Fast, Expressive SE(n)(n) Equivariant Networks through Weight-Sharing in Position-Orientation Space

Based on the theory of homogeneous spaces we derive geometrically optimal edge attributes to be used within the flexible message-passing framework. We formalize the notion of weight sharing in convolutional networks as the sharing of message functions over point-pairs that should be treated equally. We define equivalence classes of point-pairs that are identical up to a transformation in the group and derive attributes that uniquely identify these classes. Weight sharing is then obtained by conditioning message functions on these attributes. As an application of the theory, we develop an efficient equivariant group convolutional network for processing 3D point clouds. The theory of homogeneous spaces tells us how to do group convolutions with feature maps over the homogeneous space of positions $\mathbb{R}^3$, position and orientations $\mathbb{R}^3 {\times} S^2$, and the group $SE(3)$ itself. Among these, $\mathbb{R}^3 {\times} S^2$ is an optimal choice due to the ability to represent directional information, which $\mathbb{R}^3$ methods cannot, and it significantly enhances computational efficiency compared to indexing features on the full $SE(3)$ group. We support this claim with state-of-the-art results -- in accuracy and speed -- on five different benchmarks in 2D and 3D, including interatomic potential energy prediction, trajectory forecasting in N-body systems, and generating molecules via equivariant diffusion models.Comment: Our code is publicly available at https://github.com/ebekkers/ponita . Published at ICLR 202

Inference for Heterogeneous Graphical Models using Doubly High-Dimensional Linear-Mixed Models

Motivated by the problem of inferring the graph structure of functional connectivity networks from multi-level functional magnetic resonance imaging data, we develop a valid inference framework for high-dimensional graphical models that accounts for group-level heterogeneity. We introduce a neighborhood-based method to learn the graph structure and reframe the problem as that of inferring fixed effect parameters in a doubly high-dimensional linear mixed model. Specifically, we propose a LASSO-based estimator and a de-biased LASSO-based inference framework for the fixed effect parameters in the doubly high-dimensional linear mixed model, leveraging random matrix theory to deal with challenges induced by the identical fixed and random effect design matrices arising in our setting. Moreover, we introduce consistent estimators for the variance components to identify subject-specific edges in the inferred graph. To illustrate the generality of the proposed approach, we also adapt our method to account for serial correlation by learning heterogeneous graphs in the setting of a vector autoregressive model. We demonstrate the performance of the proposed framework using real data and benchmark simulation studies

Moduli difference of inverse logarithmic coefficients of univalent functions

Let $f$ be analytic in the unit disk and $\mathcal{S}$ be the subclass of normalized univalent functions with $f(0) = 0$, and $f'(0) = 1$. Let $F$ be the inverse function of $f$, given by $F(w)=w+\sum_{n=2}^{\infty}A_nw^n$ defined on some disk $|w|\le r_0(f)$. The inverse logarithmic coefficients $\Gamma_n$, $n \in \mathbb{N}$, of $f$ are defined by the equation $\log(F(w)/w)=2\sum_{n=1}^{\infty}\Gamma_{n}w^{n},\,|w|<1/4.$ In this paper, we find the sharp upper and lower bounds for moduli difference of second and first inverse logarithmic coefficients, {\em i.e.,} $|\Gamma_2|-|\Gamma_1|$ for functions in class $\mathcal{S}$ and for functions in some important subclasses of univalent functions

Topological frequency conversion in rhombohedral multilayer graphene

We show that rhombohedral multilayer graphene supports topological frequency conversion, whereby a fraction of electrons transfer energy between two monochromatic light sources at a quantized rate. The pristine nature and gate tunability of these materials, along with a Berry curvature that directly couples to electric fields, make them ideal platforms for the experimental realization of topological frequency conversion. Among the rhombohedral family, we find that Bernal bilayer graphene appears most promising for THz-scale applications due to lower dissipation. We discuss strategies to circumvent cancellations between the two valleys of graphene and to minimize dissipative losses using commensurate frequencies, thus opening a potential pathway for net amplification.Comment: 4 pages + 4 figures in the main text. 4 additional figures in the Appendi

Phased Data Augmentation for Training a Likelihood-Based Generative Model with Limited Data

Generative models excel in creating realistic images, yet their dependency on extensive datasets for training presents significant challenges, especially in domains where data collection is costly or challenging. Current data-efficient methods largely focus on GAN architectures, leaving a gap in training other types of generative models. Our study introduces "phased data augmentation" as a novel technique that addresses this gap by optimizing training in limited data scenarios without altering the inherent data distribution. By limiting the augmentation intensity throughout the learning phases, our method enhances the model's ability to learn from limited data, thus maintaining fidelity. Applied to a model integrating PixelCNNs with VQ-VAE-2, our approach demonstrates superior performance in both quantitative and qualitative evaluations across diverse datasets. This represents an important step forward in the efficient training of likelihood-based models, extending the usefulness of data augmentation techniques beyond just GANs

Field Line Curvature Scattering in the Dayside Off-Equatorial Minima Regions

Magnetic field line curvature (FLC) scattering is an effective mechanism for collisionless particle scattering. In the terrestrial magnetosphere, the FLC scattering plays an essential role in shaping the outer boundary of protons radiation belt, the rapid decay of ring current, and the formation of proton isotropic boundary (IB). However, previous studies have yet to adequately investigate the influence of FLC scattering on charged particles in the Earth's dayside magnetosphere, particularly in the off-equatorial magnetic minima regions. This study employs T89 magnetic field model to investigate the impacts of FLC scattering on ring current protons in the dayside magnetosphere, with a specific focus on the off-equatorial minimum regions. We analyze the spatial distributions of single and dual magnetic minima regions, adiabatic parameter, and pitch angle diffusion coefficients due to FLC scattering as functions of $Kp$. The results show that the effects of FLC scattering are significant not only on the dusk and dawn sides but also in the off-equatorial minima regions on the noon. Additionally, we investigate the role of dipole tilt angle in the hemispheric asymmetry of FLC scattering effects. The dipole tilt angle controls the overall displacement of the dayside magnetosphere, resulting in different FLC scattering effects in the two hemispheres. Our study holds significance for understanding the FLC scattering effects in the off-equatorial region of Earth's dayside magnetosphere and for constructing a more accurate dynamic model of particles

How to train your ears: Auditory-model emulation for large-dynamic-range inputs and mild-to-severe hearing losses

Advanced auditory models are useful in designing signal-processing algorithms for hearing-loss compensation or speech enhancement. Such auditory models provide rich and detailed descriptions of the auditory pathway, and might allow for individualization of signal-processing strategies, based on physiological measurements. However, these auditory models are often computationally demanding, requiring significant time to compute. To address this issue, previous studies have explored the use of deep neural networks to emulate auditory models and reduce inference time. While these deep neural networks offer impressive efficiency gains in terms of computational time, they may suffer from uneven emulation performance as a function of auditory-model frequency-channels and input sound pressure level, making them unsuitable for many tasks. In this study, we demonstrate that the conventional machine-learning optimization objective used in existing state-of-the-art methods is the primary source of this limitation. Specifically, the optimization objective fails to account for the frequency- and level-dependencies of the auditory model, caused by a large input dynamic range and different types of hearing losses emulated by the auditory model. To overcome this limitation, we propose a new optimization objective that explicitly embeds the frequency- and level-dependencies of the auditory model. Our results show that this new optimization objective significantly improves the emulation performance of deep neural networks across relevant input sound levels and auditory-model frequency channels, without increasing the computational load during inference. Addressing these limitations is essential for advancing the application of auditory models in signal-processing tasks, ensuring their efficacy in diverse scenarios.Comment: Accepted by IEEE/ACM Transactions on Audio, Speech and Language Processing. This version is the authors' version and may vary from the final publication in detail

NeuFlow: Real-time, High-accuracy Optical Flow Estimation on Robots Using Edge Devices

Real-time high-accuracy optical flow estimation is a crucial component in various applications, including localization and mapping in robotics, object tracking, and activity recognition in computer vision. While recent learning-based optical flow methods have achieved high accuracy, they often come with heavy computation costs. In this paper, we propose a highly efficient optical flow architecture, called NeuFlow, that addresses both high accuracy and computational cost concerns. The architecture follows a global-to-local scheme. Given the features of the input images extracted at different spatial resolutions, global matching is employed to estimate an initial optical flow on the 1/16 resolution, capturing large displacement, which is then refined on the 1/8 resolution with lightweight CNN layers for better accuracy. We evaluate our approach on Jetson Orin Nano and RTX 2080 to demonstrate efficiency improvements across different computing platforms. We achieve a notable 10x-80x speedup compared to several state-of-the-art methods, while maintaining comparable accuracy. Our approach achieves around 30 FPS on edge computing platforms, which represents a significant breakthrough in deploying complex computer vision tasks such as SLAM on small robots like drones. The full training and evaluation code is available at https://github.com/neufieldrobotics/NeuFlow

Learning of Nash Equilibria in Risk-Averse Games

This paper considers risk-averse learning in convex games involving multiple agents that aim to minimize their individual risk of incurring significantly high costs. Specifically, the agents adopt the conditional value at risk (CVaR) as a risk measure with possibly different risk levels. To solve this problem, we propose a first-order risk-averse leaning algorithm, in which the CVaR gradient estimate depends on an estimate of the Value at Risk (VaR) value combined with the gradient of the stochastic cost function. Although estimation of the CVaR gradients using finitely many samples is generally biased, we show that the accumulated error of the CVaR gradient estimates is bounded with high probability. Moreover, assuming that the risk-averse game is strongly monotone, we show that the proposed algorithm converges to the risk-averse Nash equilibrium. We present numerical experiments on a Cournot game example to illustrate the performance of the proposed method

On well-posedness of the leak localization problem in parallel pipe networks

With the advent of integrated sensor technology (smart flow meters and pressure sensors), various new numerical algorithms for leak localization (a core element of water distribution system operation) have been developed. However, there is a lack of theory regarding the limitations of leak localization. In this work, we contribute to the development of such a theory by introducing an example water network structure with parallel pipes that is tractable for analytical treatment. We define the leak localization problem for this structure and show how many sensors and what conditions are needed for the well-posedness of the problem. We present a formula for the leak position as a function of measurements from these sensors. However, we also highlight the risk of finding false but plausible leak positions in the multiple pipes. We try to answer the questions of how and when the leaking pipe can be isolated. In particular, we show that nonlinearities in the pipes' head loss functions are essential for the well-posedness of the isolation problem. We propose procedures to get around the pitfall of multiple plausible leak positions.Comment: Submitted to Automatic