Related papers: Introduction to Set Shaping Theory
We consider a one-dimensional discrete-space birth process with a bounded number of particle per site. Under the assumptions of the finite range of interaction, translation invariance, and non-degeneracy, we prove a shape theorem. We also…
This article surveys the mathematics of the cut and project method as applied to point sets, called here {\em model sets}. It covers the geometric, arithmetic, and analytical sides of this theory as well as diffraction and the connection…
What makes sets, or more precisely, the category {\bf Set} important in Mathematics are the well known {\it two} specific ways in which arbitrary mappings $f : X \longrightarrow Y$ between any two sets $X, Y$ can {\it fail} to be…
Many important questions about a model cannot be answered just by explaining how much each feature contributes to its output. To answer a broader set of questions, we generalize a popular, mathematically well-grounded explanation technique,…
This paper is concerned with a rate-distortion theory for sequences of i.i.d. random variables with general distribution supported on general sets including manifolds and fractal sets. Manifold structures are prevalent in data science,…
This paper describes various approaches to modeling a random process with a given rational power spectral density. The main attention is paid to the spectral form of mathematical description, which allows one to obtain a relation for the…
Dealing with massive data is a challenging task for machine learning. An important aspect of machine learning is function approximation. In the context of massive data, some of the commonly used tools for this purpose are sparsity,…
Recent research in machine teaching has explored the instruction of any concept expressed in a universal language. In this compositional context, new experimental results have shown that there exist data teaching sets surprisingly shorter…
Leveraging the compositional nature of our world to expedite learning and facilitate generalization is a hallmark of human perception. In machine learning, on the other hand, achieving compositional generalization has proven to be an…
In this paper I want to suggest a new solution to the problem of musical tuning. On one hand, I see it as a generalization of Just Intonation (JI) to inharmonic timbers, on another, as a unification of spectral interference and harmonicity…
Many scientific and industrial processes produce data that is best analysed as vectors of relative values, often called compositions or proportions. The Dirichlet distribution is a natural distribution to use for composition or proportion…
To any finite ordered subset and any finite partition of a group a set of tuples of positive integers, named as configurations, is associated that describes the group's behavior. The present paper provides an exposition of this notion and…
Sentence function is an important linguistic feature referring to a user's purpose in uttering a specific sentence. The use of sentence function has shown promising results to improve the performance of conversation models. However, there…
This work proposes kernel transform learning. The idea of dictionary learning is well known; it is a synthesis formulation where a basis is learnt along with the coefficients so as to generate or synthesize the data. Transform learning is…
String theory compactifications come with numerous $U(1)$ factors, implying the presence of many hidden photons in the low-energy EFT. One may call this the ``string photoverse''. We argue that, generically, these hidden photons are…
Organising the relevant literature and by letting statistical convergence play the main role in the theory of compactness, a variant of compactness called statistical compactness has been achieved. As in case of sequential compactness, one…
Choice functions constitute a simple, direct and very general mathematical framework for modelling choice under uncertainty. In particular, they are able to represent the set-valued choices that typically arise from applying decision rules…
While the utility of well-chosen abstractions for understanding and predicting the behaviour of complex systems is well appreciated, precisely what an abstraction $\textit{is}$ has so far has largely eluded mathematical formalization. In…
This article introduces a general statistical modeling principle called "Density Sharpening" and applies it to the analysis of discrete count data. The underlying foundation is based on a new theory of nonparametric approximation and…
Deep neural networks exhibit a simplicity bias, a well-documented tendency to favor simple functions over complex ones. In this work, we cast new light on this phenomenon through the lens of the Minimum Description Length principle,…