There is no generally accepted way to define wavelets on permutations. We address this issue by introducing the notion of coset based multiresolution analysis (CMRA) on the symmetric group, find the corresponding wavelet functions, and describe a fast wavelet transform for sparse signals. We discuss potential applications in ranking, sparse approximation, and multi-object tracking.

Multiresolution Matrix Factorization (MMF) was recently introduced as an alternative to the dominant low-rank paradigm in order to capture structure in matrices at multiple different scales. Using ideas from multiresolution analysis (MRA), MMF teased out hierarchical structure in symmetric matrices by constructing a sequence of wavelet bases. While effective for such matrices, there is plenty of data that is more naturally represented as nonsymmetric matrices (e.g. directed graphs), but nevertheless has similar hierarchical structure. In this paper, we explore techniques for extending MMF to any square matrix. We validate our approach on numerous matrix compression tasks, demonstrating its efficacy compared to low-rank methods. Moreover, we also show that a combined low-rank and MMF approach, which amounts to removing a small global-scale component of the matrix and then extracting hierarchical structure from the residual, is even more effective than each of the two complementary methods for matrix compression.

In this paper, we propose a generic framework for devising an adaptive approximation scheme for value function approximation in reinforcement learning, which introduces multiscale approximation. The two basic ingredients are multiresolution analysis as well as tree approximation. Starting from simple refinable functions, multiresolution analysis enables us to construct a wavelet system from which the basis functions are selected adaptively, resulting in a tree structure. Furthermore, we present the convergence rate of our multiscale approximation which does not depend on the regularity of basis functions.

There is no generally accepted way to define wavelets on permutations. We address this issue by introducing the notion of coset based multiresolution analysis (CMRA) on the symmetric group; find the corresponding wavelet functions; and describe a fast wavelet transform of O(n^p) complexity with small p for sparse signals (in contrast to the O(n^q n!) complexity typical of FFTs). We discuss potential applications in ranking, sparse approximation, and multi-object tracking.