Paper calculates ball number of links using Lorentz geometry and circle packing.
problem Calculating the minimum number of balls needed to represent a link.
method Lorentz geometry and circle packing theorem applied to ball packings.
result Shows ball(L)≤5cr(L) for any link L. In this paper, we revisit the convergence of the Heavy-ball method, and present improved convergence complexity results in the convex setting. We provide the first non-ergodic O(1/k) rate result of the Heavy-ball algorithm with constant step size for coercive objective functions. For objective functions satisfying a re…
In this paper we study the projective automorphism group of domains in real, complex, and quaternionic projective space and present two new characterizations of the unit ball in terms of the size of the automorphism group and the regularity of the boundary.
Isoperimetric regions minimize the size of their boundaries among all regions with the same volume. In Euclidean and Hyperbolic space, isoperimetric regions are round balls. We show that isoperimetric regions in two and three-dimensional nonpositively curved manifolds are not necessarily balls, and need not even be con…
This paper begins the study of relations between Riemannian geometry and contact topology in any dimension and continues this study in dimension 3. Specifically we provide a lower bound for the radius of a geodesic ball in a contact manifold that can be embedded in the standard contact structure on Euclidean space, tha…
We study compressing empirical measures in finite RKHSs using convex optimization.
problem Efficiently approximating empirical measures in high-dimensional spaces.
method Convex optimization and lower bounds on ball size.
result High probability lower bounds on ball size under various conditions.
Mogami introduced in 1995 a large class of triangulated 3-dimensional pseudomanifolds, henceforth called "Mogami pseudomanifolds". He proved an exponential bound for the size of this class in terms of the number of tetrahedra. The question of whether all 3-balls are Mogami has remained open since, a positive answer wou…
New method shows stochastic momentum can converge quickly on optimization problems.
problem Improving convergence of stochastic optimization methods.
method Stochastic heavy ball momentum with minibatching.
result Stochastic heavy ball momentum retains fast linear rate on quadratic problems.
Three new efficient algorithms project vectors onto weighted l1 ball.
problem Sparse system identification and feature selection.
method Projected gradient descent algorithms with linear or highly competitive quadratic worst case complexities.
result Efficient tools for machine learning methods like compress sensing and feature selection.
We solve robust optimization problems using Wasserstein balls and apply it to mean-CVaR optimization.
problem Distributionally robust optimization with Wasserstein ambiguity sets.
method Transformed robust optimization into non-robust with penalty term, selecting ambiguity set size.
result Impressive results in robust mean-CVaR optimization compared to other strategies.
In this paper, we prove that there exists a universal constant C, depending only on positive integers n≥3 and p≤n−1, such that if Mn is a compact free boundary submanifold of dimension n immersed in the Euclidean unit ball Bn+k whose size of the traceless second fundamental form is less…
Improved SHB method for faster convergence on strongly-convex quadratics.
problem Understanding and improving the theoretical and practical advantages of SHB.
method Noise-adaptive multi-stage algorithm for SHB with accelerated convergence.
result SHB can achieve accelerated convergence with larger mini-batch sizes.
Modeling default contagion and systemic risk using a balls-and-bins approach.
problem Understanding and quantifying systemic risk in financial networks.
method Tractable model, balls-and-bins representation, type space classification, limit theorems.
result Asymptotic Gaussian fluctuations in the final size of default cascades.
This is the second paper of two in a series under the same title ([CRX]); both study the quantitative volume space form rigidity conjecture: a closed n-manifold of Ricci curvature at least (n−1)H, H=±1 or 0 is diffeomorphic to a H-space form if for every ball of definite size on M, the lifting ball on th…
A new optimization method, BPM, converges linearly in non-convex, non-smooth problems.
problem Non-smooth and non-convex optimization challenges.
method Ball-Proximal Point Method (BPM), inspired by Proximal Point Method (PPM).
result BPM converges linearly and in a finite number of steps in non-convex, non-smooth problems.
A central result in statistical theory is Pinsker's theorem, which characterizes the minimax rate in the normal means model of nonparametric estimation. In this paper, we present an extension to Pinsker's theorem where estimation is carried out under storage or communication constraints. In particular, we place limits …
We define a capacity which measures the size of Weinstein tubular neighbourhoods of Lagrangian submanifolds. In symplectic vector spaces this leads to bounds on the codisc radius for any closed Lagrangian submanifold in terms of Viterbo's isoperimetric inequality. Moreover, we prove a generalization of Gromov's packing…
We provide upper bounds on the size of the homology of a closed aspherical Riemannian manifold that only depend on the systole and the volume of balls. Further, we show that linear growth of mod p Betti numbers or exponential growth of torsion homology imply that a closed aspherical manifold is "large".
We consider the problem of online linear regression on individual sequences. The goal in this paper is for the forecaster to output sequential predictions which are, after T time rounds, almost as good as the ones output by the best linear predictor in a given ℓ1-ball in Rd. We consider both the cases wher…
Ball trajectory data are one of the most fundamental and useful information in the evaluation of players' performance and analysis of game strategies. Although vision-based object tracking techniques have been developed to analyze sport competition videos, it is still challenging to recognize and position a high-speed …
Standard optimizers perform as well as LARS and LAMB at large batch sizes.
problem Comparing optimizers for neural network training at large batch sizes.
method Used standard optimizers like Nesterov momentum and Adam to match or exceed LARS and LAMB results.
result Standard optimizers can match or exceed LARS and LAMB at large batch sizes.
Paper proves SHB convergence with biased gradients and approximate step sizes.
problem Establishing convergence of SHB with biased gradients and approximate step sizes.
method Generalizes SHB convergence conditions for biased gradients, approximate step sizes, and block updating.
result Proves convergence of SHB with new conditions for biased gradients and approximate step sizes.
The study examines how average scalar curvature influences geometric properties of Riemannian manifolds.
problem Investigating the geometric properties of Riemannian manifolds influenced by average scalar curvature.
method Analyzing the conjugate radius, average area of geodesic spheres, average volume of metric balls, and total volume of closed manifolds.
result Improves the Bishop-Gromov estimate on the average volume of metric balls and proves monotone decreasing properties of certain geometric integrals.
We study the steady state solutions of a generalized logistic type equation on a complete Riemannian manifold. We provide sufficient conditions for existence, respectively non-existence of positive solutions, which depend on the relative size of the coefficients and their mutual interaction with the geometry of the man…
The integer hull of a polyhedron is the convex hull of the integer points contained in it. We show that the vertices of the integer hulls of a rational family of polyhedra of size O(n) have quasipolynomial coordinates. As a corollary, we show that the stable commutator length of elements in a surgery family is a ratio …
We show that accelerated gradient descent, averaged gradient descent and the heavy-ball method for non-strongly-convex problems may be reformulated as constant parameter second-order difference equation algorithms, where stability of the system is equivalent to convergence at rate O(1/n 2), where n is the number of ite…
3-balls in 4-sphere become isotopic in 5-ball.
problem Whether 3-balls in 4-sphere become isotopic in 5-ball.
method Analyzing the embedding of 3-balls in 4-sphere and 5-ball.
result Affirmative answer to Gay, Hughes, Kim, and Miller's question.
K-means clustering improved for robustness to outliers and distribution shifts.
problem K-means is brittle to outliers, distribution shifts, and limited samples.
method Developed a distributionally robust variant using Wasserstein-2 ball around the empirical distribution.
result Substantial gains in outlier detection and robustness to noise demonstrated.
Study tests uniformity of categorical data against missing-ball alternatives, finding chi-squared test outperforms.
problem Testing uniformity of categorical data against missing-ball alternatives.
method Characterizes minimax risk, uses collisions and chi-squared test, reduces to structured subset of alternatives.
result Minimax test outperforms chi-squared test under least favorable alternative.
Study on ball widths and minimal submanifolds in space forms.
problem Understanding widths of balls and minimal submanifolds.
method Analyzing the area of equatorial balls and related bounds for minimal submanifolds.
result Lower bounds for the area of free boundary minimal submanifolds.
Identifies bilinear systems from a single trajectory with optimal sample complexity.
problem Learning bilinear systems from a single trajectory of states and inputs.
method Uses a mild marginal mean-square stability assumption and martingale small-ball condition.
result Sample complexity and statistical error rates are optimal.
Study on covering probability of random balls in bounded open sets.
problem Probability of covering a bounded open set E with random balls of radius δ. method Geometric conditions and partition of E; lower bounds using good partitions. result Lower bounds to the probability of covering E with balls tend to 1 as Nexp(−δn). This paper describes a method to construct standard 4-balls from homotopy 4-balls in C2.
problem The problem is whether every homotopy 4-ball in S4 is standard. method The approach is to use Stein surfaces and pseudoconvex domains to construct a diffeomorphic domain that is the union of three pseudoconvex domains, ensuring it is a standard 4-ball.
result The construction method ensures that the domain is a standard 4-ball, providing a compelling reimbedding construction for homotopy 4-balls in C2. New step-size methods improve SHB convergence for stochastic optimization.
problem Tuning step-size and momentum parameters in SHB is challenging.
method Proposed MomSPSmax, MomDecSPS, and MomAdaSPS for SHB. result Convergence guarantees for SHB to solution neighborhoods and exact minimizers.
Two minimal hypersurfaces in a ball intersect in any half-ball.
problem Intersection properties of minimal hypersurfaces in a ball.
method Analyzing the intersection of two minimal hypersurfaces in a unit Euclidean ball.
result Intersection point in any half-ball, strong Frankel property.
New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.
problem Defining and analyzing surface area measures for ball-convex bodies.
method Introducing Lp relative surface areas, proving invariance and inequalities, and using geometric interpretations. result Established inequalities and a new notion of entropy for ball-convex bodies.
Sharp lower bound found for geodesic ball eigenvalues.
problem Finding the minimum eigenvalue for geodesic balls.
method Applied Li-Schoen's uniform Poincare inequality for non-negative Ricci curvature manifolds.
result Sharp lower bound of the first Dirichlet eigenvalue for geodesic balls.
Quantifies nearly spherical subsets in complex ball geometry.
problem Isoperimetric inequality for nearly spherical domains in Bergman ball.
method Proves a quantitative isoperimetric inequality for nearly spherical subsets of Bergman ball.
result First result on isoperimetric phenomenon in Bergman ball.
New model learns from random graph samples to estimate graph parameters.
problem Scalability issues in graph learning methods for large graphs.
method Develops a graph classification model working on randomly sampled subgraphs.
result Validates mini-batch learning on graphs and provides generalization bounds.
We completely solve the symplectic packing problem with equally sized balls for any rational, ruled, symplectic 4-manifolds. We give explicit formulae for the packing numbers, the generalized Gromov widths, the stability numbers, and the corresponding obstructing exceptional classes. As a corollary, we give explicit va…
Stochastic momentum methods trade compute efficiency for serial runtime.
problem Stochastic momentum methods trade compute efficiency for serial runtime.
method Stochastic HB and ASGD for consistent linear regression with Gaussian covariates.
result HB preserves SGD-level CE over a larger batch-size window, allowing larger batches to reduce serial runtime until HB reaches its deterministic accelerated scale.
Sharp geometric inequalities for free boundary hypersurfaces in balls.
problem Understanding geometric properties of free boundary hypersurfaces in balls.
method Proving a family of sharp geometric inequalities.
result Family of sharp geometric inequalities for free boundary hypersurfaces in balls.
New minimal surfaces found in ball with boundary constraints.
problem Finding minimal surfaces with boundary conditions.
method Equivariant differential geometry approach.
result A family of free boundary minimal surfaces in the unit ball.
New examples show non-locally-flat PL-disk bounds in rational homology balls but not in integer homology balls.
problem Characterizing knots that bound PL-disks in integer homology balls.
method Involutive Heegaard Floer homology formal properties.
result Found infinitely many manifold-knot pairs (Y, J) where J does not bound a PL-disk in an integer homology ball but does in a rational homology ball.
Study constructs disks with curved boundaries in a 3D ball.
problem Constructing non-planar free boundary disks in a unit ball.
method Infinite family of non-planar disks with non-positive Gaussian curvature.
result Constructs disks with curved boundaries in a unit ball.
Study shows no smooth embeddings of rational homology balls into complex projective plane.
problem Embedding rational homology balls into complex projective plane.
method Elementary arguments to prove non-existence of almost complex embeddings.
result No smooth embeddings of rational homology balls into complex projective plane.
Sharp inequality outside ball proved using Neumann method.
problem Anisotropic isoperimetric inequality for domains outside an Euclidean ball.
method Applied ABP method to Neumann boundary value problem.
result Proved sharp anisotropic isoperimetric inequality.
Fourth-order problem on half-ball with corner behavior.
problem Fourth-order problem with corner behavior on half-ball.
method Conformal mapping to isolate corner effect.
result Gauss-Bonnet formula simplifies to constant term at corner.