Related papers: The Shift-Dimension of Multipersistence Modules
In the covariate shift learning scenario, the training and test covariate distributions differ, so that a predictor's average loss over the training and test distributions also differ. In this work, we explore the potential of extreme…
In 2009, Chazal et al. introduced $\epsilon$-interleavings of persistence modules. $\epsilon$-interleavings induce a pseudometric $d_I$ on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of…
In this paper, we study relative metric regularity of set-valued mappings with emphasis on directional metric regularity. We establish characterizations of relative metric regularity without assuming the completeness of the image spaces, by…
Deep learning models have seen significant successes in numerous applications, but their inner workings remain elusive. The purpose of this work is to quantify the learning process of deep neural networks through the lens of a novel…
We extend Auslander and Buchsbaum's Euler characteristic from the category of finitely generated modules of finite projective dimension to the category of modules of finite G-dimension using Avramov and Martsinkovsky's notion of relative…
We prove the basic properties of determinantal semi-invariants for presentation spaces over any finite dimensional hereditary algebra over any field. These include the virtual generic decomposition theorem, stability theorem and the…
The stability theorem for persistent homology is a central result in topological data analysis. While the original formulation of the result concerns the persistence barcodes of $\mathbb{R}$-valued functions, the result was later cast in a…
In current research, machine and deep learning solutions for the classification of temporal data are shifting from single-channel datasets (univariate) to problems with multiple channels of information (multivariate). The majority of these…
For dynamical systems that switch between different modes of operation, parameter variation can cause periodic solutions to lose or acquire new switching events. When this causes the eigenvalues (stability multipliers) associated with the…
Persistence modules serve as the algebraic foundation for topological data analysis, typically studied as representations of posets over a field. This article extends the structural and decomposition theory of persistence modules to the…
We extend some recent results on the Hausdorff convergence of level-sets for total variation regularized linear inverse problems. Dimensions higher than two and measurements in Banach spaces are considered. We investigate the relation…
The work is devoted to the variety of $2$-dimensional algebras over an algebraically closed field. Firstly, we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of principal algebra series in the…
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 and resolutions relative to interval-decomposable modules)…
In this paper we introduce a new invariant (the distant degree) for difference field extensions of finite transcendence degree, and we explore some of its properties. We also discuss a generalisation of this invariant and of the limit…
We prove a result that can be applied to determine the finite-dimensional simple Poisson modules over a Poisson algebra and apply it to numerous examples. In the discussion of the examples, the emphasis is on the correspondence with the…
We study the behavior of multidegrees in families and the existence of numerical criteria to detect integral dependence. We show that mixed multiplicities of modules are upper semicontinuous functions when taking fibers and that projective…
Moduli spaces of hyperbolic surfaces may be endowed with a symplectic structure via the Weil-Petersson form. Mirzakhani proved that Weil-Petersson volumes exhibit polynomial behaviour and that their coefficients store intersection numbers…
We analyze the stability of the moduli at the quantum level in an open-string model realizing the $\mathcal{N}=2\to \mathcal{N}=0$ spontaneous breaking of supersymmetry in four-dimensional Minkowski spacetime. In the region of moduli space…
This paper attempts to study the continuity of the Hurwitz metric in arbitrary proper subdomains of the complex plane and to introduce a new invariant metric bi-Lipschitz equivalent to the Hurwitz metric in hyperbolic domains. The lower…
Typical dualities in arbitrary dimensions are understood through a Hilbert-space extension method. By these results, we rigorously dualize the quantum ingappabilities to discrete height model in one dimension which is inaccessible by…