Data-Free Learning of Reduced-Order Kinematics

Physical systems ranging from elastic bodies to kinematic linkages are defined on high-dimensional configuration spaces, yet their typical low-energy configurations are concentrated on much lower-dimensional subspaces. This work addresses the challenge of identifying such subspaces automatically: given as input an energy function for a high-dimensional system, we produce a low-dimensional map whose image parameterizes a diverse yet low-energy submanifold of configurations. The only additional input needed is a single seed configuration for the system to initialize our procedure; no dataset of trajectories is required. We represent subspaces as neural networks that map a low-dimensional latent vector to the full configuration space, and propose a training scheme to fit network parameters to any system of interest. This formulation is effective across a very general range of physical systems; our experiments demonstrate not only nonlinear and very low-dimensional elastic body and cloth subspaces, but also more general systems like colliding rigid bodies and linkages. We briefly explore applications built on this formulation, including manipulation, latent interpolation, and sampling.

Deep Augmentation: Enhancing Self-Supervised Learning through Transformations in Higher Activation Space

We introduce Deep Augmentation, an approach to data augmentation using dropout to dynamically transform a targeted layer within a neural network, with the option to use the stop-gradient operation, offering significant improvements in model performance and generalization. We demonstrate the efficacy of Deep Augmentation through extensive experiments on contrastive learning tasks in computer vision and NLP domains, where we observe substantial performance gains with ResNets and Transformers as the underlying models. Our experimentation reveals that targeting deeper layers with Deep Augmentation outperforms augmenting the input data, and the simple network- and data-agnostic nature of this approach enables its seamless integration into computer vision and NLP pipelines.

Self-Consistent Velocity Matching of Probability Flows

We present a discretization-free scalable framework for solving a large class of mass-conserving partial differential equations (PDEs), including the time-dependent Fokker-Planck equation and the Wasserstein gradient flow. The main observation is that the time-varying velocity field of the PDE solution needs to be self-consistent: it must satisfy a fixed-point equation involving the flow characterized by the same velocity field. By parameterizing the flow as a time-dependent neural network, we propose an end-to-end iterative optimization framework called self-consistent velocity matching to solve this class of PDEs. Compared to existing approaches, our method does not suffer from temporal or spatial discretization, covers a wide range of PDEs, and scales to high dimensions. Experimentally, our method recovers analytical solutions accurately when they are available and achieves comparable or better performance in high dimensions with less training time compared to recent large-scale JKO-based methods that are designed for solving a more restrictive family of PDEs.

Sampling with Mollified Interaction Energy Descent

Sampling from a target measure whose density is only known up to a normalization constant is a fundamental problem in computational statistics and machine learning. In this paper, we present a new optimization-based method for sampling called mollified interaction energy descent (MIED). MIED minimizes a new class of energies on probability measures called mollified interaction energies (MIEs). These energies rely on mollifier functions -- smooth approximations of the Dirac delta originated from PDE theory. We show that as the mollifier approaches the Dirac delta, the MIE converges to the chi-square divergence with respect to the target measure and the gradient flow of the MIE agrees with that of the chi-square divergence. Optimizing this energy with proper discretization yields a practical first-order particle-based algorithm for sampling in both unconstrained and constrained domains. We show experimentally that for unconstrained sampling problems our algorithm performs on par with existing particle-based algorithms like SVGD, while for constrained sampling problems our method readily incorporates constrained optimization techniques to handle more flexible constraints with strong performance compared to alternatives.

Outlier-Robust Group Inference via Gradient Space Clustering

Traditional machine learning models focus on achieving good performance on the overall training distribution, but they often underperform on minority groups. Existing methods can improve the worst-group performance, but they can have several limitations: (i) they require group annotations, which are often expensive and sometimes infeasible to obtain, and/or (ii) they are sensitive to outliers. Most related works fail to solve these two issues simultaneously as they focus on conflicting perspectives of minority groups and outliers. We address the problem of learning group annotations in the presence of outliers by clustering the data in the space of gradients of the model parameters. We show that data in the gradient space has a simpler structure while preserving information about minority groups and outliers, making it suitable for standard clustering methods like DBSCAN. Extensive experiments demonstrate that our method significantly outperforms state-of-the-art both in terms of group identification and downstream worst-group performance.

Riemannian Metric Learning via Optimal Transport

We introduce an optimal transport-based model for learning a metric tensor from cross-sectional samples of evolving probability measures on a common Riemannian manifold. We neurally parametrize the metric as a spatially-varying matrix field and efficiently optimize our model's objective using backpropagation. Using this learned metric, we can nonlinearly interpolate between probability measures and compute geodesics on the manifold. We show that metrics learned using our method improve the quality of trajectory inference on scRNA and bird migration data at the cost of little additional cross-sectional data.

Symmetric Volume Maps

Although shape correspondence is a central problem in geometry processing, most methods for this task apply only to two-dimensional surfaces. The neglected task of volumetric correspondence--a natural extension relevant to shapes extracted from simulation, medical imaging, volume rendering, and even improving surface maps of boundary representations--presents unique challenges that do not appear in the two-dimensional case. In this work, we propose a method for mapping between volumes represented as tetrahedral meshes. Our formulation minimizes a distortion energy designed to extract maps symmetrically, i.e., without dependence on the ordering of the source and target domains. We accompany our method with theoretical discussion describing the consequences of this symmetry assumption, leading us to select a symmetrized ARAP energy that favors isometric correspondences. Our final formulation optimizes for near-isometry while matching the boundary. We demonstrate our method on a diverse geometric dataset, producing low-distortion matchings that align to the boundary.

Log-Euclidean Signatures for Intrinsic Distances Between Unaligned Datasets

The need for efficiently comparing and representing datasets with unknown alignment spans various fields, from model analysis and comparison in machine learning to trend discovery in collections of medical datasets. We use manifold learning to compare the intrinsic geometric structures of different datasets by comparing their diffusion operators, symmetric positive-definite (SPD) matrices that relate to approximations of the continuous Laplace-Beltrami operator from discrete samples. Existing methods typically compare such operators in a pointwise manner or assume known data alignment. Instead, we exploit the Riemannian geometry of SPD matrices to compare these operators and define a new theoretically-motivated distance based on a lower bound of the log-Euclidean metric. Our framework facilitates comparison of data manifolds expressed in datasets with different sizes, numbers of features, and measurement modalities. Our log-Euclidean signature (LES) distance recovers meaningful structural differences, outperforming competing methods in various application domains.

Rewiring with Positional Encodings for Graph Neural Networks

Several recent works use positional encodings to extend the receptive fields of graph neural network (GNN) layers equipped with attention mechanisms. These techniques, however, extend receptive fields to the complete graph, at substantial computational cost and risking a change in the inductive biases of conventional GNNs, or require complex architecture adjustments. As a conservative alternative, we use positional encodings to expand receptive fields to any r-ring. Our method augments the input graph with additional nodes/edges and uses positional encodings as node and/or edge features. Thus, it is compatible with many existing GNN architectures. We also provide examples of positional encodings that are non-invasive, i.e., there is a one-to-one map between the original and the modified graphs. Our experiments demonstrate that extending receptive fields via positional encodings and a virtual fully-connected node significantly improves GNN performance and alleviates over-squashing using small r. We obtain improvements across models, showing state-of-the-art performance even using older architectures than recent Transformer models adapted to graphs.

Learning Proximal Operators to Discover Multiple Optima

Finding multiple solutions of non-convex optimization problems is a ubiquitous yet challenging task. Typical existing solutions either apply single-solution optimization methods from multiple random initial guesses or search in the vicinity of found solutions using ad hoc heuristics. We present an end-to-end method to learn the proximal operator across a family of non-convex problems, which can then be used to recover multiple solutions for unseen problems at test time. Our method only requires access to the objectives without needing the supervision of ground truth solutions. Notably, the added proximal regularization term elevates the convexity of our formulation: by applying recent theoretical results, we show that for weakly-convex objectives and under mild regularity conditions, training of the proximal operator converges globally in the over-parameterized setting. We further present a benchmark for multi-solution optimization including a wide range of applications and evaluate our method to demonstrate its effectiveness.