3arXiv cs.LG (Machine Learning)·
EntroPath: Maximum entropy path ensemble method for manifold learning and geodesic geometry recovery
EntroPath is a new manifold learning method that uses maximum entropy random walks (MERW) to build free-energy dissimilarities that converge to squared geodesic distances, addressing concentration bias in locally-normalized random walks and sensitivity to shortcut edges in shortest-path methods. The method provides a diffusion depth parameter that interpolates between local and global geometry, admits an exact Gram factorization connecting it to kernel methods, and includes scalable landmark projection extensions. Evaluations on synthetic manifolds and single-cell genomics benchmarks show EntroPath matches or outperforms diffusion- and shortest-path-based methods, with strongest gains on non-uniformly sampled manifolds and branching trajectories.
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PTL-Diffusion: Diffusion framework with periodic terminal laws for manifold-aware generation
PTL-Diffusion is a new diffusion modeling framework that replaces the standard single Gaussian terminal distribution with a periodic family of Gaussian terminal laws, embedding phase structure directly into the forward noising dynamics rather than only in the denoising network. The authors derive closed-form forward marginals and reverse posteriors for a periodically forced Ornstein-Uhlenbeck process, enabling standard noise-prediction training. Experiments on torus, cylinder, and face datasets show improvements in manifold-level distributional matching over DDPM baselines. The work is a proof-of-concept motivating structured terminal reference laws as a direction for geometry-aware generative modeling.
Global Convergence Theory for Wasserstein Policy Gradient in Entropy-Regularized RL
This paper establishes the first global convergence theory for Wasserstein Policy Gradient (WPG), a continuous-control RL optimization method that uses optimal-transport geometry over action distributions. The authors show that the Bellman recursion structure of entropy-regularized RL induces a Polyak–Łojasiewicz (PL) geometry that substitutes for classical convexity, enabling global convergence analysis. Key technical contributions include a statewise KL representation of the soft Bellman residual, a Bellman resolvent identity linking value improvement to relative Fisher information, and a uniform log-Sobolev inequality for the evolving Gibbs policy family. The result yields geometric contraction up to discretization bias, providing theoretical grounding for WPG in continuous-action settings.
Solver-dependent Nash equilibrium selection on zero-sum polytopes: regularized methods select max-entropy members
A new arXiv preprint investigates whether different Nash equilibrium solvers systematically select different members of the Nash polytope in two-player zero-sum games. Using six analytically tractable games including Kuhn poker, the authors find that regularized last-iterate methods (R-NaD, magnetic mirror descent) converge to the maximum-entropy Nash equilibrium — interpretable as an information projection — while regret-averaging methods (CFR, CFR+, fictitious play) drift to lower-entropy boundary solutions. The distinction has downstream consequences for performance against sub-optimal opponents in games with sequential or hidden-information structure, with implications for multi-agent AI training and game-solving pipelines.
Theoretical unification of discrete diffusion model objectives via Oracle Distance theorem
A new arXiv preprint provides a rigorous theoretical framework for understanding what discrete diffusion models learn, proving the 'Oracle Distance' theorem: the negative ELBO exactly equals data entropy plus the path KL from the oracle reverse process to the learned one. The work shows that denoiser, score ratio, and bridge plug-in parameterizations are the same object in different coordinates, with closed-form conversions among them. It unifies several existing discrete diffusion losses (MDM, UDM, SEDD, GIDD) as special cases and identifies practical consequences, such as why denoiser parameterization causes the uniform ELBO to diverge at initialization. All identities are verified numerically on an exactly solvable model.
Manifold Power Iteration redesigns MoE routers by aligning rows with expert singular directions
A new arXiv preprint proposes Manifold Power Iteration (MPI), a principled redesign of Mixture-of-Experts router matrices that aligns each router row with the principal singular direction of its associated expert. The method uses a 'Power-then-Retract' paradigm to enforce norm constraints while driving convergence toward these singular directions. Empirical validation spans MoE pretraining at scales from 1B to 11B parameters, showing improved model effectiveness.
SURGE: Approximation-free Training-Free Particle Filter for Diffusion Surrogate
The paper introduces URGE (Unbiased Resampling via Girsanov Estimation), a derivative-free inference-time scaling algorithm for diffusion models that performs path-wise importance reweighting using a Girsanov change of measure. Unlike existing inference-time guidance methods, URGE requires no score, Hessian, or PDE evaluations, attaching multiplicative weights to simulated trajectories and periodically resampling. The authors establish a theoretical equivalence between path-wise and particle-wise sequential Monte Carlo (SMC), guaranteeing unbiased terminal distributions. Empirically, URGE outperforms existing inference-time guidance baselines on synthetic tests and diffusion-model benchmarks while being simpler to implement.
Beyond Isotropy in JEPAs: Hamiltonian Geometry and Symplectic Prediction
This paper critiques the standard practice of regularizing Joint-Embedding Predictive Architecture (JEPA) encoders toward isotropic Gaussian marginals, showing that this Euclidean symmetry assumption incurs a quantifiable 'price of isotropy' and that no geometry-independent fixed marginal target is universally canonical. The authors prove that oracle one-view marginals do not identify the view-to-view predictive coupling, arguing structural bias should enter the cross-view coupling instead. They introduce HamJEPA, which encodes views as phase-space states and uses a learned Hamiltonian leapfrog map for view-to-view prediction, with symplectic coupling identified as the key driver of gains. HamJEPA outperforms SIGReg on CIFAR-100 by up to +6.45 kNN@20 and +10.64 linear-probe points at 80 epochs, with similar improvements on ImageNet-100.
Error-Conditioned Neural Solvers (ENS) improve PDE surrogate accuracy by feeding residual fields as inputs
A new arXiv preprint introduces Error-Conditioned Neural Solvers (ENS), a method for neural PDE surrogates that passes the PDE residual field directly as input to the network at each iteration, enabling iterative self-correction rather than gradient-based residual minimization. The authors demonstrate theoretically and empirically that residual minimization is an unreliable proxy for reconstruction accuracy in ill-conditioned systems, explaining failures of prior hybrid methods. ENS achieves up to 10× accuracy gains on turbulent Kolmogorov flow across four PDE families, with lower compute cost than hybrid approaches and generalization under distribution shift including zero-shot cross-equation transfer.