arXiv research

The Stone-Weierstrass theorem aids in solving inverse problems on specific manifolds.

The abstract discusses parallels between Galois theory and Stone-Weierstrass theorem in various fields.

Complex of cuts reveals full automorphism group for certain Stone spaces.

Consistency of k-NN rule proven in sigma-finite dimensional metric spaces.

New bound on neural nets complexity for approximating functions.

Functional input neural networks approximate continuous functions on weighted spaces.

The paper proves deep neural networks with analytic activation can approximate any function.

Improves consistency of weighted averaging estimators in active learning.

The $k$-nearest neighbour ($k$-NN) classifier is one of the oldest and most important supervised learning algorithms for classifying datasets. Traditionally the Euclidean norm is used as the distance for the $k$-NN classifier. In this thesis we investigate the use of alternative distances for the $k$-NN classifier. We …

The paper extends Stone duality to topological convexity spaces.

Graph Neural Networks (GNN) come in many flavors, but should always be either invariant (permutation of the nodes of the input graph does not affect the output) or equivariant (permutation of the input permutes the output). In this paper, we consider a specific class of invariant and equivariant networks, for which we …

Classifies homeomorphism groups of countable Stone spaces up to coarse equivalence.

Study of profinite quandles with constructions and characterizations.

UNIPoint universally approximates point process intensities.

In this paper, we prove that the mean curvature blows up at the same rate as the second fundamental form at the first singular time $T$ of any compact, Type I mean curvature flow. For the mean curvature flow of surfaces, we obtain similar result provided that the Gaussian density is less than two. Our proofs are based …

Generalizes Gelfand's spectrum to monogenic spinor fields on compact Riemannian manifolds.

A new method uses active learning to improve bile duct stone evaluation.

The importance of the power law has been well realized in econophysics over the last decade. For instance, the distribution of the rate of stock price variation and of personal assets show the power law. While these results reveal the striking scale invariance of financial markets, the behaviour of price in real econom…

Classifies when homeomorphism groups of stable surfaces have automatic continuity.

The study introduces Cayley--Abels--Rosendal graphs for Polish groups.

Let $Ω$ be a smooth compact oriented 3-dimensional Riemannian manifold with boundary. A quaternion field is a pair $q=\{α,u\}$ of a function $α$ and a vector field $u$ on $Ω$. A field $q$ is {\it harmonic} if $α, u$ are continuous in $Ω$ and $\nablaα={\rm rot\,}u,\,{\rm div\,}u=0$ holds into $Ω$. The space ${\mathscr Q…

We present an idea of unifying small scale (topology, proximity spaces, uniform spaces) and large scale (coarse spaces, large scale spaces). It relies on an analog of multilinear forms from Linear Algebra. Each form has a large scale compactification and those include all well-known compactifications: Higson corona, Gr…

Mackey showed that for a compact Lie group $K$, the pair $(K,C^{0}(K))$ has a unique non-trivial irreducible covariant pair of representations. We study the relevance of this result to the unitary equivalence of quantizations for an infinite-dimensional family of $K\times K$ invariant polarizations on $T^{\ast}K$. The …

Theory of ends of spaces using linear algebra.

Improved mean estimation for symmetric distributions with finite-sample guarantees.

Machine learning techniques are presented for automatic recognition of the historical letters (XI-XVIII centuries) carved on the stoned walls of St.Sophia cathedral in Kyiv (Ukraine). A new image dataset of these carved Glagolitic and Cyrillic letters (CGCL) was assembled and pre-processed for recognition and predictio…

Simplified Milnor-Schwarz lemma for geometric group theory.

This paper outlines an approach to the non-abelian theta functions of the $SU(2)$-Chern-Simons theory with the methods used by A. Weil for studying classical theta functions. First we translate in knot theoretic language classical theta functions, the action of the finite Heisenberg group, and the discrete Fourier tran…

We study an economic model where agents trade a variety of products by using one of three competing rules: "need", "greed" and "noise". We find that the optimal strategy for any agent depends on both product composition in the overall market and composition of strategies in the market. In particular, a strategy that do…

For the cotangent bundle $T^{*}K$ of a compact Lie group $K$, we study the complex-time evolution of the vertical tangent bundle and the associated geometric quantization Hilbert space $L^{2}(K)$ under an infinite-dimensional family of Hamiltonian flows. For each such flow, we construct a generalized coherent state tra…

Standard economic theory, starting with Adam Smith's invisible hand, holds that those who trade for their own selfish motives of maximizing their private preferences may contribute more to the public wealth than those who claim altruistic motives. Under restrictive conditions, this has been shown to result from a self-…

Defines Floer homology with DG coefficients for symplectic manifolds.

New recursive algorithm estimates conditional kernel mean embeddings in Hilbert space.

This thesis consists of two independent parts: random matrices, which form the first one-third of this thesis, and machine learning, which constitutes the remaining part. The main results of this thesis are as follows: a necessary and sufficient condition for the inverse moments of $(m,n,β)$-Laguerre matrices and compo…

We describe a new method for combinatorially computing the transverse invariant in knot Floer homology. Previous work of the authors and Stone used braid diagrams to combinatorially compute knot Floer homology of braid closures. However, that approach was unable to explicitly identify the invariant of transverse links …

Researchers extend asymptotic analysis to Bergman projections with Gevrey weights.

Paper introduces NICc for fast cluster-based validation of prediction models.

We present sufficient conditions for the cohomology of a closed aspherical manifold to be proper Lipschitz in sense of Connes-Gromov-Moscovici [CGM]. The conditions are stated in terms of the Stone-Čech compactification of the universal cover of a manifold. We show that these conditions are formally weaker than the suf…

We study prismatics sets analogously to simplical sets except that realization involves prisms, i.e., products of simplices rather than just simplices. Particular examples are the prismatic subdivision of a simplicial set S and the prismatic star of S. Both have the same homotopy type as S and in particular the latter …

It has been recently conjectured by Boyer-Gordon-Watson that a closed, orientable, irreducible $3$-manifold $M$ is a Heegaard Floer $L$-space if and only if $π_1(M)$ is not left-orderable. In this article, we study this conjecture from the point of view of lattice cohomology, an invariant introduced by Némethi which is…

Recent work in reinforcement learning demonstrated that learning solely through self-play is not only possible, but could also result in novel strategies that humans never would have thought of. However, optimization methods cast as a game between two players require careful tuning to prevent suboptimal results. Hence,…

Strategic Workforce Planning is a company process providing best in class, economically sound, workforce management policies and goals. Despite the abundance of literature on the subject, this is a notorious challenge in terms of implementation. Reasons span from the youth of the field itself to broader data integratio…

Proposes a new resampling method for off-policy evaluation in stochastic control.

A topology on a set $X$ is the same as a projection (i.e. an idempotent linear operator) $cl:2^X\to 2^X$ satisfying $A\subset cl(A)$ for all $A\subset X$. That's a good way to summarize Kuratowski's closure operator. Basic geometry on a set $X$ is a dot product $\cdot:2^X\times 2^X\to 2^Y$. Its equivalent form is an or…

Partial-monitoring games constitute a mathematical framework for sequential decision making problems with imperfect feedback: The learner repeatedly chooses an action, opponent responds with an outcome, and then the learner suffers a loss and receives a feedback signal, both of which are fixed functions of the action a…

BiPE blends intra-segment and inter-segment encodings for better length extrapolation.

We consider the problem of recovering a common latent source with independent components from multiple views. This applies to settings in which a variable is measured with multiple experimental modalities, and where the goal is to synthesize the disparate measurements into a single unified representation. We consider t…

The manufacturing sector is envisioned to be heavily influenced by artificial intelligence-based technologies with the extraordinary increases in computational power and data volumes. A central challenge in manufacturing sector lies in the requirement of a general framework to ensure satisfied diagnosis and monitoring …