Self-Supervised Learning with Lie Symmetries for Partial Differential Equations

Machine learning for differential equations paves the way for computationally efficient alternatives to numerical solvers, with potentially broad impacts in science and engineering. Though current algorithms typically require simulated training data tailored to a given setting, one may instead wish to learn useful information from heterogeneous sources, or from real dynamical systems observations that are messy or incomplete. In this work, we learn general-purpose representations of PDEs from heterogeneous data by implementing joint embedding methods for self-supervised learning (SSL), a framework for unsupervised representation learning that has had notable success in computer vision. Our representation outperforms baseline approaches to invariant tasks, such as regressing the coefficients of a PDE, while also improving the time-stepping performance of neural solvers. We hope that our proposed methodology will prove useful in the eventual development of general-purpose foundation models for PDEs.

A Cookbook of Self-Supervised Learning

Self-supervised learning, dubbed the dark matter of intelligence, is a promising path to advance machine learning. Yet, much like cooking, training SSL methods is a delicate art with a high barrier to entry. While many components are familiar, successfully training a SSL method involves a dizzying set of choices from the pretext tasks to training hyper-parameters. Our goal is to lower the barrier to entry into SSL research by laying the foundations and latest SSL recipes in the style of a cookbook. We hope to empower the curious researcher to navigate the terrain of methods, understand the role of the various knobs, and gain the know-how required to explore how delicious SSL can be.

Self-supervised learning of Split Invariant Equivariant representations

Recent progress has been made towards learning invariant or equivariant representations with self-supervised learning. While invariant methods are evaluated on large scale datasets, equivariant ones are evaluated in smaller, more controlled, settings. We aim at bridging the gap between the two in order to learn more diverse representations that are suitable for a wide range of tasks. We start by introducing a dataset called 3DIEBench, consisting of renderings from 3D models over 55 classes and more than 2.5 million images where we have full control on the transformations applied to the objects. We further introduce a predictor architecture based on hypernetworks to learn equivariant representations with no possible collapse to invariance. We introduce SIE (Split Invariant-Equivariant) which combines the hypernetwork-based predictor with representations split in two parts, one invariant, the other equivariant, to learn richer representations. We demonstrate significant performance gains over existing methods on equivariance related tasks from both a qualitative and quantitative point of view. We further analyze our introduced predictor and show how it steers the learned latent space. We hope that both our introduced dataset and approach will enable learning richer representations without supervision in more complex scenarios.

The Robustness Limits of SoTA Vision Models to Natural Variation

Recent state-of-the-art vision models introduced new architectures, learning paradigms, and larger pretraining data, leading to impressive performance on tasks such as classification. While previous generations of vision models were shown to lack robustness to factors such as pose, it's unclear the extent to which this next generation of models are more robust. To study this question, we develop a dataset of more than 7 million images with controlled changes in pose, position, background, lighting, and size. We study not only how robust recent state-of-the-art models are, but also the extent to which models can generalize variation in factors when they're present during training. We consider a catalog of recent vision models, including vision transformers (ViT), self-supervised models such as masked autoencoders (MAE), and models trained on larger datasets such as CLIP. We find out-of-the-box, even today's best models are not robust to common changes in pose, size, and background. When some samples varied during training, we found models required a significant portion of diversity to generalize -- though eventually robustness did improve. When diversity is only seen for some classes however, we found models did not generalize to other classes, unless the classes were very similar to those seen varying during training. We hope our work will shed further light on the blind spots of SoTA models and spur the development of more robust vision models.

RankMe: Assessing the downstream performance of pretrained self-supervised representations by their rank

Joint-Embedding Self Supervised Learning (JE-SSL) has seen a rapid development, with the emergence of many method variations and few principled guidelines that would help practitioners to successfully deploy those methods. The main reason for that pitfall actually comes from JE-SSL's core principle of not employing any input reconstruction. Without any visual clue, it becomes extremely cryptic to judge the quality of a learned representation without having access to a labelled dataset. We hope to correct those limitations by providing a single -- theoretically motivated -- criterion that reflects the quality of learned JE-SSL representations: their effective rank. Albeit simple and computationally friendly, this method -- coined RankMe -- allows one to assess the performance of JE-SSL representations, even on different downstream datasets, without requiring any labels, training or parameters to tune. Through thorough empirical experiments involving hundreds of repeated training episodes, we demonstrate how RankMe can be used for hyperparameter selection with nearly no loss in final performance compared to the current selection method that involve dataset labels. We hope that RankMe will facilitate the use of JE-SSL in domains with little or no labeled data.

Guillotine Regularization: Improving Deep Networks Generalization by Removing their Head

One unexpected technique that emerged in recent years consists in training a Deep Network (DN) with a Self-Supervised Learning (SSL) method, and using this network on downstream tasks but with its last few layers entirely removed. This usually skimmed-over trick is actually critical for SSL methods to display competitive performances. For example, on ImageNet classification, more than 30 points of percentage can be gained that way. This is a little vexing, as one would hope that the network layer at which invariance is explicitly enforced by the SSL criterion during training (the last layer) should be the one to use for best generalization performance downstream. But it seems not to be, and this study sheds some light on why. This trick, which we name Guillotine Regularization (GR), is in fact a generically applicable form of regularization that has also been used to improve generalization performance in transfer learning scenarios. In this work, through theory and experiments, we formalize GR and identify the underlying reasons behind its success in SSL methods. Our study shows that the use of this trick is essential to SSL performance for two main reasons: (i) improper data-augmentations to define the positive pairs used during training, and/or (ii) suboptimal selection of the hyper-parameters of the SSL loss.

On the duality between contrastive and non-contrastive self-supervised learning

Recent approaches in self-supervised learning of image representations can be categorized into different families of methods and, in particular, can be divided into contrastive and non-contrastive approaches. While differences between the two families have been thoroughly discussed to motivate new approaches, we focus more on the theoretical similarities between them. By designing contrastive and non-contrastive criteria that can be related algebraically and shown to be equivalent under limited assumptions, we show how close those families can be. We further study popular methods and introduce variations of them, allowing us to relate this theoretical result to current practices and show how design choices in the criterion can influence the optimization process and downstream performance. We also challenge the popular assumptions that contrastive and non-contrastive methods, respectively, need large batch sizes and output dimensions. Our theoretical and quantitative results suggest that the numerical gaps between contrastive and noncontrastive methods in certain regimes can be significantly reduced given better network design choice and hyperparameter tuning.

Visualizing hierarchies in scRNA-seq data using a density tree-biased autoencoder

Single cell RNA sequencing (scRNA-seq) data makes studying the development of cells possible at unparalleled resolution. Given that many cellular differentiation processes are hierarchical, their scRNA-seq data is expected to be approximately tree-shaped in gene expression space. Inference and representation of this tree-structure in two dimensions is highly desirable for biological interpretation and exploratory analysis. Our two contributions are an approach for identifying a meaningful tree structure from high-dimensional scRNA-seq data, and a visualization method respecting the tree-structure. We extract the tree structure by means of a density based minimum spanning tree on a vector quantization of the data and show that it captures biological information well. We then introduce DTAE, a tree-biased autoencoder that emphasizes the tree structure of the data in low dimensional space. We compare to other dimension reduction methods and demonstrate the success of our method experimentally. Our implementation relying on PyTorch and Higra is available at github.com/hci-unihd/DTAE.