Unveiling The Tree: A Convex Framework for Sparse Problems
Abstract
This paper presents a general framework for generating greedy algorithms for solving convex constraint satisfaction problems for sparse solutions by mapping the satisfaction problem into one of graph traversal on a rooted tree of unknown topology. For every pre-walk of the tree an initial set of generally dense feasible solutions is processed in such a way that the sparsity of each solution increases with each generation unveiled. The specific computation performed at any particular child node is shown to correspond to an embedding of a polytope into the polytope received from that nodes parent. Several issues related to pre-walk order selection, computational complexity and tractability, and the use of heuristic and/or side information is discussed. An example of a single-path, depth-first algorithm on a tree with randomized vertex reduction and a run-time path selection algorithm is presented in the context of sparse lowpass filter design.
Cite
@article{arxiv.1502.01220,
title = {Unveiling The Tree: A Convex Framework for Sparse Problems},
author = {Tarek A. Lahlou and Alan V. Oppenheim},
journal= {arXiv preprint arXiv:1502.01220},
year = {2015}
}