We discuss general notions of metrics and of Finsler structures which we call weak metrics and weak Finsler structures. Any convex domain carries a canonical weak Finsler structure, which we call its tautological weak Finsler structure. We compute distances in the tautological weak Finsler structure of a domain and we …
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48 results for “weak convexity”
Study uses weak transport for non-convex costs in fixed-income markets.
problem Characterizing optimal caplet pricing in fixed-income markets.
method Introduced weak optimal transport for non-convex costs, reduced general costs to convex problems.
result Established robust super-replication results for fixed-income markets.
Study weak super Ricci flow through neckpinch in metric measure spaces.
problem Understanding Ricci flow behavior at singularities.
method Introduce weak super Ricci flow and show conditions for continuation.
result Weak super Ricci flow properties at singularities.
We are analysing the convexity and continuity properties of the Mabuchi functional along weak geodesics. The key technical point in our paper is the global approximation of weak geodesics obtained via a well-chosen family of Monge-Ampère equations.
Boosting is a popular way to derive powerful learners from simpler hypothesis classes. Following previous work (Mason et al., 1999; Friedman, 2000) on general boosting frameworks, we analyze gradient-based descent algorithms for boosting with respect to any convex objective and introduce a new measure of weak learner p…
Boosts weak online learners to strong ones with sublinear regret.
problem Online learning agnostic setting without strong guarantees.
method Reduction to online convex optimization, boosting via marginally-better-than-trivial regret guarantees.
result First agnostic online boosting algorithm with sublinear regret.
The paper tackles performative risk optimization under weak convexity assumptions.
problem Optimizing performative risk in a closed-loop prediction system with weak convexity.
method Relaxing convexity assumptions to maintain optimization feasibility.
result Iterative optimization methods remain applicable even with weakened convexity conditions.
Weak Convergence of Approximate reflection coupling and its Application to Non-convex Optimizationmath.PR
Paper proposes a weak approximation of reflection coupling for non-convex optimization.
problem Non-convex optimization problems with different drift terms.
method Proposes an approximate reflection coupling (ARC) for stochastic differential equations (SDEs).
result ARC converges weakly to the reflection coupling and can be applied to non-convex optimization.
Characterizes Anosov representations and strongly convex cocompact groups with eigenvalue gaps.
problem Understanding Anosov representations and their properties.
method Characterizations via equivariant limit maps, Cartan property, and uniform gap summation.
result Characterizations of Anosov representations and strongly convex cocompact subgroups.
We propose a projected semi-stochastic gradient descent method with mini-batch for improving both the theoretical complexity and practical performance of the general stochastic gradient descent method (SGD). We are able to prove linear convergence under weak strong convexity assumption. This requires no strong convexit…
Unified proofs of weak holomorphic Morse inequalities using Bergman kernel functions.
problem Weak holomorphic Morse inequalities on various types of manifolds.
method Asymptotic Bergman kernel functions and Bochner-Kodaira-Nakano formulas.
result Unified proofs of weak holomorphic Morse inequalities.
A novel notion of barycenter for probability distributions based on optimal weak mass transportstat.ML
A new method for averaging probability distributions based on optimal weak mass transport.
problem Averaging probability distributions in a geometric way.
method Weak barycenters based on optimal weak mass transport.
result Extracts common geometric information shared by all input distributions.
The paper introduces a new method for risk measurement using weak optimal transport.
problem Risk measurement in insurance and financial contexts.
method Convex risk measures with weak optimal transport penalties, explicit representation via nonlinear transform, computational aspects, and approximations using neural networks.
result Explicit representation and computational methods for risk measures.
In this paper, we discuss a Donaldson's version of the modified -energy associated to the Calabi's extremal metrics on toric manifolds and prove the existence of the weak solution for extremal metrics in the sense of convex functions which minimizes the modified -energy.
We introduce a notion of a weak Poisson structure on a manifold modeled on a locally convex space. This is done by specifying a Poisson bracket on a subalgebra $\cA \subeq C^\infty(M)$ which has to satisfy a non-degeneracy condition (the differentials of elements of $\cA$ separate tangent vectors) and we postulate …
We establish the convexity of Mabuchi's K-energy functional along weak geodesics in the space of Kahler potentials on a compact Kahler manifold thus confirming a conjecture of Chen and give some applications in Kahler geometry, including a proof of the uniqueness of constant scalar curvature metrics (or more generally …
Boosting improves online decision-making for large expert sets.
problem Online convex optimization with many experts is infeasible.
method Generalizes online boosting to online convex optimization and bandit linear optimization settings.
result Near-optimal regret guarantees for various feedback models.
The paper is motivated by the study of graded representations of Takiff algebras, cominuscule parabolics, and their generalizations. We study certain special subsets of the set of weights (and of their convex hull) of the generalized Verma modules (or GVM's) of a semisimple Lie algebra $\lie g$. In particular, we exten…
We consider Blackwell approachability, a very powerful and geometric tool in game theory, used for example to design strategies of the uninformed player in repeated games with incomplete information. We extend this theory to "generalized quitting games" , a class of repeated stochastic games in which each player may ha…
In this paper we study integer multiplicity rectifiable currents carried by the subgradient (subdifferential) graphs of semi-convex functions on a -dimensional convex domain, and show a weak continuity theorem with respect to pointwise convergence for such currents. As an application, the -Hessian measures are ca…
Extends boosting to multiclass online agnostic classification.
problem Online multiclass classification with weak learners.
method Reduces multiclass online agnostic boosting to online convex optimization.
result First boosting algorithm for online agnostic multiclass classification.
We study the evolution of hypersurfaces in spacetime initial data sets by their null mean curvature. A theory of weak solutions is developed using the level-set approach. Starting from an arbitrary mean convex, outer untapped hypersurface , we show that there exists a weak solution to the null mean curvatu…
The Lebesgue property (order-continuity) of a monotone convex function on a solid vector space of measurable functions is characterized in terms of (1) the weak inf-compactness of the conjugate function on the order-continuous dual space, (2) the attainment of the supremum in the dual representation by order-continuous…
In this paper we generalize the framework of the feasible descent method (FDM) to a randomized (R-FDM) and a coordinate-wise random feasible descent method (RC-FDM) framework. We show that the famous SDCA algorithm for optimizing the SVM dual problem, or the stochastic coordinate descent method for the LASSO problem, f…
Study compares synthetic and distributional Ricci curvature bounds.
problem Comparing synthetic and distributional approaches to lower Ricci curvature bounds.
method Analyzes synthetic via weak displacement convexity and distributional via non-negativity of Ricci-tensor.
result Distributional bounds imply entropy bounds for metrics and vice versa for under convergence condition.
David Hilbert discovered in 1895 an important metric that is canonically associated to any convex domain in the Euclidean (or projective) space. This metric is known to be Finslerian, and the usual proof assumes a certain degree of smoothness of the boundary of and refers to a theorem by Busemann and Mayer that…
We study weak geodesics in the space of potentials for the deformed Hermitian-Yang-Mills equation. The geodesic equation can be formulated as a degenerate elliptic equation, allowing us to employ nonlinear Dirichlet duality theory, as developed by Harvey-Lawson. By exploiting the convexity of the level sets of the Lagr…
Explain convexity of K-energy leading to unique metrics.
problem Uniqueness of constant scalar curvature Kahler metrics and extremal metrics.
method Convexity of K-energy along weak geodesics in Kahler potentials.
result Uniqueness of extremal metrics up to automorphisms.
Paper generalizes Andreev's theorem with obtuse angles.
problem Characterizing hyperbolic polyhedra with obtuse angles.
method Established discrete analog of weak solution/regularity theory.
result Generalized Andreev's Theorem to include obtuse angles.
Study convexity of Mabuchi functional in big cohomology classes.
problem Convexity of Mabuchi functional in big cohomology classes.
method Defined an invariant related to transcendental Fujita approximations and established convexity under vanishing of this invariant.
result Established almost convexity along weak geodesics in big cohomology classes.
New algorithm solves saddle point problems in Banach spaces.
problem Solving saddle point problems in real reflexive Banach spaces.
method Stochastic Bregman Primal-Dual Splitting Algorithm with relative smoothness and strong convexity assumptions.
result Almost sure convergence to saddle points under various conditions.
This paper is a follow up to the previous author's paper on convex optimization. In that paper we began the process of adjusting greedy-type algorithms from nonlinear approximation for finding sparse solutions of convex optimization problems. We modified there three the most popular in nonlinear approximation in Banach…
New guarantees for Group LASSO in sparse convex optimization.
problem Sparse convex optimization with vector-valued features.
method Group LASSO regularization and analysis of gradient norms.
result Group LASSO selects the same features as Orthogonal Matching Pursuit.
Study shows non-compact convex hulls in certain metric spaces.
problem Compactness of convex hulls in weakly non-positive curvature spaces.
method Introduced a conical geodesic bicombing and used it to construct a counterexample.
result Existence of a metric space with a finite subset whose convex hull is not compact.
For incomplete preference relations that are represented by multiple priors and/or multiple -- possibly multivariate -- utility functions, we define a certainty equivalent as well as the utility buy and sell prices and indifference price bounds as set-valued functions of the claim. Furthermore, we motivate and introduc…
Large sectors of the recent optimization literature focused in the last decade on the development of optimal stochastic first order schemes for constrained convex models under progressively relaxed assumptions. Stochastic proximal point is an iterative scheme born from the adaptation of proximal point algorithm to nois…
Study on Mabuchi functional's convexity using ε-geodesics.
problem Understanding the convexity of the Mabuchi functional.
method Analysis of ε-geodesics to study the Mabuchi functional's convexity.
result Uniform fiberwise non-degeneracy of geodesics when Mabuchi functional is ε-affine.
Convex surfaces derived from specific Riemannian manifolds with high regularity.
problem Proving convexity of surfaces derived from Riemannian manifolds.
method Analyzing solutions to the very weak Monge-Ampère equation.
result Proved convexity of weakly regular surfaces with nonnegative intrinsic curvature.
In the present contribution we characterize law determined convex risk measures that have convex level sets at the level of distributions. By relaxing the assumptions in Weber (2006), we show that these risk measures can be identified with a class of generalized shortfall risk measures. As a direct consequence, we are …
New method estimates model performance bounds without ground truth labels.
problem Evaluation of weakly supervised models without direct access to ground truth labels.
method Formulates model evaluation as a partial identification problem and uses Fréchet bounds for performance estimation.
result Derives accurate and computationally efficient bounds for key metrics like accuracy, precision, recall, and F1-score.
This paper concerns the questions of flexibility and rigidity of solutions to the Monge-Ampère equation which arises as a natural geometrical constraint in prestrained nonlinear elasticity. In particular, we focus on anomalous i.e. "flexible" weak solutions that can be constructed through methods of convex integration …
Monetary Risk Measuresq-fin.MF
The paper explores non-convex risk measures and their characterizations.
problem Characterizing non-convex risk measures without convexity or weak convexity.
method Characterizes monetary risk measures as lower envelopes of families of convex or coherent risk measures, considering law-invariance and SSD-consistency.
result Unified representation theorems for law-invariant risk measures, including VaR.
Flows with surgery revisitedmath.AP
New method for geometric flows with surgery without smooth estimates.
problem Existence of geometric flows with surgery.
method Hybrid compactness theorem for weak limits.
result Existence of geometric flows with surgery in mean-convex surfaces.
Symmetric graphs flow without singularities on their axis.
problem Preserving symmetry in mean curvature flow.
method Weak solution approach, introducing 'vanity', mean curvature flow approximation.
result Singularities occur only on the axis of symmetry.
We study the task of online boosting--combining online weak learners into an online strong learner. While batch boosting has a sound theoretical foundation, online boosting deserves more study from the theoretical perspective. In this paper, we carefully compare the differences between online and batch boosting, and pr…
New method improves IV estimation with many weak and invalid instruments.
problem Identification in linear IV models with unknown validity.
method Non-convex penalized approaches, surrogate sparsest penalty.
result Advantages over other IV estimators in selection consistency and weak IV strength conditions.
Let be a polyhedron. It was conjectured that if is weakly convex (i. e. its vertices lie on the boundary of a strictly convex domain) and decomposable (i. e. can be triangulated without adding new vertices), then it is infinitesimally rigid. We prove this conjecture under a weak additional assu…
In this paper, we prove that the transverse Mabuchi K-energy functional is convex along the weak geodesic in the space of Sasakian metrics. As an application, we obtain the uniqueness of constant scalar curvature Sasakian metrics modulo automorphisms for the transverse holomorphic structure.