This paper initiates a systematic approach to handling mathematical data structured as (truncated) infinite $q-series, or equivalently, infinite series of integers, and demonstrates that neural networks can reliably extract essential topological information, such as homology class and underlying graph structure, directly from the $q$-series coefficients.
Abstract
Machine learning and data analysis techniques have recently emerged as powerful tools for identifying patterns and formulating conjectures in mathematical research, most notably in the field of low-dimensional topology. In this paper, we initiate a systematic approach to handling mathematical data structured as (truncated) infinite $q$-series, or equivalently, infinite series of integers. To apply this data analysis pipeline, we construct a comprehensive dataset of $\widehat{Z}$-invariants (homological blocks) for plumbed 3-manifolds. We demonstrate that neural networks can reliably extract essential topological information, such as homology class and underlying graph structure, directly from the $q$-series coefficients. A central feature of our methodology is a focus on interpretability; by contrasting local gradient sensitivity with global feature relevance, we reveal that the networks learn to bypass complex topological rules in favor of specific spectral and geometric proxies. Finally, we apply this pipeline to probe homology cobordism, discovering a high-accuracy predictive relationship between the $\widehat{Z}$-invariant exponents and the Heegaard Floer $d$-invariant (correction term). These results suggest that $\widehat{Z}$-invariants capture subtle geometric information regarding cobordism equivalences, warranting a new direction for the study of quantum invariants.
In order to gain an understanding of the underlying structure and relationships of complex and high-dimensional information, topological approaches provide effective tools. the use of topological techniques in data analysis, with an emphasis on how they relate to challenges involving machine learning. We start by introducing fundamental ideas from algebraic topology, including homology, persistent homology, and simplicial complexes. We next go into how these ideas can be used to describe and analyze data. Several machine learning applications of topological techniques, such as anomaly detection, clustering, classification, and dimensionality reduction. Researchers can identify significant aspects of the data that are difficult to identify using conventional techniques by utilizing topological descriptors like persistent homology. We use realworld examples to explain these ideas and show how well they work to reveal hidden patterns and structures in a variety of datasets. Keywords: Algebraic topology, topological techniques, data analysis, machine learning, and simplicial complexes.
Dr.Manisha Rajput· International Journal of Eng...· 0 citations
Finite groups are rigid algebraic objects, whose Cayley graphs expose a rich network geometry through which group-theoretic structure can be measured, compared, and learned. In this paper, a dataset of $131{,}406$ Cayley graphs is constructed, covering all groups of order at most $767$ except order $512$, recording exact algebraic labels for group properties together with a broad collection of graph, cycle, distance, and spectral statistics. This census aims to provide novel benchmarks for studying how finite-group properties are reflected in Cayley graph observables. It also yields new enumerative contributions: alongside recovering known OEIS sequences for standard group classes, new sequences for monolithic groups and for groups generated by at most three, four, and five elements are contributed to the OEIS. The accompanying network analysis identifies several empirical regularities and formulates testable conjectures, including relationships involving square clustering, Cayley graph diameter, average graph disorder, and spectral eigengaps of nilpotent groups. Finally, a comparison between classical models, an MLP, and graph neural network architectures is performed for predicting algebraic group properties directly from Cayley graph data. The results show that engineered graph statistics are highly informative, while GNNs, especially GIN and in some fixed-order settings GCN, can recover substantial structural signal directly from the graph. Such that graph-aware architectures show phases of optimality on these group-theoretic graph representations.
Rashid Barket, Enrico Grimaldi, Yacoub Hendi et al.· 0 citations
Algebraic statistics characterizes statistical models through polynomial constraints, but it has mainly been used for analytically specified model classes. This paper studies the inverse problem: identifying probabilistic structure from vanishing binomials observed in empirical probability tensors. We treat the vanishing binomials of a toric model as its algebraic signature, and turn the ideal-variety correspondence of algebraic statistics into an operational procedure for structural learning that identifies a model by signature matching without parameter estimation. By restricting attention to a computationally tractable class of configuration matrices, which we call {\it the Kronecker-stack class}, we make these signatures explicitly enumerable. Within this class we define minimum invariant constraint (MIC) as the atomic unit characterizing each signature and generalizing the notion of independence. We tested this approach employing MICs on synthetic data as well as on corpus-scale real language data. The results suggested the utility of the method, revealing that the identified rank-one structures correspond to interpretable sets of words. These results open up a new avenue for applying algebraic statistics to computational linguistics.
Akihiro Maeda, Shohei Hidaka, S. Aoki· 0 citations
Topological data analysis (TDA) uses topological techniques to extract meaningful shape-based information from complex datasets. Clustering is a central problem in data analysis, and there has been considerable recent interest in understanding how TDA can inform it. Existing approaches either cluster persistence diagrams directly under Wasserstein-type distances, which is computationally expensive, or use vector representations of diagrams. We propose a kernel $k$-means algorithm built on a convex combination of sliced Wasserstein (SW) kernels, one for each homology under consideration. Unlike other vector representations of persistence diagrams, the SW kernel is both stable and discriminative with respect to the $1$-Wasserstein distance. The method outperforms the baselines on two benchmark datasets and remains competitive on a third synthetic dataset, while being computationally efficient. It also outperforms both a single SW kernel on the union of all homology groups and an SW kernel computed directly on the point clouds. The convex combination assigns an interpretable weight to each $q$-homology kernel. We further validate that the weights identify the discriminating homology.
Vikram T. Aithal, Ajit Kumar, Ambika Sharma· 0 citations
Kolmogorov-Arnold Networks (KANs) replace fixed activations in deep architectures with learnable univariate edge functions, making the choice of edge parametrisation central. Existing variants rely on fixed bases such as splines, polynomials, or Fourier features, which impose a function-space geometry before data are observed. We introduce geometry-constrained KANs, a family of edge activations derived from Banach duality maps in which the geometry itself is learned through a scalar exponent $p>1$ per edge. This exponent controls the qualitative response: sub-Euclidean values produce sharp, threshold-like behaviour reminiscent of the $\ell_1$ (LASSO) geometry, $p = 2$ recovers the linear regime, and larger values produce flatter responses near the origin. Across 50 symbolic-regression targets ($40$ from the AI Feynman benchmark plus $10$ synthetic stress tests), geometry-constrained KANs match or beat every fixed-basis baseline on median NRMSE (Banach-KAN $0.030$, tying Chebyshev and improving on splines); on average rank Banach-KAN is best on the $18$-equation core ($2.00$) and statistically tied with the strongest spline on the full benchmark ($2.32$ vs. $2.34$). The clearest gains appear under measurement noise: as $\sigma$ grows from $0$ to $1$, $\ell^p$-KAN degrades only $3.7\times$ -- below even a cross-validated spline ($\approx 11\times$) -- while an unregularised spline degrades $21.6\times$; Banach-KAN degrades $8.8\times$, comparable to a tuned spline but far more stable than the unregularised one. Banach-KAN also takes the most per-equation wins in the small-sample regime, with fixed-basis models catching up only as the training set grows. Learned exponents provide an interpretable, relative signal: at a fixed initialisation they reveal a consistent, target-dependent geometric ordering across equation families and input dimensions.
Over the last decade, neural networks have been applied to an increasingly diverse range of applications, including data with rich geometric, topological, or symmetry-related structure. As a result, researchers have increasingly drawn inspiration from topology, algebra, and geometry. Despite this rich algorithmic development, the supporting software ecosystem remains fragmented. Many important methods exist only as research prototypes in unmaintained repositories. We address this by introducing Topology, Algebra, and Geometry Torch (TAGTorch), an open-source, PyTorch-based library that unifies tools inspired by topology, algebra, and geometry, including data-preprocessing methods, architectures, training techniques, and model analysis tools. We describe the design philosophy of TAGTorch and then discuss its current architecture and capabilities, highlighting areas where it can fill gaps in the current software ecosystem. We conclude with a discussion of our future development priorities for the library.
Brendan Kennedy, T. Emerson, G. Roek et al.· 0 citations