Testing Sparsity over Known and Unknown Bases

Sparsity is a basic property of real vectors that is exploited in a wide variety of ma- chine learning applications. In this work, we describe property testing algorithms for spar- sity that observe a low-dimensional projec- tion of the input. We consider two settings. In the first setting, we test sparsity with re- spect to an unknown basis: given input vec- tors $y_1 ,...,y_p \in R^d$ whose concatenation as columns forms $Y \in R^{d \times p}$ , does $Y = AX$ for matrices $A \in R^{d\times m}$ and $X \in R^{m \times p}$ such that each column of $X$ is $k$-sparse, or is $Y$ “far” from having such a decomposition? In the second setting, we test sparsity with re- spect to a known basis: for a fixed design ma- trix $A \in R^{d \times m}$ , given input vector $y \in R^d$ , is $y = Ax$ for some $k$-sparse vector $x$ or is $y$ “far” from having such a decomposition? We analyze our algorithms using tools from high-dimensional geometry and probability.