Dictionary LASSO: Guaranteed Sparse Recovery under Linear Transformation

We consider the following signal recovery problem: given a measurement matrix $Φ\in \mathbb{R}^{n\times p}$ and a noisy observation vector $c\in \mathbb{R}^{n}$ constructed from $c = Φθ^* + ε$ where $ε\in \mathbb{R}^{n}$ is the noise vector whose entries follow i.i.d. centered sub-Gaussian distribution, how to recover the signal $θ^*$ if $Dθ^*$ is sparse {\rca under a linear transformation} $D\in\mathbb{R}^{m\times p}$? One natural method using convex optimization is to solve the following problem: $\min_θ {1\over 2}\|Φθ- c\|^2 + λ\|Dθ\|_1.$ This paper provides an upper bound of the estimate error and shows the consistency property of this method by assuming that the design matrix $Φ$ is a Gaussian random matrix. Specifically, we show 1) in the noiseless case, if the condition number of $D$ is bounded and the measurement number $n\geq Ω(s\log(p))$ where $s$ is the sparsity number, then the true solution can be recovered with high probability; and 2) in the noisy case, if the condition number of $D$ is bounded and the measurement increases faster than $s\log(p)$, that is, $s\log(p)=o(n)$, the estimate error converges to zero with probability 1 when $p$ and $s$ go to infinity. Our results are consistent with those for the special case $D=\bold{I}_{p\times p}$ (equivalently LASSO) and improve the existing analysis. The condition number of $D$ plays a critical role in our analysis. We consider the condition numbers in two cases including the fused LASSO and the random graph: the condition number in the fused LASSO case is bounded by a constant, while the condition number in the random graph case is bounded with high probability if $m\over p$ (i.e., $#text{edge}\over #text{vertex}$) is larger than a certain constant. Numerical simulations are consistent with our theoretical results.

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