The study classifies Riemannian manifolds with specific Hessian properties.
problem Classifying Riemannian manifolds with a special Hessian structure.
method Analyzing manifolds with a non-identically vanishing function f whose Hessian is minus f times the Ricci tensor.
result Partial classification of manifolds with this Hessian structure.
We consider the Dolbeault operator of K1/2 -- the square root of the canonical line bundle which determines the spin structure of a compact Hermitian spin surface (M,g,J). We prove that the Dolbeault cohomology groups of K1/2 vanish if the scalar curvature of g is non-negative and non-identically zero. Moreov…
The paper provides guarantees for learning nonlinear representations from multiple non-identically distributed data sources.
problem Learning from non-identically distributed and dependent data.
method Established statistical guarantees for learning general nonlinear representations from multiple data sources.
result The excess risk of the estimated function decays as a function of the sample complexity and task diversity.
VRL-SGD reduces communication complexity in non-identical data settings.
problem Training machine learning models with non-identical data distribution.
method VRL-SGD, which eliminates gradient variance dependency and achieves linear speedup with lower communication complexity.
result VRL-SGD reduces communication complexity from $O(T^{ rac{3}{4}} N^{ rac{3}{4}})$ to $O(T^{ rac{1}{2}} N^{ rac{3}{2}})$.
This work examines how non-identical data distributions affect Federated Learning performance.
problem The impact of non-identical data distributions on Federated Learning performance.
method Synthesized datasets with varying degrees of data distribution similarity, evaluated Federated Averaging algorithm performance, proposed server momentum mitigation.
result Performance of Federated Learning degrades as data distributions differ more, and a mitigation strategy improves accuracy.
The paper analyzes ridge regression with random features for non-identically distributed data.
problem Analyzing ridge regression performance for data with heterogeneous variance profiles.
method Combining linear-plus-chaos approximation and operator-valued free probability.
result Derives asymptotic equivalents for training and test risks under non-identically distributed data.
This note displays an interesting phenomenon for percentiles of independent but non-identical random variables. Let X1,⋯,Xn be independent random variables obeying non-identical continuous distributions and X(1)≥⋯≥X(n) be the corresponding order statistics. For any p∈(0,1), we investig…
Study ridge regression for non-identically distributed data with varying variances.
problem Investigate high-dimensional regression with non-identical data variance.
method Propose a random effect model and use tools from random matrix theory.
result Highlight the double descent phenomenon in high-dimensional regression for certain variance profiles.
We strengthen the results of \cite{A1}, consequently, we improve the claims of \cite{A2} obtaining the best possible results. Namely, we prove that if a subgroup Γ of Diff+(I) contains a free semigroup on two generators then Γ is not C0-discrete. Using this, we extend the Hölder's Theorem in $\math…
New class of heavy-tailed distributions shows weighted averages dominate individual variables.
problem Understanding and comparing risks in heavy-tailed distributions.
method Introducing a new class of heavy-tailed distributions and proving stochastic dominance relations.
result Weighted averages of random variables in this class are stochastically larger than individual variables.
Novel framework for data sharing and coordinated exploration in concurrent RL with non-identical environments.
problem Learning more data-efficient and better policies in concurrent RL with non-identical environments.
method Proposes a novel algorithmic framework that leverages causal inference via ANM-MM to extract model parameters and a new data sharing scheme based on similarity measures.
result Demonstrates superior learning speeds on various tasks and effectiveness of diverse action selection.
Macbeath gave a formula for the number of fixed points for each non-identity element of a cyclic group of automorphisms of a compact Riemann surface in terms of the universal covering transformation group of the cyclic group. We observe that this formula generalizes to determine the fixed-point set of each non-identity…
We give effective proofs of residual finiteness and conjugacy separability for finitely generated nilpotent groups. In particular, we give precise asymptotic bounds for a function introduced by Bou-Rabee that measures how large the quotients that are need to separate non-identity elements of bounded length from the ide…
Meta-analysis improves interpretation and efficiency across similar but non-identical datasets.
problem Meta-analysis of heterogeneous data in high dimensions.
method Integrative sparse regression with a global parameter for adaptability and anonymity.
result Superior identification of global parameter for high-dimensional linear models.
We prove that if Γis subgroup of Diff_{+}^{1+ε}(I) and N is a natural number such that every non-identity element of Γhas at most N fixed points then Γis solvable. If in addition Γis a subgroup of Diff_{+}^{2}(I) then we can claim that Γis metaabelian.
This paper tackles computational bottlenecks in federated learning on mobile devices.
problem Computationally heterogeneous mobile devices hinder federated learning efficiency.
method Proposes efficient algorithms to schedule mobile devices based on data heterogeneity.
result Achieves up to 100x speedup and 7% accuracy gain in federated learning.
The paper solves a problem related to curvature in complex geometry.
problem Resolving the prescribed Chern scalar curvature problem.
method Divided into three cases based on the sign of the Gauduchon degree, analyzed separately.
result Proves that certain functions are Chern scalar curvatures of conformal metrics.
The study examines property testing and estimation under non-identically distributed samples, finding necessary and sufficient sample complexities.
problem Property testing and estimation under non-identically distributed samples.
method Analysis of distributional property testing and estimation in settings with heterogeneous entities.
result Necessary and sufficient sample complexities for property testing and estimation under non-identically distributed samples.
Undirected graphs are often used to describe high dimensional distributions. Under sparsity conditions, the graph can be estimated using ℓ1 penalization methods. However, current methods assume that the data are independent and identically distributed. If the distribution, and hence the graph, evolves over time t…
FedProx tackles heterogeneity in federated learning networks.
problem Significant variability in systems characteristics and non-identically distributed data in federated networks.
method FedProx is a framework that generalizes and re-parametrizes FedAvg, introducing modifications to handle both systems and statistical heterogeneity.
result FedProx demonstrates significantly more stable and accurate convergence behavior than FedAvg, improving test accuracy by 22% on average in highly heterogeneous settings.
In [13], it is proved that any subgroup of Diff+ω(I) (the group of orientation preserving analytic diffeomorphisms of the interval) is either metaabelian or does not satisfy a law. A stronger question is asked whether or not the Girth Alternative holds for subgroups of Diff+ω(I). In th…
New vanishing theorems for harmonic and pluriharmonic functions on Kähler and quaternionic Kähler manifolds.
problem Vanishing theorems for harmonic and pluriharmonic functions on Kähler and quaternionic Kähler manifolds.
method Utilized refined Kato type inequalities and Böchner technique to generalize results to Lp-integrable pluriharmonic functions and harmonic 1-forms. result Proved vanishing property of pluriharmonic functions with finite Lp energy on complete Kähler manifolds. LD-SGD improves communication in decentralized SGD.
problem Efficiently combining local updates and decentralized communication.
method Proposes LD-SGD integrating local updates and decentralized SGD, with a convergence analysis.
result LD-SGD converges to a critical point for non-convex objectives with non-identically distributed data.
In this paper we give some sufficient conditions for the vanishing of the genus-2 G-function, which was introduced by B. Dubrovin, S. Liu and Y. Zhang in [DLZ]. As a corollary we prove their conjecture for the vanishing of the genus-2 G-function for ADE singularities.
Study on algebraic curves' invariants and vanishing criteria.
problem Vanishing criteria for Griffiths infinitesimal invariants of algebraic curves.
method Analysis of moduli space of smooth genus 4 curves, study of normal functions.
result Vanishing criteria for the Griffiths infinitesimal invariants of Ceresa normal function.
Study large deviations in life insurance portfolios without identical distributions.
problem Large deviations in life insurance portfolios with bounded losses and variances.
method Upper bound from standard large deviations, counterexample for full large deviation principle.
result Exponential bound for average loss exceeding a threshold.
The paper tackles learning optimal predictions from a single trajectory of a stochastic dynamical system.
problem Learning from a single finite trajectory of an ergodic stochastic dynamical system.
method The approach involves estimating the optimal one-step prediction function using nonlinear least squares and deriving high-probability guarantees.
result The study provides high-probability guarantees for the optimal prediction function, accounting for the non-independent and non-identically distributed nature of trajectory data.
We show that under very general assumptions the partial Bergman kernel function of sections vanishing along an analytic hypersurface has exponential decay in a neighborhood of the vanishing locus. Considering an ample line bundle, we obtain a uniform estimate of the Bergman kernel function associated to a singular metr…
The portfolio optimization problem in which the variances of the return rates of assets are not identical is analyzed in this paper using the methodology of statistical mechanical informatics, specifically, replica analysis. We define two characteristic quantities of an optimal portfolio, namely, minimal investment ris…
Proves equations for high-dimensional gradient-based methods from Gaussian data.
problem High-dimensional asymptotics of gradient-based learning algorithms.
method Closed-form equations derived from dynamical mean-field theory.
result Equations match those from discretized DMFT for gradient flow.
Each compact Riemannian manifold with no conjugate points admits a family of functions whose integrals vanish exactly when central Busemann functions split linearly. These functions vanish when all central Busemann functions are sub- or superharmonic. When central Busemann functions are convex or concave, they must be …
We solve Euler equations on graph manifolds, classifying steady flows with Morse-Bott Bernoulli functions.
problem Classifying steady Euler flows with Morse-Bott Bernoulli functions.
method Constructing non-vanishing steady solutions using integrable systems and topology.
result Steady Euler flows with Morse-Bott Bernoulli functions exist only on graph three-manifolds.
In his book "Differential Geometry of Spray and Finsler spaces", page 177, Zhongmin Shen asks "wether or not there always exist non-trivial Funk functions on a spray space". In this note, we will prove that the answer is negative for the geodesic spray of a finslerian function of non-vanishing scalar flag curvature.
Optimal rates for learning hidden tree structures are determined.
problem Learning hidden tree structures from noisy data.
method Study of the (noisy) information threshold and the Chow-Liu algorithm.
result Optimal rates for structure recovery are inversely proportional to the information threshold squared.
Investigates VaR behavior for sums of one-sided random variables, showing impossibilities and conditions for super-additivity.
problem Investigates the behavior of Value-at-Risk (VaR) for sums of one-sided random variables.
method Analyzes the extremal aggregation behavior of VaR, introduces structural conditions for super-additivity.
result Characterizes when VaR is fully super-additive and provides unified framework for various dependence structures.
In many machine learning problems, labeled training data is limited but unlabeled data is ample. Some of these problems have instances that can be factored into multiple views, each of which is nearly sufficent in determining the correct labels. In this paper we present a new algorithm for probabilistic multi-view lear…
Zeta functions extended to nonorientable surfaces, order of vanishing computed.
problem Computing dynamical zeta functions for nonorientable surfaces.
method Simple argument extending microlocal proofs to nonorientable case.
result Order of vanishing of zeta function is the first Betti number.
Improved perceptron design mitigates vanishing gradient problem.
problem Vanishing gradient problem in deep multilayer perceptrons.
method Auto-rotating perceptron (ARP) design with dynamic activation region.
result Neural networks with ARP units achieve better learning performance.
Constructs Morse homology for complex algebraic varieties.
problem Homology of vanishing cycles in complex algebraic varieties.
method Morse homology groups associated with a perturbed function on a compactified variety.
result Morse homology groups are isomorphic to the homology of vanishing cycles.
Defines metrics and Einstein tensors on Riemannian manifolds, proving vanishing for non-commutative two-torus.
problem Defining metrics and Einstein tensors on Riemannian manifolds.
method Defines bilinear functionals of vector fields and differential forms, generalizing to non-commutative geometry.
result Proves the vanishing of the Einstein functional for the conformally rescaled geometry of the noncommutative two-torus.
Client adaptation improves federated learning performance with non-IID data.
problem Improving model performance in federated learning with non-identically and non-independently distributed data.
method Simulates heterogeneous clients to learn client-specific conditioning using a conditional gated activation unit.
result Client adaptation enhances model performance across balanced and imbalanced data sets from audio and image domains.
New energy functional and fields for Yang-Mills theory, proving monotonicity and vanishing theorems.
problem Developing new mathematical tools for Yang-Mills theory.
method Introducing normalized exponential Yang-Mills energy functional, deriving monotonicity formula and vanishing theorem.
result Monotonicity and vanishing theorems for exponential Yang-Mills fields.
Vanishing result for cohomology leads to extension theorem for pluriharmonic functions.
problem Extension of pluriharmonic functions on complex manifolds.
method Vanishing result for Bott-Chern cohomology combined with Ehrenpreis technique.
result Hartogs extension theorem for pluriharmonic functions on cohomologically (n−1)-complete manifolds. New neural network approach mitigates vanishing/exploding gradients.
problem Vanishing and exploding gradients in neural networks.
method Gaussian-Poincaré normalized functions and orthogonal weight matrices.
result High-dimensional probability theory shows gradients disappear with high probability in wide neural networks.
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.
The paper studies vanishing cycles in matrix singularities.
problem Understanding the topology of singular Milnor fibers of matrix families.
method Definition and analysis of vanishing cycles, proof of conjectures, study of monodromy.
result Proof of an extended Damon-Pike μ=τ conjecture for special matrix families.
We prove that any smooth action of Zm−1,m≥3 on an m-dimensional manifold that preserves a measure such that all non-identity elements of the suspension have positive entropy is essentially algebraic, i.e. isomorphic up to a finite permutation to an affine action on the torus or its factor by $\pm\Id$…
A new GAN model α-GAN with tunable loss function addresses gradient vanishing and mode collapse issues.
problem Addressing vanishing gradients and mode collapse in GANs.
method Introduced a tunable GAN α-GAN using a supervised α-loss function. result Holistic understanding of α-GAN related to Arimoto divergence and convergence properties.