A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models

High-dimensional data commonly lies on low-dimensional submanifolds, and estimating the local intrinsic dimension (LID) of a datum -- i.e. the dimension of the submanifold it belongs to -- is a longstanding problem. LID can be understood as the number of local factors of variation: the more factors of variation a datum has, the more complex it tends to be. Estimating this quantity has proven useful in contexts ranging from generalization in neural networks to detection of out-of-distribution data, adversarial examples, and AI-generated text. The recent successes of deep generative models present an opportunity to leverage them for LID estimation, but current methods based on generative models produce inaccurate estimates, require more than a single pre-trained model, are computationally intensive, or do not exploit the best available deep generative models, i.e. diffusion models (DMs). In this work, we show that the Fokker-Planck equation associated with a DM can provide a LID estimator which addresses all the aforementioned deficiencies. Our estimator, called FLIPD, is compatible with all popular DMs, and outperforms existing baselines on LID estimation benchmarks. We also apply FLIPD on natural images where the true LID is unknown. Compared to competing estimators, FLIPD exhibits a higher correlation with non-LID measures of complexity, better matches a qualitative assessment of complexity, and is the only estimator to remain tractable with high-resolution images at the scale of Stable Diffusion.

Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections

In recent years there has been increased interest in understanding the interplay between deep generative models (DGMs) and the manifold hypothesis. Research in this area focuses on understanding the reasons why commonly-used DGMs succeed or fail at learning distributions supported on unknown low-dimensional manifolds, as well as developing new models explicitly designed to account for manifold-supported data. This manifold lens provides both clarity as to why some DGMs (e.g. diffusion models and some generative adversarial networks) empirically surpass others (e.g. likelihood-based models such as variational autoencoders, normalizing flows, or energy-based models) at sample generation, and guidance for devising more performant DGMs. We carry out the first survey of DGMs viewed through this lens, making two novel contributions along the way. First, we formally establish that numerical instability of high-dimensional likelihoods is unavoidable when modelling low-dimensional data. We then show that DGMs on learned representations of autoencoders can be interpreted as approximately minimizing Wasserstein distance: this result, which applies to latent diffusion models, helps justify their outstanding empirical results. The manifold lens provides a rich perspective from which to understand DGMs, which we aim to make more accessible and widespread.

A Geometric Explanation of the Likelihood OOD Detection Paradox

Likelihood-based deep generative models (DGMs) commonly exhibit a puzzling behaviour: when trained on a relatively complex dataset, they assign higher likelihood values to out-of-distribution (OOD) data from simpler sources. Adding to the mystery, OOD samples are never generated by these DGMs despite having higher likelihoods. This two-pronged paradox has yet to be conclusively explained, making likelihood-based OOD detection unreliable. Our primary observation is that high-likelihood regions will not be generated if they contain minimal probability mass. We demonstrate how this seeming contradiction of large densities yet low probability mass can occur around data confined to low-dimensional manifolds. We also show that this scenario can be identified through local intrinsic dimension (LID) estimation, and propose a method for OOD detection which pairs the likelihoods and LID estimates obtained from a pre-trained DGM. Our method can be applied to normalizing flows and score-based diffusion models, and obtains results which match or surpass state-of-the-art OOD detection benchmarks using the same DGM backbones. Our code is available at https://github.com/layer6ai-labs/dgm_ood_detection.

Data-Efficient Multimodal Fusion on a Single GPU

The goal of multimodal alignment is to learn a single latent space that is shared between multimodal inputs. The most powerful models in this space have been trained using massive datasets of paired inputs and large-scale computational resources, making them prohibitively expensive to train in many practical scenarios. We surmise that existing unimodal encoders pre-trained on large amounts of unimodal data should provide an effective bootstrap to create multimodal models from unimodal ones at much lower costs. We therefore propose FuseMix, a multimodal augmentation scheme that operates on the latent spaces of arbitrary pre-trained unimodal encoders. Using FuseMix for multimodal alignment, we achieve competitive performance -- and in certain cases outperform state-of-the art methods -- in both image-text and audio-text retrieval, with orders of magnitude less compute and data: for example, we outperform CLIP on the Flickr30K text-to-image retrieval task with $\sim \! 600\times$ fewer GPU days and $\sim \! 80\times$ fewer image-text pairs. Additionally, we show how our method can be applied to convert pre-trained text-to-image generative models into audio-to-image ones. Code is available at: https://github.com/layer6ai-labs/fusemix.

Exposing flaws of generative model evaluation metrics and their unfair treatment of diffusion models

We systematically study a wide variety of image-based generative models spanning semantically-diverse datasets to understand and improve the feature extractors and metrics used to evaluate them. Using best practices in psychophysics, we measure human perception of image realism for generated samples by conducting the largest experiment evaluating generative models to date, and find that no existing metric strongly correlates with human evaluations. Comparing to 16 modern metrics for evaluating the overall performance, fidelity, diversity, and memorization of generative models, we find that the state-of-the-art perceptual realism of diffusion models as judged by humans is not reflected in commonly reported metrics such as FID. This discrepancy is not explained by diversity in generated samples, though one cause is over-reliance on Inception-V3. We address these flaws through a study of alternative self-supervised feature extractors, find that the semantic information encoded by individual networks strongly depends on their training procedure, and show that DINOv2-ViT-L/14 allows for much richer evaluation of generative models. Next, we investigate data memorization, and find that generative models do memorize training examples on simple, smaller datasets like CIFAR10, but not necessarily on more complex datasets like ImageNet. However, our experiments show that current metrics do not properly detect memorization; none in the literature is able to separate memorization from other phenomena such as underfitting or mode shrinkage. To facilitate further development of generative models and their evaluation we release all generated image datasets, human evaluation data, and a modular library to compute 16 common metrics for 8 different encoders at https://github.com/layer6ai-labs/dgm-eval.

TR0N: Translator Networks for 0-Shot Plug-and-Play Conditional Generation

We propose TR0N, a highly general framework to turn pre-trained unconditional generative models, such as GANs and VAEs, into conditional models. The conditioning can be highly arbitrary, and requires only a pre-trained auxiliary model. For example, we show how to turn unconditional models into class-conditional ones with the help of a classifier, and also into text-to-image models by leveraging CLIP. TR0N learns a lightweight stochastic mapping which "translates" between the space of conditions and the latent space of the generative model, in such a way that the generated latent corresponds to a data sample satisfying the desired condition. The translated latent samples are then further improved upon through Langevin dynamics, enabling us to obtain higher-quality data samples. TR0N requires no training data nor fine-tuning, yet can achieve a zero-shot FID of 10.9 on MS-COCO, outperforming competing alternatives not only on this metric, but also in sampling speed -- all while retaining a much higher level of generality. Our code is available at https://github.com/layer6ai-labs/tr0n.

Denoising Deep Generative Models

Likelihood-based deep generative models have recently been shown to exhibit pathological behaviour under the manifold hypothesis as a consequence of using high-dimensional densities to model data with low-dimensional structure. In this paper we propose two methodologies aimed at addressing this problem. Both are based on adding Gaussian noise to the data to remove the dimensionality mismatch during training, and both provide a denoising mechanism whose goal is to sample from the model as though no noise had been added to the data. Our first approach is based on Tweedie's formula, and the second on models which take the variance of added noise as a conditional input. We show that surprisingly, while well motivated, these approaches only sporadically improve performance over not adding noise, and that other methods of addressing the dimensionality mismatch are more empirically adequate.

Relating Regularization and Generalization through the Intrinsic Dimension of Activations

Given a pair of models with similar training set performance, it is natural to assume that the model that possesses simpler internal representations would exhibit better generalization. In this work, we provide empirical evidence for this intuition through an analysis of the intrinsic dimension (ID) of model activations, which can be thought of as the minimal number of factors of variation in the model's representation of the data. First, we show that common regularization techniques uniformly decrease the last-layer ID (LLID) of validation set activations for image classification models and show how this strongly affects generalization performance. We also investigate how excessive regularization decreases a model's ability to extract features from data in earlier layers, leading to a negative effect on validation accuracy even while LLID continues to decrease and training accuracy remains near-perfect. Finally, we examine the LLID over the course of training of models that exhibit grokking. We observe that well after training accuracy saturates, when models ``grok'' and validation accuracy suddenly improves from random to perfect, there is a co-occurent sudden drop in LLID, thus providing more insight into the dynamics of sudden generalization.

CaloMan: Fast generation of calorimeter showers with density estimation on learned manifolds

Precision measurements and new physics searches at the Large Hadron Collider require efficient simulations of particle propagation and interactions within the detectors. The most computationally expensive simulations involve calorimeter showers. Advances in deep generative modelling - particularly in the realm of high-dimensional data - have opened the possibility of generating realistic calorimeter showers orders of magnitude more quickly than physics-based simulation. However, the high-dimensional representation of showers belies the relative simplicity and structure of the underlying physical laws. This phenomenon is yet another example of the manifold hypothesis from machine learning, which states that high-dimensional data is supported on low-dimensional manifolds. We thus propose modelling calorimeter showers first by learning their manifold structure, and then estimating the density of data across this manifold. Learning manifold structure reduces the dimensionality of the data, which enables fast training and generation when compared with competing methods.

The Union of Manifolds Hypothesis and its Implications for Deep Generative Modelling

Deep learning has had tremendous success at learning low-dimensional representations of high-dimensional data. This success would be impossible if there was no hidden low-dimensional structure in data of interest; this existence is posited by the manifold hypothesis, which states that the data lies on an unknown manifold of low intrinsic dimension. In this paper, we argue that this hypothesis does not properly capture the low-dimensional structure typically present in data. Assuming the data lies on a single manifold implies intrinsic dimension is identical across the entire data space, and does not allow for subregions of this space to have a different number of factors of variation. To address this deficiency, we put forth the union of manifolds hypothesis, which accommodates the existence of non-constant intrinsic dimensions. We empirically verify this hypothesis on commonly-used image datasets, finding that indeed, intrinsic dimension should be allowed to vary. We also show that classes with higher intrinsic dimensions are harder to classify, and how this insight can be used to improve classification accuracy. We then turn our attention to the impact of this hypothesis in the context of deep generative models (DGMs). Most current DGMs struggle to model datasets with several connected components and/or varying intrinsic dimensions. To tackle these shortcomings, we propose clustered DGMs, where we first cluster the data and then train a DGM on each cluster. We show that clustered DGMs can model multiple connected components with different intrinsic dimensions, and empirically outperform their non-clustered counterparts without increasing computational requirements.