Equivariant score-based generative models provably learn distributions with symmetries efficiently

Symmetry is ubiquitous in many real-world phenomena and tasks, such as physics, images, and molecular simulations. Empirical studies have demonstrated that incorporating symmetries into generative models can provide better generalization and sampling efficiency when the underlying data distribution has group symmetry. In this work, we provide the first theoretical analysis and guarantees of score-based generative models (SGMs) for learning distributions that are invariant with respect to some group symmetry and offer the first quantitative comparison between data augmentation and adding equivariant inductive bias. First, building on recent works on the Wasserstein-1 ($\mathbf{d}_1$) guarantees of SGMs and empirical estimations of probability divergences under group symmetry, we provide an improved $\mathbf{d}_1$ generalization bound when the data distribution is group-invariant. Second, we describe the inductive bias of equivariant SGMs using Hamilton-Jacobi-Bellman theory, and rigorously demonstrate that one can learn the score of a symmetrized distribution using equivariant vector fields without data augmentations through the analysis of the optimality and equivalence of score-matching objectives. This also provides practical guidance that one does not have to augment the dataset as long as the vector field or the neural network parametrization is equivariant. Moreover, we quantify the impact of not incorporating equivariant structure into the score parametrization, by showing that non-equivariant vector fields can yield worse generalization bounds. This can be viewed as a type of model-form error that describes the missing structure of non-equivariant vector fields. Numerical simulations corroborate our analysis and highlight that data augmentations cannot replace the role of equivariant vector fields.

OmniRe: Omni Urban Scene Reconstruction

We introduce OmniRe, a holistic approach for efficiently reconstructing high-fidelity dynamic urban scenes from on-device logs. Recent methods for modeling driving sequences using neural radiance fields or Gaussian Splatting have demonstrated the potential of reconstructing challenging dynamic scenes, but often overlook pedestrians and other non-vehicle dynamic actors, hindering a complete pipeline for dynamic urban scene reconstruction. To that end, we propose a comprehensive 3DGS framework for driving scenes, named OmniRe, that allows for accurate, full-length reconstruction of diverse dynamic objects in a driving log. OmniRe builds dynamic neural scene graphs based on Gaussian representations and constructs multiple local canonical spaces that model various dynamic actors, including vehicles, pedestrians, and cyclists, among many others. This capability is unmatched by existing methods. OmniRe allows us to holistically reconstruct different objects present in the scene, subsequently enabling the simulation of reconstructed scenarios with all actors participating in real-time (~60Hz). Extensive evaluations on the Waymo dataset show that our approach outperforms prior state-of-the-art methods quantitatively and qualitatively by a large margin. We believe our work fills a critical gap in driving reconstruction.

Learning heavy-tailed distributions with Wasserstein-proximal-regularized $α$-divergences

In this paper, we propose Wasserstein proximals of $\alpha$-divergences as suitable objective functionals for learning heavy-tailed distributions in a stable manner. First, we provide sufficient, and in some cases necessary, relations among data dimension, $\alpha$, and the decay rate of data distributions for the Wasserstein-proximal-regularized divergence to be finite. Finite-sample convergence rates for the estimation in the case of the Wasserstein-1 proximal divergences are then provided under certain tail conditions. Numerical experiments demonstrate stable learning of heavy-tailed distributions -- even those without first or second moment -- without any explicit knowledge of the tail behavior, using suitable generative models such as GANs and flow-based models related to our proposed Wasserstein-proximal-regularized $\alpha$-divergences. Heuristically, $\alpha$-divergences handle the heavy tails and Wasserstein proximals allow non-absolute continuity between distributions and control the velocities of flow-based algorithms as they learn the target distribution deep into the tails.

360-Degree Panorama Generation from Few Unregistered NFoV Images

360$^\circ$ panoramas are extensively utilized as environmental light sources in computer graphics. However, capturing a 360$^\circ$ $\times$ 180$^\circ$ panorama poses challenges due to the necessity of specialized and costly equipment, and additional human resources. Prior studies develop various learning-based generative methods to synthesize panoramas from a single Narrow Field-of-View (NFoV) image, but they are limited in alterable input patterns, generation quality, and controllability. To address these issues, we propose a novel pipeline called PanoDiff, which efficiently generates complete 360$^\circ$ panoramas using one or more unregistered NFoV images captured from arbitrary angles. Our approach has two primary components to overcome the limitations. Firstly, a two-stage angle prediction module to handle various numbers of NFoV inputs. Secondly, a novel latent diffusion-based panorama generation model uses incomplete panorama and text prompts as control signals and utilizes several geometric augmentation schemes to ensure geometric properties in generated panoramas. Experiments show that PanoDiff achieves state-of-the-art panoramic generation quality and high controllability, making it suitable for applications such as content editing.

Statistical Guarantees of Group-Invariant GANs

Group-invariant generative adversarial networks (GANs) are a type of GANs in which the generators and discriminators are hardwired with group symmetries. Empirical studies have shown that these networks are capable of learning group-invariant distributions with significantly improved data efficiency. In this study, we aim to rigorously quantify this improvement by analyzing the reduction in sample complexity for group-invariant GANs. Our findings indicate that when learning group-invariant distributions, the number of samples required for group-invariant GANs decreases proportionally with a power of the group size, and this power depends on the intrinsic dimension of the distribution's support. To our knowledge, this work presents the first statistical estimation for group-invariant generative models, specifically for GANs, and it may shed light on the study of other group-invariant generative models.

On the Implicit Bias of Linear Equivariant Steerable Networks: Margin, Generalization, and Their Equivalence to Data Augmentation

We study the implicit bias of gradient flow on linear equivariant steerable networks in group-invariant binary classification. Our findings reveal that the parameterized predictor converges in direction to the unique group-invariant classifier with a maximum margin defined by the input group action. Under a unitary assumption on the input representation, we establish the equivalence between steerable networks and data augmentation. Furthermore, we demonstrate the improved margin and generalization bound of steerable networks over their non-invariant counterparts.

Sample Complexity of Probability Divergences under Group Symmetry

We rigorously quantify the improvement in the sample complexity of variational divergence estimations for group-invariant distributions. In the cases of the Wasserstein-1 metric and the Lipschitz-regularized $\alpha$-divergences, the reduction of sample complexity is proportional to an ambient-dimension-dependent power of the group size. For the maximum mean discrepancy (MMD), the improvement of sample complexity is more nuanced, as it depends on not only the group size but also the choice of kernel. Numerical simulations verify our theories.

SpecNet2: Orthogonalization-free spectral embedding by neural networks

Spectral methods which represent data points by eigenvectors of kernel matrices or graph Laplacian matrices have been a primary tool in unsupervised data analysis. In many application scenarios, parametrizing the spectral embedding by a neural network that can be trained over batches of data samples gives a promising way to achieve automatic out-of-sample extension as well as computational scalability. Such an approach was taken in the original paper of SpectralNet (Shaham et al. 2018), which we call SpecNet1. The current paper introduces a new neural network approach, named SpecNet2, to compute spectral embedding which optimizes an equivalent objective of the eigen-problem and removes the orthogonalization layer in SpecNet1. SpecNet2 also allows separating the sampling of rows and columns of the graph affinity matrix by tracking the neighbors of each data point through the gradient formula. Theoretically, we show that any local minimizer of the new orthogonalization-free objective reveals the leading eigenvectors. Furthermore, global convergence for this new orthogonalization-free objective using a batch-based gradient descent method is proved. Numerical experiments demonstrate the improved performance and computational efficiency of SpecNet2 on simulated data and image datasets.

Representation-Agnostic Shape Fields

3D shape analysis has been widely explored in the era of deep learning. Numerous models have been developed for various 3D data representation formats, e.g., MeshCNN for meshes, PointNet for point clouds and VoxNet for voxels. In this study, we present Representation-Agnostic Shape Fields (RASF), a generalizable and computation-efficient shape embedding module for 3D deep learning. RASF is implemented with a learnable 3D grid with multiple channels to store local geometry. Based on RASF, shape embeddings for various 3D shape representations (point clouds, meshes and voxels) are retrieved by coordinate indexing. While there are multiple ways to optimize the learnable parameters of RASF, we provide two effective schemes among all in this paper for RASF pre-training: shape reconstruction and normal estimation. Once trained, RASF becomes a plug-and-play performance booster with negligible cost. Extensive experiments on diverse 3D representation formats, networks and applications, validate the universal effectiveness of the proposed RASF. Code and pre-trained models are publicly available https://github.com/seanywang0408/RASF

Disentangling modes with crossover instantaneous frequencies by synchrosqueezed chirplet transforms, from theory to application

Analysis of signals with oscillatory modes with crossover instantaneous frequencies is a challenging problem in time series analysis. One way to handle this problem is lifting the 2-dimensional time-frequency representation to a 3-dimensional representation, called time-frequency-chirp rate (TFC) representation, by adding one extra chirp rate parameter so that crossover frequencies are disentangles in higher dimension. The chirplet transform is an algorithm for this lifting idea. However, in practice we found that it has a stronger "blurring" effect in the chirp rate axis, which limits its application in real world data. Moreover, to our knowledge, we have limited mathematical understanding of the chirplet transform in the literature. Motivated by real world data challenges, in this paper, we propose the synchrosqueezed chirplet transform (SCT) that gives a concentrated TFC representation that the contrast is enhanced so that one can distinguish different modes even with crossover instantaneous frequencies. We also analyze chirplet transform and provide theoretical guarantee of SCT.