Generative adversarial networks (GANs) learn the distribution of observed samples through a zero-sum game between two machine players, a generator and a discriminator. While GANs achieve great success in learning the complex distribution of image, sound, and text data, they perform suboptimally in learning multi-modal distribution-learning benchmarks including Gaussian mixture models (GMMs). In this paper, we propose Generative Adversarial Training for Gaussian Mixture Models (GAT-GMM), a minimax GAN framework for learning GMMs. Motivated by optimal transport theory, we design the zero-sum game in GAT-GMM using a random linear generator and a softmax-based quadratic discriminator architecture, which leads to a non-convex concave minimax optimization problem. We show that a Gradient Descent Ascent (GDA) method converges to an approximate stationary minimax point of the GAT-GMM optimization problem. In the benchmark case of a mixture of two symmetric, well-separated Gaussians, we further show this stationary point recovers the true parameters of the underlying GMM. We numerically support our theoretical findings by performing several experiments, which demonstrate that GAT-GMM can perform as well as the expectation-maximization algorithm in learning mixtures of two Gaussians.
Compressive Sensing via Low-Rank Gaussian Mixture Models
Bell Labs, Alcatel-Lucent（阿尔卡特朗讯贝尔实验室）
提出了一种基于高斯混合模型的压缩感知（CS）反演算法。虽然在我们的无透镜相机中实现的整个图像上全局执行压缩感知，但对局部图像块施加低秩GMM。该低秩GMM是经由在测量数据的投影上训练的GMM的特征值阈值化导出的，因此学习了{ \em in situ}。GMM和测量数据的投影在重建期间迭代地更新。我们的GMM算法退化到分段线性估计（PLE），如果每个补丁是由一个单一的高斯模型。受此启发，低秩PLE算法也开发CS反演，构成本文的额外贡献。仿真数据和无透镜相机捕获的实际数据的广泛结果表明所提出的算法的有效性。此外，我们比较了CS重建结果使用我们的算法与JPEG压缩。仿真结果表明，当有限的带宽是可用的（少量的测量），我们的算法可以达到与JPEG相当的结果。
HRL Laboratories, LLC（HRL实验室有限责任公司）；University of Virginia（弗吉尼亚大学）
Gaussian mixture models (GMM) are powerful parametric tools with many applications in machine learning and computer vision. Expectation maximization (EM) is the most popular algorithm for estimating the GMM parameters. However, EM guarantees only convergence to a stationary point of the log-likelihood function, which could be arbitrarily worse than the optimal solution. Inspired by the relationship between the negative log-likelihood function and the Kullback-Leibler (KL) divergence, we propose an alternative formulation for estimating the GMM parameters using the sliced Wasserstein distance, …
TerraNIS；University of Toulouse（图卢兹大学）；IRIT-INP-ENSEEIHT（图卢兹国立综合理工学院-ENSEEIHT-图卢兹计算机研究所）；TéSA（航空航天电信实验室）；IRIT-UPS（图卢兹第三大学-图卢兹计算机研究所）
Missing data is a recurrent problem in remote sensing, mainly due to cloud coverage for multispectral images and acquisition problems. This can be a critical issue for crop monitoring, especially for applications relying on machine learning techniques, which generally assume that the feature matrix does not have missing values. This paper proposes a Gaussian Mixture Model (GMM) for the reconstruction of parcel-level features extracted from multispectral images. A robust version of the GMM is also investigated, since datasets can be contaminated by inaccurate samples or features (e.g., wrong cr…
Northwestern Polytechnical University（西北工业大学）；Université de Technologie de Compiègne（贡比涅技术大学）；Institut Universitaire de France（法国大学研究院）
The Gaussian mixture model (GMM) provides a simple yet principled framework for clustering, with properties suitable for statistical inference. In this paper, we propose a new model-based clustering algorithm, called EGMM (evidential GMM), in the theoretical framework of belief functions to better characterize cluster-membership uncertainty. With a mass function representing the cluster membership of each object, the evidential Gaussian mixture distribution composed of the components over the powerset of the desired clusters is proposed to model the entire dataset. The parameters in EGMM are e…
An $\varepsilon$-coreset for a given set $D$ of $n$ points, is usually a small weighted set, such that querying the coreset \emph{provably} yields a $(1+\varepsilon)$-factor approximation to the original (full) dataset, for a given family of queries. Using existing techniques, coresets can be maintained for streaming, dynamic (insertion/deletions), and distributed data in parallel, e.g. on a network, GPU or cloud. We suggest the first coresets that approximate the negative log-likelihood for $k$-Gaussians Mixture Models (GMM) of arbitrary shapes (ratio between eigenvalues of their covariance m…
We show that a collection of Gaussian mixture models (GMMs) in $R^{n}$ can be optimally classified using $O(n)$ neurons in a neural network with two hidden layers (deep neural network), whereas in contrast, a neural network with a single hidden layer (shallow neural network) would require at least $O(\exp(n))$ neurons or possibly exponentially large coefficients. Given the universality of the Gaussian distribution in the feature spaces of data, e.g., in speech, image and text, our result sheds light on the observed efficiency of deep neural networks in practical classification problems.
This work presents a mathematical treatment of the relation between Self-Organizing Maps (SOMs) and Gaussian Mixture Models (GMMs). We show that energy-based SOM models can be interpreted as performing gradient descent, minimizing an approximation to the GMM log-likelihood that is particularly valid for high data dimensionalities. The SOM-like decrease of the neighborhood radius can be understood as an annealing procedure ensuring that gradient descent does not get stuck in undesirable local minima. This link allows to treat SOMs as generative probabilistic models, giving a formal justificatio…
Sublinear Variational Optimization of Gaussian Mixture Models with Millions to Billions of Parameters
Faculty VI Carl von Ossietzky University Oldenburg（卡尔·奥西埃茨基大学第六学院）；Artificial Intelligence Lab, Department of Computer Science, Faculty of Mathematics, Computer Science and Physics University of Innsbruck（人工智能实验室，计算机科学系，数学、计算机科学与物理学学院因斯布鲁克大学）；Data Analytics and AI, Faculty 3 Frankfurt University of Applied Sciences（数据分析与人工智能，第三学院法兰克福应用科学大学）
Growth mixture models (GMMs) incorporate both conventional random effects growth modeling and latent trajectory classes as in finite mixture modeling; therefore, they offer a way to handle the unobserved heterogeneity between subjects in their development. GMMs with Gaussian random effects dominate the literature. When the data are asymmetric and/or have heavier tails, more than one latent class is required to capture the observed variable distribution. Therefore, a GMM with continuous non-elliptical distributions is proposed to capture skewness and heavier tails in the data set. Specifically,…
In this work we solve the problem of robustly learning a high-dimensional Gaussian mixture model with $k$ components from $ε$-corrupted samples up to accuracy $\widetilde{O}(ε)$ in total variation distance for any constant $k$ and with mild assumptions on the mixture. This robustness guarantee is optimal up to polylogarithmic factors. The main challenge is that most earlier works rely on learning individual components in the mixture, but this is impossible in our setting, at least for the types of strong robustness guarantees we are aiming for. Instead we introduce a new framework which we cal…
Discrete hidden Markov models (HMM) are often applied to malware detection and classification problems. However, the continuous analog of discrete HMMs, that is, Gaussian mixture model-HMMs (GMM-HMM), are rarely considered in the field of cybersecurity. In this paper, we use GMM-HMMs for malware classification and we compare our results to those obtained using discrete HMMs. As features, we consider opcode sequences and entropy-based sequences. For our opcode features, GMM-HMMs produce results that are comparable to those obtained using discrete HMMs, whereas for our entropy-based features, GM…
Gromov-Wasserstein-like Distances in the Gaussian Mixture Models Space
ENS Paris-Saclay（巴黎萨克雷高等师范学校）；CNRS（法国国家科学研究中心）；Centre Borelli（博雷利中心）；Université de Paris（巴黎大学）；MAP5（数学及其应用实验室）；Institut Universitaire de France（法国大学研究院）
Learning a Gaussian Mixture Model (GMM) is hard when the number of parameters is too large given the amount of available data. As a remedy, we propose restricting the GMM to a Gaussian Markov Random Field Mixture Model (GMRF-MM), as well as a new method for estimating the latter's sparse precision (i.e., inverse covariance) matrices. When the sparsity pattern of each matrix is known, we propose an efficient optimization method for the Maximum Likelihood Estimate (MLE) of that matrix. When it is unknown, we utilize the popular Graphical Least Absolute Shrinkage and Selection Operator (GLASSO) t…
Classifying Overlapping Gaussian Mixtures in High Dimensions: From Optimal Classifiers to Neural Nets
Raymond and Beverly Sackler School of Physics and Astronomy（雷蒙德和贝弗利·萨克勒物理与天文学院）；Tel-Aviv University（特拉维夫大学）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
In probabilistic classification, a discriminative model based on the softmax function has a potential limitation in that it assumes unimodality for each class in the feature space. The mixture model can address this issue, although it leads to an increase in the number of parameters. We propose a sparse classifier based on a discriminative GMM, referred to as a sparse discriminative Gaussian mixture (SDGM). In the SDGM, a GMM-based discriminative model is trained via sparse Bayesian learning. Using this sparse learning framework, we can simultaneously remove redundant Gaussian components and r…
Gaussian mixture models (GMMs) are fundamental tools in statistical and data sciences. We study the moments of multivariate Gaussians and GMMs. The $d$-th moment of an $n$-dimensional random variable is a symmetric $d$-way tensor of size $n^d$, so working with moments naively is assumed to be prohibitively expensive for $d>2$ and larger values of $n$. In this work, we develop theory and numerical methods for \emph{implicit computations} with moment tensors of GMMs, reducing the computational and storage costs to $\mathcal{O}(n^2)$ and $\mathcal{O}(n^3)$, respectively, for general covariance ma…
We propose a framework, named Aggregated Wasserstein, for computing a dissimilarity measure or distance between two Hidden Markov Models with state conditional distributions being Gaussian. For such HMMs, the marginal distribution at any time spot follows a Gaussian mixture distribution, a fact exploited to softly match, aka register, the states in two HMMs. We refer to such HMMs as Gaussian mixture model-HMM (GMM-HMM). The registration of states is inspired by the intrinsic relationship of optimal transport and the Wasserstein metric between distributions. Specifically, the components of the …
Functional data analysis (FDA) is an important modern paradigm for handling infinite-dimensional data. An important task in FDA is model-based clustering, which organizes functional populations into groups via subpopulation structures. The most common approach for model-based clustering of functional data is via mixtures of linear mixed-effects models. The mixture of linear mixed-effects models (MLMM) approach requires a computationally intensive algorithm for estimation. We provide a novel Gaussian mixture model (GMM) characterization of the model-based clustering problem. We demonstrate that…
Sony Computer Science Laboratories Inc（索尼计算机科学实验室股份有限公司）；CSIRO’s Data61（英联邦科学与工业研究组织数据61研究所）
We define a novel class of distances between statistical multivariate distributions by modeling an optimal transport problem on their marginals with respect to a ground distance defined on their conditionals. These new distances are metrics whenever the ground distance between the marginals is a metric, generalize both the Wasserstein distances between discrete measures and a recently introduced metric distance between statistical mixtures, and provide an upper bound for jointly convex distances between statistical mixtures. By entropic regularization of the optimal transport, we obtain a fast…
We present an approach for efficiently training Gaussian Mixture Model (GMM) by Stochastic Gradient Descent (SGD) with non-stationary, high-dimensional streaming data. Our training scheme does not require data-driven parameter initialization (e.g., k-means) and can thus be trained based on a random initialization. Furthermore, the approach allows mini-batch sizes as low as 1, which are typical for streaming-data settings. Major problems in such settings are undesirable local optima during early training phases and numerical instabilities due to high data dimensionalities. We introduce an adapt…
The Hebrew University of Jerusalem（耶路撒冷希伯来大学）；Tel Aviv University（特拉维夫大学）
This paper studies the sample complexity of learning the $k$ unknown centers of a balanced Gaussian mixture model (GMM) in $\mathbb{R}^d$ with spherical covariance matrix $σ^2\mathbf{I}$. In particular, we are interested in the following question: what is the maximal noise level $σ^2$, for which the sample complexity is essentially the same as when estimating the centers from labeled measurements? To that end, we restrict attention to a Bayesian formulation of the problem, where the centers are uniformly distributed on the sphere $\sqrt{d}\mathcal{S}^{d-1}$. Our main results characterize the e…
We show that $k$-means (Lloyd's algorithm) is obtained as a special case when truncated variational EM approximations are applied to Gaussian Mixture Models (GMM) with isotropic Gaussians. In contrast to the standard way to relate $k$-means and GMMs, the provided derivation shows that it is not required to consider Gaussians with small variances or the limit case of zero variances. There are a number of consequences that directly follow from our approach: (A) $k$-means can be shown to increase a free energy associated with truncated distributions and this free energy can directly be reformulat…
Johns Hopkins University（约翰斯·霍普金斯大学）；University of Maryland（马里兰大学）
We present a new subspace-based method to construct probabilistic models for high-dimensional data and highlight its use in anomaly detection. The approach is based on a statistical estimation of probability density using densities of random subspaces combined with geometric averaging. In selecting random subspaces, equal representation of each attribute is used to ensure correct statistical limits. Gaussian mixture models (GMMs) are used to create the probability densities for each subspace with techniques included to mitigate singularities allowing for the ability to handle both numerical an…
Failures or breakdowns in factory machinery can be costly to companies, so there is an increasing demand for automatic machine inspection. Existing approaches to acoustic signal-based unsupervised anomaly detection, such as those using a deep autoencoder (DA) or Gaussian mixture model (GMM), have poor anomaly-detection performance. In this work, we propose a new method based on a deep autoencoding Gaussian mixture model with hyper-parameter optimization (DAGMM-HO). In our method, the DAGMM-HO applies the conventional DAGMM to the audio domain for the first time, with the idea that its total op…
We derive an asymptotic expansion for the log likelihood of Gaussian mixture models (GMMs) with equal covariance matrices in the low signal-to-noise regime. The expansion reveals an intimate connection between two types of algorithms for parameter estimation: the method of moments and likelihood optimizing algorithms such as Expectation-Maximization (EM). We show that likelihood optimization in the low SNR regime reduces to a sequence of least squares optimization problems that match the moments of the estimate to the ground truth moments one by one. This connection is a stepping stone toward …
Isolating individual instruments in a musical mixture has a myriad of potential applications, and seems imminently achievable given the levels of performance reached by recent deep learning methods. While most musical source separation techniques learn an independent model for each instrument, we propose using a common embedding space for the time-frequency bins of all instruments in a mixture inspired by deep clustering and deep attractor networks. Additionally, an auxiliary network is used to generate parameters of a Gaussian mixture model (GMM) where the posterior distribution over GMM comp…
Parkinson's disease is a neurodegenerative disorder characterized by the presence of different motor impairments. Information from speech, handwriting, and gait signals have been considered to evaluate the neurological state of the patients. On the other hand, user models based on Gaussian mixture models - universal background models (GMM-UBM) and i-vectors are considered the state-of-the-art in biometric applications like speaker verification because they are able to model specific speaker traits. This study introduces the use of GMM-UBM and i-vectors to evaluate the neurological state of Par…
Differentially Private Distribution Release of Gaussian Mixture Models via KL-Divergence Minimization
State Key Laboratory of Internet of Things for Smart City and the Department of Electrical and Computer Engineering, University of Macau（物联网智能城市国家重点实验室和澳门大学电子与计算机工程系）；Department of Electrical and Computer Engineering, Cornell Tech, Cornell University（电气与计算机工程系，康奈尔科技，康奈尔大学）；Computing Sciences Research, Lawrence Berkeley National Laboratory（计算科学研究所，劳伦斯伯克利国家实验室）
Indian Institute of Space Science and Technology（印度空间科学与技术学院）
Detecting stellar clusters have always been an important research problem in Astronomy. Although images do not convey very detailed information in detecting stellar density enhancements, we attempt to understand if new machine learning techniques can reveal patterns that would assist in drawing better inferences from the available image data. This paper describes an unsupervised approach in detecting star clusters using Deep Variational Autoencoder combined with a Gaussian Mixture Model. We show that our method works significantly well in comparison with state-of-the-art detection algorithm in…
Institut FEMTO-ST（FEMTO-ST研究所）；Université Bourgogne Franche-Comté（勃艮第-弗朗什-孔泰大学）；Université de technologie de Compiègne（贡比涅技术大学）；CNRS（法国国家科学研究中心）；Heudiasyc（Heudiasyc实验室）；Institut universitaire de France（法兰西大学研究院）
The interpretation of unlabeled acoustic emission (AE) data classically relies on general-purpose clustering methods. While several external criteria have been used in the past to select the hyperparameters of those algorithms, few studies have paid attention to the development of dedicated objective functions in clustering methods able to cope with the specificities of AE data. We investigate how to explicitly represent clusters onsets in mixture models in general, and in Gaussian Mixture Models (GMM) in particular. By modifying the internal criterion of such models, we propose the first clus…
The Chinese University of Hong Kong（香港中文大学）；SpeechX Limited（SpeechX有限公司）
This work investigates the vulnerability of Gaussian Mixture Model (GMM) i-vector based speaker verification systems to adversarial attacks, and the transferability of adversarial samples crafted from GMM i-vector based systems to x-vector based systems. In detail, we formulate the GMM i-vector system as a scoring function of enrollment and testing utterance pairs. Then we leverage the fast gradient sign method (FGSM) to optimize testing utterances for adversarial samples generation. These adversarial samples are used to attack both GMM i-vector and x-vector systems. We measure the system vuln…
TU Berlin（柏林工业大学）；Univ. Bordeaux（波尔多大学）；Bordeaux INP（波尔多国立理工学院）；CNRS（法国国家科学研究中心）
Despite the rapid development of computational hardware, the treatment of large and high dimensional data sets is still a challenging problem. This paper provides a twofold contribution to the topic. First, we propose a Gaussian Mixture Model in conjunction with a reduction of the dimensionality of the data in each component of the model by principal component analysis, called PCA-GMM. To learn the (low dimensional) parameters of the mixture model we propose an EM algorithm whose M-step requires the solution of constrained optimization problems. Fortunately, these constrained problems do not d…
Institut de Mathématiques de Bordeaux, CNRS, Université de Bordeaux, Bordeaux INP（波尔多数学研究所、法国国家科学研究中心、波尔多大学、波尔多国立理工学院）；University of California, San Diego（加利福尼亚大学圣地亚哥分校）
Patch priors have become an important component of image restoration. A powerful approach in this category of restoration algorithms is the popular Expected Patch Log-Likelihood (EPLL) algorithm. EPLL uses a Gaussian mixture model (GMM) prior learned on clean image patches as a way to regularize degraded patches. In this paper, we show that a generalized Gaussian mixture model (GGMM) captures the underlying distribution of patches better than a GMM. Even though GGMM is a powerful prior to combine with EPLL, the non-Gaussianity of its components presents major challenges to be applied to a comp…
Community detection tasks have received a lot of attention across statistics, machine learning, and information theory with a large body of work concentrating on theoretical guarantees for the stochastic block model. One line of recent work has focused on modeling the spectral embedding of a network using Gaussian mixture models (GMMs) in scaling regimes where the ability to detect community memberships improves with the size of the network. However, these regimes are not very realistic. This paper provides tractable methodology motivated by new theoretical results for networks with non-vanish…
