New bound for neural networks with full-rank weights, independent of network width.
problem Understanding generalization of neural networks with full-rank weight matrices.
method Using Koopman operators to derive a tighter generalization bound for full-rank weight matrices.
result The bound is tighter than existing norm-based bounds when condition numbers are small.
The study analyzes weighted manifolds with curvature bounds, proving eigenvalue estimates and inequalities.
problem Analyzing geometric properties of weighted manifolds under Ricci curvature bounds.
method Develops geometric analysis techniques on weighted Riemannian manifolds with lower 0-weighted Ricci curvature bounds. result Proves eigenvalue estimates for Steklov and ABP inequalities on weighted manifolds.
Upper bounds for eigenvalues on submanifolds in weighted manifolds.
problem Eigenvalue bounds for submanifolds in weighted Riemannian manifolds.
method Proving upper bounds for divergence-type operators and Steklov problems on submanifolds.
result Reilly-type upper bounds for eigenvalues.
The paper studies heat kernels on weighted Riemannian manifolds with curvature bounds.
problem Estimating heat kernels on weighted Riemannian manifolds with lower Ricci curvature bounds.
method Establishing parabolic Harnack inequalities, proving Gaussian bounds for heat kernels, and constructing Li-Yau-type gradient estimates.
result Gaussian upper and lower bounds for the heat kernel, Liouville theorem, uniqueness property, and eigenvalue bounds.
Novel analysis improves weighted majority vote in multiclass classification.
problem Improving the performance of weighted majority vote in multiclass classification.
method Analyzes expected risk of weighted majority vote, considering prediction correlations and provides a bound for efficient minimization.
result Minimization of the new bound typically does not degrade the test error of the ensemble.
Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.
problem Geometric and topological properties of Finsler metric measure manifolds with integral weighted Ricci curvature bounds.
method Establish Laplacian comparison theorem, volume comparison theorems, volume growth estimate, Gromov pre-compactness, local Dirichlet isoperimetric constant estimate.
result First Dirichlet eigenvalue estimate and gradient estimate for harmonic functions.
Study extends compactness theorems to weighted manifolds with integral curvature bounds.
problem Estimating diameter of weighted manifolds under curvature constraints.
method Extended Sprouse's compactness theorems to weighted manifolds with integral curvature bounds. Used ε-range to handle specific cases. Extended segment inequality to weighted manifolds.
result Proved theorems for weighted manifolds with effective dimension ≤ 1 and ≥ dimension.
In recent studies, several asymptotic upper bounds on generalization errors on deep neural networks (DNNs) are theoretically derived. These bounds are functions of several norms of weights of the DNNs, such as the Frobenius and spectral norms, and they are computed for weights grouped according to either input and outp…
We study Riemannian manifolds with boundary under a lower weighted Ricci curvature bound. We consider a curvature condition in which the weighted Ricci curvature is bounded from below by the density function. Under the curvature condition, and a suitable condition for the weighted mean curvature for the boundary, we ob…
Weighted SVM (or fuzzy SVM) is the most widely used SVM variant owning its effectiveness to the use of instance weights. Proper selection of the instance weights can lead to increased generalization performance. In this work, we extend the span error bound theory to weighted SVM and we introduce effective hyperparamete…
Upper bounds found for eigenvalues of weighted Steklov and (p,q)-Laplacian problems.
problem Finding upper bounds for eigenvalues of weighted Steklov and (p,q)-Laplacian problems.
method Proving upper bounds using the weighted p-Laplace operator and (p,q)-Laplacian on submanifolds. result Reilly-type upper bounds for the first eigenvalues of Steklov and (p,q)-Laplacian problems.
The standard interpretation of importance-weighted autoencoders is that they maximize a tighter lower bound on the marginal likelihood than the standard evidence lower bound. We give an alternate interpretation of this procedure: that it optimizes the standard variational lower bound, but using a more complex distribut…
Proves equivalence of two types of Ricci curvature bounds.
problem Equivalence of distributional and synthetic Ricci curvature bounds.
method Analyzes weighted Riemannian manifolds with specific smoothness conditions.
result Proves equivalence of Ricci curvature bounds under given conditions.
In arrangements of pseudocircles (Jordan curves) the weight of a vertex (intersection point) is the number of pseudocircles that contain the vertex in its interior. We give improved upper bounds on the number of vertices of weight <=k in certain arrangements of pseudocircles in the plane. In particular, forbidding cert…
Paper extends Steklov eigenvalue estimate to weighted graphs.
problem Steklov eigenvalue estimation on weighted graphs.
method Extended Perrin's estimate to general weighted graphs.
result Characterized rigidity of the extended estimate.
Upper bounds on constants for Brownian motion with sticky boundary.
problem Bounding constants for Brownian motion with sticky boundary.
method Interpolation approach based on energy interactions and Reilly formula.
result Upper bounds on Poincaré and Logarithmic Sobolev constants.
We study Riemannian manifolds with boundary under a lower N-weighted Ricci curvature bound for N at most 1, and under a lower weighted mean curvature bound for the boundary. We examine rigidity phenomena in such manifolds with boundary. We conclude a volume growth rigidity theorem for the metric neighborhoods of …
We give an explicit algorithm and source code for combining alpha streams via bounded regression. In practical applications typically there is insufficient history to compute a sample covariance matrix (SCM) for a large number of alphas. To compute alpha allocation weights, one then resorts to (weighted) regression ove…
New algorithm for learning mixtures with mostly uniform weights, improving on previous bounds.
problem Learning mixtures of Gaussians with uniform weights and mostly uniform component weights.
method Statistical Query (SQ) lower bound and quasi-polynomial upper bound for testing.
result Quasi-polynomial upper bound for testing mixtures with mostly uniform weights.
Study on stable minimal hypersurfaces under Ricci curvature constraints.
problem Stability of weighted minimal hypersurfaces under Ricci curvature bounds.
method Derive geometric consequences and prove a Schoen-Yau type criterion.
result Structure theorem for three-dimensional weighted manifolds of non-negative Ricci curvature.
New lower bounds of the first nonzero eigenvalue of the weighted p-Laplacian are established on compact smooth metric measure spaces with or without boundaries. Under the assumption of positive lower bound for the m-Bakry--Émery Ricci curvature, the Escober--Lichnerowicz--Reilly type estimates are proved; under the…
Reweighting improves risk bounds in certain data regions.
problem Improving risk bounds in classification and heteroscedastic regression.
method Weighted empirical risk minimization with a data-dependent weight function.
result A weighted ERM estimator can achieve superior performance in specific sub-regions.
Improved multiclass classification with class-weighted nearest neighbors.
problem Multiclass classification with large or imbalanced classes.
method Class-weighted k-nearest neighbors algorithm, derived bounds on accuracy and risk.
result Optimized classification metrics like F1 score or Matthew's Correlation Coefficient.
The study establishes inequalities on Finsler manifolds with weighted Ricci curvature.
problem Investigating inequalities on Finsler manifolds with weighted Ricci curvature.
method Volume comparison, Bonnet-Myers theorem, Poincaré-Lichnerowicz inequality.
result Sharp lower bound for the first eigenvalue on Finsler manifolds.
Lower bounds for delta invariant of weighted hypersurfaces proved for K-stability.
problem Proving K-stability of weighted hypersurfaces.
method Abban-Zhuang method and study of linear systems on flags of weighted hypersurfaces.
result Proves K-stability of a large class of quasi-smooth Fano hypersurfaces and all smooth Fano weighted hypersurfaces.
Study complete manifolds with weighted Poincaré inequality and Ricci curvature bounds.
problem Understanding the structure of complete manifolds with specific curvature and inequality conditions.
method Analyzing manifolds with weighted Poincaré inequality and Ricci curvature bounds.
result Obtained splitting results for manifolds with non-zero weight function limit at infinity.
Study compares manifolds with boundary under weighted Ricci curvature bounds.
problem Understand geometric properties of manifolds with boundary under lower weighted Ricci curvature bounds.
method Use lower N-weighted Ricci curvature bounds with ε-range to study comparison geometry. result Conclude splitting theorems and comparison geometric results for inscribed radius, volume, and eigenvalues.
Paper extends Aronson-Bénilan estimates for porous medium equations on manifolds with negative curvature.
problem Estimating gradients for porous medium equations on manifolds with negative curvature.
method Develops Aronson-Bénilan gradient estimates for porous medium equations under lower bounds of N-weighted Ricci curvature with N<0. result Generalizes gradient estimates for porous medium equations to manifolds with negative curvature.
New invariants help solve existence of weighted cscK metrics.
problem Existence of weighted cscK metrics in K-stability.
method Introduced weighted analytic delta invariant and beta invariant.
result Sufficient condition for existence of weighted cscK metrics.
This paper compares gradient estimators in importance-weighted VI and justifies the superiority of DREP over REP.
problem Understanding the impact of gradient estimators on importance-weighted VI algorithms.
method Unified theoretical comparison of reparameterized and doubly-reparameterized gradient estimators tied to IWAE, VR, and VR-IWAE bounds.
result Formally justifies the superiority of doubly-reparameterized gradient estimators over reparameterized ones in importance-weighted VI.
Survey on gluing constructions under lower curvature bounds.
problem Understanding lower curvature bounds in various geometric contexts.
method Analyzes gluing constructions in smooth and non-smooth settings.
result Provides conjectures and theorems on synthetic lower Ricci curvature bounds.
The paper proves lower bounds for Gaussian-weighted curvature integrals of self-shrinkers.
problem Proving lower bounds for Gaussian-weighted \(L^2\)-curvature integrals of self-shrinkers.
method Combining normal coordinate functions with weighted Poincaré inequalities and first-eigenvalue estimates.
result Explicit lower bounds in terms of entropy for closed self-shrinkers, leading to curvature gaps.
Sharp upper bounds derived for Alexandrov-Fenchel deficit using weighted Minkowski integral formulas.
problem Deriving upper bounds for the Alexandrov-Fenchel deficit.
method Using weighted Minkowski integral formulas and an integral formula for the deficit in Jensen's inequality.
result Quantitative estimates under weaker convexity assumptions, including a distance term.
Paper investigates rigidity phenomena for weighted Ricci curvature bounds with Laplacian comparison theorem.
problem Investigating rigidity phenomena for weighted Ricci curvature bounds.
method Derived comparison geometric estimates and generalized for non-symmetric Laplacian.
result Obtained rigidity results for Laplacian comparison theorem, diameter comparisons, and volume comparisons.
In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion free connection introduced recently by the last two authors. We develop two new tools for studying weighted sectional curvature bounds: a new weighted Rauch comparison theorem and a modifie…
Positive weights improve kernel quadrature's accuracy.
problem Improving kernel quadrature weights to be positive and stable.
method Using convex geometry to approximate the kernel mean embedding with positive weights.
result Positive weights lead to improved kernel quadrature bounds with Monte-Carlo-beating rates.
VOQL optimizes RL with sparse rewards using weighted bounds.
problem Sparse rewards and non-linear function approximation in RL.
method VOQL combines Q-learning with weighted bounds for optimal regret. result Achieves asymptotically optimal regret for linear function approximation.
In this article, we study properly immersed complete noncompact submanifolds in a complete shrinking gradient Ricci soliton with weighted mean curvature vector bounded in norm. We prove that such a submanifold must have polynomial volume growth under some mild assumption on the potential function. On the other hand, if…
New algorithm for weighted low rank approximation with provable guarantees.
problem Weighted low rank approximation (WLRA) is computationally hard.
method Reweights the low rank solution using the weight matrix itself.
result Provably optimal approximation guarantees for WLRA.
The conjecture of Kosniowski asserts that if the circle acts on a compact unitary manifold M with a non-empty fixed point set and M does not bound a unitary manifold equivariantly, then the dimension of the manifold is bounded above by a linear function on the number of fixed points. We confirm the conjecture for a…
New algorithm improves accuracy of importance weights for diverse applications.
problem Improving accuracy of importance weights for various applications.
method Formulated multicalibrated partitions and developed an efficient algorithm.
result Algorithm significantly improves accuracy of importance weights.
New bounds for SGD show improved performance in various settings.
problem Improving convergence bounds for SGD with random permutations.
method Analyzing convergence of SGD with random reshuffling and arbitrary permutations.
result Tighter lower bounds for weighted average iterates in both convex and strongly-convex cases.
Paper proves Liouville theorems for harmonic functions under specific curvature bounds.
problem Analyzing harmonic functions on manifolds with lower bounds of N-weighted Ricci curvature. method Uses Moser's iteration procedure to prove Liouville theorems.
result Establishes Liouville theorems for harmonic functions with sublinear growth and under weaker bounds of N-weighted Ricci curvature. We discuss optimal lower bounds for eigenvalues of Laplacians on weighted graphs. These bounds are formulated in terms of the geometry and, more specifically, the inradius of subsets of the graph. In particular, we study the first non-zero eigenvalue in the finite volume case and the first eigenvalue of the Dirichlet L…
We introduce higher-order Poincar'e constants for compact weighted manifolds and estimate them from above in terms of subsets. These estimates imply upper bounds for eigenvalues of the weighted Laplacian and the first nontrivial eigenvalue of the p-Laplacian. In the case of the closed eigenvalue problem and the Neuma…
We study geometry of complete Riemannian manifolds endowed with a weighted measure, where the weight function is of quadratic growth. Assuming the associated Bakry-Emery curvature is bounded from below, we derive a new Laplacian comparison theorem and establish various sharp volume upper and lower bounds. We also obtai…
New bound improves on weighted majority vote risk estimation.
problem Improving risk estimation for weighted majority vote.
method Novel Chebyshev-Cantelli inequality and PAC-Bayes-Bennett inequality.
result New bounds improve on existing methods.
Characterizes a new curvature bound with convexity of entropies.
problem Lowering Ricci curvature bounds with ε-range.
method Characterization through convexity of entropies over Wasserstein space.
result Derives various interpolation and functional inequalities.