Gaussian Mixture Models

Imbalanced classification remains a pervasive challenge in machine learning, particularly when minority samples are too scarce to provide a robust discriminative boundary. In such extreme scenarios, conventional models often suffer from unstable decision boundaries and a lack of reliable error control. To bridge the gap between generative modeling and discriminative classification, we propose a two-stage framework \textbf{VAE-Inf} that integrates deep representation learning with statistically interpretable hypothesis testing. In the first stage, we adopt a one-class modeling perspective by training a variational autoencoder (VAE) exclusively on majority-class data to capture the underlying reference distribution. The resulting latent posteriors are aggregated via a Wasserstein barycenter to construct a global Gaussian reference model, providing a geometrically principled baseline for the majority class. In the second stage, we transform this generative foundation into a discriminative classifier by fine-tuning the encoder with limited minority samples. This is achieved through a novel distribution-aware loss that enforces probabilistic separation between classes based on variance-normalized projection statistics. For inference, we introduce a projection-based score that admits a natural hypothesis testing interpretation, allowing for a distribution-free calibration procedure. This approach yields exact finite-sample control of the Type-I error (false positive rate) without relying on restrictive parametric assumptions. Extensive experiments on diverse real-world benchmarks demonstrate that our framework achieves competitive performance against other approaches. The codes are available upon request.

The Gaussian mixture model is widely used in unsupervised learning, owing to its simplicity and interpretability. However, a fundamental limitation of the classical Gaussian mixture model is that it forces each observation to belong to exactly one component. In many practical applications, such as genetics, social network analysis, and text mining, an observation may naturally belong to multiple components or exhibit partial membership in several latent components. To overcome this limitation, we propose the mixed membership sub-Gaussian model, which extends the classical Gaussian mixture framework by allowing each observation to belong to multiple components. This model inherits the interpretability of the classical Gaussian mixture model while offering greater flexibility for capturing complex overlapping structures. We develop an efficient spectral algorithm to estimate the mixed membership of each individual observation, and under mild separation conditions on the component centres, we prove that the estimation error of the per-individual membership vector can be made arbitrarily small with high probability. To our knowledge, this is the first work to provide a computationally efficient estimator with such a vanishing-error guarantee for a mixed-membership extension of the Gaussian mixture model. Extensive experimental studies demonstrate that our method outperforms existing approaches that ignore mixed memberships.

Dimensionality reduction is a critical preprocessing step for clustering high-dimensional data, yet comprehensive evaluation of its impact across diverse methods and data types remains limited. In this study, we systematically assess the influence of five dimensionality reduction techniques - Principal Component Analysis (PCA), Kernel Principal Component Analysis (Kernel PCA), Variational Autoencoder (VAE), Isometric Mapping (Isomap), and Multidimensional Scaling (MDS) - on the performance of four popular clustering algorithms - k-means, Agglomerative Hierarchical Clustering (AHC), Gaussian Mixture Models (GMM), and Ordering Points to Identify the Clustering Structure (OPTICS). We evaluate clustering quality using the Adjusted Rand Index (ARI), comparing results without and with dimensionality reduction at different reduction levels recommended in the literature (i.e., k-1, where k is the number of clusters, and 25% and 50% of the original number of dimensions). Our findings underscore the importance of a careful selection of the dimensionality reduction technique and the dimensionality reduction level that should be tailored to intrinsic data geometry and clustering algorithm(s) under consideration.

In the automatic reassembly of fragments acquired using laser scanners to reconstruct objects, a crucial step is the matching of fractured surfaces. In this paper, we propose a novel local descriptor that uses the Gaussian Mixture Model (GMM) to fit the distribution of points, allowing for the description and matching of fractured surfaces of fragments. Our method involves dividing a local surface patch into concave and convex regions for estimating the k value of GMM. Then the final Gaussian Mixture Descriptor (GMD) of the fractured surface is formed by merging the regional GMDs. To measure the similarities between GMDs for determining adjacent fragments, we employ the L2 distance and align the fragments using Random Sample Consensus (RANSAC) and Iterative Closest Point (ICP). The extensive experiments on real-scanned public datasets and Terracotta datasets demonstrate the effectiveness of our approach; furthermore, the comparisons with several existing methods also validate the advantage of the proposed method.

Estimating the number of components is a fundamental challenge in unsupervised learning, particularly when dealing with high-dimensional data with many components or severely imbalanced component sizes. This paper addresses this challenge for classical Gaussian mixture models. The proposed estimator is simple: center the data, compute the singular values of the centered matrix, and count those above a threshold. No iterative fitting, no likelihood calculation, and no prior knowledge of the number of components are required. We prove that, under a mild separation condition on the component centers, the estimator consistently recovers the true number of components. The result holds in high-dimensional settings where the dimension can be much larger than the sample size. It also holds when the number of components grows to the smaller of the dimension and the sample size, even under severe imbalance among component sizes. Computationally, the method is extremely fast: for example, it processes ten million samples in one hundred dimensions within one minute. Extensive experimental studies confirm its accuracy in challenging settings such as high dimensionality, many components, and severe class imbalance.

Traffic forecasting is a challenging spatio-temporal modeling task and a critical component of urban transportation management. Current studies mainly focus on deterministic predictions, with limited considerations on the uncertainty and stochasticity in traffic dynamics. Therefore, this paper proposes an elegant yet universal approach that transforms existing models into probabilistic predictors by replacing only the final output layer with a novel Gaussian Mixture Model (GMM) layer. The modified model requires no changes to the training pipeline and can be trained using only the Negative Log-Likelihood (NLL) loss, without any auxiliary or regularization terms. Experiments on multiple traffic datasets show that our approach generalizes from classic to modern model architectures while preserving deterministic performance. Furthermore, we propose a systematic evaluation procedure based on cumulative distributions and confidence intervals, and demonstrate that our approach is considerably more accurate and informative than unimodal or deterministic baselines. Finally, a more detailed study on a real-world dense urban traffic network is presented to examine the impact of data quality on uncertainty quantification and to show the robustness of our approach under imperfect data conditions. Code available at https://github.com/Weijiang-Xiong/OpenSkyTraffic

We study model-order selection and component-mean estimation for multidimensional Gaussian mixture models with a known common covariance matrix. Using empirical characteristic-function measurements, we construct Fourier covariance matrices whose population counterparts have rank equal to the number of mixture components. We establish a minimax lower bound showing that distinguishing a separated $k$-component mixture from the class of $(k-1)$-component mixtures requires $Ω(Δ^{-(4k-4)})$ samples. We then develop an oracle spectral-thresholding estimator with a sufficient sample size of order $Δ^{-(8k-8)}$ for fixed $k$, together with a practical singular-value-ratio estimator. Given the model order, we estimate the component means by score-initialized gradient descent on a MUSIC-type projection objective. Under an explicit sample-size condition, a qualifying sample initialization lies in a certified attraction region with high probability, after which the iterates converge linearly. For fixed positive component separation, the resulting mean estimates achieve the parametric rate $\mathcal{O}_p(n^{-1/2})$. Numerical experiments demonstrate competitive accuracy and lower computational cost than expectation-maximization across a range of multidimensional settings.

Reliable long-term decoding of gestures from surface electromyography (EMG) is hindered by signal drift caused by electrode displacement, muscle fatigue, and/or posture changes. Although modern models achieve high intra-session accuracy, their performance often degrades substantially across recording sessions. Existing approaches to mitigate this problem typically rely on large training datasets or computationally intensive pipelines that are unsuitable for energy-efficient wearable devices. We propose a lightweight test-time adaptation framework for EMG decoding. The framework includes three complementary adaptation strategies: (i) causal adaptive batch normalization for online statistical alignment, (ii) Gaussian Mixture Model alignment with experience replay to mitigate forgetting, and (iii) meta-learning for rapid few-shot calibration. We evaluate these methods on the multi-session NinaPro DB6 dataset. All approaches substantially improve inter-session robustness relative to a non-adaptive baseline while maintaining low computational overhead. Replay-regularized statistical alignment provides the most stable adaptation under limited data, while meta-learning achieves the highest accuracy when sparse calibration labels are available. Overall, our self-supervised test-time adaptation methods reach up to 82% inter-session accuracy, significantly improving upon prior approaches while maintaining resource-efficient operation. These results demonstrate that lightweight test-time adaptation can enable robust, long-term EMG decoding for wearable or prosthetic applications.

Mixture models, such as Gaussian mixture models, are widely used in machine learning to represent complex data distributions. A key challenge, especially in high-dimensional settings, is to determine the mixture order and estimate the mixture parameters. We study the problem of approximating a target measure, available only through finitely many of its moments, by a mixture of distributions from a parametric family (e.g., Gaussian, exponential, Poisson), with approximation quality measured by the 2-Wasserstein or the total variation distance. Unlike many existing approaches, the parameter set is not assumed to be finite; it is modeled as a compact basic semi-algebraic set. We introduce a hierarchy of semidefinite relaxations with asymptotic convergence to the desired optimal value. In addition, when a certain rank condition is satisfied, the convergence is even finite and recovery of an optimal mixing measure is obtained. We also present an application to clustering, where our framework serves either as a stand-alone method or as a preprocessing step that yields both the number of clusters and strong initial parameter estimates, thereby accelerating convergence of standard (local) clustering algorithms.

Gaussian Mixture Models (GMMs) range among the most frequently used models in machine learning. However, training large, general GMMs becomes computationally prohibitive for data sets that have many data points $N$ of high-dimensionality $D$. For GMMs with arbitrary covariances, we here derive a highly efficient variational approximation, which is then integrated with mixtures of factor analyzers (MFAs). For GMMs with $C$ components, our proposed algorithm substantially reduces runtime complexity from $\mathcal{O}(NCD^2)$ per iteration to a complexity scaling linearly with $D$ and sublinearly with $NC$. In numerical experiments, we first validate that the complexity reduction results in a sublinear scaling for the entire GMM optimization process. Second, we show on large-scale benchmarks that the sublinear algorithm results in speed-ups of an order-of-magnitude compared to the state-of-the-art. Third, as a proof of concept, we finally train GMMs with over 10 billion parameters on about 100 million images, observing training times of less than nine hours on a single state-of-the-art CPU. Finally, and fourth, we demonstrate the effectiveness of large-scale GMMs on the task of zero-shot image denoising, where sublinear training results in state-of-the-art denoising times while competitive denoising performance is maintained.

In this paper, we study the problem of learning one-dimensional Gaussian mixture models (GMMs) with a specific focus on estimating both the model order and the mixing distribution from independent and identically distributed (i.i.d.) samples. This paper establishes the optimal sampling complexity for model order estimation in one-dimensional Gaussian mixture models. We prove a fundamental lower bound on the number of samples required to correctly identify the number of components with high probability, showing that this limit depends critically on the separation between component means and the total number of components. We then propose a Fourier-based approach to estimate both the model order and the mixing distribution. Our algorithm utilizes Fourier measurements constructed from the samples, and our analysis demonstrates that its sample complexity matches the established lower bound, thereby confirming its optimality. Numerical experiments further show that our method outperforms conventional techniques in terms of efficiency and accuracy.

In this paper, we consider the problem of partitioning a small data sample of size $n$ drawn from a mixture of $2$ sub-gaussian distributions in $\mathbb{R}^p$. We consider semidefinite programming relaxations of an integer quadratic program that is formulated essentially as finding the maximum cut on a graph, where edge weights in the cut represent dissimilarity scores between two nodes based on their $p$ features. We define the signal-to-noise ratio (SNR) as $s^2 := \min\{n p γ^2, Δ^2\}$, where $Δ^2 := p γ$ denotes the $\ell_2^2$ distance between the two cluster centers. Our contributions are twofold. First, we provide a unified framework for analyzing three computationally efficient algorithms: SDP1, BalancedSDP, and Spectral clustering, yielding universal polynomial-rate misclassification guarantees for all three algorithms. Moreover, our theory allows for partial recovery (success rate $< 100\%$) as long as $s^2$ is lower bounded by a constant. Second, we prove that the misclassification errors for SDP1 and BalancedSDP decay exponentially with respect to the SNR $s^2$ and the BalancedSDP requires no explicit debiasing when the two clusters have equal sizes. To our knowledge, this is the first time such results are obtained for semidefinite relaxations of MAX CUT in population clustering. We provide simulation evidence illuminating the theoretical predictions.