arXiv subjects

Yasuaki Hiraoka

Publications and source records attributed to Yasuaki Hiraoka.

At least 19 recordsLinked to original sources

arXiv2025

Geometric construction of Kashiwara crystals on multiparameter persistence

We establish a geometric construction of Kashiwara crystals on the irreducible components of the varieties of multiparameter persistence modules. Our approach differs from the seminal work of Kashiwara and Saito, as well as subsequent related works, by emphasizing commutative relations rather than preprojective relations for a given quiver. Furthermore, we provide explicit descriptions of the Kashiwara operators in the fundamental cases of 1- and 2-parameter persistence modules, offering concrete insights into the crystal structure in these settings.

math.RT↗

arXiv2024

Persistence module and Schubert calculus

A multiplication on persistence diagrams is introduced by means of Schubert calculus. The key observation behind this multiplication comes from the fact that the representation space of persistence modules has the structure of the Schubert decomposition of a flag. In particular, isomorphism classes of persistence modules correspond to Schubert cells, thereby the Schubert calculus naturally defines a multiplication on persistence diagrams. The meaning of the multiplication on persistence diagrams is carried over from that on Schubert calculus, i.e., algebro-geometric intersections of varieties of persistence modules.

math.AT↗

arXiv2024

Curse of Dimensionality on Persistence Diagrams

The stability of persistent homology has led to wide applications of the persistence diagram as a trusted topological descriptor in the presence of noise. However, with the increasing demand for high-dimension and low-sample-size data processing in modern science, it is questionable whether persistence diagrams retain their reliability in the presence of high-dimensional noise. This work aims to study the reliability of persistence diagrams in the high-dimension low-sample-size data setting. By analyzing the asymptotic behavior of persistence diagrams for high-dimensional random data, we show that persistence diagrams are no longer reliable descriptors of low-sample-size data under high-dimensional noise perturbations. We refer to this loss of reliability of persistence diagrams in such data settings as the curse of dimensionality on persistence diagrams. Next, we investigate the possibility of using normalized principal component analysis as a method for reducing the dimensionality of the high-dimensional observed data to resolve the curse of dimensionality. We show that this method can mitigate the curse of dimensionality on persistence diagrams. Our results shed some new light on the challenges of processing high-dimension low-sample-size data by persistence diagrams and provide a starting point for future research in this area.

math.ST↗

arXiv2023

Refinement of Interval Approximations for Fully Commutative Quivers

A fundamental challenge in multiparameter persistent homology is the absence of a complete and discrete invariant. To address this issue, we propose an enhanced framework that realizes a holistic understanding of a fully commutative quiver's representation via synthesizing interpretations obtained from intervals. Additionally, it provides a mechanism to tune the balance between approximation resolution and computational complexity. This framework is evaluated on commutative ladders of both finite-type and infinite-type. For the former, we discover an efficient method for the indecomposable decomposition leveraging solely one-parameter persistent homology. For the latter, we introduce a new invariant that reveals persistence in the second parameter by connecting two standard persistence diagrams using interval approximations. We subsequently present several models for constructing commutative ladder filtrations, offering fresh insights into random filtrations and demonstrating our toolkit's effectiveness in analyzing the topology of materials.

math.AT↗

arXiv2023

Topological Node2vec: Enhanced Graph Embedding via Persistent Homology

Node2vec is a graph embedding method that learns a vector representation for each node of a weighted graph while seeking to preserve relative proximity and global structure. Numerical experiments suggest Node2vec struggles to recreate the topology of the input graph. To resolve this we introduce a topological loss term to be added to the training loss of Node2vec which tries to align the persistence diagram (PD) of the resulting embedding as closely as possible to that of the input graph. Following results in computational optimal transport, we carefully adapt entropic regularization to PD metrics, allowing us to measure the discrepancy between PDs in a differentiable way. Our modified loss function can then be minimized through gradient descent to reconstruct both the geometry and the topology of the input graph. We showcase the benefits of this approach using demonstrative synthetic examples.

stat.ML↗

arXiv2022

Large deviation principle for persistence diagrams of random cubical filtrations

The objective of this article is to investigate the asymptotic behavior of the persistence diagrams of a random cubical filtration as the window size tends to infinity. Here, a random cubical filtration is an increasing family of random cubical sets, which are the union of randomly generated higher-dimensional unit cubes with integer coordinates in a Euclidean space. We first prove the strong law of large numbers for the persistence diagrams, inspired by the work of Hiraoka, Shirai, and Trinh, where the persistence diagram of a filtration of random geometric complexes is considered. As opposed to prior papers treating limit theorems for persistence diagrams, the present article aims to further study the large deviation behavior of persistence diagrams. We prove a large deviation principle for the persistence diagrams of a class of random cubical filtrations, and show that the rate function is given as the Fenchel--Legendre transform of the limiting logarithmic moment generating function. In the proof, we also establish a general method of lifting a large deviation principle for the tuples of persistent Betti numbers to persistence diagrams for broad application.

math.PR↗

arXiv2022

Mapping Firms' Locations in Technological Space: A Topological Analysis of Patent Statistics

Where do firms innovate? Mapping their locations and directions in technological space is challenging due to its high dimensionality. We propose a new method to characterize firms' inventive activities via topological data analysis (TDA) that represents high-dimensional data in a shape graph. Applying this method to 333 major firms' patents in 1976--2005 reveals substantial heterogeneity: some firms remain undifferentiated; others develop unique portfolios. Firms with unique trajectories, which we define and measure graph-theoretically as "flares" in the Mapper graph, perform better. This association is statistically and economically significant, and continues to hold after we control for portfolio size, firm survivorship, industry classification, and firm fixed effects. By contrast, existing techniques -- such as principal component analysis (PCA) and Jaffe's (1989) clustering method -- struggle to track these firm-level dynamics.

econ.EM↗

arXiv2021

Persistent Homology Analysis for Materials Research and Persistent Homology Software: HomCloud

This paper introduces persistent homology, which is a powerful tool to characterize the shape of data using the mathematical concept of topology. We explain the fundamental idea of persistent homology from scratch using some examples. We also review some applications of persistent homology to materials researches and software for persistent homology data analysis. HomCloud, one of persistent homology software, is especially featured in this paper.

math.AT↗

arXiv2021

Algebraic stability theorem for derived categories of zigzag persistence modules

We study distances on zigzag persistence modules from the viewpoint of derived categories and Auslander--Reiten quivers. The derived category of ordinary persistence modules is derived equivalent to that of arbitrary zigzag persistence modules, depending on a classical tilting module. Through this derived equivalence, we define and compute distances on the derived category of arbitrary zigzag persistence modules and prove an algebraic stability theorem. We also compare our distance with the distance for purely zigzag persistence modules introduced by Botnan--Lesnick and the sheaf-theoretic convolution distance due to Kashiwara--Schapira.

math.RT↗

arXiv2021

Girth, magnitude homology, and phase transition of diagonality

This paper studies the magnitude homology of graphs focusing mainly on the relationship between its diagonality and the girth. Magnitude and magnitude homology are formulations of the Euler characteristic and the corresponding homology, respectively, for finite metric spaces, first introduced by Leinster and Hepworth-Willerton. Several authors study them restricting to graphs with path metric, and some properties which are similar to the ordinary homology theory have come to light. However, the whole picture of their behavior is still unrevealed, and it is expected that they catch some geometric properties of graphs. In this article, we show that the girth of graphs partially determines magnitude homology, that is, the larger girth a graph has, the more homologies near the diagonal part vanish. Furthermore, applying this result to a typical random graph, we investigate how the diagonality of graphs varies statistically as the edge density increases. In particular, we show that there exists a phase transition phenomenon for the diagonality.

math.AT↗

arXiv2020

Protein folding analysis using features obtained by persistent homology

Understanding the protein folding process is an outstanding issue in biophysics; recent developments in molecular dynamics simulation have provided insights into this phenomenon. However, the large freedom of atomic motion hinders the understanding of this process. In this study, we applied persistent homology, an emerging methods to analyze topological features in a dataset, to reveal protein folding dynamics. We developed a new method to characterize protein structure based on persistent homology and applied this method to molecular dynamics simulations of chignolin. Using principle component analysis or non-negative matrix factorization, our analysis method revealed two stable states and one saddle state, corresponding to the native, misfolded, and transition states, respectively. We also identified an unfolded state with slow dynamics in the reduced space. Our method serves as a promising tool to understand the protein folding process.

physics.comp-ph↗

arXiv2020

Zeta functions of periodic cubical lattices and cyclotomic-like polynomials

Zeta functions of periodic cubical lattices are explicitly derived by computing all the eigenvalues of the adjacency operators and their characteristic polynomials. We introduce cyclotomic-like polynomials to give factorization of the zeta function in terms of them and count the number of orbits of the Galois action associated with each cyclotomic-like polynomial to obtain its further factorization. We also give a necessary and sufficient condition for such a polynomial to be irreducible and discuss its irreducibility from this point of view.

math.CO↗

arXiv2019

Percolation on Homology Generators in Codimension One

This paper introduces a new percolation model motivated from polymer materials. The mathematical model is defined over a random cubical set in the $d$-dimensional space $\mathbb{R}^d$ and focuses on generations and percolations of $(d-1)$-dimensional holes as higher dimensional topological objects. Here, the random cubical set is constructed by the union of unit faces in dimension $d-1$ which appear randomly and independently with probability $p$, and holes are formulated by the homology generators. Under this model, the upper and lower estimates of the critical probability $p_c^{\rm hole}$ of the hole percolation are shown in this paper, implying the existence of the phase transition. The uniqueness of infinite hole cluster is also proven. This result shows that, when $p > p_c^{\rm hole}$, the probability $P_p(x^*\overset{\rm hole}{\longleftrightarrow} y^*)$ that two points in the dual lattice $(\mathbb{Z}^d)^*$ belong to the same hole cluster is uniformly greater than 0.

math.PR↗

arXiv2018

Matrix Method for Persistence Modules on Commutative Ladders of Finite Type

The theory of persistence modules on the commutative ladders $CL_n(τ)$ provides an extension of persistent homology. However, an efficient algorithm to compute the generalized persistence diagrams is still lacking. In this work, we view a persistence module $M$ on $CL_n(τ)$ as a morphism between zigzag modules, which can be expressed in a block matrix form. For the representation finite case ($n\leq 4)$, we provide an algorithm that uses certain permissible row and column operations to compute a normal form of the block matrix. In this form an indecomposable decomposition of $M$, and thus its persistence diagram, is obtained.

math.RT↗

arXiv2018

Limit theorems for random cubical homology

This paper studies random cubical sets in $\mathbb{R}^d$. Given a cubical set $X\subset \mathbb{R}^d$, a random variable $ω_Q\in[0,1]$ is assigned for each elementary cube $Q$ in $X$, and a random cubical set $X(t)$ is defined by the sublevel set of $X$ consisting of elementary cubes with $ω_Q\leq t$ for each $t\in[0,1]$. Under this setting, the main results of this paper show the limit theorems (law of large numbers and central limit theorem) for Betti numbers and lifetime sums of random cubical sets and filtrations. In addition to the limit theorems, the positivity of the limiting Betti numbers is also shown.

math.PR↗

arXiv2017

Persistence Diagrams with Linear Machine Learning Models

Persistence diagrams have been widely recognized as a compact descriptor for characterizing multiscale topological features in data. When many datasets are available, statistical features embedded in those persistence diagrams can be extracted by applying machine learnings. In particular, the ability for explicitly analyzing the inverse in the original data space from those statistical features of persistence diagrams is significantly important for practical applications. In this paper, we propose a unified method for the inverse analysis by combining linear machine learning models with persistence images. The method is applied to point clouds and cubical sets, showing the ability of the statistical inverse analysis and its advantages.

math.AT↗

arXiv2017

Kernel method for persistence diagrams via kernel embedding and weight factor

Topological data analysis is an emerging mathematical concept for characterizing shapes in multi-scale data. In this field, persistence diagrams are widely used as a descriptor of the input data, and can distinguish robust and noisy topological properties. Nowadays, it is highly desired to develop a statistical framework on persistence diagrams to deal with practical data. This paper proposes a kernel method on persistence diagrams. A theoretical contribution of our method is that the proposed kernel allows one to control the effect of persistence, and, if necessary, noisy topological properties can be discounted in data analysis. Furthermore, the method provides a fast approximation technique. The method is applied into several problems including practical data in physics, and the results show the advantage compared to the existing kernel method on persistence diagrams.

stat.ML↗

arXiv2016

Limit theorems for persistence diagrams

The persistent homology of a stationary point process on ${\bf R}^N$ is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence diagram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to infinity and give a sufficient condition for the limiting persistence diagram to have the full support. We also discuss a central limit theorem for persistent Betti numbers.

math.PR↗