arXiv research

Efficient algorithms find optimal monotone transforms for calibration under strictly convex losses.

We analyze bias-variance of margin losses.

Proposes a new loss function for robust learning.

Paper introduces a new $G^\star$ regret measure for online convex optimization with smooth losses.

We analyze the optimization landscape of α-loss in logistic models.

The paper studies quasi-$X$-convex functions and their applications in optimization.

The study shows how strictly convex domains in Euclidean spaces are rigid.

This study of properly or strictly convex real projective manifolds introduces notions of parabolic, horosphere and cusp. Results include a Margulis lemma and in the strictly convex case a thick-thin decomposition. Finite volume cusps are shown to be projectively equivalent to cusps of hyperbolic manifolds. This is pro…

We introduce a new family of affine metrics on a locally strictly convex surface $M$ in affine 4-space. Then, we define the symmetric and antisymmetric equiaffine planes associated with each metric. We show that if $M$ is immersed in a locally strictly convex hyperquadric, then the symmetric and the antisymmetric plane…

In this paper, it is shown that a Wulff shape is strictly convex if and only if its convex integrand is of class $C^1$. Moreover, applications of this result are given.

Characterizes holonomies of convex projective cusps.

Cooper and Long generalised Epstein and Penner's Euclidean cell decomposition of cusped hyperbolic manifolds of finite volume to non-compact strictly convex projective manifolds of finite volume. We show that Weeks' algorithm to compute this decomposition for a hyperbolic surface generalises to strictly convex projecti…

This paper extends a 3D result to higher dimensions for manifolds with positive curvature.

We obtain $C^2$ a priori estimates for solutions of the nonlinear second-order elliptic equation related to the geometric problem of finding a strictly locally convex hypersurface with prescribed curvature and boundary in a space form. Under the assumption of a strictly locally convex subsolution, we establish existenc…

Entropy rigidity proven for 3D and higher convex projective manifolds.

This paper is devoted to $C^2$ a priori estimates for strictly locally convex radial graphs with prescribed Weingarten curvature and boundary in space forms. By constructing two-step continuity process and applying degree theory arguments, existence results in space forms are established for prescribed Gauss curvature …

This paper gives the first example of a unipotent group that is not virtually abelian and preserves a strictly convex domain.

It is widely conjectured that the reason that training algorithms for neural networks are successful because all local minima lead to similar performance, for example, see (LeCun et al., 2015, Choromanska et al., 2015, Dauphin et al., 2014). Performance is typically measured in terms of two metrics: training performanc…

The existence of a smooth complete strictly locally convex hypersurface with prescribed scalar curvature and asymptotic boundary at infinity in $\mathbb{H}^{3}$ is proved under the assumption that there exists a strictly locally convex subsolution.

In this paper we find strictly locally convex hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed curvature and boundary. The main result is that if the given data admits a strictly locally convex radial graph as a subsolution, we can find a radial graph realizing the prescribed curvature and boundary. As an applicatio…

We prove that any smooth Riemannian manifold of non-negative scalar curvature and with a strictly mean convex and compact boundary component can be (C^2) extended beyond the component to have non-negative scalar curvature and to enjoy anyone of the following three types of (new) boundary: strictly convex, totally geode…

Paper investigates curvature problems and existence of solutions.

In this paper, we investigate simultaneous properties of a convex integrand $γ$ and its dual $δ$. The main results are the following three. (1) For a $C^\infty$ convex integrand $γ: S^n\to \mathbb{R}_+$, its dual convex integrand $δ: S^n\to \mathbb{R}_+$ is of class $C^\infty$ if and only if $γ$ is a strictly convex in…

The paper proves the existence of specific spacelike hypersurfaces in Minkowski space.

New capillary Christoffel-Minkowski problem solved for half-space.

The goal of online prediction with expert advice is to find a decision strategy which will perform almost as well as the best expert in a given pool of experts, on any sequence of outcomes. This problem has been widely studied and $O(\sqrt{T})$ and $O(\log{T})$ regret bounds can be achieved for convex losses (\cite{zin…

We introduce a tunable loss function called $α$-loss, parameterized by $α\in (0,\infty]$, which interpolates between the exponential loss ($α= 1/2$), the log-loss ($α= 1$), and the 0-1 loss ($α= \infty$), for the machine learning setting of classification. Theoretically, we illustrate a fundamental connection between $…

Let M be a compact manifold of dimension n with a strictly convex projective structure. We consider the geodesic flow of the Hilbert metric on it, which is known to be Anosov. We prove that its topological entropy is less than n-1, with equality if and only if the structure is Riemannian, that is hyperbolic. As a corol…

In this article, we study convex affine domains which can cover a compact affine manifold. For this purpose, we first show that every strictly convex quasi-homogeneous projective domain has at least $C^1$ boundary and it is an ellipsoid if its boundary is twice differentiable. And then we show that an n-dimensional par…

New guarantees for Group LASSO in sparse convex optimization.

Compact foliations preserve entropy if leaves are strictly convex projective.

Proof that convex structures on manifolds are open and closed.

Gradient descent implicitly follows regularization for general losses.

Consider a compact Riemannian manifold of dimension $\geq 3$ with strictly convex boundary, such that the manifold admits a strictly convex function. We show that the attenuated ray transform in the presence of an arbitrary connection and Higgs field is injective modulo the natural obstruction for functions and one-for…

The paper extends geometric results from negatively-curved spaces to strictly convex Hilbert geometry.

We show that associating the Euclidean cell decomposition due to Cooper and Long to each point of the moduli space of framed strictly convex real projective structures of finite volume on the once-punctured torus gives this moduli space a natural cell decomposition. The proof makes use of coordinates due to Fock and Go…

The study confirms Liouville-type theorems for positive harmonic functions on manifolds with nonnegative Ricci curvature and strictly convex boundary.

Paper relaxes optimal transport using convex functions for data science.

A theorem of Tits - Vinberg allows to build an action of a Coxeter group $Γ$ on a properly convex open set $Ω$ of the real projective space, thanks to the data $P$ of a polytope and reflection across its facets. We give sufficient conditions for such action to be of finite covolume, convex-cocompact or geometrically fi…

Wider neural networks have predominantly positive curvature, aiding optimization.

In this paper we show that bending a finite volume hyperbolic $d$-manifold $M$ along a totally geodesic hypersurface $Σ$ results in a properly convex projective structure on $M$ with finite volume. We also discuss various geometric properties of bent manifolds and algebraic properties of their fundamental groups. We th…

In this paper, we demonstrate that the complete hyperbolic structure of various two-bridge knots and links cannot be deformed to an inequivalent strictly convex projective structure. We also prove a complementary result showing that under certain rigidity hypotheses, branched covers of amphicheiral knots admit non-triv…

The study proves Liouville theorems on curved manifolds with convex boundaries.

Study inverse curvature flows for capillary hypersurfaces in a unit ball.

Local invertibility of ray transforms on convex manifolds.

Paper reconstructs compact Riemannian manifolds from travel time data.

We consider strictly convex hypersurfaces with the boundary which meets a strictly convex cone perpendicularly. We prove that if these hypersurfaces expand inside this cone, driven by the power of the Gauss curvature, then the evolution exists for all the time and the evolving hypersurfaces converge smoothly to a piece…

Proposes $φ$-balancing for more balanced expert utilization in MoE models.