Exact signal reconstruction from highly incomplete nonlinear system

Abstract

L1 regularization has long been used to obtain sparse representations of data, but is it possible to extend the convergence conditions from the linear to the nonlinear case? L1 regularization does not guarantee accurate reconstruction of a sparse state from incomplete data, even under linear constraints. However, once the amount of data exceeds a certain threshold, the probability of reconstruction failure with random matrices will be negligible. Developing a theory to determine the probability of success in the nonlinear case seems useful. This requires defining various classes of random nonlinear functions and examining how their convergence conditions vary. In particular, linear processes are often affected by nonlinear distortions, which are commonly modeled as noise. Identifying the physics and chemical kinetics of these distortions and incorporating them into the model will help improve the probability of successful reconstruction and enable the identification of factors that influence these distortions. This is especially important in biology, where understanding not only the balance of internal processes but also the factors influencing the stability of the system is crucial