On the Sample Complexity of One Hidden Layer Networks with Equivariance, Locality and Weight Sharing

Weight sharing, equivariance, and local filters, as in convolutional neural networks, are believed to contribute to the sample efficiency of neural networks. However, it is not clear how each one of these design choices contribute to the generalization error. Through the lens of statistical learning theory, we aim to provide an insight into this question by characterizing the relative impact of each choice on the sample complexity. We obtain lower and upper sample complexity bounds for a class of single hidden layer networks. It is shown that the gain of equivariance is directly manifested in the bound, while getting a similar increase for weight sharing depends on the sharing mechanism. Among our results, we obtain a completely dimension-free bound for equivariant networks for a class of pooling operations. We show that the bound depends merely on the norm of filters, which is tighter than using the spectral norm of the respective matrix. We also characterize the trade-off in sample complexity between the parametrization of filters in spatial and frequency domains, particularly when spatial filters are localized as in vanilla convolutional neural networks.

Multi-Draft Speculative Sampling: Canonical Architectures and Theoretical Limits

We consider multi-draft speculative sampling, where the proposal sequences are sampled independently from different draft models. At each step, a token-level draft selection scheme takes a list of valid tokens as input and produces an output token whose distribution matches that of the target model. Previous works have demonstrated that the optimal scheme (which maximizes the probability of accepting one of the input tokens) can be cast as a solution to a linear program. In this work we show that the optimal scheme can be decomposed into a two-step solution: in the first step an importance sampling (IS) type scheme is used to select one intermediate token; in the second step (single-draft) speculative sampling is applied to generate the output token. For the case of two identical draft models we further 1) establish a necessary and sufficient condition on the distributions of the target and draft models for the acceptance probability to equal one and 2) provide an explicit expression for the optimal acceptance probability. Our theoretical analysis also motives a new class of token-level selection scheme based on weighted importance sampling. Our experimental results demonstrate consistent improvements in the achievable block efficiency and token rates over baseline schemes in a number of scenarios.

Reinforcement Learning of Adaptive Acquisition Policies for Inverse Problems

A promising way to mitigate the expensive process of obtaining a high-dimensional signal is to acquire a limited number of low-dimensional measurements and solve an under-determined inverse problem by utilizing the structural prior about the signal. In this paper, we focus on adaptive acquisition schemes to save further the number of measurements. To this end, we propose a reinforcement learning-based approach that sequentially collects measurements to better recover the underlying signal by acquiring fewer measurements. Our approach applies to general inverse problems with continuous action spaces and jointly learns the recovery algorithm. Using insights obtained from theoretical analysis, we also provide a probabilistic design for our methods using variational formulation. We evaluate our approach on multiple datasets and with two measurement spaces (Gaussian, Radon). Our results confirm the benefits of adaptive strategies in low-acquisition horizon settings.

Variational Learning ISTA

Compressed sensing combines the power of convex optimization techniques with a sparsity-inducing prior on the signal space to solve an underdetermined system of equations. For many problems, the sparsifying dictionary is not directly given, nor its existence can be assumed. Besides, the sensing matrix can change across different scenarios. Addressing these issues requires solving a sparse representation learning problem, namely dictionary learning, taking into account the epistemic uncertainty of the learned dictionaries and, finally, jointly learning sparse representations and reconstructions under varying sensing matrix conditions. We address both concerns by proposing a variant of the LISTA architecture. First, we introduce Augmented Dictionary Learning ISTA (A-DLISTA), which incorporates an augmentation module to adapt parameters to the current measurement setup. Then, we propose to learn a distribution over dictionaries via a variational approach, dubbed Variational Learning ISTA (VLISTA). VLISTA exploits A-DLISTA as the likelihood model and approximates a posterior distribution over the dictionaries as part of an unfolded LISTA-based recovery algorithm. As a result, VLISTA provides a probabilistic way to jointly learn the dictionary distribution and the reconstruction algorithm with varying sensing matrices. We provide theoretical and experimental support for our architecture and show that our model learns calibrated uncertainties.

Probabilistic and Differentiable Wireless Simulation with Geometric Transformers

Modelling the propagation of electromagnetic signals is critical for designing modern communication systems. While there are precise simulators based on ray tracing, they do not lend themselves to solving inverse problems or the integration in an automated design loop. We propose to address these challenges through differentiable neural surrogates that exploit the geometric aspects of the problem. We first introduce the Wireless Geometric Algebra Transformer (Wi-GATr), a generic backbone architecture for simulating wireless propagation in a 3D environment. It uses versatile representations based on geometric algebra and is equivariant with respect to E(3), the symmetry group of the underlying physics. Second, we study two algorithmic approaches to signal prediction and inverse problems based on differentiable predictive modelling and diffusion models. We show how these let us predict received power, localize receivers, and reconstruct the 3D environment from the received signal. Finally, we introduce two large, geometry-focused datasets of wireless signal propagation in indoor scenes. In experiments, we show that our geometry-forward approach achieves higher-fidelity predictions with less data than various baselines.

Simulating, Fast and Slow: Learning Policies for Black-Box Optimization

In recent years, solving optimization problems involving black-box simulators has become a point of focus for the machine learning community due to their ubiquity in science and engineering. The simulators describe a forward process $f_{\mathrm{sim}}: (\psi, x) \rightarrow y$ from simulation parameters $\psi$ and input data $x$ to observations $y$, and the goal of the optimization problem is to find parameters $\psi$ that minimize a desired loss function. Sophisticated optimization algorithms typically require gradient information regarding the forward process, $f_{\mathrm{sim}}$, with respect to the parameters $\psi$. However, obtaining gradients from black-box simulators can often be prohibitively expensive or, in some cases, impossible. Furthermore, in many applications, practitioners aim to solve a set of related problems. Thus, starting the optimization ``ab initio", i.e. from scratch, each time might be inefficient if the forward model is expensive to evaluate. To address those challenges, this paper introduces a novel method for solving classes of similar black-box optimization problems by learning an active learning policy that guides a differentiable surrogate's training and uses the surrogate's gradients to optimize the simulation parameters with gradient descent. After training the policy, downstream optimization of problems involving black-box simulators requires up to $\sim$90\% fewer expensive simulator calls compared to baselines such as local surrogate-based approaches, numerical optimization, and Bayesian methods.

Vision-Assisted Digital Twin Creation for mmWave Beam Management

In the context of communication networks, digital twin technology provides a means to replicate the radio frequency (RF) propagation environment as well as the system behaviour, allowing for a way to optimize the performance of a deployed system based on simulations. One of the key challenges in the application of Digital Twin technology to mmWave systems is the prevalent channel simulators' stringent requirements on the accuracy of the 3D Digital Twin, reducing the feasibility of the technology in real applications. We propose a practical Digital Twin creation pipeline and a channel simulator, that relies only on a single mounted camera and position information. We demonstrate the performance benefits compared to methods that do not explicitly model the 3D environment, on downstream sub-tasks in beam acquisition, using the real-world dataset of the DeepSense6G challenge

Algebraic Topological Networks via the Persistent Local Homology Sheaf

In this work, we introduce a novel approach based on algebraic topology to enhance graph convolution and attention modules by incorporating local topological properties of the data. To do so, we consider the framework of sheaf neural networks, which has been previously leveraged to incorporate additional structure into graph neural networks' features and construct more expressive, non-isotropic messages. Specifically, given an input simplicial complex (e.g. generated by the cliques of a graph or the neighbors in a point cloud), we construct its local homology sheaf, which assigns to each node the vector space of its local homology. The intermediate features of our networks live in these vector spaces and we leverage the associated sheaf Laplacian to construct more complex linear messages between them. Moreover, we extend this approach by considering the persistent version of local homology associated with a weighted simplicial complex (e.g., built from pairwise distances of nodes embeddings). This i) solves the problem of the lack of a natural choice of basis for the local homology vector spaces and ii) makes the sheaf itself differentiable, which enables our models to directly optimize the topology of their intermediate features.

Neural Lattice Reduction: A Self-Supervised Geometric Deep Learning Approach

Lattice reduction is a combinatorial optimization problem aimed at finding the most orthogonal basis in a given lattice. In this work, we address lattice reduction via deep learning methods. We design a deep neural model outputting factorized unimodular matrices and train it in a self-supervised manner by penalizing non-orthogonal lattice bases. We incorporate the symmetries of lattice reduction into the model by making it invariant and equivariant with respect to appropriate continuous and discrete groups.

Transformer-Based Neural Surrogate for Link-Level Path Loss Prediction from Variable-Sized Maps

Estimating path loss for a transmitter-receiver location is key to many use-cases including network planning and handover. Machine learning has become a popular tool to predict wireless channel properties based on map data. In this work, we present a transformer-based neural network architecture that enables predicting link-level properties from maps of various dimensions and from sparse measurements. The map contains information about buildings and foliage. The transformer model attends to the regions that are relevant for path loss prediction and, therefore, scales efficiently to maps of different size. Further, our approach works with continuous transmitter and receiver coordinates without relying on discretization. In experiments, we show that the proposed model is able to efficiently learn dominant path losses from sparse training data and generalizes well when tested on novel maps.