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Georgios Batzolis, Jan Stanczuk, Carola-Bibiane Schönlieb

Variational auto-encoders (VAEs) are one of the most popular approaches to deep generative modeling. Despite their success, images generated by VAEs are known to suffer from blurriness, due to a highly unrealistic modeling assumption that the conditional data distribution $ p(\textbf{x} | \textbf{z})$ can be approximated as an isotropic Gaussian. In this work we introduce a principled approach to modeling the conditional data distribution $p(\textbf{x} | \textbf{z})$ by incorporating a diffusion model. We show that it is possible to create a VAE-like deep latent variable model without making the Gaussian assumption on $ p(\textbf{x} | \textbf{z}) $ or even training a decoder network. A trained encoder and an unconditional diffusion model can be combined via Bayes' rule for score functions to obtain an expressive model for $ p(\textbf{x} | \textbf{z}) $. Our approach avoids making strong assumptions on the parametric form of $ p(\textbf{x} | \textbf{z}) $, and thus allows to significantly improve the performance of VAEs.

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Georgios Batzolis, Jan Stanczuk, Carola-Bibiane Schönlieb

In this work, we propose a novel framework for estimating the dimension of the data manifold using a trained diffusion model. A trained diffusion model approximates the gradient of the log density of a noise-corrupted version of the target distribution for varying levels of corruption. If the data concentrates around a manifold embedded in the high-dimensional ambient space, then as the level of corruption decreases, the score function points towards the manifold, as this direction becomes the direction of maximum likelihood increase. Therefore, for small levels of corruption, the diffusion model provides us with access to an approximation of the normal bundle of the data manifold. This allows us to estimate the dimension of the tangent space, thus, the intrinsic dimension of the data manifold. Our method outperforms linear methods for dimensionality detection such as PPCA in controlled experiments.

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Georgios Batzolis, Jan Stanczuk, Carola-Bibiane Schönlieb, Christian Etmann

Diffusion models have emerged as one of the most promising frameworks for deep generative modeling. In this work, we explore the potential of non-uniform diffusion models. We show that non-uniform diffusion leads to multi-scale diffusion models which have similar structure to this of multi-scale normalizing flows. We experimentally find that in the same or less training time, the multi-scale diffusion model achieves better FID score than the standard uniform diffusion model. More importantly, it generates samples $4.4$ times faster in $128\times 128$ resolution. The speed-up is expected to be higher in higher resolutions where more scales are used. Moreover, we show that non-uniform diffusion leads to a novel estimator for the conditional score function which achieves on par performance with the state-of-the-art conditional denoising estimator. Our theoretical and experimental findings are accompanied by an open source library MSDiff which can facilitate further research of non-uniform diffusion models.

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Tolou Shadbahr, Michael Roberts, Jan Stanczuk, Julian Gilbey, Philip Teare, Sören Dittmer, Matthew Thorpe, Ramon Vinas Torne, Evis Sala, Pietro Lio, Mishal Patel, AIX-COVNET Collaboration, James H. F. Rudd, Tuomas Mirtti, Antti Rannikko, John A. D. Aston, Jing Tang, Carola-Bibiane Schönlieb

Classifying samples in incomplete datasets is a common aim for machine learning practitioners, but is non-trivial. Missing data is found in most real-world datasets and these missing values are typically imputed using established methods, followed by classification of the now complete, imputed, samples. The focus of the machine learning researcher is then to optimise the downstream classification performance. In this study, we highlight that it is imperative to consider the quality of the imputation. We demonstrate how the commonly used measures for assessing quality are flawed and propose a new class of discrepancy scores which focus on how well the method recreates the overall distribution of the data. To conclude, we highlight the compromised interpretability of classifier models trained using poorly imputed data.

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Georgios Batzolis, Jan Stanczuk, Carola-Bibiane Schönlieb, Christian Etmann

Score-based diffusion models have emerged as one of the most promising frameworks for deep generative modelling. In this work we conduct a systematic comparison and theoretical analysis of different approaches to learning conditional probability distributions with score-based diffusion models. In particular, we prove results which provide a theoretical justification for one of the most successful estimators of the conditional score. Moreover, we introduce a multi-speed diffusion framework, which leads to a new estimator for the conditional score, performing on par with previous state-of-the-art approaches. Our theoretical and experimental findings are accompanied by an open source library MSDiff which allows for application and further research of multi-speed diffusion models.

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Jan Stanczuk, Christian Etmann, Lisa Maria Kreusser, Carola-Bibiane Schönlieb

Wasserstein GANs are based on the idea of minimising the Wasserstein distance between a real and a generated distribution. We provide an in-depth mathematical analysis of differences between the theoretical setup and the reality of training Wasserstein GANs. In this work, we gather both theoretical and empirical evidence that the WGAN loss is not a meaningful approximation of the Wasserstein distance. Moreover, we argue that the Wasserstein distance is not even a desirable loss function for deep generative models, and conclude that the success of Wasserstein GANs can in truth be attributed to a failure to approximate the Wasserstein distance.

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