arXiv research

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

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

Proves Gerber statistic is always non-negative.

Efficiently constructs prediction bands with minimal assumptions.

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…

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

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…

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

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

Paper develops Riemannian geometry for SPSD matrices with DA applications.

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

Method learns SDEs from data snapshots.

Paper tackles clustering with ordinal comparisons, achieving near-optimal results.

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.

Introduce Collapsed Effective Operators for higher-order structures.

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

Unified approach to Bayesian inference with guarantees on covariance matrices.

Efficient PAC learning for contrastive linear representations is achieved.

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

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…

We propose an SDP relaxation for the Gromov-Wasserstein distance, providing globally optimal solutions.

New method for symmetric matrix completion using ReLU sampling.

New distances measure mixtures of Gaussians, useful in machine learning.

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…

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

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 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. …

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…

New regularizer for machine learning using private data.

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

Algorithm learns halfspaces in noisy data efficiently.

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…

Fundamental weight systems identified as quantum states.

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…

The paper calculates involutive Heegaard Floer homology for specific 3-manifolds.

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…

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

The framework of Integral Quadratic Constraints (IQC) reduces the computation of upper bounds on the convergence rate of several optimization algorithms to a semi-definite program (SDP). In the case of over-relaxed Alternating Direction Method of Multipliers (ADMM), an explicit and closed form solution to this SDP was …

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…

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…

Establishes metrics with positive curvature on projective line bundles.

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…

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

Community detection is a fundamental unsupervised learning problem for unlabeled networks which has a broad range of applications. Many community detection algorithms assume that the number of clusters $r$ is known apriori. In this paper, we propose an approach based on semi-definite relaxations, which does not require…

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

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…

We consider the problem of estimating the phases of K mixed complex signals from a multichannel observation, when the mixing matrix and signal magnitudes are known. This problem can be cast as a non-convex quadratically constrained quadratic program which is known to be NP-hard in general. We propose three approaches t…