Neural network-based odometry using accelerometer and gyroscope readings from a single IMU can achieve robust, and low-drift localization capabilities, through the use of _neural displacement priors (NDPs)_. These priors learn to produce denoised displacement measurements but need to ignore data variations due to specific IMU mount orientation and motion directions, hindering generalization. This work introduces EqNIO, which addresses this challenge with _canonical displacement priors_, i.e., priors that are invariant to the orientation of the gravity-aligned frame in which the IMU data is expressed. We train such priors on IMU measurements, that are mapped into a learnable canonical frame, which is uniquely defined via three axes: the first is gravity, making the frame gravity aligned, while the second and third are predicted from IMU data. The outputs (displacement and covariance) are mapped back to the original gravity-aligned frame. To maximize generalization, we find that these learnable frames must transform equivariantly with global gravity-preserving roto-reflections from the subgroup $O_g(3)\subset O(3)$, acting on the trajectory, rendering the NDP $O(3)$-_subequivariant_. We tailor specific linear, convolutional, and non-linear layers that commute with the actions of the group. Moreover, we introduce a bijective decomposition of angular rates into vectors that transform similarly to accelerations, allowing us to leverage both measurement types. Natively, angular rates would need to be inverted upon reflection, unlike acceleration, which hinders their joint processing. We highlight EqNIO's flexibility and generalization capabilities by applying it to both filter-based (TLIO), and end-to-end (RONIN) architectures, and outperforming existing methods that use _soft equivariance from auxiliary losses or data augmentation on various datasets. We believe this work paves the way for low-drift and generalizable neural inertial odometry on edge devices. The project details and code can be found at [https://github.com/RoyinaJayanth/EqNIO](https://github.com/RoyinaJayanth/EqNIO).

Incorporating inductive bias by embedding geometric entities (such as rays) as input has proven successful in multi-view learning. However, the methods adopting this technique typically lack equivariance, which is crucial for effective 3D learning. Equivariance serves as a valuable inductive prior, aiding in the generation of robust multi-view features for 3D scene understanding. In this paper, we explore the application of equivariant multi-view learning to depth estimation, not only recognizing its significance for computer vision and robotics but also addressing the limitations of previous research. Most prior studies have either overlooked equivariance in this setting or achieved only approximate equivariance through data augmentation, which often leads to inconsistencies across different reference frames. To address this issue, we propose to embed $SE(3)$ equivariance into the Perceiver IO architecture. We employ Spherical Harmonics for positional encoding to ensure 3D rotation equivariance, and develop a specialized equivariant encoder and decoder within the Perceiver IO architecture. To validate our model, we applied it to the task of stereo depth estimation, achieving state of the art results on real-world datasets without explicit geometric constraints or extensive data augmentation.

3D reconstruction and novel view rendering can greatly benefit from geometric priors when the input views are not sufficient in terms of coverage and inter-view baselines. Deep learning of geometric priors from 2D images requires each image to be represented in a $2D$ canonical frame and the prior to be learned in a given or learned $3D$ canonical frame. In this paper, given only the relative poses of the cameras, we show how to learn priors from multiple views equivariant to coordinate frame transformations by proposing an $SE(3)$-equivariant convolution and transformer in the space of rays in 3D. We model the ray space as a homogeneous space of $SE(3)$ and introduce the $SE(3)$-equivariant convolution in ray space. Depending on the output domain of the convolution, we present convolution-based $SE(3)$-equivariant maps from ray space to ray space and to $\mathbb{R}^3$. Our mathematical framework allows us to go beyond convolution to $SE(3)$-equivariant attention in the ray space. We showcase how to tailor and adapt the equivariant convolution and transformer in the tasks of equivariant $3D$ reconstruction and equivariant neural rendering from multiple views. We demonstrate $SE(3)$-equivariance by obtaining robust results in roto-translated datasets without performing transformation augmentation.

We introduce a unified framework for group equivariant networks on homogeneous spaces derived from a Fourier perspective. We consider tensor-valued feature fields, before and after a convolutional layer. We present a unified derivation of kernels via the Fourier domain by leveraging the sparsity of Fourier coefficients of the lifted feature fields. The sparsity emerges when the stabilizer subgroup of the homogeneous space is a compact Lie group. We further introduce a nonlinear activation, via an elementwise nonlinearity on the regular representation after lifting and projecting back to the field through an equivariant convolution. We show that other methods treating features as the Fourier coefficients in the stabilizer subgroup are special cases of our activation. Experiments on $SO(3)$ and $SE(3)$ show state-of-the-art performance in spherical vector field regression, point cloud classification, and molecular completion.

We propose an image synthesis approach that provides stratified navigation in the latent code space. With a tiny amount of partial or very low-resolution image, our approach can consistently out-perform state-of-the-art counterparts in terms of generating the closest sampled image to the ground truth. We achieve this through scale-independent editing while expanding scale-specific diversity. Scale-independence is achieved with a nested scale disentanglement loss. Scale-specific diversity is created by incorporating a progressive diversification constraint. We introduce semantic persistency across the scales by sharing common latent codes. Together they provide better control of the image synthesis process. We evaluate the effectiveness of our proposed approach through various tasks, including image outpainting, image superresolution, and cross-domain image translation.

Several popular approaches to 3D vision tasks process multiple views of the input independently with deep neural networks pre-trained on natural images, where view permutation invariance is achieved through a single round of pooling over all views. We argue that this operation discards important information and leads to subpar global descriptors. In this paper, we propose a group convolutional approach to multiple view aggregation where convolutions are performed over a discrete subgroup of the rotation group, enabling, thus, joint reasoning over all views in an equivariant (instead of invariant) fashion, up to the very last layer. We further develop this idea to operate on smaller discrete homogeneous spaces of the rotation group, where a polar view representation is used to maintain equivariance with only a fraction of the number of input views. We set the new state of the art in several large scale 3D shape retrieval tasks, and show additional applications to panoramic scene classification.