More Like this

1. On the hardness of learning under symmetries

   Kiani, Bobak; Le, Thien; Lawrence, Hannah; Jegelka, Stefanie; Weber, Melanie (May 2024, International Conference on Learning Representations (ICLR), 2024)

   We study the problem of learning equivariant neural networks via gradient descent. The incorporation of known symmetries ("equivariance") into neural nets has empirically improved the performance of learning pipelines, in domains ranging from biology to computer vision. However, a rich yet separate line of learning theoretic research has demonstrated that actually learning shallow, fully-connected (i.e. non-symmetric) networks has exponential complexity in the correlational statistical query (CSQ) model, a framework encompassing gradient descent. In this work, we ask: are known problem symmetries sufficient to alleviate the fundamental hardness of learning neural nets with gradient descent? We answer this question in the negative. In particular, we give lower bounds for shallow graph neural networks, convolutional networks, invariant polynomials, and frame-averaged networks for permutation subgroups, which all scale either superpolynomially or exponentially in the relevant input dimension. Therefore, in spite of the significant inductive bias imparted via symmetry, actually learning the complete classes of functions represented by equivariant neural networks via gradient descent remains hard. 

   more » « less
2. Approximately Equivariant Networks for Imperfectly Symmetric Dynamics

   Wang, Rui; Walters, Robin; Yu, Rose (January 2022, Proceedings of the 39th International Conference on Machine Learning, PMLR)

   Incorporating symmetry as an inductive bias into neural network architecture has led to improvements in generalization, data efficiency, and physical consistency in dynamics modeling. Methods such as CNNs or equivariant neural networks use weight tying to enforce symmetries such as shift invariance or rotational equivariance. However, despite the fact that physical laws obey many symmetries, real-world dynamical data rarely conforms to strict mathematical symmetry either due to noisy or incomplete data or to symmetry breaking features in the underlying dynamical system. We explore approximately equivariant networks which are biased towards preserving symmetry but are not strictly constrained to do so. By relaxing equivariance constraints, we find that our models can outperform both baselines with no symmetry bias and baselines with overly strict symmetry in both simulated turbulence domains and real-world multi-stream jet flow. 

   more » « less
3. Image to Sphere: Learning Equivariant Features for Efficient Pose Prediction

   Klee, David M; Biza, Ondrej; Platt, Robert; Robin Walters (January 2023, The Eleventh International Conference on Learning Representations)

   Predicting the pose of objects from a single image is an important but difficult computer vision problem. Methods that predict a single point estimate do not predict the pose of objects with symmetries well and cannot represent uncertainty. Alternatively, some works predict a distribution over orientations in SO(3). However, training such models can be computation- and sample-inefficient. Instead, we propose a novel mapping of features from the image domain to the 3D rotation manifold. Our method then leverages SO(3) equivariant layers, which are more sample efficient, and outputs a distribution over rotations that can be sampled at arbitrary resolution. We demonstrate the effectiveness of our method at object orientation prediction, and achieve state-of-the-art performance on the popular PASCAL3D+ dataset. Moreover, we show that our method can model complex object symmetries, without any modifications to the parameters or loss function. Code is available at https://dmklee.github.io/image2sphere/ 

   more » « less
4. Residual Pathway Priors for Soft Equivariance Constraints

   Finzi, Marc; Benton, Gregory; Wilson, Andrew G. (January 2021, Advances in neural information processing systems)

   Models such as convolutional neural networks restrict the hypothesis space to a set of functions satisfying equivariance constraints, and improve generalization in problems by capturing relevant symmetries. However, symmetries are often only partially respected, preventing models with restriction biases from fitting the data. We introduce Residual Pathway Priors (RPPs) as a method for converting hard architectural constraints into soft priors, guiding models towards structured solutions while retaining the ability to capture additional complexity. RPPs are resilient to approximate or misspecified symmetries, and are as effective as fully constrained models even when symmetries are exact. We show that RPPs provide compelling performance on both model-free and model-based reinforcement learning problems, where contact forces and directional rewards violate the assumptions of equivariant networks. Finally, we demonstrate that RPPs have broad applicability, including dynamical systems, regression, and classification. 

   more » « less
5. Learning Materials Interatomic Potentials via Hybrid Invariant-Equivariant Architectures

   Yan, Keqiang; Bohde, Montgomery; Kryvenko, Andrii; Xiang, Ziyu; Zhao, Kaiji; Zhu, Siya; Kolachina, Saagar; Sarıtürk, Doğuhan; Xie, Jianwen; Arroyave, Raymundo; et al (November 2025, Transactions on machine learning research)

   Machine learning interatomic potentials (MLIPs) can predict energy, force, and stress of materials and enable a wide range of downstream discovery tasks. A key design choice in MLIPs involves the trade-off between invariant and equivariant architectures. Invariant models offer computational efficiency but may not perform as well, especially when predicting high-order outputs. In contrast, equivariant models can capture high-order symmetries, but are computationally expensive. In this work, we propose HIENet, a hybrid invariant-equivariant materials interatomic potential model that integrates both invariant and equivariant message passing layers. Furthermore, we show that HIENet provably satisfies key physical constraints. HIENet achieves state-of-the-art performance with considerable computational speedups over prior models. Experimental results on both common benchmarks and downstream materials discovery tasks demonstrate the efficiency and effectiveness of HIENet. Finally, additional ablations further demonstrate that our hybrid invariant-equivariant approach scales well across model sizes and works with different equivariant model architectures, providing powerful insights into future MLIP designs. 

   more » « less