The Hidden Linear Structure in Score-Based Models and its Application

Score-based models have achieved remarkable results in the generative modeling of many domains. By learning the gradient of smoothed data distribution, they can iteratively generate samples from complex distribution e.g. natural images. However, is there any universal structure in the gradient field that will eventually be learned by any neural network? Here, we aim to find such structures through a normative analysis of the score function. First, we derived the closed-form solution to the scored-based model with a Gaussian score. We claimed that for well-trained diffusion models, the learned score at a high noise scale is well approximated by the linear score of Gaussian. We demonstrated this through empirical validation of pre-trained images diffusion model and theoretical analysis of the score function. This finding enabled us to precisely predict the initial diffusion trajectory using the analytical solution and to accelerate image sampling by 15-30\% by skipping the initial phase without sacrificing image quality. Our finding of the linear structure in the score-based model has implications for better model design and data pre-processing.

Diffusion Models Generate Images Like Painters: an Analytical Theory of Outline First, Details Later

How do diffusion generative models convert pure noise into meaningful images? We argue that generation involves first committing to an outline, and then to finer and finer details. The corresponding reverse diffusion process can be modeled by dynamics on a (time-dependent) high-dimensional landscape full of Gaussian-like modes, which makes the following predictions: (i) individual trajectories tend to be very low-dimensional; (ii) scene elements that vary more within training data tend to emerge earlier; and (iii) early perturbations substantially change image content more often than late perturbations. We show that the behavior of a variety of trained unconditional and conditional diffusion models like Stable Diffusion is consistent with these predictions. Finally, we use our theory to search for the latent image manifold of diffusion models, and propose a new way to generate interpretable image variations. Our viewpoint suggests generation by GANs and diffusion models have unexpected similarities.

On the Level Sets and Invariance of Neural Tuning Landscapes

Visual representations can be defined as the activations of neuronal populations in response to images. The activation of a neuron as a function over all image space has been described as a "tuning landscape". As a function over a high-dimensional space, what is the structure of this landscape? In this study, we characterize tuning landscapes through the lens of level sets and Morse theory. A recent study measured the in vivo two-dimensional tuning maps of neurons in different brain regions. Here, we developed a statistically reliable signature for these maps based on the change of topology in level sets. We found this topological signature changed progressively throughout the cortical hierarchy, with similar trends found for units in convolutional neural networks (CNNs). Further, we analyzed the geometry of level sets on the tuning landscapes of CNN units. We advanced the hypothesis that higher-order units can be locally regarded as isotropic radial basis functions, but not globally. This shows the power of level sets as a conceptual tool to understand neuronal activations over image space.

High-performance Evolutionary Algorithms for Online Neuron Control

Recently, optimization has become an emerging tool for neuroscientists to study neural code. In the visual system, neurons respond to images with graded and noisy responses. Image patterns eliciting highest responses are diagnostic of the coding content of the neuron. To find these patterns, we have used black-box optimizers to search a 4096d image space, leading to the evolution of images that maximize neuronal responses. Although genetic algorithm (GA) has been commonly used, there haven't been any systematic investigations to reveal the best performing optimizer or the underlying principles necessary to improve them. Here, we conducted a large scale in silico benchmark of optimizers for activation maximization and found that Covariance Matrix Adaptation (CMA) excelled in its achieved activation. We compared CMA against GA and found that CMA surpassed the maximal activation of GA by 66% in silico and 44% in vivo. We analyzed the structure of Evolution trajectories and found that the key to success was not covariance matrix adaptation, but local search towards informative dimensions and an effective step size decay. Guided by these principles and the geometry of the image manifold, we developed SphereCMA optimizer which competed well against CMA, proving the validity of the identified principles. Code available at https://github.com/Animadversio/ActMax-Optimizer-Dev

On the use of Cortical Magnification and Saccades as Biological Proxies for Data Augmentation

Self-supervised learning is a powerful way to learn useful representations from natural data. It has also been suggested as one possible means of building visual representation in humans, but the specific objective and algorithm are unknown. Currently, most self-supervised methods encourage the system to learn an invariant representation of different transformations of the same image in contrast to those of other images. However, such transformations are generally non-biologically plausible, and often consist of contrived perceptual schemes such as random cropping and color jittering. In this paper, we attempt to reverse-engineer these augmentations to be more biologically or perceptually plausible while still conferring the same benefits for encouraging robust representation. Critically, we find that random cropping can be substituted by cortical magnification, and saccade-like sampling of the image could also assist the representation learning. The feasibility of these transformations suggests a potential way that biological visual systems could implement self-supervision. Further, they break the widely accepted spatially-uniform processing assumption used in many computer vision algorithms, suggesting a role for spatially-adaptive computation in humans and machines alike. Our code and demo can be found here.

The Geometry of Deep Generative Image Models and its Applications

Generative adversarial networks (GANs) have emerged as a powerful unsupervised method to model the statistical patterns of real-world data sets, such as natural images. These networks are trained to map random inputs in their latent space to new samples representative of the learned data. However, the structure of the latent space is hard to intuit due to its high dimensionality and the non-linearity of the generator, which limits the usefulness of the models. Understanding the latent space requires a way to identify input codes for existing real-world images (inversion), and a way to identify directions with known image transformations (interpretability). Here, we use a geometric framework to address both issues simultaneously. We develop an architecture-agnostic method to compute the Riemannian metric of the image manifold created by GANs. The eigen-decomposition of the metric isolates axes that account for different levels of image variability. An empirical analysis of several pretrained GANs shows that image variation around each position is concentrated along surprisingly few major axes (the space is highly anisotropic) and the directions that create this large variation are similar at different positions in the space (the space is homogeneous). We show that many of the top eigenvectors correspond to interpretable transforms in the image space, with a substantial part of eigenspace corresponding to minor transforms which could be compressed out. This geometric understanding unifies key previous results related to GAN interpretability. We show that the use of this metric allows for more efficient optimization in the latent space (e.g. GAN inversion) and facilitates unsupervised discovery of interpretable axes. Our results illustrate that defining the geometry of the GAN image manifold can serve as a general framework for understanding GANs.