Related papers: Optimizing persistent homology based functions
A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter…
This article aims to study the topological invariant properties encoded in node graph representational embeddings by utilizing tools available in persistent homology. Specifically, given a node embedding representation algorithm, we…
Persistent homology is a central methodology in topological data analysis that has been successfully implemented in many fields and is becoming increasingly popular and relevant. The output of persistent homology is a persistence diagram --…
This paper presents an algorithmic framework for solving unconstrained stochastic optimization problems using only stochastic function evaluations. We employ central finite-difference based gradient estimation methods to approximate the…
We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done. The persistent homology algorithm is usually viewed as a procedure which starts with a filtered…
Persistent homology is a popular tool in Topological Data Analysis. It provides numerical characteristics of data sets which reflect global geometric properties. In order to be useful in practice, for example for feature generation in…
We study the grokking phenomenon through the lens of topology. Using persistent homology on point clouds derived from the embedding matrices of a range of models trained on modular arithmetic with varying primes, we identify a clear and…
We propose a novel approach for preserving topological structures of the input space in latent representations of autoencoders. Using persistent homology, a technique from topological data analysis, we calculate topological signatures of…
We extend the persistence algorithm, viewed as an algorithm computing the homology of a complex of free persistence or graded modules, to complexes of modules that are not free. We replace persistence modules by their presentations and…
Artificial neural networks can learn complex, salient data features to achieve a given task. On the opposite end of the spectrum, mathematically grounded methods such as topological data analysis allow users to design analysis pipelines…
Persistence diagrams, the most common descriptors of Topological Data Analysis, encode topological properties of data and have already proved pivotal in many different applications of data science. However, since the (metric) space of…
The proximal gradient descent method, well-known for composite optimization, can be completely described by the concept of proximal gradient mapping. In this paper, we highlight our previous two discoveries of proximal gradient…
Recently a new feature representation and data analysis methodology based on a topological tool called persistent homology (and its corresponding persistence diagram summary) has started to attract momentum. A series of methods have been…
The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space.…
In this work, we present a generalization of extended persistent homology to filtrations of graded sub-groups by defining relative homology in this setting. Our work provides a more comprehensive and flexible approach to get an algebraic…
Recent studies have shown that many nonconvex machine learning problems satisfy a generalized-smooth condition that extends beyond traditional smooth nonconvex optimization. However, the existing algorithms are not fully adapted to such…
Persistent homology is a technique recently developed in algebraic and computational topology well-suited to analysing structure in complex, high-dimensional data. In this paper, we exposit the theory of persistent homology from first…
A stochastic-gradient-based interior-point algorithm for minimizing a continuously differentiable objective function (that may be nonconvex) subject to bound constraints is presented, analyzed, and demonstrated through experimental results.…
A suitable feature representation that can both preserve the data intrinsic information and reduce data complexity and dimensionality is key to the performance of machine learning models. Deeply rooted in algebraic topology, persistent…
We propose an approach to learning with graph-structured data in the problem domain of graph classification. In particular, we present a novel type of readout operation to aggregate node features into a graph-level representation. To this…