arXiv research

Paper proposes a new covariance estimator ensuring positive semi-definite matrices.

Proves Gerber statistic is always non-negative.

New approach to analyze matrix denoising using gradient flow and fixed point equations.

The paper characterizes Einstein 4-manifolds with semi-definite curvature and derives inequalities.

A new method for deep Wishart processes improves kernel-based models.

In machine learning or statistics, it is often desirable to reduce the dimensionality of a sample of data points in a high dimensional space $\mathbb{R}^d$. This paper introduces a dimensionality reduction method where the embedding coordinates are the eigenvectors of a positive semi-definite kernel obtained as the sol…

Paper develops Riemannian geometry for SPSD matrices with DA applications.

Introduce Collapsed Effective Operators for higher-order structures.

Unified approach to Bayesian inference with guarantees on covariance matrices.

A new imputation method estimates missing values by matching observed marginals from masked data.

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…

Upper bound found for dimensions of subspaces where holomorphic sectional curvature vanishes.

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.

Paper introduces a new kernel model for PSD-valued functions with theoretical guarantees and applications.

This paper proposes an efficient method for sampling from stochastic differential equations using PSD models.

New method for symmetric matrix completion using ReLU sampling.

The paper presents two schemes for sampling matrices from specific distributions on a manifold.

A family of probability distributions parametrized by an open domain $Λ$ in $R^n$ 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.

Establishes metrics with positive curvature on projective line bundles.

The paper tackles feature cross search for linear models, providing approximation algorithms and structural results.

Motivated by the results of B. Berndtsson, in this memoir we use the new estimates developed by W. He to extend a theorem of the second author on the existence of weak $C^{1,1}$ geodesics between two smooth non-degenerate Kähler potentials to the case where the metrics on the end points may have singularities on some a…

Efficiently constructs prediction bands with minimal assumptions.

As is well known, a metric on a manifold determines a unique symmetric connection for which the metric is parallel: the Levi-Civita connection. In this paper we investigate the inverse problem: to what extent is the metric of a Riemannian manifold determined by its Levi-Civita connection? It is shown that for a generic…

Cuspidal edges and swallowtails are typical non-degenerate singular points on wave fronts in the Euclidean $3$-space. Their first fundamental forms belong to a class of positive semi-definite metrics called "Kossowski metrics". A point where a Kossowski metric is not positive definite is called a singular point or a se…

A characterization of the proximal normal cone is obtained and a separation theorem for convex subsets of Riemannian manifolds is established. Moreover, the convexity of the distance function $d_S$ for a convex subset $S$ in the cases where the boundary of $S$ contains a geodesic segment, the boundary of $S$ is $C^2$ o…

Topological Data Analysis (TDA) is a recent and growing branch of statistics devoted to the study of the shape of the data. In this work we investigate the predictive power of TDA in the context of supervised learning. Since topological summaries, most noticeably the Persistence Diagram, are typically defined in comple…

New convergence rates for SGD under heavy-tailed noise with infinite variance.

Study of $H$-eigenvalues for complex tensors and their applications in differential geometry.

Extended elliptical slice sampling for infinite-dimensional spaces, proving reversibility.

A new method for hierarchical clustering is presented. It combines treelets, a particular multiscale decomposition of data, with a projection on a reproducing kernel Hilbert space. The proposed approach, called kernel treelets (KT), effectively substitutes the correlation coefficient matrix used in treelets with a symm…

Learning representation from relative similarity comparisons, often called ordinal embedding, gains rising attention in recent years. Most of the existing methods are based on semi-definite programming (\textit{SDP}), which is generally time-consuming and degrades the scalability, especially confronting large-scale dat…

We propose a new input perturbation mechanism for publishing a covariance matrix to achieve $(ε,0)$-differential privacy. Our mechanism uses a Wishart distribution to generate matrix noise. In particular, We apply this mechanism to principal component analysis. Our mechanism is able to keep the positive semi-definitene…

Paper proposes a new algorithm for graph learning with covariance constraints.

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 $\mathcal{X}$, we consider the secant set $S(\mathcal{X})$ that consists of all pairwise difference vectors of $\mathcal{X}$, normalized to lie on the unit sphere. …

The paper analyzes Karcher means on restricted PSD matrices with statistical guarantees.

We study the minimization of a convex function $f(X)$ over the set of $n\times n$ positive semi-definite matrices, but when the problem is recast as $\min_U g(U) := f(UU^\top)$, with $U \in \mathbb{R}^{n \times r}$ and $r \leq n$. We study the performance of gradient descent on $g$---which we refer to as Factored Gradi…

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 …

This paper proposes a variant of the method of Guédon and Verhynin for estimating the cluster matrix in the Mixture of Gaussians framework via Semi-Definite Programming. A clustering oriented embedding is deduced from this estimate. The procedure is suitable for very high dimensional data because it is based on pairwis…

PSD models simplify probability density estimation.

The paper proves that a fiberwise Kähler-Ricci flow is positive for all time on a family of bounded strongly pseudoconvex domains.

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…

A projective parameter of a geodesic on a Finsler space is defined to be solution of a certain ODE. Using projective parameter and Funk metric, one can construct a projectively invariant intrinsic pseudo-distance on a Finsler space. In the present work, solutions of the projective parameter's ODE are characterized with…

CLuP achieves near optimal ground state energies for positive and negative Hopfield models.

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…

Gaussian process (GP) audio source separation is a time-domain approach that circumvents the inherent phase approximation issue of spectrogram based methods. Furthermore, through its kernel, GPs elegantly incorporate prior knowledge about the sources into the separation model. Despite these compelling advantages, the c…

In this paper, we present a novel two-stage metric learning algorithm. We first map each learning instance to a probability distribution by computing its similarities to a set of fixed anchor points. Then, we define the distance in the input data space as the Fisher information distance on the associated statistical ma…

Study proves Kählerness criteria for Hermitian surfaces under specific curvature conditions.