Related papers: Uniform approximation of vectors using adaptive ra…
We prove a general result demonstrating the power of Lagrangian relaxation in solving constrained maximization problems with arbitrary objective functions. This yields a unified approach for solving a wide class of {\em subset selection}…
Large language model (LLM) embeddings are increasingly used to estimate dimensional structure in psychological item pools prior to data collection, yet current applications treat embeddings as static, cross-sectional representations. This…
The Matrix-based Renyi's entropy enables us to directly measure information quantities from given data without the costly probability density estimation of underlying distributions, thus has been widely adopted in numerous statistical…
It is impossible to recover a vector from $\mathbb{R}^m$ with less than $m$ linear measurements, even if the measurements are chosen adaptively. Recently, it has been shown that one can recover vectors from $\mathbb{R}^m$ with arbitrary…
The subspace approximation problem Subspace($k$,$p$) asks for a $k$-dimensional linear subspace that fits a given set of points optimally, where the error for fitting is a generalization of the least squares fit and uses the $\ell_{p}$ norm…
We obtain hardness of approximation results for the $\ell_p$-Shortest Path problem, a variant of the classic Shortest Path problem with vector costs. For every integer $p \in [2,\infty)$, we show a hardness of $\Omega(p(\log n / \log^2\log…
We present and study approximate notions of dimensional and margin complexity, which correspond to the minimal dimension or norm of an embedding required to approximate, rather then exactly represent, a given hypothesis class. We show that…
We present two Monte Carlo sampling algorithms for probabilistic inference that guarantee polynomial-time convergence for a larger class of network than current sampling algorithms provide. These new methods are variants of the known…
We study adaptive approximation algorithms for general multivariate linear problems where the sets of input functions are non-convex cones. While it is known that adaptive algorithms perform essentially no better than non-adaptive…
Online variants of the Expectation Maximization (EM) algorithm have recently been proposed to perform parameter inference with large data sets or data streams, in independent latent models and in hidden Markov models. Nevertheless, the…
We study the approximation of arbitrary distributions $P$ on $d$-dimensional space by distributions with log-concave density. Approximation means minimizing a Kullback--Leibler-type functional. We show that such an approximation exists if…
Reinforcement learning (RL) problems are fundamental in online decision-making and have been instrumental in finding an optimal policy for Markov decision processes (MDPs). Function approximations are usually deployed to handle large or…
We study the approximation of expectations $\E(f(X))$ for Gaussian random elements $X$ with values in a separable Hilbert space $H$ and Lipschitz continuous functionals $f \colon H \to \R$. We consider restricted Monte Carlo algorithms,…
This paper is devoted to the performance analysis of the detectors proposed in the companion paper where a comprehensive design framework is presented for the adaptive detection of subspace signals. The framework addresses four variations…
We study active sampling algorithms for linear regression, which aim to query only a few entries of a target vector $b\in\mathbb R^n$ and output a near minimizer to $\min_{x\in\mathbb R^d} \|Ax-b\|$, for a design matrix $A\in\mathbb R^{n…
We construct Monte Carlo methods for the $L^2$-approximation in Hilbert spaces of multivariate functions sampling no more than $n$ function values of the target function. Their errors catch up with the rate of convergence and the…
We study approximation by arbitrary linear combinations of $n$ translates of a single function of periodic functions. We construct some linear methods of this approximation for univariate functions in the class induced by the convolution…
It is a challenging problem that solving the \textit{multivariate linear model} (MLM) $\mathbf{A}\mathbf{x}=\mathbf{b}$ with the $\ell_1 $-norm approximation method such that $||\mathbf{A}\mathbf{x}-\mathbf{b}||_1$, the $\ell_1$-norm of the…
We study the integration of functions with respect to an unknown density. We compare the simple Monte Carlo method (which is almost optimal for a certain large class of inputs) and compare it with the Metropolis algorithm (based on a…
Mixtures of linear mixed models (MLMMs) are useful for clustering grouped data and can be estimated by likelihood maximization through the EM algorithm. The conventional approach to determining a suitable number of components is to compare…