In this paper, we present a novel neural architecture for 5G channel estimation (CE), the Recurrent Equivariant UERS Estimation Network (ReQuestNet). It incorporates several practical considerations in wireless communication systems, such as ability to handle variable number of resource block (RB), dynamic number of transmit layers, physical resource block groups (PRGs) bundling size (BS), demodulation reference signal (DMRS) patterns with a single unified model, thereby, drastically simplifying the CE pipeline. Besides it addresses several limitations of the legacy linear minimum mean squared error (MMSE) solutions, for example, by being independent of other reference signals and particularly by jointly processing multiple input multiple output (MIMO) layers and differently precoded channels, unknown at the receiver. ReQuestNet comprises of two sub-units, CoarseNet followed by RefinementNet. CoarseNet performs per PRG, per transmit-receive (Tx-Rx) stream channel estimation, while RefinementNet refines the CoarseNet channel estimate by incorporating correlations across differently precoded PRGs, and correlation across MIMO channel spatial dimensions (cross-MIMO). The simulation results show that ReQuestNet outperforms genie MMSE across various channel profiles as well as unseen channel profiles during training, achieving up to 10dB gain at high signal-to-noise ration (SNR).

Steerable convolutional neural networks (SCNNs) enhance task performance by modelling geometric symmetries through equivariance constraints on weights. Yet, unknown or varying symmetries can lead to overconstrained weights and decreased performance. To address this, this paper introduces a probabilistic method to learn the degree of equivariance in SCNNs. We parameterise the degree of equivariance as a likelihood distribution over the transformation group using Fourier coefficients, offering the option to model layer-wise and shared equivariance. These likelihood distributions are regularised to ensure an interpretable degree of equivariance across the network. Advantages include the applicability to many types of equivariant networks through the flexible framework of SCNNs and the ability to learn equivariance with respect to any subgroup of any compact group without requiring additional layers. Our experiments reveal competitive performance on datasets with mixed symmetries, with learnt likelihood distributions that are representative of the underlying degree of equivariance.

The Poisson Matching Lemma (PML) introduced by Li & Anantharam (IT-Trans 2021) is a powerful technique for one-shot analysis of a variety of multi-terminal source and channel coding problems. In this work we make use of PML to derive one-shot achievability results for unequal message protection with a fixed number of message classes. Our analysis involves revisiting the proof of the PML to account for the error associated with each codebook at the decoder. Our approach leads to compact bounds on the error probability for each message class for arbitrary input distributions and channels. For the example of binary erasure channel, we compare our bounds numerically with prior work [1] and demonstrate improvements in the achievable rate.

Steerable convolutional neural networks (CNNs) provide a general framework for building neural networks equivariant to translations and transformations of an origin-preserving group $G$, such as reflections and rotations. They rely on standard convolutions with $G$-steerable kernels obtained by analytically solving the group-specific equivariance constraint imposed onto the kernel space. As the solution is tailored to a particular group $G$, implementing a kernel basis does not generalize to other symmetry transformations, complicating the development of general group equivariant models. We propose using implicit neural representation via multi-layer perceptrons (MLPs) to parameterize $G$-steerable kernels. The resulting framework offers a simple and flexible way to implement Steerable CNNs and generalizes to any group $G$ for which a $G$-equivariant MLP can be built. We prove the effectiveness of our method on multiple tasks, including N-body simulations, point cloud classification and molecular property prediction.

Equivariant networks capture the inductive bias about the symmetry of the learning task by building those symmetries into the model. In this paper, we study how equivariance relates to generalization error utilizing PAC Bayesian analysis for equivariant networks, where the transformation laws of feature spaces are deter- mined by group representations. By using perturbation analysis of equivariant networks in Fourier domain for each layer, we derive norm-based PAC-Bayesian generalization bounds. The bound characterizes the impact of group size, and multiplicity and degree of irreducible representations on the generalization error and thereby provide a guideline for selecting them. In general, the bound indicates that using larger group size in the model improves the generalization error substantiated by extensive numerical experiments.

Cryo-Electron Microscopy (Cryo-EM) is an important imaging method which allows high-resolution reconstruction of the 3D structures of biomolecules. It produces highly noisy 2D images by projecting a molecule's 3D density from random viewing directions. Because the projection directions are unknown, estimating the images' poses is necessary to perform the reconstruction. We focus on this task and study it under the group synchronization framework: if the relative poses of pairs of images can be approximated from the data, an estimation of the images' poses is given by the assignment which is most consistent with the relative ones.In particular, by studying the symmetries of cryo-EM, we show that relative poses in the group O(2) provide sufficient constraints to identify the images' poses, up to the molecule's chirality. With this in mind, we improve the existing multi-frequency vector diffusion maps (MFVDM) method: by using O(2) relative poses, our method not only predicts the similarity between the images' viewing directions but also recovers their poses. Hence, we can leverage all input images in a 3D reconstruction algorithm by initializing the poses with our estimation rather than just clustering and averaging the input images. We validate the recovery capabilities and robustness of our method on randomly generated synchronization graphs and a synthetic cryo-EM dataset.

The big empirical success of group equivariant networks has led in recent years to the sprouting of a great variety of equivariant network architectures. A particular focus has thereby been on rotation and reflection equivariant CNNs for planar images. Here we give a general description of E(2)-equivariant convolutions in the framework of Steerable CNNs. The theory of Steerable CNNs thereby yields constraints on the convolution kernels which depend on group representations describing the transformation laws of feature spaces. We show that these constraints for arbitrary group representations can be reduced to constraints under irreducible representations. A general solution of the kernel space constraint is given for arbitrary representations of the Euclidean group E(2) and its subgroups. We implement a wide range of previously proposed and entirely new equivariant network architectures and extensively compare their performances. E(2)-steerable convolutions are further shown to yield remarkable gains on CIFAR-10, CIFAR-100 and STL-10 when used as drop in replacement for non-equivariant convolutions.