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1. Generalization in diffusion models arises from geometry-adaptive harmonic representations

   Kadkhodaie, Zahra; Guth, Florentin; Simoncelli, Eero; Mallat, Stéphane (May 2024, International Conference on Learning Representations 2024)

   Deep neural networks (DNNs) trained for image denoising are able to generate high-quality samples with score-based reverse diffusion algorithms. These impressive capabilities seem to imply an escape from the curse of dimensionality, but recent reports of memorization of the training set raise the question of whether these networks are learning the "true" continuous density of the data. Here, we show that two DNNs trained on non-overlapping subsets of a dataset learn nearly the same score function, and thus the same density, when the number of training images is large enough. In this regime of strong generalization, diffusion-generated images are distinct from the training set, and are of high visual quality, suggesting that the inductive biases of the DNNs are well-aligned with the data density. We analyze the learned denoising functions and show that the inductive biases give rise to a shrinkage operation in a basis adapted to the underlying image. Examination of these bases reveals oscillating harmonic structures along contours and in homogeneous regions. We demonstrate that trained denoisers are inductively biased towards these geometry-adaptive harmonic bases since they arise not only when the network is trained on photographic images, but also when it is trained on image classes supported on low-dimensional manifolds for which the harmonic basis is suboptimal. Finally, we show that when trained on regular image classes for which the optimal basis is known to be geometry-adaptive and harmonic, the denoising performance of the networks is near-optimal 

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2. Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data

   Chen, Minshuo; Huang, Kaixuan; Zhao, Tuo; Wang, Mengdi (July 2023, Proceedings of Machine Learning Research)

   Diffusion models achieve state-of-the-art performance in various generation tasks. However, their theoretical foundations fall far behind. This paper studies score approximation, estimation, and distribution recovery of diffusion models, when data are supported on an unknown low-dimensional linear subspace. Our result provides sample complexity bounds for distribution estimation using diffusion models. We show that with a properly chosen neural network architecture, the score function can be both accurately approximated and efficiently estimated. Further, the generated distribution based on the estimated score function captures the data geometric structures and converges to a close vicinity of the data distribution. The convergence rate depends on subspace dimension, implying that diffusion models can circumvent the curse of data ambient dimensionality. 

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3. A scale-dependent measure of system dimensionality

   https://doi.org/10.1016/j.patter.2022.100  

   Recanatesi, S.; Bradde, S.; Balasubramanian, V.; Steinmetz, N.; Shea-Brown, E. (January 2022, Patterns)

   A fundamental problem in science is uncovering the effective number of degrees of freedom in a complex system: its dimensionality. A system’s dimensionality depends on its spatiotemporal scale. Here, we introduce a scale-dependent generalization of a classic enumeration of latent variables, the participation ratio. We demonstrate how the scale-dependent participation ratio identifies the appropriate dimension at local, intermediate, and global scales in several systems such as the Lorenz attractor, hidden Markov models, and switching linear dynamical systems. We show analytically how, at different limiting scales, the scale-dependent participation ratio relates to well-established measures of dimensionality. This measure applied in neural population recordings across multiple brain areas and brain states shows fundamental trends in the dimensionality of neural activity—for example, in behaviorally engaged versus spontaneous states. Our novel method unifies widely used measures of dimensionality and applies broadly to multivariate data across several fields of science. 

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4. Adaptivity of Diffusion Models to Manifold Structures

   Tang, Rong; Yang, Yun (May 2024, Proceedings of Machine Learning Research)

   Empirical studies have demonstrated the effectiveness of (score-based) diffusion models in generating high-dimensional data, such as texts and images, which typically exhibit a low-dimensional manifold nature. These empirical successes raise the theoretical question of whether score-based diffusion models can optimally adapt to low-dimensional manifold structures. While recent work has validated the minimax optimality of diffusion models when the target distribution admits a smooth density with respect to the Lebesgue measure of the ambient data space, these findings do not fully account for the ability of diffusion models in avoiding the the curse of dimensionality when estimating high-dimensional distributions. This work considers two common classes of diffusion models: Langevin diffusion and forward-backward diffusion. We show that both models can adapt to the intrinsic manifold structure by showing that the convergence rate of the inducing distribution estimator depends only on the intrinsic dimension of the data. Moreover, our considered estimator does not require knowing or explicitly estimating the manifold. We also demonstrate that the forward-backward diffusion can achieve the minimax optimal rate under the Wasserstein metric when the target distribution possesses a smooth density with respect to the volume measure of the low-dimensional manifold. 

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5. Evaluation, Tuning, and Interpretation of Neural Networks for Working with Images in Meteorological Applications

   https://doi.org/10.1175/BAMS-D-20-0097.1  

   Ebert-Uphoff, Imme; Hilburn, Kyle (December 2020, Bulletin of the American Meteorological Society)

   null (Ed.)

   Abstract The method of neural networks (aka deep learning) has opened up many new opportunities to utilize remotely sensed images in meteorology. Common applications include image classification, e.g., to determine whether an image contains a tropical cyclone, and image-to-image translation, e.g., to emulate radar imagery for satellites that only have passive channels. However, there are yet many open questions regarding the use of neural networks for working with meteorological images, such as best practices for evaluation, tuning, and interpretation. This article highlights several strategies and practical considerations for neural network development that have not yet received much attention in the meteorological community, such as the concept of receptive fields, underutilized meteorological performance measures, and methods for neural network interpretation, such as synthetic experiments and layer-wise relevance propagation. We also consider the process of neural network interpretation as a whole, recognizing it as an iterative meteorologist-driven discovery process that builds on experimental design and hypothesis generation and testing. Finally, while most work on neural network interpretation in meteorology has so far focused on networks for image classification tasks, we expand the focus to also include networks for image-to-image translation. 

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