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We present a detailed analysis of the unconstrained $\ell_1$-weighted LASSOmethod for recovery of sparse data from its observation by randomly generatedmatrices, 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 LASSOminimizer, $s$, maintains a comparable sparsity, $s\leq C_\delta k$. Forexample, 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 highlightprecise dependence on $k$ and on the LASSO parameter $\lambda$, before theerror is driven below the scale of negligible measurement/ and compressiblityerrors.

On the sparsity of LASSO minimizers in sparse data recovery