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 Abbas Kazemipour (UMD) Expanders October 22, 2015 2 / 8
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
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 θ s . Abbas Kazemipour (UMD) Expanders October 22, 2015 4 / 8
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 θ s . Abbas Kazemipour (UMD) Expanders October 22, 2015 4 / 8
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 θ s . Abbas Kazemipour (UMD) Expanders October 22, 2015 4 / 8
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 θ s . Abbas Kazemipour (UMD) Expanders October 22, 2015 4 / 8
Bipartite Graphs 1 0 = θ 1 ≤ θ 2 ≤ · · · ≤ θ N . 2 The samller the better. θ ks ≤ ( k − 1) θ 2 s + θ s . 3 4 E ( K ; J ) = edges going out of K ending in R ( J ). J, K ⊂ L , disjoint, | J | + | K |≤ s | E ( K ; J ) |≤ θ s ds. Abbas Kazemipour (UMD) Expanders October 22, 2015 5 / 8
Bipartite Graphs 1 0 = θ 1 ≤ θ 2 ≤ · · · ≤ θ N . 2 The samller the better. θ ks ≤ ( k − 1) θ 2 s + θ s . 3 4 E ( K ; J ) = edges going out of K ending in R ( J ). J, K ⊂ L , disjoint, | J | + | K |≤ s | E ( K ; J ) |≤ θ s ds. Abbas Kazemipour (UMD) Expanders October 22, 2015 5 / 8
Bipartite Graphs 1 0 = θ 1 ≤ θ 2 ≤ · · · ≤ θ N . 2 The samller the better. θ ks ≤ ( k − 1) θ 2 s + θ s . 3 4 E ( K ; J ) = edges going out of K ending in R ( J ). J, K ⊂ L , disjoint, | J | + | K |≤ s | E ( K ; J ) |≤ θ s ds. Abbas Kazemipour (UMD) Expanders October 22, 2015 5 / 8
Bipartite Graphs 1 0 = θ 1 ≤ θ 2 ≤ · · · ≤ θ N . 2 The samller the better. θ ks ≤ ( k − 1) θ 2 s + θ s . 3 4 E ( K ; J ) = edges going out of K ending in R ( J ). J, K ⊂ L , disjoint, | J | + | K |≤ s | E ( K ; J ) |≤ θ s ds. Abbas Kazemipour (UMD) Expanders October 22, 2015 5 / 8
Bipartite Graphs 1 E ′ ( S ) = { jiE ( S ) : j � = l ( i ) } . | E ′ ( S ) ≤ θ s ds | . 2 R 1 ( S ) = vertices in R ( S ) which are connected to a unique vertex in S . | R 1 ( S ) ≥ (1 − 2 θ s ) 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 � eN � � eN � d = ⌈ 1 θ ln ⌉ ; m ≥ c θ s ln . ǫs ǫs 3 Abbas Kazemipour (UMD) Expanders October 22, 2015 6 / 8
Bipartite Graphs 1 E ′ ( S ) = { jiE ( S ) : j � = l ( i ) } . | E ′ ( S ) ≤ θ s ds | . 2 R 1 ( S ) = vertices in R ( S ) which are connected to a unique vertex in S . | R 1 ( S ) ≥ (1 − 2 θ s ) 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 � eN � � eN � d = ⌈ 1 θ ln ⌉ ; m ≥ c θ s ln . ǫs ǫs 3 Abbas Kazemipour (UMD) Expanders October 22, 2015 6 / 8
Bipartite Graphs 1 E ′ ( S ) = { jiE ( S ) : j � = l ( i ) } . | E ′ ( S ) ≤ θ s ds | . 2 R 1 ( S ) = vertices in R ( S ) which are connected to a unique vertex in S . | R 1 ( S ) ≥ (1 − 2 θ s ) 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 � eN � � eN � d = ⌈ 1 θ ln ⌉ ; m ≥ c θ s ln . ǫs ǫs 3 Abbas Kazemipour (UMD) Expanders October 22, 2015 6 / 8
Bipartite Graphs Adjacency Matrix Adjacency matrix of a bipartite graph an m × N is a binary matrix A with the property A ij = 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 (1 θ ) 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
Bipartite Graphs Adjacency Matrix Adjacency matrix of a bipartite graph an m × N is a binary matrix A with the property A ij = 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 (1 θ ) 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
Bipartite Graphs Adjacency Matrix Adjacency matrix of a bipartite graph an m × N is a binary matrix A with the property A ij = 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 (1 θ ) 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
Bipartite Graphs RIP-1 and BP If θ 2 s < 1 / 6 then a solution of basis pursuit satisfies: x � 1 ≤ 2(1 − θ ) 4 � x − � (1 − 6 θ ) � x − x S � 1 + (1 − 6 θ ) dη 1 2 Similar procedure requires θ 3 s < 1 / 12 for OMP and thresholding algorithms. Abbas Kazemipour (UMD) Expanders October 22, 2015 8 / 8
Bipartite Graphs RIP-1 and BP If θ 2 s < 1 / 6 then a solution of basis pursuit satisfies: x � 1 ≤ 2(1 − θ ) 4 � x − � (1 − 6 θ ) � x − x S � 1 + (1 − 6 θ ) dη 1 2 Similar procedure requires θ 3 s < 1 / 12 for OMP and thresholding algorithms. Abbas Kazemipour (UMD) Expanders October 22, 2015 8 / 8
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