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In this paper, we consider higher order paired symmetric tensors and strongly paired symmetric tensors. Elasticity tensors and higher order elasticity tensors in solid mechanics are strongly paired symmetric tensors. A (strongly) paired…
Principal component analysis (PCA) is a well-known linear dimension-reduction method that has been widely used in data analysis and modeling. It is an unsupervised learning technique that identifies a suitable linear subspace for the input…
We present a data analysis pipeline for CMB polarization experiments, running from multi-frequency maps to the power spectra. We focus mainly on component separation and, for the first time, we work out the covariance matrix accounting for…
While any symmetric and positive semidefinite mapping can be the non-centered covariance of a Gaussian random field, it is known that these conditions are no longer sufficient when the random field is valued in a two-point set. The question…
Conformal Prediction (CP) stands out as a robust framework for uncertainty quantification, which is crucial for ensuring the reliability of predictions. However, common CP methods heavily rely on data exchangeability, a condition often…
Current cosmological tensions motivate investigating extensions to the standard $\Lambda$CDM model. Additional model parameters are typically varied one or two at a time, in a series of separate tests. The purpose of this paper is to…
Probabilistic principal component analysis (PPCA) seeks a low dimensional representation of a data set in the presence of independent spherical Gaussian noise. The maximum likelihood solution for the model is an eigenvalue problem on the…
We study the estimation of the covariance matrix $\Sigma$ of a $p$-dimensional normal random vector based on $n$ independent observations corrupted by additive noise. Only a general nonparametric assumption is imposed on the distribution of…
It is well known that sparse approximation problem is \textsf{NP}-hard under general dictionaries. Several algorithms have been devised and analyzed in the past decade under various assumptions on the \emph{coherence} $\mu$ of the…
Motivated mainly by applications to partial differential equations with random coefficients, we introduce a new class of Monte Carlo estimators, called Toeplitz Monte Carlo (TMC) estimator for approximating the integral of a multivariate…
Sparse PCA is the optimization problem obtained from PCA by adding a sparsity constraint on the principal components. Sparse PCA is NP-hard and hard to approximate even in the single-component case. In this paper we settle the computational…
Conformal prediction (CP) offers distribution-free marginal coverage guarantees under an exchangeability assumption, but these guarantees can fail if the data distribution shifts. We analyze the use of pseudo-calibration as a tool to…
Conformal prediction (CP) constructs uncertainty sets for model outputs with finite-sample coverage guarantees. A candidate output is included in the prediction set if its non-conformity score is not considered extreme relative to the…
Predictive posterior densities (PPDs) are of interest in approximate Bayesian inference. Typically, these are estimated by simple Monte Carlo (MC) averages using samples from the approximate posterior. We observe that the signal-to-noise…
This paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions. Here, sparsistency refers to the property that all…
The robust PCA of covariance matrices plays an essential role when isolating key explanatory features. The currently available methods for performing such a low-rank plus sparse decomposition are matrix specific, meaning, those algorithms…
This paper addresses the following three topics: positive semidefinite (psd) matrix completions, universal rigidity of frameworks, and the Strong Arnold Property (SAP). We show some strong connections among these topics, using semidefinite…
We study the problem of quantifying epistemic predictive uncertainty (EPU) -- that is, uncertainty faced at prediction time due to the existence of multiple plausible predictive models -- within the framework of conformal prediction (CP).…
In this paper, we propose a computationally simple estimator of the asymptotic covariance matrix of the Principal Components (PC) factors valid in the presence of cross-correlated idiosyncratic components. The proposed estimator of the…
We propose a new method for statistical inference in generalized linear models. In the overparameterized regime, Principal Component Regression (PCR) reduces variance by projecting high-dimensional data to a low-dimensional principal…