A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
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.
This note generalizes the visual angle to convex sets in 3D space.
problem Analyzing geometric properties of convex sets in 3D space.
method Generalizing the visual angle to convex sets in Euclidean space and expressing geometric quantities in terms of integrals of functions related to the solid angle.
result Invariant quantities of the original convex set can be expressed by integrals of functions related to the solid angle.
Optimal hidden-target learning for online inventory optimization on general convex sets.
problem Online inventory optimization (OIO) on arbitrary bounded convex capacity sets.
method Maintaining a hidden target and projecting it onto the feasible order-up-to set.
result The method improves the best known regret guarantee for OIO on general convex sets from inverse to inverse-square-root dependence on the common-demand probability.
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…
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.
Paper extends capillary convex body results to anisotropic setting with Alexandrov-Fenchel inequalities.
problem Extending capillary convex body results to anisotropic setting.
method Developed theory for anisotropic capillary convex bodies in half-space and established Alexandrov-Fenchel inequality for mixed volumes.
result Established a general Alexandrov-Fenchel inequality for mixed volumes of anisotropic capillary convex bodies, weakening and extending previous results.
In this paper we have generalized the notion of λ-radial contraction in complete Riemannian manifold and developed the concept of pλ-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…
Deep learning models generalize by extending decision boundaries outside the convex hull of training data.
problem Understanding how deep learning models generalize beyond their training data.
method Investigation of decision boundaries inside and outside the convex hull of training sets, using various neural network architectures and training regimes.
result Over-parameterization is necessary for deep learning models to extend decision boundaries outside the convex hull of their training data.
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 …
Paper revisits DP-SCO in Euclidean and ℓpd spaces, focusing on constrained and bounded sets.
problem Differentially private stochastic convex optimization in constrained and bounded sets in Euclidean and ℓpd spaces.
method Proposes methods achieving excess population risks dependent on Gaussian width of the constraint set, and novel algorithms for unconstrained and heavy-tailed data.
result Theoretical results for DP-SCO in ℓpd spaces, including optimal bounds for strongly convex functions.
Stochastic gradient descent is the method of choice for large scale optimization of machine learning objective functions. Yet, its performance is greatly variable and heavily depends on the choice of the stepsizes. This has motivated a large body of research on adaptive stepsizes. However, there is currently a gap in o…
The paper studies invariant convex sets in representations with nontrivial copolarity.
problem Understanding the face structure of invariant convex sets in representations with nontrivial copolarity.
method Proves that the face structure of an invariant convex set is determined by its intersection with a fat section, and that a face is exposed if and only if the corresponding face of the intersection is exposed.
result The face structure of invariant convex sets is completely determined by their intersections with fat sections, and exposed faces are preserved.
Convex geometry has recently attracted great attention as a framework to formulate general probabilistic theories. In this framework, convex sets and affine maps represent the state spaces of physical systems and the possible dynamics, respectively. In the first part of this paper, we present a result on separation of …
As a generalization of geodesic function, in the present paper, we introduce the notion of geodesic φ-convex function and deduce some basic properties of φ-convex function and geodesic φ-convex function. We also introduce the concept of geodesic φ-convex set and φ-epigraph and in…
Greedy optimization methods such as Matching Pursuit (MP) and Frank-Wolfe (FW) algorithms regained popularity in recent years due to their simplicity, effectiveness and theoretical guarantees. MP and FW address optimization over the linear span and the convex hull of a set of atoms, respectively. In this paper, we cons…
We prove two new estimates for the level set flow of mean convex domains in Riemannian manifolds. Our estimates give control - exponential in time - for the infimum of the mean curvature, and the ratio between the norm of the second fundamental form and the mean curvature. In particular, the estimates remove a stumblin…
We consider the homogeneous and the non-homogeneous convex relaxations for combinatorial penalty functions defined on support sets. Our study identifies key differences in the tightness of the resulting relaxations through the notion of the lower combinatorial envelope of a set-function along with new necessary conditi…
Convexity and convex functions play an important role in theoretical physics. To initiate a study of the possible uses of convex functions in General Relativity, we discuss the consequences of a spacetime (M,gμν) or an initial data set (Σ,hij,Kij) admitting a suitably defined convex function. We show how…
Convexity and convex functions play an important role in theoretical physics. To initiate a study of the possible uses of convex functions in General Relativity, we discuss the consequences of a spacetime (M,gμν) or an initial data set (Σ,hij,Kij) admitting a suitably defined convex function. We show how…