Dynamic Negative Guidance of Diffusion Models

Negative Prompting (NP) is widely utilized in diffusion models, particularly in text-to-image applications, to prevent the generation of undesired features. In this paper, we show that conventional NP is limited by the assumption of a constant guidance scale, which may lead to highly suboptimal results, or even complete failure, due to the non-stationarity and state-dependence of the reverse process. Based on this analysis, we derive a principled technique called Dynamic Negative Guidance, which relies on a near-optimal time and state dependent modulation of the guidance without requiring additional training. Unlike NP, negative guidance requires estimating the posterior class probability during the denoising process, which is achieved with limited additional computational overhead by tracking the discrete Markov Chain during the generative process. We evaluate the performance of DNG class-removal on MNIST and CIFAR10, where we show that DNG leads to higher safety, preservation of class balance and image quality when compared with baseline methods. Furthermore, we show that it is possible to use DNG with Stable Diffusion to obtain more accurate and less invasive guidance than NP.

Losing dimensions: Geometric memorization in generative diffusion

Generative diffusion processes are state-of-the-art machine learning models deeply connected with fundamental concepts in statistical physics. Depending on the dataset size and the capacity of the network, their behavior is known to transition from an associative memory regime to a generalization phase in a phenomenon that has been described as a glassy phase transition. Here, using statistical physics techniques, we extend the theory of memorization in generative diffusion to manifold-supported data. Our theoretical and experimental findings indicate that different tangent subspaces are lost due to memorization effects at different critical times and dataset sizes, which depend on the local variance of the data along their directions. Perhaps counterintuitively, we find that, under some conditions, subspaces of higher variance are lost first due to memorization effects. This leads to a selective loss of dimensionality where some prominent features of the data are memorized without a full collapse on any individual training point. We validate our theory with a comprehensive set of experiments on networks trained both in image datasets and on linear manifolds, which result in a remarkable qualitative agreement with the theoretical predictions.

Manifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusion

In this paper, we investigate the latent geometry of generative diffusion models under the manifold hypothesis. To this purpose, we analyze the spectrum of eigenvalues (and singular values) of the Jacobian of the score function, whose discontinuities (gaps) reveal the presence and dimensionality of distinct sub-manifolds. Using a statistical physics approach, we derive the spectral distributions and formulas for the spectral gaps under several distributional assumptions and we compare these theoretical predictions with the spectra estimated from trained networks. Our analysis reveals the existence of three distinct qualitative phases during the generative process: a trivial phase; a manifold coverage phase where the diffusion process fits the distribution internal to the manifold; a consolidation phase where the score becomes orthogonal to the manifold and all particles are projected on the support of the data. This `division of labor' between different timescales provides an elegant explanation on why generative diffusion models are not affected by the manifold overfitting phenomenon that plagues likelihood-based models, since the internal distribution and the manifold geometry are produced at different time points during generation.

Robust and highly scalable estimation of directional couplings from time-shifted signals

The estimation of directed couplings between the nodes of a network from indirect measurements is a central methodological challenge in scientific fields such as neuroscience, systems biology and economics. Unfortunately, the problem is generally ill-posed due to the possible presence of unknown delays in the measurements. In this paper, we offer a solution of this problem by using a variational Bayes framework, where the uncertainty over the delays is marginalized in order to obtain conservative coupling estimates. To overcome the well-known overconfidence of classical variational methods, we use a hybrid-VI scheme where the (possibly flat or multimodal) posterior over the measurement parameters is estimated using a forward KL loss while the (nearly convex) conditional posterior over the couplings is estimated using the highly scalable gradient-based VI. In our ground-truth experiments, we show that the network provides reliable and conservative estimates of the couplings, greatly outperforming similar methods such as regression DCM.

The statistical thermodynamics of generative diffusion models

Generative diffusion models have achieved spectacular performance in many areas of generative modeling. While the fundamental ideas behind these models come from non-equilibrium physics, in this paper we show that many aspects of these models can be understood using the tools of equilibrium statistical mechanics. Using this reformulation, we show that generative diffusion models undergo second-order phase transitions corresponding to symmetry breaking phenomena. We argue that this lead to a form of instability that lies at the heart of their generative capabilities and that can be described by a set of mean field critical exponents. We conclude by analyzing recent work connecting diffusion models and associative memory networks in view of the thermodynamic formulations.

Stationarity without mean reversion: Improper Gaussian process regression and improper kernels

Gaussian processes (GP) regression has gained substantial popularity in machine learning applications. The behavior of a GP regression depends on the choice of covariance function. Stationary covariance functions are favorite in machine learning applications. However, (non-periodic) stationary covariance functions are always mean reverting and can therefore exhibit pathological behavior when applied to data that does not relax to a fixed global mean value. In this paper, we show that it is possible to use improper GP prior with infinite variance to define processes that are stationary but not mean reverting. To this aim, we introduce a large class of improper kernels that can only be defined in this improper regime. Specifically, we introduce the Smooth Walk kernel, which produces infinitely smooth samples, and a family of improper Mat\'ern kernels, which can be defined to be $j$-times differentiable for any integer $j$. The resulting posterior distributions can be computed analytically and it involves a simple correction of the usual formulas. By analyzing both synthetic and real data, we demonstrate that these improper kernels solve some known pathologies of mean reverting GP regression while retaining most of the favourable properties of ordinary smooth stationary kernels.

In search of dispersed memories: Generative diffusion models are associative memory networks

Hopfield networks are widely used in neuroscience as simplified theoretical models of biological associative memory. The original Hopfield networks store memories by encoding patterns of binary associations, which result in a synaptic learning mechanism known as Hebbian learning rule. Modern Hopfield networks can achieve exponential capacity scaling by using highly non-linear energy functions. However, the energy function of these newer models cannot be straightforwardly compressed into binary synaptic couplings and it does not directly provide new synaptic learning rules. In this work we show that generative diffusion models can be interpreted as energy-based models and that, when trained on discrete patterns, their energy function is equivalent to that of modern Hopfield networks. This equivalence allows us to interpret the supervised training of diffusion models as a synaptic learning process that encodes the associative dynamics of a modern Hopfield network in the weight structure of a deep neural network. Accordingly, in our experiments we show that the storage capacity of a continuous modern Hopfield network is identical to the capacity of a diffusion model. Our results establish a strong link between generative modeling and the theoretical neuroscience of memory, which provide a powerful computational foundation for the reconstructive theory of memory, where creative generation and memory recall can be seen as parts of a unified continuum.

Spontaneous symmetry breaking in generative diffusion models

Generative diffusion models have recently emerged as a leading approach for generating high-dimensional data. In this paper, we show that the dynamics of these models exhibit a spontaneous symmetry breaking that divides the generative dynamics into two distinct phases: 1) A linear steady-state dynamics around a central fixed-point and 2) an attractor dynamics directed towards the data manifold. These two "phases" are separated by the change in stability of the central fixed-point, with the resulting window of instability being responsible for the diversity of the generated samples. Using both theoretical and empirical evidence, we show that an accurate simulation of the early dynamics does not significantly contribute to the final generation, since early fluctuations are reverted to the central fixed point. To leverage this insight, we propose a Gaussian late initialization scheme, which significantly improves model performance, achieving up to 3x FID improvements on fast samplers, while also increasing sample diversity (e.g., racial composition of generated CelebA images). Our work offers a new way to understand the generative dynamics of diffusion models that has the potential to bring about higher performance and less biased fast-samplers.

Closing the gap: Exact maximum likelihood training of generative autoencoders using invertible layers

In this work, we provide an exact likelihood alternative to the variational training of generative autoencoders. We show that VAE-style autoencoders can be constructed using invertible layers, which offer a tractable exact likelihood without the need for any regularization terms. This is achieved while leaving complete freedom in the choice of encoder, decoder and prior architectures, making our approach a drop-in replacement for the training of existing VAEs and VAE-style models. We refer to the resulting models as Autoencoders within Flows (AEF), since the encoder, decoder and prior are defined as individual layers of an overall invertible architecture. We show that the approach results in strikingly higher performance than architecturally equivalent VAEs in term of log-likelihood, sample quality and denoising performance. In a broad sense, the main ambition of this work is to close the gap between the normalizing flow and autoencoder literature under the common framework of invertibility and exact maximum likelihood.

Embedded-model flows: Combining the inductive biases of model-free deep learning and explicit probabilistic modeling

Normalizing flows have shown great success as general-purpose density estimators. However, many real world applications require the use of domain-specific knowledge, which normalizing flows cannot readily incorporate. We propose embedded-model flows (EMF), which alternate general-purpose transformations with structured layers that embed domain-specific inductive biases. These layers are automatically constructed by converting user-specified differentiable probabilistic models into equivalent bijective transformations. We also introduce gated structured layers, which allow bypassing the parts of the models that fail to capture the statistics of the data. We demonstrate that EMFs can be used to induce desirable properties such as multimodality, hierarchical coupling and continuity. Furthermore, we show that EMFs enable a high performance form of variational inference where the structure of the prior model is embedded in the variational architecture. In our experiments, we show that this approach outperforms state-of-the-art methods in common structured inference problems.