Abstract
We revisit the asymptotic analysis of probabilistic construction of adjacency matrices of expander graphs proposed in Bah and Tanner [2013]. With better bounds we derived a new reduced sample complexity for d, the number of nonzeros per column of these matrices (or equivalently the left-degree of the underlying expander graph). Precisely d = O(log(N/s) /log(s)); as opposed to the standard d = O(\log(N/s)), where N is the number of columns of the matrix (also the cardinality of set of left vertices of the expander graph) or the ambient dimension of the signals that can be sensed by such matrices. This gives insights into why using such sensing matrices with small d performed well in numerical compressed sensing experiments. Furthermore, we derive quantitative sampling theorems for our constructions which show our construction outperforming the existing state-of-the-art. We also used our results to compare performance of sparse recovery algorithms where these matrices are used for linear sketching.
Highlights
Before stating the main results of the paper, we summarise here how we improve upon existing results in Bah and Tanner [1], which is generally true of standard random construction of such matrices
The main result of this work is formalized in Theorem 3.1, which is an asymptotic result, where the dimensions grow while their ratios remain bounded
We point out that it has been observed in practice that applying these matrices, which necessitate having finite dimensions, produces results that agree with the theory
Summary
Say A ∈ {0, 1}n×N, with n ≪ N are widely used in applications including graph sketching [2, 3], network tomography [4, 5], data streaming [6, 7], breaking privacy of databases via aggregate queries [8], compressed imaging of intensity patterns [9], and more generally combinatorial compressed sensing [10,11,12,13,14,15], linear sketching [7], and group testing [16, 17]. For computational advantages, such as faster application and smaller storage, it is advantageous to use sparse A in application [1, 14, 18]
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