Optimal bounds found for torus curvatures in high dimensions.
problem Finding optimal bounds on normal curvatures of tori.
method Analyzing immersed n-torus in a Euclidean ball of large dimension.
result Optimal bounds on normal curvatures of tori established.
POTD estimates SDR subspace using optimal transport for binary response.
problem Insufficient performance of existing SDR methods for categorical responses.
method Principal optimal transport direction (POTD) using optimal transport coupling.
result POTD exclusively estimates SDR subspace for error-free class labels.
Optimizes dimension estimate for holomorphic functions on Kähler manifolds.
problem Determining the optimal dimension for holomorphic functions with polynomial growth.
method Analyzes Kähler manifolds with non-negative holomorphic bisectional curvature.
result Identifies the specific gap and optimal dimension for maximal volume growth.
New research determines the optimal sample complexity for multiclass and list learning.
problem Determining the optimal sample complexity for multiclass classification.
method Algebraic characterization of multiclass hypothesis classes in terms of their DS dimension.
result Proves a longstanding conjecture and determines the optimal dependence of sample complexity on DS dimension.
Characterizes statistical complexity of realizable regression in PAC and online learning.
problem Understanding the statistical complexity of realizable regression in both PAC and online learning settings.
method Introduces minimax instance optimal learners, novel and combinatorial dimensions to characterize learnability.
result Characterizes which classes of real-valued predictors are learnable and provides necessary conditions for learnability.
Quantum codes with optimal distance and dimension for n-dimensional space.
problem Designing efficient quantum codes in high dimensions.
method Combining asymptotically good codes, manifold construction, and embedding theorem.
result Optimal quantum codes with distance and dimension for n-dimensional space.
Gradient methods struggle with high dimensions in convex optimization.
problem The generalization performance of gradient methods in high-dimensional stochastic convex optimization.
method Construction of learning problems in high dimensions to analyze gradient methods' performance.
result Gradient methods require exponentially more training examples in high dimensions to achieve non-trivial test error.
AdaScale-TuRBO improves high-dimensional Bayesian optimization by dynamically scaling the GP lengthscale.
problem Inappropriate lengthscale design in TuRBO's local GP model causes suboptimal performance in high dimensions.
method Proposes AdaScale-TuRBO, which scales the GP lengthscale with both problem dimension and trust region size.
result AdaScale-TuRBO robustly outperforms standard TuRBO and other methods on synthetic and real-world tasks.
Optimizes algorithms for non-concave bandit problems.
problem Optimizing algorithms for non-concave bandit problems.
method Unified zeroth-order optimization paradigm.
result Minimax-optimal algorithms in the dimension for low-rank generalized linear bandit problems.
Study shows how optimal transport behaves in higher dimensions.
problem Characterizing optimal transport in higher dimensions with Euclidean distance.
method Investigates the small regularization limit of entropic optimal transport.
result The limiting transport plan is supported on transport rays and uniquely minimizes a relative entropy functional.
We determine the optimal structure of couplings for the \emph{Martingale transport problem} between radially symmetric initial and terminal laws μ , ν μ, ν μ , ν on R d \R^d R d and show the uniqueness of optimizer. Here optimality means that such solutions will minimize the functional $\E |X-Y|^p$ where 0 < p ≤ 1 0<p \leq 1 0 < p ≤ 1 , and the dimensio…
Random Function Descent improves optimization in high dimensions.
problem Lack of effective optimization methods in high-dimensional spaces.
method Introducing a 'random function' framework to optimize classical optimization problems.
result Random Function Descent (RFD) is a scalable optimization method that bridges Bayesian and classical optimization.
Bayesian optimization techniques have been successfully applied to robotics, planning, sensor placement, recommendation, advertising, intelligent user interfaces and automatic algorithm configuration. Despite these successes, the approach is restricted to problems of moderate dimension, and several workshops on Bayesia…
Vanilla Bayesian optimization performs well in high dimensions.
problem Bayesian optimization's poor performance in high-dimensional problems.
method Identified and addressed degeneracies, proposed scaling of Gaussian process lengthscale prior.
result Vanilla Bayesian optimization outperforms existing algorithms in high-dimensional tasks.
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 O ( 1 / T ) O(1/\sqrt{T}) O ( 1/ T ) convergence rate. result Achieved the optimal O ( 1 / T ) O(1/\sqrt{T}) O ( 1/ T ) convergence rate, closing the gap in one dimension. The study reveals optimal early stopping behaviors in deep learning models.
problem Understanding optimal early stopping in deep learning models.
method Theoretical analysis of linear models and experimental validation.
result Two distinct behaviors of optimal early stopping time depending on model dimension relative to dataset features.
SubRiemannian structures fail to meet Riemannian Brunn--Minkowski inequalities.
problem SubRiemannian structures do not satisfy Riemannian Brunn--Minkowski inequalities.
method The proof relies on the method used for the Heisenberg group and new investigations by Agrachev, Barillari, and Rizzi on subRiemannian structures.
result No Brunn--Minkowski inequality can be satisfied by strictly subRiemannian structures.
Optimal persuasion involves projecting state vectors onto lower-dimensional 'optimal information manifolds'.
problem Optimal persuasion of another agent observing multi-dimensional data.
method Performing non-linear dimension reduction by projecting state vectors onto the 'optimal information manifold'.
result Optimal information design splits information into 'good' and 'bad' components, revealing only the direction of good information.
Characterizes kernel interpolation in large dimensions, revealing optimal and sub-optimal regions.
problem Understanding the phase diagram of kernel interpolation in large dimensions.
method Characterization of variance and bias under various source conditions.
result Determined the ( s , γ ) (s,γ) ( s , γ ) -phase diagram of large-dimensional kernel interpolation. This work improves understanding of dimension reduction algorithms and their probabilistic embeddings.
problem Improving theoretical understanding of non-linear dimension reduction algorithms.
method Analytical investigation of a generalized multidimensional scaling optimization problem.
result Probabilistic formulation of the problem leads to deterministic embeddings, contrary to standard implementations.
Bayesian optimization improves with nonstationary covariance functions.
problem Stationary covariance functions fail to capture prior information in high dimensions.
method Proposes nonstationary covariance functions to encode prior information and adaptively promote local exploration.
result Nonstationary covariance functions increase sample efficiency in high dimensions.
The paper explores how to reduce classification tasks to optimization problems in Euclidean space.
problem Understanding the minimum dimension needed for reducing classification tasks to optimization problems.
method Developed a generalization of the Borsuk-Ulam Theorem to analyze the expressivity of reductions.
result The minimum Euclidean dimension required can be exponentially larger than the VC dimension, even for slightly non-trivial reductions.
New BE dimension measure reveals rich RL problems with sample-efficient algorithms.
problem Finding sample-efficient algorithms for complex RL problems.
method Introducing Bellman Eluder (BE) dimension and designing GOLF and OLIVE algorithms.
result GOLF and OLIVE algorithms learn near-optimal policies for low BE dimension problems with polynomial samples.
Bayesian optimization (BO) has been broadly applied to computational expensive problems, but it is still challenging to extend BO to high dimensions. Existing works are usually under strict assumption of an additive or a linear embedding structure for objective functions. This paper directly introduces a supervised dim…
Optimal pinching results on Einstein manifolds with positive Yamabe invariant.
problem Understanding the rigidity of Einstein manifolds with positive Yamabe invariant.
method Optimal pinching results and bounds on scalar curvature and Weyl tensor norms.
result Improved bounds on the Yamabe invariant and scalar curvature.
New theory explains how chaotic training improves neural network generalization.
problem Understanding how chaotic training improves neural network generalization.
method Representing stochastic optimizers as random dynamical systems and introducing a new dimension concept.
result Generalization in chaotic training depends on the complete Hessian spectrum and partial determinants.
New algorithm reduces sketching dimension to effective problem size.
problem Solving L2-regularized least-squares problems efficiently.
method Randomized algorithm using Gaussian and SRHT embeddings.
result Preserves convergence guarantees with reduced embedding dimension.
New method speeds up Bayesian optimization in high dimensions.
problem High-dimensional expensive function optimization struggles.
method Structured automatic differentiation for kernel matrices.
result First-order Bayesian optimization scalable to high dimensions.
Proves existence of maximizers for eigenvalue optimization on manifolds.
problem Eigenvalue optimization on Riemannian manifolds of dimension m ≥ 3 m \geq 3 m ≥ 3 . method Use of topological tensor products to analyze eigenvalue functionals.
result Absolutely continuous maximizers are induced by p p p -harmonic maps into spheres. Paper improves learning efficiency by focusing on effective dimensionality.
problem Dimensionality bottleneck in modern learning tasks.
method Developed tools to reduce dimensional costs using effective dimensionality.
result Uniform concentration bounds involving effective dimensionality, improving over existing results.
Estimates expected information gain using density approximations and dimension reduction.
problem Estimating expected information gain in nonlinear and non-Gaussian settings.
method Flexible transport-based schemes for EIG estimation, optimal sample allocation, and gradient-based upper bounds on mutual information.
result Optimal sample allocation and dimension reduction schemes improve EIG estimation accuracy and convergence rate.
A new method for faster optimization in high dimensions.
problem Slow convergence in high-dimensional optimization problems.
method Subspace cubic regularized Newton method within Krylov subspace.
result Achieves a dimension-independent convergence rate of O(1/mk + 1/k^2).
Sparse OSEs achieve optimal embedding dimension of O(d).
problem Achieving optimal embedding dimension for sparse OSEs.
method Random sparsified matrix with m ≥ ( 1 + θ ) d m \geq (1+θ)d m ≥ ( 1 + θ ) d non-zeros per column. result Sparse OSEs can achieve embedding dimension m = O ( d ) m=O(d) m = O ( d ) , improving on previous m = O ( d log ( d ) ) m=O(d\log(d)) m = O ( d log ( d )) . LCBO tackles constrained optimization in high dimensions, offering a polynomial convergence rate.
problem Bayesian optimization for high-dimensional constrained problems.
method LCBO uses local descent and uncertainty-driven exploration, proving polynomial convergence rate.
result LCBO achieves a polynomial convergence rate for KKT residuals in high dimensions.
Study on computable online learning with new conditions and complexities.
problem Characterizing optimal online learning under varying optimality requirements.
method Introduced anytime optimal (a-optimal) online learning and explored computational separations.
result Found a computational separation between a-optimal and optimal online learning.
Parametric shape optimization aims at minimizing an objective function f(x) where x are CAD parameters. This task is difficult when f is the output of an expensive-to-evaluate numerical simulator and the number of CAD parameters is large. Most often, the set of all considered CAD shapes resides in a manifold of lower e…
The paper tackles noisy labels in high-dimensional data, showing low-dimensional intuitions fail and proposing an optimized method.
problem Noisy labels in high-dimensional data classification.
method Linear classifier with a label noisiness aware loss function, using random matrix theory and Gaussian mixture data model.
result The performance of the linear classifier in high-dimension converges to a limit involving scalar statistics of the data, and the optimal classifier in low-dimension fails.
New algorithm reduces dimensionality in stochastic optimization.
problem Stochastic optimization in high-dimensional problems.
method Proposes a sparsity-inducing stochastic gradient-free (SI-SGF) algorithm.
result Proves dimension-free query complexity in convex and strongly convex cases.
Generative adversarial networks benefit from optimal input dimension and adaptive generator architecture.
problem Minimizing generalization error in GANs through optimal input dimension.
method Introducing generalized GANs (G-GANs) with group penalty and architecture penalty for adaptive dimensionality reduction and network architecture identification.
result G-GANs achieve superior performance with 40%+ improvements in maximum mean discrepancy or Frechet inception distance compared to off-the-shelf methods.
Study eigenvalues and eigenfunctions of fourth-order operators in annuli, proving optimal estimates and non-radiality.
problem Eigenvalue and eigenfunction analysis of fourth-order operators in degenerating annuli.
method Optimal estimates and non-radiality results for eigenfunctions in annuli.
result Nigh optimal estimate for the first eigenvalue and non-radiality of eigenfunctions in degenerating annuli.
Optimal sample complexity for autoregressive chain-of-thought learning proven.
problem Determining the minimum number of samples needed for accurate autoregressive chain-of-thought learning.
method Proved upper bound on sample complexity using Daniely-Shalev-Shwartz dimension and roll-out stable parity dimension.
result The sample complexity is bounded by the local next-token class rate, with no dependence on rollout length.
New scalable algorithm estimates barycenters of measures in high dimensions.
problem Estimating barycenters of measures in high-dimensional settings.
method Optimizes generative models to estimate barycenters, scaling by introducing inductive biases.
result First scalable method to estimate barycenters in thousands of dimensions.
A method for high-dimensional Bayesian optimization reduces dimensionality using EDR and Gaussian process.
problem Extending Bayesian optimization to high-dimensional settings.
method Two-step framework: EDR subspace identification followed by Gaussian process optimization.
result Algorithm converges in high-dimensional contexts, validated by numerical experiments.
New estimators for intrinsic dimension and Wasserstein distance improve OT accuracy.
problem Intrinsic dimension estimation and Wasserstein distance estimation in large-scale OT.
method Introduces novel estimators for intrinsic dimension and Wasserstein distance.
result Simple, tuning-free estimator of OT and fast intrinsic dimension estimator.
A new method reduces both input and output dimensions for better goal-oriented analysis.
problem Simultaneous reduction of input and output dimensions for more accurate analysis.
method Coupled input-output dimension reduction, optimizing gradient-based bounds.
result Determine most informative sensors and influential parameters efficiently.
A new DDR framework learns low-dimensional data representations using dynamical systems.
problem Learning efficient low-dimensional data representations.
method DDR framework based on nonlinear dynamical systems, using linear combinations of functions and regularization.
result DDR method outperforms other methods on synthetic and real datasets.
New rates for GLD and SGLD in infinite-dimensional spaces without dimensionality issues.
problem Gradient Langevin dynamics and SGLD convergence rates in high-dimensional spaces.
method Analysis of GLD and SGLD in infinite-dimensional Hilbert spaces, using stochastic differential equations and Markov chains.
result Derivation of dimension-free convergence rates for GLD and SGLD.
Optimal scaling for proximal MALA in high dimensions confirmed.
problem Optimizing sampling efficiency in high-dimensional target densities.
method Introduced and analyzed the proximal MALA algorithm, showing it maintains optimal scaling.
result Proximal MALA achieves optimal scaling in high dimensions with an average acceptance probability of 0.574.