Related papers: On Sets That Encode Themselves
Current approaches for predicting sets from feature vectors ignore the unordered nature of sets and suffer from discontinuity issues as a result. We propose a general model for predicting sets that properly respects the structure of sets…
Learning representations that generalize to novel compositions of known concepts is crucial for bridging the gap between human and machine perception. One prominent effort is learning object-centric representations, which are widely…
One says that a property $P$ of sets of natural numbers can be made into itself iff there is a numbering $\alpha_0,\alpha_1,\ldots$ of all left-r.e. sets such that the index set $\{e: \alpha_e$ satisfies $P\}$ has the property $P$ as well.…
We consider the arithmetic complexity of index sets of uniformly computably enumerable families learnable under different learning criteria. We determine the exact complexity of these sets for the standard notions of finite learning,…
Recent work in computability theory has focused on various notions of asymptotic computability, which capture the idea of a set being "almost computable." One potentially upsetting result is that all four notions of asymptotic computability…
In this paper, we have compared r.e. sets based on their enumeration orders with Turing machines. Accordingly, we have defined novel concept uniformity for Turing machines and r.e. sets and have studied some relationships between uniformity…
Let $A$ be a finite or countable alphabet and let $\theta$ be literal (anti)morphism onto $A^*$ (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under…
The problem of learning a manifold structure on a dataset is framed in terms of a generative model, to which we use ideas behind autoencoders (namely adversarial/Wasserstein autoencoders) to fit deep neural networks. From a machine learning…
Do word embeddings converge to learn similar things over different initializations? How repeatable are experiments with word embeddings? Are all word embedding techniques equally reliable? In this paper we propose evaluating methods for…
A conjecture of Graham (repeated by Erd\H{o}s) asserts that for any set $A \subseteq \mathbb{F}_p \setminus \{0\}$, there is an ordering $a_1, \ldots, a_{|A|}$ of the elements of $A$ such that the partial sums $a_1, a_1+a_2, \ldots,…
In estimating the complexity of objects, in particular of graphs, it is common practice to rely on graph- and information-theoretic measures. Here, using integer sequences with properties such as Borel normality, we explain how these…
A pattern of a sequence is a sequence of integer indices with each index describing the order of first occurrence of the respective symbol in the original sequence. In a recent paper, tight general bounds on the block entropy of patterns of…
Several machine learning models are defined for inputs of any size, such as graphs with different numbers of nodes and point clouds containing varying numbers of points. The universality properties of such any-dimensional models remain…
We introduce a concept of approximately invertible elements in non-unital normed algebras which is, on one side, a natural generalization of invertibility when having approximate identities at hand, and, on the other side, it is a direct…
We study the modular representation theory of the symmetric and alternating groups. One of the most natural ways to label the irreducible representations of a given group or algebra in the modular case is to show the unitriangularity of the…
We introduce a uniform representation of general objects that captures the regularities with respect to their structure. It allows a representation of a general class of objects including geometric patterns and images in a sparse, modular,…
The coherence of an individual quantum state can be meaningfully discussed only when referring to a preferred basis. This arbitrariness can however be lifted when considering sets of quantum states. Here we introduce the concept of set…
We study four (families of) sets of algebraic integers of degree less than or equal to three. Apart from being simply defined, we show that they share two distinctive characteristics: almost uniformity and arithmetical independence. Here,…
Given a countable graph, we say a set $A$ of its vertices is \emph{universal} if it contains every countable graph as an induced subgraph, and $A$ is \emph{weakly universal} if it contains every finite graph as an induced subgraph. We show…
A beautiful theorem of Zeckendorf states that every integer can be written uniquely as the sum of non-consecutive Fibonacci numbers $\{ F_i \}_{i = 1}^{\infty}$. A set $S \subset \mathbb{Z}$ is said to satisfy Benford's law if the density…