In the Gaussian setting the Wasserstein error of score-matching-plus-diffusion sampling equals a kernel norm of the data power spectrum whose kernel is determined by the four error sources and the algorithm parameters.
Wasserstein Riemannian Geometry of Positive Definite Matrices
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abstract
The Wasserstein distance on multivariate non-degenerate Gaussian densities is a Riemannian distance. After reviewing the properties of the distance and the metric geodesic, we present an explicit form of the Riemannian metrics on positive-definite matrices and compute its tensor form with respect to the trace inner product. The tensor is a matrix which is the solution to a Lyapunov equation. We compute the explicit formula for the Riemannian exponential, the normal coordinates charts and the Riemannian gradient. Finally, the Levi-Civita covariant derivative is computed in matrix form together with the differential equation for the parallel transport. While all computations are given in matrix form, nonetheless we discuss also the use of a special moving frame.
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First-order asymptotic expansions of weak and Fréchet discretization errors in diffusion sampling are derived, explicit under Gaussian data through covariance geometry and robust to other data geometries.
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From Score Matching to Diffusion: A Fine-Grained Error Analysis in the Gaussian Setting
In the Gaussian setting the Wasserstein error of score-matching-plus-diffusion sampling equals a kernel norm of the data power spectrum whose kernel is determined by the four error sources and the algorithm parameters.
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Geometry-Aware Discretization Error of Diffusion Models
First-order asymptotic expansions of weak and Fréchet discretization errors in diffusion sampling are derived, explicit under Gaussian data through covariance geometry and robust to other data geometries.