AUTHOR=Bah Bubacarr , Tanner Jared 

TITLE=On the Construction of Sparse Matrices From Expander Graphs

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=4

YEAR=2018

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2018.00039

DOI=10.3389/fams.2018.00039

ISSN=2297-4687

ABSTRACT=<p>We revisit the asymptotic analysis of probabilistic construction of adjacency matrices of expander graphs proposed in Bah and Tanner [<xref ref-type="bibr" rid="B1">1</xref>]. With better bounds we derived a new reduced sample complexity for <italic>d</italic>, the number of non-zeros per column of these matrices (or equivalently the left-degree of the underlying expander graph). Precisely <inline-formula><mml:math id="M1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="-tex-caligraphic">O</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mo class="qopname">log</mml:mo></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>N</mml:mi><mml:mo>/</mml:mo><mml:mi>s</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula>; as opposed to the standard <inline-formula><mml:math id="M2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="-tex-caligraphic">O</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mo class="qopname">log</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>N</mml:mi><mml:mo>/</mml:mo><mml:mi>s</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula>, where <italic>N</italic> 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 <italic>d</italic> 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.</p>