Related papers: Sharp failure rates for the bootstrap particle fil…
It has been widely realized that Monte Carlo methods (approximation via a sample ensemble) may fail in large scale systems. This work offers some theoretical insight into this phenomenon in the context of the particle filter. We demonstrate…
Recently developed particle flow algorithms provide an alternative to importance sampling for drawing particles from a posterior distribution, and a number of particle filters based on this principle have been proposed. Samples are drawn…
We derive a scale-free bound on the density of the maximum of a centered Gaussian vector. The basic bound is non-uniform, depends logarithmically on the dimension, and allows any covariance matrix. When the largest marginal variance is…
Gaussian time-series models are often specified through their spectral density. Such models present several computational challenges, in particular because of the non-sparse nature of the covariance matrix. We derive a fast approximation of…
We study the optimal sample complexity of variable selection in linear regression under general design covariance, and show that subset selection is optimal while under standard complexity assumptions, efficient algorithms for this problem…
Particle filters are a popular and flexible class of numerical algorithms to solve a large class of nonlinear filtering problems. However, standard particle filters with importance weights have been shown to require a sample size that…
The Bonferroni adjustment, or the union bound, is commonly used to study rate optimality properties of statistical methods in high-dimensional problems. However, in practice, the Bonferroni adjustment is overly conservative. The extreme…
Under certain conditions, the largest eigenvalue of a sample covariance matrix undergoes a well-known phase transition when the sample size $n$ and data dimension $p$ diverge proportionally. In the subcritical regime, this eigenvalue has…
The filtering distribution captures the statistics of the state of a dynamical system from partial and noisy observations. Classical particle filters provably approximate this distribution in quite general settings; however they behave…
In this paper, we use a new approach to prove that the largest eigenvalue of the sample covariance matrix of a normally distributed vector is bigger than the true largest eigenvalue with probability 1 when the dimension is infinite. We…
The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semi-circle law. If the Gaussian entries are all shifted by a constant amount c/Sqrt(2N), where N is the size of the matrix, in the large…
In this paper, we derive moment conditions for particle filter importance weights, which ensure that the particle filter estimates of the expectations of bounded Borel functions converge in mean square and $L^4$ sense, and that the…
We consider fully connected and feedforward deep neural networks with dependent and possibly heavy-tailed weights, as introduced in [26], to address limitations of the standard Gaussian prior. It has been proved in [26] that, as the number…
In the context of principal components analysis (PCA), the bootstrap is commonly applied to solve a variety of inference problems, such as constructing confidence intervals for the eigenvalues of the population covariance matrix $\Sigma$.…
Particle filters (PFs), which are successful methods for approximating the solution of the filtering problem, can be divided into two types: weighted and unweighted PFs. It is well known that weighted PFs suffer from the weight degeneracy…
This paper derives central limit and bootstrap theorems for probabilities that sums of centered high-dimensional random vectors hit hyperrectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for…
This paper presents a neural network filter method based on contraction operators to address model collapse in recursive training of generative models. Unlike \cite{xu2024probabilistic}, which requires superlinear sample growth…
For sample covariance matrices with iid entries with sub-Gaussian tails, when both the number of samples and the number of variables become large and the ratio approaches to one, it is a well-known result of A. Soshnikov that the limiting…
We consider the numerical approximation of the filtering problem in high dimensions, that is, when the hidden state lies in $\mathbb{R}^d$ with $d$ large. For low dimensional problems, one of the most popular numerical procedures for…
A Gaussian elimination algorithm is presented that reveals the numerical rank of a matrix by yielding small entries in the Schur complement. The algorithm uses the maximum volume concept to find a square nonsingular submatrix of maximum…