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Randomization has shown catalyzing effects in linear algebra with promising perspectives for tackling computational challenges in large-scale problems. For solving a system of linear equations, we demonstrate the convergence of a broad class of algorithms that at each step pick a subset of $n$ equations at random and update the iterate with the orthogonal projection to the subspace those equations represent. We identify, in this context, a specific degree-$n$ polynomial that non-linearly transforms the singular values of the system towards equalization. This transformation to singular values and the corresponding condition number then characterizes the expected convergence rate of iterations. As a consequence, our results specify the convergence rate of the stochastic gradient descent algorithm, in terms of the mini-batch size $n$, when used for solving systems of linear equations.

We introduce a procedure for conditional density estimation under logarithmic loss, which we call SMP (Sample Minmax Predictor). This estimator minimizes a new general excess risk bound for statistical learning. On standard examples, this bound scales as $d/n$ with $d$ the model dimension and $n$ the sample size, and critically remains valid under model misspecification. Being an improper (out-of-model) procedure, SMP improves over within-model estimators such as the maximum likelihood estimator, whose excess risk degrades under misspecification. Compared to approaches reducing to the sequential problem, our bounds remove suboptimal $\log n$ factors and can handle unbounded classes. For the Gaussian linear model, the predictions and risk bound of SMP are governed by leverage scores of covariates, nearly matching the optimal risk in the well-specified case without conditions on the noise variance or approximation error of the linear model. For logistic regression, SMP provides a non-Bayesian approach to calibration of probabilistic predictions relying on virtual samples, and can be computed by solving two logistic regressions. It achieves a non-asymptotic excess risk of $O((d + B^2R^2)/n)$, where $R$ bounds the norm of features and $B$ that of the comparison parameter; by contrast, no within-model estimator can achieve better rate than $\min({B R}/{\sqrt{n}}, {d e^{BR}}/{n} )$ in general. This provides a more practical alternative to Bayesian approaches, which require approximate posterior sampling, thereby partly addressing a question raised by Foster et al. (2018).

This paper provides some extended results on estimating the parameter matrix of high-dimensional regression model when the covariate or response possess weaker moment condition. We investigate the $M$-estimator of Fan et al. (Ann Stat 49(3):1239--1266, 2021) for matrix completion model with $(1+\epsilon)$-th moments. The corresponding phase transition phenomenon is observed. When $\epsilon \geq 1$, the robust estimator possesses the same convergence rate as previous literature. While $1> \epsilon>0$, the rate will be slower. For high dimensional multiple index coefficient model, we also apply the element-wise truncation method to construct a robust estimator which handle missing and heavy-tailed data with finite fourth moment.

It is often of interest to estimate regression functions non-parametrically. Penalized regression (PR) is one statistically-effective, well-studied solution to this problem. Unfortunately, in many cases, finding exact solutions to PR problems is computationally intractable. In this manuscript, we propose a mesh-based approximate solution (MBS) for those scenarios. MBS transforms the complicated functional minimization of NPR, to a finite parameter, discrete convex minimization; and allows us to leverage the tools of modern convex optimization. We show applications of MBS in a number of explicit examples (including both uni- and multi-variate regression), and explore how the number of parameters must increase with our sample-size in order for MBS to maintain the rate-optimality of NPR. We also give an efficient algorithm to minimize the MBS objective while effectively leveraging the sparsity inherent in MBS.

A phase-type distribution is the distribution of the time until absorption in a finite state-space time-homogeneous Markov jump process, with one absorbing state and the rest being transient. These distributions are mathematically tractable and conceptually attractive to model physical phenomena due to their interpretation in terms of a hidden Markov structure. Three recent extensions of regular phase-type distributions give rise to models which allow for heavy tails: discrete- or continuous-scaling; fractional-time semi-Markov extensions; and inhomogeneous time-change of the underlying Markov process. In this paper, we present a unifying theory for heavy-tailed phase-type distributions for which all three approaches are particular cases. Our main objective is to provide useful models for heavy-tailed phase-type distributions, but any other tail behavior is also captured by our specification. We provide relevant new examples and also show how existing approaches are naturally embedded. Subsequently, two multivariate extensions are presented, inspired by the univariate construction which can be considered as a matrix version of a frailty model. We provide fully explicit EM-algorithms for all models and illustrate them using synthetic and real-life data.

Estimation of linear functionals from observed data is an important task in many subjects. Juditsky & Nemirovski [The Annals of Statistics 37.5A (2009): 2278-2300] propose a framework for non-parametric estimation of linear functionals in a very general setting, with nearly minimax optimal confidence intervals. They compute this estimator and the associated confidence interval by approximating the saddle-point of a function. While this optimization problem is convex, it is rather difficult to solve using existing off-the-shelf optimization software. Furthermore, this computation can be expensive when the estimators live in a high-dimensional space. We propose a different algorithm to construct this estimator. Our algorithm can be used with existing optimization software and is much cheaper to implement even when the estimators are in a high-dimensional space, as long as the Hellinger affinity (or the Bhattacharyya coefficient) for the chosen parametric distribution can be efficiently computed given the parameters. We hope that our algorithm will foster the adoption of this estimation technique to a wider variety of problems with relative ease.

Univariate and multivariate general linear regression models, subject to linear inequality constraints, arise in many scientific applications. The linear inequality restrictions on model parameters are often available from phenomenological knowledge and motivated by machine learning applications of high-consequence engineering systems (Agrell, 2019; Veiga and Marrel, 2012). Some studies on the multiple linear models consider known linear combinations of the regression coefficient parameters restricted between upper and lower bounds. In the present paper, we consider both univariate and multivariate general linear models subjected to this kind of linear restrictions. So far, research on univariate cases based on Bayesian methods is all under the condition that the coefficient matrix of the linear restrictions is a square matrix of full rank. This condition is not, however, always feasible. Another difficulty arises at the estimation step by implementing the Gibbs algorithm, which exhibits, in most cases, slow convergence. This paper presents a Bayesian method to estimate the regression parameters when the matrix of the constraints providing the set of linear inequality restrictions undergoes no condition. For the multivariate case, our Bayesian method estimates the regression parameters when the number of the constrains is less than the number of the regression coefficients in each multiple linear models. We examine the efficiency of our Bayesian method through simulation studies for both univariate and multivariate regressions. After that, we illustrate that the convergence of our algorithm is relatively faster than the previous methods. Finally, we use our approach to analyze two real datasets.

Unlike univariate extreme value theory, multivariate extreme value distributions cannot be specified through a finite-dimensional parameter family of distributions. Instead, the many facets of multivariate extremes are mirrored in the inherent dependence structure of component-wise maxima which must be dissociated from the limiting extreme behaviour of its marginal distribution functions before a probabilistic characterisation of an extreme value quality can be determined. Mechanisms applied to elicit extremal dependence typically rely on standardisation of the unknown marginal distribution functions from which pseudo-observations for either Pareto or Fr\'echet marginals result. The relative merits of both of these choices for transformation of marginals have been discussed in the literature, particularly in the context of domains of attraction of an extreme value distribution. This paper is set within this context of modelling penultimate dependence as it proposes a unifying class of estimators for the residual dependence index that eschews consideration of choice of marginals. In addition, a reduced bias variant of the new class of estimators is introduced and their asymptotic properties are developed. The pivotal role of the unifying marginal transform in effectively removing bias is borne by a comprehensive simulation study. The leading application in this paper comprises an analysis of asymptotic independence between rainfall occurrences originating from monsoon-related events at several locations in Ghana.

Gaussian covariance graph model is a popular model in revealing underlying dependency structures among random variables. A Bayesian approach to the estimation of covariance structures uses priors that force zeros on some off-diagonal entries of covariance matrices and put a positive definite constraint on matrices. In this paper, we consider a spike and slab prior on off-diagonal entries, which uses a mixture of point-mass and normal distribution. The point-mass naturally introduces sparsity to covariance structures so that the resulting posterior from this prior renders covariance structure learning. Under this prior, we calculate posterior model probabilities of covariance structures using Laplace approximation. We show that the error due to Laplace approximation becomes asymptotically marginal at some rate depending on the posterior convergence rate of covariance matrix under the Frobenius norm. With the approximated posterior model probabilities, we propose a new framework for estimating a covariance structure. Since the Laplace approximation is done around the mode of conditional posterior of covariance matrix, which cannot be obtained in the closed form, we propose a block coordinate descent algorithm to find the mode and show that the covariance matrix can be estimated using this algorithm once the structure is chosen. Through a simulation study based on five numerical models, we show that the proposed method outperforms graphical lasso and sample covariance matrix in terms of root mean squared error, max norm, spectral norm, specificity, and sensitivity. Also, the advantage of the proposed method is demonstrated in terms of accuracy compared to our competitors when it is applied to linear discriminant analysis (LDA) classification to breast cancer diagnostic dataset.

Discrete random structures are important tools in Bayesian nonparametrics and the resulting models have proven effective in density estimation, clustering, topic modeling and prediction, among others. In this paper, we consider nested processes and study the dependence structures they induce. Dependence ranges between homogeneity, corresponding to full exchangeability, and maximum heterogeneity, corresponding to (unconditional) independence across samples. The popular nested Dirichlet process is shown to degenerate to the fully exchangeable case when there are ties across samples at the observed or latent level. To overcome this drawback, inherent to nesting general discrete random measures, we introduce a novel class of latent nested processes. These are obtained by adding common and group-specific completely random measures and, then, normalising to yield dependent random probability measures. We provide results on the partition distributions induced by latent nested processes, and develop an Markov Chain Monte Carlo sampler for Bayesian inferences. A test for distributional homogeneity across groups is obtained as a by product. The results and their inferential implications are showcased on synthetic and real data.