Gradient flows and randomised thresholding: sparse inversion and classification*

Sparse inversion and classification problems are ubiquitous in modern data science imaging. They often formulated as non-smooth minimisation problems. In sparse inversion, we minimise, e.g., the sum of a fidelity term an L1/LASSO regulariser. classification, consider, Ginzburg--Landau energy. Standard (sub)gradient descent methods have shown to be inefficient when approaching such Splitting techniques much more useful: here, target function is partitioned into two subtarget functions -- each which can efficiently optimised. proceeds by performing optimisation steps alternately with respect functions. this work, study splitting from stochastic continuous-time perspective. Indeed, define differential inclusion that follows one function's negative subdifferential at point time. The choice controlled binary Markov process. resulting dynamical system approximation underlying subgradient flow. We investigate for L1-regularised flow discrete Allen-Cahn equation minimising both cases, longtime behaviour its ability approximate any accuracy. illustrate our theoretical findings simple estimation problem also low- high-dimensional