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Given a computable sequence of natural numbers, it is a natural task to find a G\"odel number of a program that generates this sequence. It is easy to see that this problem is neither continuous nor computable. In algorithmic learning theory this problem is well studied from several perspectives and one question studied there is for which sequences this problem is at least learnable in the limit. Here we study the problem on all computable sequences and we classify the Weihrauch complexity of it. For this purpose we can, among other methods, utilize the amalgamation technique known from learning theory. As a benchmark for the classification we use closed and compact choice problems and their jumps on natural numbers, and we argue that these problems correspond to induction and boundedness principles, as they are known from the Kirby-Paris hierarchy in reverse mathematics. We provide a topological as well as a computability-theoretic classification, which reveal some significant differences.

In this paper, we study an intelligent reflecting surface (IRS) assisted communication system with single-antenna transmitter and receiver, under imperfect channel state information (CSI). More specifically, we deal with the robust selection of binary (on/off) states of the IRS elements in order to maximize the worst-case energy efficiency (EE), given a bounded CSI uncertainty, while satisfying a minimum signal-to-noise ratio (SNR). The IRS phase shifts are adjusted so as to maximize the ideal SNR (i.e., without CSI error), based only on the estimated channels. First, we derive a closed-form expression of the worst-case SNR, and then formulate the robust (discrete) optimization problem. Moreover, we design and analyze a dynamic programming (DP) algorithm that is theoretically guaranteed to achieve the global maximum with polynomial complexity $O(L \log L)$, where $L$ is the number of IRS elements. Finally, numerical simulations confirm the theoretical results. In particular, the proposed algorithm shows identical performance with the exhaustive search, and significantly outperforms a baseline scheme, namely, the activation of all IRS elements.

We discuss two approaches to solving the parametric (or stochastic) eigenvalue problem. One of them uses a Taylor expansion and the other a Chebyshev expansion. The parametric eigenvalue problem assumes that the matrix $A$ depends on a parameter $\mu$, where $\mu$ might be a random variable. Consequently, the eigenvalues and eigenvectors are also functions of $\mu$. We compute a Taylor approximation of these functions about $\mu_{0}$ by iteratively computing the Taylor coefficients. The complexity of this approach is $O(n^{3})$ for all eigenpairs, if the derivatives of $A(\mu)$ at $\mu_{0}$ are given. The Chebyshev expansion works similarly. We first find an initial approximation iteratively which we then refine with Newton's method. This second method is more expensive but provides a good approximation over the whole interval of the expansion instead around a single point. We present numerical experiments confirming the complexity and demonstrating that the approaches are capable of tracking eigenvalues at intersection points. Further experiments shed light on the limitations of the Taylor expansion approach with respect to the distance from the expansion point $\mu_{0}$.

In literature on imprecise probability little attention is paid to the fact that imprecise probabilities are precise on some events. We call these sets system of precision. We show that, under mild assumptions, the system of precision of a lower and upper probability form a so-called (pre-)Dynkin-system. Interestingly, there are several settings, ranging from machine learning on partial data over frequential probability theory to quantum probability theory and decision making under uncertainty, in which a priori the probabilities are only desired to be precise on a specific underlying set system. At the core of all of these settings lies the observation that precise beliefs, probabilities or frequencies on two events do not necessarily imply this precision to hold for the intersection of those events. Here, (pre-)Dynkin-systems have been adopted as systems of precision, too. We show that, under extendability conditions, those pre-Dynkin-systems equipped with probabilities can be embedded into algebras of sets. Surprisingly, the extendability conditions elaborated in a strand of work in quantum physics are equivalent to coherence in the sense of Walley (1991, Statistical reasoning with imprecise probabilities, p. 84). Thus, literature on probabilities on pre-Dynkin-systems gets linked to the literature on imprecise probability. Finally, we spell out a lattice duality which rigorously relates the system of precision to credal sets of probabilities. In particular, we provide a hitherto undescribed, parametrized family of coherent imprecise probabilities.

We consider the standard broadcast setup with a single server broadcasting information to a number of clients, each of which contains local storage (called cache) of some size, which can store some parts of the available files at the server. The centralized coded caching framework, consists of a caching phase and a delivery phase, both of which are carefully designed in order to use the cache and the channel together optimally. In prior literature, various combinatorial structures have been used to construct coded caching schemes. One of the chief drawbacks of many of these existing constructions is the large subpacketization level, which denotes the number of times a file should be split for the schemes to provide coding gain. In this work, using a new binary matrix model, we present several novel constructions for coded caching based on the various types of combinatorial designs and their $q$-analogs, which are also called subspace designs. While most of the schemes constructed in this work (based on existing designs) have a high cache requirement, they provide a rate that is either constant or decreasing, and moreover require competitively small levels of subpacketization, which is an extremely important feature in practical applications of coded caching. We also apply our constructions to the distributed computing framework of MapReduce, which consists of three phases, the Map phase, the Shuffle phase and the Reduce phase. Using our binary matrix framework, we present a new simple generic coded data shuffling scheme. Employing our designs-based constructions in conjunction with this new shuffling scheme, we obtain new coded computing schemes which have low file complexity, with marginally higher communication load compared to the optimal scheme for equivalent parameters. We show that our schemes can neatly extend to the scenario with full and partial stragglers also.

The multivariate Hawkes process is a past-dependent point process used to model the relationship of event occurrences between different phenomena.Although the Hawkes process was originally introduced to describe excitation interactions, which means that one event increases the chances of another occurring, there has been a growing interest in modelling the opposite effect, known as inhibition.In this paper, we focus on how to infer the parameters of a multidimensional exponential Hawkes process with both excitation and inhibition effects. Our first result is to prove the identifiability of this model under a few sufficient assumptions. Then we propose a maximum likelihood approach to estimate the interaction functions, which is, to the best of our knowledge, the first exact inference procedure in the frequentist framework.Our method includes a variable selection step in order to recover the support of interactions and therefore to infer the connectivity graph.A benefit of our method is to provide an explicit computation of the log-likelihood, which enables in addition to perform a goodness-of-fit test for assessing the quality of estimations.We compare our method to standard approaches, which were developed in the linear framework and are not specifically designed for handling inhibiting effects.We show that the proposed estimator performs better on synthetic data than alternative approaches. We also illustrate the application of our procedure to a neuronal activity dataset, which highlights the presence of both exciting and inhibiting effects between neurons.

This paper proposes a cognitive radio enabled LEO SatCom using RSMA radio access technique with the coexistence of GEO SatCom network. In particular, this work aims to maximize the sum rate of LEO SatCom by simultaneously optimizing the power budget over different beams, RSMA power allocation for users over each beam, and subcarrier user assignment while restricting the interference temperature to GEO SatCom. The problem of sum rate maximization is formulated as non-convex, where the global optimal solution is challenging to obtain. Thus, an efficient solution can be obtained in three steps: first we employ a successive convex approximation technique to reduce the complexity and make the problem more tractable. Second, for any given resource block user assignment, we adopt KKT conditions to calculate the transmit power over different beams and RSMA power allocation of users over each beam. Third, using the allocated power, we design an efficient algorithm based on the greedy approach for resource block user assignment. Numerical results demonstrate the benefits of the proposed optimization scheme compared to the benchmark schemes.

Gradient coding schemes effectively mitigate full stragglers in distributed learning by introducing identical redundancy in coded local partial derivatives corresponding to all model parameters. However, they are no longer effective for partial stragglers as they cannot utilize incomplete computation results from partial stragglers. This paper aims to design a new gradient coding scheme for mitigating partial stragglers in distributed learning. Specifically, we consider a distributed system consisting of one master and N workers, characterized by a general partial straggler model and focuses on solving a general large-scale machine learning problem with L model parameters using gradient coding. First, we propose a coordinate gradient coding scheme with L coding parameters representing L possibly different diversities for the L coordinates, which generates most gradient coding schemes. Then, we consider the minimization of the expected overall runtime and the maximization of the completion probability with respect to the L coding parameters for coordinates, which are challenging discrete optimization problems. To reduce computational complexity, we first transform each to an equivalent but much simpler discrete problem with N\llL variables representing the partition of the L coordinates into N blocks, each with identical redundancy. This indicates an equivalent but more easily implemented block coordinate gradient coding scheme with N coding parameters for blocks. Then, we adopt continuous relaxation to further reduce computational complexity. For the resulting minimization of expected overall runtime, we develop an iterative algorithm of computational complexity O(N^2) to obtain an optimal solution and derive two closed-form approximate solutions both with computational complexity O(N). For the resultant maximization of the completion probability, we develop an iterative algorithm of...

The interconnected smart devices and industrial internet of things devices require low-latency communication to fulfill control objectives despite limited resources. In essence, such devices have a time-critical nature but also require a highly accurate data input based on its significance. In this paper, we investigate various coordinated and distributed semantic scheduling schemes with a data significance perspective. In particular, novel algorithms are proposed to analyze the benefit of such schemes for the significance in terms of estimation accuracy. Then, we derive the bounds of the achievable estimation accuracy. Our numerical results showcase the superiority of semantic scheduling policies that adopt an integrated control and communication strategy. In essence, such policies can reduce the weighted sum of mean squared errors compared to traditional policies.

This paper proposes a physically consistent Gaussian Process (GP) enabling the identification of uncertain Lagrangian systems. The function space is tailored according to the energy components of the Lagrangian and the differential equation structure, analytically guaranteeing physical and mathematical properties such as energy conservation and quadratic form. The novel formulation of Cholesky decomposed matrix kernels allow the probabilistic preservation of positive definiteness. Only differential input-to-output measurements of the function map are required while Gaussian noise is permitted in torques, velocities, and accelerations. We demonstrate the effectiveness of the approach in numerical simulation.