Latent Abstractions in Generative Diffusion Models

In this work we study how diffusion-based generative models produce high-dimensional data, such as an image, by implicitly relying on a manifestation of a low-dimensional set of latent abstractions, that guide the generative process. We present a novel theoretical framework that extends NLF, and that offers a unique perspective on SDE-based generative models. The development of our theory relies on a novel formulation of the joint (state and measurement) dynamics, and an information-theoretic measure of the influence of the system state on the measurement process. According to our theory, diffusion models can be cast as a system of SDE, describing a non-linear filter in which the evolution of unobservable latent abstractions steers the dynamics of an observable measurement process (corresponding to the generative pathways). In addition, we present an empirical study to validate our theory and previous empirical results on the emergence of latent abstractions at different stages of the generative process.

Information Theoretic Text-to-Image Alignment

Diffusion models for Text-to-Image (T2I) conditional generation have seen tremendous success recently. Despite their success, accurately capturing user intentions with these models still requires a laborious trial and error process. This challenge is commonly identified as a model alignment problem, an issue that has attracted considerable attention by the research community. Instead of relying on fine-grained linguistic analyses of prompts, human annotation, or auxiliary vision-language models to steer image generation, in this work we present a novel method that relies on an information-theoretic alignment measure. In a nutshell, our method uses self-supervised fine-tuning and relies on point-wise mutual information between prompts and images to define a synthetic training set to induce model alignment. Our comparative analysis shows that our method is on-par or superior to the state-of-the-art, yet requires nothing but a pre-trained denoising network to estimate MI and a lightweight fine-tuning strategy.

S$Ω$I: Score-based O-INFORMATION Estimation

The analysis of scientific data and complex multivariate systems requires information quantities that capture relationships among multiple random variables. Recently, new information-theoretic measures have been developed to overcome the shortcomings of classical ones, such as mutual information, that are restricted to considering pairwise interactions. Among them, the concept of information synergy and redundancy is crucial for understanding the high-order dependencies between variables. One of the most prominent and versatile measures based on this concept is O-information, which provides a clear and scalable way to quantify the synergy-redundancy balance in multivariate systems. However, its practical application is limited to simplified cases. In this work, we introduce S$\Omega$I, which allows for the first time to compute O-information without restrictive assumptions about the system. Our experiments validate our approach on synthetic data, and demonstrate the effectiveness of S$\Omega$I in the context of a real-world use case.

MINDE: Mutual Information Neural Diffusion Estimation

In this work we present a new method for the estimation of Mutual Information (MI) between random variables. Our approach is based on an original interpretation of the Girsanov theorem, which allows us to use score-based diffusion models to estimate the Kullback Leibler divergence between two densities as a difference between their score functions. As a by-product, our method also enables the estimation of the entropy of random variables. Armed with such building blocks, we present a general recipe to measure MI, which unfolds in two directions: one uses conditional diffusion process, whereas the other uses joint diffusion processes that allow simultaneous modelling of two random variables. Our results, which derive from a thorough experimental protocol over all the variants of our approach, indicate that our method is more accurate than the main alternatives from the literature, especially for challenging distributions. Furthermore, our methods pass MI self-consistency tests, including data processing and additivity under independence, which instead are a pain-point of existing methods.

Multi-modal Latent Diffusion

Multi-modal data-sets are ubiquitous in modern applications, and multi-modal Variational Autoencoders are a popular family of models that aim to learn a joint representation of the different modalities. However, existing approaches suffer from a coherence-quality tradeoff, where models with good generation quality lack generative coherence across modalities, and vice versa. We discuss the limitations underlying the unsatisfactory performance of existing methods, to motivate the need for a different approach. We propose a novel method that uses a set of independently trained, uni-modal, deterministic autoencoders. Individual latent variables are concatenated into a common latent space, which is fed to a masked diffusion model to enable generative modeling. We also introduce a new multi-time training method to learn the conditional score network for multi-modal diffusion. Our methodology substantially outperforms competitors in both generation quality and coherence, as shown through an extensive experimental campaign.

One-Line-of-Code Data Mollification Improves Optimization of Likelihood-based Generative Models

Generative Models (GMs) have attracted considerable attention due to their tremendous success in various domains, such as computer vision where they are capable to generate impressive realistic-looking images. Likelihood-based GMs are attractive due to the possibility to generate new data by a single model evaluation. However, they typically achieve lower sample quality compared to state-of-the-art score-based diffusion models (DMs). This paper provides a significant step in the direction of addressing this limitation. The idea is to borrow one of the strengths of score-based DMs, which is the ability to perform accurate density estimation in low-density regions and to address manifold overfitting by means of data mollification. We connect data mollification through the addition of Gaussian noise to Gaussian homotopy, which is a well-known technique to improve optimization. Data mollification can be implemented by adding one line of code in the optimization loop, and we demonstrate that this provides a boost in generation quality of likelihood-based GMs, without computational overheads. We report results on image data sets with popular likelihood-based GMs, including variants of variational autoencoders and normalizing flows, showing large improvements in FID score.

Continuous-Time Functional Diffusion Processes

We introduce functional diffusion processes (FDPs), which generalize traditional score-based diffusion models to infinite-dimensional function spaces. FDPs require a new mathematical framework to describe the forward and backward dynamics, and several extensions to derive practical training objectives. These include infinite-dimensional versions of the Girsanov theorem, in order to be able to compute an ELBO, and of the sampling theorem, in order to guarantee that functional evaluations in a countable set of points are equivalent to infinite-dimensional functions. We use FDPs to build a new breed of generative models in function spaces, which do not require specialized network architectures, and that can work with any kind of continuous data. Our results on synthetic and real data illustrate the advantages of FDPs in simplifying the design requirements of diffusion models.

How Much is Enough? A Study on Diffusion Times in Score-based Generative Models

Score-based diffusion models are a class of generative models whose dynamics is described by stochastic differential equations that map noise into data. While recent works have started to lay down a theoretical foundation for these models, an analytical understanding of the role of the diffusion time T is still lacking. Current best practice advocates for a large T to ensure that the forward dynamics brings the diffusion sufficiently close to a known and simple noise distribution; however, a smaller value of T should be preferred for a better approximation of the score-matching objective and higher computational efficiency. Starting from a variational interpretation of diffusion models, in this work we quantify this trade-off, and suggest a new method to improve quality and efficiency of both training and sampling, by adopting smaller diffusion times. Indeed, we show how an auxiliary model can be used to bridge the gap between the ideal and the simulated forward dynamics, followed by a standard reverse diffusion process. Empirical results support our analysis; for image data, our method is competitive w.r.t. the state-of-the-art, according to standard sample quality metrics and log-likelihood.

A Unified View of Stochastic Hamiltonian Sampling

In this work, we revisit the theoretical properties of Hamiltonian stochastic differential equations (SDEs) for Bayesian posterior sampling, and we study the two types of errors that arise from numerical SDE simulation: the discretization error and the error due to noisy gradient estimates in the context of data subsampling. We consider overlooked results describing the ergodic convergence rates of numerical integration schemes, and we produce a novel analysis for the effect of mini-batches through the lens of differential operator splitting. In our analysis, the stochastic component of the proposed Hamiltonian SDE is decoupled from the gradient noise, for which we make no normality assumptions. This allows us to derive interesting connections among different sampling schemes, including the original Hamiltonian Monte Carlo (HMC) algorithm, and explain their performance. We show that for a careful selection of numerical integrators, both errors vanish at a rate $\mathcal{O}(\eta^2)$, where $\eta$ is the integrator step size. Our theoretical results are supported by an empirical study on a variety of regression and classification tasks for Bayesian neural networks.

Isotropic SGD: a Practical Approach to Bayesian Posterior Sampling

In this work we define a unified mathematical framework to deepen our understanding of the role of stochastic gradient (SG) noise on the behavior of Markov chain Monte Carlo sampling (SGMCMC) algorithms. Our formulation unlocks the design of a novel, practical approach to posterior sampling, which makes the SG noise isotropic using a fixed learning rate that we determine analytically, and that requires weaker assumptions than existing algorithms. In contrast, the common traits of existing \sgmcmc algorithms is to approximate the isotropy condition either by drowning the gradients in additive noise (annealing the learning rate) or by making restrictive assumptions on the \sg noise covariance and the geometry of the loss landscape. Extensive experimental validations indicate that our proposal is competitive with the state-of-the-art on \sgmcmc, while being much more practical to use.