arXiv research

New method tackles online DR-submodular maximization with improved regret guarantees.

In this paper, we study fundamental problems of maximizing DR-submodular continuous functions that have real-world applications in the domain of machine learning, economics, operations research and communication systems. It captures a subclass of non-convex optimization that provides both theoretical and practical guar…

Diminishing-returns (DR) submodular optimization is an important field with many real-world applications in machine learning, economics and communication systems. It captures a subclass of non-convex optimization that provides both practical and theoretical guarantees. In this paper, we study the fundamental problem of…

This paper considers stochastic optimization problems for a large class of objective functions, including convex and continuous submodular. Stochastic proximal gradient methods have been widely used to solve such problems; however, their applicability remains limited when the problem dimension is large and the projecti…

DR-submodular continuous functions are important objectives with wide real-world applications spanning MAP inference in determinantal point processes (DPPs), and mean-field inference for probabilistic submodular models, amongst others. DR-submodularity captures a subclass of non-convex functions that enables both exact…

In 1979, building on S. Lie's theory of symmetries of (partial) differrential equations, P.J. Olver formulated inductive formulas which are appropriate for the computation of the prolongations of an infinitesimal Lie symmetry to jet spaces, for an arbitrary number n\geq 1 of independent variables (x^1, ..., x^n) and fo…

Every convex set in a generic Riemannian manifold has peculiar properties.

Study shows non-symmetric convex sets have full boundary limits.

The paper explores different smooth map notions on convex sets and their relationships.

Classifies geodetically convex sets and functions on Heisenberg group.

Study minimal freezing sets in convex digital disks.

On a flat plane, convexity of a set is preserved by both radial expansion and contraction of the set about any point inside it. Using the Poincaré disk model of hyperbolic geometry, we prove that radial expansion of a hyperbolic convex set about a point inside it always preserves hyperbolic convexity. Using stereograph…

This note generalizes the visual angle to convex sets in 3D space.

In an earlier paper we showed that the radial expansion of a hyperbolic convex set in the Poincaré disk about any point inside it results in a hyperbolic convex set. In this work, we generalize this result by showing that the asymmetric expansion of a hyperbolic convex set about any point inside it also results in a hy…

We introduce the new notion of Bianchi-convex sets, a generalization of convex sets of algebraic curvature tensors inspired by the second Bianchi identity. It turns out that Hamilton's maximum principle for the Ricci flow can be generalized for Bianchi-convex sets.

The paper explores connections between perimeter, area, and visual angle of convex sets.

Unified framework for robust risk measures beyond convexity.

Study finds a non-locally contractible $r$-convex set.

In this article a class of closed convex sets in the Euclidean $n$-space which are the convex hull of their profiles is described. Thus a generalization of Krein-Milman theorem\cite{Lay:1982} to a class of closed non-compact convex sets is obtained. Sufficient and necessary conditions for convexity, affinity and starsh…

We define a class of L-convex-concave subsets of $\Bbb{R}P^n$, where L is a projective subspace of dimension l in $\Bbb{R}P^n$. These are sets whose sections by any (l+1)-dimensional space L' containing L are convex and concavely depend on L'. We introduce an L-duality for these sets, and prove that the L-dual to an L-…

Study contractibility of boundaries in convex sets and limit sets of subgroups.

A mean-convex set can be regarded as a barrier for the construction of minimal surfaces. Namely, if we are given a mean-convex set and a null-homotopic Jordan curve on its boundary, then there exists an embedded minimal disk with boundary the given curve contained in the starting mean-convex set. Does a mean-convex set…

Solves equality case in isoperimetric inequality for non-convex domains.

An open convex set in real projective space is called divisible if there exists a discrete group of projective automorphisms which acts co-compactly. There are many examples of such sets and a theorem of Benoist implies that many of these examples are strictly convex, have $C^1$ boundary, and have word hyperbolic divid…

Study shows singular set of distance functions is delta-convex.

New algorithms for differentially private optimization in convex and non-convex settings with near-optimal rates.

The paper studies stability and singularities of a two-convex level set flow.

Paper solves Minkowski problem for non-compact convex sets with asymptotic boundary conditions.

The paper characterizes sets with infinite hyperbolic convex hull volume.

A spherical set is called convex if for every pair of its points there is at least one minimal geodesic segment that joins these points and lies in the set. We prove that for n >= 3 a complete locally-convex (topological) immersion of a connected (n-1)-manifold into the n-sphere is a surjection onto the boundary of a c…

The usual notion of set-convexity, valid in the classical Euclidean context, metamorphoses into several distinct convexity types in the more general Riemannian setting. By studying this phenomenon in reverse, we characterize complete manifolds for which certain convexity types are assumed a priori to coincide.

Characterizes convex cocompact actions in projective space with dynamical properties.

Optimal hidden-target learning for online inventory optimization on general convex sets.

Generalizes smoothness conditions for optimization methods.

The paper proves a rigidity theorem for non-compact convex sets in hyperbolic 3-space.

Paper extends capillary convex body results to anisotropic setting with Alexandrov-Fenchel inequalities.

The paper studies invariant convex sets in representations with nontrivial copolarity.

Efficient algorithms for online convex optimization with limited switching decisions.

For the minimal graph defined on a convex ring in the space form with nonnegative curvature, we obtain the regularity and the strict convexity about its level sets by the continuity method.

Two groups with specific limit sets in hyperbolic spaces are identified.

In this paper we have generalized the notion of $λ$-radial contraction in complete Riemannian manifold and developed the concept of $p^λ$-convex function. We have also given a counter example proving the fact that in general $λ$-radial contraction of a geodesic is not necessarily a geodesic. We have also deduced some r…

In this paper we consider the problem of minimizing area subject to a volume constraint in a given convex set.

We prove some results concerning the boundary of a convex set in $\H^n$. This includes the convergence of curvature measures under Hausdorff convergence of the sets, the study of normal points, and, for convex surfaces, a generalized Gauss equation and some natural characterizations of the regular part of the Gaussian …

Uniform convexity in divisible domains leads to hyperbolic geometry.

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…

The paper studies convexity of products of squared Euclidean distances.

New algorithm exploits curvature of feasible sets for fast online convex optimization.

Non-compact convex sets in hyperbolic 3-space are rigid under isometries.