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In this article, we study curvature-like feature value of data sets in Euclidean spaces. First we formulate such curvature functions with desirable properties under the manifold hypothesis. Then we make a test property for the validity of the curvature function by the law of large numbers, and check it for the function we construct by numerical experiments. These experiments also suggest us to conjecture that mean of the curvature of sample manifolds coincides with the curvature of the mean manifold. Our construction is based on the dimension estimation by the principal component analysis and the Gaussian curvature of hypersurfaces. Our function depends on provisional parameters $\varepsilon, \delta$, and we suggest to deal with the resulting functions as a function of these parameters to get some robustness. As an application, we propose a method to decompose data sets into some parts reflecting local structure. For this, we embed the data sets into higher dimensional Euclidean space by using curvature values and cluster them in the embedded space. We also give some computational experiments that support effectiveness of our methods.

Yang et al. (2016) proved that the symmetric random walk Metropolis--Hastings algorithm for Bayesian variable selection is rapidly mixing under mild high-dimensional assumptions. We propose a novel MCMC sampler using an informed proposal scheme, which we prove achieves a much faster mixing time that is independent of the number of covariates, under the same assumptions. To the best of our knowledge, this is the first high-dimensional result which rigorously shows that the mixing rate of informed MCMC methods can be fast enough to offset the computational cost of local posterior evaluation. Motivated by the theoretical analysis of our sampler, we further propose a new approach called "two-stage drift condition" to studying convergence rates of Markov chains on general state spaces, which can be useful for obtaining tight complexity bounds in high-dimensional settings. The practical advantages of our algorithm are illustrated by both simulation studies and real data analysis.

We develop a new primitive for stochastic optimization: a low-bias, low-cost estimator of the minimizer $x_\star$ of any Lipschitz strongly-convex function. In particular, we use a multilevel Monte-Carlo approach due to Blanchet and Glynn to turn any optimal stochastic gradient method into an estimator of $x_\star$ with bias $\delta$, variance $O(\log(1/\delta))$, and an expected sampling cost of $O(\log(1/\delta))$ stochastic gradient evaluations. As an immediate consequence, we obtain cheap and nearly unbiased gradient estimators for the Moreau-Yoshida envelope of any Lipschitz convex function, allowing us to perform dimension-free randomized smoothing. We demonstrate the potential of our estimator through four applications. First, we develop a method for minimizing the maximum of $N$ functions, improving on recent results and matching a lower bound up logarithmic factors. Second and third, we recover state-of-the-art rates for projection-efficient and gradient-efficient optimization using simple algorithms with a transparent analysis. Finally, we show that an improved version of our estimator would yield a nearly linear-time, optimal-utility, differentially-private non-smooth stochastic optimization method.

Mixture transition distribution time series models build high-order dependence through a weighted combination of first-order transition densities for each one of a specified number of lags. We present a framework to construct stationary transition mixture distribution models that extend beyond linear, Gaussian dynamics. We study conditions for first-order strict stationarity which allow for different constructions with either continuous or discrete families for the first-order transition densities given a pre-specified family for the marginal density, and with general forms for the resulting conditional expectations. Inference and prediction are developed under the Bayesian framework with particular emphasis on flexible, structured priors for the mixture weights. Model properties are investigated both analytically and through synthetic data examples. Finally, Poisson and Lomax examples are illustrated through real data applications.

Stochastic Approximation (SA) is a popular approach for solving fixed-point equations where the information is corrupted by noise. In this paper, we consider an SA involving a contraction mapping with respect to an arbitrary norm, and show its finite-sample error bounds while using different stepsizes. The idea is to construct a smooth Lyapunov function using the generalized Moreau envelope, and show that the iterates of SA have negative drift with respect to that Lyapunov function. Our result is applicable in Reinforcement Learning (RL). In particular, we use it to establish the first-known convergence rate of the V-trace algorithm for off-policy TD-learning. Moreover, we also use it to study TD-learning in the on-policy setting, and recover the existing state-of-the-art results for $Q$-learning. Importantly, our construction results in only a logarithmic dependence of the convergence bound on the size of the state-space.

Large-scale modern data often involves estimation and testing for high-dimensional unknown parameters. It is desirable to identify the sparse signals, ``the needles in the haystack'', with accuracy and false discovery control. However, the unprecedented complexity and heterogeneity in modern data structure require new machine learning tools to effectively exploit commonalities and to robustly adjust for both sparsity and heterogeneity. In addition, estimates for high-dimensional parameters often lack uncertainty quantification. In this paper, we propose a novel Spike-and-Nonparametric mixture prior (SNP) -- a spike to promote the sparsity and a nonparametric structure to capture signals. In contrast to the state-of-the-art methods, the proposed methods solve the estimation and testing problem at once with several merits: 1) an accurate sparsity estimation; 2) point estimates with shrinkage/soft-thresholding property; 3) credible intervals for uncertainty quantification; 4) an optimal multiple testing procedure that controls false discovery rate. Our method exhibits promising empirical performance on both simulated data and a gene expression case study.

Assessing homogeneity of distributions is an old problem that has received considerable attention, especially in the nonparametric Bayesian literature. To this effect, we propose the semi-hierarchical Dirichlet process, a novel hierarchical prior that extends the hierarchical Dirichlet process of Teh et al. (2006) and that avoids the degeneracy issues of nested processes recently described by Camerlenghi et al. (2019a). We go beyond the simple yes/no answer to the homogeneity question and embed the proposed prior in a random partition model; this procedure allows us to give a more comprehensive response to the above question and in fact find groups of populations that are internally homogeneous when I greater or equal than 2 such populations are considered. We study theoretical properties of the semi-hierarchical Dirichlet process and of the Bayes factor for the homogeneity test when I = 2. Extensive simulation studies and applications to educational data are also discussed.

In information retrieval (IR) and related tasks, term weighting approaches typically consider the frequency of the term in the document and in the collection in order to compute a score reflecting the importance of the term for the document. In tasks characterized by the presence of training data (such as text classification) it seems logical that the term weighting function should take into account the distribution (as estimated from training data) of the term across the classes of interest. Although `supervised term weighting' approaches that use this intuition have been described before, they have failed to show consistent improvements. In this article we analyse the possible reasons for this failure, and call consolidated assumptions into question. Following this criticism we propose a novel supervised term weighting approach that, instead of relying on any predefined formula, learns a term weighting function optimised on the training set of interest; we dub this approach \emph{Learning to Weight} (LTW). The experiments that we run on several well-known benchmarks, and using different learning methods, show that our method outperforms previous term weighting approaches in text classification.

Using the 6,638 case descriptions of societal impact submitted for evaluation in the Research Excellence Framework (REF 2014), we replicate the topic model (Latent Dirichlet Allocation or LDA) made in this context and compare the results with factor-analytic results using a traditional word-document matrix (Principal Component Analysis or PCA). Removing a small fraction of documents from the sample, for example, has on average a much larger impact on LDA than on PCA-based models to the extent that the largest distortion in the case of PCA has less effect than the smallest distortion of LDA-based models. In terms of semantic coherence, however, LDA models outperform PCA-based models. The topic models inform us about the statistical properties of the document sets under study, but the results are statistical and should not be used for a semantic interpretation - for example, in grant selections and micro-decision making, or scholarly work-without follow-up using domain-specific semantic maps.

Robust estimation is much more challenging in high dimensions than it is in one dimension: Most techniques either lead to intractable optimization problems or estimators that can tolerate only a tiny fraction of errors. Recent work in theoretical computer science has shown that, in appropriate distributional models, it is possible to robustly estimate the mean and covariance with polynomial time algorithms that can tolerate a constant fraction of corruptions, independent of the dimension. However, the sample and time complexity of these algorithms is prohibitively large for high-dimensional applications. In this work, we address both of these issues by establishing sample complexity bounds that are optimal, up to logarithmic factors, as well as giving various refinements that allow the algorithms to tolerate a much larger fraction of corruptions. Finally, we show on both synthetic and real data that our algorithms have state-of-the-art performance and suddenly make high-dimensional robust estimation a realistic possibility.