Geometric Neural Diffusion Processes
- URL: http://arxiv.org/abs/2307.05431v1
- Date: Tue, 11 Jul 2023 16:51:38 GMT
- Title: Geometric Neural Diffusion Processes
- Authors: Emile Mathieu, Vincent Dutordoir, Michael J. Hutchinson, Valentin De
Bortoli, Yee Whye Teh, Richard E. Turner
- Abstract summary: We extend the framework of diffusion models to incorporate a series of geometric priors in infinite-dimension modelling.
We show that with these conditions, the generative functional model admits the same symmetry.
- Score: 55.891428654434634
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Denoising diffusion models have proven to be a flexible and effective
paradigm for generative modelling. Their recent extension to infinite
dimensional Euclidean spaces has allowed for the modelling of stochastic
processes. However, many problems in the natural sciences incorporate
symmetries and involve data living in non-Euclidean spaces. In this work, we
extend the framework of diffusion models to incorporate a series of geometric
priors in infinite-dimension modelling. We do so by a) constructing a noising
process which admits, as limiting distribution, a geometric Gaussian process
that transforms under the symmetry group of interest, and b) approximating the
score with a neural network that is equivariant w.r.t. this group. We show that
with these conditions, the generative functional model admits the same
symmetry. We demonstrate scalability and capacity of the model, using a novel
Langevin-based conditional sampler, to fit complex scalar and vector fields,
with Euclidean and spherical codomain, on synthetic and real-world weather
data.
Related papers
- Latent diffusion models for parameterization and data assimilation of facies-based geomodels [0.0]
Diffusion models are trained to generate new geological realizations from input fields characterized by random noise.
Latent diffusion models are shown to provide realizations that are visually consistent with samples from geomodeling software.
arXiv Detail & Related papers (2024-06-21T01:32:03Z) - von Mises Quasi-Processes for Bayesian Circular Regression [57.88921637944379]
We explore a family of expressive and interpretable distributions over circle-valued random functions.
The resulting probability model has connections with continuous spin models in statistical physics.
For posterior inference, we introduce a new Stratonovich-like augmentation that lends itself to fast Markov Chain Monte Carlo sampling.
arXiv Detail & Related papers (2024-06-19T01:57:21Z) - Diffusion posterior sampling for simulation-based inference in tall data settings [53.17563688225137]
Simulation-based inference ( SBI) is capable of approximating the posterior distribution that relates input parameters to a given observation.
In this work, we consider a tall data extension in which multiple observations are available to better infer the parameters of the model.
We compare our method to recently proposed competing approaches on various numerical experiments and demonstrate its superiority in terms of numerical stability and computational cost.
arXiv Detail & Related papers (2024-04-11T09:23:36Z) - Scaling Riemannian Diffusion Models [68.52820280448991]
We show that our method enables us to scale to high dimensional tasks on nontrivial manifold.
We model QCD densities on $SU(n)$ lattices and contrastively learned embeddings on high dimensional hyperspheres.
arXiv Detail & Related papers (2023-10-30T21:27:53Z) - Metropolis Sampling for Constrained Diffusion Models [11.488860260925504]
Denoising diffusion models have recently emerged as the predominant paradigm for generative modelling on image domains.
We introduce an alternative, simple noretisation scheme based on the reflected Brownian motion.
arXiv Detail & Related papers (2023-07-11T17:05:23Z) - Data-driven reduced-order modelling for blood flow simulations with
geometry-informed snapshots [0.0]
A data-driven surrogate model is proposed for the efficient prediction of blood flow simulations on similar but distinct domains.
A non-intrusive reduced-order model for geometrical parameters is constructed using proper decomposition.
A radial basis function interpolator is trained for predicting the reduced coefficients of the reduced-order model.
arXiv Detail & Related papers (2023-02-21T21:18:17Z) - Score-based Diffusion Models in Function Space [140.792362459734]
Diffusion models have recently emerged as a powerful framework for generative modeling.
We introduce a mathematically rigorous framework called Denoising Diffusion Operators (DDOs) for training diffusion models in function space.
We show that the corresponding discretized algorithm generates accurate samples at a fixed cost independent of the data resolution.
arXiv Detail & Related papers (2023-02-14T23:50:53Z) - Unveiling the Latent Space Geometry of Push-Forward Generative Models [24.025975236316846]
Many deep generative models are defined as a push-forward of a Gaussian measure by a continuous generator, such as Generative Adversarial Networks (GANs) or Variational Auto-Encoders (VAEs)
This work explores the latent space of such deep generative models.
A key issue with these models is their tendency to output samples outside of the support of the target distribution when learning disconnected distributions.
arXiv Detail & Related papers (2022-07-21T15:29:35Z) - Riemannian Score-Based Generative Modeling [56.20669989459281]
We introduce score-based generative models (SGMs) demonstrating remarkable empirical performance.
Current SGMs make the underlying assumption that the data is supported on a Euclidean manifold with flat geometry.
This prevents the use of these models for applications in robotics, geoscience or protein modeling.
arXiv Detail & Related papers (2022-02-06T11:57:39Z) - Tensor lattice field theory with applications to the renormalization
group and quantum computing [0.0]
We discuss the successes and limitations of statistical sampling for a sequence of models studied in the context of lattice QCD.
We show that these lattice models can be reformulated using tensorial methods where the field integrations in the path-integral formalism are replaced by discrete sums.
We derive Hamiltonians suitable to perform quantum simulation experiments, for instance using cold atoms, or to be programmed on existing quantum computers.
arXiv Detail & Related papers (2020-10-13T16:46:34Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.