The matrix-based companion to Horn’s 1987 quaternion method, deriving the optimal rotation as the orthonormal factor in the polar decomposition of the cross-covariance matrix via eigendecomposition of a 3x3 symmetric matrix.
Corrects a flaw in prior SVD-based alignment methods (Arun et al., Horn et al.) that could produce reflections instead of rotations under noisy data, and provides a complete closed-form solution for similarity transformations in arbitrary dimensions.
This paper introduces consistency models, a new family of generative models that map any point on a Probability Flow ODE trajectory to its origin. They support fast one-step generation by design, while allowing multi-step sampling for improved quality and zero-shot editing tasks like inpainting and colorization.
Derives the optimal rotation between two 3D point sets as the eigenvector of a 4x4 symmetric matrix built from cross-covariance sums, using unit quaternions to enforce the orthogonality constraint.
A foundational 1976 short communication presenting a direct, non-iterative method for finding the best rotation matrix between two point sets via eigendecomposition of a cross-covariance matrix.
A theoretical paper that introduces displacement interpolation (optimal transport) to establish a new convexity principle for energy functionals. It proves the uniqueness of ground states for interacting gases and generalizes the Brunn-Minkowski inequality, providing mathematical tools later used in flow matching and optimal transport-based generative models.
Proposes ‘InterFlow’, a method to learn continuous normalizing flows between arbitrary densities using stochastic interpolants. It avoids ODE backpropagation by minimizing a quadratic objective on the velocity field, enabling scalable ODE-based generation. On CIFAR-10, NLL matches ScoreSDE (2.99 bits per dim) with simulation-free training, though FID (10.27) trails dedicated image models (ScoreSDE: 2.92); the primary strength is tractable likelihood with efficient training cost.
Introduces Flow Matching, a scalable method for training CNFs by regressing vector fields of conditional probability paths. It generalizes diffusion and enables Optimal Transport paths for straighter, more efficient sampling.
This paper replaces discrete network layers with continuous ordinary differential equations (ODEs), allowing for adaptive computation depth and constant memory cost during training via the adjoint sensitivity method. It introduces Continuous Normalizing Flows and latent ODEs for time-series.
Introduces ‘Rectified Flow,’ a method to transport distributions via ODEs with straight paths. Uses a ‘reflow’ procedure to iteratively straighten trajectories, enabling high-quality 1-step generation with optional lightweight distillation.
This paper provides a rigorous probabilistic foundation for Denoising Autoencoders by proving they are mathematically equivalent to Score Matching on a kernel-smoothed data distribution. It derives a specific energy function for DAEs and justifies the use of tied weights.
This 1994 handbook chapter serves as a practical user guide for the Embedded-Atom Method (EAM). It details the theoretical derivation from density-functional theory, synthesizes related methods like the Glue Model, and provides a complete tutorial on fitting potentials, illustrated with a specific implementation for the Ni-Al-B system.