A new method for spectral positional encodings in directed graphs using Hermitian block Krylov subspaces.
problem Challenges in spectral positional encodings for directed graphs, including computational complexity and gauge invariance issues.
method Learnable spectral positional encodings of the form hθ(Aq)R, computed in a Hermitian block Krylov subspace from sparse matrix-vector products. result The method is gauge-invariant and converges to the exact eigendecomposition oracle as the depth grows.
New method for directed graphs using learnable spectral positional encodings.
problem Challenges in magnetic Laplacians and unitary gauge invariance for directed graphs.
method Learnable spectral PEs of the form hθ(Aq)R, computed in Hermitian block Krylov subspace.
result Gauge-invariant and computationally efficient solution for directed graphs.
Enhances gradient estimates for Hermitian Monge-Ampère equations.
problem Improving estimates for Hermitian Monge-Ampère equations.
method Improves gradient estimates using Evans-Krylov and third derivatives estimates.
result Enhanced estimates for second and third order derivatives.
Kaczmarz++ accelerates convergence for ill-conditioned systems.
problem Solving ill-conditioned linear systems efficiently.
method Adaptive momentum acceleration, Tikhonov-regularized projections, and memoization.
result Kaczmarz++ converges faster than Krylov methods on ill-conditioned systems.
Proposes a new regularizer for semi-supervised learning on multilayer graphs.
problem Semi-supervised learning on multilayer graphs with labeled and unlabeled data.
method Generalized matrix mean regularizer and matrix-free numerical scheme.
result The regularizer outperforms state-of-the-art methods numerically.
New lower bounds for sampling from log-concave distributions in higher dimensions.
problem Proving lower bounds for sampling from log-concave distributions in higher dimensions.
method Multiscale construction inspired by geometric measure theory and reduction to block Krylov algorithms.
result Query lower bounds for sampling from log-concave distributions in higher dimensions are established.
A new diffusion sampling method combines Krylov subspace and diffusion models for faster and more efficient inverse problems.
problem Efficiently solving large-scale inverse problems in high-performance computing.
method Proposes a novel diffusion sampling strategy that integrates Krylov subspace methods with diffusion models.
result Demonstrates significant speedup (80x faster inference time) and improved reconstruction quality on real-world medical imaging problems.
We establish Evans-Krylov estimates for certain nonconvex fully nonlinear elliptic and parabolic equations by exploiting partial Legendre transformations. The equations under consideration arise in part from the study of the "pluriclosed flow" introduced by the first author and Tian
Computes isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
problem Computing isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
method Algorithm for solving a matrix equation to compute isotropy subgroups.
result Computed isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
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.
Note on advancements in nonlinear elliptic equations' regularity theory.
problem Nonlinear elliptic equations and their regularity.
method De Giorgi-Nash-Moser theory, Krylov-Safonov theory, Evans-Safonov theory.
result Contributions to Hilbert's 19th problem and fully nonlinear equations.
We study k-SVD that is to obtain the first k singular vectors of a matrix A. Recently, a few breakthroughs have been discovered on k-SVD: Musco and Musco [1] proved the first gap-free convergence result using the block Krylov method, Shamir [2] discovered the first variance-reduction stochastic method, and Bhoj…
We compared the regular Singular Value Decomposition (SVD), truncated SVD, Krylov method and Randomized PCA, in terms of time and space complexity. It is well-known that Krylov method and Randomized PCA only performs well when k << n, i.e. the number of eigenpair needed is far less than that of matrix size. We compared…
KBB algorithm reduces sample complexity for policy evaluation in general state spaces.
problem Policy evaluation in large state spaces with high sample complexity.
method Alternates between fitting Bellman residual and estimating value function via adaptive feature set growth.
result Super-linear convergence rates demonstrated, with reductions in sample complexity.
Semisimplicity proven for conformal blocks representations.
problem Semisimplicity of conformal blocks representations.
method Theory of extensions in non-Abelian Hodge theory and Ocneanu rigidity.
result Semisimplicity of braid group and mapping class group representations.
Parallel computing has played an important role in speeding up convex optimization methods for big data analytics and large-scale machine learning (ML). However, the scalability of these optimization methods is inhibited by the cost of communicating and synchronizing processors in a parallel setting. Iterative ML metho…
Recently, neural network based approaches have achieved significant improvement for solving large, complex, graph-structured problems. However, their bottlenecks still need to be addressed, and the advantages of multi-scale information and deep architectures have not been sufficiently exploited. In this paper, we theor…
We describe how to use the perturbation theory of Caffarelli to prove Evans-Krylov type C2,α estimates for solutions of nonlinear elliptic equations in complex geometry, assuming a bound on the Laplacian of the solution. Our results can be used to replace the various Evans-Krylov type arguments in the complex geom…
Hessian-free training has become a popular parallel second or- der optimization technique for Deep Neural Network training. This study aims at speeding up Hessian-free training, both by means of decreasing the amount of data used for training, as well as through reduction of the number of Krylov subspace solver iterati…
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 O(ε−1) convergence rate and efficiency in synthetic and deep learning tasks. Diagonal Frog: High-order positivity-preserving FD schemes for anisotropic Fokker-Planck equations
problem Positivity-preserving discretizations for anisotropic Fokker-Planck equations
method Diagonal Frog discretization
result Second-order accuracy and mass conservation
In this paper, we propose a second order optimization method to learn models where both the dimensionality of the parameter space and the number of training samples is high. In our method, we construct on each iteration a Krylov subspace formed by the gradient and an approximation to the Hessian matrix, and then use a …
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).
Using the concept of a cohesive module defined by Block, we use the theory of superconnections in the sense of Quillen to construct natural superconnections on Hermitian cohesive modules. By the Chern-Weil construction, we obtain characteristic classes with values in Bott-Chern cohomology which refines the usual deRham…
In this paper, we study the solvability of a general class of fully nonlinear curvature equations, which can be viewed as generalizations of the equations for Christoffel-Minkowski problem in convex geometry. We will also study the Dirichlet problem of the corresponding degenerate equations as an extension of the equat…
AI-driven framework optimizes MCMC-based preconditioners for faster linear system solving.
problem Slow convergence of Krylov subspace solvers for ill-conditioned matrices.
method Graph neural surrogate and Bayesian optimization for AI-tuned MCMC parameters.
result 50% reduction in iterations to convergence on unseen system.
Efficiently computes matrix square roots and their inverses for large matrices.
problem Computing matrix square roots and inverses for large matrices efficiently.
method Combines Krylov subspace methods with rational approximation for quadratic-time computation.
result Achieves 4 decimal places of accuracy with fewer than 100 matrix-vector multiplications.
Paper solves Hessian equations on Kähler manifolds.
problem Solving Hessian equations on Kähler manifolds.
method Combines elementary symmetric functions; provides sufficient and necessary condition.
result Generalizes results for Hessian and Hessian quotient equations.
The memory capacity of linear echo state networks is accurately calculated using new numerical methods.
problem Numerical evaluations of memory capacity in recurrent neural networks often contradict theoretical bounds.
method Developed robust numerical approaches exploiting MC neutrality with respect to the input mask matrix.
result Memory curves fully agree with theory when using the proposed methods.
Conservation laws, heirarchies, scattering theory and Bäcklund transformations are known to be the building blocks of integrable partial differential equations. We identify these as facets of a theory of Poisson group actions, and apply the theory to the ZS-AKNS nxn heirarchy (which includes the non-linear Schrödinger …
Unified framework for multi-view learning with orthogonal projections.
problem Learning individual orthogonal projections for multiple views.
method Successive approximations via eigenvectors, iterative Krylov subspace method.
result Consistently competitive and often better than existing methods.
Paper proposes a new method for efficient second-order neural network training.
problem Infeasibility of Hessian calculation and noisy second-order information in deep learning.
method Adopting complex-step directional derivative (CSFD) for accurate Hessian computation and designing an effective Newton Krylov procedure.
result Our method outperforms existing methods and often converges one-order faster.
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.
New Bethe-Hessian method improves community detection in sparse networks.
problem Detect communities in sparse networks efficiently.
method Spectral clustering using the Bethe-Hessian matrix.
result Bethe-Hessian consistently estimates block number above Kesten-Stigum threshold.
Operator-theoretic analysis of nonlinear dynamical systems has attracted much attention in a variety of engineering and scientific fields, endowed with practical estimation methods using data such as dynamic mode decomposition. In this paper, we address a lifted representation of nonlinear dynamical systems with random…
The class of non-rigid registration methods proposed in the framework of PDE-constrained Large Deformation Diffeomorphic Metric Mapping is a particularly interesting family of physically meaningful diffeomorphic registration methods. PDE-constrained LDDMM methods are formulated as constrained variational problems, wher…
This paper constructs Brownian motion on complex flag manifolds and finds joint distribution of stochastic areas.
problem Modeling stochastic areas on complex partial flag manifolds.
method Constructs Brownian motion on complex partial flag manifolds and uses it to find joint distribution of stochastic areas.
result Limit law of stochastic areas is a multivariate Cauchy distribution.
New method speeds up kernel-based machine learning for force field reconstruction.
problem Scalability issues in kernel-based machine learning for force field reconstruction.
method Nyström-type methods to construct preconditioners based on low-rank approximations of the kernel matrix.
result Effective preconditioners lead to super-linear convergence in kernel-based machine learning.
This paper considers exponential utility indifference pricing for a multidimensional non-traded assets model subject to inter-temporal default risk, and provides a semigroup approximation for the utility indifference price. The key tool is the splitting method, whose convergence is proved based on the Barles-Souganidis…
Efficiently maps indoor magnetic fields with SKI and D-SKI.
problem Computing large-scale magnetic field maps in indoor environments.
method Structured kernel interpolation (SKI) with derivatives (D-SKI) for Gaussian process regression.
result Achieves better accuracy and faster computation than state-of-the-art methods.
The study proves inequalities for complex operators on curved spaces.
problem Establishing inequalities for nonlocal operators on curved spaces.
method Defining and analyzing nonlocal Pucci operators on manifolds with nonnegative sectional curvatures, proving Harnack inequalities and Holder estimates.
result Harnack inequalities and Holder estimates for nonlocal operators on manifolds with nonnegative sectional curvatures.
Local Neural Operators enable efficient system-level analysis of complex PDEs.
problem System-level analysis of large-scale dynamical systems using neural operators.
method Integrating local Neural Operators with Krylov subspace iterative methods for stability and bifurcation analysis.
result Demonstrated effectiveness of local Neural Operators in fixed-point, stability, and bifurcation analysis of nonlinear PDEs.
This paper tackles unpaired data in multi-view learning, proposing a new framework and models.
problem Handling unpaired data in multi-view learning, which is more common than paired data.
method Generalized uncorrelated multi-view subspace learning framework with successive alternating approximation (SAA) method.
result Proposed models perform competitively or better than baselines in multi-view feature extraction and multi-modality classification.
Holomorphic maps are a special case of Hermitian pluriharmonic maps between almost Hermitian manifolds.
problem Characterizing maps between almost Hermitian manifolds.
method Introducing Hermitian pluriharmonic maps and proving their properties.
result Holomorphic or anti-holomorphic maps are Hermitian pluriharmonic.
The paper proves Liouville theorems for holomorphic maps on pseudo-Hermitian manifolds.
problem Proving Liouville theorems for holomorphic maps on pseudo-Hermitian manifolds.
method Analyzing maps between different classes of pseudo-Hermitian manifolds, using curvature assumptions.
result Holomorphic maps are constant under certain curvature conditions.
Construct a Hermitian metric on non-Hermitian Yang--Mills moduli spaces near the Hermitian locus.
problem Moduli space of non-Hermitian Yang--Mills connections over a compact Kähler manifold
method Using normalized harmonic metrics
result Near the Hermitian locus, the unobstructed locus carries an almost hypercomplex structure compatible with the associated Riemannian metric.
Paper proves solutions for deformed Hermitian-Yang-Mills equation on almost Hermitian manifolds.
problem Existence of solutions for deformed Hermitian-Yang-Mills equation on almost Hermitian manifolds.
method Derive a priori estimates under the existence of an admissible C-subsolution; prove existence of solutions under the condition of existence of a supersolution. result Proves existence of solutions for the deformed Hermitian-Yang-Mills equation.
We study geometric realization questions of curvature in the affine, Riemannian, almost Hermitian, almost para Hermitian, almost hyper Hermitian, almost hyper para Hermitian, Hermitian, and para Hermitian settings. We also express questions in Ivanov-Petrova geometry, Osserman geometry, and curvature homogeneity in ter…