arXiv research

Derives a sharp inequality for trace-free matrices with applications to hypersurfaces.

The paper proves new Harnack inequalities for various nonlinear heat equations on manifolds.

We derive exponential tail inequalities for sums of random matrices with no dependence on the explicit matrix dimensions. These are similar to the matrix versions of the Chernoff bound and Bernstein inequality except with the explicit matrix dimensions replaced by a trace quantity that can be small even when the dimens…

This paper gives new concentration inequalities for the spectral norm of a wide class of matrix martingales in continuous time. These results extend previously established Freedman and Bernstein inequalities for series of random matrices to the class of continuous time processes. Our analysis relies on a new supermarti…

Sharp inequality found on three-balls for fourth order Sobolev traces.

Shampoo optimizes tensor spaces with faster convergence.

Derives sharp Sobolev trace inequalities for higher order derivatives on balls and half-spaces.

The paper introduces boundary operators for Poincaré-Einstein manifolds and proves higher order trace inequalities.

Sharp inequalities in unit ball with constraints on moments.

We generalize Hamilton's matrix Li-Yau-type Harnack estimate for the Ricci flow by considering the space of all LYH (Li-Yau-Hamilton) quadratics that arise as curvature tensors of space-time connections satisfying the Ricci flow with respect to the natural space-time degenerate metric. As a special case, we employ scal…

This paper deals with the trace regression model where $n$ entries or linear combinations of entries of an unknown $m_1\times m_2$ matrix $A_0$ corrupted by noise are observed. We propose a new nuclear norm penalized estimator of $A_0$ and establish a general sharp oracle inequality for this estimator for arbitrary val…

We give a geometric interpretation of the linear trace Harnack inequality for the Ricci flow.

Study of Gaussian distributions using entropic Gromov-Wasserstein and inner product Gromov-Wasserstein.

Sharp inequality on Siegel domain involving weighted norms and sub-Laplacian.

Introduces a new model for mapping matrices to matrices, subsuming linear regression.

This is a continuation of our previous work arXiv:1601.05617 on trace and inverse trace of Steklov eigenvalues. More new inequalities for the trace and inverse trace of Steklov eigenvalues are obtained.

The paper classifies minimizers for a specific inequality in Euclidean balls.

Improved MMWU algorithm achieves instance-optimal regret bound for matrix LEA.

Generalizes Escobar-Riemann mapping problem for smooth metric measure spaces.

We consider the problem of unveiling the implicit network structure of node interactions (such as user interactions in a social network), based only on high-frequency timestamps. Our inference is based on the minimization of the least-squares loss associated with a multivariate Hawkes model, penalized by $\ell_1$ and t…

We establish a one-parameter family of Harnack inequalities connecting the constrained trace Li-Yau differential Harnack inequality for a nonlinear parabolic equation to the constrained trace Chow-Hamilton Harnack inequality for this nonlinear equation with respect to evolving metrics related to Ricci flow on a 2-dimen…

The paper introduces a penalized matrix estimation procedure aiming at solutions which are sparse and low-rank at the same time. Such structures arise in the context of social networks or protein interactions where underlying graphs have adjacency matrices which are block-diagonal in the appropriate basis. We introduce…

The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.

Develops a new nonparametric trace regression model for high-dimensional data.

In this paper, we proved the mass angular momentum inequality\cite{D1}\cite{ChrusLiWe}\cite{SZ} for axisymmetric, asymptotically flat, vacuum constraint data sets with small trace. Given an initial data set with small trace, we construct a boost evolution spacetime of the Einstein vacuum equations as \cite{ChOM}. Then …

The paper introduces new boundary operators and proves higher order CR Sobolev trace inequalities for Siegel domain and complex ball.

We present a new trace estimator of the matrix whose explicit form is not given but its matrix multiplication to a vector is available. The form of the estimator is similar to the Hutchison stochastic trace estimator, but instead of the random noise vectors in Hutchison estimator, we use small number of probing vectors…

We solve robust regression and matrix completion problems with sparse and low-rank models.

We present a formula for the trace of any symmetric power of a $n\times n$ matrix (with coefficients in a field) in terms of the ordinary powers of the matrix, an arbitrarily chosen linear function which vanishes on the identity matrix, and $n-2$ polynomial functions defined recursively.

We introduce a new family of matrix norms, the "local max" norms, generalizing existing methods such as the max norm, the trace norm (nuclear norm), and the weighted or smoothed weighted trace norms, which have been extensively used in the literature as regularizers for matrix reconstruction problems. We show that this…

The paper proves inequalities for isometries in loxodromic Kleinian groups.

We prove a linear trace Li-Yau-Hamilton inequality for the Kaehler-Ricci flow. We then use this sharp differential inequality to study the Liouville properties of the plurisubharmonic functions on complete Kaehler manifolds with nonnegative bisectional curvature.

New boundary operators for sixth-order GJMS operator on manifolds.

A new method for optimizing deep neural networks using TKFAC.

We consider the problem of approximately reconstructing a partially-observed, approximately low-rank matrix. This problem has received much attention lately, mostly using the trace-norm as a surrogate to the rank. Here we study low-rank matrix reconstruction using both the trace-norm, as well as the less-studied max-no…

Sharp inequalities for matrix means with unknown variance.

Saha seeks physics models to manage income inequality.

Study trace systoles on surfaces, finding optimal bounds and implications.

Study bounds for Brownian motion on manifolds with sticky boundary conditions.

We prove certain generalization of Hardy's inequality where the "boundary defining function" is replaced by a polynomial defining a singular algebraic variety. An application is given on the existence of a small time heat trace expansion for a Schrödinger operator with mild singularities along this algebraic set.

Paper tackles clipped matrix recovery from scientific areas with theoretical and practical methods.

We show a connection between the linear trace Li-Yau-Hamilton inequality for the Kaehler-Ricci flow and the monotonicity formula for the positive currents. As an application of the linear trace Li-Yau-Hamilton stated in this paper and the one proved by Chow-Hamiltonm, we give another proof on the classification of the …

We establish sharp Sobolev inequalities of order four on Euclidean d-balls for d greater than or equal to four. When d=4, our inequality generalizes the classical second order Lebedev-Milin inequality on Euclidean 2-balls. Our method relies on the use of scattering theory on hyperbolic d-balls. As an application, we ch…

The paper extends log-Sobolev inequalities to matrix-valued settings using combinatorial methods.

Develops interpolation methods for matrix functions in statistics and machine learning.

Estimates matrix trace optimization with statistical learning theory.

We propose a novel hierarchical model for multitask bipartite ranking. The proposed approach combines a matrix-variate Gaussian process with a generative model for task-wise bipartite ranking. In addition, we employ a novel trace constrained variational inference approach to impose low rank structure on the posterior m…

Paper analyzes and improves non-convex optimization for low rank matrix factorization.