# Convex Reconstruction of Structured Matrix Signals from Random Linear Measurements (I): Theoretical Results

We investigate the problem of reconstructing n-by-n column-wise sparse matrix signal X=(x1,...,xn) via convex programming, where each column xj is a vector of s-sparsity. The regularizer is matrix norm |||X|||1:=maxj|xj|1 where |.|1 is the l1-norm in vector space. We take the convex geometric approach in random measurement setting and establish sufficient conditions on dimensions of measurement spaces for robust reconstruction in noise and some necessary conditions for accurate reconstruction. For example, for the m-by-m measurement Y=AXB+E where E is bounded noise and A, B are m-by-n random matrices, one of the established sufficient conditions for X to be reconstructed robustly with respect to Frobenius norm is m2 > C(n2-r(n-slog2(C1n2r))+C2n) when A, B are both sub-Gaussian matrices, where r and s are signal's structural parameters, i.e., s is the maximum number of nonzero entries in each column and r is the number of columns which l1-norms are maximum among all columns. In particular, when r = n the sufficient condition reduces to m2 > Cnslog2(Cn3). This bound is relatively tight because a provable necessary condition is m2 > C1nslog(C2n/s)

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