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This paper considers the performance of Reed-Muller (RM) codes transmitted over binary memoryless symmetric (BMS) channels under bitwise maximum-a-posteriori (bit-MAP) decoding. Its main result is that, for a fixed BMS channel, the family of binary RM codes can achieve a vanishing bit-error probability at rates approaching the channel capacity. This partially resolves a long-standing open problem that connects information theory and error-correcting codes. In contrast with the earlier result for the binary erasure channel, the new proof does not rely on hypercontractivity. Instead, it combines a nesting property of RM codes with new information inequalities relating the generalized extrinsic information transfer function and the extrinsic minimum mean-squared error.

In this work we propose an extension of physics informed supervised learning strategies to parametric partial differential equations. Indeed, even if the latter are indisputably useful in many applications, they can be computationally expensive most of all in a real-time and many-query setting. Thus, our main goal is to provide a physics informed learning paradigm to simulate parametrized phenomena in a small amount of time. The physics information will be exploited in many ways, in the loss function (standard physics informed neural networks), as an augmented input (extra feature employment) and as a guideline to build an effective structure for the neural network (physics informed architecture). These three aspects, combined together, will lead to a faster training phase and to a more accurate parametric prediction. The methodology has been tested for several equations and also in an optimal control framework.

In this work we propose tailored model order reduction for varying boundary optimal control problems governed by parametric partial differential equations. With varying boundary control, we mean that a specific parameter changes where the boundary control acts on the system. This peculiar formulation might benefit from model order reduction. Indeed, fast and reliable simulations of this model can be of utmost usefulness in many applied fields, such as geophysics and energy engineering. However, varying boundary control features very complicated and diversified parametric behaviour for the state and adjoint variables. The state solution, for example, changing the boundary control parameter, might feature transport phenomena. Moreover, the problem loses its affine structure. It is well known that classical model order reduction techniques fail in this setting, both in accuracy and in efficiency. Thus, we propose reduced approaches inspired by the ones used when dealing with wave-like phenomena. Indeed, we compare standard proper orthogonal decomposition with two tailored strategies: geometric recasting and local proper orthogonal decomposition. Geometric recasting solves the optimization system in a reference domain simplifying the problem at hand avoiding hyper-reduction, while local proper orthogonal decomposition builds local bases to increase the accuracy of the reduced solution in very general settings (where geometric recasting is unfeasible). We compare the various approaches on two different numerical experiments based on geometries of increasing complexity.

This paper addresses the communication issues when estimating hyper-gradients in decentralized federated learning (FL). Hyper-gradients in decentralized FL quantifies how the performance of globally shared optimal model is influenced by the perturbations in clients' hyper-parameters. In prior work, clients trace this influence through the communication of Hessian matrices over a static undirected network, resulting in (i) excessive communication costs and (ii) inability to make use of more efficient and robust networks, namely, time-varying directed networks. To solve these issues, we introduce an alternative optimality condition for FL using an averaging operation on model parameters and gradients. We then employ Push-Sum as the averaging operation, which is a consensus optimization technique for time-varying directed networks. As a result, the hyper-gradient estimator derived from our optimality condition enjoys two desirable properties; (i) it only requires Push-Sum communication of vectors and (ii) it can operate over time-varying directed networks. We confirm the convergence of our estimator to the true hyper-gradient both theoretically and empirically, and we further demonstrate that it enables two novel applications: decentralized influence estimation and personalization over time-varying networks.

Hybrid dynamical systems, i.e. systems that have both continuous and discrete states, are ubiquitous in engineering, but are difficult to work with due to their discontinuous transitions. For example, a robot leg is able to exert very little control effort while it is in the air compared to when it is on the ground. When the leg hits the ground, the penetrating velocity instantaneously collapses to zero. These instantaneous changes in dynamics and discontinuities (or jumps) in state make standard smooth tools for planning, estimation, control, and learning difficult for hybrid systems. One of the key tools for accounting for these jumps is called the saltation matrix. The saltation matrix is the sensitivity update when a hybrid jump occurs and has been used in a variety of fields including robotics, power circuits, and computational neuroscience. This paper presents an intuitive derivation of the saltation matrix and discusses what it captures, where it has been used in the past, how it is used for linear and quadratic forms, how it is computed for rigid body systems with unilateral constraints, and some of the structural properties of the saltation matrix in these cases.

We approximate the d complex zeros of a univariate polynomial p(x) of a degree d or those zeros that lie in a fixed region of interest on the complex plane such as a disc or a square. Our divide and conquer algorithm of STOC 1995 supports solution of this problem in optimal Boolean time (up to a poly-logarithmic factor), that is, runs nearly as fast as one can access the coefficients of p with the precision necessary to support required accuracy of the output. That record complexity has not been matched by any other algorithm yet, but our root-finder of 1995 is quite involved and has never been implemented. We present alternative nearly optimal root-finders based on our novel variants of the classical subdivision iterations. Unlike our predecessor of 1995, we require randomization of Las Vegas type, allowing us to detect any output error at a dominated computational cost, but our new root-finders are much simpler to implement than their predecessor of 1995. According to the results of extensive test with standard test polynomials for their preliminary version, which incorporates only a part of our novel techniques, the new root-finders compete and for a large class of inputs significantly supersedes the package of root-finding subroutines MPSolve, which for decades has been user's choice package. Unlike our predecessor of 1995 and all known fast algorithms for the cited tasks of polynomial root-finding, our new algorithms can be also applied to a polynomial given by a black box oracle for its evaluation rather than by its coefficients. This makes our root-finders particularly efficient for polynomials p(x) that can be evaluated fast such as the Mandelbrot polynomials or those given by the sum of a small number of shifted monomials. Our algorithm can be readily extended to fast approximation of the eigenvalues of a matrix or a matrix polynomial.

In this paper, we initiate the study of rate-splitting multiple access (RSMA) for a mono-static integrated sensing and communication (ISAC) system, where the dual-functional base station (BS) simultaneously communicates with multiple users and detects multiple moving targets. We aim at optimizing the ISAC waveform to jointly maximize the max-min fairness (MMF) rate of the communication users and minimize the largest eigenvalue of the Cram\'er-Rao bound (CRB) matrix for unbiased estimation. The CRB matrix considered in this work is general as it involves the estimation of angular direction, complex reflection coefficient, and Doppler frequency for multiple moving targets. Simulation results demonstrate that RSMA maintains a larger communication and sensing trade-off than conventional space-division multiple access (SDMA) and it is capable of detecting multiple targets with a high detection accuracy. The finding highlights the potential of RSMA as an effective and powerful strategy for interference management in the general multi-user multi-target ISAC systems.

The Tucker tensor decomposition is a natural extension of the singular value decomposition (SVD) to multiway data. We propose to accelerate Tucker tensor decomposition algorithms by using randomization and parallelization. We present two algorithms that scale to large data and many processors, significantly reduce both computation and communication cost compared to previous deterministic and randomized approaches, and obtain nearly the same approximation errors. The key idea in our algorithms is to perform randomized sketches with Kronecker-structured random matrices, which reduces computation compared to unstructured matrices and can be implemented using a fundamental tensor computational kernel. We provide probabilistic error analysis of our algorithms and implement a new parallel algorithm for the structured randomized sketch. Our experimental results demonstrate that our combination of randomization and parallelization achieves accurate Tucker decompositions much faster than alternative approaches. We observe up to a 16X speedup over the fastest deterministic parallel implementation on 3D simulation data.

We consider the problem of solving linear least squares problems in a framework where only evaluations of the linear map are possible. We derive randomized methods that do not need any other matrix operations than forward evaluations, especially no evaluation of the adjoint map is needed. Our method is motivated by the simple observation that one can get an unbiased estimate of the application of the adjoint. We show convergence of the method and then derive a more efficient method that uses an exact linesearch. This method, called random descent, resembles known methods in other context and has the randomized coordinate descent method as special case. We provide convergence analysis of the random descent method emphasizing the dependence on the underlying distribution of the random vectors. Furthermore we investigate the applicability of the method in the context of ill-posed inverse problems and show that the method can have beneficial properties when the unknown solution is rough. We illustrate the theoretical findings in numerical examples. One particular result is that the random descent method actually outperforms established transposed-free methods (TFQMR and CGS) in examples.

In this paper we investigate formal verification problems for Neural Network computations. Various reachability problems will be in the focus, such as: Given symbolic specifications of allowed inputs and outputs in form of Linear Programming instances, one question is whether valid inputs exist such that the given network computes a valid output? Does this property hold for all valid inputs? The former question's complexity has been investigated recently by S\"alzer and Lange for nets using the Rectified Linear Unit and the identity function as their activation functions. We complement their achievements by showing that the problem is NP-complete for piecewise linear functions with rational coefficients that are not linear, NP-hard for almost all suitable activation functions including non-linear ones that are continuous on an interval, complete for the Existential Theory of the Reals $\exists \mathbb R$ for every non-linear polynomial and $\exists \mathbb R$-hard for the exponential function and various sigmoidal functions. For the completeness results, linking the verification tasks with the theory of Constraint Satisfaction Problems turns out helpful.