419 citations · 451 across the 16 of their papers we have counts for
17 papers
On Angle-optimization and Simplification of Degree-1 Homology Representatives
Emerson G. Escolar, Yuta Shimada
In topological data analysis, in particular persistent homology analysis, extracting "optimal" representatives for homology classes is crucial for identifying geometric regions of…
Barcoding Invariants and Their Comparison
Emerson G. Escolar, Woojin Kim
The persistence barcode, which can be obtained from the interval decomposition of a persistence module, plays a pivotal role in applications of persistent homology. For multi-param…
A topological analysis of the space of recipes
Emerson G. Escolar, Yuta Shimada, Masahiro Yuasa
In recent years, the use of data-driven methods has provided insights into underlying patterns and principles behind culinary recipes. In this exploratory work, we introduce the us…
Bipath Persistence
Toshitaka Aoki, Emerson G. Escolar, Shunsuke Tada
In persistent homology analysis, interval modules play a central role in describing the birth and death of topological features across a filtration. In this work, we extend this se…
Summand-injectivity of interval covers and monotonicity of interval resolution global dimensions
Toshitaka Aoki, Emerson G. Escolar, Shunsuke Tada
Recently, there is growing interest in the use of relative homology algebra to develop invariants using interval covers and interval resolutions (i.e., right minimal approximations…
Approximation by interval-decomposables and interval resolutions of persistence modules
Hideto Asashiba, Emerson G. Escolar, Ken Nakashima +1
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…
The Whole in the Parts: Putting D Persistence Modules Inside Indecomposable D Ones
Mickaël Buchet, Emerson G. Escolar
Multidimensional persistence has been proposed to study the persistence of topological features in data indexed by multiple parameters. In this work, we further explore its algebra…
Interleavings and Matchings as Representations
Emerson G. Escolar, Killian Meehan, Michio Yoshiwaki
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,…
On Approximation of D Persistence Modules by Interval-decomposables
Hideto Asashiba, Emerson G. Escolar, Ken Nakashima +1
In this work, we propose a new invariant for D persistence modules called the compressed multiplicity and show that it generalizes the notions of the dimension vector and the ra…
Mapping Firms' Locations in Technological Space: A Topological Analysis of Patent Statistics
Emerson G. Escolar, Yasuaki Hiraoka, Mitsuru Igami +1
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'…
Every 1D Persistence Module is a Restriction of Some Indecomposable 2D Persistence Module
Mickaël Buchet, Emerson G. Escolar
A recent work by Lesnick and Wright proposed a visualisation of D persistence modules by using their restrictions onto lines, giving a family of D persistence modules. We giv…
On Interval Decomposability of 2D Persistence Modules
Hideto Asashiba, Mickaël Buchet, Emerson G. Escolar +2
In the persistent homology of filtrations, the indecomposable decompositions provide the persistence diagrams. However, in almost all cases of multidimensional persistence, the cla…
Realizations of Indecomposable Persistence Modules of Arbitrarily Large Dimension
Mickaël Buchet, Emerson G. Escolar
While persistent homology has taken strides towards becoming a wide-spread tool for data analysis, multidimensional persistence has proven more difficult to apply. One reason is th…
Matrix Method for Persistence Modules on Commutative Ladders of Finite Type
Hideto Asashiba, Emerson G. Escolar, Yasuaki Hiraoka +1
The theory of persistence modules on the commutative ladders provides an extension of persistent homology. However, an efficient algorithm to compute the generalized pers…
Persistent Homology and Many-Body Atomic Structure for Medium-Range Order in the Glass
Takenobu Nakamura, Yasuaki Hiraoka, Akihiko Hirata +2
Characterization of medium-range order in amorphous materials and its relation to short-range order is discussed. A new topological approach is presented here to extract a hierarch…
Hierarchical structures of amorphous solids characterized by persistent homology
Yasuaki Hiraoka, Takenobu Nakamura, Akihiko Hirata +3
This article proposes a topological method that extracts hierarchical structures of various amorphous solids. The method is based on the persistence diagram (PD), a mathematical to…
Persistence Modules on Commutative Ladders of Finite Type
Emerson G. Escolar, Yasuaki Hiraoka
We study persistence modules defined on commutative ladders. This class of persistence modules frequently appears in topological data analysis, and the theory and algorithm propose…