arXiv research

The paper extends inequalities for projection bodies to arbitrary measures.

The paper studies projections of asset prices under equivalent martingale measures.

Unique entropy measure found for convex projective manifolds.

A result about projections of Gibbs measures from a particular class arising in economic modeling is proved.

We show that on any compact Riemann surface with variable negative curvature there exists a measure which is invariant and ergodic under the geodesic flow and whose projection to the base manifold is 2-dimensional and singular with respect to the 2-dimensional Lebesgue measure.

Entropy study of geodesic flow on convex projective surfaces.

Proposes a new method to rank risky investments based on Omega measure.

A new distance measure balances projection exploration and informativeness.

Estimates dimensions of maximal simplices for rational and irrational trees in Outer space.

In this paper, for $μ$ and $ν$ two probability measures on $\mathbb{R}^d$ with finite moments of order $ρ\ge 1$, we define the respective projections for the $W_ρ$-Wasserstein distance of $μ$ and $ν$ on the sets of probability measures dominated by $ν$ and of probability measures larger than $μ$ in the convex order. Th…

We introduce a natural stratification of the space of projective classes of measured laminations on a complete hyperbolic surface of finite area. We prove a rigidity result, namely, the group of self-homeomorphisms of the space of projective measured laminations that preserve such a stratification is in general identif…

Paper establishes robust no-arbitrage conditions under projective determinacy.

A solution of Hilberts fourth problem lead to integral equation of the type generalized cosine transform. The present paper considers the solution that integral equation by integral geometry methods and propose an inversion formula for reconstruction of Crofton measures from projective smooth Finsler metrics in R3.

Let $S^{n}$ be the $n$-sphere of constant positive curvature. For $n \geq 2$, we will show that a measure on the unit tangent bundle of $S^{2n}$, which is even and invariant under the geodesic flow, is not uniquely determined by its projection to $S^{2n}$.

New findings on geometric flows and equidistribution in Hilbert geometry.

A new indicator measures project risk from activity durations.

Develops efficient projections for multivariate probability measures.

We study several new invariants associated to a holomorphic projective structure on a Riemann surface of finite analytic type: the Lyapunov exponent of its holonomy which is of probabilistic/dynamical nature and was introduced in our previous work; the degree which measures the asymptotic covering rate of the developin…

Paper develops a new method for differential privacy sampling using Wasserstein distance.

In this paper, we show that the Euler characteristic of an even dimensional closed projectively flat manifold is equal to the total measure which is induced from a probability Borel measure on RP^n invariant under the holonomy action, and then discuss its consequences and applications. As an application, we show that t…

We design a new sparse projection method for a set of vectors that guarantees a desired average sparsity level measured leveraging the popular Hoyer measure (an affine function of the ratio of the $\ell_1$ and $\ell_2$ norms). Existing approaches either project each vector individually or require the use of a regulariz…

Study curvatures of diffeomorphisms on non-orientable surfaces.

We study the limits of sequences of spheres and complex projective spaces with unbounded dimensions. A sequence of spheres (resp. complex projective spaces) either is a Levy family, infinitely dissipates, or converges to (resp. the Hopf quotient of) a virtual infinite-dimensional Gaussian space, depending on the size o…

Characterizes closures of mapping class group orbits on non-orientable surfaces.

Modified neural network enhances unsupervised anomaly detection.

The study connects projective codes to the distribution of zeros of odd maps.

This paper analyzes quantiles of heavy-tailed distributions, separating projection direction and quantile threshold effects.

We study the geometry of the cuspidal edge $M$ in $\mathbb R^3$ derived from its contact with planes and lines (referred to as flat geometry). The contact of $M$ with planes is measured by the singularities of the height functions on $M$. We classify submersions on a model of $M$ by diffeomorphisms and recover the cont…

Convex optimization method recovers low-rank matrices from rank-one projections efficiently.

Paper proposes PRWB and RPRWB for Wasserstein barycenters.

Improves point-cloud reconstruction by optimizing projections with self-attention.

We calculate a projective space of essential measured laminations in a surface pair, which will be used in another paper to help describe spaces of "finite height laminations."

The study constructs universal invariants for non-Archimedean metrics on projective varieties.

In this paper, we study the recovery of a signal from a set of noisy linear projections (measurements), when such projections are unlabeled, that is, the correspondence between the measurements and the set of projection vectors (i.e., the rows of the measurement matrix) is not known a priori. We consider a special case…

A new method approximates the Sliced-Wasserstein distance without random projections.

Measuring conditional dependence is an important topic in statistics with broad applications including graphical models. Under a factor model setting, a new conditional dependence measure based on projection is proposed. The corresponding conditional independence test is developed with the asymptotic null distribution …

We study the problem of determining the optimal low dimensional projection for maximising the separability of a binary partition of an unlabelled dataset, as measured by spectral graph theory. This is achieved by finding projections which minimise the second eigenvalue of the graph Laplacian of the projected data, whic…

Computing optimal transport (OT) between measures in high dimensions is doomed by the curse of dimensionality. A popular approach to avoid this curse is to project input measures on lower-dimensional subspaces (1D lines in the case of sliced Wasserstein distances), solve the OT problem between these reduced measures, a…

We extend the Besicovitch-Federer projection theorem to transversal families of mappings. As an application we show that on a certain class of Riemann surfaces with constant negative curvature and with boundary, there exist natural 2-dimensional measures invariant under the geodesic flow having 2-dimensional supports s…

Since their introduction by Thurston, measured geodesic laminations on hyperbolic surfaces occur in many contexts. In [Mor], we have introduced a notion of flat laminations on surfaces endowed with a half-translation structure (that is a singular flat surface with holonomy {+/-Id}, similar to geodesic laminations on hy…

We consider actions of non-compact simple Lie groups preserving an analytic rigid geometric structure of algebraic type on a compact manifold. The structure is not assumed to be unimodular, so an invariant measure may not exist. Ergodic stationary measures always exist, and when such a measure has full support, we show…

Defines a measure of knot concordance using cobordism distance.

Investigates projections onto explicit subspaces and their variance effects.

We show that grafting any fixed hyperbolic surface defines a homeomorphism from the space of measured laminations to Teichmuller space, complementing a result of Scannell-Wolf on grafting by a fixed lamination. This result is used to study the relationship between the complex-analytic and geometric coordinate systems f…

A convex projective surface is the quotient of a properly convex open $Ω$ of $\mathbb{P}(\R)$ by a discret subgroup $Γ$ of $\mathrm{SL}_3(\R)$. We give some caracterisations of the fact that a convex projective surface is of finite volume for the Busemann's measure. We deduce of this that if $Ω$ is not a triangle then …

Proposes variance reduction techniques for sliced Wasserstein distance estimation.

Paper quantifies distortion risk measures' robustness to distributional uncertainty.

A new method optimizes projection directions for sliced Wasserstein distances.