Muon with Newton-Schulz converges to the same stationary point as SVD-polar, up to a constant factor.
problem Improving the convergence rate of Muon optimizer.
method Using Newton-Schulz steps for momentum orthogonalization, proving convergence rate and constant factor.
result Muon with Newton-Schulz converges to the same stationary point as SVD-polar, up to a constant factor.
NS-RGS improves orthogonal group synchronization with faster convergence.
problem Orthogonal group synchronization from pairwise measurements.
method Newton-Schulz iteration for Riemannian gradient optimization.
result NS-RGS achieves linear convergence and near-optimal accuracy.
The paper proves stability of minimal embeddings in spheres and relates it to Yau's conjecture.
problem Stability of minimal embeddings in spheres and Yau's conjecture.
method Analyzes stability index and solves differential equation to relate to Yau's conjecture.
result Stability index of minimal hypersurfaces is at least n^2+4n+3 and Yau's conjecture holds under specific conditions.
HyperINF improves influence function estimation for large models with better accuracy and efficiency.
problem Inaccurate and computationally expensive influence function estimation for large-scale models.
method HyperINF leverages Schulz's iterative algorithm and GFIM for low-rank approximation of Hessian matrix.
result HyperINF achieves superior accuracy and performance compared to existing methods on LoRA-tuned models.
The paper refines the stability index for a specific minimal hypersurface and verifies Yau's conjecture.
problem Stability of minimal hypersurfaces in spheres and eigenvalue multiplicity.
method Analytical and numerical methods to study eigenvalues and stability indices.
result The multiplicity of the eigenvalue for the Carlotto-Schulz minimal embedding is at least 2n+1+n^2.
Method recovers complex-valued signals from speckle-noised measurements.
problem Recovering complex-valued signals from speckle-noised measurements.
method Bagged Deep Image Priors integrated with projected gradient descent and Newton-Schulz algorithm.
result Achieves state-of-the-art performance in MSE reduction.
AuON is a linear-time optimizer that improves upon Muon's performance without approximate orthogonal matrices.
problem High memory and computational costs of orthogonal momentum updates.
method AuON uses normalized nonlinear scaling and a 'emergency brake' to handle exploding attention logits.
result AuON achieves strong performance without approximate orthogonal matrices, preserving structural alignment and reconditioning.
MuonEq improves training of matrix-valued parameters by rebalancing momentum before orthogonalization.
problem Training matrix-valued parameters with orthogonalized-update optimizers like Muon.
method MuonEq introduces three lightweight pre-orthogonalization equilibration schemes: two-sided row/column normalization (RC), row normalization (R), and column normalization (C).
result Row/column normalization acts as a zeroth-order surrogate for whitening and improves the geometry seen by orthogonalization.
Paper proves unique conical tangent flows in all dimensions.
problem Proving uniqueness of conical singularities in mean curvature flows.
method Based on Chodosh-Schulze's work in hypersurfaces, extended to all codimensions.
result Uniqueness of asymptotically conical tangent flows in all codimensions.
The paper addresses speckle noise in coherent imaging systems.
problem Speckle noise degrades image quality in coherent imaging systems.
method Theoretical and algorithmic analysis of likelihood-based approaches for multilook coherent imaging.
result Established the first theoretical upper bound on MSE of the maximum likelihood estimator.
Proves uniqueness of blowups for forced mean curvature flow.
problem Proving uniqueness of blowups for forced mean curvature flow.
method Adapting methods from Euclidean space mean curvature flow to handle forcing term and blow-up limits.
result Uniqueness of tangent cones for forced mean curvature flow at self-shrinkers and cylindrical self-shrinkers.
We study and solve the Dirichlet problem for graphs of prescribed mean curvature in Rn+1 over general domains Ω without requiring a mean convexity assumption. By using pieces of nodoids as barriers we first give sufficient conditions for the solvability in case of zero boundary values. Applying a result …
Three-manifolds with non-negative pinched Ricci curvature have complete Ricci flows.
problem Proving Hamilton's pinching conjecture for three-manifolds.
method Ricci flow with scale-invariant curvature decay and pinching preservation.
result Hamilton's pinching conjecture is proven without additional hypotheses.
Study proves Łojasiewicz inequalities for self-shrinkers, aiding in their uniqueness.
problem Proving uniqueness of self-shrinkers in codimension.
method Analyzes product of round shrinking spheres, proving Łojasiewicz inequalities.
result Explicit Łojasiewicz inequalities near self-shrinkers, leading to convergence rates.
New foliations found for critical surfaces of Hawking energy, resolving discrepancies.
problem Finding consistent critical surfaces for the Hawking energy in non-totally geodesic spacelike hypersurfaces.
method Constructing a unique local foliation of area constrained critical surfaces of the Hawking energy in the general case of non-totally geodesic spacelike hypersurfaces.
result Discrepancy found in the small sphere limit of the Hawking energy, explained and resolved.
Explains mapping properties of elliptic operators in conical spaces.
problem Understanding mapping properties of geometric elliptic operators in conical spaces.
method Develops an approach based on B.-W. Schulze's work.
result Illustrates versatility of results in Geometric Analysis.
Study shows expanding Ricci solitons from specific metric cones.
problem Analyzing Ricci flows from weakly PIC1 metric cones.
method Complete weakly PIC1 Ricci flows with Euclidean volume growth.
result Ricci flows must be expanding gradient Ricci solitons.
Space partitions of Rd underlie a vast and important class of fast nearest neighbor search (NNS) algorithms. Inspired by recent theoretical work on NNS for general metric spaces [Andoni, Naor, Nikolov, Razenshteyn, Waingarten STOC 2018, FOCS 2018], we develop a new framework for building space partitions re…
Perturbs area-minimizing hypersurfaces to reduce singular set's dimension.
problem Reduces the dimension of the singular set of area-minimizing hypersurfaces.
method Perturbs a smooth hypersurface to minimize the Minkowski dimension of the singular set.
result The singular set of the perturbed minimizing current has Minkowski dimension less than n-9.
We prove the existence of self-similar expanding solutions of the curvature flow on planar networks where the initial configuration is any number of half-lines meeting at the origin. This generalizes recent work by Schnürer and Schulze which treats the case of three half-lines. There are multiple solutions, and these a…
Study stabilizes translating solitons in hyperbolic space for MCF.
problem Stability of translating solitons in hyperbolic space.
method Developed theory, constructed rotationally invariant translators, used avoidance principle and maximum principle.
result Horospheres are dynamically stable as radial graphical solutions to MCF.
We study the area preserving Willmore flow in an asymptotic region of an asymptotically flat manifold which is C3−close to Schwarzschild. It was shown by Lamm, Metzger and Schulze that such an end is foliated by spheres of Willmore type. In this paper, we prove that the leaves of this foliation are stable under sm…
The paper finds new constant mean curvature hypersurfaces in spheres.
problem Finding new constant mean curvature hypersurfaces in spheres.
method Analyzing hypersurfaces of specific types in spheres with given symmetries.
result Existence of new compact embedded CMC-hypersurfaces in spheres.
This paper concerns the evolution of a closed convex hypersurface in Rn+1, in direction of its inner unit normal vector, where the speed is given by a smooth function depending only on the mean curvature, and satisfies some further restrictions, without requiring homogeneity. It is shown that the flow e…
Authors construct hypertori with constant negative mean curvature in a sphere.
problem Constructing constant mean curvature hypertori in a sphere.
method Constructing two different constant mean curvature (2n−1)-dimensional hypertori in a 2n-dimensional sphere. result Two different constant mean curvature (2n−1)-dimensional hypertori with negative mean curvature in a 2n-dimensional sphere. New proof shows 3-manifolds with Ricci curvature bound are either flat or grow non-Euclidean.
problem Proving properties of 3-manifolds with Ricci curvature bounds.
method Inverse mean curvature flow approach.
result Proves that 3-manifolds are either flat or have non-Euclidean volume growth.
Proves mean curvature flow from conical singularities to shrinkers.
problem Proving mean curvature flow from conical singularities to shrinkers.
method Ważewski box argument
result Existence of embedded closed hypersurface with mean curvature flow.
This paper extends 3D results to higher dimensions, proving compactness for PIC1 pinched manifolds.
problem Proving compactness for higher-dimensional manifolds with specific curvature conditions.
method Constructing Ricci flows for non-compact PIC1 pinched manifolds to prove compactness.
result Proves that PIC1 pinched manifolds of non-negative complex sectional curvature must be flat or compact.
A 3-manifold's Ricci pinching condition implies it's flat if it has Euclidean volume growth.
problem Understanding the flatness of 3-manifolds under Ricci pinching conditions.
method Alternative proof using potential theory.
result If a 3-manifold has Euclidean volume growth and satisfies the Ricci pinching condition, it is flat.
New theory maps neural network weights to optimize faster and scale.
problem Optimizing neural networks for speed and scalability.
method Constructing a duality map using layer-wise operator norms.
result Derived GPU-friendly algorithms for various layers.
The paper shows mean curvature flow keeps diameter bounded under certain conditions.
problem Proving the bounded diameter of hypersurfaces under mean curvature flow.
method Use of Lojasiewicz inequalities and solution of mean-convex neighbourhood conjecture.
result The intrinsic diameter stays uniformly bounded as the flow approaches the first singular time.
In this paper we establish stability results for symmetric spaces of noncompact type under Ricci flow, i.e. we will show that any small perturbation of the symmetric metric is flown back to the original metric under an appropriately rescaled Ricci flow. It will be important for us which smallness assumptions we have to…
Paper proves short-time existence for network flow, providing detailed insights.
problem Short-time existence for the flow of a network of curves in the plane.
method Direct PDE approach, handling singularities at vertices using self-similar expanding solutions.
result Substantially more detailed information about network resolution into a regular one.
Proves existence and uniqueness of curvature motion for regular networks.
problem Existence and uniqueness of motion by curvature for regular networks.
method Proves existence and uniqueness using $W^{2- rac{2}{p}}_p$ initial data and investigates regularization effects.
result Proves existence and uniqueness of motion by curvature for regular networks.
The paper studies eigenvalues and stability of hypersurfaces in spheres.
problem Finding eigenvalues and stability of hypersurfaces in spheres.
method Derives equations for mean curvature and uses numerical methods to compute eigenvalues.
result Numerical computation of eigenvalues and stability indices for specific hypersurfaces.
Study on network flow singularities, focusing on Type-0 singularities.
problem Understanding singularities in network flow evolution.
method Analysis of curvature evolution and junction behavior.
result Bounded curvature for Type-0 singularities in network flow.
Improved tensor GLM estimation for complex data.
problem Complex tensor data in GLMs leads to high-dimensional, ill-posed estimation.
method Proposed LSRTR-M algorithm using Muon updates for faster convergence and lower errors.
result LSRTR-M converges faster and achieves lower errors than LSRTR.
Proves multiplicity one conjecture for surface flows, with applications in flow regularity.
problem Proving multiplicity one for mean curvature flows of surfaces.
method Analyzing blow-up limits and using level set flow properties.
result Blow-up limits of mean curvature flows have multiplicity one.
New calculus for pseudodifferential operators on manifolds with cylindrical ends.
problem Analyzing layer potentials on manifolds with cylindrical ends.
method Introducing and studying two classes of pseudodifferential operators.
result Spectrally invariant property of the 'essentially translation invariant calculus'.
The study finds translators for higher order mean curvature flows in Euclidean and hyperbolic spaces.
problem Finding translators for higher order mean curvature flows in different spaces.
method Analyzing velocity functions of translators to r-mean curvature flows in RnimesR and HnimesR. result Existence and uniqueness of translators, including bowl-type, catenoid-type, and Grim Reaper-type translators.
Improved iterative methods for risk parity portfolio weights.
problem Solving for portfolio weights in risk parity allocation.
method Enhanced CCD and Newton methods, including a rescaling step and improved initial guess.
result Improved CCD method is the best, three times faster with 40% fewer iterations.
We describe a novel optimization method for finite sums (such as empirical risk minimization problems) building on the recently introduced SAGA method. Our method achieves an accelerated convergence rate on strongly convex smooth problems. Our method has only one parameter (a step size), and is radically simpler than o…
A new method combines Laplace and Variational Bayes for scalable inference.
problem Complex models and large datasets make exact inference infeasible.
method Low-Rank Variational Bayes Correction (VBC) using Laplace method and Variational Bayes correction in a lower dimension.
result The method ensures scalability in both model complexity and data size.
Unified framework for model explanation methods based on feature removal.
problem Unclear relationships and preferences among various model explanation methods.
method Characterizes removal-based explanations along three dimensions.
result Unified 26 existing methods, including widely used approaches.
This work reviews and evaluates methods for predicting prediction intervals in regression problems.
problem Calibration of prediction intervals in regression problems.
method Four classes of methods: Bayesian, ensemble, direct interval estimation, and conformal prediction.
result Conformal prediction can be used as a general calibration procedure.
Derives kernel PCA with Nyström method for scalability.
problem Scalability of kernel PCA.
method Nyström method for kernel PCA.
result Provides scalable alternative to full kernel PCA.
In this paper, the author considers the numerical computation of CVA for large systems by Mote Carlo methods. He introduces two types of stochastic mesh methods for the computations of CVA. In the first method, stochastic mesh method is used to obtain the future value of the derivative contracts. In the second method, …
New method combines spectral and sparse methods for Gaussian processes.
problem Efficiently fitting Gaussian processes to large datasets.
method Orthogonally decoupled variational Fourier features.
result Competitive performance on synthetic and real-world data.