# The Incremental Multiresolution Matrix Factorization Algorithm

## TL;DR

This paper introduces an incremental multiresolution matrix factorization algorithm that uncovers hierarchical structures in large symmetric matrices, enhancing analysis in vision tasks and deep network representations.

## Contribution

The paper presents a scalable, feature-by-feature algorithm for multiresolution matrix factorization that reveals hierarchical structures in large matrices, advancing analysis in vision and deep learning.

## Key findings

- Effective in medical imaging regression tasks
- Reveals semantic relationships in deep network features
- Scalable to large matrices

## Abstract

Multiresolution analysis and matrix factorization are foundational tools in computer vision. In this work, we study the interface between these two distinct topics and obtain techniques to uncover hierarchical block structure in symmetric matrices -- an important aspect in the success of many vision problems. Our new algorithm, the incremental multiresolution matrix factorization, uncovers such structure one feature at a time, and hence scales well to large matrices. We describe how this multiscale analysis goes much farther than what a direct global factorization of the data can identify. We evaluate the efficacy of the resulting factorizations for relative leveraging within regression tasks using medical imaging data. We also use the factorization on representations learned by popular deep networks, providing evidence of their ability to infer semantic relationships even when they are not explicitly trained to do so. We show that this algorithm can be used as an exploratory tool to improve the network architecture, and within numerous other settings in vision.