On the sparsity of LASSO minimizers in sparse data recovery

Abstract

We present a detailed analysis of the unconstrained $\ell_1$-weighted LASSO method for recovery of sparse data from its observation by randomly generated matrices, satisfying the Restricted Isometry Property (RIP) with constant $\delta<1$, and subject to negligible measurement and compressibility errors. We prove that if the data is $k$-sparse, then the size of support of the LASSO minimizer, $s$, maintains a comparable sparsity, $s\leq C_\delta k$. For example, if $\delta=0.7$ then $s< 11k$ and a slightly smaller $\delta=0.4$ yields $s< 4k$. We also derive new $\ell_2/\ell_1$ error bounds which highlight precise dependence on $k$ and on the LASSO parameter $\lambda$, before the error is driven below the scale of negligible measurement/ and compressiblity errors.