arXiv research

Sharp estimates for Finsler metrics in convex domains.

The paper examines hyperbolicity in bounded strongly minimally convex domains in R^d.

Almost all local minima in neural networks are strongly convex.

In this paper, we prove that a strongly convex complex Finsler metric $F$ on a domain $D\subset\mathbb{C}^n$ is projectively flat (resp. dually flat) if and only if $F$ comes from a strongly convex complex Minkowski metric.

We prove that Kobayashi isometries between strongly convex domains are holomorphic or anti-holomorphic. More precisely, let $n_1, n_2$ be positive integers and let $Ω_i \subset \C^{n_i}, \ i=1,2$, be bounded $C^3$ strongly convex domains. If $φ: (Ω_1, d^K_{Ω_1}) \rightarrow (Ω_2, d^K_{Ω_2})$ is an isometry, i.e. $ d^K_…

In this paper we establish a gap theorem for the complex geometry of smoothly bounded convex domains which informally says that if the complex geometry near the boundary is close to the complex geometry of the unit ball, then the domain must be strongly pseudoconvex. One consequence of our general result is the followi…

We give a necessary complex geometric condition for a bounded smooth convex domain in Cn, endowed with the Kobayashi distance, to be Gromov hyperbolic. More precisely, we prove that if a smooth bounded convex domain contains an analytic disk in its boundary, then the domain is not Gromov hyperbolic for the Kobayashi di…

The paper proves Gromov hyperbolicity of certain metrics using isoperimetric inequalities.

The paper proves properties of complex Finsler metrics on specific domains.

New study shows acceleration in hyperbolic spaces is impossible for strongly geodesically convex functions.

In this article, we prove a Lichnerowicz estimate for a compact convex domain of a Kähler manifold whose Ricci curvature satisfies $\Ric \ge k$ for some constant $k>0$. When equality is achieved, the boundary of the domain is totally geodesic and there exists a nontrivial holomorphic vector field. We show that a ball o…

Investigates properties of a pseudometric on domains in Euclidean space, linking it to hyperbolic geometry.

The Bergman kernel's minimal point determines domain properties.

Negative curvature restricts the gap between the first and second eigenvalues of convex domains.

We study how the existence of a negatively pinched Kähler metric on a domain in complex Euclidean space restricts the geometry of its boundary. In particular, we show that if a convex domain admits a complete Kähler metric, with pinched negative holomorphic bisectional curvature outside a compact set, then the boundary…

The paper studies complex Finsler metrics and their equivalence to the Kobayashi metric.

Smoothly bounded domains have special functions that are plurisubharmonic.

The paper characterizes complex Finsler metrics invariant under U(n) and their properties.

New geometric conditions ensure compactness of $\bar{\partial}$-Neumann problem.

We consider Hilbert and Funk geometries on a strongly convex domain in the Euclidean space. We show that, with respect to the Lebesgue measure on the domain, Hilbert (resp. Funk) metric has the bounded (resp. constant negative) weighted Ricci curvature. As one of corollaries, these metric measure spaces satisfy the cur…

Strongly convex bodies can be approximated by smooth ones.

Improved online learning with time-varying constraints for complex domains.

Study of pseudometric properties on domains in Nagano spaces.

New algorithm tackles heterogeneous curvature in online convex optimization.

The paper simplifies strongly convex problems to simplicial structures.

Smooth families of biholomorphisms between strongly pseudoconvex domains are shown to be smooth.

We consider the problem of minimizing the sum of an average function of a large number of smooth convex components and a general, possibly non-differentiable, convex function. Although many methods have been proposed to solve this problem with the assumption that the sum is strongly convex, few methods support the non-…

We propose an optimization method for minimizing the finite sums of smooth convex functions. Our method incorporates an accelerated gradient descent (AGD) and a stochastic variance reduction gradient (SVRG) in a mini-batch setting. Unlike SVRG, our method can be directly applied to non-strongly and strongly convex prob…

Log-concavity of eigenfunctions on curved surfaces is proven, leading to fundamental gap estimates.

Unified approach for first-order methods with Markovian noise in stochastic optimization and variational inequalities.

GenFlow optimizes faster, avoiding saddle points in fixed time.

Many classical algorithms are found until several years later to outlive the confines in which they were conceived, and continue to be relevant in unforeseen settings. In this paper, we show that SVRG is one such method: being originally designed for strongly convex objectives, it is also very robust in non-strongly co…

Fewer data weight updates lead to faster convergence in machine learning models.

This paper is a sequel to \cite{Choi} in Math. Ann. In that paper we studied the subharmonicity of Kähler-Einstein metrics on strongly pseudoconvex domains of dimension greater than or equal to $3$. In this paper, we study the variations Kähler-Einstein metrics on bounded strongly pseudoconvex domains of dimension $2$.…

For $J$-holomorphic mappings for a strongly pseudo-convex manifold, we prove elliptic regularity by the argument of boots-strapping.

Characterizes Anosov representations and strongly convex cocompact groups with eigenvalue gaps.

The Adam algorithm has become extremely popular for large-scale machine learning. Under convexity condition, it has been proved to enjoy a data-dependant $O(\sqrt{T})$ regret bound where $T$ is the time horizon. However, whether strong convexity can be utilized to further improve the performance remains an open problem…

The paper proves a Schwarz lemma for weakly Kähler-Finsler manifolds.

In this paper, we study locally strongly convex centroaffine hypersurfaces with parallel cubic form with respect to the Levi-Civita connection of the centroaffine metric. As the main result, we obtain a complete classification of such centroaffine hypersurfaces. The result of this paper is a centroaffine version of the…

In this paper, we establish a general inequality for locally strongly convex centroaffine hypersurfaces in $\mathbb{R}^{n+1}$ involving the norm of the covariant derivatives of both the difference tensor $K$ and the Tchebychev vector field $T$. Our result is optimal in that, applying our recent classification for local…

Stochastic gradient algorithms estimate the gradient based on only one or a few samples and enjoy low computational cost per iteration. They have been widely used in large-scale optimization problems. However, stochastic gradient algorithms are usually slow to converge and achieve sub-linear convergence rates, due to t…

Establishes a lower bound for Kähler hyperbolicity modulus in hyperconvex domains and bounded strongly pseudoconvex domains.

A multiobjective optimization problem is simplicial if the Pareto set and front are homeomorphic to a simplex and, under the homeomorphisms, each face of the simplex corresponds to the Pareto set and front of a subproblem. In this paper, we show that strongly convex problems are simplicial under a mild assumption on th…

Paper improves privacy and utility of SGD with bounded domain and smooth losses.

Improved SGD for non-strongly-convex regression with faster convergence.

Linear optimization is many times algorithmically simpler than non-linear convex optimization. Linear optimization over matroid polytopes, matching polytopes and path polytopes are example of problems for which we have simple and efficient combinatorial algorithms, but whose non-linear convex counterpart is harder and …

Paper introduces a new Poisson kernel for strongly pseudoconvex domains.

New lower bounds for gradient methods in strongly convex finite-sum optimization.