Large Rectangular Matrices Research Articles

Uncovering block structures in large rectangular matrices

In this article, we proposed a conceptually simple, efficient, and easily implemented approach, named Block-structure Analysis on Rectangular Matrix (BARM), for learning the block structure in a large rectangular data matrix corrupted with random effects and white noise. With the possible unknown order of row or column variables, their group structures can be directly uncovered based on the singular values and singular vectors of the scaled data matrix. We also established the asymptotic properties of the proposed approach under regular conditions. Extensive experimental evaluations also demonstrate the reliability and robustness of the proposed approach.

Binary matrix factorization on special purpose hardware.

Many fundamental problems in data mining can be reduced to one or more NP-hard combinatorial optimization problems. Recent advances in novel technologies such as quantum and quantum-inspired hardware promise a substantial speedup for solving these problems compared to when using general purpose computers but often require the problem to be modeled in a special form, such as an Ising or quadratic unconstrained binary optimization (QUBO) model, in order to take advantage of these devices. In this work, we focus on the important binary matrix factorization (BMF) problem which has many applications in data mining. We propose two QUBO formulations for BMF. We show how clustering constraints can easily be incorporated into these formulations. The special purpose hardware we consider is limited in the number of variables it can handle which presents a challenge when factorizing large matrices. We propose a sampling based approach to overcome this challenge, allowing us to factorize large rectangular matrices. In addition to these methods, we also propose a simple baseline algorithm which outperforms our more sophisticated methods in a few situations. We run experiments on the Fujitsu Digital Annealer, a quantum-inspired complementary metal-oxide-semiconductor (CMOS) annealer, on both synthetic and real data, including gene expression data. These experiments show that our approach is able to produce more accurate BMFs than competing methods.

Transposition of large rectangular matrices

Transposition of large rectangular matrices

A simple algorithm for transposing large rectangular matrices without an auxiliary array. Application to the processing 2D NMR data

A simple algorithm for transposing large rectangular matrices without an auxiliary array. Application to the processing 2D NMR data

A Generalization of Eklundh's Algorithm for Transposing Large Matrices

The purpose of this paper is to present an algorithm for transposing large rectangular matrices. This is basically a generalization of Eklundh's algorithm. Eklundh's method is designed to handle arrays in place of using powers-of-2 approach wherein partitions of the matrix have dimensions which are powers of 2. The algorithm presented here does not have this restriction. The choice of partitions is posed as an optimal control problem amenable to solution using Bellman's principle of optimality. The optimal partitioning of the matrix can be determined depending on the computer configuration (i.e., core size, whether moving-head or fixed-head disk drives are used, number of words per track, etc.) and the matrix dimensions. Simple modifications are noted for transposing complex/double precision and packed integer matrices. Experimental results are presented comparing the times needed for transposition using optimal partitioning and powers-of-2 partitioning. It is found in the cases tested that a speed gain of up to 5.55 is obtained by optimizing the partition sizes.