!

text: E

text: E

text: nm

text: inc

text: (n

text: m)!

text: is

text: a

text: scaling

text: factor

text: with

text: E

text: being

text: the

text: magnitude

text: of

text: the

text: incident

text: ﬁeld.

text: The

text: expansion

text: inc

text: coeﬃcients

text: a

text: (ω)

text: and

text: b

text: (ω)

text: are

text: called

text: scattering

text: coeﬃcients.

text: The

text: expansion

text: coeﬃcients

text: nm

text: nm

text: p

text: (ω)

text: and

text: q

text: (ω)

text: describe

text: the

text: illumination.

text: They

text: are

text: analytically

text: known

text: for

text: a

text: plane

text: nm

text: nm

text: wave

text: or

text: a

text: Gaussian

text: beam,

text: but

text: they

text: can

text: be

text: also

text: kept

text: entirely

text: free

text: to

text: expand

text: an

text: arbitrary

text: illumination

text: close

text: to

text: the

text: scattering

text: object.

text: The

text: link

text: between

text: the

text: scattering

text: coeﬃcients

text: and

text: the

text: incident

text: coeﬃcients

text: is

text: given

text: by

text: the

text: T-matrix

text: of

text: the

text: object.

text: This

text: matrix

text: contains

text: the

text: entire

text: information

text: on

text: how

text: a

text: speciﬁc

text: objects

text: responds

text: to

text: an

text: external

text: electromagnetic

text: ﬁeld

text: and

text: reads

text: as

text: a

text: p

text: T

text: b

text: q

text: a

text: and

text: b

text: are

text: concatenated

text: vectors

text: containing

text: the

text: scattering

text: coeﬃcients

text: of

text: the

text: outgoing

text: wave

text: and

text: p

text: and

text: q

text: are

text: concatenated

text: vectors

text: containing

text: the

text: coeﬃcients

text: of

text: the

text: incident

text: wave.

text: The

text: T-matrix

text: links

text: them.

text: ##LTLine##


text: λ [nm] 700 600 500 400 25 δ=36/4 20 δ=23/5 δ=16/6 ideal X 15 s 10 5 0 −5 −10 400 500 600 700 800 Frequency [THz] FIG. 1: Surface reactance calculated for diﬀerent aspect ratios δ = a /a of the z x constituting ellipsoids and the ideal surface reactance that is required to cancel the scattering. Spectral positions where both surface reactances are equal correspond to the operational frequencies for diﬀerent aspect ratios. Here, β(ω) describes the interaction of the nanoparticles in the array while being exposed to an external excitation [30], d deﬁnes the distance of the particles, and k is the wave number of the incident plane wave. The polarizability α(ω) of an ellipsoid is analytically known to be [31] − ε (ω) ε α (ω) = p h− , (2) z ε + N (ε (ω) ε ) h z p h where ε (ω) is the permittivity of the ellipsoid obtained from experimental data [32], p ε = 1 is the host medium, in our case air, and N is the depolarization factor along z which h z depends on the aspect ratio δ = a /a of the ellipsoid. The axes are chosen such that the z x volume V = 4πa2a stays constant. 3 x z 5 ]Ω[ X s ##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##

text: λ

label: paragraph

font: default

text: [nm]

text: ideal

text: X

text: s

text: −5

text: Frequency

text: [THz]

text: FIG.

text: Surface

label: caption

text: reactance

text: calculated

text: for

text: diﬀerent

text: aspect

text: ratios

text: δ

text: a

text: /a

text: of

text: the

text: z

text: x

text: constituting

text: ellipsoids

text: and

text: the

text: ideal

text: surface

text: reactance

text: that

text: is

text: required

text: to

text: cancel

text: the

text: scattering.

text: Spectral

text: positions

text: where

text: both

text: surface

text: reactances

text: are

text: equal

text: correspond

text: to

text: the

text: operational

text: frequencies

text: for

text: diﬀerent

text: aspect

text: ratios.

text: Here,

text: β(ω)

text: describes

text: the

text: interaction

text: of

text: the

text: nanoparticles

text: in

text: the

text: array

text: while

text: being

text: exposed

text: to

text: an

text: external

text: excitation

text: d

text: deﬁnes

text: the

text: distance

text: of

text: the

text: particles,

text: and

text: k

text: is

text: the

text: wave

text: number

text: of

text: the

text: incident

text: plane

text: wave.

text: The

text: polarizability

text: α(ω)

text: of

text: an

text: ellipsoid

text: is

text: analytically

text: known

text: to

text: be

text: ε

text: (ω)

text: ε

text: α

text: (ω)

text: p

text: h−

text: z

text: ε

text: N

text: (ε

text: (ω)

text: ε

text: h

text: z

text: p

text: h

text: where

text: ε

text: (ω)

text: is

text: the

text: permittivity

text: of

text: the

text: ellipsoid

text: obtained

text: from

text: experimental

text: data

text: p

text: ε

text: is

text: the

text: host

text: medium,

text: in

text: our

text: case

text: air,

text: and

text: N

text: is

text: the

text: depolarization

text: factor

text: along

text: z

text: which

text: h

text: z

text: depends

text: on

text: the

text: aspect

text: ratio

text: δ

text: a

text: /a

text: of

text: the

text: ellipsoid.

text: The

text: axes

text: are

text: chosen

text: such

text: that

text: the

text: z

text: x

text: volume

text: V

text: stays

text: constant.

text: x

text: z

text: ]Ω[

text: X

text: s

text: ##LTFigure##

text: ##LTLine##

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text: 12 bicubic interpolation, the OMP-based super-resolution tech- resolution (RAISR) algorithm [53]. In this method, different nique of [68] (with 3 iterations) and its version with IHT and ﬁlters are trained for different types of patches in natural LIPGD (replacing OMP). It can be seen clearly that IHT leads images. This leads to improved quality in the attained up- to inferior results compared to OMP since it does not converge scaled images with only minor overhead in the computational in 3 iterations. LIPGD improves over both IHT and OMP as cost, leading to a very efﬁcient super-resolution technique. the training of the network allows it to provide good sparse approximation with only 3 iterations. This demonstrates the IX. CONCLUSION efﬁciency of the proposed LIPGD technique, which has the In this work we suggested an approach to trade-off between same computational complexity of both OMP and IHT. approximation error and convergence speed. This is accom- plished by approximating complicated projections by inexact VIII. LEARNING THE PROJECTION – LISTA MIXTURE ones that are computationally efﬁcient. We provided theory MODEL for the convergence of an iterative algorithm that uses such an approximate projection and showed that at the cost of an Though the theory in this paper applies directly only to error in the projection one may achieve faster convergence in (35) (with some constraints on A and U that stem from the ﬁrst iterations. The larger the error the smaller the number the constraints on p), the fast convergence of LISTA may be of iterations that enjoy fast convergence. This suggests that explained by the resemblance of the two methods. The success if we have a budget for only a small number of iterations of LISTA may be interpreted as learning to approximate the K (with a given complexity), then it may be worthwhile to use set ˆ in an indirect way by learning the linear operators A and inexact projections which can result in a worse solution in U. In other words, it can be viewed as a method for learning a S the long term but make better use of the given computational linear operator that together with the proximal mapping f,λ constraints. Moreover, we showed that even when we can approximates a more accurate proximal mapping of a true afford a larger number of iterations, it may be worthwhile to unknown function fˆ that leads to much faster convergence. use inexact projections in the ﬁrst iterations and then change With this understanding, we argue that using multiple inex- to more accurate ones at latter stages. act projections may lead to faster convergence as each can Our theory offers an explanation to the recent success of approximate in a more accurate way different parts of the K neural networks for approximating the solution of certain min- set ˆ. In order to show this, we propose a LISTA mixture imization problems. These networks achieve similar accuracy model (MM), similar to the Gaussian mixture model proposed to iterative techniques developed for such problems (e.g., ISTA in [67], in which we train several LISTA networks, one for for (cid:96) minimization) but with much smaller computational each part of the dataset. Then, once we get a new vector, we 1 cost. We demonstrate the usage of this method for the problem apply all the networks on it in parallel (and therefore with of image super-resolution. In addition, our analysis provides negligible impact on the latency, which is very important in a technique for estimating the solution of these minimization many applications) and chose the one that attains the smallest problems by using multiple networks but with fewer layers in value in the objective of the minimization problem (3). each of them. We test this strategy on the house image by extracting from × it patches of size 5 5, adding random Gaussian noise to APPENDIX A each of them with variance σ2 = 25 and then removing the PROOF OF THEOREM 5.1 DC and normalizing each. We take 7/9 of the patches for training and 1/9 for validation and testing. We train LISTA The proof of Theorem 5.1 relies on the following lemma. to minimize directly the objective (3) as in [55] and stop the optimization after the error of the validation set increases. For Lemma A.1: Under the same conditions of Theorem 5.1 the LISTA-MM we use 6 LISTA networks such that we train the ﬁrst one on the whole data. We then remove 1/6 of the (cid:107)P − ∗ − (cid:107) (p (I µM M) (z x)) (36) data whose objective value in (3) is the closest to the one ISTA C t ≤ (cid:107) − (cid:107) (cid:107) (cid:107) attains after 1000 iterations. We use this LISTA network as the ρp(C) zt x + (cid:15)(1 + ρp(C)) x . initialization of the next one that is trained on the rest of the data. We repeat this process by removing in the same way P Proof: Since z = K(pv) for a certain vector v, we have the part of the data with the smallest relative error and then t train the next network. After training 6 networks we cluster (cid:107)P − ∗ − (cid:107) the data points by selecting for each patch the network that (p (I µM M) (z x)) (37) C t leads to the smallest objective error for it in (3) and ﬁne tune ≤ (cid:107)P − ∗ P − (cid:107) sup (p (I µM M) ( K(pv) x)) each network for its corresponding group of patches. We repeat ∈R C (cid:107)P v −d(cid:107)s≤.t.(cid:107) − (cid:107) this process 5 times. The objective error of (3) as a function of K(pv) x zt x (cid:107)P − ∗ P − (cid:107) the number of iterations/depth of the networks is presented in = sup (p (I µM M) (pv x)) , C D ∈R Fig. 6. Indeed, it can be seen that partitioning the data, which (cid:107)P v− d(cid:107)s≤.t.(cid:107) − (cid:107) leads to a better approximation, accelerates convergence. D(pv x) zt x Our proposed LISTA-MM strategy bears some resemblence where the last equality follows from Lemma 2.3. Using the to the recently proposed rapid and accurate image super triangle inequality with (37) leads to

text: bicubic

text: interpolation,

text: the

text: OMP-based

text: super-resolution

text: tech-

text: resolution

text: (RAISR)

text: algorithm

text: In

text: this

text: method,

text: different

text: nique

text: of

text: (with

text: iterations)

text: and

text: its

text: version

text: with

text: IHT

text: and

text: ﬁlters

text: are

text: trained

text: for

text: different

text: types

text: of

text: patches

text: in

text: natural

text: LIPGD

text: (replacing

text: OMP).

text: It

text: can

text: be

text: seen

text: clearly

text: that

text: IHT

text: leads

text: images.

text: This

text: leads

text: to

text: improved

text: quality

text: in

text: the

text: attained

text: up-

text: to

text: inferior

text: results

text: compared

text: to

text: OMP

text: since

text: it

text: does

text: not

text: converge

text: scaled

text: images

text: with

text: only

text: minor

text: overhead

text: in

text: the

text: computational

text: in

text: iterations.

text: LIPGD

text: improves

text: over

text: both

text: IHT

text: and

text: OMP

text: as

text: cost,

text: leading

text: to

text: a

text: very

text: efﬁcient

text: super-resolution

text: technique.

text: the

text: training

text: of

text: the

text: network

text: allows

text: it

text: to

text: provide

text: good

text: sparse

text: approximation

text: with

text: only

text: iterations.

text: This

text: demonstrates

text: the

text: IX.

text: CONCLUSION

text: efﬁciency

text: of

text: the

text: proposed

text: LIPGD

text: technique,

text: which

text: has

text: the

text: In

text: this

text: work

text: we

text: suggested

text: an

text: approach

text: to

text: trade-off

text: between

text: same

text: computational

text: complexity

text: of

text: both

text: OMP

text: and

text: IHT.

text: approximation

text: error

text: and

text: convergence

text: speed.

text: This

text: is

text: accom-

text: plished

text: by

text: approximating

text: complicated

text: projections

text: by

text: inexact

text: VIII.

text: LEARNING

text: THE

text: PROJECTION

text: LISTA

text: MIXTURE

text: ones

text: that

text: are

text: computationally

text: efﬁcient.

text: We

text: provided

text: theory

text: MODEL

text: for

text: the

text: convergence

text: of

text: an

text: iterative

text: algorithm

text: that

text: uses

text: such

text: an

text: approximate

text: projection

text: and

text: showed

text: that

text: at

text: the

text: cost

text: of

text: an

text: Though

text: the

text: theory

text: in

text: this

text: paper

text: applies

text: directly

text: only

text: to

text: error

text: in

text: the

text: projection

text: one

text: may

text: achieve

text: faster

text: convergence

text: in

text: (with

text: some

text: constraints

text: on

text: A

text: and

text: U

text: that

text: stem

text: from

text: the

text: ﬁrst

text: iterations.

text: The

text: larger

text: the

text: error

text: the

text: smaller

text: the

text: number

text: the

text: constraints

text: on

text: p),

text: the

text: fast

text: convergence

text: of

text: LISTA

text: may

text: be

text: of

text: iterations

text: that

text: enjoy

text: fast

text: convergence.

text: This

text: suggests

text: that

text: explained

text: by

text: the

text: resemblance

text: of

text: the

text: two

text: methods.

text: The

text: success

text: if

text: we

text: have

text: a

text: budget

text: for

text: only

text: a

text: small

text: number

text: of

text: iterations

text: of

text: LISTA

text: may

text: be

text: interpreted

text: as

text: learning

text: to

text: approximate

text: the

text: K

text: (with

text: a

text: given

text: complexity),

text: then

text: it

text: may

text: be

text: worthwhile

text: to

text: use

text: set

text: ˆ

text: in

text: an

text: indirect

text: way

text: by

text: learning

text: the

text: linear

text: operators

text: A

text: and

text: inexact

text: projections

text: which

text: can

text: result

text: in

text: a

text: worse

text: solution

text: in

text: U.

text: In

text: other

text: words,

text: it

text: can

text: be

text: viewed

text: as

text: a

text: method

text: for

text: learning

text: a

text: S

text: the

text: long

text: term

text: but

text: make

text: better

text: use

text: of

text: the

text: given

text: computational

text: linear

text: operator

text: that

text: together

text: with

text: the

text: proximal

text: mapping

text: f,λ

text: constraints.

text: Moreover,

text: we

text: showed

text: that

text: even

text: when

text: we

text: can

text: approximates

text: a

text: more

text: accurate

text: proximal

text: mapping

text: of

text: a

text: true

text: afford

text: a

text: larger

text: number

text: of

text: iterations,

text: it

text: may

text: be

text: worthwhile

text: to

text: unknown

text: function

text: fˆ

text: that

text: leads

text: to

text: much

text: faster

text: convergence.

text: use

text: inexact

text: projections

text: in

text: the

text: ﬁrst

text: iterations

text: and

text: then

text: change

text: With

text: this

text: understanding,

text: we

text: argue

text: that

text: using

text: multiple

text: inex-

text: to

text: more

text: accurate

text: ones

text: at

text: latter

text: stages.

text: act

text: projections

text: may

text: lead

text: to

text: faster

text: convergence

text: as

text: each

text: can

text: Our

text: theory

text: offers

text: an

text: explanation

text: to

text: the

text: recent

text: success

text: of

text: approximate

text: in

text: a

text: more

text: accurate

text: way

text: different

text: parts

text: of

text: the

text: K

text: neural

text: networks

text: for

text: approximating

text: the

text: solution

text: of

text: certain

text: min-

text: set

text: ˆ.

text: In

text: order

text: to

text: show

text: this,

text: we

text: propose

text: a

text: LISTA

text: mixture

text: imization

text: problems.

text: These

text: networks

text: achieve

text: similar

text: accuracy

text: model

text: (MM),

text: similar

text: to

text: the

text: Gaussian

text: mixture

text: model

text: proposed

text: to

text: iterative

text: techniques

text: developed

text: for

text: such

text: problems

text: (e.g.,

text: ISTA

text: in

text: in

text: which

text: we

text: train

text: several

text: LISTA

text: networks,

text: one

text: for

text: for

text: minimization)

text: but

text: with

text: much

text: smaller

text: computational

text: each

text: part

text: of

text: the

text: dataset.

text: Then,

text: once

text: we

text: get

text: a

text: new

text: vector,

text: we

text: cost.

text: We

text: demonstrate

text: the

text: usage

text: of

text: this

text: method

text: for

text: the

text: problem

text: apply

text: all

text: the

text: networks

text: on

text: it

text: in

text: parallel

text: (and

text: therefore

text: with

text: of

text: image

text: super-resolution.

text: In

text: addition,

text: our

text: analysis

text: provides

text: negligible

text: impact

text: on

text: the

text: latency,

text: which

text: is

text: very

text: important

text: in

text: a

text: technique

text: for

text: estimating

text: the

text: solution

text: of

text: these

text: minimization

text: many

text: applications)

text: and

text: chose

text: the

text: one

text: that

text: attains

text: the

text: smallest

text: problems

text: by

text: using

text: multiple

text: networks

text: but

text: with

text: fewer

text: layers

text: in

text: value

text: in

text: the

text: objective

text: of

text: the

text: minimization

text: problem

text: each

text: of

text: them.

text: We

text: test

text: this

text: strategy

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text: the

text: house

text: image

text: by

text: extracting

text: from

text: it

text: patches

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text: APPENDIX

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text: variance

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text: PROOF

text: OF

text: THEOREM

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text: each.

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text: following

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text: the

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text: then

text: t

text: train

text: the

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text: network.

text: After

text: training

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text: the

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text: selecting

text: for

text: each

text: patch

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text: (p

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text: The

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text: K(pv)

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text: the

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text: the

text: networks

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text: ∈R

text: Fig.

text: Indeed,

text: it

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text: partitioning

text: the

text: data,

text: which

text: v−

text: leads

text: to

text: a

text: better

text: approximation,

text: accelerates

text: convergence.

text: D(pv

text: x)

text: zt

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text: proposed

text: LISTA-MM

text: strategy

text: bears

text: some

text: resemblence

text: where

text: the

text: last

text: equality

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text: from

text: Lemma

text: Using

text: the

text: to

text: the

text: recently

text: proposed

text: rapid

text: and

text: accurate

text: image

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text: triangle

text: inequality

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text: leads

text: to


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text: space

text: methods

text: for

text: incoherence

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text: in

text: optimization.

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text: Academy

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text: Sciences

text: and

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text: analysis

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text: Hegde,

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text: L.

text: Schmidt.

text: Approximation-tolerant

text: model-

text: nonlinear

text: regression:

text: Parameter

text: estimation

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text: asymptotic

text: inference

text: based

text: compressive

text: sensing.

text: In

text: ACM

text: Symposium

text: on

text: Discrete

text: Algorithms

text: under

text: nonconvexity.

text: In

text: International

text: Conference

text: on

text: Machine

text: Learning

text: (SODA),

text: (ICML),

text: A.

text: O.

text: Hero,

text: R.

text: Piramuthu,

text: J.

text: A.

text: Fessler,

text: and

text: S.

text: R.

text: Titus.

text: Minimax

text: G.

text: Yu,

text: G.

text: Sapiro,

text: and

text: S.

text: Mallat.

text: Solving

text: inverse

text: problems

text: with

text: emission

text: computed

text: tomography

text: using

text: high-resolution

text: anatomical

text: side

text: piecewise

text: linear

text: estimators:

text: From

text: Gaussian

text: mixture

text: models

text: to

text: structured

text: information

text: and

text: b-spline

text: models.

text: IEEE

text: Trans.

text: Inf.

text: Theory,

text: sparsity.

text: IEEE

text: Trans.

text: on

text: Image

text: Processing,

text: may

text: Apr.

text: M.

text: A.

text: Khajehnejad,

text: W.

text: Xu,

text: A.

text: S.

text: Avestimehr,

text: and

text: B.

text: Hassibi.

text: Weighted

text: R.

text: Zeyde,

text: M.

text: Elad,

text: and

text: M.

text: Protter.

text: On

text: single

text: image

text: scale-up

text: using

text: rIencfoorvmeraytiownitThhperoiroyr

text: (iInSfIoTr)m,

text: aptaigoens.

text: June

text: Springer-Verlag.

text: A.

text: Kipnis,

text: Y.

text: C.

text: Eldar,

text: and

text: A.

text: J.

text: Goldsmith.

text: Sampling

text: stationary

text: signals

text: subject

text: to

text: bitrate

text: constraints.

text: P.

text: Machart,

text: S.

text: Anthoine,

text: and

text: L.

text: Baldassarre.

text: Optimal

text: computational

text: trade-off

text: of

text: inexact

text: proximal

text: methods.

text: In

text: Multi-Trade-offs

text: in

text: Machine

text: Learning

text: (NIPS

text: workshop),

text: S.

text: Mallat

text: and

text: Z.

text: Zhang.

text: Matching

text: pursuits

text: with

text: time-frequency

text: dictionaries.

text: IEEE

text: Trans.

text: Signal

text: Process.,

text: T.

text: Moreau

text: and

text: J.

text: Bruna.

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text: neural

text: sparse

text: coding

text: with

text: matrix

text: factorization.

text: ICLR,

text: Y.

text: Nesterov.

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text: of

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text: rate

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text: tradeoffs

text: for

text: linear

text: inverse

text: problems.

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text: J.

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text: programs

text: with

text: sharp

text: guarantees.

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text: Theory,

text: M.

text: Pilanci

text: and

text: M.

text: J.

text: Wainwright.

text: Iterative

text: hessian

text: sketch:

text: Fast

text: and

text: accurate

text: solution

text: approximation

text: for

text: constrained

text: least-squares.

text: Journal

text: of

text: Machine

text: Learning

text: Research

text: (JMLR),

text: Y.

text: Plan

text: and

text: R.

text: Vershynin.

text: Robust

text: compressed

text: sensing

text: and

text: sparse

text: logistic

text: regression:

text: A

text: convex

text: programming

text: approach.

text: IEEE

text: Trans.

text: Inf.

text: Theory,

text: Jan.

text: T.

text: Remez,

text: O.

text: Litany,

text: and

text: A.

text: M.

text: Bronstein.

text: A

text: picture

text: is

text: worth

text: a

text: billion

text: bits:

text: Real-time

text: image

text: reconstruction

text: from

text: dense

text: binary

text: pixels.

text: In

text: International

text: Conference

text: on

text: Computational

text: Photography

text: (ICCP),

text: Y.

text: Romano,

text: J.

text: Isidoro,

text: and

text: P.

text: Milanfar.

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text: Rapid

text: and

text: accurate

text: image

text: super

text: resolution.

text: IEEE

text: Trans.

text: on

text: Computational

text: Imaging,

text: Mar.

text: S.

text: Shalev-Shwartz

text: and

text: N.

text: Srebro.

text: Svm

text: optimization:

text: Inverse

text: dependence

text: on

text: training

text: set

text: size.

text: In

text: International

text: Conference

text: on

text: Machine

text: Learning

text: (ICML),

text: pages

text: P.

text: Sprechmann,

text: A.

text: M.

text: Bronstein,

text: and

text: G.

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text: sparse

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text: low

text: rank

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text: Analysis

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text: Machine

text: Intelligence,

text: Sept.

text: W.

text: Su,

text: S.

text: Boyd,

text: and

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text: J.

text: Cande`s.

text: A

text: differential

text: equation

text: for

text: modeling

text: nesterov’s

text: accelerated

text: gradient

text: method:

text: Theory

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text: insights.

text: In

text: Advances

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text: Neural

text: Information

text: Processing

text: Systems

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text: T.

text: Tirer

text: and

text: R.

text: Giryes.

text: Generalizing

text: cosamp

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text: a

text: union

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text: low

text: dimensional

text: linear

text: subspaces.

text: arXiv

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text: Sprechmann,

text: and

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text: Perlin.

text: Accelerating

text: Eulerian

text: ﬂuid

text: simulation

text: with

text: convolutional

text: networks.

text: Proceedings

text: of

text: the

text: International

text: Conference

text: on

text: Machine

text: Learning

text: (ICML),

text: E.

text: Treister

text: and

text: I.

text: Yavneh.

text: A

text: multilevel

text: iterated-shrinkage

text: approach

text: to

text: penalized

text: least-squares

text: minimization.

text: IEEE

text: Trans.

text: Signal

text: Process.,

text: Dec.


text: 9 100 100 10-1 10-1 PGD K PGD K PGD tree PGD tree IPGD 1 tree level IPGD 1 tree level IPGD 2 tree levels IPGD 2 tree levels IPGD 3 tree levels IPGD 3 tree levels IPGD 4 tree levels IPGD 4 tree levels 10-2 IPGD changing levels 10-2 IPGD changing levels 0 5 10 15 20 25 30 35 40 45 50 0 1 2 3 4 5 6 t (Iteration Number) Running Time (sec) 10-3 (a) Recovery error as a function of iteration number (b) Recovery error as a function of running time (in sec) 2.4 2.4 2.2 2.2 2 2 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1 1 PGD K PGD K 0.8 0.8 PGD tree PGD tree IPGD 1 tree level IPGD 1 tree level IPGD 2 tree levels IPGD 2 tree levels IPGD 3 tree levels IPGD 3 tree levels 0.6 IPGD 4 tree levels 0.6 IPGD 4 tree levels IPGD changing levels IPGD changing levels 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 1.2 t (Iteration Number) Running Time (sec) 10-3 (c) Recovery error as a function of iteration number zoomed (d) Recovery error as a function of running time (in sec) zoomed Fig. 3. Reconstruction error as a function of the iterations (left) and the running time (right) for recovering a sparse vector with tree structure. Since we (cid:107) (cid:107) initialize all algorithms with the zero vector, the error at iteration/time zero is x . Zoomed version of the ﬁrst 10 iterations and ﬁrst 1ms appears in the bottom row. This ﬁgure demonstrates the convergence rate of PGD with projections onto the sparse set and sparse tree set compared to IPGD with p that projects onto a certain number of levels of the tree and IPGD with changing p that projects onto an increasing number of levels as the iterations proceed. Note that while PGD with a projection onto a tree structure converges faster than IPGD as a function of the number of iterations (left ﬁgure), it converges slower than IPGD if we take into account the actual run time of each iteration, as shown in the right ﬁgure, due to the higher complexity of the PGD projections. only onto the ﬁrst layer, then the algorithm does not converge last line in Fig. 3 demonstrates this strategy, where at the ﬁrst as the resulting approximation error (cid:15) is too large. However, iteration p is set to be a projection onto the ﬁrst two levels, starting from the second layer, we get a faster convergence at and then every four iterations another tree level is added to the ﬁrst iterations with p that projects onto a smaller set, which the projection until it becomes a projection onto all the tree yields a smaller ρ. As the number of iterations increases, the levels (in this case IPGD coincides with PGD). Note that more accurate projections achieve a lower reconstruction error, IPGD converges faster than PGD also when the projection in where the plateau attained is proportional to the approximation it becomes onto all the tree levels. This can be explained by error of p as predicted by our theory. the fact that typically convergence of non-linear optimization techniques depends on the initialization point [8]. This tradeoff can be used to further accelerate the conver- While here we arbitrarily chose to add another level every gence by changing the projection in IPGD over the iterations. ﬁxed number of iterations, in general, a control set can be used Thus, in the ﬁrst iterations we enjoy the fast convergence of for setting the number of iterations to be performed in each the coarser projections and in the later ones we use more training level. We demonstrate this strategy in Section . K accurate projections that allow achieving a lower plateau. 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text: 3 100 400 100 10−1 (cid:15)2/3k−5/3 (a) |ρ| (d) |ρ| 10−1 10−2 10−3 10−2 10−4 ∼ k−5/3 10−3 10−5 10−6 (a) 00 k 160 0 k 160 10−4 101 240 100 100 (b) Ei (e) Ei k k 10−1 η (cid:15)−1/3k−5/3 t = 0.3 10−2 t = 2 10−1 10−3 t = 5 t = 8 10−4 (b) t = 10 10−1500 101 102 103 2400 10−2 k (c) Ec (f) Ec k k FIG. 2. (Color online) Spectrum of the (a) incompressible 10−3 kinetic energy, and (b) helicity in the 20483 GPE run. At large scales both follow a scaling compatible with a classical dual cascade (thick dashed lines). At scales smaller than the ≈ 0 10−4 intervortex scale (k(cid:96) 80) a second range compatible with 0 k 90 0 k 90 a Kelvin wave cascade is observed in Ei (thick dash-dotted k line). The helicity spectrum broadens in time indicating a FIG. 4. (Color online) Spatio-temporal power spectra for the direct transfer. 5123 GPE run between t = 0 and 2 for (a) the mass density ρ, and zooms between k = 0 and 100 for (b) the incompressible ∈ and (c) compressible velocity. Same for late times (t [8, 10]) 101 t = 6 are shown in (d), (e), and (f). The solid (green) curve is the t = 7 dispersion relation of Kelvin waves, while the dotted (white) 100 t = 8 curve corresponds to sound waves. t = 9 10−1 Average 100 101 102 Ψx,y,z = exp{i[A sin(kx) m/(cid:126)]y + i[A cos(kx) m/(cid:126)]z}, A,k k where [a] stands for the nearest integer to a to enforce periodicity. The wavefunction of the quantum ABC ﬂow FIG. 3. (Color online) Compensated helicity spectra in is then obtained as Ψ(k) = Ψx,y,z × Ψy,z,x × Ψz,x,y. the 20483 GPE run. The spectrum is compatible with ABC A,k B,k C,k η(cid:15)−1/3k−5/3 scaling. The time-averaged spectrum is also Finally, ΨABC = Ψ(A1B)C × Ψ(A2B)C corresponds to the ini- shown. tial ﬂow v . In practice, to correctly set the initial ABC density with defects along the vortex lines and to cor- rect frustration errors arising from periodicity, following [24, 25] we ﬁrst evolve Ψ using the advected real ABC with ∇− · u = 0, using 5123 points and viscosity ν = Guinzburg-Landau equation 1, whose stationary solu- 6.5 × 10 4. All equations were integrated using GHOST tions are solutions of the GPE with minimal emission of [30], a pseudospectral code with periodic boundary condi- acoustic energy. tions, fourth order Runge-Kutta to compute time deriva- III. RESULTS tives, and the 2/3 rule for dealiasing. As initial condition we use a superposition of k = 1 In Fig. 1 we show the evolution of the incompressible and k = 2 basic ABC ﬂows: vABC = vA(1B)C + vA(2B)C, with kinetic energy Eki and of the regularized helicity Hr for the 10243 and 20483 GPE runs, and for the NS equa- tion (with H ≡ H). In all cases, Ei and H remain v(k) = [B cos(ky) + C sin(kz)] xˆ + [C cos(kz)+ approximatelyr constant for the ﬁrst turknover timres while ABC A sin(kx)] yˆ + [A cos(kx) + B sin(ky)] zˆ, (9) turbulence develops (the so-called “inviscid” phase in the decay of classical ﬂows). The total vortex length (cid:96) peaks v at the end of this phase, which ends slightly earlier for √ and (A, B, C) = (0.9, 1, 1.1)/ 3. The basic ABC Hr than for Eki . Afterwards, Hr and Eki decrease in what ﬂow is a 2π-periodic stationary solution of the Euler equation with maximal helicity. To build its quantum version we ﬁrst take the ﬂow with B = C = 0 and use Madelung transformation to obtain a wavefunction 1 ∂tΨ = 2(cid:126)m∇2Ψ + (gmρ0 − g|Ψ|2 − mv2A2(cid:126)BC )Ψ − ivABC · ∇Ψ )k(kiE )k(H 3/5k3/1(cid:15)1−η)k(H ω ω ω ##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## 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text: (a)

text: |ρ|

text: (d)

text: |ρ|

text: (a)

text: k

text: k

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text: Ei

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text: Ei

text: k

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text: η

text: t

text: t

text: t

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text: Ec

text: (f)

text: Ec

text: k

text: k

text: FIG.

text: (Color

text: online)

text: Spectrum

text: of

text: the

text: (a)

text: incompressible

text: kinetic

text: energy,

text: and

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text: helicity

text: in

text: the

text: GPE

text: run.

text: At

text: large

text: scales

text: both

text: follow

text: a

text: scaling

text: compatible

text: with

text: a

text: classical

text: dual

text: cascade

text: (thick

text: dashed

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text: At

text: scales

text: smaller

text: than

text: the

text: intervortex

text: scale

text: a

text: second

text: range

text: compatible

text: with

text: k

text: k

text: a

text: Kelvin

text: wave

text: cascade

text: is

text: observed

text: in

text: Ei

text: (thick

text: dash-dotted

text: k

text: line).

text: The

text: helicity

text: spectrum

text: broadens

text: in

text: time

text: indicating

text: a

text: FIG.

text: (Color

text: online)

text: Spatio-temporal

text: power

text: spectra

text: for

text: the

text: direct

text: transfer.

text: GPE

text: run

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text: t

text: and

text: for

text: (a)

text: the

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text: the

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text: velocity.

text: Same

text: for

text: late

text: times

text: (t

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text: shown

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text: and

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text: The

text: solid

text: (green)

text: curve

text: is

text: the

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text: dispersion

text: relation

text: of

text: Kelvin

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text: while

text: the

text: dotted

text: (white)

text: t

text: curve

text: corresponds

text: to

text: sound

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text: t

text: Average

text: Ψx,y,z

text: exp{i[A

text: sin(kx)

text: i[A

text: cos(kx)

text: A,k

text: k

text: where

text: [a]

text: stands

text: for

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text: nearest

text: integer

text: to

text: a

text: to

text: enforce

text: periodicity.

text: The

text: wavefunction

text: of

text: the

text: quantum

text: ABC

text: ﬂow

text: FIG.

text: (Color

text: online)

text: Compensated

text: helicity

text: spectra

text: in

text: is

text: then

text: obtained

text: as

text: Ψ(k)

text: Ψx,y,z

text: Ψy,z,x

text: Ψz,x,y.

text: the

text: GPE

text: run.

text: The

text: spectrum

text: is

text: compatible

text: with

text: ABC

text: A,k

text: B,k

text: C,k

text: scaling.

text: The

text: time-averaged

text: spectrum

text: is

text: also

text: Finally,

text: ΨABC

text: corresponds

text: to

text: the

text: ini-

text: shown.

text: tial

text: ﬂow

text: v

text: In

text: practice,

text: to

text: correctly

text: set

text: the

text: initial

text: ABC

text: density

text: with

text: defects

text: along

text: the

text: vortex

text: lines

text: and

text: to

text: cor-

text: rect

text: frustration

text: errors

text: arising

text: from

text: periodicity,

text: following

text: we

text: ﬁrst

text: evolve

text: Ψ

text: using

text: the

text: advected

text: real

text: ABC

text: with

text: u

text: using

text: points

text: and

text: viscosity

text: ν

text: Guinzburg-Landau

text: equation

text: whose

text: stationary

text: solu-

text: All

text: equations

text: were

text: integrated

text: using

text: GHOST

text: tions

text: are

text: solutions

text: of

text: the

text: GPE

text: with

text: minimal

text: emission

text: of

text: a

text: pseudospectral

text: code

text: with

text: periodic

text: boundary

text: condi-

text: acoustic

text: energy.

text: tions,

text: fourth

text: order

text: Runge-Kutta

text: to

text: compute

text: time

text: deriva-

text: III.

text: RESULTS

text: tives,

text: and

text: the

text: rule

text: for

text: dealiasing.

text: As

text: initial

text: condition

text: we

text: use

text: a

text: superposition

text: of

text: k

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text: Fig.

text: we

text: show

text: the

text: evolution

text: of

text: the

text: incompressible

text: and

text: k

text: basic

text: ABC

text: ﬂows:

text: vABC

text: with

text: kinetic

text: energy

text: Eki

text: and

text: of

text: the

text: regularized

text: helicity

text: Hr

text: for

text: the

text: and

text: GPE

text: runs,

text: and

text: for

text: the

text: NS

text: equa-

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text: H

text: H).

text: In

text: all

text: cases,

text: Ei

text: and

text: H

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text: v(k)

text: [B

text: cos(ky)

text: C

text: sin(kz)]

text: xˆ

text: [C

text: cos(kz)+

text: approximatelyr

text: constant

text: for

text: the

text: ﬁrst

text: turknover

text: timres

text: while

text: ABC

text: A

text: sin(kx)]

text: yˆ

text: [A

text: cos(kx)

text: B

text: sin(ky)]

text: zˆ,

text: turbulence

text: develops

text: (the

text: so-called

text: “inviscid”

text: phase

text: in

text: the

text: decay

text: of

text: classical

text: ﬂows).

text: The

text: total

text: vortex

text: length

text: peaks

text: v

text: at

text: the

text: end

text: of

text: this

text: phase,

text: which

text: ends

text: slightly

text: earlier

text: for

text: and

text: (A,

text: B,

text: C)

text: The

text: basic

text: ABC

text: Hr

text: than

text: for

text: Eki

text: Afterwards,

text: Hr

text: and

text: Eki

text: decrease

text: in

text: what

text: ﬂow

text: is

text: a

text: stationary

text: solution

text: of

text: the

text: Euler

text: equation

text: with

text: maximal

text: helicity.

text: To

text: build

text: its

text: quantum

text: version

text: we

text: ﬁrst

text: take

text: the

text: ﬂow

text: with

text: B

text: C

text: and

text: use

text: Madelung

text: transformation

text: to

text: obtain

text: a

text: wavefunction

text: ∂tΨ

text: )Ψ

text: ivABC

text: ∇Ψ

text: )k(kiE

text: )k(H

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