Analysis of SGD for Gaussian mixture classification using dynamical mean-field theory.
problem Learning dynamics of SGD for a neural network classifying Gaussian mixture.
method Applying dynamical mean-field theory to track SGD dynamics in high dimensions.
result Reveals how SGD navigates the non-convex loss landscape.
We analyze training dynamics in Gaussian mixture models using a comparison theorem.
problem Analyzing training algorithms with Gaussian mixture data.
method Applying a Gaussian comparison theorem to a specific family of training algorithms.
result Validated dynamic mean-field expressions and provided iterative refinement schemes.
Proposes a deep generative model for robust forecasting on sparse multivariate time series.
problem Forecasting on sparse multivariate time series with suboptimal results when sparsity is high.
method Dynamic Gaussian Mixture distribution for modeling latent clusters, using neural networks and gating mechanism.
result Demonstrates robust modeling of sparse multivariate time series with improved accuracy.
Deep learning models converge to Gaussian dynamics with mixed structured inputs.
problem Understanding neural network dynamics with complex input distributions.
method Extended hidden manifold model to Gaussian mixtures, analyzed via SGD.
result Learning dynamics with mixed inputs converge to Gaussian behavior.
New GM layers improve neural network performance.
problem Improving neural network performance.
method Employing Gaussian mixture models and Wasserstein gradient flows.
result GM layers achieve comparable performance to two-layer networks.
A new method for steering large agent populations efficiently.
problem Controlling the configuration of a swarm of identical, interacting cooperative agents.
method Mean-Field Schrodinger Bridges with Gaussian Mixture Models.
result A highly efficient parameterization to approximate optimal solutions of the MFSB problem in closed form.
Paper analyzes Langevin dynamics for multimodal Gaussian mixtures, controlling errors across dimensions.
problem Challenges in obtaining stable diffusion-based samplers in high- and infinite-dimensional settings.
method Study of preconditioned Annealed Langevin Dynamics (ALD) for Gaussian mixtures, focusing on Euler-Maruyama (EM) and exponential-integrator schemes.
result Proves dimension-uniform KL bounds for the exponential-integrator scheme, allowing arbitrarily small divergence with dimension.
A novel Bayesian method for dynamic sparsity in Gaussian dynamic linear regression.
problem Variable selection and shrinkage in time-varying regression models.
method Time-varying sparsity via Markov switching priors for coefficients' variances, extending spike-and-slab priors.
result Induces smoothness or shrinkage towards zero at each time point, leading to improved model performance.
Neural networks outperform kernel methods in classifying high-dimensional Gaussian mixtures.
problem Classifying high-dimensional Gaussian mixtures using kernel methods and neural networks.
method Theoretical analysis and derivation of learning dynamics for 2LNN and comparison with kernel methods.
result 2LNN can achieve near-optimal performance on high-dimensional Gaussian mixture classification tasks, surpassing kernel methods.
The paper proposes a Gaussian mixture model for Hilbert-space-valued data.
problem Challenges in characterizing probability measures for infinite-dimensional random objects.
method Gaussian mixture framework based on kernel mean embeddings.
result The proposed algorithm yields a dense class of approximations in infinite-dimensional spaces.
Transformers converge linearly to optimal models for Gaussian mixtures classification.
problem Theoretical understanding of transformers' in-context classification.
method Gradient descent training of a single-layer transformer for Gaussian mixtures classification.
result Transformers converge linearly to globally optimal models for Gaussian mixtures classification.
Paper analyzes Annealed Langevin Dynamics for multimodal sampling stability.
problem Ensuring stability of Annealed Langevin Dynamics across dimensions.
method Uniform-in-dimension analysis of ALD for Gaussian-mixture targets.
result ALD achieves prescribed accuracy in KL divergence with spectral conditions.
A new MFG framework for evolving clusters from Gaussian mixtures.
problem Evolutionary clustering of time-dependent Gaussian mixtures.
method Control-theoretic framework based on Mean Field Games (MFG) with coupled HJB and Fokker-Planck systems.
result MFG dynamics recover classical EM algorithm trajectories with mass conservation.
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 …
This paper tackles online reinforcement learning for unseen tasks with unknown boundaries.
problem Real-world tasks violate assumptions of task distributions, independence, and clear task delineations.
method A mixture of Gaussian Processes models different dynamics, and a transition prior handles temporal dependencies.
result The approach reliably handles task distribution shifts and outperforms alternatives in non-stationary tasks.
Method for initializing Gaussian mixtures for variational inference with multi-modal distributions.
problem Challenges in variational inference with Gaussian mixtures due to multimodality and nonconvex loss functions.
method Optimization to find local maxima, local Gaussian approximations, and constrained least squares regression.
result Robust initialization improves variational inference performance and scalability.
Langevin Dynamics fails to sample from mixture distributions efficiently.
problem Analyzing Langevin Dynamics for sampling from mixture distributions.
method Theoretical analysis of Langevin Dynamics and proposing Chained-Langevin Dynamics.
result Langevin Dynamics fails to sample from mixture distributions efficiently.
Efficient inference for multimodal Gaussian mixture models of interacting dynamical systems.
problem Efficient inference for multimodal distributions in stochastic dynamical systems.
method Graph neural networks with moment matching for sample-free inference and structured covariance approximations.
result Sample-free inference with improved efficiency and stability compared to Monte Carlo alternatives.
Transformers can cluster data from Gaussian mixtures without supervision.
problem Clustering data from Gaussian mixtures without labeled data.
method Theoretical analysis of attention-based layers, focusing on a simplified two-head attention layer and an identity matrix attention layer.
result Attention-based layers can align with true mixture centroids and adapt to input-specific distributions.
Improved VB algorithm for NIG mixtures outperforms Gaussian mixtures for non-Gaussian data.
problem Clustering non-Gaussian data, especially heavy-tailed and asymmetric.
method Proposed an improved VB algorithm for NIG mixture models and extended Dirichlet process mixture models.
result Outperforms Gaussian mixtures and existing NIG mixture models, especially for highly non-normative data.
We propose a new blind source separation algorithm based on mixtures of alpha-stable distributions. Complex symmetric alpha-stable distributions have been recently showed to better model audio signals in the time-frequency domain than classical Gaussian distributions thanks to their larger dynamic range. However, infer…
The paper uses Gaussian mixture models for Bayesian networks and proposes an optimization algorithm.
problem Modeling nodes in Bayesian networks with complex distributions.
method Gaussian mixture models combined with double iteration algorithm.
result The double iteration algorithm optimizes Gaussian mixture models effectively.
The study bounds the stability of Gaussian mixtures under small perturbations.
problem Stability of Gaussian mixtures under small changes in distribution.
method Deriving an explicit bound on parameter stability of spherical Gaussian Mixture Models (sGMM) in a pre-defined model class.
result Upper bound on parameter distance of close sGMMs to the original sGMM, dependent only on the original model.
Spatially constrained Gaussian mixture models reduce covariance complexity.
problem High dimensionality in finite mixture models for spatial data.
method Spatial covariance constraint with only four free parameters.
result Improves clustering of multi-way spatial data and inference of spatial patterns.
Deep neural networks converge to Gaussian mixtures as layer width increases.
problem Understanding the distribution of outputs from deep neural networks.
method Proof and experiments with a simple model showing the convergence of neural network outputs to Gaussian mixtures.
result Neural networks converge to Gaussian mixtures as the width of the last hidden layer increases.
Optimal transport for vector Gaussian mixtures improves efficiency and structure preservation.
problem Optimal mass transport for vector-valued Gaussian mixtures.
method Vectorizing Gaussian mixture models and studying optimal mass transport problems.
result Computational efficiency and structure preservation in optimal mass transport.
Learning representation on graph plays a crucial role in numerous tasks of pattern recognition. Different from grid-shaped images/videos, on which local convolution kernels can be lattices, however, graphs are fully coordinate-free on vertices and edges. In this work, we propose a Gaussian-induced convolution (GIC) fra…
Gradient EM converges globally for over-parameterized Gaussian mixtures.
problem Recovering ground truth Gaussian mixtures with over-parameterized models.
method Gradient EM with over-parameterization, using Hermite polynomials and tensor decomposition.
result Gradient EM globally converges to ground truth with n=Ω(mlogm) over-parameterization. Theoretical analysis of entropy approximation for Gaussian mixtures.
problem Lack of theoretical guarantees for entropy approximation of Gaussian mixtures.
method Theoretical analysis of the error between true and approximate entropy.
result The error converges to zero as the ratios of means to variances tend to infinity, providing a guarantee for high-dimensional problems.
Paper uses Gaussian mixture models and Wasserstein distance for schema matching.
problem Schema matching between different datasets.
method Gaussian mixture models and Wasserstein distance for comparison.
result Derives an approximation for Wasserstein distance between Gaussian mixture models.
Algorithm estimates mixtures of arbitrary Gaussians robustly in presence of corruptions.
problem Estimating mixtures of arbitrary Gaussians in the presence of a constant fraction of arbitrary corruptions.
method Polynomial-time algorithm using partial clustering and tensor decomposition.
result Resolves the main open problem in several previous works on algorithmic robust statistics.
We address two shortcomings in online travel time estimation methods for congested urban traffic. The first shortcoming is related to the determination of the number of mixture modes, which can change dynamically, within day and from day to day. The second shortcoming is the wide-spread use of Gaussian probability dens…
Estimates Gaussian mixtures from weighted samples efficiently.
problem Estimating Gaussian mixtures from weighted samples with correct weight treatment.
method Density interpretation and expectation-maximization method considering weights.
result Correctly estimates Gaussian mixtures with weighted samples.
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, wh…
Study uniform rates for estimating Gaussian mixtures without separation assumption.
problem Estimating parameters in two-component Gaussian mixtures without separation.
method Uniform convergence rates derived using minimax lower bounds and careful analysis of polynomial equalities.
result Phase transition in optimal estimation rate based on mixture balance.
Study learns Gaussian mixtures from censored data.
problem Learning Gaussian mixtures with incomplete data.
method Proposes an algorithm to estimate weights and means with minimal samples.
result Achieves accurate estimation with very few samples.
GGMPs improve non-Gaussian conditional density estimation.
problem Multimodality, heteroscedasticity, and strong non-Gaussianity in conditional density estimation.
method GGMP combines local Gaussian mixture fitting, cross-input component alignment, and per-component heteroscedastic GP training.
result GGMPs improve distributional approximation on synthetic and real-world datasets.
Study proposes a new metric for comparing Gaussian mixtures in RKHS.
problem Comparing complex multimodal densities in RKHS.
method Wasserstein-type metric for kernel Gaussian mixtures.
result Enhanced capability to model multimodal densities.
Heavy-tailed distributions are widely used in robust mixture modelling due to possessing thick tails. As a computationally tractable subclass of the stable distributions, sub-Gaussian α-stable distribution received much interest in the literature. Here, we introduce a type of expectation maximization algorithm that e…
Improved sample efficiency for private learning of Gaussian mixtures.
problem Learning mixtures of Gaussians with differential privacy.
method Inverse sensitivity mechanism, sample compression, sumset volume bounds.
result Proved optimal sample complexity for private learning of mixtures of Gaussians.
The mixture of Gaussian distributions, a soft version of k-means , is considered a state-of-the-art clustering algorithm. It is widely used in computer vision for selecting classes, e.g., color, texture, and shapes. In this algorithm, each class is described by a Gaussian distribution, defined by its mean and covarianc…
Sharp inequality between TV and Hellinger distances for Gaussian mixtures.
problem Understanding the relationship between total variation and Hellinger distances for Gaussian mixtures.
method Established a general upper bound on Hellinger distance in terms of TV distance raised to a power, demonstrating sharpness with specific examples.
result The Hellinger distance between two Gaussian mixtures is bounded by the TV distance raised to a power 1−o(1), where o(1) is of order 1/loglog(1/TV). Tensor decomposition recovers Gaussian mixtures from moments.
problem Recovering Gaussian mixture models from datasets.
method Symmetric tensor decomposition of moment tensors built from empirical moments.
result Identifiable tensors with interpolation degree less than half their order.
A novel optimization-based Gaussian mixture reduction method using composite transportation divergence.
problem Exponential increase in Gaussian mixture order leads to intractable inference.
method Optimization-based Gaussian mixture reduction (GMR) using composite transportation divergence (CTD).
result Unified framework for selecting optimal cost function in various applications.
New method for analyzing learning dynamics in singular models.
problem Challenges in analyzing learning of singular models with no one-to-one parameter space.
method Relative reparameterization technique to extract regular sub-models.
result Demonstrated differences in convergence behavior due to algorithmic and intrinsic aspects.
This paper formed part of a preliminary research report for a risk consultancy and academic research. Stochastic Programming models provide a powerful paradigm for decision making under uncertainty. In these models the uncertainties are represented by a discrete scenario tree and the quality of the solutions obtained i…
New findings show Gaussian universality breaks down in high-dimensional linear factor mixtures.
problem The limitations of Gaussian universality in high-dimensional classification.
method Characterization of empirical risk minimization for classification under linear factor mixture models.
result Gaussian universality breaks down under high-dimensional linear factor mixtures.
Modeling glucose distribution changes over time using neural ODEs.
problem Analyzing how continuous glucose distribution changes over time in diabetic patients.
method Combines Gaussian mixture, MMD, and Neural ODE to model temporal evolution of glucose distribution.
result Highly interpretable model detects subtle distribution shifts and remains computationally efficient.