For a bounded domain in a complete Riemannian manifold , we study estimates for lower order eigenvalues of a clamped plate problem. We obtain universal inequalities for lower order eigenvalues. We would like to remark that our results are sharp.
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In this paper, we consider lower order eigenvalues of Laplacian operator with any order in Euclidean domains. By choosing special rectangular coordinates, we obtain two estimates for lower order eigenvalues.
We consider the lower order eigenvalues of poly-Laplacian with any order on spherical domains. We obtain universal inequalities for them and show that our results are optimal.
Lower bounds for higher-order methods in non-convex optimization.
problem Proving lower bounds for higher-order methods in smooth non-convex finite-sum optimization.
method Analyzing deterministic and randomized algorithms, proposing a new smoothness assumption.
result Proves optimal lower bounds for simulating pth-order regularized methods on the whole function.
Lower Complexity Bounds for Nonconvex-Strongly-Convex Bilevel Optimization with First-Order Oraclescs.LG
New lower bounds for bilevel optimization with first-order oracles.
problem Complexity of bilevel optimization with first-order oracles.
method Development of hard instances and proof of lower bounds.
result Nontrivial lower bounds for first-order zero-respecting algorithms.
We prove a universal recursive formulas for Branson's -curvature of order eight in terms of lower-order -curvatures, lower-order GJMS-operators and holographic coefficients. The results prove a special case of a conjecture in {arXiv:0905.3992}.
Additive Poisson Process: Learning Intensity of Higher-Order Interaction in Stochastic Processesstat.ML
New model estimates higher-order interactions in stochastic processes using lower-dimensional projections.
problem Estimating higher-order interaction effects in stochastic processes with limited data.
method Additive Poisson Process (APP) combines information geometry and generalized additive models to model intensity functions in lower dimensions.
result The model can estimate higher-order intensity functions with sparse data.
We prove universal recursive formulas for Branson's -curvatures in terms of respective lower-order -curvatures, lower-order GJMS-operators and holographic coefficients.
New algorithms avoid a dominant lower-order term in heavy-tailed loss settings.
problem Prediction with heavy-tailed losses without prior knowledge.
method Adaptive algorithms that avoid the maximum of losses as a lower-order term in regret.
result Improved regret bounds of and .
The study sets limits on the complexity of Klein geometries.
problem Understanding the complexity of Klein geometries.
method Simple upper and lower bounds for the order of Klein geometries.
result Established upper and lower bounds for the order of Klein geometries.
New algorithm finds approximate stationary points in non-convex optimization.
problem Finding approximate stationary points in non-convex stochastic optimization.
method Design of an algorithm using stochastic gradient and Hessian-vector products.
result Optimal rate of for finding -approximate stationary points, matching lower bounds.
In recent years, Streets and Tian introduced a series of curvature flows to study non-Kähler geometry. In this paper, we study how to construct second order curvature flows in a uniform way, under some natural assumptions which holds in Streets and Tian's works. As a result, by classifying the lower order tensors, we c…
Paper improves stochastic bilevel optimization methods for highly-smooth problems.
problem Finding -stationary points in stochastic bilevel optimization.
method Proposes FSA- methods using th-order finite differences for hyper-gradient approximation.
result Achieves upper complexity bound of for th-order smooth problems.
Paper establishes tight lower bounds for minimizing certain smooth and convex functions.
problem Minimizing high-order Hölder smooth and uniformly convex functions.
method Analyzes two asymmetric cases of and using worst-case oracle complexities.
result Establishes worst-case oracle complexities for reaching an ε-approximate solution.
New curves share invariant up to any fixed order.
problem Finite-order rondle invariants and lower central series.
method Formulation and construction of infinite sequences of curves.
result Infinitely many curves share invariants up to any fixed order.
The study confirms positivity of Q-curvatures for specific conformal metrics.
problem Analyzing positivity of Q-curvatures for conformal metrics with certain properties.
method Examined a specific form of conformal metric and used properties of Q-curvatures to prove positivity.
result Lower order Q-curvatures are also positive under certain conditions.
In this paper, we study eigenvalues of the poly-Laplacian with arbitrary order on a bounded domain in an -dimensional Euclidean space and obtain a lower bound for eigenvalues, which gives an important improvement of results due to Levine and Protter. In particular, the result of Melas is included here.
The paper examines gradient and Riesz transform estimates under Ricci lower bounds.
problem Investigating estimates for solutions of the Poisson equation under Ricci lower bounds.
method Analyzes estimates for gradient and Riesz transforms under Ricci lower bounds, providing counterexamples and bounds.
result Valid estimates for gradient and Riesz transforms under Ricci lower bounds, with conditions on injectivity radius and curvature.
New algorithm reduces regret in infinite MDPs with optimal variance-dependent bounds.
problem Infinite horizon MDPs lack optimal algorithms with low regret.
method Developed a UCB-style algorithm for average-reward and γ-regret.
result Achieved optimal variance-dependent regret bounds for both objectives.
Paper estimates the order of vertices in random recursive trees.
problem Estimating the order of arrival of vertices in random recursive trees.
method Proposes an order estimator based on the Jordan centrality measure and defines risk measures.
result Establishes a nearly optimal estimator for the problem.
Study non-standard bi-orders on punctured torus bundles, matching standard ones in key subgroups.
problem Investigate non-standard bi-orders on punctured torus bundles.
method Analyze various bi-orderings and compare them to standard ones formed by the lower central series.
result For every bi-ordering, the largest and second largest proper convex subgroups match those of a standard bi-ordering. Third largest subgroup matches if it exists.
For a bounded domain with a piecewise smooth boundary in an -dimensional Euclidean space , we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. First we give a general inequality for eigenvalues of the Laplacian. As an application, we study lower order eigenvalues of the Lap…
New algorithms optimize convex functions with high-order derivatives.
problem Optimizing convex functions with high-order derivatives under various norms.
method Developed a non-Euclidean inexact accelerated proximal point method using an inexact uniformly convex regularizer.
result Showed nearly optimal algorithms for high dimensions in the black-box oracle model for -settings and all .
We study the information-theoretic lower bound of the sample complexity of the correct recovery of diffusion network structures. We introduce a discrete-time diffusion model based on the Independent Cascade model for which we obtain a lower bound of order , for directed graphs of nodes, and at most …
A sharp lower bound on the mean curvature integral with critical power for integral varifoldsmath.DG
Sharp lower bound found for integral varifolds' mean curvature.
problem Finding a sharp lower bound for the mean curvature integral of integral varifolds.
method Developed a new approach using integral varifolds and mean curvature.
result A sharp lower bound on the mean curvature integral with critical power for integral varifolds.
This work sets a universal lower bound for learning causal DAGs with atomic interventions.
problem Learning causal DAGs using only observational data results in a Markov equivalence class, requiring interventions to fully orient.
method Developed CBSP orderings and used them to prove a universal lower bound on the number of single-node interventions needed.
result The universal lower bound is within a factor of two of the minimum number of single-node interventions required to fully orient a given Markov equivalence class.
Novel methods for accelerating optimization in complex bilevel and minimax problems.
problem Optimization challenges in bilevel and minimax problems, especially when strong convexity assumptions are not met.
method Accelerated fully first-order methods for Bilevel Optimization (BLO) and Minimax Optimization (NCSC).
result State-of-the-art complexity for finding approximate second-order stationary points in BLO and NCSC.
In this paper an explicit formula for a lower bound on the volume of a hyperbolic orbifold, dependent on dimension and the maximal order of torsion in the orbifolds' fundamental group, is constructed.
New methods solve complex optimization problems without strong convexity assumptions.
problem Complex bilevel optimization problems with minimax lower-level structures.
method Penalty-based first-order methods for bilevel minimax optimization.
result Achieves -KKT point with improved oracle complexity.
Grid homology shows knot unknotting lower bound.
problem Knot unknotting number determination
method Grid homology analysis
result Torsion homology classes order bounds unknotting number
What's in a Smoothness Constant? Tighter Rates for Local SGD with Bounded Second-order Heterogeneitycs.LG
Improved Local SGD convergence for general convex objectives with bounded second-order heterogeneity.
problem Understanding when and why Local SGD outperforms alternatives in distributed optimization.
method Established improved convergence guarantees for Local SGD on general convex objectives under bounded second-order heterogeneity.
result Upper bounds for Local SGD are nearly tight, providing a sharper convergence theory.
In principle, higher-order networks that have multiple edge types are more informative than their lower-order counterparts. In practice, however, excessively rich information may be algorithmically infeasible to extract. It requires an algorithm that assumes a high-dimensional model and such an algorithm may perform po…
This paper has been withdrawn since the results are not satisfied.
We study involuntary micro-movements of the eye for biometric identification. While prior studies extract lower-frequency macro-movements from the output of video-based eye-tracking systems and engineer explicit features of these macro-movements, we develop a deep convolutional architecture that processes the raw eye-t…
Researchers recover Riemannian manifolds and lower order terms from travel time data.
problem Recovering Riemannian manifolds and lower order terms from travel time data.
method Adaptation of the Boundary Control method to recover lower order terms.
result Complete Riemannian manifolds and lower order terms can be uniquely recovered from a local source to solution map.
For certain classes of knots we define geometric invariants called higher-order genera. Each of these invariants is a refinement of the slice genus of a knot. We find lower bounds for the higher-order genera in terms of certain von Neumann -invariants, which we call higher-order signatures. The higher-order genera o…
A new autoregressive model learns the order of graph generation tasks.
problem Generating graphs in a meaningful order when the canonical order is not obvious.
method Introduces a variant of autoregressive models that dynamically decides the autoregressive order based on data.
result Achieves state-of-the-art results on molecular graph generation benchmarks.
We prove a lower bound for the -th Steklov eigenvalues in terms of an isoperimetric constant called the -th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Ste…
Study shows marketable order routing to wholesalers benefits all traders, leading to lower market depth and price volatility.
problem Determining the preference of retail traders for marketable order routing.
method Two models: one for market makers competing for retail order flow (Bertrand model) and another for price-taking competitive liquidity providers (open exchange model).
result Routing marketable orders to wholesalers is preferred by all traders, leading to mean reverting inventories and lower market depth.
State-of-the-art methods in convex and non-convex optimization employ higher-order derivative information, either implicitly or explicitly. We explore the limitations of higher-order optimization and prove that even for convex optimization, a polynomial dependence on the approximation guarantee and higher-order smoothn…
Improved regret bounds for bandits with expert advice.
problem Optimizing decision-making in environments with expert advice.
method Proved lower and upper bounds for regret in restricted and standard feedback models.
result Proved a new upper bound of order for the worst-case regret, matching a previously known lower bound.
New algorithm optimizes convex functions with noisy evaluations in one dimension.
problem Optimizing convex functions with noisy zero-order evaluations in one dimension.
method Proposed a computationally efficient algorithm achieving convergence rate.
result Achieved the optimal convergence rate, closing the gap in one dimension.
This paper studies eigenvalues of the buckling problem of arbitrary order on bounded domains in Euclidean spaces and spheres. We prove universal bounds for the k-th eigenvalue in terms of the lower ones independent of the domains. Our results strengthen the recent work in [28] and generalize Cheng-Yang's recent estimat…
In this paper, we investigate the Dirchlet eigenvalue problems of poly-Laplacian with any order and quadratic polynomial operator of the Laplacian. We give some estimates for lower bounds of the sums of their first eigenvalues which improve the previous results.
A Generic First-Order Algorithmic Framework for Bi-Level Programming Beyond Lower-Level Singletoncs.LG
New framework tackles bi-level optimization without LLS condition.
problem Bi-level optimization problems without LLS condition.
method Bi-level Descent Aggregation (BDA) framework, proving convergence without LLS condition.
result Proves convergence of BDA without LLS condition.
Estimates lower bounds for isoperimetric profiles and improves on previous estimates for specific manifolds.
problem Estimating lower bounds for isoperimetric profiles of specific Riemannian manifolds.
method Explicit lower bounds for isoperimetric profiles of Riemannian product manifolds.
result Improved lower bounds for isoperimetric profiles and Yamabe constants.
New algorithm for identifying optimal arms in stochastic bandit problems.
problem Optimal arm identification in stochastic bandit problems with many arms.
method Characterized optimal learning rates and provided algorithms with matching bounds.
result Lower bounds and matching upper bounds for cumulative regret and best-arm identification.
The genus of knots is a one of the fundamental invariant and can be seen as a complexity of knots. In this paper, we give a lower bound of genus using Dehornoy floor, which is a measure of complexity of braids in terms of braid ordering.