Conditional Sampling With Monotone GANs

Baptista, R.; Hosseini, Bamdad; Kovachki, Nikola B.; Marzouk, Youssef

doi:10.48550/arxiv.2006.06755

Cited by 8 publications

(12 citation statements)

References 29 publications

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“…Starting with the last coordinate k = d, we set B d = A d . Suppose for the first k dimensions (k > 1), we have a coefficient tensor B k ∈ R r k−1 ×n k ×r k that defines a marginal function p ≤k (x ≤k ) as in (26). The following procedure can be used to obtain the coefficient tensor…”

Section: Algorithm 3 Computing B and Zmentioning

confidence: 99%

“…As suggested by [1,26,41], the above triangular structure permits one to easily build conditional map T Θ|Y =y (u Θ ) := T Θ (T −1 Y (y), u Θ ). This conditional map is such that, for any observed information y, the random variable T Θ|Y =y (U Θ ) with U Θ ∼ ρ Θ follows the approximate conditional pdf p Θ|Y =y .…”

Section: Introductionmentioning

confidence: 99%

“…Related work. We highlight relevant works in transport maps [1,4,26,37,43,44] and normalising flows [6,7,10,27,36,38], in which the map is identified by minimizing the Kullback-Leibler (KL) divergence between the target density and its approximation over some class of tractable triangular maps. By and large, these approaches adopt a density estimation perspective in which one only has access to samples from the target pdf, and so minimizing the KL divergence boils down to maximum likelihood estimation.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Scalable conditional deep inverse Rosenblatt transports using tensor-trains and gradient-based dimension reduction

Cui¹,

Dolgov²,

Zahm³

2021

Preprint

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We present a novel offline-online method to mitigate the computational burden of the characterization of conditional beliefs in statistical learning. In the offline phase, the proposed method learns the joint law of the belief random variables and the observational random variables in the tensor-train (TT) format. In the online phase, it utilizes the resulting order-preserving conditional transport map to issue real-time characterization of the conditional beliefs given new observed information. Compared with the state-of-the-art normalizing flows techniques, the proposed method relies on function approximation and is equipped with thorough performance analysis. This also allows us to further extend the capability of transport maps in challenging problems with high-dimensional observations and high-dimensional belief variables. On the one hand, we present novel heuristics to reorder and/or reparametrize the variables to enhance the approximation power of TT. On the other, we integrate the TT-based transport maps and the parameter reordering/reparametrization into layered compositions to further improve the performance of the resulting transport maps. We demonstrate the efficiency of the proposed method on various statistical learning tasks in ordinary differential equations (ODEs) and partial differential equations (PDEs).Preprint. Under review.

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Section: Algorithm 3 Computing B and Zmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Scalable conditional deep inverse Rosenblatt transports using tensor-trains and gradient-based dimension reduction

Cui¹,

Dolgov²,

Zahm³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…A close relative of transport maps in machine learning are normalizing flows (see Kobyzev et al, 2020, for a review), where triangular layers are used to ensure easy evaluation and inversion of likelihood objectives. Kovachki et al (2020) also design generative adversarial networks with triangular generators that allow easy conditional sampling. Our approach can be viewed as a Bayesian autoencoder (e.g., Goodfellow et al, 2016, Ch.…”

Section: The Posterior Mapmentioning

confidence: 99%

“…Such draws, which maintain the large-scale features in the held-out (98th) test field but allow for newly sampled fine-scale features, are shown in Figure 10. This is related to the supervised conditional sampling ideas in Kovachki et al (2020), with their inputs given by our first i ordered test observations.…”

Section: Climate-data Applicationmentioning

confidence: 99%

Scalable Bayesian transport maps for high-dimensional non-Gaussian spatial fields

Katzfuß¹,

Schäfer²

2021

Preprint

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A multivariate distribution can be described by a triangular transport map from the target distribution to a simple reference distribution. We propose Bayesian nonparametric inference on the transport map by modeling its components using Gaussian processes. This enables regularization and accounting for uncertainty in the map estimation, while still resulting in a closed-form and invertible posterior map. We then focus on inferring the distribution of a nonstationary spatial field from a small number of replicates. We develop specific transport-map priors that are highly flexible and are motivated by the behavior of a large class of stochastic processes. Our approach is scalable to high-dimensional fields due to data-dependent sparsity and parallel computations. We also discuss extensions, including Dirichlet process mixtures for marginal non-Gaussianity. We present numerical results to demonstrate the accuracy, scalability, and usefulness of our methods, including statistical emulation of non-Gaussian climate-model output.

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Bayesian Nonparametric Generative Modeling of Large Multivariate Non-Gaussian Spatial Fields

Wiemann,

Katzfuss

2023

JABES

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Multivariate spatial fields are of interest in many applications, including climate model emulation. Not only can the marginal spatial fields be subject to nonstationarity, but the dependence structure among the marginal fields and between the fields might also differ substantially. Extending a recently proposed Bayesian approach to describe the distribution of a nonstationary univariate spatial field using a triangular transport map, we cast the inference problem for a multivariate spatial field for a small number of replicates into a series of independent Gaussian process (GP) regression tasks with Gaussian errors. Due to the potential nonlinearity in the conditional means, the joint distribution modeled can be non-Gaussian. The resulting nonparametric Bayesian methodology scales well to high-dimensional spatial fields. It is especially useful when only a few training samples are available, because it employs regularization priors and quantifies uncertainty. Inference is conducted in an empirical Bayes setting by a highly scalable stochastic gradient approach. The implementation benefits from minibatching and could be accelerated with parallel computing. We illustrate the extended transport-map model by studying hydrological variables from non-Gaussian climatemodel output.

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Conditional Sampling With Monotone GANs

Cited by 8 publications

References 29 publications

Scalable conditional deep inverse Rosenblatt transports using tensor-trains and gradient-based dimension reduction

Scalable conditional deep inverse Rosenblatt transports using tensor-trains and gradient-based dimension reduction

Scalable Bayesian transport maps for high-dimensional non-Gaussian spatial fields

Bayesian Nonparametric Generative Modeling of Large Multivariate Non-Gaussian Spatial Fields

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