In all dimensions and arbitrary signature, we demonstrate the existence of a new local potential -- a double (2,3)-form -- for the Weyl curvature tensor, and more generally for all tensors with the symmetry properties of the Weyl curvature tensor. The classical four-dimensional Lanczos potential for a Weyl tensor -- a …
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48 results for “Lanczos Potential”
We exploit four-dimensional tensor identities to give a very simple proof of the existence of a Lanczos potential for a Weyl tensor in four dimensions with any signature, and to show that the potential satisfies a simple linear second order differential equation, e.g., a wave equation in Lorentz signature. Furthermore,…
LanczosNet uses Lanczos algorithm for efficient graph convolution.
problem Efficiently capturing multi-scale information in graph data.
method Lanczos algorithm for low-rank graph Laplacian approximation, learnable spectral filters.
result LanczosNet achieves state-of-the-art performance on citation networks and QM8.
New insights into gravitational waves and Riemann/Weyl operators using computer algebra.
problem Understanding the existence and structure of gravitational waves in General Relativity.
method Explicit computation of extension modules for differential sequences.
result Revisits C. Lanczos's work on Riemann/Weyl operators and their adjoints.
Paper introduces fast botnet detection from web server logs.
problem Botnets cause various malicious activities; fast detection is needed.
method Inspired by PCA, uses Lanczos method to improve detection speed.
result Lanczos method significantly reduces time complexity for botnet detection.
Efficiently differentiate functions of large matrices using new adjoint systems.
problem Differentiating functions of large matrices in scientific and probabilistic machine learning models.
method Deriving and implementing new adjoint systems for Lanczos and Arnoldi iterations in JAX.
result Efficient differentiation of PDEs, Gaussian process models, and Bayesian neural networks.
A new method speeds up Gaussian process covariance computations.
problem Efficiently computing predictive covariance in Gaussian processes.
method Lanczos algorithm for rapid predictive covariance matrix approximation.
result 2,000 times faster computation of predictive covariances.
A new method tackles bilevel optimization using Lanczos process for efficient hyper-gradient computation.
problem Efficiently solving large-scale bilevel optimization problems with gradient-based methods.
method Constructing low-dimensional approximate Krylov subspaces with the Lanczos process to approximate the Hessian inverse vector product.
result Demonstrates a convergence rate and efficiency in synthetic and deep learning tasks.
New scalable methods for log determinant computations speed up Gaussian process kernel learning.
problem Prohibitive computational cost of log determinant calculations for Gaussian process kernel learning.
method Stochastic approximations based on Chebyshev, Lanczos, and surrogate models.
result Lanczos method is superior for kernel learning, and surrogate models are highly efficient and accurate.
We introduce a weighted de Rham operator which acts on arbitrary tensor fields by considering their structure as r-fold forms. We can thereby define associated superpotentials for all tensor fields in all dimensions and, from any of these superpotentials, we deduce in a straightforward and natural manner the existence …
The study examines algebraic structures of specific tensor forms in four-dimensional spacetimes.
problem Investigating algebraic features of certain tensor forms in spacetimes.
method General treatment followed by specialization to four-dimensional spacetimes, focusing on invariant subspaces and generalizing relations.
result Generalized relations such as the Ruse-Lanczos identity, Bel-Matte decomposition, and Lovelock-like quadratic identities.
The main purpose of this paper is to revisit the well known potentials, called stress functions, needed in order to study the parametrizations of the stress equations, respectively provided by G.B. Airy (1863) for 2-dimensional elasticity, then by E. Beltrami (1892), J.C. Maxwell (1870) and G. Morera (1892) for 3-dimen…
A Matrix--free Likelihood Method for Exploratory Factor Analysis of High-dimensional Gaussian Datastat.ME
A new method speeds up factor analysis for high-dimensional data.
problem Estimating covariance parameters in high-dimensional Gaussian data with limited observations.
method Matrix-free likelihood method using implicitly restarted Lanczos and limited-memory quasi-Newton algorithms.
result Our method is faster than EM without sacrificing accuracy.
New recommendations improve Gaussian process accuracy and stability.
problem Numerical instabilities and poor test likelihoods in iterative Gaussian process learning.
method Investigated CG tolerance, preconditioner rank, and Lanczos decomposition rank. Recommended small CG tolerance and large root decomposition size.
result L-BFGS-B optimizer achieves convergence with fewer gradient updates, improving Gaussian process accuracy.
The runtime for Kernel Partial Least Squares (KPLS) to compute the fit is quadratic in the number of examples. However, the necessity of obtaining sensitivity measures as degrees of freedom for model selection or confidence intervals for more detailed analysis requires cubic runtime, and thus constitutes a computationa…
The paper analyzes contraction rates for GP regression approximations.
problem Computational infeasibility of exact GP posterior in large-scale applications.
method Lanczos and conjugate gradient approximations of the posterior mean.
result Minimax contraction rates for these approximations in large-scale applications.
Deep Curvature Suitestat.ML
Deep Curvature Suite offers a PyTorch package for neural network curvature analysis.
problem Insufficient use of curvature information in neural networks.
method Implementation of Lanczos algorithm for neural network curvature analysis.
result Our package outperforms existing methods for similar purposes.
The purpose if this master's thesis is to study and develop a new algorithmic framework for Collaborative Filtering to produce recommendations in the top-N recommendation problem. Thus, we propose Lanczos Latent Factor Recommender (LLFR); a novel "big data friendly" collaborative filtering algorithm for top-N recommend…
A simple sketch improves online eigenvector and SDP problems.
problem Online eigenvector and semidefinite programming problems.
method Randomized mirror projection and mirror descent analysis.
result Regret bounds similar to MMW with reduced complexity.
Scalable Krylov Subspace Methods for Generalized Mixed-Effects Models with Crossed Random Effectsstat.ME
New Krylov subspace methods speed up mixed-effects models with crossed random effects.
problem Slow computations for high-dimensional crossed random effects in mixed-effects models.
method Krylov subspace-based methods for generalized mixed-effects models with cross effects.
result Speedups by factors of up to 10,000 in computations for mixed-effects models.
We develop and analyze efficient "coordinate-wise" methods for finding the leading eigenvector, where each step involves only a vector-vector product. We establish global convergence with overall runtime guarantees that are at least as good as Lanczos's method and dominate it for slowly decaying spectrum. Our methods a…
A Novel Skew-Hamiltonian Isotropic Lanczos Algorithm for Spectral Conformal Parameterizationsmath.NA
Numerous methods for computing conformal mesh paramterizations has been developed due to the vast applications in the field of geometry processing. Spectral conformal parameterization (SCP) is one of these methods to computing a quality conformal parameterization based on the spectral technique. SCP focus on a generali…
Fast matrix-vector products for large graph Laplacians using NFFT.
problem Efficient computation of matrix-vector products for large, densely populated graph Laplacians.
method Combining NFFT with Krylov methods and iterative solvers.
result Significant speedup in computations for graph Laplacians.
We analyze the Hessian spectra of large models up to 100B parameters.
problem Accurate Hessian spectra of large foundation models are difficult to obtain.
method We use shard-local finite-difference Hessian vector products and stochastic Lanczos quadrature.
result We produce the first large-scale spectral density estimates of foundation models.
On Gauss-Bonnet Curvaturesmath.DG
The -th Gauss-Bonnet curvature is a generalization to higher dimensions of the -dimensional Gauss-Bonnet integrand, it coincides with the usual scalar curvature for . The Gauss-Bonnet curvatures are used in theoretical physics to describe gravity in higher dimensional space times where they are known a…
Quadratic model surprisingly predicts optimization dynamics in large neural networks.
problem Complexity of neural network loss landscapes and optimization dynamics.
method Stress testing the quadratic model, Taylor expansion, Lanczos quadrature, and local linear stability analysis.
result The quadratic model can accurately predict optimization dynamics over long windows in large neural networks.
A simple power iteration with momentum achieves optimal PCA in stochastic settings.
problem Accelerating PCA in the stochastic setting with limited data.
method A simple variant of the power iteration with momentum.
result Achieves optimal sample and iteration complexity of .
Match van Stockum dust to vacuum metrics with a single parameter.
problem Matching van Stockum dust to vacuum metrics.
method 1-parametric family of non-static Papapetrou vacuum metrics, Ehlers and Kramer--Neugebauer transformations.
result Explicit examples of matching, including Bonnor metric and Lanczos--van Stockum dust metric.
A new method reduces log determinant evaluation cost from cubic to quadratic.
problem Efficiently evaluating log determinants in machine learning.
method Variational Bayesian approximation with complexity O(n^2).
result State-of-the-art performance on synthetic and real-world datasets.
Study critical metrics on Riemannian manifolds, finding new minimizers and rigidity results.
problem Investigate critical metrics of higher-order curvature functionals on compact Riemannian manifolds.
method Develop variational framework using double forms and generalize Lanczos identity.
result Critical -Thorpe and -anti-Thorpe metrics are absolute minimizers of in the critical dimension .
A scalable end-to-end Gaussian process adapter for irregularly sampled time series classificationstat.ML
We present a general framework for classification of sparse and irregularly-sampled time series. The properties of such time series can result in substantial uncertainty about the values of the underlying temporal processes, while making the data difficult to deal with using standard classification methods that assume …
HessFormer enables distributed Hessian computation for large models.
problem Limited Hessian computation for large deep learning models.
method Distributed stochastic Lanczos quadrature algorithm.
result Investigated Hessian spectral density of a 70 billion parameter model.
HybridSVD combines user and item info for efficient, flexible recommendations.
problem Lack of effective methods for incorporating both user and item side information in collaborative filtering.
method Hybrid algorithm using PureSVD with generalized singular value decomposition and cold start solution.
result Superior performance compared to similar hybrid models on various datasets.
Beyond Accuracy: A Unified Random Matrix Theory Diagnostic Framework for Crash Classification Modelscs.LG
Unified framework detects overfitting in crash classification models.
problem Evaluation metrics fail to detect overfitting in crash classification models.
method Random Matrix Theory and Heavy-Tailed Self-Regularization framework applied to various model types.
result Power-law exponent α reliably distinguishes well-regularized from overfit models.
New methods improve anomaly detection from large streaming data.
problem Anomaly detection in large streaming data fails with current methods.
method Two novel randomized algorithms (rPS and gPS) for better detection of correlated anomalies.
result High and balanced recall and estimated accuracy for anomaly detection.
The purpose of this paper is to revisit the Bianchi identities existing for the Riemann and Weyl tensors in the combined framework of the formal theory of systems of partial differential equations (Spencer cohomology, differential systems, formal integrability) and Algebraic Analysis (homological algebra, differential …
Efficient approximations reduce computation of matrix-based Renyi's entropy.
problem High computational complexity of matrix-based Renyi's entropy.
method Taylor, Chebyshev, and Lanczos approximations to reduce complexity.
result Reduced complexity to significantly less than with negligible accuracy loss.
Convergence Rate of Block-Coordinate Maximization Burer-Monteiro Method for Solving Large SDPsmath.OC
New algorithm solves large SDPs with provable convergence guarantees.
problem Solving large SDPs with diagonal constraints efficiently.
method Block-coordinate maximization applied to Burer-Monteiro method.
result Global convergence to first-order stationary point with sublinear rate.
New insights into learning rates and batch sizes for neural networks using random matrix theory.
problem Understanding how batch size affects learning rates in neural networks.
method Random matrix theory applied to spiked, field-dependent random matrices.
result Analytical expressions for maximal learning rates as a function of batch size.
A fast spectral algorithm estimates mean of heavy-tailed vectors efficiently.
problem Estimating the mean of heavy-tailed random vectors with optimal error bound.
method Spectral algorithm using eigenvector computations and novel hyperplane connection.
result Achieves optimal sub-gaussian error bound with improved runtime.
This paper deals with finding an -dimensional solution to a system of quadratic equations of the form for , which is also known as phase retrieval and is NP-hard in general. We put forth a novel procedure for minimizing the amplitude-based least-squares empirical los…
In this article we discuss the distribution of asset price movements by the market potential function. From the principle of free energy minimization we analyze two different kinds of market potentials. We obtain a U-shaped potential when market reversion (i.e. contrarian investors) is dominant. On the other hand, if t…
Develops potential theory for WZW equation in Kähler potentials space.
problem Solving the Wess--Zumino--Witten equation in Kähler potentials.
method Introduces -harmonicity on graphs to characterize the WZW equation and uses subharmonic distance.
result Shows solvability of Dirichlet problem and approximation by finite-dimensional maps.
The paper examines stability of harmonic and symphonic maps with forms and potentials.
problem Stability of harmonic and symphonic maps with forms and potentials.
method Analyzes stability of -harmonic and -symphonic maps with forms and potentials.
result Stability conditions for harmonic and symphonic maps are established.
Extracts interpretable potential energy from Hamiltonian systems.
problem Learning an interpretable potential energy function from Hamiltonian systems.
method Constructs a neural network model of the potential and applies equation discovery to extract a closed-form algebraic expression.
result Close agreement between learned neural potentials and ground truth potentials, including correct effective potential for a central force problem.
The paper examines stability of subelliptic harmonic maps with potential.
problem Stability of subelliptic harmonic maps with potential.
method Derived first and second variation formulas, proved stability conditions, and gave instability results.
result Subelliptic harmonic maps with potential are stable under certain curvature and potential conditions.
The paper describes flat Hessian metrics on surfaces and their potentials.
problem Understanding Hessian metrics on surfaces.
method Theoretical description and explicit construction using integrable systems.
result Explicit construction of potentials for flat Hessian metrics on surfaces.
A hyperKähler potential is a function rho that is a Kähler potential for each complex structure compatible with the hyperKähler structure. Nilpotent orbits in a complex simple Lie algebra are known to carry hyperKähler metrics admitting such potentials. In this paper, we explicitly calculate the hyperKähler potential w…