Related papers: Quantization for a condensation system
We discuss the properties of an ideal relativistic gas of events possessing Bose-Einstein statistics. We find that the mass spectrum of such a system is bounded by $\mu \leq m\leq 2M/\mu _K,$ where $\mu $ is the usual chemical potential,…
The infinite (in both directions) sequence of the distributions $\mu^{(k)}$ of the stochastic integrals $\int_0^{\infty-}c^{-N_{t-}^{(k)}} dL_t^{(k)}$ for integers $k$ is investigated. Here $c>1$ and $(N_t^{(k)},L_t^{(k)})$, $t\geq0$, is a…
Under the name prime decomposition (pd), a unique decomposition of an arbitrary $N$-dimensional density matrix $\rho$ into a sum of seperable density matrices with dimensions given by the coprime factors of $N$ is introduced. For a class of…
If $\mu$ is a distribution over the $d$-dimensional Boolean cube $\{0,1\}^d$, our goal is to estimate its mean $p\in[0,1]^d$ based on $n$ iid draws from $\mu$. Specifically, we consider the empirical mean estimator $\hat p_n$ and study the…
A tri-objective optimal Micro Phasor Measurement Units ({\mu}-PMUs) Placement method is presented, with a focus on minimizing the following three parameters: i) the total number of {\mu}-PMU channels, (ii) the maximum state estimation…
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…
We introduce notions of dimension of an infinite group, or more generally, a metric space, defined using percolation. Roughly speaking, the percolation dimension $pdim(G)$ of a group $G$ is the fastest rate of decay of a symmetric…
We develop a rigorous framework for quantifying quantum coherence in finite-dimensional systems by applying the Schur-Horn majorization theorem to relate eigenvalue distributions and diagonal entries of density matrices. Building on this…
A method to compute the optimal success probability of discrimination of N arbitrary quantum states is presented, based on the decomposition of any N-outcome measurement into sequences of nested two-outcome ones. In this way the…
Let $ r, s>0 $. For a given probability measure $P$ on $\mathbb{R}^d$, let $(\alpha_n)_{n \geq 1}$ be a sequence of (asymptotically) $L^r(P)$- optimal quantizers. For all $\mu \in \mathbb{R}^d $ and for every $\theta >0$, one defines the…
We show that the method of distributed noise-shaping beta-quantization offers superior performance for the problem of spectral super-resolution with quantization whenever there is redundancy in the number of measurements. More precisely, if…
Modern computers enable methods for design optimization that account for uncertainty in the system---so-called optimization under uncertainty. We propose a metric for OUU that measures the distance between a designer-specified probability…
In this paper it is shown that if $\mu$ is a finite Radon measure in $\mathbb R^d$ which is $n$-rectifiable and $1\leq p\leq 2$, then $$\int_0^\infty \beta_{\mu,p}^n(x,r)^2\,\frac{dr}r<\infty \quad {for $\mu$-a.e. $x\in\mathbb R^d$,}$$…
Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. If in the quantization some of the elements in the support are…
We suggest that the curse of dimensionality affecting the similarity-based search in large datasets is a manifestation of the phenomenon of concentration of measure on high-dimensional structures. We prove that, under certain geometric…
We consider the problem of distributed mean estimation (DME), in which $n$ machines are each given a local $d$-dimensional vector $x_v \in \mathbb{R}^d$, and must cooperate to estimate the mean of their inputs $\mu = \frac 1n\sum_{v = 1}^n…
We discuss in detail the uniform discretization approach to the quantization of totally constrained theories. This approach allows to construct the continuum theory of interest as a well defined, controlled, limit of well behaved discrete…
Several important measures of quantum correlations of a state of a finite-dimensional composite system are defined as linear combinations of marginal entropies of this state. This paper is devoted to the infinite-dimensional generalizations…
In recent years Deep Neural Networks (DNNs) have been rapidly developed in various applications, together with increasingly complex architectures. The performance gain of these DNNs generally comes with high computational costs and large…
Fidelity is the standard measure for quantifying the similarity between two quantum states. It is equal to the square of the minimum Bhattacharyya coefficient between the probability distributions induced by quantum measurements on the two…