Related papers: Metric complexity is a Bryant--Tupper diversity
In this work, we develop a new complexity metric for an important class of low-rank matrix optimization problems in both symmetric and asymmetric cases, where the metric aims to quantify the complexity of the nonconvex optimization…
Data Science and Machine Learning have become fundamental assets for companies and research institutions alike. As one of its fields, supervised classification allows for class prediction of new samples, learning from given training data.…
The uncertainty relation for continuous variables due to Byalinicki-Birula and Mycielski expresses the complementarity between two $n$-uples of canonically conjugate variables $(x_1,x_2,\cdots x_n)$ and $(p_1,p_2,\cdots p_n)$ in terms of…
We give the first sample complexity characterizations for outcome indistinguishability, a theoretical framework of machine learning recently introduced by Dwork, Kim, Reingold, Rothblum, and Yona (STOC 2021). In outcome…
The notion of multiplicity of a module first arose as consequence of Hilbert's work on commutative algebra, relating the dimension of rings with the degree of certain polynomials. For noncommutative rings, the notion of multiplicity first…
A recent work of Abbasi et al. [FOCS 2023] introduced the notion of $\varepsilon$-scatter dimension of a metric space and showed a general framework for efficient parameterized approximation schemes (so-called EPASes) for a wide range of…
Gravity is understood as a geometrization of spacetime. But spacetime is also the manifold of the boundary values of the spinless point particle in a variational approach. Since all known matter, baryons, leptons and gauge bosons are…
We consider dynamical systems for which the spatial extension plays an important role. For these systems, the notions of attractor, epsilon-entropy and topological entropy per unit time and volume have been introduced previously. In this…
WWe define the notion of a random metric space and prove that with probability one such a space is isometricto the Urysohn universal metric space. The main technique is the study of universal and random distance matrices; we relate the…
Let $G$ be a finitely presented group. A new complexity called \textit{Karoubi-Weibel complexity} or \textit{covering type}, is defined for $G$. The construction is inspired by recent work of Karoubi and Weibel \cite{KW}, initially applied…
For a continuous map $T$ of a compact metrizable space $X$ with finite topological entropy, the order of accumulation of entropy of $T$ is a countable ordinal that arises in the context of entropy structure and symbolic extensions. We show…
The universal concept of complexity by the dynamic redundance paradigm and the ensuing concept of extended dynamic fractality (physics/9806002) are applied here to higher levels of complexity corresponding to living systems. After recalling…
For a cardinal $\kappa > \omega$ a metric space $X$ is called to be $\kappa$-superuniversal whenever for every metric space $Y$ with $|Y| < \kappa$ every partial isometry from a subset of $Y$ into $X$ can be extended over the whole space…
Kolmogorov complexity of a finite binary word reflects both algorithmic structure and the empirical distribution of symbols appearing in the word. Words with symbol frequencies far from one half have smaller combinatorial richness and…
We study topological properties of random metric spaces which arise by Lambda-coalescents. These are stochastic processes, which start with an infinite number of lines and evolve through multiple mergers in an exchangeable setting. We show…
Magnitude is a numerical isometric invariant of metric spaces, whose definition arises from a precise analogy between categories and metric spaces. Despite this exotic provenance, magnitude turns out to encode many invariants from integral…
In this article, we introduce the concept of lexicographic metric space and, after discussing some basic properties of these metric spaces, such as completeness, boundedness, compactness and separability, we obtain a formula for the metric…
We show from a categorical point of view that probability measures on certain measurable or topological spaces arise canonically as the extension of probability distributions on countable sets. We do this by constructing probability monads…
Characterisations of metrizable topological spaces or metrizable uniform spaces are well known. A natural counterpart to being metrizable for topological spaces can be expressed in terms of probabilistic metrizability for approach spaces.…
The unreduced, universally nonperturbative analysis of arbitrary interaction process, described by a quite general equation, provides the truly complete, "dynamically multivalued" general solution that leads to dynamically derived,…