Minkovskian Gradient for Sparse Optimization

摘要

Information geometry is used to elucidate convex optimization problems under $L_{1}$ constraint. A convex function induces a Riemannian metric and two dually coupled affine connections in the manifold of parameters of interest. A generalized Pythagorean theorem and projection theorem hold in such a manifold. An extended LARS algorithm, applicable to both under-determined and over-determined cases, is studied and properties of its solution path are given. The algorithm is shown to be a Minkovskian gradient-descent method, which moves in the steepest direction of a target function under the Minkovskian $L_{1}$ norm. Two dually coupled affine coordinate systems are useful for analyzing the solution path.