Incorporates matrix exponential into generative flows for improved performance.
problem Improving generative flow models for better density estimation.
method Integrates matrix exponential into generative flows, proposing new layers and modifying network architecture.
result The proposed model achieves great performance on density estimation.
NSA-Flow optimizes matrix representations for interpretability in complex data.
problem Balancing interpretability and model flexibility in high-dimensional data.
method Non-negative Stiefel Approximating Flow (NSA-Flow) unifies sparse matrix factorization and orthogonalization.
result NSA-Flow yields sparse, stable, and interpretable representations.
New approach to analyze matrix denoising using gradient flow and fixed point equations.
problem Positive semi-definite matrix denoising in extensive-rank and high-dimensional settings.
method Gradient flow and fixed point equations derived from linear pencil techniques of random matrix theory.
result Continuous phase transitions in the extensive-rank and high-dimensional regime.
This work interprets diffusion score matching using normalizing flows for better model training and evaluations.
problem Limitations of diffusion score matching when dealing with certain types of distributions.
method The approach involves interpreting the diffusion matrix using normalizing flows to provide better interpretation and usage of diffusion score matching.
result Diffusion score matching is equivalent to the original score matching evaluated in the transformed space defined by the normalizing flow.
Proves estimates for Kähler-Ricci flow solutions.
problem Positive solutions to Kähler-Ricci flow.
method Matrix Li-Yau-Hamilton estimates coupled with flow.
result Monotonicity formula derived.
Study finds weak solutions for complex map flows with optimal lifespan.
problem Existence of weak solutions for two-phase matrix-valued harmonic map flows.
method Modified minimizing movement scheme, discretizing time and interpolating solutions.
result Existence of weak solutions with optimal lifespan for the limiting system.
Gradient flow solves optimal mass transport for covariance matrices.
problem Optimal mass transport for covariance matrices.
method Gradient flow on fiber bundle structure.
result Global convergence to polar decomposition.
New matrix completion method for arbitrary sampling patterns using network flows.
problem Matrix completion under arbitrary sampling patterns.
method Network flow approach to matrix completion.
result Minimax optimal estimation for individual entries.
CoreFlow models matrix-valued distributions efficiently, preserving shared low-rank structure.
problem Learning matrix-valued distributions from high-dimensional and incomplete data.
method Low-rank flow model that learns shared row/column subspaces and trains a normalizing flow on the core.
result CoreFlow improves generation quality in few-sample regimes and remains competitive in data-rich settings.
Since Li and Yau obtained the gradient estimate for the heat equation, related estimates have been extensively studied. With additional curvature assumptions, matrix estimates that generalize such estimates have been discovered for various time-dependent settings, including the heat equation on a Kähler manifold, Ricci…
A new framework uses matrix flows to unify frequentist and Bayesian approaches for sparse GGMs.
problem Challenges in studying conditional independence among many variables with few observations.
method General framework for variational inference with matrix-variate Normalizing Flow in Gaussian Graphical Models.
result Unified benefits of frequentist and Bayesian frameworks for sparse GGMs.
Paper proves estimates for heat and conjugate heat equations under Ricci flow, leading to monotonicity of parabolic frequencies.
problem Establishing estimates for heat and conjugate heat equations under Ricci flow.
method Proving matrix Li-Yau-Hamilton estimates for positive solutions to the heat and conjugate heat equations coupled with Ricci flow.
result Monotonicity of parabolic frequencies established up to correction factors.
New paradigm for Neural ODEs stabilizes training and improves model performance.
problem Gradient vanishing-explosion problem in training deep neural networks.
method ODEtoODE: Nested system of flows with orthogonal group constraints.
result Strong convergence results and improved downstream models in reinforcement learning and supervised learning.
Unified treatment of eigenvalue processes using Riemannian geometry.
problem Eigenvalue processes in various settings.
method Riemannian submersion and gradient flow of isospectral orbits.
result Eigenvalue processes are projections of Brownian motion through Riemannian submersions.
We give a geometric interpretation of Hamilton's matrix Harnack inequality for the Ricci flow as the curvature of a connection on space-time.
Random matrix theory explains transient signal detectability in early-stopped gradient flow.
problem Transient signal detectability in early-stopped gradient flow.
method Random matrix theory applied to gradient flow in a linear teacher-student setting.
result Transient Baik-Ben Arous-Péché (BBP) transition in learning dynamics due to anisotropy and noise.
The study quantizes ancient flows in cylinders, revealing their asymptotic behavior.
problem Analyzing ancient mean curvature flows with cylindrical tangent profiles.
method Proved asymptotic behavior of cylindrical profile functions using spectral quantization.
result Asymptotic behavior of cylindrical profile functions quantized to eigenvalues 0 or -sqrt(2(n-k))/4.
We introduce two methods for estimating the density matrix for a quantum system: Quantum Maximum Likelihood and Quantum Variational Inference. In these methods, we construct a variational family to model the density matrix of a mixed quantum state. We also introduce quantum flows, the quantum analog of normalizing flow…
We describe several algorithms for matrix completion and matrix approximation when only some of its entries are known. The approximation constraint can be any whose approximated solution is known for the full matrix. For low rank approximations, similar algorithms appears recently in the literature under different name…
We generalize Hamilton's matrix Li-Yau-type Harnack estimate for the Ricci flow by considering the space of all LYH (Li-Yau-Hamilton) quadratics that arise as curvature tensors of space-time connections satisfying the Ricci flow with respect to the natural space-time degenerate metric. As a special case, we employ scal…
Nonnegative sectional curvature linked to matrix displacement convexity.
problem Nonnegative sectional curvature in Riemannian manifolds.
method Matrix displacement convexity as a criterion for nonnegative sectional curvature.
result Entropy functional matrix displacement convexity implies nonnegative sectional curvature.
Constructs coordinates to diagonalize Toda flow on matrices with simple spectrum.
problem Diagonalizing the Toda flow on matrices with simple spectrum.
method Lie theoretic methods applied to complex semisimple Lie algebras and their real forms.
result Decouples the Toda vector field into simpler components.
Discussing curvature flows and their applications.
problem Analyzing expanding curvature flows.
method Classical aspects of expanding curvature flows.
result First applications of curvature flows.
Gradient flow with infinitesimal initialization converges to Greedy Low-Rank Learning for matrix factorization.
problem Understanding implicit regularization in gradient descent for matrix factorization.
method Theoretical and empirical analysis of gradient flow with infinitesimal initialization and Greedy Low-Rank Learning.
result Gradient flow with infinitesimal initialization is mathematically equivalent to Greedy Low-Rank Learning for depth-2 matrix factorization under reasonable assumptions.
Muon dynamics study uses spectral Wasserstein flow for optimization stability.
problem Optimizing deep learning models with gradient normalization.
method Introduces Spectral Wasserstein distances for matrix flows, proving equivalence with Benamou--Brenier formulation.
result Gradient-flow interpretation of mean-field normalized training dynamics.
Gradient flow in softmax models tends to produce low-entropy outputs.
problem Understanding the training dynamics of softmax-based models.
method Analysis of gradient flow dynamics in the value-softmax model.
result Gradient flow drives optimization towards low-entropy solutions.
We derive an interpolation version of constrained matrix Li-Yau-Hamilton estimate on Kähler manifolds. As a result, we first get a constrained matrix Li-Yau-Hamilton estimate for heat equation on a Kähler manifold with fixed Kähler metric. Secondly, we get a corresponding estimate for forward conjugate heat equation on…
The space of probability densities is an infinite-dimensional Riemannian manifold, with Riemannian metrics in two flavors: Wasserstein and Fisher--Rao. The former is pivotal in optimal mass transport (OMT), whereas the latter occurs in information geometry---the differential geometric approach to statistics. The Rieman…
Develops a gradient flow for Muon optimizer, a method for optimization.
problem Optimization of complex systems with matrix-valued parameters.
method Gradient flow on probability measures induced by regularized Muon optimizer.
result Derives continuous-time limits and proves Hamiltonian dissipation.
We prove constrained trace, matrix and constrained matrix Harnack inequalities for the nonlinear heat equation ωt=Δω+aωlnω on closed manifolds. We also derive a new interpolated Harnack inequality for the equation ωt=Δω−ωlnω+εRω on closed surfaces under the ε-Ricci flow. Finally we prove…
Derives a Hamiltonian model for 3D axially symmetric magnetohydrodynamics.
problem Modeling of 3D axially symmetric magnetohydrodynamics.
method Hamiltonian formulation and matrix discretization.
result First discrete model for 3D magnetohydrodynamics compatible with underlying Lie-Poisson structure.
We introduce an atlas adapted to the Toda flow on the manifold of full flags of any non-compact real semisimple Lie algebra, and on its Hessenberg-type submanifolds. In our local coordinates the Toda flow becomes linear. We use these new coordinates to show that the Toda flow on the manifold of full flags is Morse-Smal…
Generative model learns from simpler distributions on Lie groups.
problem Learning from complex Lie group data.
method Substituting exponential curves for line segments on Lie groups.
result Simple, intrinsic, and fast implementation for generative modelling.
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.
Develops a new sampling method for gauge theories.
problem Sampling from SU(N) gauge theories. method Gauge-equivariant flows for SU(N) variables. result Constructs a class of flows respecting matrix conjugation symmetry.
We investigate solutions of the elliptic sinh-Gordon equation of spectral genus g<3. These solutions are parametrized by complex matrix-valued polynomials called potentials. On the space of these potentials there act two commuting flows. The orbits of these flows are called Polynomial Killing fields and are double peri…
Bounds on chemical reaction network relaxation rates using convex analysis.
problem Understanding relaxation dynamics in chemical reaction networks.
method Convex analysis, generalized gradient flows, singular values of stoichiometric matrix.
result Bounds on Kullback-Leibler divergence to equilibrium for CRNs.
It is known that the Schrödinger flow on a complex Grassmann manifold is equivalent to the matrix non-linear Schrödinger equation and the Ferapontov flow on a principal Adjoint U(n)-orbit is equivalent to the n-wave equation. In this paper, we give a systematic method to construct integrable geometric curve flows on …
Gradient flow on softmax attention minimizes nuclear norm of weight matrices.
problem Classification with separate key and query weight matrices.
method Gradient flow on exponential loss, separability assumption, reparameterization, approximate KKT conditions.
result Gradient flow implicitly minimizes nuclear norm of weight matrices, contrasting with Frobenius norm minimization.
Based on a suggestion of Richard Hamilton, we give an alternate proof of his matrix Harnack inequality for solutions of the Ricci flow with positive curvature operator. This Harnack inequality says that a certain endomorphism, consisting of an expression in the curvature and its first two covariant derivatives, of the …
We define Discrete Quasi-Einstein metrics (DQE-metrics) as the critical points of discrete total curvature functional on triangulated 3-manifolds. We study DQE-metrics by introducing some combinatorial curvature flows. We prove that these flows produce solutions which converge to discrete quasi-Einstein metrics when th…
Study uses Google matrix analysis to show how COVID-19 changed international trade flows.
problem Impact of COVID-19 on international trade patterns.
method Google matrix analysis of World Trade Network (WTN), including PageRank, CheiRank, and reduced Google matrix.
result Significant changes in international trade flows due to the pandemic, affecting export and import balances.
New proof shows coupling-based flows converge linearly to diagonalize data covariance.
problem Understanding convergence of coupling-based normalizing flows to arbitrary data distributions.
method Proved linear convergence rate for whitening of data distribution.
result Coupling-based flows achieve linear convergence to diagonalize data covariance.
We show that a steady-state stock-flow consistent macro-economic model can be represented as a Constraint Satisfaction Problem (CSP).The set of solutions is a polytope, which volume depends on the constraintsapplied and reveals the potential fragility of the economic circuit,with no need to study the dynamics. Several …
The notion of quasi-Einstein metric in physics is equivalent to the notion of Ricci soliton in Riemannian spaces. Quasi-Einstein metrics serve also as solution to the Ricci flow equation. Here, the Riemannian metric is replaced by a Hessian matrix derived from a Finsler structure and a quasi-Einstein Finsler metric is …
Analyzes the generalization and training errors of the random feature model over time.
problem Understanding the temporal behavior of generalization and training errors in deep learning.
method Uses Cauchy complex integral representations and random matrix methods based on linear pencils.
result Analytical solution of the full time-evolution path of generalization and training errors.
A new coordinate system for SPD matrices simplifies computations and generative modeling.
problem Computing and modeling SPD matrices
method Reverse telescoping coordinate system
result Significantly reduces computational complexity and facilitates generative modeling.
Study examines money flow network among firms' accounts in a Japanese region.
problem Understanding the relationship between money flow and economic activities of firms.
method Employed exhaustive bank transfer data, network statistics, Hodge decomposition, and non-negative matrix factorization.
result Identified a 'walnut' structure with core and upstream/downstream components, correlated with economic activities.