Random hyperbolic surfaces have nearly optimal spectral gaps.
problem Proving the nearly optimal spectral gap conjecture for random Belyi surfaces.
method Using the Brooks-Makover model, the authors show a spectral gap greater than 1/4 - c/log(n).
result A random hyperbolic surface in the Brooks-Makover model has a spectral gap greater than 1/4 - c/log(n).
Spectral Inference Networks learn eigenfunctions from data using optimization.
problem Learning eigenfunctions of linear operators from data.
method Spectral Inference Networks generalize Slow Feature Analysis to generic symmetric operators and use stochastic optimization.
result Spectral Inference Networks accurately recover eigenfunctions and discover interpretable representations from video data.
Optimal spectral method found for inhomogeneous spiked Wigner model.
problem Structured noise in learning scenarios.
method Random matrix theory and spectral analysis.
result Optimal threshold for phase transition in block-structured Wigner model.
New method uses Chebyshev expansions to compute unbiased stochastic gradients for spectral functions.
problem Computing gradients of spectral functions is expensive and challenging.
method Combining randomized trace estimators with Chebyshev expansions for unbiased stochastic gradients.
result Developed methods for optimizing objectives involving spectral-sums with fast and stable convergence.
EGO-MDA identifies optimal spectral-bands for process discrimination.
problem Optimal spectral-bands for process discrimination.
method EGO-MDA, an unsupervised method using EGO and Mixture Discriminant Analysis.
result EGO-MDA achieves at least 70% improvement in median deviance.
Muon optimizer simplifies matrix optimization with spectral orthogonalization.
problem Matrix optimization challenges, especially with large condition numbers.
method Simplified Muon optimizer using spectral orthogonalization of gradients.
result Simplified Muon converges linearly with independent scalar sequences, outperforming gradient descent and Adam.
New algorithms optimize spectral risk measures, improving interpolation between average and worst-case performance.
problem Optimizing spectral risk measures for learning systems.
method Developed stochastic algorithms to optimize spectral risk measures by characterizing their subdifferential and addressing challenges like biasedness of subgradient estimates and non-smoothness.
result Our approach outperforms out-of-the-box stochastic subgradient and dual averaging methods in optimizing spectral risk measures.
We study Spectral Measures of Risk from the perspective of portfolio optimization. We derive exact results which extend to general Spectral Measures M_phi the Pflug--Rockafellar--Uryasev methodology for the minimization of alpha--Expected Shortfall. The minimization problem of a spectral measure is shown to be equivale…
Accelerates optimal transport computation by 10x with spectral insights.
problem Exponential slow-down of convergence in Entropic Optimal Transport as regularization weakens.
method Spectral insights and spectral warm-start strategy to mitigate convergence issues.
result Faster convergence compared to the reference method Sinkhorn algorithm.
Study shows optimal spectral gaps diminish in large genus surfaces.
problem Optimizing spectral gaps in large genus surfaces.
method Analysis of Weil-Petersson probability and eigenvalues of Laplacian.
result Probability of optimal spectral gaps vanishes as genus increases.
Paper optimizes Laplacian regularization for sparse network clustering.
problem Improving spectral clustering in sparse networks.
method Formally determines optimal Laplacian regularization.
result Proper regularization is closely tied to state-of-the-art techniques.
Optimizes risk measures given known marginal distributions of two unknown factors.
problem Determining an upper bound for spectral risk measures with unknown joint distribution.
method Introduces Maximum Spectral Measure (MSP) as a worst-case risk measure, formulated as an optimization problem with a more general objective function.
result Characterizes the continuity properties of the optimal value function and optimal solution set with respect to marginal distributions.
Study optimizes dividend strategies for risk processes with Lévy jumps.
problem Optimizing dividend payments in risk processes with Lévy jumps.
method Analyzes spectrally positive and negative Lévy processes, using scale functions.
result Periodic barrier strategy is optimal for spectrally negative Lévy processes with completely monotone Lévy density.
New method optimizes portfolios for non-stationary markets.
problem Inadequate classical portfolio optimization for non-stationary markets.
method Reformulate portfolio optimization in spectral domain, using complex statistics.
result Time-varying optimal capital allocations for non-stationary markets.
Spectral algorithms solve optimal community detection and related problems.
problem Optimal detection of community structures and related substructures.
method Spectral algorithms applied to various planted substructures.
result Spectral algorithms achieve optimal performance for a wide range of planted substructures.
Self-distillation optimally improves model performance in spiked covariance models.
problem Improving model performance in spiked covariance models.
method Developed spectral shrinkage estimators and analyzed self-distillation.
result Self-distillation achieves optimal performance among spectral shrinkage estimators for spiked covariance matrices.
New spectral approach improves space-filling designs in high dimensions.
problem Designing high-quality space-filling sample designs in arbitrary dimensions.
method Quantifying space-filling property, connecting spatial and spectral domains, deriving optimal designs, and developing an optimization framework.
result Proposed space-filling spectral designs significantly outperform existing methods, especially in high dimensions.
Study spectral learning for odeco tensors, addressing initialization bottlenecks.
problem Recovering orthogonally decomposable tensors under noise.
method Investigates perturbation bounds, non-convex optimization, and initialization strategies.
result Initialization is the main bottleneck for efficient algorithms.
We develop spectral spanners for vectors and use them to create efficient core-sets for determinant maximization.
problem Maximizing determinants of vector sets.
method Spectral spanners and greedy algorithm.
result Almost optimal composable core-sets for determinant maximization.
Optimal spectral estimators and AMP combine for efficient weak recovery in orthogonally invariant GLMs.
problem Parameter estimation from generalized linear models with complex correlation structures.
method Spectral initialization and approximate message passing (AMP) algorithm.
result Established rigorous performance guarantees for spectral initialization and AMP.
Optimizes spectral density estimation for stationary and nonstationary processes.
problem Estimating spectral density of time series with complex structure.
method Optimally adaptive Bayesian spectral density estimation using smoothing spline covariance structure.
result Optimal eigendecomposition provides superior performance compared to alternative covariance functions.
The paper proves optimal estimates and inequalities for spectral functions on certain manifolds.
problem Optimal estimates and inequalities for spectral functions on weakly 1-complete manifolds.
method Establishes optimal fundamental estimates and weak Morse inequalities for lower energy forms.
result Optimal fundamental estimates and weak Morse inequalities are proven for lower energy forms on weakly 1-complete manifolds.
The optimal dividend problem by De Finetti (1957) has been recently generalized to the spectrally negative Lévy model where the implementation of optimal strategies draws upon the computation of scale functions and their derivatives. This paper proposes a phase-type fitting approximation of the optimal strategy. We con…
Paper explores SNN for learning spectral geometric info from data.
problem Challenges in applying traditional eigensolvers to big data.
method Introduces Spectral Neural Networks (SNN) as an alternative.
result Investigates tradeoffs and optimization landscape of SNN.
New method accelerates smooth games using spectral shape analysis.
problem Accelerating optimization in smooth games with complex numerical challenges.
method Matrix iteration theory and spectral shape analysis to characterize and manipulate acceleration.
result Identified a continuum of optimization strategies from convex minimization to gradient descent.
Study optimal rates for spectral algorithms in Hilbert spaces.
problem Regression problems over separable Hilbert spaces with square loss.
method Investigate spectral/regularized algorithms including ridge, principal component, and gradient methods.
result Prove optimal, high-probability convergence results in terms of norms.
Muon optimizer improves deep learning with spectral norm constraints.
problem Improving optimization algorithms in deep learning.
method Theoretical analysis of Muon optimizer within the Lion- K \mathcal{K} K family. result Muon implicitly solves an optimization problem enforcing spectral norm constraints.
Paper proposes a new method for sparse spectral clustering on Stiefel manifold.
problem Sparse spectral clustering on Stiefel manifold with nonsmooth and nonconvex objective.
method Proposes a manifold proximal linear method (ManPL) to solve the original SSC formulation.
result Demonstrates the advantage of ManPL over existing methods on single-cell RNA sequencing data.
New SAM method improves model robustness with spectral inner perturbation and Muon optimizer.
problem Improving model robustness to small parameter perturbations.
method Introducing a spectral inner perturbation step in SAM combined with Muon optimizer.
result Spectral inner perturbation combined with Muon optimizer achieves best validation accuracy on ImageNet-1K.
This research optimizes Andrews plots for better visual clarity in high-dimensional data.
problem Visualizing high-dimensional datasets with clarity and aesthetics.
method Developed a method to add spectral smoothing to Andrews plots to reduce visual clutter.
result Optimal spatial-spectral smoothing leads to more aesthetically pleasing and clutter-free visualizations.
Optimizes graph spectral density learning for large networks.
problem Ad-hoc kernel function and bandwidth selection in graph spectral techniques.
method Maximum Entropy approach to learn a smooth graph spectral density.
result Outperforms comparable iterative spectral approaches on synthetic and real graphs.
The paper finds optimal threshold strategies for insurance companies with a positive terminal value at creeping ruin.
problem Optimizing dividend payments in an insurance company's surplus process with a positive terminal value at creeping ruin.
method Using fluctuation theory, the paper derives explicit formulas for the objective function and shows the optimality of threshold strategies.
result Threshold strategies are optimal for the dividend optimization problem under certain conditions.
Improves learning of spectral mixture kernels with approximate Bayesian inference.
problem Difficult optimization of large number of SM kernel parameters.
method Approximate Bayesian inference using variational distribution of spectral points and random Fourier features.
result Accelerates convergence and leads to better optimal parameters.
We propose a new reinforcement learning algorithm for partially observable Markov decision processes (POMDP) based on spectral decomposition methods. While spectral methods have been previously employed for consistent learning of (passive) latent variable models such as hidden Markov models, POMDPs are more challenging…
Spectral algorithms improve under covariate shift with novel weighted techniques.
problem Improving spectral algorithms' performance under covariate shift.
method Analysis of spectral algorithms in non-parametric regression over RKHS, proposing a weighted spectral algorithm with clipped weights.
result Normalized weighted spectral algorithm achieves optimal capacity-independent convergence rates, and clipped weights can approach optimal capacity-dependent rates.
We propose a new reinforcement learning algorithm for partially observable Markov decision processes (POMDP) based on spectral decomposition methods. While spectral methods have been previously employed for consistent learning of (passive) latent variable models such as hidden Markov models, POMDPs are more challenging…
Stochastic optimization problems often involve the expectation in its objective. When risk is incorporated in the problem description as well, then risk measures have to be involved in addition to quantify the acceptable risk, often in the objective. For this purpose it is important to have an adjusted, adapted and eff…
Rewiring GNNs to optimize community and feature alignment improves their performance.
problem Improving GNNs' performance by addressing over-squashing and generalization issues.
method Three rewiring strategies: ComMa, FeaSt, and ComFy, targeting community structure, node labels, and their alignment.
result Rewiring strategies enhance GNNs' performance by optimizing label-community alignment.
Study minimizes risk in MDPs with spectral measures.
problem Minimizing risk in MDPs with spectral measures.
method Splitting into inner and outer minimization problems; solving inner as MDP; proving existence for outer.
result Existence and solution methods for the outer minimization problem.
The paper analyzes the generalization performance of spectral clustering algorithms and proposes new methods to improve their effectiveness.
problem Theoretical analysis of spectral clustering's generalization performance.
method Theoretical analysis and development of new spectral clustering algorithms.
result The excess risk bounds of spectral clustering algorithms have a O ( 1 / n ) \mathcal{O}(1/\sqrt{n}) O ( 1/ n ) convergence rate. The paper analyzes how sampling design affects machine learning model generalization.
problem The impact of sampling properties on machine learning model generalization.
method Spectral analysis of the generalization error in Euclidean space using Fourier analysis.
result Estimation of expected error bounds and convergence rates for various sampling patterns.
We introduce a new parameterization method for deep learning layers using spectral tensor train decomposition.
problem Efficiency and stability in deep learning models with weight matrix compression.
method Spectral Tensor Train Parameterization (STTP) of weight matrices.
result Improved compression and training stability in neural networks.
This paper provides theoretical guarantees for spectral clustering using graph cuts.
problem Lack of performance guarantees for spectral clustering.
method Convex relaxation of graph cuts, spectral proximity condition, algebraic connectivity, inter-cluster connectivity.
result Deterministic bounds for successful spectral clustering are derived.
Develops an ℓ_p theory for PCA and spectral clustering.
problem Lack of precise characterizations of PCA scores for low-dimensional embedding.
method An ℓ_p perturbation theory for PCA in Hilbert spaces, analyzing eigenvectors and Gram matrix.
result Optimal recovery results for Gaussian mixture and stochastic block models.
This note is devoted to optimal spectral estimates for Schrödinger operators on compact connected Riemannian manifolds without boundary. These estimates are based on the use of appropriate interpolation inequalities and on some recent rigidity results for nonlinear elliptic equations on those manifolds.
Study confirms learning rates for vector-valued spectral algorithms, proving consistency.
problem Theoretical confirmation of learning rates for vector-valued spectral algorithms.
method Rigorous analysis of learning rates for various vector-valued spectral algorithms, including kernel ridge regression and gradient descent.
result Upper and lower bounds on learning rates for vector-valued spectral algorithms, proving minimax optimality in various scenarios.
In this paper we consider a modified version of the classical optimal dividends problem of de Finetti in which the dividend payments subject to a penalty at ruin. We assume that the risk process is modeled by a general spectrally positive Levy process before dividends are deducted. Using the fluctuation theory of spect…
Gradient descent with small random init mimics spectral methods for low-rank matrix recovery.
problem Reconstructing a low-rank matrix from few measurements.
method Gradient descent with small random initialization followed by a few iterations.
result Gradient descent from small random init converges to a well-generalizing solution.