Unveiling The Tree: A Convex Framework for Sparse Problems

TL;DR

This paper introduces a convex framework that transforms sparse solution problems into graph traversal tasks on trees, enabling the design of greedy algorithms with improved sparsity and computational efficiency.

Contribution

It proposes a novel tree-based convex framework for greedy algorithms in sparse problems, including a depth-first algorithm with randomized vertex reduction.

Findings

Framework effectively increases solution sparsity with each tree generation

Algorithm demonstrates practical application in sparse filter design

Discussion on complexity and heuristic improvements

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.