In many applications, we desire neural networks to exhibit invariance or equivari-ance to certain groups due to symmetries inherent in the data. Recently, frame-averaging methods emerged to be a unified framework for attaining symmetries efficiently by averaging over input-dependent subsets of the group, i.e., frames. What we currently lack is a principled understanding of the design of frames.

In many applications, we desire neural networks to exhibit invariance or equivariance to certain groups due to symmetries inherent in the data. Recently, frame-averaging methods emerged to be a unified framework for attaining symmetries efficiently by averaging over input-dependent subsets of the group, i.e., frames. What we currently lack is a principled understanding of the design of frames. In this work, we introduce a canonization perspective that provides an essential and complete view of the design of frames. Canonization is a classic approach for attaining invariance by mapping inputs to their canonical forms. We show that there exists an inherent connection between frames and canonical forms. Leveraging this connection, we can efficiently compare the complexity of frames as well as determine the optimality of certain frames. Guided by this principle, we design novel frames for eigenvectors that are strictly superior to existing methods -- some are even optimal -- both theoretically and empirically. The reduction to the canonization perspective further uncovers equivalences between previous methods. These observations suggest that canonization provides a fundamental understanding of existing frame-averaging methods and unifies existing equivariant and invariant learning methods.

A true interpreting agent not only understands sign language and translates to text, but also understands text and translates to signs. Much of the AI work in sign language translation to date has focused mainly on translating from signs to text. Towards the latter goal, we propose a text-to-sign translation model, SignNet, which exploits the notion of similarity (and dissimilarity) of visual signs in translating. This module presented is only one part of a dual-learning two task process involving text-to-sign (T2S) as well as sign-to-text (S2T). We currently implement SignNet as a single channel architecture so that the output of the T2S task can be fed into S2T in a continuous dual learning framework. By single channel, we refer to a single modality, the body pose joints. In this work, we present SignNet, a T2S task using a novel metric embedding learning process, to preserve the distances between sign embeddings relative to their dissimilarity. We also describe how to choose positive and negative examples of signs for similarity testing. From our analysis, we observe that metric embedding learning-based model perform significantly better than the other models with traditional losses, when evaluated using BLEU scores. In the task of gloss to pose, SignNet performed as well as its state-of-the-art (SoTA) counterparts and outperformed them in the task of text to pose, by showing noteworthy enhancements in BLEU 1 - BLEU 4 scores (BLEU 1: 31->39; ~26% improvement and BLEU 4: 10.43->11.84; ~14\% improvement) when tested on the popular RWTH PHOENIX-Weather-2014T benchmark dataset

We introduce SignNet and BasisNet -- new neural architectures that are invariant to two key symmetries displayed by eigenvectors: (i) sign flips, since if $v$ is an eigenvector then so is $-v$; and (ii) more general basis symmetries, which occur in higher dimensional eigenspaces with infinitely many choices of basis eigenvectors. We prove that under certain conditions our networks are universal, i.e., they can approximate any continuous function of eigenvectors with the desired invariances. When used with Laplacian eigenvectors, our networks are provably more expressive than existing spectral methods on graphs; for instance, they subsume all spectral graph convolutions, certain spectral graph invariants, and previously proposed graph positional encodings as special cases. Experiments show that our networks significantly outperform existing baselines on molecular graph regression, learning expressive graph representations, and learning neural fields on triangle meshes. Our code is available at https://github.com/cptq/SignNet-BasisNet .