Related papers: VC density and dp rank
In this work we introduce a viscosity-based notion of solution for general approximation schemes associated with partial differential equations, such as dynamic programming principles~(DPPs). A key feature of our approach is that it…
Nonparametric density estimation is an unsupervised learning problem. In this work we propose a two-step procedure that casts the density estimation problem in the first step into a supervised regression problem. The advantage is that we…
We introduce the following variant of the VC-dimension. Given $S \subseteq \{0, 1\}^n$ and a positive integer $d$, we define $\mathbb{U}_d(S)$ to be the size of the largest subset $I \subseteq [n]$ such that the projection of $S$ on every…
The index $n$ of a power law power spectrum of primordial density fluctuations, $P(k) \propto k^n$, has been estimated using many different techniques. The most precise compare the COBE DMR large angular scale $\Delta T$ to the amplitude of…
A characteristic-dependent linear rank inequality is a linear inequality that holds by ranks of subspaces of a vector space over a finite field of determined characteristic, and does not in general hold over other characteristics. In this…
Measuring and quantifying dependencies between random variables (RV's) can give critical insights into a data-set. Typical questions are: `Do underlying relationships exist?', `Are some variables redundant?', and `Is some target variable…
This paper focuses on density-based clustering, particularly the Density Peak (DP) algorithm and the one based on density-connectivity DBSCAN; and proposes a new method which takes advantage of the individual strengths of these two methods…
Denote by $\mathbb{N}$ and $\mathbb{P}$ the set of all positive integers and prime numbers, respectively. Let $\mathbb{P}=\{p_1<p_2<\dots <p_n<\dots\}$, where $p_n$ is the $n$-th prime number. For $k\in\mathbb{N}$ we recursively define…
We show that any formula with two free variables in a VC-minimal theory has VC-codensity at most two. Modifying the argument slightly, we give a new proof of the fact that, in a VC-minimal theory where acl = dcl, the VC-codensity of a…
Dense vector retrieval is an important building block of modern machine learning systems, underlying applications ranging from semantic search to retrieval-augmented generation and knowledge-intensive reasoning. Beyond retrieving items that…
The independence density of a finite hypergraph is the probability that a subset of vertices, chosen uniformly at random contains no hyperedges. Independence densities can be generalized to countable hypergraphs using limits. We show that,…
Clustering multi-dimensional points is a fundamental task in many fields, and density-based clustering supports many applications as it can discover clusters of arbitrary shapes. This paper addresses the problem of Density-Peaks Clustering…
Hyperuniformity is an emergent property, whereby the structure factor of the density $n$ scales as $S(q) \sim q^\alpha$, with $\alpha>0$. We show that for the conserved directed percolation (CDP) class, to which the Manna model belongs,…
Principal component regression (PCR) is a simple, but powerful and ubiquitously utilized method. Its effectiveness is well established when the covariates exhibit low-rank structure. However, its ability to handle settings with noisy,…
Estimating the true rank of a noisy data matrix is a fundamental problem underlying techniques such as principal component analysis, matrix completion, etc. Existing rank estimation criteria, including information-based and cross-validation…
The entropy power inequality for independent random vectors is a foundational result of information theory, with deep connections to probability and geometric functional analysis. Several extensions of the entropy power inequality have been…
The Probability Ranking Principle states that the document set with the highest values of probability of relevance optimizes information retrieval effectiveness given the probabilities are estimated as accurately as possible. The key point…
VC-dimension and VC-density are measures of combinatorial complexity of set systems. VC-dimension was first introduced in the context of statistical learning theory, and is tightly related to the sample complexity in PAC learning.…
The notion of a completely saturated packing [Fejes Toth, Kuperberg and Kuperberg, Highly saturated packings and reduced coverings, Monats. Math. 125 (1998) 127-145] is a sharper version of maximum density, and the analogous notion of a…
Accurate approximation of a real-valued function depends on two aspects of the available data: the density of inputs within the domain of interest and the variation of the outputs over that domain. There are few methods for assessing…