ICA reveals deep learning's feature learning mechanisms from non-Gaussian data.
problem Understanding feature learning from non-Gaussian inputs in deep neural networks.
method Investigates ICA and SGD on synthetic and real data.
result FastICA requires n≳d4 samples for single non-Gaussian direction recovery, while SGD outperforms and optimised SGD reaches n≳d2. The paper analyzes how gradient descent learns convolutional filters for non-Gaussian inputs.
problem Learning convolutional filters with ReLU for non-Gaussian input distributions.
method Analysis of gradient descent convergence for ReLU activation with polynomial time complexity.
result Gradient descent can learn convolutional filters in polynomial time, with convergence rate dependent on input distribution smoothness and patch similarity.
The ACCRU framework improves probabilistic forecasts by capturing input-dependent uncertainty.
problem Uncertainty in deterministic predictions, especially for skewed and non-Gaussian errors.
method Neural network trained with a loss function balancing accuracy and reliability to learn input-dependent, non-Gaussian uncertainty distributions.
result Improves probabilistic forecasts relative to existing methods, capturing skewed and non-Gaussian errors.
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.
Random neural networks with ReLU activations are non-Gaussian processes.
problem Understanding the behavior of neural networks with random initialization and rectified linear units.
method Proving these networks are non-Gaussian processes and deriving their properties.
result These networks can converge to non-Gaussian processes under certain conditions.
Robust method estimates state, input, and parameters of linear systems online.
problem Joint estimation of state, input, and parameters in noisy or outlier-prone measurements.
method Combines recursive, alternating, and iteratively-reweighted least squares into a single algorithm.
result Good performance in presence of outliers and compared to state-of-the-art methods.
Extends Gaussian process regression for non-Gaussian data.
problem Inadequate modeling of uncertainty and over-smoothing in non-Gaussian datasets.
method Time-changed Gaussian processes with Lévy processes.
result Improved modeling of heavy-tailed non-Gaussian behaviors.
Gaussian Process (GP) regression models typically assume that residuals are Gaussian and have the same variance for all observations. However, applications with input-dependent noise (heteroscedastic residuals) frequently arise in practice, as do applications in which the residuals do not have a Gaussian distribution. …
Learning rate needs to decrease with higher data moments for effective ICA in high dimensions.
problem Slower convergence of ICA in high-dimensional data with high-order moments.
method High-dimensional ODE analysis of ICA algorithm under controlled moment structure.
result Critical learning rate threshold for effective ICA when moments are high.
The discovery of non-linear causal relationship under additive non-Gaussian noise models has attracted considerable attention recently because of their high flexibility. In this paper, we propose a novel causal inference algorithm called least-squares independence regression (LSIR). LSIR learns the additive noise model…
This paper improves GP for learning complex data distributions.
problem Vanilla Gaussian processes struggle with complex data distributions.
method Introduces scalable GP paradigms with latent variables and variational inference.
result Scalable modulated GPs, especially latent GPs, learn diverse data distributions better.
A new geostatistical model using deep learning for spatial prediction.
problem Traditional kriging assumptions and limitations.
method Nearest-Neighbor Neural Network (4N) process integrating deep learning.
result 4N process outperforms existing geostatistical methods for non-Gaussian data.
The statistical dependencies which independent component analysis (ICA) cannot remove often provide rich information beyond the linear independent components. It would thus be very useful to estimate the dependency structure from data. While such models have been proposed, they usually concentrated on higher-order corr…
New framework models neural systems with random architecture on manifolds.
problem Complex, uncertain systems with non-Gaussian outputs.
method Latent random field on compact manifold generates neural architecture and weights.
result Synthetic neural systems can produce stochastic outputs for deterministic inputs.
The Kalman filter is extensively used for state estimation for linear systems under Gaussian noise. When non-Gaussian Lévy noise is present, the conventional Kalman filter may fail to be effective due to the fact that the non-Gaussian Lévy noise may have infinite variance. A modified Kalman filter for linear systems wi…
This work extracts stochastic dynamical systems with α-stable Lévy noise.
problem Extracting data-driven governing laws of dynamical systems with non-Gaussian noise.
method End-to-end deep learning approach for learning drift and diffusion coefficients for α-stable Lévy noise. result Effectiveness of the method confirmed by numerical experiments.
LNGCA extends ICA to non-Gaussian signals and noise, improving estimation and testing.
problem Modeling multivariate data with non-Gaussian components and Gaussian noise.
method Linear latent factor model, simultaneous estimation of non-Gaussian and Gaussian components, discrepancy maximization, resampling-based test.
result Improved estimation and testing of non-Gaussian components over competing methods.
The paper tackles Bayesian inference with small datasets using manifold learning.
problem Small datasets challenge Bayesian inference with non-Gaussian models.
method Manifold learning and sampling for Bayesian posterior approximation.
result The method effectively samples Bayesian posteriors with non-Gaussian models from small datasets.
Researchers derive exact priors for finite Bayesian neural networks.
problem Understanding non-Gaussian priors in finite Bayesian neural networks.
method Analytical derivation of function space priors for finite fully-connected feedforward networks.
result Exact solutions for priors of finite networks, including Meijer G-function for linear networks and mixtures for ReLU networks.
The paper analyzes the non-Gaussian behavior of inflation and unemployment over 70 years using multifractal methods.
problem Capturing unusual fluctuations in inflation and unemployment over long periods.
method Coupled multifractal approach to analyze non-Gaussian distributions of inflation and unemployment over 70 years.
result The non-Gaussianity of unemployment is noticeable only for periods smaller than 1 year, while inflation's non-Gaussianity persists across all time scales.
Finite-width neural networks use non-Gaussian priors, extending Gaussian process theory.
problem Understanding the behavior of neural networks with finite width.
method Perturbative extension of Gaussian process theory to finite-width neural networks, tracking preactivation distributions.
result Non-Gaussian processes as priors in finite-width neural networks.
Paper uses optimal transport for Bayesian filtering, deriving new EnKF and FPF formulations.
problem Bayesian filtering for nonlinear systems with non-Gaussian observations.
method Optimal transport theory applied to Bayes' law, constructing Brenier maps.
result New variational formulations of EnKF and FPF for non-Gaussian settings.
Paper proposes a generalized precision matrix for t-Student distributions to improve portfolio optimization.
problem Limitations of inverse covariance matrix in non-Gaussian settings.
method Exploits local dependence function to define generalized precision matrix (GPM) for multivariate t-Student distribution.
result GPM leads to statistically significant lower out-of-sample variances in minimum-variance portfolios.
Improves graph-based active learning for non-Gaussian models.
problem Efficiently selecting data points for labeling in graph-based semi-supervised learning.
method Approximates non-Gaussian distributions, introduces rank-one update and model change acquisition function.
result Enhanced active learning for graph-based SSL under non-Gaussian models.
This research develops an evolutionary approach to discover non-Gaussian stochastic dynamical systems.
problem Discovering explicit governing equations of stochastic dynamical systems with Lévy noise from data.
method ESSR approach using genetic programming, sparse regression, and nonlocal Kramers-Moyal formulas.
result The approach effectively extracts non-Gaussian stochastic dynamical systems from sample path data.
The problem of Non-Gaussian Component Analysis (NGCA) is about finding a maximal low-dimensional subspace E in Rn so that data points projected onto E follow a non-gaussian distribution. Although this is an appropriate model for some real world data analysis problems, there has been little progress on t…
Extends option pricing theory for informed traders.
problem Empirical evidence for non-Gaussian returns, long-range dependence, volatility clustering, and asymmetric information.
method Extended option pricing theory to account for these factors.
result Improved understanding of option pricing for informed traders.
New algorithm for learning causal structures with disjoint cycles in linear non-Gaussian models.
problem Learning causal structures with cycles in linear non-Gaussian models.
method Characterizing when graphs determine the same model, using quadratic and cubic polynomial relations, and a strategy of decorrelating cycles and multivariate regression.
result Consistent and computationally efficient algorithm for learning causal structures with disjoint cycles.
Study improves regression models for non-Gaussian noise and outliers using correntropy.
problem Improving regression models for non-Gaussian noise and outliers.
method Introduces mixture of symmetric stable noise and uses correntropy for regression.
result Establishes asymptotic optimal learning rates for correntropy based regression.
The paper finds non-Gaussian directions in high-dimensional data using Wasserstein distance.
problem Locating interesting non-Gaussian features in high-dimensional data.
method Projection pursuit using 2-Wasserstein distance to maximize the difference from Gaussian.
result Statistical guarantees for accurately approximating an unknown low-dimensional non-Gaussian subspace.
Proposes efficient Gaussian approximations for non-Gaussian likelihoods.
problem Computational challenges in learning and inference with non-Gaussian likelihoods.
method Variational inference and moment matching in transformed bases.
result Good approximation quality for binary and multiclass classification.
A new vine copula mixture model improves clustering accuracy for non-Gaussian data.
problem Finite mixture models struggle with asymmetric tail dependencies and non-elliptical clusters.
method Proposes a vine copula mixture model for clustering non-Gaussian data, addressing model selection and parameter estimation.
result Significant improvement in clustering accuracy for data with asymmetric tail dependencies or non-Gaussian margins.
We use Gaussian processes to estimate conditional distributions with latent variables.
problem Challenging task of estimating conditional distributions with model complexity and overfitting trade-offs.
method Extend model input with latent variables and use Gaussian processes for mapping.
result Bayesian approach allows for modeling small datasets and applying to big data.
This paper presents a method for efficient density estimation in nonlinear systems.
problem Accurate representation of non-Gaussian distributions in nonlinear dynamical systems is challenging.
method Uses Seminonparametric (SNP) densities with probabilists' Hermite polynomial basis and Monte Carlo approximation for maximum likelihood estimation.
result Demonstrates that the method can accurately capture non-Gaussian density structure and compute quantiles using fewer samples than raw Monte Carlo.
Paper solves NGCA for discrete distributions using LLL method.
problem Learning hidden non-Gaussian components in discrete distributions.
method Utilizes LLL lattice basis reduction method.
result Sample and computationally efficient algorithm for NGCA in discrete distributions.
Max-margin classifiers' behavior is studied in high dimensions with non-Gaussian features.
problem Understanding the role of featurization maps and high-dimensional misclassification error.
method High-dimensional asymptotics, Gaussian model, support vector representation.
result Asymptotic behavior of max-margin classifiers is determined by feature covariance and label covariance.
Sparse non-Gaussian component analysis (SNGCA) is an unsupervised method of extracting a linear structure from a high dimensional data based on estimating a low-dimensional non-Gaussian data component. In this paper we discuss a new approach to direct estimation of the projector on the target space based on semidefinit…
Algorithm learns non-Gaussian graphical models via Hessian scores and triangular transport.
problem Learning graph structure from non-Gaussian data.
method Score based on integrated Hessian information, coupled with triangular transport map.
result Algorithm successfully recovers graph structure for non-Gaussian data.
Wavelet scattering spectra model non-Gaussian time-series, proving scale invariance for self-similar processes.
problem Modeling non-Gaussian time-series with stationary increments.
method Complex wavelet transform for scale variations, joint correlation matrix for scale dependencies, second wavelet transform for diagonalization, maximum entropy models conditioned by scattering spectra coefficients.
result Scattering spectra of self-similar processes are scale invariant, allowing statistical testing and generation of new time-series.
The paper examines non-Gaussian models for financial data.
problem Modeling financial data with non-Gaussian distributions.
method Analysis of multivariate non-Gaussian models focusing on parsimony, dependence structure, and computational aspects.
result Characterization and calibration of models for financial log-returns.
Transformer with denoising diffusion improves probabilistic density estimation.
problem Estimating non-Gaussian and multimodal probability distributions for regression problems.
method Training a denoising diffusion head on top of a Transformer model.
result The model provides reasonable probability density estimation for high-dimensional inputs.
New method certifies anti-concentration for various non-Gaussian distributions.
problem Efficiently certifying anti-concentration for non-Gaussian distributions.
method Sum-of-Squares relaxation of integer program for anti-concentration.
result Quasi-polynomial time certificates for non-Gaussian distributions.
Non-Gaussian component analysis (NGCA) is an unsupervised linear dimension reduction method that extracts low-dimensional non-Gaussian "signals" from high-dimensional data contaminated with Gaussian noise. NGCA can be regarded as a generalization of projection pursuit (PP) and independent component analysis (ICA) to mu…
This work extends Tweedie's formulae to non-Gaussian processes for better diffusion model generation.
problem Limited exploration of non-Gaussian diffusion models and corresponding Tweedie's formulae.
method Extended Tweedie's formulae to geometric Brownian motion, squared Bessel, and Cox-Ingersoll-Ross processes.
result Demonstrated potential of non-Gaussian models in image and financial time series generation.
Develops a novel method to estimate non-Gaussian hydraulic conductivities efficiently.
problem Estimation of non-Gaussian hydraulic conductivity fields in subsurface flow models.
method Integrates adversarial autoencoders with residual dense convolutional networks for parameterization and surrogate modeling.
result Significantly reduces computation time for accurate inversion results.
Study large deviations in fractional volatility models with non-Gaussian volatility.
problem Large deviations in fractional volatility models with non-Gaussian volatility.
method Established a small-noise large deviation principle for log-price.
result Logarithmic call price asymptotics for large strikes in a special case.
New bounds for kernel regression under non-Gaussian noise.
problem Uncertainty quantification for function estimates from noisy observations.
method Novel non-asymptotic probabilistic uniform error bounds for kernel-based regression.
result Proposed bounds apply to a broad class of non-Gaussian noise distributions.
Model for stock prices using non-Gaussian path integral.
problem Fit stock price dynamics with a small number of parameters.
method Generalized Ilinski's path integral model with a different action.
result Provides excellent fits for stock prices and indices.