A novel update rule of HALS algorithm for nonnegative matrix factorization and Zangwill’s global convergence

Nonnegative Matrix Factorization (NMF) has attracted a great deal of attention as an effective technique for dimensionality reduction of large-scale nonnegative data. Given a nonnegative matrix, NMF aims to obtain two low-rank nonnegative factor matrices by solving a constrained optimization problem. The Hierarchical Alternating Least Squares (HALS) algorithm is a well-known and widely-used iterative method for solving such optimization problems. However, the original update rule used in the HALS algorithm is not well defined. In this paper, we propose a novel well-defined update rule of the HALS algorithm, and prove its global convergence in the sense of Zangwill. Unlike conventional globally-convergent update rules, the proposed one allows variables to take the value of zero and hence can obtain sparse factor matrices. We also present two stopping conditions that guarantee the finite termination of the HALS algorithm. The practical usefulness of the proposed update rule is shown through experiments using real-world datasets

Design of continuous-time recurrent neural networks with piecewise-linear activation function for generation of prescribed sequences of bipolar vectors

A recurrent neural network (RNN) can generate a sequence of patterns as the temporal evolution of the output vector. This paper focuses on a continuous-time RNN model with a piecewise-linear activation function that has neither external inputs nor hidden neurons, and studies the problem of finding the parameters of the model so that it generates a given sequence of bipolar vectors. First, a sufficient condition for the model to generate the desired sequence is derived, which is expressed as a system of linear inequalities in the parameters. Next, three approaches to finding solutions of the system of linear inequalities are proposed: One is formulated as a convex quadratic programming problem and others are linear programming problems. Then, two types of sequences of bipolar vectors that can be generated by the model are presented. Finally, the case where the model generates a periodic sequence of bipolar vectors is considered, and a sufficient condition for the trajectory of the state vector to converge to a limit cycle is provided

HIP KINEMATICS AND MUSCLE ACTIVITY DURING INSIDE SOCCER KICK IN PLAYERS WITH A HISTORY OF GROIN PAIN

The purpose of this study was to clarify the kinematic characteristics of groin pain (GP). In addition, we investigated the correlations between the kinematics of inside soccer kick movement and muscle activities in the lower extremities. Twenty-four male soccer players (control group, 13; GP group, I1 ) were instructed to perform maximum inside kick. Our results showed that the adductor muscle activity was maintained from the back swing to the leg acceleration phase in the in GP group but was decreased from the peak value at the back swing to the leg acceleration phase in the control group. In the leg acceleration phase, the adductor muscle activity was significantly higher in the GP group than in the control group. The GP group showed faster adduction/abduction velocity of the kicking leg in all kicking phases than the control group

Identifying Context-Specific Categories for Visualizing Livability of Cities—a Case Study of Malmö

Livability is a concept being applied to cities, even though it is vague. Worldwide, there are several livable city ranking schemes in use, which compare the livability of cities by making use of standardized indicator sets. The research presented here recognizes, as a point of departure, that each city is unique, implying that comparisons of cities by standardized categories only does not adequately reflect the reality of each city. A qualitative approach to identify context-specific categories of livability is proposed and employed to the case of Malmo ̈ in Sweden. Through interviews, nine context-specific categories were identified and visualized. The findings of the study demonstrate that a qualitative approach enables a more in-depth description of livability categories because it can capture and illustrate relationships among the categories. An explicit awareness of such relationships may provide a more holistic perspective to city officials and planners as they aim to improve the livability of their cities. The study concludes that a qualitative approach in identifying context-specific categories can complement existing assessment schemes and allow a better grasp of livability challenges to cities