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Sparse Recovery Using Sparse (Random) Matrices

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Sparse Recovery Using Sparse (Random) Matrices

Piotr Indyk

MIT

Joint work with: Radu Berinde, Anna Gilbert, Howard Karloff, Martin Strauss and Milan Ruzic

“Stay awake until the end”

Ax

A

=

x

x

x*

k=2

- Setup:
- Data/signal in n-dimensional space : x
E.g., x is an 1000x1000 image n=1000,000

- Goal: compress x into a “sketch” Ax ,
where A is a m x n “sketch matrix”, m << n

- Data/signal in n-dimensional space : x
- Requirements:
- Plan A: want to recover x from Ax
- Impossible: undetermined system of equations

- Plan B: want to recover an “approximation” x* of x
- Sparsity parameter k
- Want x* such that ||x*-x||p C(k) ||x’-x||q ( lp/lq guarantee )
over all x’ that are k-sparse (at most k non-zero entries)

- The best x* contains k coordinates of x with the largest abs value

- Plan A: want to recover x from Ax
- Want:
- Good compression (small m)
- Efficient algorithms for encoding and recovery

- Why linear compression ?

destination

source

- Data stream algorithms
(e.g. for network monitoring)

- Efficient increments:
A(x+) = Ax + A

- Efficient increments:
- Single pixel camera
[Wakin, Laska, Duarte, Baron, Sarvotham, Takhar, Kelly, Baraniuk’06]

- Pooling, Microarray Experiments [Kainkaryam, Woolf], [Hassibi et al], [Dai-Sheikh, Milenkovic, Baraniuk]

- Choose encoding matrix Aat random
(the algorithms for recovering x* are more complex)

- Sparse matrices:
- Data stream algorithms
- Coding theory (LDPCs)

- Dense matrices:
- Compressed sensing
- Complexity/learning theory
(Fourier matrices)

- Sparse matrices:
- “Traditional” tradeoffs:
- Sparse: computationally more efficient, explicit
- Dense: shorter sketches

- Goal: the “best of both worlds”

“state

of art”

Scale:

Excellent

Very Good

Good

Fair

- Restricted Isometry Property (RIP) *- sufficient property of a dense matrix A:
is k-sparse ||||2 ||A||2 C ||||2

- Holds w.h.p. for:
- Random Gaussian/Bernoulli: m=O(k log (n/k))
- Random Fourier: m=O(k logO(1) n)

- Consider random m x n 0-1 matrices with d ones per column
- Do they satisfy RIP ?
- No, unless m=(k2)[Chandar’07]

- However, they can satisfy the following RIP-1 property [Berinde-Gilbert-Indyk-Karloff-Strauss’08]:
is k-sparse d (1-) ||||1 ||A||1 d||||1

- Sufficient (and necessary) condition: the underlying graph is a
( k, d(1-/2) )-expander

* Other (weaker) conditions exist, e.g., “kernel of A is a near-Euclidean subspace of l1”. More later.

- A bipartite graph is a (k,d(1-))-expander if for any left set S, |S|≤k, we have |N(S)|≥(1-)d |S|
- Constructions:
- Randomized: m=O(k log (n/k))
- Explicit: m=k quasipolylog n

- Plenty of applications in computer science, coding theory etc.
- In particular, LDPC-like techniques yield good algorithms for exactly k-sparse vectors x
[Xu-Hassibi’07, Indyk’08, Jafarpour-Xu-Hassibi-Calderbank’08]

N(S)

d

S

m

n

- Want to show that for any k-sparse we have
d (1-) ||||1 ||A ||1 d||||1

- RHS inequality holds for any
- LHS inequality:
- W.l.o.g. assume
|1|≥… ≥|k| ≥ |k+1|=…= |n|=0

- Consider the edges e=(i,j) in a lexicographic order
- For each edge e=(i,j) define r(e) s.t.
- r(e)=-1 if there exists an edge (i’,j)<(i,j)
- r(e)=1 if there is no such edge

- W.l.o.g. assume
- Claim 1: ||A||1 ≥∑e=(i,j) |i|re
- Claim 2: ∑e=(i,j) |i|re ≥ (1-) d||||1

d

m

n

A

x*-x

Ax-Ax*

- Iterative algorithm: given current approximation x* :
- Find (possibly several) i s. t. Ai “correlates” with Ax-Ax* . This yields i and z s. t.
||x*+zei-x||p << ||x* - x||p

- Update x*
- Sparsify x* (keep only k largest entries)
- Repeat

- Find (possibly several) i s. t. Ai “correlates” with Ax-Ax* . This yields i and z s. t.
- Norms:
- p=2 : CoSaMP, SP, IHTetc (dense matrices)
- p=1 : SMP, this paper (sparse matrices)
- p=0 : LDPC bit flipping (sparse matrices)

=

i

i

- Algorithm:
- x*=0
- Repeat T times
- Repeat S=O(k) times
- Find i and zthat minimize* ||A(x*+zei)-b||1
- x* = x*+zei

- Sparsify x*
(set all but k largest entries of x* to 0)

- Repeat S=O(k) times

- Similar to SMP, but updates done sequentially

A

i

N(i)

Ax-Ax*

x-x*

* Set z=median[ (Ax*-Ax)N(i) ]. Instead, one could first optimize (gradient) i and then z [ Fuchs’09]

- Algorithm:
- x*=0
- Repeat T times
- For each i=1…n compute* zi that achieves
Di=minz ||A(x*+zei)-b||1

and store Di in a heap

- Repeat S=O(k) times
- Pick i,z that yield the best gain
- Update x* = x*+zei
- Recompute and store Di’ for all i’ such that
N(i) and N(i’) intersect

- Sparsify x*
(set all but k largest entries of x* to 0)

- For each i=1…n compute* zi that achieves

- Running time:
T [ n(d+log n) + k nd/m*d (d+log n)]

= T [ n(d+log n) + nd (d+log n)] = T [ nd (d+log n)]

A

i

Ax-Ax*

x-x*

x

a2

a1

x

a1

a2

Supports of a1anda2have small overlap (typically)

- Want to find k-sparse x* that minimizes ||x-x*||1
- By RIP1, this is approximately the same as minimizing ||Ax-Ax*||1
- Need to show we can do it greedily

256x256

SSMP is ran with S=10000,T=20. SMP is ran for 100 iterations. Matrix sparsity is d=8.

- Even better algorithms for sparse approximation (using sparse matrices)
- State of the art: can do 2 out of 3:
- Near-linear encoding/decoding
- O(k log (n/k)) measurements
- Approximation guarantee with respect to L2/L1 norm

- Questions:
- 3 out of 3 ?
- “Kernel of A is a near-Euclidean subspace of l1” [Kashin et al]
- Constructions of sparse A exist [Guruswami-Lee-Razborov, Guruswami-Lee-Wigderson]
- Challenge: match the optimal measurement bound

- Explicit constructions

- 3 out of 3 ?

Thanks!

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