Related papers: Dimension and the Structure of Complexity Classes
We introduce new combinatorial quantities for concept classes, and prove lower and upper bounds for learning complexity in several models of query learning in terms of various combinatorial quantities. Our approach is flexible and powerful…
VC-dimension and $\varepsilon$-nets are key concepts in Statistical Learning Theory. Intuitively, VC-dimension is a measure of the size of a class of sets. The famous $\varepsilon$-net theorem, a fundamental result in Discrete Geometry,…
Large language models (LLMs) have achieved remarkable progress in natural language generation, yet they continue to display puzzling behaviors -- such as repetition and incoherence -- even when exhibiting low perplexity. This highlights a…
We study the complexity of geometric problems on spaces of low fractal dimension. It was recently shown by [Sidiropoulos & Sridhar, SoCG 2017] that several problems admit improved solutions when the input is a pointset in Euclidean space…
We propose a novel probabilistic dimensionality reduction framework that can naturally integrate the generative model and the locality information of data. Based on this framework, we present a new model, which is able to learn a smooth…
We formulate the conditional Kolmogorov complexity of x given y at precision r, where x and y are points in Euclidean spaces and r is a natural number. We demonstrate the utility of this notion in two ways. 1. We prove a point-to-set…
The dimension of a partial order $P$ is the minimum number of linear orders whose intersection is $P$. There are efficient algorithms to test if a partial order has dimension at most $2$. In 1982 Yannakakis showed that for $k\geq 3$ to test…
Evaluating conjunctive queries and solving constraint satisfaction problems are fundamental problems in database theory and artificial intelligence, respectively. These problems are NP-hard, so that several research efforts have been made…
Dimension theory is a branch of topology concerned with defining and analyzing dimensions of geometric and topological spaces in purely topological terms. In this work, we adapt the classical notion of topological dimension (Lebesgue…
In this article, we present a fresh perspective on language, combining ideas from various sources, but mixed in a new synthesis. As in the minimalist program, the question is whether we can formulate an elegant formalism, a universal…
In this article we undertake a study of extension complexity from the perspective of formal languages. We define a natural way to associate a family of polytopes with binary languages. This allows us to define the notion of extension…
Sentence embeddings encode natural language sentences as low-dimensional dense vectors. A great deal of effort has been put into using sentence embeddings to improve several important natural language processing tasks. Relation extraction…
We investigate (quantifier-free) spatial constraint languages with equality, contact and connectedness predicates as well as Boolean operations on regions, interpreted over low-dimensional Euclidean spaces. We show that the complexity of…
We present and study approximate notions of dimensional and margin complexity, which correspond to the minimal dimension or norm of an embedding required to approximate, rather then exactly represent, a given hypothesis class. We show that…
While concepts and tools from Theoretical Computer Science are regularly applied to, and significantly support, software development for discrete problems, Numerical Engineering largely employs recipes and methods whose correctness and…
Abductive explanations (AXp's) are widely used for understanding decisions of classifiers. Existing definitions are suitable when features are independent. However, we show that ignoring constraints when they exist between features may lead…
Generalized contextuality is a hallmark of nonclassical theories like quantum mechanics. Yet, three fundamental computational problems concerning its decidability and complexity remain open. First, determining the complexity of deciding if…
There are various notions of dimension in fractal geometry to characterise (random and non-random) subsets of $\mathbb R^d$. In this expository text, we discuss their analogues for infinite subsets of $\mathbb Z^d$ and, more generally, for…
This paper defines a constraint-based model dedicated to multidimensional databases. The model we define represents data through a constellation of facts (subjects of analyse) associated to dimensions (axis of analyse), which are possibly…
We build a bridge from density combinatorics to dimension theory of continued fractions. We establish a fractal transference principle that transfers common properties of subsets of $\mathbb N$ with positive upper density to properties of…