The paper extends curvature bounds for nonsymmetric diffusion operators.
problem Characterizing lower bounds for nonsymmetric diffusion operators.
method Using convexity of entropy on the L2-Wasserstein space and curvature-dimension condition. result New curvature bounds and estimates for nonsymmetric diffusion operators.
We formulate an approach to the geometry of Riemann-Cartan spaces provided with nonholonomic distributions defined by generic off-diagonal and nonsymmetric metrics inducing effective nonlinear and affine connections. Such geometries can be modelled by moving nonholonomic frames on (pseudo) Riemannian manifolds and desc…
New method learns nonsymmetric DPPs for better modeling diverse sets.
problem Nonsymmetric DPPs for better modeling diverse sets.
method Maximum likelihood estimation with a specific kernel decomposition.
result Improved predictive performance compared to symmetric DPPs.
We provide a proof that nonholonomically constrained Ricci flows of (pseudo) Riemannian metrics positively result into nonsymmetric metrics (as explicit examples, we consider flows of some physically valuable exact solutions in general relativity). There are constructed and analyzed three classes of solutions of Ricci …
We show the existence of nonsymmetric homogeneous spin Riemannian manifolds whose Dirac operator is like that on a Riemannian symmetric spin space. Such manifolds are exactly the homogeneous spin Riemannian manifolds (M,g) which are traceless cyclic with respect to some quotient expression M=G/K and reductive decom…
We argue that the Einstein gravity theory can be reformulated in almost Kahler (nonsymmetric) variables with effective symplectic form and compatible linear connection uniquely defined by a (pseudo) Riemannian metric. A class of nonsymmetric theories of gravitation (NGT) on manifolds enabled with nonholonomic distribut…
The paper presents the Einstein connection for nonsymmetric pseudo-Riemannian manifolds with specific torsion conditions.
problem Tackles the Einstein connection in nonsymmetric pseudo-Riemannian manifolds with non-degenerate skew-symmetric tensor.
method Explicitly presents the Einstein connection for nonsymmetric pseudo-Riemannian manifolds with f2-torsion condition. result Reduces to M.Prvanović's solution in the almost Hermitian case.
We simplify symmetric NMF by transforming it into a nonsymmetric problem, enabling faster and more efficient solutions.
problem Efficiently solving symmetric nonnegative matrix factorization (NMF).
method Transforming symmetric NMF into a nonsymmetric problem, applying fast alternating algorithms, and rigorously proving convergence.
result Fast algorithms for symmetric NMF can converge to a critical point at least at a sublinear rate.
The paper presents the Einstein connection for nonsymmetric pseudo-Riemannian manifolds.
problem The challenge is to define the Einstein connection for nonsymmetric pseudo-Riemannian manifolds.
method The approach involves using a weak almost contact structure and a linear connection with torsion.
result Explicit formulas for the Einstein connection are provided.
This work tackles scalable sampling for nonsymmetric DPPs.
problem Scalability issue in existing DPP sampling algorithms for nonsymmetric DPPs.
method Developed a linear-time algorithm for kernels with low-rank structure and a sublinear-time rejection sampling algorithm.
result Bounded rejection rate for kernels with structural constraints.
New Einstein solvmanifolds constructed without using nilsolitons.
problem Constructing Einstein solvmanifolds not based on nilsolitons.
method Using nonsymmetric derivations and a computer algorithm to classify and find solutions up to dimension 9.
result Einstein solvmanifolds of dimensions ≤ 5 are not isometric to standard extensions of nilsolitons.
New scalable MCMC sampling for nonsymmetric DPPs speeds up computations.
problem Efficient sampling for nonsymmetric DPPs with low-rank kernels.
method Enhanced rejection sampling with efficient proposal distribution construction.
result Sublinear runtime for scalable MCMC sampling of k-NDPPs. Extends MMF to nonsymmetric matrices for hierarchical structure.
problem Capturing hierarchical structure in nonsymmetric matrices.
method Multiresolution Matrix Factorization (MMF) extended to nonsymmetric matrices.
result Effective for matrix compression tasks, outperforming low-rank methods.
New algorithm scales NDPP learning and inference to large item collections.
problem Memory and runtime limitations in existing NDPP learning and inference algorithms.
method Introduced a new NDPP kernel decomposition for learning and a linear-complexity MAP inference algorithm.
result Our algorithms scale linearly in M, matching prior work's predictive performance. Certain solvable extensions of H-type groups provide noncompact counterexamples to the so-called Lichnerowicz conjecture, which asserted that ``harmonic'' Riemannian spaces must be rank 1 symmetric spaces.
Unified framework for multi-view diffusion geometries using intertwined diffusion trajectories.
problem Constructing multi-view diffusion geometries with flexible view interaction and fusion.
method Intertwined multi-view diffusion trajectories (MDTs) as a class of inhomogeneous diffusion processes.
result Established theoretical properties and derived diffusion distances and embeddings.
New algorithms for online MAP inference and learning for NDPPs.
problem Online inference and learning for nonsymmetric determinantal point processes.
method Single-pass algorithms with sub-linear memory usage.
result Comparable performance to offline algorithms with multiple passes.
The central problem of strip theory is the calculation of potential flowaround 2D sections. One particular method of solutions to this problem is conformal mapping of the body section to the unit circle over which a solution of potential flow is available. Here, a new multiparameter conformal mapping method is presente…
New method solves blind inverse problems by optimizing both operator and image parameters.
problem Solving blind inverse problems with known forward operator.
method Parallel reverse diffusion guided by gradients from intermediate stages.
result State-of-the-art performance on blind deblurring and imaging through turbulence.
A new method splits diffusion operators on principal bundles, leading to disintegration theorems.
problem Diffusion operators on principal bundles with constant rank.
method Defining semi-connections and splitting diffusion operators into horizontal and vertical components.
result A disintegration theorem for the law of diffusion operators on principal bundles.
Study shows neural operators can efficiently solve complex reaction-diffusion systems.
problem Efficiently solving nonlinear reaction-diffusion systems using neural operators.
method Laplacian-based neural operators applied to a generalized Gierer-Meinhardt system.
result Explicit approximation error bounds established for neural operators in terms of network parameters.
Paper extends Green-Osher inequality for convex bodies at dilation position.
problem Extending Green-Osher inequality for specific geometric configurations.
method Analyzes strictly convex bodies at dilation position and derives necessary and sufficient conditions.
result Establishes extended Green-Osher inequality with conditions for equality.
This paper tackles infinite-dimensional diffusion bridge simulation using operator learning.
problem Challenges in simulating diffusion bridges for modeling natural data due to intractable drift terms and continuous data representations.
method Merges score matching techniques with operator learning to directly learn infinite-dimensional bridges.
result Demonstrates high efficacy in simulating diffusion bridges for various applications, including real-world biological data.
We classify noncompact homogeneous spaces which are Einstein and asymptotically harmonic. This completes the classification of Riemannian harmonic spaces in the homogeneous case: Any simply connected homogeneous harmonic space is flat, or rank-one symmetric, or a nonsymmetric Damek-Ricci space. Independently, Y. Nikola…
Study mass transport in low-diffusivity using Lagrangian coordinates.
problem Mass preserving transport of passive tracers in low-diffusivity limit.
method Lagrangian coordinates, time-averaged diffusion equation, weighted manifold structure.
result Leading order asymptotics extend to dominant nontrivial singular value in low-diffusivity limit.
Paper proposes a new method for training diffusion models using Markov operators.
problem Training efficiency and accuracy in diffusion models.
method Operator-informed score matching using spectral decomposition of Markov operators.
result Improved score matching for both low and high-dimensional distributions.
Discretizes diffusions and harmonic functions on covering spaces.
problem Harmonic functions on covering spaces with bounded growth.
method Lyons-Sullivan discretizations of diffusion operators.
result Equivalence of discretized and continuous harmonic functions.
Improved diffusion sampling for inverse problems with faster and more robust inference.
problem High computational cost and lack of robustness in diffusion posterior sampling.
method Amortized variational inference with explicit likelihood guidance.
result Improved trade-off between inference speed and robustness to unseen degradations.
FKEE estimates expectations without samples, using diffusion bridges and PINNs.
problem Estimating expectations without large sample sizes.
method Diffusion bridge models and Feynman-Kac operator approximation using PINNs.
result Significantly reduces variance and improves efficiency.
Compositional diffusion models simulate coupled PDEs efficiently.
problem Efficiently simulating long-horizon coupled PDE systems.
method Diffusion models trained on decoupled data are composed at inference time.
result Compositional diffusion models recover coupled trajectories with low error.
PolyNSD improves Neural Sheaf Diffusion with polynomial operators and spectral rescaling.
problem Limitations of common Neural Sheaf Diffusion implementations, including scalability and stability issues.
method Introduces Polynomial Neural Sheaf Diffusion (PolyNSD) with a degree-K polynomial propagation operator and spectral rescaling.
result PolyNSD achieves state-of-the-art results on both homophilic and heterophilic benchmarks with reduced runtime and memory requirements.
ParPIC clusters directed graphs using random walks and diffusion operators.
problem Challenges in vertex-level clustering for directed graphs due to edge directionality.
method Parametrized Power-Iteration Clustering (ParPIC) based on reversible random walks and diffusion operators.
result ParPIC achieves competitive clustering accuracy with improved scalability compared to spectral and teleportation-based methods.
We consider a nonlinear extension of the generalized network flow model, with the flow leaving an arc being an increasing concave function of the flow entering it, as proposed by Truemper and Shigeno. We give a polynomial time combinatorial algorithm for solving corresponding flow maximization problems, finding an epsi…
Finslerian graph neural networks recover nonlinear diffusion geometry
problem Graph neural networks on point clouds
method Estimates of the Finsler Laplacian
result Recovery of Finsler geometry
Study hypocoercive estimates for diffusion on foliations, focusing on velocity spherical Brownian motion.
problem Proving hypocoercive estimates for diffusion on non-geodesic foliations.
method Developing generalized Γ-calculus for hypoelliptic operators, studying velocity spherical Brownian motion.
result Convergence to equilibrium in H1 and L2 for velocity spherical Brownian motion. New method uses diffusions to measure sample quality in multivariate targets.
problem Measuring convergence to multivariate continuous targets.
method Ito diffusions and explicit multivariate Stein factor bounds.
result Established near-linear relationship between diffusion Stein discrepancies and Wasserstein distances.
Improved spectral convergence bounds for diffusion maps on tori.
problem Weak theoretical error bounds for diffusion maps.
method Spatial Hardy space estimates, PDE spectral stability, Sinkhorn weights.
result Matched pointwise error bounds for spectral data and operator convergence.
Let L=Δ−∇φ⋅∇ be a symmetric diffusion operator with an invariant measure dμ=e−φdx on a complete Riemannian manifold. In this paper we prove Li-Yau gradient estimates for weighted elliptic equations on the complete manifold with ∣∇φ∣≤θ and ∞-dimensional Bakry-Émer…
Develops a smooth operator framework for analyzing neural network representations.
problem Analyzing the geometry of feedforward neural network representations.
method Introduces a smooth operator-theoretic approach based on diffusion Markov operators derived from feature clouds.
result Establishes a stable operator-geometric framework for tracking training, width, and perturbation stability.
We study a natural Lie algebra structure on the free vector space generated by all rooted planar trees as the associated Lie algebra of the nonsymmetric operad (non-Σ operad, preoperad) of rooted planar trees. We determine whether the Lie algebra and some related Lie algebras are finitely generated or not, and prove …
DeepONets improve surrogate modeling for engineering systems.
problem Accurately modeling complex PDEs for engineering systems.
method DeepONets specialize in approximating mathematical operators for PDEs.
result DeepONets achieve high prediction accuracy and zero-shot capability.
Study gradient bounds for Kolmogorov type diffusions using coupling and Γ-calculus.
problem Gradient bounds for Kolmogorov type diffusions.
method Coupling techniques and Γ-calculus.
result Advantages and drawbacks of each method discussed.
New method selects diffusion scales for graph wavelets.
problem Choosing optimal diffusion scales for graph wavelets.
method Proposes an unsupervised method using information theory.
result Method selects diffusion scales for graph wavelets.
Study non-symmetric diffusions on RCD spaces, proving their convergence.
problem Analyzing non-symmetric diffusion processes on RCD spaces.
method Constructing diffusion processes with Dirichlet forms, investigating conservativeness and weak convergence.
result Established convergence of diffusion laws under geometric and coefficient convergences.
Efficient diffusion model for symmetric manifolds reduces training and computation costs.
problem Heat kernel computations for manifold diffusion models are computationally expensive and infeasible.
method Spatially-varying covariance diffusion model, efficient objective derived via Ito's Lemma.
result Our model reduces training time and arithmetic operations by orders of magnitude.
Infinite-dimensional SBDMs improve image generation across multiple resolutions.
problem Efficient image generation at high resolutions and across different levels.
method Developed SBDMs in infinite-dimensional setting, using trace class operators and operator networks.
result Improved efficiency and generalization across resolution levels.
Study describes how national credit operations emerge from subnational data.
problem Understanding national credit dynamics from subnational data.
method Proposed diffusion process to aggregate subnational credit operations.
result National credit dynamics accurately described with proposed model.
The paper develops a new probabilistic framework for denoising diffusion models using free entropy and stochastic analysis.
problem Developing a mathematical framework for denoising diffusion models in noncommutative settings.
method Formulating diffusion and reverse processes governed by operator-valued stochastic dynamics, using tools from free stochastic analysis.
result Establishing an information-geometric link between entropy production, transport, and deconvolution.