# Sparse Recovery (Using Sparse Matrices)

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Sparse Recovery (Using Sparse Matrices)
Piotr Indyk MIT

HHpφ (x) = { i: |xi| ≥ φ ||x||p }
Heavy Hitters Also called frequent elements and elephants Define HHpφ (x) = { i: |xi| ≥ φ ||x||p } Lp Heavy Hitter Problem: Parameters: φ and φ’ (often φ’ = φ-ε) Goal: return a set S of coordinates s.t. S contains HHpφ (x) S is included in HHpφ’ (x) Lp Point Query Problem: Parameter:  Goal: at the end of the stream, given i, report x*i=xi   ||x||p Can solve L2 point query problem, with parameter  and failure probability P by storing O(1/2 log(1/P)) numbers sketches [A-Gibbons-M-S’99], [Gilbert-Kotidis-Muthukrishnan-Strauss’01]

||x*-x||p C(k) minx’ ||x’-x||q
Sparse Recovery (approximation theory, learning Fourier coeffs, linear sketching, finite rate of innovation, compressed sensing...) Setup: Data/signal in n-dimensional space : x E.g., x is an 256x256 image  n=65536 Goal: compress x into a “sketch” Ax , where A is a m x n “sketch matrix”, m << n Requirements: Plan A: want to recover x from Ax Impossible: underdetermined system of equations Plan B: want to recover an “approximation” x* of x Sparsity parameter k Informally: want to recover largest k coordinates of x Formally: want x* such that ||x*-x||p C(k) minx’ ||x’-x||q over all x’ that are k-sparse (at most k non-zero entries) Want: Good compression (small m=m(k,n)) Efficient algorithms for encoding and recovery Why linear compression ? = A x Ax

Application I: Monitoring Network Traffic Data Streams
Router routs packets Where do they come from ? Where do they go to ? Ideally, would like to maintain a traffic matrix x[.,.] Easy to update: given a (src,dst) packet, increment xsrc,dst Requires way too much space! (232 x 232 entries) Need to compress x, increment easily Using linear compression we can: Maintain sketch Ax under increments to x, since A(x+) = Ax + A Recover x* from Ax destination source x

Applications, ctd. Single pixel camera
[Wakin, Laska, Duarte, Baron, Sarvotham, Takhar, Kelly, Baraniuk’06] Pooling Experiments [Kainkaryam, Woolf’08], [Hassibi et al’07], [Dai-Sheikh, Milenkovic, Baraniuk], [Shental-Amir-Zuk’09],[Erlich-Shental-Amir-Zuk’09]

Sparse matrices: Data stream algorithms Coding theory (LDPCs) Dense matrices: Compressed sensing Complexity/learning theory (Fourier matrices) “Traditional” tradeoffs: Sparse: computationally more efficient, explicit Dense: shorter sketches Recent results: the “best of both worlds”

Results Scale: Excellent Very Good Good Fair Paper R/D Sketch length
Encode time Column sparsity Recovery time Approx [CCF’02], [CM’06] R k log n n log n log n l2 / l2 k logc n n logc n logc n [CM’04] l1 / l1 [CRT’04] [RV’05] D k log(n/k) nk log(n/k) nc l2 / l1 [GSTV’06] [GSTV’07] k2 logc n [BGIKS’08] n log(n/k) log(n/k) [GLR’08] k lognlogloglogn kn1-a n1-a [NV’07], [DM’08], [NT’08], [BD’08], [GK’09], … nk log(n/k) * log n log n * log emphasize arbitrary signals [IR’08], [BIR’08],[BI’09] D k log(n/k) n log(n/k) log(n/k) n log(n/k)* log l1 / l1 [GLSP’09] R k log(n/k) n logc n logc n k logc n l2 / l1 Caveats: (1) most “dominated” results not shown (2) only results for general vectors x are displayed (3) sometimes the matrix type matters (Fourier, etc)

||x-x*||1≤ C Err1 , where Err1=mink-sparse x’ ||x-x’||1
Part I Paper R/D Sketch length Encode time Column sparsity Recovery time Approx [CM’04] R k log n n log n log n l1 / l1 Theorem: There is a distribution over mxn matrices A, m=O(k logn), such that for any x, given Ax, we can recover x* such that ||x-x*||1≤ C Err1 , where Err1=mink-sparse x’ ||x-x’||1 with probability 1-1/n. The recovery algorithm runs in O(n log n) time. This talk: Assume x≥0 – this simplifies the algorithm and analysis; see the original paper for the general case Prove the following l∞/l1 guarantee: ||x-x*||∞≤ C Err1 /k This is actually stronger than the l1/l1 guarantee (cf. [CM’06], see also the Appendix) Note: [CM’04] originally proved a weaker statement where ||x-x*||∞≤ C||x||1 /k. The stronger guarantee follows from the analysis of [CCF’02] (cf. [GGIKMS’02]) who applied it to Err2

First attempt Matrix view: Hashing view: Estimator: xi*=Zh(i)
A 0-1 wxn matrix A, with one 1 per column The i-th column has 1 at position h(i), where h(i) be chosen uniformly at random from {1…w} Hashing view: Z=Ax h hashes coordinates into “buckets” Z1…Zw Estimator: xi*=Zh(i) xi xi* Z1 ………..Z(4/)k Closely related: [Estan-Varghese’03], “counting” Bloom filters

∑t not in S u {i}, h(i)=h(t) xt >  Err/k
Analysis x We show xi* ≤ xi   Err/k with probability >1/2 Assume |xi1| ≥ … ≥ |xim| and let S={i1…ik} (“elephants”) When is xi* > xi   Err/k ? Event 1: S and i collide, i.e., h(i) in h(S-{i}) Probability: at most k/(4/)k = /4<1/4 (if <1) Event 2: many “mice” collide with i., i.e., ∑t not in S u {i}, h(i)=h(t) xt >  Err/k Probability: at most ¼ (Markov inequality) Total probability of “bad” events <1/2 xi1 xi2 xik xi Z1 ………..Z(4/)k

Second try Algorithm: Analysis:
Maintain d functions h1…hd and vectors Z1…Zd Estimator: Xi*=mint Ztht(i) Analysis: Pr[|xi*-xi| ≥ α Err/k ] ≤ 1/2d Setting d=O(log n) (and thus m=O(k log n) ) ensures that w.h.p |x*i-xi|< α Err/k

Part II Paper R/D Sketch length Encode time Column sparsity
Recovery time Approx [CRT’04] [RV’05] D k log(n/k) nk log(n/k) nc l2 / l1 k logc n n log n [BGIKS’08] D k log(n/k) n log(n/k) log(n/k) nc l1 / l1 [IR’08], [BIR’08],[BI’09] D k log(n/k) n log(n/k) log(n/k) n log(n/k)* log l1 / l1

Compressed Sensing A.k.a. compressive sensing or compressive sampling
[Candes-Romberg-Tao’04; Donoho’04] Signal acquisition/processing framework: Want to acquire a signal x=[x1…xn] Acquisition proceeds by computing Ax+noise of dimension m<<n We ignore noise in this lecture From Ax we want to recover an approximation x* of x Note: x* does not have to be k-sparse in general Method: solve the following program: minimize ||x*||1 subject to Ax*=Ax Guarantee: for some C>1 (1) ||x-x*||1 ≤ C mink-sparse x” ||x-x”||1 (2) ||x-x*||2 ≤ C mink-sparse x” ||x-x”||1 /k1/2 Notes: Sparsity in different basis -

Linear Programming Recovery: This is a linear program:
minimize ||x*||1 subject to Ax*=Ax This is a linear program: minimize ∑i ti subject to -ti ≤ x*i ≤ ti Ax*=Ax Can solve in nΘ(1) time

Intuition LP: On the first sight, somewhat mysterious
minimize ||x*||1 subject to Ax*=Ax On the first sight, somewhat mysterious But the approach has a long history in signal processing, statistics, etc. Intuition: The actual goal is to minimize ||x*||0 But this comes down to the same thing (if A is “nice”) The choice of L1 is crucial (L2 does not work) Ax*=Ax n=2, m=1, k=1

Restricted Isometry Property
A matrix A satisfies RIP of order k with constant δ if for any k-sparse vector x we have (1-δ) ||x||2 ≤ ||Ax||2 ≤ (1+δ) ||x||2 Theorem 1: If each entry of A is i.i.d. as N(0,1), and m=Θ(k log(n/k)), then A satisfies (k,1/3)-RIP w.h.p. Theorem 2: (O(k),1/3)-RIP implies L2/L1 guarantee [CRT’04] [RV’05] D k log(n/k) nk log(n/k) nc l2 / l1 k logc n n log n

dense vs. sparse Restricted Isometry Property (RIP) [Candes-Tao’04]
 is k-sparse  ||||2 ||A||2  C ||||2 Consider 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

 is k-sparse  d (1-) ||||1 ||A||1  d||||1
Expanders n m A bipartite graph is a (k,d(1-))-expander if for any left set S, |S|≤k, we have |N(S)|≥(1-)d |S| Objects well-studied in theoretical computer science and coding theory Constructions: Probabilistic: m=O(k log (n/k)) Explicit: m=k quasipolylog n High expansion implies RIP-1:  is k-sparse  d (1-) ||||1 ||A||1  d||||1 [Berinde-Gilbert-Indyk-Karloff-Strauss’08] S N(S) d m n

Proof: d(1-/2)-expansion  RIP-1
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 Claim 1: ||A||1 ≥∑e=(i,j) |i|re Claim 2: ∑e=(i,j) |i|re ≥ (1-) d||||1 d m n

Recovery: algorithms

L1 minimization Algorithm: Theoretical performance:
minimize ||x*||1 subject to Ax=Ax* Theoretical performance: RIP1 replaces RIP2 L1/L1 guarantee Experimental performance Thick line: Gaussian matrices Thin lines: prob. levels for sparse matrices (d=10) Same performance! k/m m/n [BGIKS’08] D k log(n/k) n log(n/k) log(n/k) nc l1 / l1

||x*+zei-x||p << ||x* - x||p
Matching Pursuit(s) A x*-x Ax-Ax* = i i 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 Norms: p=2 : CoSaMP, SP, IHT etc (RIP) p=1 : SMP, SSMP (RIP-1) p=0 : LDPC bit flipping (sparse matrices)

Sequential Sparse Matching Pursuit [Berinde-Indyk’09]
Algorithm: x*=0 Repeat T times Repeat S=O(k) times Find i and z that minimize* ||A(x*+zei)-Ax||1 x* = x*+zei Sparsify x* (set all but k largest entries of x* to 0) Similar to SMP, but updates done sequentially A i N(i) color code copies ? if Ax contains copies of k, then we can reverse this process. If we look at each coordinate on the left, and its neighbors, then we can perform a voting process uses insights developed by l2/l1 Emphasize the algorithm works for *any* signals x Ax-Ax* x-x* * Set z=median[ (Ax*-Ax)N(I).Instead, one could first optimize (gradient) i and then z [ Fuchs’09]

Experiments 256x256 SSMP is ran with S=10000,T=20. SMP is ran for 100 iterations. Matrix sparsity is d=8.

Conclusions Sparse approximation using sparse matrices
State of the art: deterministically can do 2 out of 3: Near-linear encoding/decoding O(k log (n/k)) measurements Approximation guarantee with respect to L2/L1 norm Open problems: 3 out of 3 ? Explicit constructions ? This talk

References Survey: A. Gilbert, P. Indyk, “Sparse recovery using sparse matrices”, Proceedings of IEEE, June 2010. Courses: “Streaming, sketching, and sub-linear space algorithms”, Fall’07 “Sub-linear algorithms” (with Ronitt Rubinfeld), Fall’10 Blogs: Nuit blanche: nuit-blanche.blogspot.com/

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