CMF is a technique for simultaneously learning low-rank representations based on a collection of matrices with shared entities. A typical example is the joint modeling of useritem, item-property, and user-feature matrices in a recommender system. The key idea in CMF is that the embeddings are shared across the matrices, which enables transferring information between them. The existing solutions...

A recent challenge in data analysis for science and engineering is that data are often represented in a structured way. In particular, many data mining tasks have to deal with group-structured prior information, where features or data items are organized into groups. In this paper, we develop group sparsity regularization methods for nonnegative matrix factorization (NMF). NMF is an effective d...

Given electroencephalogram (EEG) data measured from several subjects under the same conditions, our goal is to estimate common task-related bases in a linear model that capture intra-subject variations as well as inter-subject variations. Such bases capture the common phenomenon in group data, which is a core of group analysis. In this paper we present a method of nonnegative matrix factorizati...

Introduction. A long-standing problem in group theory is to determine the number of non-isomorphic groups of a given order. The inverse problem–determining the orders for which there are a given number of groups–has received considerably less attention. In this note, we will give a characterization of those positive integers n for which there exist exactly 2 distinct groups of order n (up to is...

A well known unresolved conjecture states that every Cayley graph on a solvable group G has a 1-factorization. We show that if the commutator subgroup of of such a group is an elementary abelian p-group, then every quartic Cayley graph on G has a 1-factorization.

We give a detailed description of the arithmetic Fuchsian group of the Bolza surface and the associated quaternion order. This description enables us to show that the corresponding principal congruence covers satisfy the bound sys(X) > 4 3 log g(X) on the systole, where g is the genus. We also exhibit the Bolza group as a congruence subgroup, and calculate out a few examples of “Bolza twins” (u...

This paper presents a full rank factorization of a 2 × 2 block matrix without any restriction concerning the group inverse. Applying this factorization, we obtain an explicit representation of the group inverse in terms of four individual blocks of the partitioned matrix without certain restriction. We also derive some important coincidence theorems, including the expressions of the group inver...