arXiv research

The paper studies the correlation of Hilbert lengths for convex projective surfaces.

The paper studies critical points and flows of a $G_2$-Hilbert functional on manifolds with circle actions.

A strictly convex real projective orbifold is equipped with a natural Finsler metric called the Hilbert metric. In the case that the projective structure is hyperbolic, the Hilbert metric and the hyperbolic metric coincide. We prove that the marked Hilbert length spectrum determines the projective structure only up to …

We prove a sharp inequality between the Blaschke and Hilbert distance on a proper convex domain: for any two points $x$ and $y$, \[d^B(x,y) < d^H(x,y) +1.\] We obtain two interesting consequences: the first one is the volume entropy rigidity for Hilbert geometries : for any proper convex domain of $\mathbb{R}\mathbf{P}…

Optimal inequalities found between Riemannian and Hilbert metrics in convex projective domains.

The double tetrahedron is the triangulation of the three-sphere gotten by gluing together two congruent tetrahedra along their boundaries. As a piecewise flat manifold, its geometry is determined by its six edge lengths, giving a notion of a metric on the double tetrahedron. We study notions of Einstein metrics, consta…

The study identifies and analyzes different market regimes in equity markets using advanced signal processing techniques.

Dominant representations found via Fock-Goncharov coordinates.

The intersection of a complex plane curve with a small three-sphere surrounding one of its singularities is a non-trivial link. The refined punctual Hilbert schemes of the singularity parameterize subschemes supported at the singular point of fixed length and whose defining ideals have a fixed number of generators. We …

We introduce a new distance metric for non-linear embeddings of Tempered Exponential Measures.

Paper details Hilbert-curve for high-performance data mining.

Let $U_2({\cal H})$ be the Banach-Lie group of unitary operators in the Hilbert space ${\cal H}$ which are Hilbert-Schmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit $\{upu^*: u\in U_2({\cal H})\},$ of an infinite projection $p$ in ${\cal H}$. This orbit coincides with t…

This paper investigates a novel algorithmic approach to data representation based on kernel methods. Assuming that the observations lie in a Hilbert space X, the introduced Kernel Autoencoder (KAE) is the composition of mappings from vector-valued Reproducing Kernel Hilbert Spaces (vv-RKHSs) that minimizes the expected…

We review some basic concepts related to convex real projective structures from the differential geometry point of view. We start by recalling a Riemannian metric which originates in the study of affine spheres using the Blaschke connection (work of Calabi and of Cheng-Yau) mentioning its relation with the Hilbert metr…

In this short note, we prove that the space of all admissible piecewise linear metrics parameterized by length square on a triangulated manifolds is a convex cone. We further study Regge's Einstein-Hilbert action and give a much more reasonable definition of discrete Einstein metric than our former version in \cite{G}.…

Study pressure metrics for cusped Hitchin representations.

A new Riemannian metric on curve spaces is complete and smooth.

We study the following problem: given an Einstein metric on a manifold, characterize and study all Einstein metrics which are pointwise projective to the given one. By definition, two metrics are said to be pointwise projectively related if they have the same geodesics as point sets. This is closely related to Hilbert'…

The aim of this paper is the geometric study of the symplectic operators which are a perturbation of the identity by a Hilbert-Schmidt operator. This subgroup of the symplectic group was introduced in Pierre de la Harpe's classical book of Banach-Lie groups. Throughout this paper we will endow the tangent spaces with d…

New algorithm reduces online regression error in RKHS.

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

Embeddings in machine learning are low-dimensional representations of complex input patterns, with the property that simple geometric operations like Euclidean distances and dot products can be used for classification and comparison tasks. The proposed meta-embeddings are special embeddings that live in more general in…

In the paper "Direct Images, Fields of Hilbert Spaces, and Geometric Quantization", Lempert and Szőke proved that any flat analytic Hilbert field will induce a hermitian Hilbert bundle and gave an example of a flat Hilbert field that does not induce any Hilbert bundle. In this paper, we will provide an example of an an…

The Palais-Smale condition is proven for various knot energies.

A piecewise flat manifold is a triangulated manifold given a geometry by specifying edge lengths (lengths of 1-simplices) and specifying that all simplices are Euclidean. We consider the variation of angles of piecewise flat manifolds as the geometry varies in a particular way, which we call a conformal variation. This…

The Hilbert map's image is discussed, showing when it's surjective.

New Hilbert bundles with ends defined from indexed bases.

The infinite Viterbi alignment is the limiting maximum a-posteriori estimate of the unobserved path in a hidden Markov model as the length of the time horizon grows. For models on state-space $\mathbb{R}^{d}$ satisfying a new ``decay-convexity'' condition, we develop an approach to existence of the infinite Viterbi ali…

Study examines Hilbert area of inscribed polygons in projective geometry.

Study finds Hilbert square of real surfaces can be maximal even when the surface has disconnected real locus.

Theory developed for Hilbert geometry over valued fields, linking real and non-Archimedean geometries.

The nonparametric problem of detecting existence of an anomalous interval over a one dimensional line network is studied. Nodes corresponding to an anomalous interval (if exists) receive samples generated by a distribution q, which is different from the distribution p that generates samples for other nodes. If anomalou…

A commuting $n$-tuple $(T_1, \ldots, T_n)$ of bounded linear operators on a Hilbert space $\clh$ associate a Hilbert module $\mathcal{H}$ over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \rightarrow \mathcal{H}, \quad \quad (p, h) \mapsto p(T_1, \ldots, T_n)h…

The paper introduces austere and arid submanifolds in Hilbert spaces.

Formulates Hilbert reciprocity law on 3-manifolds.

Characterizes Hilbert schemes and their geometric properties.

We give first examples of finitely generated groups having an intermediate, with values in (0,1), Hilbert space compression (which is a numerical parameter measuring the distortion required to embed a metric space into Hilbert space). These groups include certain diagram groups. In particular, we show that the Hilbert …

Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that e…

Proves existence and uniqueness of solutions for a nonlinear equation on Hilbert manifold.

Let $\{T_1, \ldots, T_n\}$ be a set of $n$ commuting bounded linear operators on a Hilbert space $\mathcal{H}$. Then the $n$-tuple $(T_1, \ldots, T_n)$ turns $\mathcal{H}$ into a module over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \raro \clh, \quad \quad …

This survey is an introduction to positive definite kernels and the set of methods they have inspired in the machine learning literature, namely kernel methods. We first discuss some properties of positive definite kernels as well as reproducing kernel Hibert spaces, the natural extension of the set of functions $\{k(x…

Defines and computes a generalized spectral action for Lorentz warped products.

The article analyzes LCE in Hilbert space, deriving new formulas and regularisation methods.

Complex embeddings handle non-metric proximity data better than traditional methods.

Metric spaces uniquely split into Hilbert and non-line-split parts.

Study geometry of tetrahedra in complex hyperbolic space and Hilbert spaces.

Study Markov chain gradient descent in Hilbert spaces for quadratic loss.

Stochastic Gradient Descent improved for various Hilbert scales and misspecified models.