arXiv subjects

Ryoya Fukasaku

Publications and source records attributed to Ryoya Fukasaku.

5 recordsLinked to original sources

arXiv2026

A curvature-based criterion for harmonic circadian waveforms

Experimental and theoretical studies of circadian rhythms have focused largely on the period, on mutants that alter it and on phase shifts, and this focus has driven the identification of clock genes and clarified how clocks entrain to light--dark cycles. The waveform of the oscillation itself, by contrast, has attracted little attention as an indicator of the properties of the underlying oscillator. To assess the waveform directly, we focus on whether the trajectory in the plane spanned by a variable and its time derivative possesses an inflection point, and we define an oscillation to be harmonic when no inflection point is present. Bioluminescence recordings from cyanobacteria and from the mammalian SCN were harmonic in this sense, as were most of the core clock components in mathematical models of the circadian clock. Numerical analysis of the Goodwin model, a minimal representation of the core circadian oscillator, yielded harmonic oscillation throughout. We confirmed this numerical trend semi-analytically using a piecewise-linearized Goodwin model. Because it evaluates the properties of a waveform without assuming a model structure, the approach we propose offers a new perspective on the waveform analysis of biological rhythms in general, not only circadian ones.

q-bio.QM↗

arXiv2025

Algebraic Approach to Ridge-Regularized Mean Squared Error Minimization in Minimal ReLU Neural Network

This paper investigates a perceptron, a simple neural network model, with ReLU activation and a ridge-regularized mean squared error (RR-MSE). Our approach leverages the fact that the RR-MSE for ReLU perceptron is piecewise polynomial, enabling a systematic analysis using tools from computational algebra. In particular, we develop a Divide-Enumerate-Merge strategy that exhaustively enumerates all local minima of the RR-MSE. By virtue of the algebraic formulation, our approach can identify not only the typical zero-dimensional minima (i.e., isolated points) obtained by numerical optimization, but also higher-dimensional minima (i.e., connected sets such as curves, surfaces, or hypersurfaces). Although computational algebraic methods are computationally very intensive for perceptrons of practical size, as a proof of concept, we apply the proposed approach in practice to minimal perceptrons with a few hidden units.

stat.ML↗

arXiv2025

Algebraic Approach for Orthomax Rotations

In exploratory factor analysis, rotation techniques are employed to derive interpretable factor loading matrices. Factor rotations deal with equality-constrained optimization problems aimed at determining a loading matrix based on measure of simplicity, such as ``perfect simple structure'' and ``Thurstone simple structure.'' Numerous criteria have been proposed, since the concept of simple structure is fundamentally ambiguous and involves multiple distinct aspects. However, most rotation criteria may fail to consistently yield a simple structure that is optimal for analytical purposes, primarily due to two challenges. First, existing optimization techniques, including the gradient projection descent method, exhibit strong dependence on initial values and frequently become trapped in suboptimal local optima. Second, multifaceted nature of simple structure complicates the ability of any single criterion to ensure interpretability across all aspects. In certain cases, even when a global optimum is achieved, other rotations may exhibit simpler structures in specific aspects. To address these issues, obtaining all equality-constrained stationary points -- including both global and local optima -- is advantageous. Fortunately, many rotation criteria are expressed as algebraic functions, and the constraints in the optimization problems in factor rotations are formulated as algebraic equations. Therefore, we can employ computational algebra techniques that utilize operations within polynomial rings to derive exact all equality-constrained stationary points. Unlike existing optimization methods, the computational algebraic approach can determine global optima and all stationary points, independent of initial values. We conduct Monte Carlo simulations to examine the properties of the orthomax rotation criteria, which generalizes various orthogonal rotation methods.

math.ST↗

arXiv2024

Algebraic approach to maximum likelihood factor analysis

In exploratory factor analysis, model parameters are usually estimated by maximum likelihood method. The maximum likelihood estimate is obtained by solving a complicated multivariate algebraic equation. Since the solution to the equation is usually intractable, it is typically computed with continuous optimization methods, such as Newton-Raphson methods. With this procedure, however, the solution is inevitably dependent on the estimation algorithm and initial value since the log-likelihood function is highly non-concave. Particularly, the estimates of unique variances can result in zero or negative, referred to as improper solutions; in this case, the maximum likelihood estimate can be severely unstable. To delve into the issue of the instability of the maximum likelihood estimate, we compute exact solutions to the multivariate algebraic equation by using algebraic computations. We provide a computationally efficient algorithm based on the algebraic computations specifically optimized for maximum likelihood factor analysis. To be specific, Gröebner basis and cylindrical decomposition are employed, powerful tools for solving the multivariate algebraic equation. Our proposed procedure produces all exact solutions to the algebraic equation; therefore, these solutions are independent of the initial value and estimation algorithm. We conduct Monte Carlo simulations to investigate the characteristics of the maximum likelihood solutions.

math.ST↗

arXiv2019

An algebraic reduction of Hedetniemi's conjecture

For a graph $G$, let $χ(G)$ denote the chromatic number. In graph theory, the following famous conjecture posed by Hedetniemi has been studied: For two graphs $G$ and $H$, $χ(G\times H)=\min\{χ(G),χ(H)\}$, where $G \times H$ is the tensor product of $G$ and $H$. In this paper, we give a reduction of Hedetniemi's conjecture to an inclusion relation problem on ideals of polynomial rings, and we demonstrate computational experiments for partial solutions of Hedetniemi's conjecture along such a strategy using Gröbner basis.

math.CO↗