joint work with Takumi Fukuda
This paper presents efficient algorithms for k-sparse recovery problem on the basis of the Continuous Exact k-Sparse Penalties (CXPs). Each CXP is defined by a non-convex but continuous function and the equality forcing the function value to be zero is known to be equivalent to the k-sparse constraint defined with the so-called l0-norm. Algorithms based on the proximal maps of the l1-norm or l2-norm based CXPs will be examined for the sparse recovery problems with/without noise.
In this talk, we consider proximal Newton-type methods for the minimization of a composite function that is the sum of a smooth function and a nonsmooth function. Proximal Newton-type methods make use of weighted proximal mappings whose calculations are much expensive. Accordingly, some researchers have proposed inexact Newton-type proximal methods, which calculate the proximal mapping inexactly. To calculate proximal mappings more easily, we propose an inexact proximal method based on memoryless quasi-Newton methods with the Broyden family. We analyze global and local convergence properties.
joint work with Akiko Takeda, Akihiro Kawana
We propose a bilevel optimization strategy for selecting the best hyperparameter value for the nonsmooth lp regularizer with 0 < p < = 1. The concerned bilevel optimization problem has a nonsmooth, possibly nonconvex, lp-regularized problem as the lower-level problem. For solving this problem, we propose a smoothing-type algorithm and show that a sequence generated by the proposed method converges to a point satisfying certain optimality conditions. We also conducted some numerical experiments to exhibit the efficiency by comparing it with Bayesian optimization.