Paper proposes a new covariance estimator ensuring positive semi-definite matrices.
problem Estimating spot covariance matrices while maintaining positive semi-definiteness.
method Modification of the Fourier covariance estimator with a symmetric positive semi-definite constraint.
result The estimator is consistent and produces accurate positive semi-definite matrices.
Paper develops Riemannian geometry for SPSD matrices with DA applications.
problem Riemannian geometry of SPSD matrices for DA.
method Closed-form expressions, approximations of geodesic path, PT, canonical representation.
result Proposes an algorithm for DA with improved performance.
A new method for deep Wishart processes improves kernel-based models.
problem Inference in deep Wishart processes is challenging due to the need for flexible distributions over positive semi-definite matrices.
method Developed a novel approach to flexible distributions over positive semi-definite matrices using the Bartlett decomposition of the Wishart probability density. Used this to create an approximate posterior for the DWP.
result Improved performance of inference in the DWP compared to DGP with equivalent prior.
The paper presents two schemes for sampling matrices from specific distributions on a manifold.
problem Sampling matrices from Gibbs distributions on the manifold of positive semi-definite matrices with fixed rank.
method Two explicit schemes based on Euler-Maruyama discretization of the Riemannian Langevin equation with Brownian motion on the manifold.
result Numerical validation of the schemes using specific energy functions and metrics.
Unified approach to Bayesian inference with guarantees on covariance matrices.
problem Approximate Bayesian inference with PSD guarantees.
method Bayes-Newton methods extending Newton's method for optimisation.
result Novel algorithms with PSD covariance matrices.
Method estimates sparse inverse covariance and partial correlation matrices efficiently.
problem Sparse high-dimensional inverse covariance and partial correlation matrix estimation.
method Two-stage estimation method using partial regression with positive semi-definiteness.
result Efficient estimation of inverse covariance and partial correlation matrices with derived non-asymptotic rates.
The paper analyzes Karcher means on restricted PSD matrices with statistical guarantees.
problem Statistical analysis of non-linear manifolds in machine learning.
method Intrinsic mean model on restricted PSD matrices, Karcher mean analysis, extrinsic signal-plus-noise model.
result Non-asymptotic statistical analysis of Karcher means with deterministic error bounds.
Proves Gerber statistic is always non-negative.
problem Verifying the positive semi-definiteness of Gerber statistic.
method Analytical proof of both forms of Gerber statistic.
result Gerber statistic is positive semi-definite.
The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. The goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank density matrices in trace regression models used in quantum state tomography (in particular, i…
Paper proposes a new algorithm for graph learning with covariance constraints.
problem Graphical models and factor analysis not jointly leveraged in graph learning processes.
method Penalized maximum likelihood estimation of an elliptical distribution with Riemannian optimization.
result Effectiveness of the proposed approach demonstrated on real-world data sets.
The (stochastic) gradient descent and the multiplicative update method are probably the most popular algorithms in machine learning. We introduce and study a new regularization which provides a unification of the additive and multiplicative updates. This regularization is derived from an hyperbolic analogue of the entr…
New geometric structures defined on SPD matrices for better understanding.
problem Understanding SPD matrices and their geometric properties.
method Introducing Finslerian and dual information-geometric structures on James' bicone domain.
result Geodesics correspond to straight lines in coordinate systems, and new dissimilarities generalize existing ones.
We introduce a stochastic process with Wishart marginals: the generalised Wishart process (GWP). It is a collection of positive semi-definite random matrices indexed by any arbitrary dependent variable. We use it to model dynamic (e.g. time varying) covariance matrices. Unlike existing models, it can capture a diverse …
Paper introduces a new kernel model for PSD-valued functions with theoretical guarantees and applications.
problem Enforcing positive semi-definiteness (PSD) in function models with good performance and theoretical guarantees.
method Kernel sum-of-squares model for PSD-valued functions, extending previous models for non-negative scalar functions.
result The model constitutes a universal approximator of PSD functions and can represent any smooth and strongly convex function.
New convergence rates for SGD under heavy-tailed noise with infinite variance.
problem Convergence analysis of SGD under heavy-tailed noise with infinite variance.
method Identifying a condition on the Hessian and providing a convergence rate for the distance to the global optimum.
result SGD can converge to the global optimum under heavy-tailed noise with infinite variance.
Paper defines conditions for feasible correlation matrices from factor structures.
problem Feasibility of option implied correlation matrices in non-FX markets.
method Quantitative and economic approaches to solve the nearest correlation matrix problem.
result Introduces methods to ensure feasible correlation matrices from factor structures.
Develops interpolation methods for matrix functions in statistics and machine learning.
problem Estimating matrix functions in statistics and machine learning.
method Interpolates log-determinant and trace of matrix powers using modified sharp bounds.
result Accuracy and performance demonstrated in numerical examples.
New approach to analyze matrix denoising using gradient flow and fixed point equations.
problem Positive semi-definite matrix denoising in extensive-rank and high-dimensional settings.
method Gradient flow and fixed point equations derived from linear pencil techniques of random matrix theory.
result Continuous phase transitions in the extensive-rank and high-dimensional regime.
The paper characterizes Einstein 4-manifolds with semi-definite curvature and derives inequalities.
problem Characterizing Einstein 4-manifolds with semi-definite sectional curvature.
method Using pointwise inequalities involving scalar curvature and Weyl curvatures.
result Closed 4-dimensional Einstein metrics saturating the pointwise inequality are completely characterized.
This paper studies geodesics between covariance matrices of different ranks using the Bures-Wasserstein metric.
problem Geodesics between covariance matrices of varying ranks.
method Analyzes the Bures-Wasserstein distance on covariance matrices, completing previous work on geodesics and providing explicit formulas.
result The set of all minimizing geodesics between two covariance matrices is parametrized by a closed unit ball in R(k−r)imes(l−r). In machine learning or statistics, it is often desirable to reduce the dimensionality of a sample of data points in a high dimensional space Rd. This paper introduces a dimensionality reduction method where the embedding coordinates are the eigenvectors of a positive semi-definite kernel obtained as the sol…
The paper proposes methods to find a shared active subspace for multivariate vector-valued functions.
problem Minimizing the deviation between function evaluations in the original and reconstructed spaces.
method Manipulating gradients or SPD matrices to identify a shared structure.
result Summing SPD matrices often identifies the best shared active subspace.
We propose two practical non-convex approaches for learning near-isometric, linear embeddings of finite sets of data points. Given a set of training points X, we consider the secant set S(X) that consists of all pairwise difference vectors of X, normalized to lie on the unit sphere. …
We study the minimization of a convex function f(X) over the set of n×n positive semi-definite matrices, but when the problem is recast as minUg(U):=f(UU⊤), with U∈Rn×r and r≤n. We study the performance of gradient descent on g---which we refer to as Factored Gradi…
Riemannian gradient descent escapes some spurious critical points on low-rank matrix manifold.
problem Spurious critical points on the boundary of low-rank matrix manifold.
method Riemannian gradient descent with dynamical low-rank approximation and rescaled gradient flow.
result Riemannian gradient descent escapes some spurious critical points on the boundary of the manifold.
Embedding complex objects as vectors in low dimensional spaces is a longstanding problem in machine learning. We propose in this work an extension of that approach, which consists in embedding objects as elliptical probability distributions, namely distributions whose densities have elliptical level sets. We endow thes…
We study the problem of column selection in large-scale kernel canonical correlation analysis (KCCA) using the Nyström approximation, where one approximates two positive semi-definite kernel matrices using "landmark" points from the training set. When building low-rank kernel approximations in KCCA, previous work mostl…
Graph neural networks improve AMG convergence for sparse systems.
problem Efficiently constructing algebraic multigrid prolongation operators for sparse linear systems.
method Train a graph neural network to learn prolongation operators from matrix classes, using an unsupervised loss function.
result Improved convergence rates compared to classical AMG methods.
Let Sm be the set of all m×m density matrices (Hermitian positively semi-definite matrices of unit trace). Consider a problem of estimation of an unknown density matrix ρ∈Sm based on outcomes of n measurements of observables X1,…,Xn∈Hm (Hm bei…
Paper improves efficiency in matrix computations for Gaussian processes.
problem Efficiency in matrix computations for Gaussian processes.
method Variance reduction via matrix factorization.
result Factorized estimator can be up to 1,000 times more efficient.
We consider whether algorithmic choices in over-parameterized linear matrix factorization introduce implicit regularization. We focus on noiseless matrix sensing over rank-r positive semi-definite (PSD) matrices in Rn×n, with a sensing mechanism that satisfies restricted isometry properties (RIP)…
AniDS improves molecular force field modeling by learning anisotropic noise.
problem Molecular force field modeling suffers from oversimplified assumptions about atomic motions.
method AniDS introduces anisotropic noise generation for better modeling of directional and structural variability.
result AniDS outperforms existing methods on benchmarks, achieving significant improvements in force prediction accuracy.
Traditional linear methods for forecasting multivariate time series are not able to satisfactorily model the non-linear dependencies that may exist in non-Gaussian series. We build on the theory of learning vector-valued functions in the reproducing kernel Hilbert space and develop a method for learning prediction func…
Most machine learning algorithms, such as classification or regression, treat the individual data point as the object of interest. Here we consider extending machine learning algorithms to operate on groups of data points. We suggest treating a group of data points as an i.i.d. sample set from an underlying feature dis…
Introduce Collapsed Effective Operators for higher-order structures.
problem Existing spectral operators decompose topology into separate ranks, leaving practitioners to fuse information back to vertices.
method Introduce Collapsed Effective Operators via Schur complementation of a graded Laplacian.
result Preserves positive semi-definiteness, lowers system energy under higher-order connectivity.
Simplified analysis of SGD for linear regression with weight averaging.
problem Understanding SGD optimization in linear regression models.
method Simplified analysis using linear algebra tools, bypassing complex operator manipulations.
result Recovery of bias and variance bounds for SGD in linear regression.
Density matrices are positively semi-definite Hermitian matrices with unit trace that describe the states of quantum systems. Many quantum systems of physical interest can be represented as high-dimensional low rank density matrices. A popular problem in {\it quantum state tomography} (QST) is to estimate the unknown l…
Kernel methods summarize and integrate posterior similarity matrices from Bayesian clustering.
problem Summarizing and integrating posterior similarity matrices from Bayesian clustering.
method Positive semi-definite PSMs, kernel matrices, kernel methods, combining kernels.
result Kernel methods effectively summarize and integrate posterior similarity matrices.
Upper bound found for dimensions of subspaces where holomorphic sectional curvature vanishes.
problem Finding upper bounds for dimensions of subspaces where holomorphic sectional curvature vanishes.
method Connection with D'Angelo's work on complex subvarieties of real algebraic varieties and decomposition of polynomials into differences of squares.
result An upper bound for the dimensions of these subspaces is found.
In this paper we present a slight modification of the Fourier estimation method of the spot volatility (matrix) process of a continuous Itô semimartingale where the estimators are always non-negative definite. Since the estimators are factorized, computational cost will be saved a lot.
A new imputation method estimates missing values by matching observed marginals from masked data.
problem Missing values in data undermine statistical and machine learning analysis.
method Estimates a distribution from masked observations using positive semi-definite kernel density estimation.
result The method yields both single and multiple imputations from the same fitted density, with statistical consistency and fast adaptive excess risk.
New method for symmetric matrix completion using ReLU sampling.
problem Symmetric positive semi-definite low-rank matrix completion with deterministic entry-dependent sampling.
method ReLU sampling, gradient descent with tailored initialization.
result Gradient descent with tailored initialization achieves global minima.
Improved variational approximation for deep Wishart process models.
problem Improving predictive performance of deep Wishart process models.
method Generalizing the Bartlett decomposition of the Wishart distribution to allow linear combinations of rows and columns.
result Better predictive performance achieved with minimal additional computation cost.
A family of probability distributions parametrized by an open domain Λ in Rn defines the Fisher information matrix on this domain which is positive semi-definite. In information geometry the standard assumption has been that the Fisher information matrix tensor is positive definite defining in this way a Riemannia…
Fundamental weight systems identified as quantum states.
problem Identifying which weight systems are quantum states.
method Analyzing the Cayley distance kernel on the symmetric group and its positivity.
result All fundamental gl(n)-weight systems are quantum states.
Models like support vector machines or Gaussian process regression often require positive semi-definite kernels. These kernels may be based on distance functions. While definiteness is proven for common distances and kernels, a proof for a new kernel may require too much time and effort for users who simply aim at prac…
Establishes metrics with positive curvature on projective line bundles.
problem Existence of complete Kähler metrics with semi-positive holomorphic sectional curvature.
method Calabi's Ansatz and product approach.
result Existence of complete Kähler metrics with many zeroes.
Tseytlin has recently proposed that an action functional exists whose gradient generates to all orders in perturbation theory the Renormalization Group (RG) flow of the target space metric in the worldsheet sigma model. The gradient is defined with respect to a metric on the space of coupling constants which is explici…