Latent Field Discovery In Interacting Dynamical Systems With Neural Fields

Systems of interacting objects often evolve under the influence of field effects that govern their dynamics, yet previous works have abstracted away from such effects, and assume that systems evolve in a vacuum. In this work, we focus on discovering these fields, and infer them from the observed dynamics alone, without directly observing them. We theorize the presence of latent force fields, and propose neural fields to learn them. Since the observed dynamics constitute the net effect of local object interactions and global field effects, recently popularized equivariant networks are inapplicable, as they fail to capture global information. To address this, we propose to disentangle local object interactions -- which are $\mathrm{SE}(n)$ equivariant and depend on relative states -- from external global field effects -- which depend on absolute states. We model interactions with equivariant graph networks, and combine them with neural fields in a novel graph network that integrates field forces. Our experiments show that we can accurately discover the underlying fields in charged particles settings, traffic scenes, and gravitational n-body problems, and effectively use them to learn the system and forecast future trajectories.

On genuine invariance learning without weight-tying

In this paper, we investigate properties and limitations of invariance learned by neural networks from the data compared to the genuine invariance achieved through invariant weight-tying. To do so, we adopt a group theoretical perspective and analyze invariance learning in neural networks without weight-tying constraints. We demonstrate that even when a network learns to correctly classify samples on a group orbit, the underlying decision-making in such a model does not attain genuine invariance. Instead, learned invariance is strongly conditioned on the input data, rendering it unreliable if the input distribution shifts. We next demonstrate how to guide invariance learning toward genuine invariance by regularizing the invariance of a model at the training. To this end, we propose several metrics to quantify learned invariance: (i) predictive distribution invariance, (ii) logit invariance, and (iii) saliency invariance similarity. We show that the invariance learned with the invariance error regularization closely reassembles the genuine invariance of weight-tying models and reliably holds even under a severe input distribution shift. Closer analysis of the learned invariance also reveals the spectral decay phenomenon, when a network chooses to achieve the invariance to a specific transformation group by reducing the sensitivity to any input perturbation.

Learned Gridification for Efficient Point Cloud Processing

Neural operations that rely on neighborhood information are much more expensive when deployed on point clouds than on grid data due to the irregular distances between points in a point cloud. In a grid, on the other hand, we can compute the kernel only once and reuse it for all query positions. As a result, operations that rely on neighborhood information scale much worse for point clouds than for grid data, specially for large inputs and large neighborhoods. In this work, we address the scalability issue of point cloud methods by tackling its root cause: the irregularity of the data. We propose learnable gridification as the first step in a point cloud processing pipeline to transform the point cloud into a compact, regular grid. Thanks to gridification, subsequent layers can use operations defined on regular grids, e.g., Conv3D, which scale much better than native point cloud methods. We then extend gridification to point cloud to point cloud tasks, e.g., segmentation, by adding a learnable de-gridification step at the end of the point cloud processing pipeline to map the compact, regular grid back to its original point cloud form. Through theoretical and empirical analysis, we show that gridified networks scale better in terms of memory and time than networks directly applied on raw point cloud data, while being able to achieve competitive results. Our code is publicly available at https://github.com/computri/gridifier.

Regular SE(3) Group Convolutions for Volumetric Medical Image Analysis

Regular group convolutional neural networks (G-CNNs) have been shown to increase model performance and improve equivariance to different geometrical symmetries. This work addresses the problem of SE(3), i.e., roto-translation equivariance, on volumetric data. Volumetric image data is prevalent in many medical settings. Motivated by the recent work on separable group convolutions, we devise a SE(3) group convolution kernel separated into a continuous SO(3) (rotation) kernel and a spatial kernel. We approximate equivariance to the continuous setting by sampling uniform SO(3) grids. Our continuous SO(3) kernel is parameterized via RBF interpolation on similarly uniform grids. We demonstrate the advantages of our approach in volumetric medical image analysis. Our SE(3) equivariant models consistently outperform CNNs and regular discrete G-CNNs on challenging medical classification tasks and show significantly improved generalization capabilities. Our approach achieves up to a 16.5% gain in accuracy over regular CNNs.

An Exploration of Conditioning Methods in Graph Neural Networks

The flexibility and effectiveness of message passing based graph neural networks (GNNs) induced considerable advances in deep learning on graph-structured data. In such approaches, GNNs recursively update node representations based on their neighbors and they gain expressivity through the use of node and edge attribute vectors. E.g., in computational tasks such as physics and chemistry usage of edge attributes such as relative position or distance proved to be essential. In this work, we address not what kind of attributes to use, but how to condition on this information to improve model performance. We consider three types of conditioning; weak, strong, and pure, which respectively relate to concatenation-based conditioning, gating, and transformations that are causally dependent on the attributes. This categorization provides a unifying viewpoint on different classes of GNNs, from separable convolutions to various forms of message passing networks. We provide an empirical study on the effect of conditioning methods in several tasks in computational chemistry.

Modelling Long Range Dependencies in N-D: From Task-Specific to a General Purpose CNN

Performant Convolutional Neural Network (CNN) architectures must be tailored to specific tasks in order to consider the length, resolution, and dimensionality of the input data. In this work, we tackle the need for problem-specific CNN architectures. We present the Continuous Convolutional Neural Network (CCNN): a single CNN able to process data of arbitrary resolution, dimensionality and length without any structural changes. Its key component are its continuous convolutional kernels which model long-range dependencies at every layer, and thus remove the need of current CNN architectures for task-dependent downsampling and depths. We showcase the generality of our method by using the same architecture for tasks on sequential ($1{\rm D}$), visual ($2{\rm D}$) and point-cloud ($3{\rm D}$) data. Our CCNN matches and often outperforms the current state-of-the-art across all tasks considered.

Towards a General Purpose CNN for Long Range Dependencies in $\mathrm{N}$D

The use of Convolutional Neural Networks (CNNs) is widespread in Deep Learning due to a range of desirable model properties which result in an efficient and effective machine learning framework. However, performant CNN architectures must be tailored to specific tasks in order to incorporate considerations such as the input length, resolution, and dimentionality. In this work, we overcome the need for problem-specific CNN architectures with our Continuous Convolutional Neural Network (CCNN): a single CNN architecture equipped with continuous convolutional kernels that can be used for tasks on data of arbitrary resolution, dimensionality and length without structural changes. Continuous convolutional kernels model long range dependencies at every layer, and remove the need for downsampling layers and task-dependent depths needed in current CNN architectures. We show the generality of our approach by applying the same CCNN to a wide set of tasks on sequential (1$\mathrm{D}$) and visual data (2$\mathrm{D}$). Our CCNN performs competitively and often outperforms the current state-of-the-art across all tasks considered.

Towards Lightweight Controllable Audio Synthesis with Conditional Implicit Neural Representations

The high temporal resolution of audio and our perceptual sensitivity to small irregularities in waveforms make synthesizing at high sampling rates a complex and computationally intensive task, prohibiting real-time, controllable synthesis within many approaches. In this work we aim to shed light on the potential of Conditional Implicit Neural Representations (CINRs) as lightweight backbones in generative frameworks for audio synthesis. Our experiments show that small Periodic Conditional INRs (PCINRs) learn faster and generally produce quantitatively better audio reconstructions than Transposed Convolutional Neural Networks with equal parameter counts. However, their performance is very sensitive to activation scaling hyperparameters. When learning to represent more uniform sets, PCINRs tend to introduce artificial high-frequency components in reconstructions. We validate this noise can be minimized by applying standard weight regularization during training or decreasing the compositional depth of PCINRs, and suggest directions for future research.

ChebLieNet: Invariant Spectral Graph NNs Turned Equivariant by Riemannian Geometry on Lie Groups

We introduce ChebLieNet, a group-equivariant method on (anisotropic) manifolds. Surfing on the success of graph- and group-based neural networks, we take advantage of the recent developments in the geometric deep learning field to derive a new approach to exploit any anisotropies in data. Via discrete approximations of Lie groups, we develop a graph neural network made of anisotropic convolutional layers (Chebyshev convolutions), spatial pooling and unpooling layers, and global pooling layers. Group equivariance is achieved via equivariant and invariant operators on graphs with anisotropic left-invariant Riemannian distance-based affinities encoded on the edges. Thanks to its simple form, the Riemannian metric can model any anisotropies, both in the spatial and orientation domains. This control on anisotropies of the Riemannian metrics allows to balance equivariance (anisotropic metric) against invariance (isotropic metric) of the graph convolution layers. Hence we open the doors to a better understanding of anisotropic properties. Furthermore, we empirically prove the existence of (data-dependent) sweet spots for anisotropic parameters on CIFAR10. This crucial result is evidence of the benefice we could get by exploiting anisotropic properties in data. We also evaluate the scalability of this approach on STL10 (image data) and ClimateNet (spherical data), showing its remarkable adaptability to diverse tasks.