Upsample Guidance: Scale Up Diffusion Models without Training

Diffusion models have demonstrated superior performance across various generative tasks including images, videos, and audio. However, they encounter difficulties in directly generating high-resolution samples. Previously proposed solutions to this issue involve modifying the architecture, further training, or partitioning the sampling process into multiple stages. These methods have the limitation of not being able to directly utilize pre-trained models as-is, requiring additional work. In this paper, we introduce upsample guidance, a technique that adapts pretrained diffusion model (e.g., $512^2$) to generate higher-resolution images (e.g., $1536^2$) by adding only a single term in the sampling process. Remarkably, this technique does not necessitate any additional training or relying on external models. We demonstrate that upsample guidance can be applied to various models, such as pixel-space, latent space, and video diffusion models. We also observed that the proper selection of guidance scale can improve image quality, fidelity, and prompt alignment.

GNRK: Graph Neural Runge-Kutta method for solving partial differential equations

Neural networks have proven to be efficient surrogate models for tackling partial differential equations (PDEs). However, their applicability is often confined to specific PDEs under certain constraints, in contrast to classical PDE solvers that rely on numerical differentiation. Striking a balance between efficiency and versatility, this study introduces a novel approach called Graph Neural Runge-Kutta (GNRK), which integrates graph neural network modules with a recurrent structure inspired by the classical solvers. The GNRK operates on graph structures, ensuring its resilience to changes in spatial and temporal resolutions during domain discretization. Moreover, it demonstrates the capability to address general PDEs, irrespective of initial conditions or PDE coefficients. To assess its performance, we benchmark the GNRK against existing neural network based PDE solvers using the 2-dimensional Burgers' equation, revealing the GNRK's superiority in terms of model size and accuracy. Additionally, this graph-based methodology offers a straightforward extension for solving coupled differential equations, typically necessitating more intricate models.

Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometry

Despite the success of diffusion models (DMs), we still lack a thorough understanding of their latent space. To understand the latent space $\mathbf{x}_t \in \mathcal{X}$, we analyze them from a geometrical perspective. Specifically, we utilize the pullback metric to find the local latent basis in $\mathcal{X}$ and their corresponding local tangent basis in $\mathcal{H}$, the intermediate feature maps of DMs. The discovered latent basis enables unsupervised image editing capability through latent space traversal. We investigate the discovered structure from two perspectives. First, we examine how geometric structure evolves over diffusion timesteps. Through analysis, we show that 1) the model focuses on low-frequency components early in the generative process and attunes to high-frequency details later; 2) At early timesteps, different samples share similar tangent spaces; and 3) The simpler datasets that DMs trained on, the more consistent the tangent space for each timestep. Second, we investigate how the geometric structure changes based on text conditioning in Stable Diffusion. The results show that 1) similar prompts yield comparable tangent spaces; and 2) the model depends less on text conditions in later timesteps. To the best of our knowledge, this paper is the first to present image editing through $\mathbf{x}$-space traversal and provide thorough analyses of the latent structure of DMs.

Tradeoff of generalization error in unsupervised learning

Finding the optimal model complexity that minimizes the generalization error (GE) is a key issue of machine learning. For the conventional supervised learning, this task typically involves the bias-variance tradeoff: lowering the bias by making the model more complex entails an increase in the variance. Meanwhile, little has been studied about whether the same tradeoff exists for unsupervised learning. In this study, we propose that unsupervised learning generally exhibits a two-component tradeoff of the GE, namely the model error and the data error -- using a more complex model reduces the model error at the cost of the data error, with the data error playing a more significant role for a smaller training dataset. This is corroborated by training the restricted Boltzmann machine to generate the configurations of the two-dimensional Ising model at a given temperature and the totally asymmetric simple exclusion process with given entry and exit rates. Our results also indicate that the optimal model tends to be more complex when the data to be learned are more complex.

Unsupervised Discovery of Semantic Latent Directions in Diffusion Models

Despite the success of diffusion models (DMs), we still lack a thorough understanding of their latent space. While image editing with GANs builds upon latent space, DMs rely on editing the conditions such as text prompts. We present an unsupervised method to discover interpretable editing directions for the latent variables $\mathbf{x}_t \in \mathcal{X}$ of DMs. Our method adopts Riemannian geometry between $\mathcal{X}$ and the intermediate feature maps $\mathcal{H}$ of the U-Nets to provide a deep understanding over the geometrical structure of $\mathcal{X}$. The discovered semantic latent directions mostly yield disentangled attribute changes, and they are globally consistent across different samples. Furthermore, editing in earlier timesteps edits coarse attributes, while ones in later timesteps focus on high-frequency details. We define the curvedness of a line segment between samples to show that $\mathcal{X}$ is a curved manifold. Experiments on different baselines and datasets demonstrate the effectiveness of our method even on Stable Diffusion. Our source code will be publicly available for the future researchers.

Mirror descent of Hopfield model

Mirror descent is a gradient descent method that uses a dual space of parametric models. The great idea has been developed in convex optimization, but not yet widely applied in machine learning. In this study, we provide a possible way that the mirror descent can help data-driven parameter initialization of neural networks. We adopt the Hopfield model as a prototype of neural networks, we demonstrate that the mirror descent can train the model more effectively than the usual gradient descent with random parameter initialization.

Scale-invariant representation of machine learning

The success of machine learning stems from its structured data representation. Similar data have close representation as compressed codes for classification or emerged labels for clustering. We observe that the frequency of the internal representation follows power laws in both supervised and unsupervised learning. The scale-invariant distribution implies that machine learning largely compresses frequent typical data, and at the same time, differentiates many atypical data as outliers. In this study, we derive how the power laws can naturally arise in machine learning. In terms of information theory, the scale-invariant representation corresponds to a maximally uncertain data grouping among possible representations that guarantee pre-specified learning accuracy.

Compression phase is not necessary for generalization in representation learning

The outstanding performance of deep learning in various fields has been a fundamental query, which can be potentially examined using information theory that interprets the learning process as the transmission and compression of information. Information plane analyses of the mutual information between the input-hidden-output layers demonstrated two distinct learning phases of fitting and compression. It is debatable if the compression phase is necessary to generalize the input-output relations extracted from training data. In this study, we investigated this through experiments with various species of autoencoders and evaluated their information processing phase with an accurate kernel-based estimator of mutual information. Given sufficient training data, vanilla autoencoders demonstrated the compression phase, which was amplified after imposing sparsity regularization for hidden activities. However, we found that the compression phase is not universally observed in different species of autoencoders, including variational autoencoders, that have special constraints on network weights or manifold of hidden space. These types of autoencoders exhibited perfect generalization ability for test data without requiring the compression phase. Thus, we conclude that the compression phase is not necessary for generalization in representation learning.

Inference of stochastic time series with missing data

Inferring dynamics from time series is an important objective in data analysis. In particular, it is challenging to infer stochastic dynamics given incomplete data. We propose an expectation maximization (EM) algorithm that iterates between alternating two steps: E-step restores missing data points, while M-step infers an underlying network model of restored data. Using synthetic data generated by a kinetic Ising model, we confirm that the algorithm works for restoring missing data points as well as inferring the underlying model. At the initial iteration of the EM algorithm, the model inference shows better model-data consistency with observed data points than with missing data points. As we keep iterating, however, missing data points show better model-data consistency. We find that demanding equal consistency of observed and missing data points provides an effective stopping criterion for the iteration to prevent overshooting the most accurate model inference. Armed with this EM algorithm with this stopping criterion, we infer missing data points and an underlying network from a time-series data of real neuronal activities. Our method recovers collective properties of neuronal activities, such as time correlations and firing statistics, which have previously never been optimized to fit.

Tractable loss function and color image generation of multinary restricted Boltzmann machine

The restricted Boltzmann machine (RBM) is a representative generative model based on the concept of statistical mechanics. In spite of the strong merit of interpretability, unavailability of backpropagation makes it less competitive than other generative models. Here we derive differentiable loss functions for both binary and multinary RBMs. Then we demonstrate their learnability and performance by generating colored face images.