Holger Boche’s research while affiliated with Technical University of Munich and other places

We consider a fractional-calculus example of a continuous hierarchy of algorithmic information in the context of its potential applications in digital twinning. Digital twinning refers to different emerging methodologies in control engineering that involve the creation of a digital replica of some physical entity. From the perspective of computability theory, the problem of ensuring the digital twin's integrity -- i.e., keeping it in a state where it matches its physical counterpart -- entails a notion of algorithmic information that determines which of the physical system's properties we can reliably deduce by algorithmically analyzing its digital twin. The present work investigates the fractional calculus of periodic functions -- particularly, we consider the Wiener algebra -- as an exemplary application of the algorithmic-information concept. We establish a continuously ordered hierarchy of algorithmic information among spaces of periodic functions -- depending on their fractional degree of smoothness -- in which the ordering relation determines whether a certain representation of some function contains ``more'' or ``less'' information than another. Additionally, we establish an analogous hierarchy among lp-spaces, which form a cornerstone of (traditional) digital signal processing. Notably, both hierarchies are (mathematically) ``dual'' to each other. From a practical perspective, our approach ultimately falls into the category of formal verification and (general) formal methods.

Molecular communication (MC) enables information transfer via molecules, making it ideal for biomedical applications where traditional methods fall short. In many such scenarios, identifying specific events is more critical than decoding full messages, motivating the use of deterministic identification (DI). This paper investigates DI over discrete-time Poisson channels (DTPCs) with inter-symbol interference (ISI), a realistic setting due to channel memory effects. We improve the known upper bound on DI capacity under power constraints from $\frac{3}{2} + \kappa$ to $\frac{1 + \kappa}{2}$. Additionally, we present the first results on deterministic identification with feedback (DIF) in this context, providing a constructive lower bound. These findings enhance the theoretical understanding of MC and support more efficient, feedback-driven biomedical systems.

In our previous work, we presented the Hypothesis Testing Lemma, a key tool that establishes sufficient conditions for the existence of good deterministic identification (DI) codes for memoryless channels with finite output, but arbitrary input alphabets. In this work, we provide a full quantum analogue of this lemma, which shows that the existence of a DI code in the quantum setting follows from a suitable packing in a modified space of output quantum states. Specifically, we demonstrate that such a code can be constructed using product states derived from this packing. This result enables us to tighten the capacity lower bound for DI over quantum channels beyond the simultaneous decoding approach. In particular, we can now express these bounds solely in terms of the Minkowski dimension of a certain state space, giving us new insights to better understand the nature of the protocol, and the separation between simultaneous and non-simultaneous codes. We extend the discussion with a particular channel example for which we can construct an optimum code.

This paper aims at computing the capacity-distortion-cost (CDC) function for continuous memoryless channels, which is defined as the supremum of the mutual information between channel input and output, constrained by an input cost and an expected distortion of estimating channel state. Solving the optimization problem is challenging because the input distribution does not lie in a finite-dimensional Euclidean space and the optimal estimation function has no closed form in general. We propose to adopt the Wasserstein proximal point method and parametric models such as neural networks (NNs) to update the input distribution and estimation function alternately. To implement it in practice, the importance sampling (IS) technique is used to calculate integrals numerically, and the Wasserstein gradient descent is approximated by pushing forward particles. The algorithm is then applied to an integrated sensing and communications (ISAC) system, validating theoretical results at minimum and maximum distortion as well as the random-deterministic trade-off.

We experimentally investigate the performance of semantically-secure physical layer security (PLS) in 5G new radio (NR) mmWave communications during the initial cell search procedure in the NR band n257 at 27 GHz. A gNB transmits PLS-encoded messages in the presence of an eavesdropper, who intercepts the communication by non-intrusively collecting channel readings in the form of IQ samples. For the message transmission, we use the physical broadcast channel (PBCH) within the synchronization signal block. We analyze different signal-to-noise ratio (SNR) conditions by progressively reducing the transmit power of the subcarriers carrying the PBCH channel, while ensuring optimal conditions for over-the-air frequency and timing synchronization. We measure the secrecy performance of the communication in terms of upper and lower bounds for the distinguishing error rate (DER) metric for different SNR levels and beam angles when performing beamsteering in indoor scenarios, such as office environments and laboratory settings.

This article shows that the minimum mean square error (MMSE) for predicting a stationary stochastic time series from its past observations is not generally Turing computable, even if the spectral density of the stochastic process is differentiable with a computable first derivative. Thus there are spectral densities with the property that for any approximation sequence that converges to the MMSE there does not exist an algorithmic stopping criterion that guarantees that the computed approximation is sufficiently close to the true value of the MMSE. Furthermore, it is shown that under the same conditions on the spectral density, the coefficients of the optimal prediction filter are not generally Turing computable. In such cases and for any sequence of computable finite-impulse response approximations of the optimal prediction filter there exists no algorithmic stopping criterion that is able to guarantee a desired approximation error.

Common randomness (CR), as a resource, is not commonly exploited in existing practical communication systems. In the CR generation framework, both the sender and receiver aim to generate a common random variable observable to both, ideally with low error probability. The availability of this CR allows us to implement correlated random protocols that can lead to faster and more efficient algorithms. Previous work focused on CR generation over perfect channels with limited capacity. In our work, we consider the problem of CR generation from independent and identically distributed (i.i.d.) samples of a correlated finite source with one-way communication over a Gaussian channel. We first derive the CR capacity for single-input single-output (SISO) Gaussian channels. This result is then used for the derivation of the CR capacity in the multiple-input multiple-output (MIMO) case. CR plays a key role in the identification scheme since it may allow a significant increase in the identification capacity of channels. In the identification framework, the decoder is interested in knowing whether a specific message of special interest to him has been sent or not, rather than knowing what the received message is. In many new applications, such as several machine-to-machine and human-to-machine systems and the tactile internet, this post-Shannon scheme is more efficient than classical transmission. In our work, we also consider a CR-assisted secure identification scheme and develop a lower bound on the corresponding secure identification capacity.

Sadly, our esteemed colleague and friend Ning Cai passed away on 25th May, 2023. In his memory, Ingo Althöfer, Holger Boche, Christian Deppe, Jens Stoye, Ulrich Tamm, Andreas Winter, and Raymond Yeung organized the “Workshop on Information Theory and Related Fields” at the Bielefeld ZiF (Center for Interdisciplinary Research). This special event was held from 24th November to 26th November, 2023. The workshop celebrated Ning Cai’s remarkable contributions to the field of information theory and provided a platform for discussing current research in related areas. Ning Cai’s work has had a significant impact on many domains, and this gathering brought together colleagues, collaborators, and young researchers who were influenced by his pioneering efforts.

Fekete’s lemma is a well known result from combinatorial mathematics that shows the existence of a limit value related to super- and subadditive sequences of real numbers. In this paper, we analyze Fekete’s lemma in view of the arithmetical hierarchy of real numbers by Xizhong Zheng and Klaus Weihrauch and fit the results into an information-theoretic context. We introduce special sets associated to super- and subadditive sequences and prove their effective equivalence to $\varSigma _1$ and $\varPi _1$. Using methods from the theory established by Xizhong Zheng and Klaus Weihrauch, we then show that the limit value emerging from Fekete’s lemma is, in general, not a computable number. Given a sequence that additionally satisfies non-negativity, we characterize under which conditions the associated limit value can be computed effectively and investigate the corresponding modulus of convergence. Subsidiarily, we prove a theorem concerning the structural differences between computable sequences of computable numbers and computable sequences of rational numbers. We close the paper by a discussion on how our findings affect common problems from information theory.

New applications in modern communications are demanding robust and ultra-reliable low-latency information exchange such as machine-to-machine and human-to-machine communications. For many of these applications, the identification approach of Ahlswede and Dueck is much more efficient than the classical message transmission scheme proposed by Shannon. Previous studies concentrate mainly on identification over discrete channels. For discrete channels, it was proved that identification is robust under channel uncertainty. Furthermore, optimal identification schemes that are secure and robust against jamming attacks have been considered. However, no results for continuous channels have yet been established. That is why we focus on the continuous case: the Gaussian channel for its known practical relevance. We deal with secure identification over Gaussian channels. Provable secure communication is of high interest for future communication systems. A key technique for implementing secure communication is the physical layer security based on information-theoretic security. We model this with the wiretap channel. In particular, we provide a suitable coding scheme for the Gaussian wiretap channel (GWC) and determine the corresponding secure identification capacity. We also consider Multiple-Input Multiple-Output (MIMO) Gaussian channels and provide an efficient signal-processing scheme. This scheme allows a separation of signal processing and Gaussian coding as in the classical case.