Related papers: A Framework for Differential Calculus on Persisten…
In probabilistic coherence spaces, a denotational model of probabilistic functional languages, morphisms are analytic and therefore smooth. We explore two related applications of the corresponding derivatives. First we show how derivatives…
This paper deals with differentiable dynamical models congruent with neural process theories that cast brain function as the hierarchical refinement of an internal generative model explaining observations. Our work extends existing…
The differentiable programming paradigm is a cornerstone of modern scientific computing. It refers to numerical methods for computing the gradient of a numerical model's output. Many scientific models are based on differential equations,…
Persistence modules and barcodes are used in symplectic topology to define various invariants of Hamiltonian diffeomorphisms, however numerical methods for computing these barcodes are not yet well developed. In this paper we define one…
A deformed differential calculus is developed based on an associative star-product. In two dimensions the Hamiltonian vector fields model the algebra of pseudo-differential operator, as used in the theory of integrable systems. Thus one…
The intrinsic connection between lattice theory and topology is fairly well established, For instance, the collection of open subsets of a topological subspace always forms a distributive lattice. Persistent homology has been one of the…
We study sublevel set and superlevel set persistent homology on discrete functions through the perspective of finite ordered sets of both linearly ordered and cyclically ordered domains. Finite ordered sets also serve as the codomain of our…
We investigate the parameterization of deep neural networks that by design satisfy the continuity equation, a fundamental conservation law. This is enabled by the observation that any solution of the continuity equation can be represented…
Extended persistence is a technique from topological data analysis to obtain global multiscale topological information from a graph. This includes information about connected components and cycles that are captured by the so-called…
We develop a system-theoretic framework for the structured analysis of distributed optimization algorithms with decomposable cost functions. We model such algorithms as a network of interacting dynamical systems and derive tests for…
Optimization of beamlines and lattices is a common problem in accelerator physics, which is usually solved with semi-analytical methods and numerical optimization routines. However, these are usually of the gradient-free or…
A graph's spectral wavelet signature determines a filtration, and consequently an associated set of extended persistence diagrams. We propose a framework that optimises the choice of wavelet for a dataset of graphs, such that their…
We show that an embedding in Euclidean space based on tropical geometry generates stable sufficient statistics for barcodes. In topological data analysis, barcodes are multiscale summaries of algebraic topological characteristics that…
Given a filtration of simplicial complexes, one usually applies persistent homology and summarizes the results in barcodes. Then, in order to extract statistical information from these barcodes, one needs to compute statistical indicators…
We present a row reduction algorithm to compute the barcode decomposition of persistence modules. This algorithm dualises the standard persistence one and clarifies the symmetry between clear and compress optimisations.
Dynamic Bayesian networks (DBNs) are a widely used framework for modeling systems whose probabilistic structure evolves over time. Standard inference methods focus on local conditional distributions and can miss larger-scale patterns in how…
We study the ring of algebraic functions on the space of persistence barcodes, with applications to pattern recognition.
Persistent homology barcodes and diagrams are a cornerstone of topological data analysis that capture the "shape" of a wide range of complex data structures, such as point clouds, networks, and functions. However, their use in statistical…
It has been recognized recently that fractional calculus is useful for handling scaling structures and processes. We begin this survey by pointing out the relevance of the subject to physical situations. Then the essential definitions and…
We have established a coherent framework for applying variational methods to partial differential equations on hypergraphs, which includes the propositions of calculus and function spaces on hypergraphs. Several results related to the…