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Lossless Expander Graphs in Compressive Sensing Abbas Kazemipour MAST Group Meeting University of Maryland. College Park kaazemi@umd.edu October 22, 2015 Abbas Kazemipour (UMD) Expanders October 22, 2015 1 / 8 Overview 1 Introduction

SLIDE 1

Lossless Expander Graphs in Compressive Sensing

Abbas Kazemipour

MAST Group Meeting University of Maryland. College Park kaazemi@umd.edu

October 22, 2015

Abbas Kazemipour (UMD) Expanders October 22, 2015 1 / 8

SLIDE 2

Overview

1 Introduction

Abbas Kazemipour (UMD) Expanders October 22, 2015 2 / 8

SLIDE 3

Bipartite Graphs

1 N = |L|= 8,m = |R|= 4

Figure : A left refular bipartite graph of left degree d = 2

Abbas Kazemipour (UMD) Expanders October 22, 2015 3 / 8

SLIDE 4

Bipartite Graphs

1 J ⊂ L 2 E(J) = set of edges going out of J 3 R(J) = set of vertices in R connected to J

|E(J)|= d|J|≥ |R(J)|. A left regular bipartite graph with left degree d is called an (s, d, θ)-lossless expander if it satisfies the expansion property |R(J)|≥ (1 − θ)d|J|, for all J with |J|≤ s.

4 Smallest such θ: restricted expansion constant θs. Abbas Kazemipour (UMD) Expanders October 22, 2015 4 / 8

SLIDE 5

Bipartite Graphs

1 J ⊂ L 2 E(J) = set of edges going out of J 3 R(J) = set of vertices in R connected to J

|E(J)|= d|J|≥ |R(J)|. A left regular bipartite graph with left degree d is called an (s, d, θ)-lossless expander if it satisfies the expansion property |R(J)|≥ (1 − θ)d|J|, for all J with |J|≤ s.

4 Smallest such θ: restricted expansion constant θs. Abbas Kazemipour (UMD) Expanders October 22, 2015 4 / 8

SLIDE 6

Bipartite Graphs

1 J ⊂ L 2 E(J) = set of edges going out of J 3 R(J) = set of vertices in R connected to J

|E(J)|= d|J|≥ |R(J)|. A left regular bipartite graph with left degree d is called an (s, d, θ)-lossless expander if it satisfies the expansion property |R(J)|≥ (1 − θ)d|J|, for all J with |J|≤ s.

4 Smallest such θ: restricted expansion constant θs. Abbas Kazemipour (UMD) Expanders October 22, 2015 4 / 8

SLIDE 7

Bipartite Graphs

1 J ⊂ L 2 E(J) = set of edges going out of J 3 R(J) = set of vertices in R connected to J

|E(J)|= d|J|≥ |R(J)|. A left regular bipartite graph with left degree d is called an (s, d, θ)-lossless expander if it satisfies the expansion property |R(J)|≥ (1 − θ)d|J|, for all J with |J|≤ s.

4 Smallest such θ: restricted expansion constant θs. Abbas Kazemipour (UMD) Expanders October 22, 2015 4 / 8

SLIDE 8

Bipartite Graphs

0 = θ1 ≤ θ2 ≤ · · · ≤ θN.

2 The samller the better.

θks ≤ (k − 1)θ2s + θs.

3 4 E(K; J) = edges going out of K ending in R(J).

J, K ⊂ L, disjoint, |J|+|K|≤ s |E(K; J)|≤ θsds.

Abbas Kazemipour (UMD) Expanders October 22, 2015 5 / 8

SLIDE 9

Bipartite Graphs

0 = θ1 ≤ θ2 ≤ · · · ≤ θN.

2 The samller the better.

θks ≤ (k − 1)θ2s + θs.

3 4 E(K; J) = edges going out of K ending in R(J).

J, K ⊂ L, disjoint, |J|+|K|≤ s |E(K; J)|≤ θsds.

Abbas Kazemipour (UMD) Expanders October 22, 2015 5 / 8

SLIDE 10

Bipartite Graphs

0 = θ1 ≤ θ2 ≤ · · · ≤ θN.

2 The samller the better.

θks ≤ (k − 1)θ2s + θs.

3 4 E(K; J) = edges going out of K ending in R(J).

J, K ⊂ L, disjoint, |J|+|K|≤ s |E(K; J)|≤ θsds.

Abbas Kazemipour (UMD) Expanders October 22, 2015 5 / 8

SLIDE 11

Bipartite Graphs

0 = θ1 ≤ θ2 ≤ · · · ≤ θN.

2 The samller the better.

θks ≤ (k − 1)θ2s + θs.

3 4 E(K; J) = edges going out of K ending in R(J).

J, K ⊂ L, disjoint, |J|+|K|≤ s |E(K; J)|≤ θsds.

Abbas Kazemipour (UMD) Expanders October 22, 2015 5 / 8

SLIDE 12

Bipartite Graphs

1 E′(S) = {jiE(S) : j = l(i)}.

|E′(S) ≤ θsds|.

2 R1(S) = vertices in R(S) which are connected to a unique vertex

in S. |R1(S) ≥ (1 − 2θs)ds|

Existence

0 < ǫ < 1/2, N, m, d given. Randomly choose edges. With probability ≥ 1 − ǫ we have an (s, d, θ)-lossless expander if d = ⌈1 θ ln eN ǫs

; m ≥ cθs ln eN ǫs

3 Abbas Kazemipour (UMD) Expanders October 22, 2015 6 / 8

SLIDE 13

Bipartite Graphs

1 E′(S) = {jiE(S) : j = l(i)}.

|E′(S) ≤ θsds|.

2 R1(S) = vertices in R(S) which are connected to a unique vertex

in S. |R1(S) ≥ (1 − 2θs)ds|

Existence

0 < ǫ < 1/2, N, m, d given. Randomly choose edges. With probability ≥ 1 − ǫ we have an (s, d, θ)-lossless expander if d = ⌈1 θ ln eN ǫs

; m ≥ cθs ln eN ǫs

3 Abbas Kazemipour (UMD) Expanders October 22, 2015 6 / 8

SLIDE 14

Bipartite Graphs

1 E′(S) = {jiE(S) : j = l(i)}.

|E′(S) ≤ θsds|.

2 R1(S) = vertices in R(S) which are connected to a unique vertex

in S. |R1(S) ≥ (1 − 2θs)ds|

Existence

0 < ǫ < 1/2, N, m, d given. Randomly choose edges. With probability ≥ 1 − ǫ we have an (s, d, θ)-lossless expander if d = ⌈1 θ ln eN ǫs

; m ≥ cθs ln eN ǫs

3 Abbas Kazemipour (UMD) Expanders October 22, 2015 6 / 8

SLIDE 15

Bipartite Graphs

Adjacency Matrix

Adjacency matrix of a bipartite graph an m × N is a binary matrix A with the property Aij = 1 iff j-th node on the left is connected to i-th node on the right.

1 2 think of an (s, d, θ)-lossless expander as a matrix A populated with

zeros and ones, with d ones per column, and such that there are at least (1θ)dk nonzero rows in any submatrix of A composed of k ≤ s columns.

3 Easier to store than Gaussian matrices Abbas Kazemipour (UMD) Expanders October 22, 2015 7 / 8

SLIDE 16

Bipartite Graphs

Adjacency Matrix

Adjacency matrix of a bipartite graph an m × N is a binary matrix A with the property Aij = 1 iff j-th node on the left is connected to i-th node on the right.

1 2 think of an (s, d, θ)-lossless expander as a matrix A populated with

zeros and ones, with d ones per column, and such that there are at least (1θ)dk nonzero rows in any submatrix of A composed of k ≤ s columns.

3 Easier to store than Gaussian matrices Abbas Kazemipour (UMD) Expanders October 22, 2015 7 / 8

SLIDE 17

Bipartite Graphs

Adjacency Matrix

Adjacency matrix of a bipartite graph an m × N is a binary matrix A with the property Aij = 1 iff j-th node on the left is connected to i-th node on the right.

1 2 think of an (s, d, θ)-lossless expander as a matrix A populated with

zeros and ones, with d ones per column, and such that there are at least (1θ)dk nonzero rows in any submatrix of A composed of k ≤ s columns.

3 Easier to store than Gaussian matrices Abbas Kazemipour (UMD) Expanders October 22, 2015 7 / 8

SLIDE 18

Bipartite Graphs

RIP-1 and BP

If θ2s < 1/6 then a solution of basis pursuit satisfies: x − x1≤ 2(1 − θ) (1 − 6θ)x − xS1+ 4 (1 − 6θ)dη

1 2 Similar procedure requires θ3s < 1/12 for OMP and thresholding

algorithms.

Abbas Kazemipour (UMD) Expanders October 22, 2015 8 / 8

SLIDE 19

Bipartite Graphs

RIP-1 and BP

If θ2s < 1/6 then a solution of basis pursuit satisfies: x − x1≤ 2(1 − θ) (1 − 6θ)x − xS1+ 4 (1 − 6θ)dη

1 2 Similar procedure requires θ3s < 1/12 for OMP and thresholding

algorithms.

Abbas Kazemipour (UMD) Expanders October 22, 2015 8 / 8