Related papers: Geometric Kolmogorov-Arnold Superposition Theorem
The paper introduces several new concepts for solving nonconvex or nonsmooth optimization problems, including convertible nonconvex function, exact convertible nonconvex function and differentiable convertible nonconvex function. It is…
We develop a unified representation theory for the categories of finite subsets and relation-preserving maps of highly homogeneous relational structures classified by Cameron. For any commutative coefficient ring $k$, we extend the…
Single-view 3D reconstruction in complex real-world scenes is challenging due to noise, object diversity, and limited dataset availability. To address these challenges, we propose MGP-KAD, a novel multimodal feature fusion framework that…
A fundamental result in representation theory is Kostant's theorem which describes the algebra of polynomials on a reductive Lie algebra as a module over its invariants. We prove a quantum analogue of this theorem for the general linear…
An abstract convergence theorem for a class of generalized descent methods that explicitly models relative errors is proved. The convergence theorem generalizes and unifies several recent abstract convergence theorems. It is applicable to…
Partial differential equations (PDEs) form a central component of scientific computing. Among recent advances in deep learning, evolutionary neural networks have been developed to successively capture the temporal dynamics of time-dependent…
Data representation techniques have made a substantial contribution to advancing data processing and machine learning (ML). Improving predictive power was the focus of previous representation techniques, which unfortunately perform rather…
We present a well-structured detailed exposition of a well-known proof of the following celebrated result solving Hilbert's 13th problem on superpositions. For functions of 2 variables the statement is as follows. Kolmogorov Theorem. There…
In the mid 1980s, while working on establishing completion theorems for equivariant Algebraic K-Theory similar to the well-known completion theorems for equivariant topological K-theory, the late Robert Thomason found the strong finiteness…
This is part II of our book on KAM theory. We start by defining functorial analysis and then switch to the particular case of Kolmogorov spaces. We develop functional calculus based on the notion of local operators. This allows to define…
Adaptive methods are extremely popular in machine learning as they make learning rate tuning less expensive. This paper introduces a novel optimization algorithm named KATE, which presents a scale-invariant adaptation of the well-known…
Incorporating group symmetry directly into the learning process has proved to be an effective guideline for model design. By producing features that are guaranteed to transform covariantly to the group actions on the inputs,…
We introduce a global equivariant refinement of algebraic K-theory; here `global equivariant' refers to simultaneous and compatible actions of all finite groups. Our construction turns a specific kind of categorical input data into a global…
We provide extension procedures for nonlinear expectations to the space of all bounded measurable functions. We first discuss a maximal extension for convex expectations which have a representation in terms of finitely additive measures.…
Kleene algebra with tests, KAT, provides a simple two-sorted algebraic framework for verifying properties of propositional while programs. Kleene algebra with domain, KAD, is a one-sorted alternative to KAT. The equational theory of KAT…
We introduce a machine free mathematical framework to get a natural formalization of some general notions of infinite computation in the context of Kolmogorov complexity. Namely, the classes Max^{X\to D}_{PR} and Max^{X\to D}_{Rec} of…
Bivariant (equivariant) K-theory is the standard setting for non-commutative topology. We may carry over various techniques from homotopy theory and homological algebra to this setting. Here we do this for some basic notions from…
Predictive modeling on web-scale tabular data with billions of instances and hundreds of heterogeneous numerical features faces significant scalability challenges. These features exhibit anisotropy, heavy-tailed distributions, and…
Random tensor networks provide useful models that incorporate various important features of holographic duality. A tensor network is usually defined for a fixed graph geometry specified by the connection of tensors. In this paper, we…
While Kolmogorov's probability axioms are widely recognized, it is less well known that in an often-overlooked 1930 note, Kolmogorov proposed an axiomatic framework for a unifying concept of the mean -- referred to as regular means. This…