This paper proposes a novel method for deep learning based on the analytical convolution of multidimensional Gaussian mixtures. In contrast to tensors, these do not suffer from the curse of dimensionality and allow for a compact representation, as data is only stored where details exist. Convolution kernels and data are Gaussian mixtures with unconstrained weights, positions, and covariance matrices. Similar to discrete convolutional networks, each convolution step produces several feature channels, represented by independent Gaussian mixtures. Since traditional transfer functions like ReLUs do not produce Gaussian mixtures, we propose using a fitting of these functions instead. This fitting step also acts as a pooling layer if the number of Gaussian components is reduced appropriately. We demonstrate that networks based on this architecture reach competitive accuracy on Gaussian mixtures fitted to the MNIST and ModelNet data sets.
McMaster University（麦克马斯特大学）；University of Toronto（多伦多大学）
We consider the problem of learning mixtures of Gaussians under the constraint of approximate differential privacy. We prove that $\widetilde{O}(k^2 d \log^{3/2}(1/δ) / α^2 \varepsilon)$ samples are sufficient to learn a mixture of $k$ axis-aligned Gaussians in $\mathbb{R}^d$ to within total variation distance $α$ while satisfying $(\varepsilon, δ)$-differential privacy. This is the first result for privately learning mixtures of unbounded axis-aligned (or even unbounded univariate) Gaussians. If the covariance matrices of each of the Gaussians is the identity matrix, we show that $\widetilde{…
We propose and analyze a new vantage point for the learning of mixtures of Gaussians: namely, the PAC-style model of learning probability distributions introduced by Kearns et al. Here the task is to construct a hypothesis mixture of Gaussians that is statistically indistinguishable from the actual mixture generating the data; specifically, the KL-divergence should be at most epsilon. In this scenario, we give a poly(n/epsilon)-time algorithm that learns the class of mixtures of any constant number of axis-aligned Gaussians in n-dimensional Euclidean space. Our algorithm makes no assumptions a…
Microsoft Research（微软研究院）；MIT EECS（麻省理工学院电子工程与计算机科学系）
Efficiently learning mixture of Gaussians is a fundamental problem in statistics and learning theory. Given samples coming from a random one out of k Gaussian distributions in Rn, the learning problem asks to estimate the means and the covariance matrices of these Gaussians. This learning problem arises in many areas ranging from the natural sciences to the social sciences, and has also found many machine learning applications. Unfortunately, learning mixture of Gaussians is an information theoretically hard problem: in order to learn the parameters up to a reasonable accuracy, the number of s…
One of the most popular algorithms for clustering in Euclidean space is the $k$-means algorithm; $k$-means is difficult to analyze mathematically, and few theoretical guarantees are known about it, particularly when the data is {\em well-clustered}. In this paper, we attempt to fill this gap in the literature by analyzing the behavior of $k$-means on well-clustered data. In particular, we study the case when each cluster is distributed as a different Gaussian -- or, in other words, when the input comes from a mixture of Gaussians. We analyze three aspects of the $k$-means algorithm under this …
Clustering mixtures of Gaussian distributions is a fundamental and challenging problem that is ubiquitous in various high-dimensional data processing tasks. While state-of-the-art work on learning Gaussian mixture models has focused primarily on improving separation bounds and their generalization to arbitrary classes of mixture models, less emphasis has been paid to practical computational efficiency of the proposed solutions. In this paper, we propose a novel and highly efficient clustering algorithm for $n$ points drawn from a mixture of two arbitrary Gaussian distributions in $\mathbb{R}^p…
Kogakuin University of Technology and Engineering（工学院大学）
Finite mixture models, typically Gaussian mixtures, are well known and widely used as model-based clustering. In practical situations, there are many non-Gaussian data that are heavy-tailed and/or asymmetric. Normal inverse Gaussian (NIG) distributions are normal-variance mean which mixing densities are inverse Gaussian distributions and can be used for both haavy-tail and asymmetry. For NIG mixture models, both expectation-maximization method and variational Bayesian (VB) algorithms have been proposed. However, the existing VB algorithm for NIG mixture have a disadvantage that the shape of th…
This paper uses Gaussian mixture model instead of linear Gaussian model to fit the distribution of every node in Bayesian network. We will explain why and how we use Gaussian mixture models in Bayesian network. Meanwhile we propose a new method, called double iteration algorithm, to optimize the mixture model, the double iteration algorithm combines the expectation maximization algorithm and gradient descent algorithm, and it performs perfectly on the Bayesian network with mixture models. In experiments we test the Gaussian mixture model and the optimization algorithm on different graphs which…
Deep learning is a hierarchical inference method formed by subsequent multiple layers of learning able to more efficiently describe complex relationships. In this work, Deep Gaussian Mixture Models are introduced and discussed. A Deep Gaussian Mixture model (DGMM) is a network of multiple layers of latent variables, where, at each layer, the variables follow a mixture of Gaussian distributions. Thus, the deep mixture model consists of a set of nested mixtures of linear models, which globally provide a nonlinear model able to describe the data in a very flexible way. In order to avoid overparam…
We quantify the parameter stability of a spherical Gaussian Mixture Model (sGMM) under small perturbations in distribution space. Namely, we derive the first explicit bound to show that for a mixture of spherical Gaussian $P$ (sGMM) in a pre-defined model class, all other sGMM close to $P$ in this model class in total variation distance has a small parameter distance to $P$. Further, this upper bound only depends on $P$. The motivation for this work lies in providing guarantees for fitting Gaussian mixtures; with this aim in mind, all the constants involved are well defined and distribution fr…
Given data drawn from a mixture of multivariate Gaussians, a basic problem is to accurately estimate the mixture parameters. We give an algorithm for this problem that has a running time, and data requirement polynomial in the dimension and the inverse of the desired accuracy, with provably minimal assumptions on the Gaussians. As simple consequences of our learning algorithm, we can perform near-optimal clustering of the sample points and density estimation for mixtures of k Gaussians, efficiently. The building blocks of our algorithm are based on the work Kalai et al. [STOC 2010] that gives …
In this conceptual work, we present Deep Convolutional Gaussian Mixture Models (DCGMMs): a new formulation of deep hierarchical Gaussian Mixture Models (GMMs) that is particularly suitable for describing and generating images. Vanilla (i.e., flat) GMMs require a very large number of components to describe images well, leading to long training times and memory issues. DCGMMs avoid this by a stacked architecture of multiple GMM layers, linked by convolution and pooling operations. This allows to exploit the compositionality of images in a similar way as deep CNNs do. DCGMMs can be trained end-to…
Identification and extraction of singing voice from within musical mixtures is a key challenge in source separation and machine audition. Recently, deep neural networks (DNN) have been used to estimate 'ideal' binary masks for carefully controlled cocktail party speech separation problems. However, it is not yet known whether these methods are capable of generalizing to the discrimination of voice and non-voice in the context of musical mixtures. Here, we trained a convolutional DNN (of around a billion parameters) to provide probabilistic estimates of the ideal binary mask for separation of v…
KTH Royal Institute of Technology（KTH皇家理工学院）；Schlumberger Cambridge Research（斯伦贝谢剑桥研究中心）
We propose a greedy variational method for decomposing a non-negative multivariate signal as a weighted sum of Gaussians, which, borrowing the terminology from statistics, we refer to as a Gaussian mixture model. Notably, our method has the following features: (1) It accepts multivariate signals, i.e. sampled multivariate functions, histograms, time series, images, etc. as input. (2) The method can handle general (i.e. ellipsoidal) Gaussians. (3) No prior assumption on the number of mixture components is needed. To the best of our knowledge, no previous method for Gaussian mixture model decomp…
This paper presents a new model called infinite mixtures of multivariate Gaussian processes, which can be used to learn vector-valued functions and applied to multitask learning. As an extension of the single multivariate Gaussian process, the mixture model has the advantages of modeling multimodal data and alleviating the computationally cubic complexity of the multivariate Gaussian process. A Dirichlet process prior is adopted to allow the (possibly infinite) number of mixture components to be automatically inferred from training data, and Markov chain Monte Carlo sampling techniques are use…
Predicting smartphone users activity using WiFi fingerprints has been a popular approach for indoor positioning in recent years. However, such a high dimensional time-series prediction problem can be very tricky to solve. To address this issue, we propose a novel deep learning model, the convolutional mixture density recurrent neural network (CMDRNN), which combines the strengths of convolutional neural networks, recurrent neural networks and mixture density networks. In our model, the CNN sub-model is employed to detect the feature of the high dimensional input, the RNN sub-model is utilized …
This work provides a computationally efficient and statistically consistent moment-based estimator for mixtures of spherical Gaussians. Under the condition that component means are in general position, a simple spectral decomposition technique yields consistent parameter estimates from low-order observable moments, without additional minimum separation assumptions needed by previous computationally efficient estimation procedures. Thus computational and information-theoretic barriers to efficient estimation in mixture models are precluded when the mixture components have means in general posit…
Diffusion Model Conditioning on Gaussian Mixture Model and Negative Gaussian Mixture Gradient
University of Macau（澳门大学）；Great Bay University（大湾区大学）；Great Bay Institute for Advanced Study（大湾区高等研究院）；University of Science and Technology of China（中国科学技术大学）
Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences（中国科学院深圳先进技术研究院）；Shenzhen College of Advanced Technology, University of Chinese Academy of Sciences（中国科学院大学深圳先进技术学院）；Institute for Computing and Information Sciences, Radboud University（拉德堡德大学计算与信息科学研究所）；Faculty of Management, Science and Technology, Open University of The Netherlands（荷兰开放大学管理、科学与技术学院）
Multi-output Gaussian processes (MOGPs) are an extension of Gaussian Processes (GPs) for predicting multiple output variables (also called channels, tasks) simultaneously. In this paper we use the convolution theorem to design a new kernel for MOGPs, by modeling cross channel dependencies through cross convolution of time and phase delayed components in the spectral domain. The resulting kernel is called Multi-Output Convolution Spectral Mixture (MOCSM) kernel. Results of extensive experiments on synthetic and real-life datasets demonstrate the advantages of the proposed kernel and its state o…
Deep unfolding showed to be a very successful approach for accelerating and tuning classical signal processing algorithms. In this paper, we propose learned Gaussian-mixture AMP (L-GM-AMP) - a plug-and-play compressed sensing (CS) recovery algorithm suitable for any i.i.d. source prior. Our algorithm builds upon Borgerding's learned AMP (LAMP), yet significantly improves it by adopting a universal denoising function within the algorithm. The robust and flexible denoiser is a byproduct of modelling source prior with a Gaussian-mixture (GM), which can well approximate continuous, discrete, as we…
Vector-valued Gaussian mixtures form an important special subset of vector-valued distributions. In general, vector-valued distributions constitute natural representations for physical entities, which can mutate or transit among alternative manifestations distributed in a given space. A key example is color imagery. In this note, we vectorize the Gaussian mixture model and study several different optimal mass transport related problems associated to such models. The benefits of using vector Gaussian mixture for optimal mass transport include computational efficiency and the ability to preserve…
In this paper we present a method for learning the parameters of a mixture of $k$ identical spherical Gaussians in $n$-dimensional space with an arbitrarily small separation between the components. Our algorithm is polynomial in all parameters other than $k$. The algorithm is based on an appropriate grid search over the space of parameters. The theoretical analysis of the algorithm hinges on a reduction of the problem to 1 dimension and showing that two 1-dimensional mixtures whose densities are close in the $L^2$ norm must have similar means and mixing coefficients. To produce such a lower bo…
We introduce Latent Vector Grammars (LVeGs), a new framework that extends latent variable grammars such that each nonterminal symbol is associated with a continuous vector space representing the set of (infinitely many) subtypes of the nonterminal. We show that previous models such as latent variable grammars and compositional vector grammars can be interpreted as special cases of LVeGs. We then present Gaussian Mixture LVeGs (GM-LVeGs), a new special case of LVeGs that uses Gaussian mixtures to formulate the weights of production rules over subtypes of nonterminals. A major advantage of using…
In data-mining applications, we are frequently faced with a large fraction of missing entries in the data matrix, which is problematic for most discriminant machine learning algorithms. A solution that we explore in this paper is the use of a generative model (a mixture of Gaussians) to compute the conditional expectation of the missing variables given the observed variables. Since training a Gaussian mixture with many different patterns of missing values can be computationally very expensive, we introduce a spanning-tree based algorithm that significantly speeds up training in these condition…
The article considers semi-supervised multitask learning on a Gaussian mixture model (GMM). Using methods from statistical physics, we compute the asymptotic Bayes risk of each task in the regime of large datasets in high dimension, from which we analyze the role of task similarity in learning and evaluate the performance gain when tasks are learned together rather than separately. In the supervised case, we derive a simple algorithm that attains the Bayes optimal performance.
Gaussian mixture models find their place as a powerful tool, mostly in the clustering problem, but with proper preparation also in feature extraction, pattern recognition, image segmentation and in general machine learning. When faced with the problem of schema matching, different mixture models computed on different pieces of data can maintain crucial information about the structure of the dataset. In order to measure or compare results from mixture models, the Wasserstein distance can be very useful, however it is not easy to calculate for mixture distributions. In this paper we derive one o…
Theoretical Error Analysis of Entropy Approximation for Gaussian Mixtures
Education and Research Center for Mathematical and Data Science, Shimane University（岛根大学数学与数据科学教育研究中心）；Graduate School of Mathematics, Nagoya University（名古屋大学数学研究生院）；Mathematical and Systems Engineering, Shizuoka University（静冈大学数学与系统工程系）；Graduate School of Biomedical Engineering, Tohoku University（东北大学生物医学工程研究生院）；AISIN SOFTWARE（爱信软件公司）
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.
Fukuoka University（福冈大学）；University of Tsukuba（筑波大学）
We give a proof that, under relatively mild conditions, fully-connected feed-forward deep random neural networks converge to a Gaussian mixture distribution as only the width of the last hidden layer goes to infinity. We conducted experiments for a simple model which supports our result. Moreover, it gives a detailed description of the convergence, namely, the growth of the last hidden layer gets the distribution closer to the Gaussian mixture, and the other layer successively get the Gaussian mixture closer to the normal distribution.
We propose a new algorithm for tensor decomposition, based on Jennrich's algorithm, and apply our new algorithmic ideas to blind deconvolution and Gaussian mixture models. Our first contribution is a simple and efficient algorithm to decompose certain symmetric overcomplete order-3 tensors, that is, three dimensional arrays of the form $T = \sum_{i=1}^n a_i \otimes a_i \otimes a_i$ where the $a_i$s are not linearly independent.Our algorithm comes with a detailed robustness analysis. Our second contribution builds on top of our tensor decomposition algorithm to expand the family of Gaussian mix…
The speed of convergence of the Expectation Maximization (EM) algorithm for Gaussian mixture model fitting is known to be dependent on the amount of overlap among the mixture components. In this paper, we study the impact of mixing coefficients on the convergence of EM. We show that when the mixture components exhibit some overlap, the convergence of EM becomes slower as the dynamic range among the mixing coefficients increases. We propose a deterministic anti-annealing algorithm, that significantly improves the speed of convergence of EM for such mixtures with unbalanced mixing coefficients. …
University of Bari（巴里大学）；Dublin City University（都柏林城市大学）
This paper describes the system proposed for the SemEval-2020 Task 1: Unsupervised Lexical Semantic Change Detection. We focused our approach on the detection problem. Given the semantics of words captured by temporal word embeddings in different time periods, we investigate the use of unsupervised methods to detect when the target word has gained or loosed senses. To this end, we defined a new algorithm based on Gaussian Mixture Models to cluster the target similarities computed over the two periods. We compared the proposed approach with a number of similarity-based thresholds. We found that…
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.
University of Toronto（多伦多大学）；Vector Institute（向量研究院）
Bayesian methods estimate a measure of uncertainty by using the posterior distribution. One source of difficulty in these methods is the computation of the normalizing constant. Calculating exact posterior is generally intractable and we usually approximate it. Variational Inference (VI) methods approximate the posterior with a distribution usually chosen from a simple family using optimization. The main contribution of this work is described is a set of update rules for natural gradient variational inference with mixture of Gaussians, which can be run independently for each of the mixture com…
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…
We study the community detection problem on a Gaussian mixture model, in which vertices are divided into $k\geq 2$ distinct communities. The major difference in our model is that the intensities for Gaussian perturbations are different for different entries in the observation matrix, and we do not assume that every community has the same number of vertices. We explicitly find the threshold for the exact recovery of the maximum likelihood estimation. Applications include the community detection on hypergraphs.
University of Cantabria（坎塔布里亚大学）；University of Sheffield（谢菲尔德大学）
In this work we introduce a mixture of GPs to address the data association problem, i.e. to label a group of observations according to the sources that generated them. Unlike several previously proposed GP mixtures, the novel mixture has the distinct characteristic of using no gating function to determine the association of samples and mixture components. Instead, all the GPs in the mixture are global and samples are clustered following "trajectories" across input space. We use a non-standard variational Bayesian algorithm to efficiently recover sample labels and learn the hyperparameters. We …
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…
Sony Computer Science Laboratories, Inc.（索尼计算机科学实验室有限公司）；Tokyo（东京）
The traditional Minkowski distances are induced by the corresponding Minkowski norms in real-valued vector spaces. In this work, we propose novel statistical symmetric distances based on the Minkowski's inequality for probability densities belonging to Lebesgue spaces. These statistical Minkowski distances admit closed-form formula for Gaussian mixture models when parameterized by integer exponents. This result extends to arbitrary mixtures of exponential families with natural parameter spaces being cones: This includes the binomial, the multinomial, the zero-centered Laplacian, the Gaussian a…
We propose a practical and scalable Gaussian process model for large-scale nonlinear probabilistic regression. Our mixture-of-experts model is conceptually simple and hierarchically recombines computations for an overall approximation of a full Gaussian process. Closed-form and distributed computations allow for efficient and massive parallelisation while keeping the memory consumption small. Given sufficient computing resources, our model can handle arbitrarily large data sets, without explicit sparse approximations. We provide strong experimental evidence that our model can be applied to lar…
Gauss--Hermite Quadrature for Gaussian-Mixture Entropy with an Action-Space Hermite Surrogate
Extreme Materials Research Center, Korea Institute of Science and Technology（韩国科学技术研究院极端材料研究中心）；Climate and Environmental Research Institute, Korea Institute of Science and Technology（韩国科学技术研究院气候与环境研究所）；Division of AI-Robotics, KIST Campus, University of Science and Technology（科学技术大学KIST校区AI机器人系）
Gaussian Mixture Models are a powerful tool in Data Science and Statistics that are mainly used for clustering and density approximation. The task of estimating the model parameters is in practice often solved by the Expectation Maximization (EM) algorithm which has its benefits in its simplicity and low per-iteration costs. However, the EM converges slowly if there is a large share of hidden information or overlapping clusters. Recent advances in manifold optimization for Gaussian Mixture Models have gained increasing interest. We introduce an explicit formula for the Riemannian Hessian for G…
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 …
Recently, the combination of robust one-dimensional convolutional neural networks (1-D CNNs) and Raman spectroscopy has shown great promise in rapid identification of unknown substances with good accuracy. Using this technique, researchers can recognize a pure compound and distinguish it from unknown substances in a mixture. The novelty of this approach is that the trained neural network operates automatically without any pre- or post-processing of data. Some studies have attempted to extend this technique to the classification of pure compounds in an unknown mixture. However, the application …
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.
University of Waterloo（滑铁卢大学）；Bar Ilan University（巴伊兰大学）；Northeastern University（东北大学）
Learning the parameters of Gaussian mixture models is a fundamental and widely studied problem with numerous applications. In this work, we give new algorithms for learning the parameters of a high-dimensional, well separated, Gaussian mixture model subject to the strong constraint of differential privacy. In particular, we give a differentially private analogue of the algorithm of Achlioptas and McSherry. Our algorithm has two key properties not achieved by prior work: (1) The algorithm's sample complexity matches that of the corresponding non-private algorithm up to lower order terms in a wi…
In this paper, we develop an unsupervised generative clustering framework that combines the Variational Information Bottleneck and the Gaussian Mixture Model. Specifically, in our approach, we use the Variational Information Bottleneck method and model the latent space as a mixture of Gaussians. We derive a bound on the cost function of our model that generalizes the Evidence Lower Bound (ELBO) and provide a variational inference type algorithm that allows computing it. In the algorithm, the coders' mappings are parametrized using neural networks, and the bound is approximated by Monte Carlo s…
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 …
