Deep Two-Way Matrix Reordering for Relational Data Analysis

TL;DR

The proposed DeepTMR provides the denoised mean matrix of a given observed matrix as an output of the trained network, which can be used to visualize the global structure of the reordered observed matrix.

Abstract

Matrix reordering is a task to permute the rows and columns of a given observed matrix such that the resulting reordered matrix shows meaningful or interpretable structural patterns. Most existing matrix reordering techniques share the common processes of extracting some feature representations from an observed matrix in a predefined manner, and applying matrix reordering based on it. However, in some practical cases, we do not always have prior knowledge about the structural pattern of an observed matrix. To address this problem, we propose a new matrix reordering method, called deep two-way matrix reordering (DeepTMR), using a neural network model. The trained network can automatically extract nonlinear row/column features from an observed matrix, which can then be used for matrix reordering. Moreover, the proposed DeepTMR provides the denoised mean matrix of a given observed matrix as an output of the trained network. This denoised mean matrix can be used to visualize the global structure of the reordered observed matrix. We demonstrate the effectiveness of the proposed DeepTMR by applying it to both synthetic and practical datasets.

Figures (9)

• Figure 1: Matrix reordering problem. Given an observed matrix $A$ (left), the proposed DeepTMR reorders the rows and columns of matrix $A$ such that the reordered input matrix (center) shows a meaningful or interpretable structure. Th proposed DeepTMR provides us with the denoised mean matrix of the reordered matrix (right) as the output of a trained network, as well as row/column ordering.
• Figure 3: Results of the LBM. Top figures: original matrix $\bar{A}$, observed matrix $A$ obtained by applying random row-column permutation to $\bar{A}$, reordered input matrix $\underline{A}$, and reordered output matrix $\underline{\hat{A}}$ (left to right). Bottom figures: Encoded row and column features $\bm{g}$ and $\bm{h}$ and reordered row and column features $\underline{\bm{g}}$ and $\underline{\bm{h}}$ (left to right).
• Figure 5: Results of the gradation block model for matrices $\bar{A}$, $A$, $\underline{A}$, $\underline{\hat{A}}$, and vectors $\bm{g}$, $\bm{h}$, $\underline{\bm{g}}$, $\underline{\bm{h}}$.
• Figure 6: Examples of matrix $\bar{A}$ for the diagonal gradation model with different levels of noise standard deviation.
• Figure 9: Examples of reordered input matrix $\underline{A}$ for the diagonal gradation model with different levels of noise standard deviation (SVD-Rank-One).