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Recursive Compressed Sensing

Pantelis Sopasakis∗

Presentation at ICTEAM – UC Louvain, Belgiumjoint work with N. Freris† and P. Patrinos‡

∗ IMT Institute for advanced studies Lucca, Italy† NYU, Abu Dhabi, United Arab Emirates

‡ ESAT, KU Leuven, Belgium

April 7, 2016

Motivation

MRI

Radio-astronomy

Holography

Seismology

Photography

RadarsFa

cial

rec

ogni

tion

Speech recognition

Fault detection

Medicalimaging

Faci

al r

ecog

nitio

n

Part

icle

phy

sics

Video processing

ECG

Encr

yptio

n

Communication networks

System identification

Compressed Sensing

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Spoiler alert!

The proposed method is an order of magnitude faster compared toother reported methods for recursive compressed sensing.

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Outline

1. Forward-Backward Splitting

2. The Forward-Backward envelope function

3. The Forward-Backward Newton method

4. Recursive compressed sensing

5. Simulations

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I. Forward-Backward Splitting

Forward-Backward Splitting

Problem structure

minimize ϕ(x) = f(x) + g(x)

where

1. f, g : Rn → R are proper, closed, convex

2. f has L-Lipschitz gradient

3. g is prox-friendly, i.e., its proximal operator

proxγg(v) := arg minz

{g(z) + 1

2‖v − z‖2}

is easily computable[1].

[1]Parikh & Boyd 2014; Combettes & Pesquette, 2010.

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Example #1

Constrained QPs

minimize 12x>Qx+ q>x︸︷︷︸

f

+ δ(x | B)︸︷︷︸g

where B is a set on which projections are easy to compute and

δ(x | B) =

{0, if x ∈ B,+∞, otherwise

Then proxγg(x) = proj(x | B).

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Example #2

LASSO problems

minimize 12‖Ax− b‖

2︸︷︷︸f

+λ‖x‖1︸︷︷︸g

Indeed,

1. f is cont. diff/ble with ∇f(x) = A>(Ax− b)2. g is prox-friendly

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Other examples

X Constrained optimal control

X Elastic net

X Sparse log-logistic regression

X Matrix completion

X Subspace identification

X Support vector machines

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FBS offers a generic framework for solving such problems using theiteration

xk+1 = proxγg(xk − γ∇f(xk)) =: Tγ(xk),

for γ < 2/L.

Features:

1. ϕ(xk)− ϕ? ∈ O(1/k)

2. with Nesterov’s extrapolation ϕ(xk)− ϕ? ∈ O(1/k2)

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The iteration

xk+1 = proxγg(xk − γ∇f(xk)),

can be written as[2]

xk+1 = arg minz

{f(xk) + 〈∇f(xk), z − xk〉+ 1

2γ ‖z − xk‖2︸︷︷︸

Qfγ(z,xk)

+g(z)},

where Qfγ(z, xk) serves as a quadratic model for f [3].

[2]Beck and Teboulle, 2010.[3]Qfγ(·, x

k) is the linearization of f at xk plus a quadratic term; moreover, Qfγ(z, xk) ≥ f(x) and Qfγ(z, z) = f(z).

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x0

ϕ(x0)

ϕ = f + g

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x0

ϕ(x0)

ϕ = f + g

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x0

ϕ(x0)

ϕ = f + g

Qfγ(z;x

0) + g(z)

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x0 x1

ϕ(x0)

ϕ(x1)

ϕ = f + g

Qfγ(z;x

0) + g(z)

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x0 x1 x2

ϕ(x0)

ϕ(x1)

ϕ(x2)

ϕ = f + g

Qfγ(z;x

1) + g(z)

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x0 x1 x2 x3

ϕ(x0)

ϕ(x1)

ϕ(x2)ϕ(x3)

ϕ = f + g

Qfγ(z;x

2) + g(z)

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Overview

Generic convex optimization problem

minimize f(x) + g(x).

The generic iteration

xk+1 = proxγg(xk − γ∇f(xk))

is a fixed-point iteration for the optimality condition

x? = proxγg(x? − γ∇f(x?))

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Overview

It generalizes several other methods

xk+1 =

xk − γ∇f(xk) gradient method, g = 0

ΠC(xk − γ∇f(xk)) gradient projection, g = δ(· | C)

proxγg(xk) proximal point algorithm, f = 0

There are several flavors of proximal gradient algorithms[4].

[4]Nesterov’s accelerated method, FISTA (Beck & Teboulle), etc.

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Shortcomings

FBS are first-order methods, therefore, they can be slow!

Overhaul. Use a better quadratic model for f [5]:

Qfγ,B(z, xk) = f(xk) + 〈∇f(xk), z − xk〉+ 12γ ‖z − x

k‖2Bk ,

where Bk is (an approximation of) ∇2f(x).

Drawback. No closed form solution of the inner problem.

[5]As in Becker & Fadili 2012; Lee et al. 2012; Tran-Dinh et al. 2013.

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II. Forward-Backward Envelope

Forward-Backward Envelope

The Forward-Backward envelope of ϕ is defined as

ϕγ(x) = minz

{f(x) + 〈∇f(x), z − x〉+ g(z) + 1

2γ ‖z − x‖2},

with γ ≤ 1/L. Let’s see how it looks...

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x

ϕ(x)

ϕγ(x)

ϕ

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x

ϕ(x)

ϕγ(x)

ϕ

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x

ϕ(x)

ϕγ(x)

ϕϕγ

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Properties of FBE

Define

Tγ(x) = proxγg(x− γ∇f(x))

Rγ(x) = γ−1(x− Tγ(x))

FBE upper bound

ϕγ(x) ≤ ϕ(x)− 12γ ‖Rγ(x)‖2

FBE lower bound

ϕγ(x) ≥ ϕ(Tγ(x)) +1−γLf

2γ ‖Rγ(x)‖2

x Tγ(x)

ϕ(x)

ϕ(Tγ(x))ϕγ(x)

ϕϕγ

x? = Tγ(x?)

ϕ(x?)

ϕϕγ

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Properties of FBE

Ergo: Minimizing ϕ is equivalent to minimizing its FBE ϕγ .

inf ϕ = inf ϕγ

arg minϕ = arg minϕγ

However, ϕγ is continuously diff/able[6] whenever f ∈ C2.

[6]More about the FBE: P. Patrinos, L. Stella and A. Bemporad, 2014.

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FBE is C2

FBE can be written as

ϕγ(x) = f(x)− γ2‖∇f(x)‖2 + gγ(x−∇f(x)),

where gγ is the Moreau envelope of g,

gγ(v) = minz{g(z) + 1

2γ ‖z − v‖2}

gγ is a smooth approximation of g with ∇gγ(x) = γ−1(x− proxγg(x)). Iff ∈ C2, then

∇ϕγ(x) = (I − γ∇2f(x))Rγ(x).

Therefore,arg minϕ = arg minϕγ = zer∇ϕγ .

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The Moreau envelope

g(x) = |x|g0.1

g10

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Forward-Backward Newton

X Since ϕγ is C1 but not C2, we may not apply a Newton method.

X The FB Newton method is a semi-smooth method for minimizing ϕγusing a notion of generalized differentiability.

X The FBN iterations are

xk+1 = xk + τkdk,

where dk is a Newton direction given by

Hkdk = −∇ϕγ(xk),

Hk ∈ ∂2Bϕγ(xk),

X ∂B is the so-called B-subdifferential (we’ll define it later)

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III. Forward-Backward Newton

Optimality conditions

LASSO problem

minimize 12‖Ax− b‖

2︸︷︷︸f

+λ‖x‖1︸︷︷︸g

.


−∇f(x?) ∈ ∂g(x?).

where ∇f(x) = A>(Ax− b) and ∂g(x)i = λ sign(xi) for xi 6= 0 and∂g(x)i = [−λ, λ] otherwise, so

−∇if(x?) = λ sign(x?i ), if xi 6= 0,

|∇if(x?)| ≤ λ, otherwise

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If we knew the set

α = {i : x?i = 0},β = {j : x?j 6= 0},

we would be able to write down the optimality conditions as

A>αAαx?α = A>α b+ λ sign(x?α)

Goal. Devise a method to determine α efficiently.

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We may write the optimality conditions as follows

x? = proxγg(x? − γ∇f(x?)),

where

proxγg(z)i = sign(zi)(|zi| − γλ)+.

ISTA and FISTA are method for the iterative solution of theseconditions. Instead, we are looking for a zero of the fixed-point residualoperator

Rγ(x) = x− proxγg(x− γ∇f(x)).

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B-subdifferential

For a function F : Rn → Rn which is almost everywhere differentiable, wedefine its B-subdifferential to be[7]

∂BF (x) :=

{B ∈ Rn×n

∣∣∣∣ ∃{xn}n : xn → x,R′γ(xn) exists and R′γ(xn)→ B

}.

[7]See Facchinei & Pang, 2004

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Rγ(x) is nonexpansive ⇒ Lipschitz ⇒ Differentiable a.e. ⇒ B-sub-differentiable (∂BRγ(x)). The proposed algorithm takes the form

xk+1 = xk − τkH−1k Rγ(xk), with Hk ∈ ∂BRγ(xk).

When close to the solution, all Hk are nonsingular. Take

Hk = I − Pk(I − γA>A),

where Pk is diagonal with (Pk)ii = 1 iff i ∈ αk, where

αk = {i : |xki − γ∇if(xki )| > γλ}

The scalar τk is computed by a simple line search method to ensureglobal convergence of the algorithm.

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The Forward-Backward Newton method can be concisely written as

xk+1 = xk + τkdk.

The Newton direction dk is determined as follows without the need toformulate Hk

dkβk = −(Rγ(xk))βk ,

γA>αkAαkdkαk

= −(Rγ(xk))αk − γA>αkAβkd

kβk.

For the method to converge globally, we compute τk so that the Armijocondition is satisfied for ϕγ

ϕγ(xk + τkdk) ≤ ϕγ(xk) + ζτk∇ϕγ(xk)>dk.

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Require: A, y, λ, x0, εγ ← 0.95/‖A‖2x← x0

while ‖Rγ(x)‖ > ε doα← {i : |xi − γ∇if(x)| > γλ}β ← {i : |xi − γ∇if(x)| ≤ γλ}dβ ← −xβsα ← sign(xα − γ∇αf(x))Solve A>αAα(xα + dα) = A>α y − λsατ ← 1while ϕγ(x+ τd) ≤ ϕγ(x) + ζτ∇ϕγ(x)>d doτ ← 1

2τend whilex← x+ τd

end while

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Speeding up FBN by Continuation

1. In applications of LASSO we have ‖x?‖0 ≤ m� n[8]

2. If λ ≥ λ0 := ‖∇f(x0)‖∞, then supp(x) = ∅3. We relax the optimization problem solving

P(λ) : minimize 12‖Ax− y‖

2 + λ‖x‖1

4. Once we have approximately solved P(λ) we update λ as

λ← max{ηλ, λ},

until eventually λ = λ.

5. This way we enforce that (i) |αk| increases smoothly, (ii) |αk| < m,(iii) A>αkAαk remains always positive definite.

[8]The zero-norm of x, ‖x‖0, is the number of its nonzeroes.

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2. If λ ≥ λ0 := ‖∇f(x0)‖∞, then supp(x) = ∅

3. We relax the optimization problem solving


2 + λ‖x‖1






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2 + λ‖x‖1






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2 + λ‖x‖1






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2 + λ‖x‖1






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Require: A, y, λ, x0, η ∈ (0, 1), ελ← max{λ, ‖∇f(x0)‖∞}, ε← ε

while λ > λ or ‖Rγ(xk; λ)‖ > ε doxk+1 ← xk + τkd

k (dk: Newton direction, τk line search)if ‖Rγ(xk; λ)‖ ≤ λε thenλ← max{λ, ηλ}ε← ηε

end ifend while

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Further speed up

When A>α is positive definite[9], we may compute a Cholesky factorizationof A>α0

Aα0 and then update the Cholesky factorization of A>αk+1Aαk+1

using the factorization of A>αkAαk .

[9]In practice, always (when the continuation heuristic is used). Furthermore, α0 = ∅.

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Further speed up

Cheap

Expensive

Expensive

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Further speed up

Cheap

Expensive

Expensive

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Overview

Why FBN?

X Fast convergence

X Very fast convergence when close to the solution

X Few, inexpensive iterations

X The FBE serves as a merit function ensuring global convergence

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IV. Recursive Compressed Sensing

Introduction

We say that a vector x ∈ Rn is s-sparse if it has at most s nonzeroes.

Assume that a sparsely-sampled signal y ∈ Rm (m�n) is produced by

y = Ax,

by an s-sparse vector x and a sampling matrix A. In reality, however,measurements will be noisy

y = Ax+ w.

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Sparse Sampling

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Sparse Sampling

We require that A satisfies the restricted isometry property[10], that is

(1− δs)‖x‖2 ≤ ‖Ax‖2 ≤ (1 + δs)‖x‖2

A typical choice is a random matrix A with entries drawn from N (0, 1m)

with m = 4s.

[10]This can be established using the Johnson-Lindenstrauss lemma.

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Decompression

Assuming that

I w ∼ N (0, σ2I),

I the smallest element of |x| is not too small (> 8σ√

2 lnn),

I λ = 4σ√

2 lnn,

the LASSO recovers the support of x[11], that is

x? = arg min 12‖Ax− y‖

2 + λ‖x‖1,

has the same support as the actual x.

[11]Candes & Plan, 2009.

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Decompression

LASSO

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Define

x(i) :=[xi xi+1 · · · xi+n−1

]>Then x(i) produces the measured signal

y(i) = A(i)x(i) + w(i).

Sampling is performed with a constant matrix A[12] and

A(0) = A,

A(i+1) = A(i)P,

where P is a permutation matrix which shifts the columns of A leftwards.

[12]For details see: N. Freris, O. Ocal and M. Vetterli, 2014.

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Require: Stream of observations, Window size n, Sparsity sλ← 4σ

√2 lnn and m← 4s

Construct A ∈ Rm×n with entries from N (0, 1m)

A(0) ← A, x(0)◦ ← 0

for i = 0, 1, . . . do1. Sample y(i) ∈ Rm

2. Support estimation (using the initial guess x(i)◦ )

x(i)? = arg min 1

2‖A(i)x(i) − y(i)‖2 + λ‖x(i)‖1

3. Perform debiasing

4. x(i+1)◦ ← P>x

(i)?

5. A(i+1) ← A(i)Pend for

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V. Simulations

Simulations

We compared the proposed methodology with

X ISTA (or proximal gradient method)

X FISTA (or accelerated ISTA)

X ADMM

X L1LS (interior point method)

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Simulations

For a 10%-sparse stream

Window size ×10 4

0.5 1 1.5 2

Avera

ge r

untim

e [s]

10 -1

10 0

10 1

FBN

FISTA

ADMM

L1LS

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Simulations

For n = 5000 varying the stream sparsity

Sparsity [%]

0 5 10 15

Avera

ge r

untim

e [s]

10 -1

10 0

FBN

FISTA

ADMM

L1LS

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References

1. S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior- pointmethod for large-scale 1 -regularized least squares,” IEEE J Select Top Sign Proc,1(4), pp. 606–617, 2007.

2. A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm forlinear inverse problems,” SIAM J Imag Sci, 2(1), pp. 183–202, 2009.

3. S. Becker and M. J. Fadili, “A quasi-Newton proximal splitting method,” inAdvances in Neural Information Processing Systems, vol. 1, pp. 2618–2626, 2012.

4. A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm forlinear inverse problems,” SIAM J Imag Sci, 2(1), pp. 183–202, 2009.

5. P. Patrinos, L. Stella and A. Bemporad, “Forward-backward truncated Newtonmethods for convex composite optimization,” arXiv:1402.6655, 2014.

6. P. Sopasakis, N. Freris and P. Patrinos, “Accelerated reconstruction of acompressively sampled data stream,” 24th European Signal Processingconference, submitted, 2016.

7. N. Freris, O. Ocal and M. Vetterli, “Recursive Compressed Sensing,”arXiv:1312.4895, 2013.

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Thank you for your attention.

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