Searcharxiv⌕ Search

arXiv subjects

Igor Mineyev

Publications and source records attributed to Igor Mineyev.

6 recordsLinked to original sources
arXiv2025

On zero-divisors and units in group rings of torsion-free CAT$(0)$ groups

This paper addresses two of Kaplansky's conjectures concerning group rings $K[G]$, where $K$ is a field and $G$ is a torsion-free group: the zero-divisor conjecture, which asserts that $K[G]$ has no non-trivial zero-divisors, and the unit conjecture, which asserts that $K[G]$ has no non-trivial units. While the zero-divisor conjecture still remains open, the unit conjecture was disproven by Gardam in 2021. The search for more counterexamples remains an open problem. Let $m$ and $n$ be the cardinality of support of two non-trivial elements $\alpha, \beta \in \mathbb{F}_2[G]$, respectively. We address these conjectures by introducing a process called \text{left alignment} and recursively constructing the taikos of size $(m,n)$ which would yield counterexamples to both conjectures over the field $\mathbb{F}_2$ if they satisfy conditions $\mathsf{T}_1-\mathsf{T}_4$ given in \cite{Mineyev2024}. We also present a computer-search method that can be utilized to search for counterexamples of a certain geometry by significantly pruning the search space. We prove that a class of CAT(0) groups with certain geometry cannot be counterexamples to these conjectures. Moreover, we prove that for $ 1\le m \le 5$ and $n$ any positive integer, there are no counterexamples to the conjectures such that the associated oriented product structures are of type $(m,n)$. With the aid of computer, we prove that, in fact, there are no such counterexamples of the length combination $(m,n)$ where $1\le m \le 13$ and $1\le n \le 13$.

math.GR↗
arXiv2022

Learning from One and Only One Shot

Humans can generalize from only a few examples and from little pretraining on similar tasks. Yet, machine learning (ML) typically requires large data to learn or pre-learn to transfer. Motivated by nativism and artificial general intelligence, we directly model human-innate priors in abstract visual tasks such as character and doodle recognition. This yields a white-box model that learns general-appearance similarity by mimicking how humans naturally ``distort'' an object at first sight. Using just nearest-neighbor classification on this cognitively-inspired similarity space, we achieve human-level recognition with only $1$--$10$ examples per class and no pretraining. This differs from few-shot learning that uses massive pretraining. In the tiny-data regime of MNIST, EMNIST, Omniglot, and QuickDraw benchmarks, we outperform both modern neural networks and classical ML. For unsupervised learning, by learning the non-Euclidean, general-appearance similarity space in a $k$-means style, we achieve multifarious visual realizations of abstract concepts by generating human-intuitive archetypes as cluster centroids.

cs.CV↗
arXiv2020

Orbit Computation for Atomically Generated Subgroups of Isometries of $\mathbb{Z}^n$

Isometries are ubiquitous in nature; isometries of discrete (quantized) objects---abstracted as the group of isometries of $\mathbb{Z}^n$ denoted by $\mathsf{ISO}(\mathbb{Z}^n)$---are important concepts in the computational world. In this paper, we compute various isometric invariances which mathematically are orbit-computation problems under various isometry-subgroup actions $H \curvearrowright \mathbb{Z}^n, H \leq \mathsf{ISO}(\mathbb{Z}^n)$. One computational challenge here is about the \emph{infinite}: in general, we can have an infinite subgroup acting on $\mathbb{Z}^n$, resulting in possibly an infinite number of orbits of possibly infinite size. In practice, we restrict the set of orbits (a partition of $\mathbb{Z}^n$) to a finite subset $Z \subseteq \mathbb{Z}^n$ (a partition of $Z$), where $Z$ is specified a priori by an application domain or a data set. Our main contribution is an efficient algorithm to solve this \emph{restricted} orbit-computation problem in the special case of \emph{atomically generated subgroups}---a new notion partially motivated from interpretable AI. The atomic property is key to preserving the \emph{semidirect-product structure}---the core structure we leverage to make our algorithm outperform generic approaches. Besides algorithmic merit, our approach enables \emph{parallel-computing} implementations in many subroutines, which can further benefit from hardware boosts. Moreover, our algorithm works efficiently for \emph{any} finite subset ($Z$) regardless of the shape (continuous/discrete, (non)convex) or location; so it is application-independent.

math.GR↗
arXiv2019

A Group-Theoretic Approach to Computational Abstraction: Symmetry-Driven Hierarchical Clustering

Abstraction plays a key role in concept learning and knowledge discovery; this paper is concerned with computational abstraction. In particular, we study the nature of abstraction through a group-theoretic approach, formalizing it as symmetry-driven---as opposed to data-driven---hierarchical clustering. Thus, the resulting clustering framework is data-free, feature-free, similarity-free, and globally hierarchical---the four key features that distinguish it from common data clustering models such as $k$-means. Beyond a theoretical foundation for abstraction, we also present a top-down and a bottom-up approach to establish an algorithmic foundation for practical abstraction-generating methods. Lastly, via both a theoretical explanation and a real-world application, we illustrate that further coupling of our abstraction framework with statistics realizes Shannon's information lattice and even further, brings learning into the picture. This not only presents one use case of our proposed computational abstraction, but also gives a first step towards a principled and cognitive way of automatic concept learning and knowledge discovery.

cs.LG↗
arXiv2005

Flows and joins of metric spaces

We introduce the functor * which assigns to every metric space X its symmetric join *X. As a set, *X is a union of intervals connecting ordered pairs of points in X. Topologically, *X is a natural quotient of the usual join of X with itself. We define an Isom(X)-invariant metric d* on *X. Classical concepts known for H^n and negatively curved manifolds are defined in a precise way for any hyperbolic complex X, for example for a Cayley graph of a Gromov hyperbolic group. We define a double difference, a cross-ratio and horofunctions in the compactification X-bar= X union bdry X. They are continuous, Isom(X)-invariant, and satisfy sharp identities. We characterize the translation length of a hyperbolic isometry g in Isom(X). For any hyperbolic complex X, the symmetric join *X-bar of X-bar and the (generalized) metric d* on it are defined. The geodesic flow space F(X) arises as a part of *X-bar. (F(X),d*) is an analogue of (the total space of) the unit tangent bundle on a simply connected negatively curved manifold. This flow space is defined for any hyperbolic complex X and has sharp properties. We also give a construction of the asymmetric join X*Y of two metric spaces. These concepts are canonical, i.e. functorial in X, and involve no `quasi'-language. Applications and relation to the Borel conjecture and others are discussed.

math.MG↗