New algorithm approximates large psd matrices from sketches.
problem Large-scale positive-semidefinite matrices from streaming data.
method Combines Nystrom approximation with rank truncation.
result Achieves prescribed relative error in Schatten 1-norm.
Improved Nystrom method reduces landmark points for better kernel matrix approximations.
problem Poor performance and lack of theoretical guarantees in standard Nystrom method.
method QR decomposition for efficient rank reduction in fixed-rank Nystrom approximations.
result Improved accuracy in many cases with nearly identical computational complexity.
New method reduces computational cost for nonnegative low rank matrix approximation.
problem Efficiently compute nonnegative low rank matrix approximation for nonnegative matrices.
method Alternating projections onto tangent spaces of fixed rank matrices manifold and nonnegative matrix manifold.
result Sequence converges linearly to optimal solutions, showing better performance in terms of computational time and accuracy.
Minimal submanifolds in matrix spaces proven for specific ranks.
problem Minimal submanifolds in matrix spaces.
method Proving semialgebraic sets of matrices are minimal.
result Rectangular, skew-symmetric, and symmetric matrices with prescribed eigenvalues are minimal.
This research solves Hermite interpolation on manifolds using retractions.
problem Interpolating data on non-Euclidean spaces with matching derivatives.
method Proposes a novel procedure using retractions for Hermite interpolation on various manifolds.
result Establishes the well-posedness of the method and extends Hermite interpolation results to manifolds.
Paper explores geometry of covariance matrices using associated bundles.
problem Geometry of fixed-rank covariance matrices.
method Associated bundle approach to Bures--Wasserstein geometry.
result Established a one-to-one correspondence between geodesics.
No hyperbolic group can have an infinite chain of free subgroups of fixed rank.
problem Infinite ascending chains of free subgroups in hyperbolic groups.
method Proof by contradiction and properties of hyperbolic groups.
result Hyperbolic groups do not contain strictly ascending chains of free quasiconvex subgroups of constant rank.
Estimates covariance matrices for matrix-variate data via core covariance geometry.
problem Estimating covariance matrices for matrix-variate data with partial isotropy.
method Fixed-rank core covariance geometry, partial-isotropy rank-r core shrinkage estimator.
result The geometry of the space of rank-r cores is a smooth manifold.
A new Riemannian framework optimizes LoRA for faster convergence and better performance.
problem Optimizing low-rank adapters in neural networks to improve convergence and performance.
method Integrates Riemannion optimizer, LoRA initialization, and efficient implementation for geometrically treating low-rank adapters.
result Consistent and noticeable improvements in convergence speed and final task performance over standard LoRA and its modifications.
New geometric framework for positive semidefinite matrices of fixed rank.
problem Statistical analysis of positive semidefinite matrices of fixed rank.
method Introducing a manifold S(n,p)∗ with Riemannian geometry and Lie group structure. result Analytical closed forms for geodesics and Fréchet means.
Riemannian gradient descent escapes some spurious critical points on low-rank matrix manifold.
problem Spurious critical points on the boundary of low-rank matrix manifold.
method Riemannian gradient descent with dynamical low-rank approximation and rescaled gradient flow.
result Riemannian gradient descent escapes some spurious critical points on the boundary of the manifold.
KReTTaH uses tensor trains and Hadamard overparameterization for fast, interpretable multi-way data imputation.
problem Multi-way data imputation for high-dimensional functional MRI and dynamic graph recovery.
method Reformulates imputation as RKHS regression with TT-constrained coefficients and Hadamard overparameterization. Optimizes TT coefficients and kernel matrices on Riemannian manifolds.
result Consistently outperforms state-of-the-art methods in modeling accuracy.
KReTTaH uses tensor trains and Hadamard overparameterization for fast, interpretable multi-way data imputation.
problem Multi-way data imputation in high-dimensional spaces.
method Reformulates imputation as RKHS regression with TT-constrained coefficients, optimized on manifold frameworks.
result Consistently outperforms state-of-the-art methods in accuracy.
RaFM improves FM performance with variable-rank embeddings.
problem Learning pairwise interactions with varying feature frequencies.
method Introduces RaFM with variable-rank embeddings for better performance.
result RaFM achieves better performance on real-world datasets.
AdaRL improves robust RL by adaptively adjusting policy complexity.
problem Handling epistemic uncertainty in environment dynamics.
method Bi-level optimization framework with adaptive rank adjustment.
result AdaRL outperforms existing methods on MuJoCo benchmarks.
We consider two Riemannian geometries for the manifold M(p,m×n) of all m×n matrices of rank p. The geometries are induced on M(p,m×n) by viewing it as the base manifold of the submersion π:(M,N)↦MNT, selecting an adequate Riemannian metric on the total space, and …
Geometrically, tensors of fixed rank form a minimal submanifold.
problem Understanding the geometric properties of tensors of fixed rank.
method Geometric analysis of tensors in Euclidean space.
result Real tensors of fixed multilinear rank form a minimal submanifold.
The matrix completion problem consists of finding or approximating a low-rank matrix based on a few samples of this matrix. We propose a new algorithm for matrix completion that minimizes the least-square distance on the sampling set over the Riemannian manifold of fixed-rank matrices. The algorithm is an adaptation of…
Study financial contagion in networks using low-rank approximations and graphons.
problem Modeling distress contagion in heterogeneous financial networks.
method Rank-K factorization, nonautonomous ODE, transport representation, graphon limits.
result Established well-posedness and stability for contagion models in various settings.
Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles, and defines a natural transformation from isomorphism classes of real vector bundles to isomorphism classes of matroid bundles, as well as a transformation from matroid b…
A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface M endowed with an anti-holomorphic involution which determines topologically the o…
Study on deformations of pre-symplectic structures using an L-infinity algebra.
problem Deformation theory of pre-symplectic structures.
method Parametrization of deformations using Koszul L-infinity algebra.
result A quotient of the Koszul L-infinity algebra is isomorphic to the L-infinity algebra controlling foliations.
Decides if elements in free groups are primitive in polynomial time.
problem Deciding if elements in free groups are primitive.
method Non-deterministic polynomial time algorithm for general r; deterministic polynomial time for r=2. result Decidability of compressed primitivity problem in free groups.
Structured sparsity is an important modeling tool that expands the applicability of convex formulations for data analysis, however it also creates significant challenges for efficient algorithm design. In this paper we investigate the generalized conditional gradient (GCG) algorithm for solving structured sparse optimi…
Clarifies the structure of quantum states using algebraic methods.
problem Unclear stratification of quantum states in physics literature.
method Analyzes the state space S(A) of a finite-dimensional C*-algebra A, focusing on unitary orbits and their properties.
result Identifies a natural Whitney stratification of the state space into matrices of fixed rank, providing a pseudo-manifold structure.
We consider numerical integrators of ODEs on homogeneous spaces (spheres, affine spaces, hyperbolic spaces). Homogeneous spaces are equipped with a built-in symmetry. A numerical integrator respects this symmetry if it is equivariant. One obtains homogeneous space integrators by combining a Lie group integrator with an…
Moduli spaces of semi-stable real and quaternionic vector bundles of a fixed topological type admit a presentation as Lagrangian quotients, and can be embedded into the symplectic quotient corresponding to the moduli variety of semi-stable holomorphic vector bundles of fixed rank and degree on a smooth complex projecti…
The paper presents two schemes for sampling matrices from specific distributions on a manifold.
problem Sampling matrices from Gibbs distributions on the manifold of positive semi-definite matrices with fixed rank.
method Two explicit schemes based on Euler-Maruyama discretization of the Riemannian Langevin equation with Brownian motion on the manifold.
result Numerical validation of the schemes using specific energy functions and metrics.
Extends variational principle to tensor Banach spaces.
problem High-dimensional partial differential equations and minimization problems.
method Describes tensor product as disjoint connected components, each modeled as a Banach manifold.
result Extension of Dirac-Frenkel variational principle to topological tensor spaces.
Paper proposes a new algorithm for graph learning with covariance constraints.
problem Graphical models and factor analysis not jointly leveraged in graph learning processes.
method Penalized maximum likelihood estimation of an elliptical distribution with Riemannian optimization.
result Effectiveness of the proposed approach demonstrated on real-world data sets.
Global geometric expressions derived for manifold embeddings.
problem Expressing geometric quantities globally on manifolds.
method Global formulas using operator-valued expressions and affine projection.
result Explicit cross-curvature results for specific metrics.
New formulas for Riemannian gradient and Hessian on manifold metrics.
problem Evaluate Riemannian gradient and Hessian for various metrics on manifolds.
method Explicit formulas derived from Levi-Civita connection and projection.
result Derives new metrics and optimization frameworks on manifolds.
New method clusters brain networks via nonlinear dependencies.
problem Capturing non-linear nodal dependencies in brain networks.
method Kernel ARMA modeling and Grassmannian mapping.
result Effective clustering framework for various brain network problems.
New algorithm reduces rank constrained optimization problems.
problem Rank constrained optimization problems in machine learning and statistics.
method Recursive Importance Sketching (RISRO) algorithm.
result RISRO offers clear advantages over existing algorithms and converges efficiently.
Boosting framework improves Factorization Machines for recommendation systems.
problem Finding informative interactions between features in recommendation systems.
method Adaptive Boosting framework of Factorization Machines (AdaFM) that adaptively searches for proper ranks.
result AdaFM outperforms state-of-the-art Factorization Machines on real-world datasets.
In higher dimensions, Schottky spaces have unique topological properties.
problem Characterize the topology of Schottky spaces in higher dimensions.
method Analyzing the fundamental group and homotopy properties of Schottky spaces in the borderline dimension.
result In the borderline dimension, the space is simply connected but has a dense open part with fundamental group a product of cyclic groups of order two.
Improved spatial prediction for massive datasets using SME model.
problem Efficiently estimating parameters in massive spatial datasets.
method Spatial Mixed Effects (SME) model with AECM algorithm for flexibility.
result Improved estimation without sacrificing prediction accuracy.
Study of correlated Wigner matrices with BBP transitions.
problem Understanding spectral transitions in correlated Wigner matrices.
method Analyzes a Wigner-type matrix with row/column correlations, decomposes into bulk and outliers, and uses integral operators to model transitions.
result Correlated Wigner matrices exhibit multiple BBP transitions at critical points.
This paper studies geodesics between covariance matrices of different ranks using the Bures-Wasserstein metric.
problem Geodesics between covariance matrices of varying ranks.
method Analyzes the Bures-Wasserstein distance on covariance matrices, completing previous work on geodesics and providing explicit formulas.
result The set of all minimizing geodesics between two covariance matrices is parametrized by a closed unit ball in R(k−r)imes(l−r). A new method matches point sets of low-rank networks via their Laplace transforms.
problem Matching nodes in unseeded, low-rank networks without known correspondences.
method Transform-based unsupervised point registration via minimizing discrepancy between Laplace transforms.
result First consistency guarantee and explicit error rate for general low-rank models.
This thesis enhances ML reliability by selectively abstaining from predictions when uncertain.
problem Improving reliability in machine learning systems, especially in high-stakes domains.
method Exploiting uncertainty signals from training trajectories to develop lightweight, post-hoc abstention methods compatible with differential privacy.
result A robust trajectory-based approach to selective prediction that maintains high accuracy under privacy noise.
A novel approach for federated learning over-the-air computation to reduce latency and improve privacy.
problem Low-latency and privacy issues in edge machine learning for intelligent devices.
method Over-the-air computation and sparse-low-rank optimization for efficient global model aggregation.
result Efficient global model aggregation with low latency and improved privacy.
Develops new approach to recover CR structures from their Levi foliations.
problem Recovering CR structures from their Levi foliations for nonregular symbols.
method Reduction to dynamical Legendrian contact structure on leaf space.
result New geometric interpretation of CR prolongation conditions.
New geometric description of matrix manifolds avoiding equivalence classes.
problem Geometric description of matrix manifolds of fixed rank.
method Introducing a new geometric description of manifolds of matrices of fixed rank, avoiding equivalence classes.
result The matrix space Rnimesm is described as an analytic manifold equipped with a topology for which the matrix rank is a continuous map. Paper connects neural network score approximation to reverse diffusion model distribution approximation.
problem Quantifying the relationship between neural network score approximation and the distribution generated by reverse diffusion models.
method Combines Hornik's universal approximation theorem, Girsanov's theorem, and data processing inequality.
result Neural network score approximation guarantees distribution approximation in reverse diffusion models.
The study provides conditions for approximating Riemannian manifolds with polyhedral metrics.
problem Approximating Riemannian manifolds with polyhedral metrics.
method Conditions on curvature tensors for Lipschitz and local polyhedral approximations.
result Conditions are sufficient for local polyhedral approximations, conjectured to be sufficient for global approximations.
Paper proposes a new adaptive multiscale value function approximation for reinforcement learning.
problem Value function approximation in reinforcement learning with varying complexity.
method Adaptive multiscale approximation using multiresolution analysis and tree approximation.
result Convergence rate of the multiscale approximation is independent of basis function regularity.
Optimal function approximation with Relu neural networks achieves minimal error.
problem Finding the minimal error in approximating convex functions with Relu networks.
method Established necessary and sufficient conditions for optimal approximations, presented neural network architectures, and proposed an algorithm for convergence.
result Proved the convergence of the proposed algorithm and validated it with experimental results.