David W. Hogg

Soledad Villar, Weichi Yao, David W. Hogg et al.

Units equivariance (or units covariance) is the exact symmetry that follows from the requirement that relationships among measured quantities of physics relevance must obey self-consistent dimensional scalings. Here, we express this symmetry in terms of a (non-compact) group action, and we employ dimensional analysis and ideas from equivariant machine learning to provide a methodology for exactly units-equivariant machine learning: For any given learning task, we first construct a dimensionless version of its inputs using classic results from dimensional analysis, and then perform inference in the dimensionless space. Our approach can be used to impose units equivariance across a broad range of machine learning methods which are equivariant to rotations and other groups. We discuss the in-sample and out-of-sample prediction accuracy gains one can obtain in contexts like symbolic regression and emulation, where symmetry is important. We illustrate our approach with simple numerical examples involving dynamical systems in physics and ecology.

Soledad Villar, David W. Hogg, Weichi Yao et al.

Any representation of data involves arbitrary investigator choices. Because those choices are external to the data-generating process, each choice leads to an exact symmetry, corresponding to the group of transformations that takes one possible representation to another. These are the passive symmetries; they include coordinate freedom, gauge symmetry, and units covariance, all of which have led to important results in physics. In machine learning, the most visible passive symmetry is the relabeling or permutation symmetry of graphs. Our goal is to understand the implications for machine learning of the many passive symmetries in play. We discuss dos and don'ts for machine learning practice if passive symmetries are to be respected. We discuss links to causal modeling, and argue that the implementation of passive symmetries is particularly valuable when the goal of the learning problem is to generalize out of sample. This paper is conceptual: It translates among the languages of physics, mathematics, and machine-learning. We believe that consideration and implementation of passive symmetries might help machine learning in the same ways that it transformed physics in the twentieth century.

Yunfeng Diao, He Wang, Tianjia Shao et al.

Human Activity Recognition (HAR) has been employed in a wide range of applications, e.g. self-driving cars, where safety and lives are at stake. Recently, the robustness of skeleton-based HAR methods have been questioned due to their vulnerability to adversarial attacks. However, the proposed attacks require the full-knowledge of the attacked classifier, which is overly restrictive. In this paper, we show such threats indeed exist, even when the attacker only has access to the input/output of the model. To this end, we propose the very first black-box adversarial attack approach in skeleton-based HAR called BASAR. BASAR explores the interplay between the classification boundary and the natural motion manifold. To our best knowledge, this is the first time data manifold is introduced in adversarial attacks on time series. Via BASAR, we find on-manifold adversarial samples are extremely deceitful and rather common in skeletal motions, in contrast to the common belief that adversarial samples only exist off-manifold. Through exhaustive evaluation, we show that BASAR can deliver successful attacks across classifiers, datasets, and attack modes. By attack, BASAR helps identify the potential causes of the model vulnerability and provides insights on possible improvements. Finally, to mitigate the newly identified threat, we propose a new adversarial training approach by leveraging the sophisticated distributions of on/off-manifold adversarial samples, called mixed manifold-based adversarial training (MMAT). MMAT can successfully help defend against adversarial attacks without compromising classification accuracy.

Jose Sosa, David Hogg

We propose a new self-supervised method for predicting 3D human body pose from a single image. The prediction network is trained from a dataset of unlabelled images depicting people in typical poses and a set of unpaired 2D poses. By minimising the need for annotated data, the method has the potential for rapid application to pose estimation of other articulated structures (e.g. animals). The self-supervision comes from an earlier idea exploiting consistency between predicted pose under 3D rotation. Our method is a substantial advance on state-of-the-art self-supervised methods in training a mapping directly from images, without limb articulation constraints or any 3D empirical pose prior. We compare performance with state-of-the-art self-supervised methods using benchmark datasets that provide images and ground-truth 3D pose (Human3.6M, MPI-INF-3DHP). Despite the reduced requirement for annotated data, we show that the method outperforms on Human3.6M and matches performance on MPI-INF-3DHP. Qualitative results on a dataset of human hands show the potential for rapidly learning to predict 3D pose for articulated structures other than the human body.

Jose Sosa, David Hogg

Obtaining labelled data to train deep learning methods for estimating animal pose is challenging. Recently, synthetic data has been widely used for pose estimation tasks, but most methods still rely on supervised learning paradigms utilising synthetic images and labels. Can training be fully unsupervised? Is a tiny synthetic dataset sufficient? What are the minimum assumptions that we could make for estimating animal pose? Our proposal addresses these questions through a simple yet effective self-supervised method that only assumes the availability of unlabelled images and a small set of synthetic 2D poses. We completely remove the need for any 3D or 2D pose annotations (or complex 3D animal models), and surprisingly our approach can still learn accurate 3D and 2D poses simultaneously. We train our method with unlabelled images of horses mainly collected for YouTube videos and a prior consisting of 2D synthetic poses. The latter is three times smaller than the number of images needed for training. We test our method on a challenging set of horse images and evaluate the predicted 3D and 2D poses. We demonstrate that it is possible to learn accurate animal poses even with as few assumptions as unlabelled images and a small set of 2D poses generated from synthetic data. Given the minimum requirements and the abundance of unlabelled data, our method could be easily deployed to different animals.

Hanh Thi Minh Tran, David Hogg

In this paper, we propose a novel method for video anomaly detection motivated by an existing architecture for sequence-to-sequence prediction and reconstruction using a spatio-temporal convolutional Long Short-Term Memory (convLSTM). As in previous work on anomaly detection, anomalies arise as spatially localised failures in reconstruction or prediction. In experiments with five benchmark datasets, we show that using prediction gives superior performance to using reconstruction. We also compare performance with different length input/output sequences. Overall, our results using prediction are comparable with the state of the art on the benchmark datasets.

Jose Sosa, Sharn Perry, Jane Alty et al.

Numerous fields, such as ecology, biology, and neuroscience, use animal recordings to track and measure animal behaviour. Over time, a significant volume of such data has been produced, but some computer vision techniques cannot explore it due to the lack of annotations. To address this, we propose an approach for estimating 2D mouse body pose from unlabelled images using a synthetically generated empirical pose prior. Our proposal is based on a recent self-supervised method for estimating 2D human pose that uses single images and a set of unpaired typical 2D poses within a GAN framework. We adapt this method to the limb structure of the mouse and generate the empirical prior of 2D poses from a synthetic 3D mouse model, thereby avoiding manual annotation. In experiments on a new mouse video dataset, we evaluate the performance of the approach by comparing pose predictions to a manually obtained ground truth. We also compare predictions with those from a supervised state-of-the-art method for animal pose estimation. The latter evaluation indicates promising results despite the lack of paired training data. Finally, qualitative results using a dataset of horse images show the potential of the setting to adapt to other animal species.

Valentino F. Foit, David W. Hogg, Soledad Villar

Many machine learning tasks in the natural sciences are precisely equivariant to particular symmetries. Nonetheless, equivariant methods are often not employed, perhaps because training is perceived to be challenging, or the symmetry is expected to be learned, or equivariant implementations are seen as hard to build. Group averaging is an available technique for these situations. It happens at test time; it can make any trained model precisely equivariant at a (often small) cost proportional to the size of the group; it places no requirements on model structure or training. It is known that, under mild conditions, the group-averaged model will have a provably better prediction accuracy than the original model. Here we show that an inexpensive group averaging can improve accuracy in practice. We take well-established benchmark machine learning models of differential equations in which certain symmetries ought to be obeyed. At evaluation time, we average the models over a small group of symmetries. Our experiments show that this procedure always decreases the average evaluation loss, with improvements of up to 37\% in terms of the VRMSE. The averaging produces visually better predictions for continuous dynamics. This short paper shows that, under certain common circumstances, there are no disadvantages to imposing exact symmetries; the ML4PS community should consider group averaging as a cheap and simple way to improve model accuracy.

Weichi Yao, Kate Storey-Fisher, David W. Hogg et al.

Physical systems obey strict symmetry principles. We expect that machine learning methods that intrinsically respect these symmetries should have higher prediction accuracy and better generalization in prediction of physical dynamics. In this work we implement a principled model based on invariant scalars, and release open-source code. We apply this Scalars method to a simple chaotic dynamical system, the springy double pendulum. We show that the Scalars method outperforms state-of-the-art approaches for learning the properties of physical systems with symmetries, both in terms of accuracy and speed. Because the method incorporates the fundamental symmetries, we expect it to generalize to different settings, such as changes in the force laws in the system.

Jeroen Audenaert, Daniel Muthukrishna, Paul F. Gregory et al.

Foundation models for structured time series data must contend with a fundamental challenge: observations often conflate the true underlying physical phenomena with systematic distortions introduced by measurement instruments. This entanglement limits model generalization, especially in heterogeneous or multi-instrument settings. We present a causally-motivated foundation model that explicitly disentangles physical and instrumental factors using a dual-encoder architecture trained with structured contrastive learning. Leveraging naturally occurring observational triplets (i.e., where the same target is measured under varying conditions, and distinct targets are measured under shared conditions) our model learns separate latent representations for the underlying physical signal and instrument effects. Evaluated on simulated astronomical time series designed to resemble the complexity of variable stars observed by missions like NASA's Transiting Exoplanet Survey Satellite (TESS), our method significantly outperforms traditional single-latent space foundation models on downstream prediction tasks, particularly in low-data regimes. These results demonstrate that our model supports key capabilities of foundation models, including few-shot generalization and efficient adaptation, and highlight the importance of encoding causal structure into representation learning for structured data.

Wilson G. Gregory, David W. Hogg, Ben Blum-Smith et al.

Machine learning methods are increasingly being employed as surrogate models in place of computationally expensive and slow numerical integrators for a bevy of applications in the natural sciences. However, while the laws of physics are relationships between scalars, vectors, and tensors that hold regardless of the frame of reference or chosen coordinate system, surrogate machine learning models are not coordinate-free by default. We enforce coordinate freedom by using geometric convolutions in three model architectures: a ResNet, a Dilated ResNet, and a UNet. In numerical experiments emulating 2D compressible Navier-Stokes, we see better accuracy and improved stability compared to baseline surrogate models in almost all cases. The ease of enforcing coordinate freedom without making major changes to the model architecture provides an exciting recipe for any CNN-based method applied to an appropriate class of problems

Soledad Villar, David W. Hogg, Kate Storey-Fisher et al.

There has been enormous progress in the last few years in designing neural networks that respect the fundamental symmetries and coordinate freedoms of physical law. Some of these frameworks make use of irreducible representations, some make use of high-order tensor objects, and some apply symmetry-enforcing constraints. Different physical laws obey different combinations of fundamental symmetries, but a large fraction (possibly all) of classical physics is equivariant to translation, rotation, reflection (parity), boost (relativity), and permutations. Here we show that it is simple to parameterize universally approximating polynomial functions that are equivariant under these symmetries, or under the Euclidean, Lorentz, and Poincaré groups, at any dimensionality $d$. The key observation is that nonlinear O($d$)-equivariant (and related-group-equivariant) functions can be universally expressed in terms of a lightweight collection of scalars -- scalar products and scalar contractions of the scalar, vector, and tensor inputs. We complement our theory with numerical examples that show that the scalar-based method is simple, efficient, and scalable.

Ningyuan Huang, David W. Hogg, Soledad Villar

Overparameterization in deep learning is powerful: Very large models fit the training data perfectly and yet often generalize well. This realization brought back the study of linear models for regression, including ordinary least squares (OLS), which, like deep learning, shows a "double-descent" behavior: (1) The risk (expected out-of-sample prediction error) can grow arbitrarily when the number of parameters $p$ approaches the number of samples $n$, and (2) the risk decreases with $p$ for $p>n$, sometimes achieving a lower value than the lowest risk for $p<n$. The divergence of the risk for OLS can be avoided with regularization. In this work, we show that for some data models it can also be avoided with a PCA-based dimensionality reduction (PCA-OLS, also known as principal component regression). We provide non-asymptotic bounds for the risk of PCA-OLS by considering the alignments of the population and empirical principal components. We show that dimensionality reduction improves robustness while OLS is arbitrarily susceptible to adversarial attacks, particularly in the overparameterized regime. We compare PCA-OLS theoretically and empirically with a wide range of projection-based methods, including random projections, partial least squares (PLS), and certain classes of linear two-layer neural networks. These comparisons are made for different data generation models to assess the sensitivity to signal-to-noise and the alignment of regression coefficients with the features. We find that methods in which the projection depends on the training data can outperform methods where the projections are chosen independently of the training data, even those with oracle knowledge of population quantities, another seemingly paradoxical phenomenon that has been identified previously. This suggests that overparameterization may not be necessary for good generalization.

Bernhard Schölkopf, David W. Hogg, Dun Wang et al.

We describe a method for removing the effect of confounders in order to reconstruct a latent quantity of interest. The method, referred to as half-sibling regression, is inspired by recent work in causal inference using additive noise models. We provide a theoretical justification and illustrate the potential of the method in a challenging astronomy application.

Sivaram Ambikasaran, Daniel Foreman-Mackey, Leslie Greengard et al.

A number of problems in probability and statistics can be addressed using the multivariate normal (Gaussian) distribution. In the one-dimensional case, computing the probability for a given mean and variance simply requires the evaluation of the corresponding Gaussian density. In the $n$-dimensional setting, however, it requires the inversion of an $n \times n$ covariance matrix, $C$, as well as the evaluation of its determinant, $\det(C)$. In many cases, such as regression using Gaussian processes, the covariance matrix is of the form $C = σ^2 I + K$, where $K$ is computed using a specified covariance kernel which depends on the data and additional parameters (hyperparameters). The matrix $C$ is typically dense, causing standard direct methods for inversion and determinant evaluation to require $\mathcal O(n^3)$ work. This cost is prohibitive for large-scale modeling. Here, we show that for the most commonly used covariance functions, the matrix $C$ can be hierarchically factored into a product of block low-rank updates of the identity matrix, yielding an $\mathcal O (n\log^2 n) $ algorithm for inversion. More importantly, we show that this factorization enables the evaluation of the determinant $\det(C)$, permitting the direct calculation of probabilities in high dimensions under fairly broad assumptions on the kernel defining $K$. Our fast algorithm brings many problems in marginalization and the adaptation of hyperparameters within practical reach using a single CPU core. The combination of nearly optimal scaling in terms of problem size with high-performance computing resources will permit the modeling of previously intractable problems. We illustrate the performance of the scheme on standard covariance kernels.

Dustin Lang, David W. Hogg, Bernhard Scholkopf

We describe a system that builds a high dynamic-range and wide-angle image of the night sky by combining a large set of input images. The method makes use of pixel-rank information in the individual input images to improve a "consensus" pixel rank in the combined image. Because it only makes use of ranks and the complexity of the algorithm is linear in the number of images, the method is useful for large sets of uncalibrated images that might have undergone unknown non-linear tone mapping transformations for visualization or aesthetic reasons. We apply the method to images of the night sky (of unknown provenance) discovered on the Web. The method permits discovery of astronomical objects or features that are not visible in any of the input images taken individually. More importantly, however, it permits scientific exploitation of a huge source of astronomical images that would not be available to astronomical research without our automatic system.