arXiv subjects

Michio Yoshiwaki

Publications and source records attributed to Michio Yoshiwaki.

10 recordsLinked to original sources

arXiv2023

On Approximation of $2$D Persistence Modules by Interval-decomposables

In this work, we propose a new invariant for $2$D persistence modules called the compressed multiplicity and show that it generalizes the notions of the dimension vector and the rank invariant. In addition, for a $2$D persistence module $M$, we propose an "interval-decomposable replacement" $δ^{\ast}(M)$ (in the split Grothendieck group of the category of persistence modules), which is expressed by a pair of interval-decomposable modules, that is, its positive and negative parts. We show that $M$ is interval-decomposable if and only if $δ^{\ast}(M)$ is equal to $M$ in the split Grothendieck group. Furthermore, even for modules $M$ not necessarily interval-decomposable, $δ^{\ast}(M)$ preserves the dimension vector and the rank invariant of $M$. In addition, we provide an algorithm to compute $δ^{\ast}(M)$ (a high-level algorithm in the general case, and a detailed algorithm for the size $2\times n$ case).

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arXiv2023

Approximation by interval-decomposables and interval resolutions of persistence modules

In topological data analysis, two-parameter persistence can be studied using the representation theory of the 2d commutative grid, the tensor product of two Dynkin quivers of type A. In a previous work, we defined interval approximations using restrictions to essential vertices of intervals together with Mobius inversion. In this work, we consider homological approximations using interval resolutions, and show that the interval resolution global dimension is finite for finite posets and that it is equal to the maximum of the interval dimensions of the Auslander-Reiten translates of the interval representations. In fact, for the latter equality, we obtained a general formula in the setting of finite-dimensional algebras and resolutions relative to a generator-cogenerator. Furthermore, in the commutative ladder case, by a suitable modification of our interval approximation, we provide a formula linking the two conceptions of approximation.

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arXiv2021

On Interval Decomposability of 2D Persistence Modules

In the persistent homology of filtrations, the indecomposable decompositions provide the persistence diagrams. However, in almost all cases of multidimensional persistence, the classification of all indecomposable modules is known to be a wild problem. One direction is to consider the subclass of interval-decomposable persistence modules, which are direct sums of interval representations. We introduce the definition of pre-interval representations, a more natural algebraic definition, and study the relationships between pre-interval, interval, and indecomposable thin representations. We show that over the ``equioriented'' commutative $2$D grid, these concepts are equivalent. Moreover, we provide a criterion for determining whether or not an $n$D persistence module is interval/pre-interval/thin-decomposable without having to explicitly compute decompositions. For $2$D persistence modules, we provide an algorithm together with a worst-case complexity analysis that uses the total number of intervals in an equioriented commutative $2$D grid. We also propose several heuristics to speed up the computation.

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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.

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arXiv2020

Field choice problem in persistent homology

This paper tackles the problem of coefficient field choice in persistent homology. When we compute a persistence diagram, we need to select a coefficient field before computation. We should understand the dependency of the diagram on the coefficient field to facilitate computation and interpretation of the diagram. We clarify that the dependency is strongly related to the torsion of $\mathbb{Z}$ relative homology in the filtration. We show the sufficient and necessary conditions of the independence of coefficient field choice. An efficient algorithm is proposed to verify the independence. In a numerical experiment with the algorithm, a persistence diagram rarely changes even when the coefficient field changes if we consider a filtration in $\mathbb{R}^3$. The experiment suggests that, in practical terms, changes in the field coefficient will not change persistence diagrams when the data are in $\mathbb{R}^3$.

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arXiv2020

Interleavings and Matchings as Representations

In order to better understand and to compare interleavings between persistence modules, we elaborate on the algebraic structure of interleavings in general settings. In particular, we provide a representation-theoretic framework for interleavings, showing that the category of interleavings under a fixed translation is isomorphic to the representation category of what we call a shoelace. Using our framework, we show that any two interleavings of the same pair of persistence modules are themselves interleaved. Furthermore, in the special case of persistence modules over $\mathbb{Z}$, we show that matchings between barcodes correspond to the interval-decomposable interleavings.

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arXiv2017

Decomposition theory of modules: the case of Kronecker algebra

Let $A$ be a finite-dimensional algebra over an algebraically closed field $\Bbbk$. For any finite-dimensional $A$-module $M$ we give a general formula that computes the indecomposable decomposition of $M$ without decomposing it, for which we use the knowledge of AR-quivers that are already computed in many cases. The proof of the formula here is much simpler than that in a prior literature by Dowbor and Mróz. As an example we apply this formula to the Kronecker algebra $A$ and give an explicit formula to compute the indecomposable decomposition of $M$, which enables us to make a computer program.

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arXiv2016

Relative derived dimensions for cotilting modules

For a Noetherian ring $R$ and a cotilting $R$-module $T$ of injective dimension at least $1$, we prove that the derived dimension of $R$ with respect to the category $\mathcal{X}_T$ is precisely the injective dimension of $T$ by applying Auslander-Buchweitz theory and Ghost Lemma. In particular, when $R$ is a commutative Noetherian local ring with a canonical module $ω_R$ and $\dim R\ge1$, the derived dimension of R with respect to the category of maximal Cohen-Macaulay modules is precisely $\dim R$.

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arXiv2013

Dimensions of triangulated categories with respect to subcategories

This paper introduces the concept of the dimension of a triangulated category with respect to a fixed full subcategory. For the bounded derived category of an abelian category, upper bounds of the dimension with respect to a contravariantly finite subcategory and a resolving subcategory are given. Our methods not only recover some known results on the dimensions of derived categories in the sense of Rouquier, but also apply to various commutative and non-commutative noetherian rings.

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arXiv2010

On selfinjective algebras of stable dimension zero

Let $A$ be a selfinjective algebra over an algebraically closed field. We study the stable dimension of $A$, which is the dimension of the stable module category of $A$ in the sense of Rouquier. Then we prove that $A$ is representation-finite if the stable dimension of $A$ is $0$.

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