Study exact minimax rates for density estimation over convex classes, extending previous work.
problem Deriving minimax rates for density estimation over convex density classes.
method Building on Le Cam's work, determine exact minimax rates using local metric entropy.
result Exact minimax rates derived for any convex density class, including nonparametric and parametric cases.
We introduce and study a new class of $\eps$-convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how $\eps$-convex bodies connect with some classical results of Convex Geometry, as Helly theorem, an…
New subgroup behavior in genus-2 mapping class group identified.
problem Understanding subgroups in genus-2 mapping class group.
method Analyzing purely pseudo-Anosov subgroups as convex cocompact.
result Finitely-generated, purely pseudo-Anosov subgroups are convex cocompact.
We strengthen the analogy between convex co-compact Kleinian groups and convex co-compact subgroups of the mapping class group of a surface (in the sense of B. Farb and L. Mosher).
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 examples show some convex-cocompact subgroups are separable.
problem Whether all convex-cocompact subgroups are separable.
method Using Manning-Mj-Sageev construction, examples of separable subgroups of arbitrary finite rank are given.
result Examples of separable convex-cocompact subgroups of arbitrary finite rank exist.
Characterizes a specific type of convex curves on a 3-sphere.
problem Understanding convex curves on a 3-sphere.
method Decomposes curves on 3-sphere into 2-sphere curves, characterizes locally convex ones.
result Completely characterized a class of convex curves on the 3-sphere.
A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, t…
We study relations of some classes of k-convex, k-visible bodies in Euclidean spaces. We introduce and study \textrm{circular projections} in normed linear spaces and classes of bodies related with families of such maps, in particular, \textrm{k-circular convex} and \textrm{k-circular visible} ones. Investigati…
Convex functions and bodies can be approximated by smoother convex functions.
problem Approximating convex functions and bodies with smoother ones.
method Using properties of convex functions and bodies, constructing smoother approximations.
result Smooth approximations of convex functions and bodies exist for any given tolerance.
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…
Convex optimization models predict outputs from inputs via optimization problems.
problem Predicting outputs from inputs using convex optimization models.
method Proposed a heuristic for learning parameters of convex optimization models from datasets.
result Demonstrated the effectiveness of the proposed method on three model classes.
The study proves the existence of k-convex hypersurfaces for specific curvature equations.
problem Proving the existence of k-convex hypersurfaces for Hessian curvature equations. method Combining a priori estimates with the continuity method, and establishing a constant rank theorem.
result Existence and uniqueness of k-convex hypersurfaces for both nonhomogeneous and homogeneous Hessian curvature equations. We prove a complete family of `cylindrical estimates' for solutions of a class of fully non-linear curvature flows, generalising the cylindrical estimate of Huisken-Sinestrari for the mean curvature flow. More precisely, we show that, for the class of flows considered, an (m+1)-convex (0≤m≤n−2) solution bec…
In this paper, it is shown that a Wulff shape is strictly convex if and only if its convex integrand is of class C1. Moreover, applications of this result are given.
New statistical convex-cocompactness found for non-orientable surfaces.
problem Understanding the dynamics of mapping class groups on non-orientable surfaces.
method Using Teichmüller space and complexity length, showing geodesics leave compact regions with exponentially low probabilities.
result Statistical convex-cocompactness of mapping class groups on non-orientable surfaces.
Investigates neural codes and their embeddings, proving conjectures and introducing new code types.
problem Analyzing neural codes and their embedding dimensions.
method Combinatorial, topological, and algebraic analysis; proving conjectures; introducing new neural code types.
result Proves conjectures about neural codes and their embeddings, introduces new code types.
This paper extends boundary embedding results to coarsely convex spaces.
problem Generalizing boundary embedding results to coarsely convex spaces.
method Generalizing Dydak and Virk's work on Gromov hyperbolic spaces to coarsely convex spaces.
result Maps between coarsely convex spaces induce continuous maps between their boundaries.
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.
In this paper, we provide near-optimal accelerated first-order methods for minimizing a broad class of smooth nonconvex functions that are strictly unimodal on all lines through a minimizer. This function class, which we call the class of smooth quasar-convex functions, is parameterized by a constant γ∈(0,1], wher…
Study shows certain subgroups of fibered 3-manifolds are convex cocompact.
problem Understanding subgroups of fibered 3-manifolds in mapping class groups.
method Used the Birman exact sequence to show convex cocompactness.
result Finitely generated pseudo-Anosov subgroups are convex cocompact.
Finding efficient and provable methods to solve non-convex optimization problems is an outstanding challenge in machine learning and optimization theory. A popular approach used to tackle non-convex problems is to use convex relaxation techniques to find a convex surrogate for the problem. Unfortunately, convex relaxat…
Pseudo-Anosov subgroups in surface bundles over tori are convex cocompact.
problem Understanding the structure of pseudo-Anosov subgroups in surface bundles over tori.
method Using the Birman exact sequence to show convex cocompactness.
result Finitely generated, purely pseudo-Anosov subgroups are convex cocompact in surface bundles over tori.
The paper studies quasi-X-convex functions and their applications in optimization.
problem Optimization problems with quasi-X-convex functions. method Definition and study of X-convex, quasi-X-convex, and related functions. result Applications of quasi-X-convex functions in optimization problems. Sharp bounds found for various risk measures using generalized FGM copulas.
problem Finding sharp bounds for risk measures in high dimensions.
method Proved that generalized FGM copulas form a convex polytope, used this structure to find bounds for risk measures.
result Sharp analytical bounds for convex risk measures in the class of generalized FGM copulas.
Least Squares Estimators are suboptimal for 5D convex functions.
problem Suboptimality of Least Squares Estimators in estimating multidimensional convex functions.
method Analysis of natural subclasses of convex functions in random and fixed design settings.
result Risk of LSE is n−2/d while minimax risk is n−4/(d+4) for d≥5. Study shows volumes of complex classes can be represented by convex bodies.
problem Understanding volumes of complex classes on Kähler manifolds.
method Approximation by partial Okounkov bodies, restricted volume properties, and bimeromorphic behavior of currents.
result Volume of transcendental big (1,1)-classes can be realized by convex bodies. It is known that every infinite index quasi-convex subgroup H of a non-elementary hyperbolic group G is a free factor in a larger quasi-convex subgroup of G. We give a probabilistic generalization of this result. That is, we show that when R is a subgroup generated by independent random walks in G, then $\lan…
Characterizes symmetric Bernoulli distributions with minimal convex sums.
problem Understanding minimal dependence among Bernoulli random vectors.
method Geometric and algebraic representations of multivariate symmetric Bernoulli distributions.
result Characterizes extremal negative dependence and builds minimal dependence copulas.
We characterize convex cocompact subgroups of the mapping class group of a surface in terms of uniform convergence actions on the zero locus of the limit set. We also construct subgroups that act as uniform convergence groups on their limit sets, but are not convex cocompact.
We propose a family of optimization methods that achieve linear convergence using first-order gradient information and constant step sizes on a class of convex functions much larger than the smooth and strongly convex ones. This larger class includes functions whose second derivatives may be singular or unbounded at th…
This paper studies quasar-convex functions to improve optimization methods.
problem Improving optimization methods for non-convex functions.
method Study of first order methods for quasar-convex functions.
result Proves complexity upper bounds similar to convex functions.
We characterize convex cocompact subgroups of mapping class groups that arise as subgroups of specially embedded right-angled Artin groups. That is, if the right-angled Artin group G in Mod(S) satisfies certain conditions that imply G is quasi-isometrically embedded in Mod(S), then a purely pseudo-Anosov subgroup H of …
New methods accelerate gradient descent for convex and strongly convex functions.
problem Improving convergence rates of gradient-based optimization methods.
method Formulated two classes of first-order algorithms with Lyapunov analyses and Hamiltonian assisted gradient method.
result Achieved accelerated convergence rates matching Nesterov's methods in strongly and general convex settings.
We study learning problems involving arbitrary classes of functions F, distributions X and targets Y. Because proper learning procedures, i.e., procedures that are only allowed to select functions in F, tend to perform poorly unless the problem satisfies some additional structural property (e.g., that F is co…
Many high dimensional sparse learning problems are formulated as nonconvex optimization. A popular approach to solve these nonconvex optimization problems is through convex relaxations such as linear and semidefinite programming. In this paper, we study the statistical limits of convex relaxations. Particularly, we con…
Study on mapping class groups of non-orientable surfaces, proving some conjectures and refuting others.
problem Analogies between Fuchsian groups and mapping class groups of non-orientable surfaces.
method Analyzing limit sets, foliations, and geometric properties.
result Established parts of a conjecture about the limit set and provided evidence for and against the analogy.
Optimal inequalities found between Riemannian and Hilbert metrics in convex projective domains.
problem Finding optimal bounds between Riemannian and Hilbert metrics in convex projective domains.
method Optimal control techniques applied to Riemannian metrics induced by centro-affine hypersurface immersions.
result Optimal inequalities between Riemannian and Hilbert metrics for a class of convex projective domains.
New study shows acceleration in hyperbolic spaces is impossible for strongly geodesically convex functions.
problem Acceleration in hyperbolic spaces for strongly geodesically convex functions is impossible.
method Perturbing hard functions with sums of bump functions chosen by a resisting oracle.
result Acceleration is unachievable for any deterministic algorithm in hyperbolic spaces for strongly geodesically convex functions.
Muon fails to converge on convex Lipschitz functions.
problem Understanding the convergence of Muon on convex and Lipschitz functions.
method Analyzing Muon's performance on convex and Lipschitz functions without error feedback.
result Muon does not converge on convex and Lipschitz functions, regardless of learning rate schedule.
The paper generalizes offset Rademacher complexities to convex and non-convex problems.
problem Improper learning and convexity in statistical learning.
method Generalization of offset Rademacher complexities to convex and non-convex problems.
result The offset complexity provides versatile analytic tools for both convex and non-convex learning.
Study geodesic distances and convexity in contact sets.
problem Understanding geodesic distances and convexity in contact sets.
method Extending results on quasi-psh functions and big cohomology classes, studying Monge-Ampère measures on contact sets.
result Convexity of the K-energy in big and nef cohomology classes.
Study finds a non-locally contractible r-convex set.
problem Find an r-convex set which is not locally contractible. method Constructs a counterexample of a non-locally contractible r-convex set. result Proves that the class of supports with positive reach of absolutely continuous distributions includes strictly the class of r-convex supports. Counting subgroups of a surface using convex core lengths.
problem Counting conjugacy classes of subgroups of fundamental groups of surfaces.
method Using half the sum of the lengths of the boundaries of the convex core of a subgroup.
result The number of conjugacy classes of subgroups is asymptotic to cL6g−6+2r. Let Wn be the class of C∞ complete simply connected n−dimensional manifolds without conjugate points. The hyperbolic space as well as Euclidean space are good examples of such manifolds. Let and let A be a subset of W. This article aims at characterization and bu…
We introduce a class of generalized relative entropies (inspired by the Bregman divergence in information theory) on the Wasserstein space over a weighted Riemannian or Finsler manifold. We prove that the convexity of all the entropies in this class is equivalent to the combination of the nonnegative weighted Ricci cur…
Paper introduces quasi-logconvex risk measures and their properties.
problem Characterizing and understanding new risk measures.
method Characterization through dual representation and properties of acceptance sets.
result Established dual representation and taxonomy of quasi-logconvex risk measures.
We investigate which jump-diffusion models are convexity preserving. The study of convexity preserving models is motivated by monotonicity results for such models in the volatility and in the jump parameters. We give a necessary condition for convexity to be preserved in several-dimensional jump-diffusion models. This …