arXiv research

Study estimates gaps in semigroup products, proving embedding properties.

The paper establishes pressure gaps for manifolds with flat subtori singularities.

The paper tackles system identification via Hankel nuclear norm regularization, improving estimation rates and singular value gaps.

The paper analyzes how random perturbations affect RSVD and its applications.

New method cleans cross-covariance matrices for better financial forecasting.

We describe two-dimensional potential Schrodinger and Dirac operators which are finite-gap at one energy level and have singular spectral curves. It appears that the singularities can be rather complicated. Such Dirac operators appear as the spectral curves of tori immersed into the three-space.

Informed by recent work on tensor singular value decomposition and circulant algebra matrices, this paper presents a new theoretical bridge that unifies the hypercomplex and tensor-based approaches to singular value decomposition and robust principal component analysis. We begin our work by extending the principal comp…

Local gaps in Ricci shrinkers depend only on dimension.

This note removes technical assumptions and characterizes relatively dominated representations.

The paper provides uniform length estimates for trajectories on flat cone surfaces.

Study spectral properties on manifolds with conical singularities, proving new inequalities.

New rigidity theorem for sharp spectral gap in nonnegatively curved spaces.

We demonstrate that almost all non-parametric dimensionality reduction methods can be expressed by a simple procedure: regularized loss minimization plus singular value truncation. By distinguishing the role of the loss and regularizer in such a process, we recover a factored perspective that reveals some gaps in the c…

The study proves unique and isolated properties of Einstein 5-manifolds via gap theorems in 4 dimensions.

The paper analyzes the latent geometry of generative diffusion models.

Study of hyperbolic directions in convex projective geometry.

Proves regularity for stable varifolds near cones, expanding previous work.

In the first part we use Gromov's K--area to define the K--area homology which stabilizes into singular homology on the category of pairs of compact smooth manifolds. The second part treats the questions of certain curvature gaps. For instance, the $L^\infty$--curvature gap of complex vector bundles on a compact manif…

Constructs coordinate systems from spectral curve sheaves.

Study establishes monodromy equivalence for Lamé-type equations and constructs cone spherical metrics.

Note on minimal maps' uniqueness via singular values.

New technique stabilizes singular values in concatenated matrices.

Study inequalities for singular values of rectangular matrices.

The paper describes correlations of spectra for higher rank Anosov representations.

Study differentiable maps on hypersurface links, finding fold maps with circle singular value sets.

Study evaluates thresholds for removing noise from DNN weights using random matrix theory.

Study shows XRP price correlates with transaction network metrics.

A low rank matrix X has been contaminated by uniformly distributed noise, missing values, outliers and corrupt entries. Reconstruction of X from the singular values and singular vectors of the contaminated matrix Y is a key problem in machine learning, computer vision and data science. In this paper we show that common…

Deterministic bounds for tensor singular values and vectors, differing from matrix cases.

We develop some estimates under the Ricci flow and use these estimates to study the blowup rates of curvatures at singularities. As applications, we obtain some gap theorems: $\displaystyle \sup_X |Ric|$ and $\displaystyle \sqrt{\sup_X |Rm|} \cdot \sqrt{\sup_X |R|}$ must blowup at least at the rate of type-I. Our estim…

We introduce the concept of singular values for the Riemann curvature tensor, a central mathematical tool in Einstein's theory of general relativity. We study the properties related to the singular values, and investigate five typical cases to show its relationship to the Ricci scalar and other invariants.

Study shows singular set of certain graphs has codimension 1.

Proves weak cosmic censorship for a specific Einstein-scalar field system in 2+1 dimensions.

Study precise rates of horizontal gap shrinkage on generic translation surfaces.

We give bounds on the gap functions of the singularities of a cuspidal plane curve of arbitrary genus, generalising recent work of Borodzik and Livingston. We apply these inequalities to unicuspidal curves whose singularity has one Puiseux pair: we prove two identities tying the parameters of the singularity, the genus…

In this paper, we prove that the mean curvature blows up at the same rate as the second fundamental form at the first singular time $T$ of any compact, Type I mean curvature flow. For the mean curvature flow of surfaces, we obtain similar result provided that the Gaussian density is less than two. Our proofs are based …

Solves initial value problem for harmonic maps on specific manifolds.

Improved singular value approximation for convolutional layers.

The paper solves conditions for extending circle-valued Morse functions.

We reveal connections between RBMs and Bosons, explaining symmetry breaking in their energy landscapes.

New framework for higher-order singular-value derivatives of rectangular matrices.

SLT explains neural network success by closing theory-practice gap.

Energy minimizing harmonic maps between manifolds are known to be smooth outside a rectifiable set of codimension $3$, called the singular set. The possibility that this set is not a manifold, but has arbitrarily many small gaps in it, is not excluded in general. Here we prove that some part of the singular set - chara…

Establishes refined singularity estimate for nonnegative n-superharmonic functions in locally conformally flat manifolds.

The problem of low-rank matrix completion has recently generated a lot of interest leading to several results that offer exact solutions to the problem. However, in order to do so, these methods make assumptions that can be quite restrictive in practice. More specifically, the methods assume that: a) the observed indic…

Study optimal liquidation with multiple regimes using BSDEs with singular terminal values.

Proves continuity and singular set dimension for 2D maps with Q values.

The complex Lie superalgebras $\mathfrak{g}$ of type $D(2,1;a)$ - also denoted by $\mathfrak{osp}(4,2;a)$ - are usually considered for "non-singular" values of the parameter $a$, for which they are simple. In this paper we introduce five suitable integral forms of $\mathfrak{g}$, that are well-defined at singular valu…