Parameter Instability Regimes in Sparse Proximal Denoising Programs

Compressed sensing theory explains why LASSO programs recover structured high-dimensional signals with minimax order-optimal error. Yet, the optimal choice of the program's governing parameter is often unknown in practice. It is still unclear how variation of the governing parameter impacts recovery error in compressed sensing, which is otherwise provably stable and robust. We establish a novel notion of instability in LASSO programs when the measurement matrix is identity. This is the proximal denoising setup. We prove asymptotic cusp-like behaviour of the risk as a function of the parameter choice, and illustrate the theory with numerical simulations. For example, a 0.1% underestimate of a LASSO parameter can increase the error significantly; and a 50% underestimate can cause the error to increase by a factor of 10
⁹
. We hope that revealing parameter instability regimes of LASSO programs helps to inform a practitioner's choice.