Related papers: Algorithmic information, plane Kakeya sets, and co…

We study $d$-dimensional simplicial complexes that are PL embeddable in $\mathbb{R}^{d+1}$. It is shown that such a complex must satisfy a certain homological condition. The existence of this obstruction allows us to provide a systematic…

We consider a system with a discrete configuration space. We show that the geometrical structures associated with such a system provide the tools necessary for a reconstruction of discrete quantum mechanics once dynamics is brought into the…

Approximation of the optimal two-part MDL code for given data, through successive monotonically length-decreasing two-part MDL codes, has the following properties: (i) computation of each step may take arbitrarily long; (ii) we may not know…

We provide space complexity lower bounds for data structures that approximate logistic loss up to $\epsilon$-relative error on a logistic regression problem with data $\mathbf{X} \in \mathbb{R}^{n \times d}$ and labels $\mathbf{y} \in…

In this paper we introduce and study discrete analogues of Lebesgue and Hausdorff dimensions for graphs. It turned out that they are closely related to well-known graph characteristics such as rank dimension and Prague (or…

Halfspace (or Tukey) depth is a fundamental and robust measure of centrality of data points in multivariate datasets. Computing the depth of a point with respect to the uniform distribution on an open convex body in $\mathbb{R}^d$ is a…

We expand the basic geometric elements of the simplex method to linear programs in locally convex topological vector spaces and provide conditions under which the method converges in value to optimality. This setting generalizes many…

The manifold hypothesis says that natural high-dimensional data lie on or around a low-dimensional manifold. The recent success of statistical and learning-based methods in very high dimensions empirically supports this hypothesis,…

Katz and Zahl used a planebrush argument to prove that Kakeya sets in $\mathbb{R}^4$ have Hausdorff dimension at least 3.059. In the special case when the Kakeya set is plany, their argument gives a better lower bound of 10/3. We give a…

We consider Calderon's inverse problem with partial data in dimensions $n \geq 3$. If the inaccessible part of the boundary satisfies a (conformal) flatness condition in one direction, we show that this problem reduces to the invertibility…

In an earlier paper (arxiv:1108.4292) we introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. For a compact metric space $K$ let $\dim_{H}K$ and $\dim_{tH} K$ denote its Hausdorff and…

The dimension of Kakeya sets can be bounded using sum-difference exponents $\SD(R;s)$ for various sets of rational slopes $R$ and output slope $s$; the arithmetic Kakeya conjecture, which implies the Kakeya conjecture in all dimensions,…

The manifold hypothesis suggests that the generalization performance of machine learning methods improves significantly when the intrinsic dimension of the input distribution's support is low. In the context of KRR, we investigate two…

The proof of the theorem, which states that the Euclidean metric on the set of random points in an $n$-dimensional Euclidean space with the distribution of a special class, converges in probability in the limit $n\rightarrow\infty$ to the…

After reviewing unnormalized and normalized information distances based on incomputable notions of Kolmogorov complexity, we discuss how Kolmogorov complexity can be approximated by data compression algorithms. We argue that optimal…

Let $E\subset\rr$ be a closed set of Hausdorff dimension $\alpha$. We prove that if $\alpha$ is sufficiently close to 1, and if $E$ supports a probabilistic measure obeying appropriate dimensionality and Fourier decay conditions, then $E$…

In this paper we study subsequences of random numbers. In Kamae (1973), selection functions that depend only on coordinates are studied, and their necessary and sufficient condition for the selected sequences to be normal numbers is given.…

This paper presents a rigorous resolution of the Borel-Kolmogorov paradox using the Maximum Entropy Principle. We construct a metric-based framework for Bayesian inference that uniquely extends conditional probability to events of null…

Let $E\subset \mathbb{R}$ be a closed set of Hausdorff dimension $\alpha\in (0, 1)$. Let $P: \mathbb{R}\to \mathbb{R}$ be a polynomial without a constant term whose degree is bigger than one. We prove that if $E$ supports a probability…

Spatial models of preference, in the form of vector embeddings, are learned by many deep learning and multiagent systems, including recommender systems. Often these models are assumed to approximate a Euclidean structure, where an…