Structural matrix-variate observations routinely arise in diverse fields such as multi-layer network analysis and brain image clustering. While data of this type have been extensively investigated with fruitful outcomes being delivered, the fundamental questions like its statistical optimality and computational limit are largely under-explored. In this paper, we propose a low-rank Gaussian mixture model (LrMM) assuming each matrix-valued observation has a planted low-rank structure. Minimax lower bounds for estimating the underlying low-rank matrix are established allowing a whole range of sample sizes and signal strength. Under a minimal condition on signal strength, referred to as the information-theoretical limit or statistical limit, we prove the minimax optimality of a maximum likelihood estimator which, in general, is computationally infeasible. If the signal is stronger than a certain threshold, called the computational limit, we design a computationally fast estimator based on spectral aggregation and demonstrate its minimax optimality. Moreover, when the signal strength is smaller than the computational limit, we provide evidences based on the low-degree likelihood ratio framework to claim that no polynomial-time algorithm can consistently recover the underlying low-rank matrix. Our results reveal multiple phase transitions in the minimax error rates and the statistical-to-computational gap. Numerical experiments confirm our theoretical findings. We further showcase the merit of our spectral aggregation method on the worldwide food trading dataset.
We derive uniform convergence rates for the maximum likelihood estimator and minimax lower bounds for parameter estimation in two-component location-scale Gaussian mixture models with unequal variances. We assume the mixing proportions of the mixture are known and fixed, but make no separation assumption on the underlying mixture components. A phase transition is shown to exist in the optimal parameter estimation rate, depending on whether or not the mixture is balanced. Key to our analysis is a careful study of the dependence between the parameters of location-scale Gaussian mixture models, a…
We study the Nonparametric Maximum Likelihood Estimator (NPMLE) for estimating Gaussian location mixture densities in $d$-dimensions from independent observations. Unlike usual likelihood-based methods for fitting mixtures, NPMLEs are based on convex optimization. We prove finite sample results on the Hellinger accuracy of every NPMLE. Our results imply, in particular, that every NPMLE achieves near parametric risk (up to logarithmic multiplicative factors) when the true density is a discrete Gaussian mixture without any prior information on the number of mixture components. NPMLEs can natural…
Gaussian mixture models are widely used to study clustering problems. These model-based clustering methods require an accurate estimation of the unknown data density by Gaussian mixtures. In Maugis and Michel (2009), a penalized maximum likelihood estimator is proposed for automatically selecting the number of mixture components. In the present paper, a collection of univariate densities whose logarithm is locally β-Hölder with moment and tail conditions are considered. We show that this penalized estimator is minimax adaptive to the β regularity of such densities in the Hellinger sense.
We propose the finite mixture of skewed sub-Gaussian stable distributions. The maximum likelihood estimator for the parameters of proposed finite mixture model is computed through the expectation-maximization algorithm. The proposed model contains the finite mixture of normal and skewed normal distributions. Since the tails of proposed model is heavier than even the Student's t distribution, it can be used as a powerful model for robust model-based clustering. Performance of the proposed model is demonstrated by clustering simulation data and two sets of real data.
University of Texas（德克萨斯大学）；University of California（加利福尼亚大学）
We provide a theoretical treatment of over-specified Gaussian mixtures of experts with covariate-free gating networks. We establish the convergence rates of the maximum likelihood estimation (MLE) for these models. Our proof technique is based on a novel notion of \emph{algebraic independence} of the expert functions. Drawing on optimal transport theory, we establish a connection between the algebraic independence and a certain class of partial differential equations (PDEs). Exploiting this connection allows us to derive convergence rates and minimax lower bounds for parameter estimation.
We propose the use of probability models for ranked data as a useful alternative to a quantitative data analysis to investigate the outcome of bioassay experiments, when the preliminary choice of an appropriate normalization method for the raw numerical responses is difficult or subject to criticism. We review standard distance-based and multistage ranking models and in this last context we propose an original generalization of the Plackett-Luce model to account for the order of the ranking elicitation process. The usefulness of the novel model is illustrated with its maximum likelihood estima…
Finite mixture of Gaussian distributions provide a flexible semi-parametric methodology for density estimation when the variables under investigation have no boundaries. However, in practical applications variables may be partially bounded (e.g. taking non-negative values) or completely bounded (e.g. taking values in the unit interval). In this case the standard Gaussian finite mixture model assigns non-zero densities to any possible values, even to those outside the ranges where the variables are defined, hence resulting in severe bias. In this paper we propose a transformation-based approach…
Ohio State University（俄亥俄州立大学）；Microsoft Research India（微软研究院印度）
In this paper we show that very large mixtures of Gaussians are efficiently learnable in high dimension. More precisely, we prove that a mixture with known identical covariance matrices whose number of components is a polynomial of any fixed degree in the dimension n is polynomially learnable as long as a certain non-degeneracy condition on the means is satisfied. It turns out that this condition is generic in the sense of smoothed complexity, as soon as the dimensionality of the space is high enough. Moreover, we prove that no such condition can possibly exist in low dimension and the problem…
Cluster-weighted modeling (CWM) is a mixture approach for modeling the joint probability of a response variable and a set of explanatory variables. The parameters are estimated by means of the expectation-maximization algorithm according to the maximum likelihood approach. Under Gaussian assumptions, we analyse the complete-data likelihood function of cluster weighted models. Further, under suitable hypotheses we show that the maximization of the likelihood function of Gaussian cluster weighted models leads to the same parameter estimates of finite mixtures of regression and finite mixtures of…
We determine the information-theoretic cutoff value on separation of cluster centers for exact recovery of cluster labels in a $K$-component Gaussian mixture model with equal cluster sizes. Moreover, we show that a semidefinite programming (SDP) relaxation of the $K$-means clustering method achieves such sharp threshold for exact recovery without assuming the symmetry of cluster centers.
This paper is concerned with an important issue in finite mixture modelling, the selection of the number of mixing components. We propose a new penalized likelihood method for model selection of finite multivariate Gaussian mixture models. The proposed method is shown to be statistically consistent in determining of the number of components. A modified EM algorithm is developed to simultaneously select the number of components and to estimate the mixing weights, i.e. the mixing probabilities, and unknown parameters of Gaussian distributions. Simulations and a real data analysis are presented t…
We show, by an explicit construction, that a mixture of univariate Gaussian densities with variance $1$ and means in $[-A,A]$ can have $Ω(A^2)$ modes. This disproves a recent conjecture of Dytso, Yagli, Poor and Shamai \cite{DYPS20} who showed that such a mixture can have at most $O(A^{2})$ modes and surmised that the upper bound could be improved to $O(A)$. Our result holds even if an additional variance constraint is imposed on the mixing distribution. Extending the result to higher dimensions, we exhibit a mixture of Gaussians in $\mathbb{R}^{d}$, with identity covariances and means inside …
Princeton University（普林斯顿大学）；University of California Berkeley（加州大学伯克利分校）
We study the problem of learning mixtures of low-rank models, i.e. reconstructing multiple low-rank matrices from unlabelled linear measurements of each. This problem enriches two widely studied settings -- low-rank matrix sensing and mixed linear regression -- by bringing latent variables (i.e. unknown labels) and structural priors (i.e. low-rank structures) into consideration. To cope with the non-convexity issues arising from unlabelled heterogeneous data and low-complexity structure, we develop a three-stage meta-algorithm that is guaranteed to recover the unknown matrices with near-optima…
Northwestern University（西北大学）；Halicioǧlu Data Science Institute, UC San Diego（加州大学圣地亚哥分校哈利西奥卢数据科学研究所）；University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；UC San Diego（加州大学圣地亚哥分校）
Mixtures of high dimensional Gaussian distributions have been studied extensively in statistics and learning theory. While the total variation distance appears naturally in the sample complexity of distribution learning, it is analytically difficult to obtain tight lower bounds for mixtures. Exploiting a connection between total variation distance and the characteristic function of the mixture, we provide fairly tight functional approximations. This enables us to derive new lower bounds on the total variation distance between pairs of two-component Gaussian mixtures that have a shared covarian…
We prove that all moment varieties of univariate Gaussian mixtures have the expected dimension. Our approach rests on intersection theory and Terracini's classification of defective surfaces. The analogous identifiability result is shown to be false for mixtures of Gaussians in dimension three and higher. Their moments up to third order define projective varieties that are defective. Our geometric study suggests an extension of the Alexander-Hirschowitz Theorem for Veronese varieties to the Gaussian setting.
For a given positive random variable $V>0$ and a given $Z\sim N(0,1)$ independent of $V$, we compute the scalar $t_0$ such that the distance between $Z\sqrt{V}$ and $Z\sqrt{t_0}$ in the $L^2(\R)$ sense, is minimal. We also consider the same problem in several dimensions when $V$ is a random positive definite matrix.
We study a dimensionality reduction technique for finite mixtures of high-dimensional multivariate response regression models. Both the dimension of the response and the number of predictors are allowed to exceed the sample size. We consider predictor selection and rank reduction to obtain lower-dimensional approximations. A class of estimators with a fast rate of convergence is introduced. We apply this result to a specific procedure, introduced in [11], where the relevant predictors are selected by the Group-Lasso.
Gaussian mixture models are central to classical statistics, widely used in the information sciences, and have a rich mathematical structure. We examine their maximum likelihood estimates through the lens of algebraic statistics. The MLE is not an algebraic function of the data, so there is no notion of ML degree for these models. The critical points of the likelihood function are transcendental, and there is no bound on their number, even for mixtures of two univariate Gaussians.
This paper investigates the minimum mean square error (MMSE) estimation of x, given the observation y = Hx+n, when x and n are independent and Gaussian Mixture (GM) distributed. The introduction of GM distributions, represents a generalization of the more familiar and simpler Gaussian signal and Gaussian noise instance. We present the necessary theoretical foundation and derive the MMSE estimator for x in a closed form. Furthermore, we provide upper and lower bounds for its mean square error (MSE). These bounds are validated through Monte Carlo simulations.
Gaussian mixture models are widely used in Statistics. A fundamental aspect of these distributions is the study of the local maxima of the density, or modes. In particular, it is not known how many modes a mixture of $k$ Gaussians in $d$ dimensions can have. We give a brief account of this problem's history. Then, we give improved lower bounds and the first upper bound on the maximum number of modes, provided it is finite.
University of California, San Diego（加利福尼亚大学圣地亚哥分校）
We resolve one of the major outstanding problems in robust statistics. In particular, if $X$ is an evenly weighted mixture of two arbitrary $d$-dimensional Gaussians, we devise a polynomial time algorithm that given access to samples from $X$ an $\eps$-fraction of which have been adversarially corrupted, learns $X$ to error $\poly(\eps)$ in total variation distance.
While several papers have investigated computationally and statistically efficient methods for learning Gaussian mixtures, precise minimax bounds for their statistical performance as well as fundamental limits in high-dimensional settings are not well-understood. In this paper, we provide precise information theoretic bounds on the clustering accuracy and sample complexity of learning a mixture of two isotropic Gaussians in high dimensions under small mean separation. If there is a sparse subset of relevant dimensions that determine the mean separation, then the sample complexity only depends …
We study the convergence behavior of the Expectation Maximization (EM) algorithm on Gaussian mixture models with an arbitrary number of mixture components and mixing weights. We show that as long as the means of the components are separated by at least $Ω(\sqrt{\min\{M,d\}})$, where $M$ is the number of components and $d$ is the dimension, the EM algorithm converges locally to the global optimum of the log-likelihood. Further, we show that the convergence rate is linear and characterize the size of the basin of attraction to the global optimum.
Nowadays in density estimation, posterior rates of convergence for location and location-scale mixtures of Gaussians are only known under light-tail assumptions; with better rates achieved by location mixtures. It is conjectured, but not proved, that the situation should be reversed under heavy tails assumptions. The conjecture is based on the feeling that there is no need to achieve a good order of approximation in regions with few data (say, in the tails), favoring location-scale mixtures which allow for spatially varying order of approximation. Here we test the previous argument on the Gaus…
The points of a moment variety are the vectors of all moments up to some order of a family of probability distributions. We study this variety for mixtures of Gaussians. Following up on Pearson's classical work from 1894, we apply current tools from computational algebra to recover the parameters from the moments. Our moment varieties extend objects familiar to algebraic geometers. For instance, the secant varieties of Veronese varieties are the loci obtained by setting all covariance matrices to zero. We compute the ideals of the 5-dimensional moment varieties representing mixtures of two uni…
Let $(X\_1,\ldots,X\_n)$ be a $d$-dimensional i.i.d sample from a distribution with density $f$. The problem of detection of a two-component mixture is considered. Our aim is to decide whether $f$ is the density of a standard Gaussian random $d$-vector ($f=ϕ\_d$) against $f$ is a two-component mixture: $f=(1-\varepsilon)ϕ\_d +\varepsilon ϕ\_d (.-μ)$ where $(\varepsilon,μ)$ are unknown parameters. Optimal separation conditions on $\varepsilon, μ, n$ and the dimension $d$ are established, allowing to separate both hypotheses with prescribed errors. Several testing procedures are proposed and two…
We wish to estimate conditional density using Gaussian Mixture Regression model with logistic weights and means depending on the covariate. We aim at selecting the number of components of this model as well as the other parameters by a penalized maximum likelihood approach. We provide a lower bound on penalty, proportional up to a logarithmic term to the dimension of each model, that ensures an oracle inequality for our estimator. Our theoretical analysis is supported by some numerical experiments.
This work represents a natural coalescence of two important lines of work: learning mixtures of Gaussians and algorithmic robust statistics. In particular we give the first provably robust algorithm for learning mixtures of any constant number of Gaussians. We require only mild assumptions on the mixing weights (bounded fractionality) and that the total variation distance between components is bounded away from zero. At the heart of our algorithm is a new method for proving dimension-independent polynomial identifiability through applying a carefully chosen sequence of differential operations …
We consider the problem of estimating means of two Gaussians in a 2-Gaussian mixture, which is not balanced and is corrupted by noise of an arbitrary distribution. We present a robust algorithm to estimate the parameters, together with upper bounds on the numbers of samples required for the estimate to be correct, where the bounds are parametrised by the dimension, ratio of the mixing coefficients, a measure of the separation of the two Gaussians, related to Mahalanobis distance, and a condition number of the covariance matrix. In theory, this is the first sample-complexity result for imbalanc…
At least seven modes in a heteroscedastic three-component bivariate Gaussian mixture
Graduate School of Science and Engineering, Kagoshima University（鹿儿岛大学理工学研究科）；School of Information and Data Sciences, Nagasaki University（长崎大学信息数据科学学院）；Institute of Statistical Mathematics（统计数学研究所）；The Graduate University for Advanced Studies, SOKENDAI（综合研究大学院大学）；RIKEN（理化学研究所）
We propose a new procedure for clustering nucleotide sequences based on the "Laplacian Eigenmaps" and Gaussian Mixture modelling. This proposal is then applied to a set of 100 DNA sequences from the mitochondrially encoded NADH dehydrogenase 3 (ND3) gene of a collection of Platyhelminthes and Nematoda species. The resulting clusters are then shown to be consistent with the gene phylogenetic tree computed using a maximum likelihood approach. This comparison shows in particular that the clustering produced by the methodology combining Laplacian Eigenmaps with Gaussian Mixture models is coherent …
The ability to quantify distinctness of a cluster structure is fundamental for certain simulation studies, in particular for those comparing performance of different classification algorithms. The intrinsic integral measure based on the overlap of corresponding mixture components is often analytically intractable. This is also the case for Gaussian mixture models with unequal covariance matrices when space dimension $d > 1$. In this work we focus on Gaussian mixture models and at the sample level we assume the class assignments to be known. We derive a measure of component overlap based on eig…
