The paper provides exact multivariate amplitude distributions for non-stationary Gaussian or algebraic fluctuations.
problem Capturing the statistical properties of fluctuating correlations in non-stationary systems.
method Developed a random matrix model to average multivariate amplitude distributions from short time scales to large time scales.
result Explicit multivariate distributions for non-stationary correlation systems are provided, capturing the degree of non-stationarity.
This paper is a step-by-step tutorial for fitting a mixture distribution to data. It merely assumes the reader has the background of calculus and linear algebra. Other required background is briefly reviewed before explaining the main algorithm. In explaining the main algorithm, first, fitting a mixture of two distribu…
New algorithms learn polytree structures from data.
problem Learning causal graphs from non-Gaussian data.
method Combines Chow-Liu algorithm with edge orientation schemes.
result Established high-dimensional consistency results.
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 new algebra for probabilistic programming improves tail behavior accuracy.
problem Inaccurate tail behavior in probabilistic models based on neural networks.
method Developed a three-parameter tail asymptotics algebra based on the generalized Gamma distribution.
result Inference algorithms using the heavy-tailed algebra achieve superior performance.
In this paper, we propose an auto-encoder based generative neural network model whose encoder compresses the inputs into vectors in the tangent space of a special Lie group manifold: upper triangular positive definite affine transform matrices (UTDATs). UTDATs are representations of Gaussian distributions and can strai…
Algorithm learns mixtures of Gaussians efficiently using diffusion models.
problem Learning mixtures of Gaussians with identity covariance.
method Analytic approach using diffusion models to learn score functions.
result Quasi-polynomial time and sample complexity for learning mixtures.
Efficiently solves inverse PDE problems with Gaussian processes.
problem Solving inverse problems in linear PDEs with noisy data.
method Gaussian process regression with algebraic priors.
result High accuracy and computational efficiency achieved.
Two new algorithms improve robust PCA and Schatten packing.
problem Robustly estimating the top eigenvector of corrupted sub-Gaussian data.
method Two iterative filtering and nearly-linear time algorithms.
result First polynomial-time algorithms for non-trivial covariance estimation.
We determine the expected curvature polynomial of random real projective varieties given as the zero set of independent random polynomials with Gaussian distribution, whose distribution is invariant under the action of the orthogonal group. In particular, the expected Euler characteristic of such random real projective…
Develops a Gaussian model to compute the Alexander polynomial of knots.
problem Computing the Alexander polynomial of knots.
method Uses perturbed Gaussian functions, Heisenberg algebra, and tensor-contraction formalism.
result Associates a Gaussian function to a knot whose partition function recovers the Alexander polynomial.
Proves a special case of the Gaussian kinematic formula using large sphere limits.
problem Proving a special case of the Gaussian kinematic formula.
method Viewing the GKF as the limit of spherical kinematic formulas for large dimension spheres.
result Proves a special case of the Gaussian kinematic formula.
We compare systematically several classes of stochastic volatility models of stock market fluctuations. We show that the long-time return distribution is either Gaussian or develops a power-law tail, while the short-time return distribution has generically a stretched-exponential form, but can assume also an algebraic …
While the Matrix Generalized Inverse Gaussian (MGIG) distribution arises naturally in some settings as a distribution over symmetric positive semi-definite matrices, certain key properties of the distribution and effective ways of sampling from the distribution have not been carefully studied. In this paper…
We present two different approaches for parameter learning in several mixture models in one dimension. Our first approach uses complex-analytic methods and applies to Gaussian mixtures with shared variance, binomial mixtures with shared success probability, and Poisson mixtures, among others. An example result is that …
We prove that the ring $\Aff{\R}{M}$ of all polynomials defined on a real algebraic variety M⊂Rn is dense in the Hilbert space $L^2(M,e^{-|x|^2}\deμ)$, where $\deμ$ denotes the volume form of M and $\deν=e^{-|x|^2}\deμ$ the Gaussian measure on M.
We algorithmically construct multi-output Gaussian process priors which satisfy linear differential equations. Our approach attempts to parametrize all solutions of the equations using Gröbner bases. If successful, a push forward Gaussian process along the paramerization is the desired prior. We consider several exampl…
The paper analyzes heavy-tailed multivariate distributions in non-stationary systems using random matrix theory.
problem Risk assessment for rare events in complex, non-stationary systems.
method Generalized scalar product between correlation matrices, model for non-stationary fluctuations.
result Formulae for multivariate distributions with reduced parameters, facilitating applications.
Novel knot polynomials from Gaussian calculus show half vanish and determine Jones polynomials.
problem Understanding and characterizing knot polynomials from Gaussian calculus.
method Gaussian calculus of generating series for noncommutative algebras, connected sum of knots.
result Half of the polynomials vanish and three polynomials are explicitly given.
Gaussian Processes (GPs) are a popular approach to predict the output of a parameterized experiment. They have many applications in the field of Computer Experiments, in particular to perform sensitivity analysis, adaptive design of experiments and global optimization. Nearly all of the applications of GPs require the …
CoLA automates efficient numerical linear algebra for complex matrix structures.
problem Efficiently solving large-scale linear algebra problems with complex matrix structures.
method Combining linear operator abstraction with compositional dispatch rules.
result Automatic and efficient numerical algorithms for various linear algebra operations.
Extends machine learning models for analytic boundary conditions in differential equations.
problem Inclusion of data in differential equations using symbolic algorithms.
method Combines computer algebra with Gaussian processes and extends to analytic boundary conditions using Gröbner and Janet bases of Weyl algebras.
result Describes divergence-free flow in domains bounded by analytic functions.
Conditional mean embeddings (CMEs) have proven themselves to be a powerful tool in many machine learning applications. They allow the efficient conditioning of probability distributions within the corresponding reproducing kernel Hilbert spaces (RKHSs) by providing a linear-algebraic relation for the kernel mean embedd…
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. …
Geometric Gaussian approximations capture any distribution.
problem Approximating complex probability distributions.
method Geometric Gaussian approximations through diffeomorphisms or exponential maps.
result Geometric Gaussian approximations are universal, capturing any distribution.
Proves that emergent algebras right-distributivity implies left-distributivity.
problem Proving the implication between emergent algebra distributivity conditions.
method Analyzing families of quasigroup operations indexed by commutative groups.
result Emergent algebras right-distributive imply left-distributive.
The accuracy of probability distributions inferred using machine-learning algorithms heavily depends on data availability and quality. In practical applications it is therefore fundamental to investigate the robustness of a statistical model to misspecification of some of its underlying probabilities. In the context of…
Study of Gaussian distributions using entropic Gromov-Wasserstein and inner product Gromov-Wasserstein.
problem Optimal transportation between Gaussian distributions with different dimensions.
method Entropic Gromov-Wasserstein and inner product Gromov-Wasserstein, with closed-form expressions and von Neumann's trace inequality.
result Closed-form expressions for the entropic IGW and its unbalanced variant between Gaussian distributions.
Extends distribution algebra concept to Lie groupoids.
problem Distribution algebra on Lie groupoids.
method Construction of convolution C_c(M)/R-bialgebra associated with Lie groupoid adjoint action.
result Extension of Cartier-Gabriel decomposition to Lie groupoids.
This paper uses second-order Poincaré inequalities to establish quantitative central limit theorems for Gaussian neural networks.
problem Establishing quantitative central limit theorems for Gaussian neural networks.
method Using second-order Poincaré inequalities to reduce the problem to computing the gradient and Hessian of the NN's output.
result Suboptimal rates of convergence for the NN's output due to the use of second-order Poincaré inequalities.
We explore the algebraic structure of the solution space of convex optimization problem Constrained Minimum Trace Factor Analysis (CMTFA), when the population covariance matrix Σx has an additional latent graphical constraint, namely, a latent star topology. In particular, we have shown that CMTFA can have either a …
Classifies non-integrable distributions with simple infinite-dimensional Lie superalgebras of symmetries.
problem Classifying non-integrable distributions with specific Lie superalgebras.
method Classification based on locality assumptions and W-grading.
result 15 series and 7 exceptional Lie superalgebras identified over C, and analogs over K of characteristic p>0. A new method combines Gaussian graphical models for better distributed Gaussian process predictions.
problem Poor results from traditional DGP due to violated conditional independence assumption.
method Proposes using Gaussian graphical models to aggregate local predictions from subsets of data.
result Our method outperforms other state-of-the-art DGP approaches on both synthetic and real datasets.
Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. Submatrices with low rank correspond to generalizations of conditional independence constraints on collections of random variables. We give a precise graph-theoretic characterization of when submatrices of the cov…
Applying the general theory about complete spacelike stationary (i.e. zero mean curvature) surfaces in 4-dimensional Lorentz space R14, we classify those regular algebraic ones with total Gaussian curvature −∫KdM=4π. Such surfaces must be oriented and be congruent to either the generalized c…
GS-B3SE improves label shift estimation by smoothing priors on a graph.
problem Label shift adaptation when source and target distributions share conditional but not marginal probabilities.
method Graph-Smoothed Bayesian Black-Box Shift Estimator (GS-B3SE) places Laplacian-Gaussian priors on log-priors and confusion-matrix columns tied by a label-similarity graph. result GS-B3SE produces a tractable posterior with HMC or Newton-CG schemes, proving identifiability, contraction, and robustness. I define higher codimensional versions of contact structures on manifolds as maximally non-integrable distributions. I call them multicontact structures. Cartan distributions on jet spaces provide canonical examples. More generally, I define higher codimensional versions of pre-contact structures as distributions on ma…
SFSVI uses Gaussian mixtures to approximate neural network outputs for continual learning.
problem Learning new tasks without forgetting old ones in neural networks.
method Sequential function-space variational inference with Gaussian mixture approximation.
result Gaussian mixture SFSVI outperforms other methods in continual learning.
Bayesian Gaussian Processes layer detects out-of-distribution data in medical imaging.
problem Detecting out-of-distribution data in medical imaging tasks.
method Parameter-efficient hierarchical convolutional Gaussian Processes in Wasserstein-2 space.
result Uncertainty estimates enable superior out-of-distribution detection compared to previous methods.
Clarifies connections between Nyström and SVGP methods for scalable GPs.
problem Lack of understanding between GP and kernel methods communities.
method Investigates Nyström and SVGP methods for scalable Gaussian processes.
result Establishes connections and equivalences between Nyström and SVGP methods.
Gaussian prior and likelihood improve bandit learning performance.
problem Improving bandit learning with misspecified Gaussian distributions.
method An agent with a bounded information ratio interacts with a Bernoulli bandit based on a Gaussian prior and likelihood.
result The regret increase is at most linear in the square-root of the time horizon for diffuse distributions.
New Stein identity for q-Gaussians reduces gradient variance in machine learning.
problem Improving gradient estimators for non-Gaussian distributions.
method Deriving a new Stein identity for bounded-support q-Gaussians and simplifying previous results.
result Gradient estimators for q-Gaussians have nearly identical forms to Gaussian ones, reducing variance.
AQFC method estimates mesh curvatures using quadratic surfaces.
problem Estimating curvatures for irregular polygonal meshes.
method Local approximation of vertices and normals by quadratic surfaces, computed as implicit surfaces.
result AQFC provides robust curvature estimation for irregular meshes.
Bayesian layer improves image segmentation and out-of-distribution detection.
problem Outlier detection in image segmentation.
method Parameter-efficient hierarchical convolutional Gaussian Processes in Wasserstein-2 space.
result Uncertainty estimates improve out-of-distribution detection.
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…
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.
The paper explores geometric and algebraic structures on Lie groups.
problem Investigating F-manifolds and Fextman-algebras on Lie groups. method Constructing a canonical connection and analyzing curvature and holonomy.
result Established the integrability of a Poisson-algebra distribution.