Related papers: Some measure-theoretic properties of U-statistics …
Incomplete U-statistics have been proposed to accelerate computation. They use only a subset of the subsamples required for kernel evaluations by complete U-statistics. This paper gives a finite sample bound in the style of Bernstein's…
Given $n$ independent random marked $d$-vectors (points) $X_i$ distributed with a common density, define the measure $\nu_n=\sum_i\xi_i$, where $\xi_i$ is a measure (not necessarily a point measure) which stabilizes; this means that $\xi_i$…
We analyze the Nystr\"om approximation of a positive definite kernel associated with a probability measure. We first prove an improved error bound for the conventional Nystr\"om approximation with i.i.d. sampling and singular-value…
We introduce and investigate in this short report the new notion of uniform measure (distribution) on the arbitrary compact metric space. We consider also some possible applications of these measures in the theory of imbedding theorems and…
Based on the theory of quantum mechanics, intrinsic randomness in measurement distinguishes quantum effects from classical ones. From the perspective of states, this quantum feature can be summarized as coherence or superposition in a…
Comparing probability distributions is a core challenge across the natural, social, and computational sciences. Existing methods, such as Maximum Mean Discrepancy (MMD), struggle in high-dimensional and non-compact domains. Here we…
In this note, we develop some of the basic theory of s-finite (measures and) kernels, a little-studied class that Staton has recently argued convincingly to be precisely the semantic counterpart of (first-order) probabilistic programs. We…
Measurements on classical systems are usually idealized and assumed to have infinite precision. In practice, however, any measurement has a finite resolution. We investigate the theory of non-ideal measurements in classical mechanics using…
We study quantum statistical inference tasks of hypothesis testing and their canonical variations, in order to review relations between their corresponding figures of merit---measures of statistical distance---and demonstrate the crucial…
A basic issue in both teaching of and practice of statistics is the interplay between modelling assumptions and inference performance. The general message conveyed is that stronger assumptions lead to better statistical performance of the…
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…
Teramoto et al. defined a new measure called the gap ratio that measures the uniformity of a finite point set sampled from $\cal S$, a bounded subset of $\mathbb{R}^2$. We generalize this definition of measure over all metric spaces by…
We connect shift-invariant characteristic kernels to infinitely divisible distributions on $\mathbb{R}^{d}$. Characteristic kernels play an important role in machine learning applications with their kernel means to distinguish any two…
Partial differential equations are central to describing many physical phenomena. In many applications these phenomena are observed through a sensor network, with the aim of inferring their underlying properties. Leveraging from certain…
We discuss the relation between p-adic numbers and kernels in view of a recent large deviation theory for mean-field spin glasses. As an application we show several fundamental properties of numerical bases in kernel language. In…
Quantum mechanics predicts the joint probability distributions of the outcomes of simultaneous measurements of commuting observables, but the current formulation lacks the operational definition of simultaneous measurements. In order to…
Measurement incompatibility stipulates the existence of quantum measurements that cannot be carried out simultaneously on single systems. We show that the set of input-output probabilities obtained from d-dimensional classical systems…
Given an isotropic probability measure $\mu$ on ${\mathbb R}^d$ with ${\rm d}\mu \left( x \right) = {\left( {\varrho \left( x \right)} \right)^{ - \alpha }}{\rm d}x$, where $\alpha > d + 1$ and $\varrho :{{\mathbb R}^d} \to \left( {0, +…
The aim of this article is to establish basic results in a conditional measure theory. The results are applied to prove that arbitrary kernels and conditional distributions are represented by measures in a conditional set theory. In…
Nonparametric two sample testing deals with the question of consistently deciding if two distributions are different, given samples from both, without making any parametric assumptions about the form of the distributions. The current…