Weighted Rank-One Binary Matrix Factorization

Abstract

Mining discrete patterns in binary data is important for many data analysis tasks, such as data sampling, compression, and clustering. An example is that replacing individual records with their patterns would greatly reduce data size and simplify subsequent data analysis tasks. As a straightforward approach, rank-one binary matrix approximation has been actively studied recently for mining discrete patterns from binary data. It factorizes a binary matrix into the multiplication of one binary pattern vector and one binary presence vector, while minimizing mismatching entries. However, this approach suffers from two serious problems. First, if all records are replaced with their respective patterns, the noise could make as much as 50% in the resulting approximate data. This is because the approach simply assumes that a pattern is present in a record as long as their matching entries are more than their mismatching entries. Second, two error types, 1-becoming-0 and 0-becoming-1, are treated evenly, while in many application domains they are discriminated. To address the two issues, we propose weighted rank-one binary matrix approximation. It enables the tradeoff between the accuracy and succinctness in approximate data and allows users to impose their personal preferences on the importance of different error types. The decision problem, however, as proved in the paper is NP-complete. To solve it, several different mathematical programming formulations are provided, from which 2-approximation algorithms are derived for some special cases. An adaptive tabu search heuristic is presented for solving the general problem, and our experimental study shows the effectiveness of the heuristic.

Average value of solutions for the bipartite boolean quadratic programs and rounding algorithms

“…In this section we consider two natural local search heuristics for BBQP and show that the solution produced could have objective function value worse than A (Q, c, d). One of the popular heuristics for BBQP is the alternating algorithm proposed by many authors [21,10,19]. The algorithm starts with a candidate solution x 0 and try to choose an optimal y 0 .…”

Efficient local search for L_1 and L_2 binary matrix factorization

“…The NMF problem as well as the K−means clustering one are known to be NP-hard [1,3]. A related problem to the one considered here, namely the weighted rank 1 binary matrix factorization, is also shown to be NPcomplete in [15]. This problem, defined for K = 1, however differs from ours in that the errors made in the reconstruction (having a 1 instead of a 0 or a 0 instead of a 1) are weighted differently (they have the same weight, 1, in our formulation).…”

“…This problem, defined for K = 1, however differs from ours in that the errors made in the reconstruction (having a 1 instead of a 0 or a 0 instead of a 1) are weighted differently (they have the same weight, 1, in our formulation). Even though this difference may seem marginal, it suffices to obtain a reduction from the maximum edge weight biclique problem, and the reduction proposed in [15] can not be directly applied to our problem. In fact, even though Problem (1) has been claimed in several places to be NP-hard, we know of no formal proof for this fact.…”

A divide-and-conquer algorithm for binary matrix completion

“…An approach combining convex quadratic programming relaxations and binary thresholding was proposed in [29]. For the special case of rank-one approximation, linear programming relaxations were proposed in [21,16,29], the first two of which also both proved that the linear programming solution yields a 2-approximation to the optimal objective. An approach to binary factorisation based on k-means clustering was considered in [9].…”