Study shows uniqueness of solutions on complex manifolds without requiring solution decay.
problem Uniqueness of solutions to Monge-Ampere equation on complex manifolds.
method Caccioppoli inequality techniques applied to Kähler manifolds with sub-quadratic volume growth.
result Uniqueness of bounded C1,1 solutions to Monge-Ampere equation without decay requirement. New findings on gradient expanding Ricci solitons with finite scalar curvature ratio.
problem Understanding the behavior of gradient expanding Ricci solitons with finite scalar curvature ratio.
method Analyzing complete gradient expanding Ricci solitons with nonnegative Ricci curvature.
result Riemann curvature tensor must have at least sub-quadratic decay for finite asymptotic scalar curvature ratio.
New analysis improves SGD for robust and quantile regression with sub-quadratic convergence.
problem Improving SGD for robust and quantile regression with sub-quadratic convergence.
method Piecewise Lyapunov function for first-order differentiable functions.
result First geometrical convergence result for sub-quadratic SGD.
New compression methods handle biased input sequences for more accurate posterior summaries.
problem Handling biased input sequences for accurate posterior summaries.
method Stein kernel thinning, low-rank SKT, Stein recombination, Stein Cholesky.
result Achieves accurate posterior summaries with biased input sequences.
We present a new algorithm, trimed, for obtaining the medoid of a set, that is the element of the set which minimises the mean distance to all other elements. The algorithm is shown to have, under certain assumptions, expected run time O(N^(3/2)) in R^d where N is the set size, making it the first sub-quadratic exact m…
We consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show the existence on such a manifold of a distance-like function with bounded gradi…
The paper tests properties of trees in graphical models using covariance queries.
problem Testing properties of trees in graphical models.
method Covariance queries model, randomized tests for tree properties.
result Efficient testing of global tree properties using sub-quadratic number of queries.
Efficient sparse modern Hopfield models are introduced for memory retrieval and learning tasks.
problem Efficient modern Hopfield models for memory retrieval and learning tasks.
method Nonparametric interpretation of Hopfield models as a regression problem.
result Sparse-structured modern Hopfield models with sub-quadratic complexity.
The covariance matrix of a p-dimensional random variable is a fundamental quantity in data analysis. Given n i.i.d. observations, it is typically estimated by the sample covariance matrix, at a computational cost of O(np2) operations. When n,p are large, this computation may be prohibitively slow. Moreover, …
We study manifolds satisfying a weighed Poincare inequality, which was first introduced by Li-Wang. We generalized one of their results by relaxing the Ricci curvature bound condition only being satisfied outside a compact set and established a finitely many ends result. We proved a vanishing result for L2 harmonic …
Study sharp convergence rates of empirical UOT for spatio-temporal point processes.
problem Statistical analysis of UOT for spatio-temporal point processes.
method Empirical plug-in estimators for Kantorovich-Rubinstein distance between intensity measures.
result Sharp convergence rates of empirical UOT in terms of intrinsic dimensions of measures.
Quadratic regression involves modeling the response as a (generalized) linear function of not only the features xj1 but also of quadratic terms xj1xj2. The inclusion of such higher-order "interaction terms" in regression often provides an easy way to increase accuracy in already-high-dimensional problem…
Improved MMD test for two-sample testing with random Fourier features.
problem Quadratic-time complexity of MMD test for large-scale analysis.
method Approximated MMD test using random Fourier features, investigating time-power trade-off.
result Sub-quadratic time complexity with same minimax separation rates as MMD test.
The so-called {\it kissing number} for hyperbolic surfaces is the maximum number of homotopically distinct systoles a surface of given genus g can have. These numbers, first studied (and named) by Schmutz Schaller by analogy with lattice sphere packings, are known to grow, as a function of genus, at least like $g^{\s…
Study calculates stable norm of slit tori using Farey sequence.
problem Computing the stable norm of slit tori.
method Explicit computations using the Farey sequence and gluing slit tori.
result Estimates the asymptotic counting of simple homology classes.
Semidefinite programs (SDP) are important in learning and combinatorial optimization with numerous applications. In pursuit of low-rank solutions and low complexity algorithms, we consider the Burer--Monteiro factorization approach for solving SDPs. We show that all approximate local optima are global optima for the pe…
We consider the problem of recovering a signal x∗∈Rn, from magnitude-only measurements yi=∣⟨ai,x∗⟩∣ for i=[m]. Also called the phase retrieval, this is a fundamental challenge in bio-,astronomical imaging and speech processing. The problem abov…
Extends Minkowski stability proof to minimal decay assumptions.
problem Global stability of Minkowski spacetime with minimal decay.
method Extends Christodoulou-Klainerman's proof to minimal decay assumptions.
result Exterior stability of Minkowski holds with borderline decay.
New method creates vacuum data at minimal and borderline decay thresholds.
problem Creating vacuum initial data at specific decay thresholds.
method Conical solution-operator method applied to vacuum asymptotically flat initial data.
result Demonstrates global and exterior stability of Minkowski spacetime.
We describe the first sub-quadratic sampling algorithm for the Multiplicative Attribute Graph Model (MAGM) of Kim and Leskovec (2010). We exploit the close connection between MAGM and the Kronecker Product Graph Model (KPGM) of Leskovec et al. (2010), and show that to sample a graph from a MAGM it suffices to sample sm…
Unique solutions found for wave-like decaying null infinity equations.
problem Wave-like decaying null infinity equations with spherically symmetric Einstein-scalar-field.
method Local and global unique solutions for small initial data.
result Sharp decaying condition for unique solutions.
New findings on flatness of certain metrics with fast decay.
problem Rigidity of positive mass theorem under fast metric decay.
method Considered metrics with nonnegative scalar curvature and rapid decay at infinity.
result Any such metric is necessarily flat in dimensions 4 and higher if decay rate exceeds Schwarzschild metric.
Study on curvature decay in steady Ricci solitons, proving dichotomy.
problem Curvature decay in steady Ricci solitons.
method Established a dichotomy for curvature decay in specific types of solitons.
result Proved a dichotomy on curvature decay for certain steady Ricci solitons.
Cautious Weight Decay modifies weight decay for better optimization.
problem Improving optimization in deep learning models.
method Applies weight decay selectively based on parameter sign alignment.
result Consistently improves model performance across various tasks and scales.
Study on scalar curvature decay on non-compact manifolds linked at infinity.
problem Understanding scalar curvature decay on non-compact manifolds with topological linking at infinity.
method Analyzing polynomial decay, developing obstruction theory, using μ--bubble exhaustions, and index theory. result Topological linking at infinity forces polynomial decay of scalar curvature on manifolds of weakly bounded geometry.
Study on massless Vlasov equation on Reissner-Nordström spacetimes, showing decay rates and non-decay phenomena.
problem Analyzing decay and non-decay rates of solutions to the massless Vlasov equation on Reissner-Nordström spacetimes.
method Quantitative analysis of geodesic flow and comparison to wave equation instability results.
result Exponential decay rates in subextremal cases and polynomial rates in extremal cases, with non-decay of transversal derivatives in extremal cases.
Empirical risk minimization (ERM) is ubiquitous in machine learning and underlies most supervised learning methods. While there has been a large body of work on algorithms for various ERM problems, the exact computational complexity of ERM is still not understood. We address this issue for multiple popular ERM problems…
Introduces gradient decay in Softmax for better generalization.
problem Improving generalization performance in neural networks.
method Gradient decay hyperparameter in Softmax for varying gradient rates based on probability.
result Gradient decay rate affects generalization performance and can be tuned for better optimization.
Study examines wave equation decay and Strichartz estimates on conic manifolds.
problem Analyzing wave equation behavior on conic spaces with critical electromagnetic potentials.
method Established decay and Strichartz estimates through localized spectral measure construction.
result Extended and improved previous results on wave equation behavior with critical potentials.
Study shows uniform decay rate for singular mean curvature flows.
problem Understanding singularities in mean curvature flows.
method Rescaled flow analysis near compact singularities.
result Uniform decay order bound for the rescaled flow.
Weight decay stabilizes training dynamics by slowing progressive sharpening.
problem Understanding how weight decay affects training stability in deep learning models.
method Analyzing weight decay effects at the Edge of Stability, developing a mathematical framework.
result Weight decay dampens oscillations and stabilizes sharpness in CNNs, causing a phase transition in MLPs.
Polynomial decay of correlations shown for curved surfaces.
problem Analyzing geodesic flows on curved surfaces.
method Proving polynomial decay of correlations for geodesic flows on nonpositively curved surfaces.
result Polynomial decay of correlations for geodesic flows on nonpositively curved surfaces.
DBSCAN is a classical density-based clustering procedure with tremendous practical relevance. However, DBSCAN implicitly needs to compute the empirical density for each sample point, leading to a quadratic worst-case time complexity, which is too slow on large datasets. We propose DBSCAN++, a simple modification of DBS…
A known failing of many popular random graph models is that the Aldous-Hoover Theorem guarantees these graphs are dense with probability one; that is, the number of edges grows quadratically with the number of nodes. This behavior is considered unrealistic in observed graphs. We define a notion of edge exchangeability …
Last SGD iterate bounds for overparameterized linear regression.
problem Analyzing the last iterate risk bounds of SGD with decaying stepsize for overparameterized linear regression.
method Problem-dependent analysis of last iterate risk bounds of SGD with geometrically decaying stepsize.
result Proved nearly matching upper and lower bounds on the excess risk for last iterate SGD with geometrically decaying stepsize.
Optimal learning rates decay to zero in easy tasks and maintain a warmup phase in hard tasks.
problem Optimizing learning rates under functional scaling laws for model training.
method Deriving optimal learning-rate schedules based on exponents s and β. result Sharp phase transition between easy and hard tasks, with different decay behaviors.
We found that factors decay over time, with momentum fitting best.
problem Understanding how factors decay over time and their impact on performance.
method Derived a hyperbolic decay model for factors, tested against linear and exponential alternatives.
result Momentum exhibits hyperbolic decay, outperforming linear and exponential models.
The family of temporal difference (TD) methods span a spectrum from computationally frugal linear methods like TD(λ) to data efficient least squares methods. Least square methods make the best use of available data directly computing the TD solution and thus do not require tuning a typically highly sensitive learning r…
The paper predicts and explains the decay of stock anomaly performance over time.
problem Predicting and explaining the drop in risk-adjusted performance of stock anomalies.
method The authors propose ex-ante characteristics based on hypotheses of out-of-sample decay and in-sample overfitting.
result The year of publication explains 30% of the variance in Sharpe decay across factors.
Sharp decay found for solutions of a specific equation in Lie groups.
problem Asymptotic decay of solutions to a Yamabe type equation.
method Analysis of a specific pseudodifferential operator in a homogeneous Lie group.
result Established sharp asymptotic decay of positive solutions.
We give conditions which imply that a complete noncompact manifold with quadratic curvature decay has finite topological type. In particular, we find links between the topology of a manifold with quadractic curvature decay and some properties of the asymptotic cones of such a manifold.
The paper extends decay estimates to graphs with positive spectrum.
problem Proving decay estimates for nonnegative functions on graphs.
method Sharp ℓ2 decay estimates for nonnegative generalized subharmonic functions. result Extends Li and Wang's result to graphs with positive Laplacian spectrum.
The paper studies harmonic map heat flow stability and decay rates.
problem Analyzing stability and decay rates of harmonic map heat flow solutions.
method Use of homogeneous Besov space B˙p,∞pd(Rd) for small initial data and self-similar decay assumption. result Decay rates for solutions of the harmonic map flow of the form ∥ablau(t)∥L∞(Rd)≤Ct−21 and self-similar decay under stronger initial conditions. In this paper, we give a description for steady Ricci solitons with a linear decay of sectional curvature. In particular, we classify all 3-dimensional steady Ricci solitons and 4-dimensional κ-noncollpased steady Ricci solitons with nonnegative sectional curvature under the linear curvature decay.
Extends global stability of Minkowski spacetime to minimal decay assumptions.
problem Global stability of Minkowski spacetime under minimal decay assumptions.
method Uses rp-weighted estimates instead of vectorfield method. result Proves global stability of Minkowski spacetime under minimal decay assumptions.
SHIFT framework identifies subgroups with large ML model performance decay.
problem Large model performance decay in subgroups when deployed.
method Subgroup-scanning Hierarchical Inference Framework (SHIFT) for performance drift.
result SHIFT identifies interpretable subgroups with large performance decay and suggests targeted actions to mitigate it.
New Sobolev inequalities found for curved spaces.
problem Sobolev inequalities in curved spaces with specific decay conditions.
method Used ABP method developed by Cabré and Brendle.
result Established Sobolev inequalities for compact domains and submanifolds.
New inequality shows energy growth and decay in geometric problems.
problem Understanding energy behavior in geometric problems.
method Introduced a symmetric (log-)epiperimetric inequality.
result Energy growth and decay observed in geometric problems.