arXiv Math @arxiv_math@qoto.org

While neural lossy compression techniques have markedly advanced the efficiency of Channel State Information (CSI) compression and reconstruction for feedback in MIMO communications, efficient algorithms for more challenging and practical tasks-such as CSI compression for future channel prediction and reconstruction with relevant side information-remain underexplored, often resulting in suboptimal performance when existing methods are extended to these scenarios. To that end, we propose a novel framework for compression with side information, featuring an encoding process with fixed-rate compression using a trainable codebook for codeword quantization, and a decoding procedure modeled as a backward diffusion process conditioned on both the codeword and the side information. Experimental results show that our method significantly outperforms existing CSI compression algorithms, often yielding over twofold performance improvement by achieving comparable distortion at less than half the data rate of competing methods in certain scenarios. These findings underscore the potential of diffusion-based compression for practical deployment in communication systems.

We say that a measure of dependence between two random variables $X$ and $Y$, denoted as $ρ(X;Y)$, satisfies the data processing property if $ρ(X;Y)\geq ρ(X';Y')$ for every $X'\rightarrow X\rightarrow Y\rightarrow Y'$, and satisfies the tensorization property if $ρ(X_1X_2;Y_1Y_2)=\max\{ρ(X_1;Y_1),ρ(X_2;Y_2)\}$ when $(X_1,Y_1)$ is independent of $(X_2,Y_2)$. It is known that measures of dependence defined based on $Φ$-entropy satisfy these properties. These measures are important because they generalize R{é}nyi's maximal correlation and the hypercontractivity ribbon. The data processing and tensorization properties are special cases of monotonicity under wirings of non-local boxes. We show that ribbons defined using $Φ$-entropic measures of dependence are monotone under wiring of non-local no-signaling boxes, generalizing an earlier result. In addition, we also discuss the evaluation of $Φ$-strong data processing inequality constant for joint distributions obtained from a $Z$-channel.

Probability theory is fundamental for modeling uncertainty, with traditional probabilities being real and non-negative. Complex probability extends this concept by allowing complex-valued probabilities, opening new avenues for analysis in various fields. This paper explores the information-theoretic aspects of complex probability, focusing on its definition, properties, and applications. We extend Shannon entropy to complex probability and examine key properties, including maximum entropy, joint entropy, conditional entropy, equilibration, and cross entropy. These results offer a framework for understanding entropy in complex probability spaces and have potential applications in fields such as statistical mechanics and information theory.

The main objective of this work is to show, through counterexamples, that some of the theorems presented in the papers of Sharma \textit{et al.} (2018) and Chauhan \textit{et al.} ( 2021) are incorrect. Although they used these theorems to establish a sufficient condition for a multi-twisted (MT) code to be linear complementary dual (LCD), we show that this condition itself remains valid. We further improve this condition by removing the restrictions on the shift constants and relaxing the required coprimality condition. We show that compared to the previous condition, the modified condition is able to identify more LCD MT codes. Furthermore, without the need for a normalized set of generators, we develop a formula to determine the dimension of any $ρ$-generator MT code.

Integrated sensing and communication (ISAC) represents a pivotal advancement for future wireless networks. This paper introduces a novel ISAC beamforming method for enhancing sensing performance while preserving communication quality by leveraging the ambiguity function (AF). We formulate an optimization problem to minimize the integrated sidelobe level ratio (ISLR) of the AF subject to the constraints of transmission power, communication signalto-interference-plus-noise ratio, and sensing gain. To address the non-convexity of the optimization problem, semidefinite relaxation is adopted. Numerical results show that our method significantly reduces range sidelobes and achieves a lower ISLR in the rangeangle domain compared to other approaches.

This study addresses the blind deconvolution problem with modulated inputs, focusing on a measurement model where an unknown blurring kernel $\boldsymbol{h}$ is convolved with multiple random modulations $\{\boldsymbol{d}_l\}_{l=1}^{L}$(coded masks) of a signal $\boldsymbol{x}$, subject to $\ell_2$-bounded noise. We introduce a more generalized framework for coded masks, enhancing the versatility of our approach. Our work begins within a constrained least squares framework, where we establish a robust recovery bound for both $\boldsymbol{h}$ and $\boldsymbol{x}$, demonstrating its near-optimality up to a logarithmic factor. Additionally, we present a new recovery scheme that leverages sparsity constraints on $\boldsymbol{x}$. This approach significantly reduces the sampling complexity to the order of $L=O(\log n)$ when the non-zero elements of $\boldsymbol{x}$ are sufficiently separated. Furthermore, we demonstrate that incorporating sparsity constraints yields a refined error bound compared to the traditional constrained least squares model. The proposed method results in more robust and precise signal recovery, as evidenced by both theoretical analysis and numerical simulations. These findings contribute to advancing the field of blind deconvolution and offer potential improvements in various applications requiring signal reconstruction from modulated inputs.

In the past over two decades, very fruitful results have been obtained in information theory in the study of the Shannon entropy. This study has led to the discovery of a new class of constraints on the Shannon entropy called non-Shannon-type inequalities. Intimate connections between the Shannon entropy and different branches of mathematics including group theory, combinatorics, Kolmogorov complexity, probability, matrix theory, etc, have been established. All these discoveries were based on a formality introduced for constraints on the Shannon entropy, which suggested the possible existence of constraints that were not previously known. We assert that the same formality can be applied to inequalities beyond information theory. To illustrate the ideas, we revisit through the lens of this formality three fundamental inequalities in mathematics: the AM-GM inequality in algebra, Markov's inequality in probability theory, and the Cauchy-Scharwz inequality for inner product spaces. Applications of this formality have the potential of leading to the discovery of new inequalities and constraints in different branches of mathematics.

A set with a group action is referred to as a $G$-set, and the set of functions that commute with this action forms a monoid under function composition. This paper examines the case where the $G$-set is finite, which implies that the monoid of $G$-equivariant functions is also finite. The document provides formulas for calculating the cardinality of this monoid, its group of units, and explores special cases of $G$-equivariant functions, known as fixing elementary collapsings. All of these results are expressed in terms of specific properties of the $G$-set, including the number of orbits and certain indices of the subgroups acting as stabilizers.

We introduce the notion of a gauge and of a tagged partition (subordinate to a given gauge) by intersections of open and closed sets of a compact metric space extending the corresponding notions in Henstock-Kurzweil integration of real-valued functions with respect to the Lebesgue measure on the unit interval. We show that, for the integration of bounded functions with respect to a normalised Borel measure $μ$ on a compact metric space, the notion of a gauge and an associated tagged partition, arise naturally from a normalised simple valuation way-below the Borel measure. Then we consider the integration of unbounded functions with respect to a normalised Borel measure on a compact metric space, for which the Lebesgue integral may fail to exist. A pair of a tagged partition and a gauge defines a simple valuation and we introduce a partial order on these pairs, emulating the partial order of simple valuations in the probabilistic power domain. We define the $D_μ$-integral of a real-valued function with respect to a Borel measure using the limit of the net of the integrals of the simple valuations induced by pairs of tagged partitions and gauges for the function. The $D_μ$-integral of functions on a compact metric space with respect to a normalised Borel measure satisfies the basic properties of an integral and generalises the Henstock-Kurzweil integral. We show that when the Lebesgue integral of the function exists then the $D_μ$-integral also exists and they have the same value. We provide a family of real-valued functions on the Cantor space that are $D_μ$-integrable but not Lebesgue integrable.

In our effort to find an arithmetically pure proof of the Bertrand postulate, we investigate and solve (using only elementary arithmetical methods) another less usual inequality in positive integers inspired by the classical proof of the postulate given by P. Erdős.

We study the structures of ordinary simple Hurwitz numbers and monotone Hurwitz numbers with varying genus. More precisely, we prove that when the ramification type is fixed and the genus is treated as a variable, the connected monotone Hurwitz number is a linear combination of products of exponentials and polynomials, and the ordinary simple Hurwitz number is a linear combination of exponentials. Using these structural properties, we also derive the large genus asymptotics of these two kinds of Hurwitz numbers. As a result, we prove one conjecture, and disprove another, both proposed by Do, He and Robertson.

In this paper we study the complexity of solving orientable quadratic equations in wreath products $A\wr B$ of finitely generated abelian groups. We give a classification of cases (depending on genus and other characteristics of a given equation) when the problem is computationally hard or feasible.

We construct finitely generated simple torsion-free groups with strong homological control. Our main result is that every subset of $\mathbb{N} \cup \{\infty\}$, with some obvious exceptions, can be realized as the set of dimensions of subgroups of a finitely generated simple torsion-free group. This is new even for basic cases such as $\{ 0, 1, 3 \}$ and $\{ 0, 1, \infty \}$, even without simplicity or finite generation, and answers a question of Talelli and disproves a conjecture of Petrosyan. Moreover, we prove that every countable group of dimension at least $2$ embeds into a finitely generated simple group of the same dimension. These are the first examples of finitely generated simple groups with dimension other than $2$ or $\infty$. Our method combines small cancellation theory with group theoretic Dehn filling, and allows to do several other exotic constructions with control on the dimension. Along the way we construct relative versions of torsion-free Tarski monsters, and construct the first uncountable family of pairwise non-measure equivalent finitely generated torsion-free groups.

This paper develops a new family of locally recoverable codes for distributed storage systems, Sequential Locally Recoverable Codes (SLRCs) constructed to handle multiple erasures in a sequential recovery approach. We propose a new connection between parallel and sequential recovery, which leads to a general construction of q-ary linear codes with information $(r, t_i, δ)$-sequential-locality where each of the $i$-th information symbols is contained in $t_i$ punctured subcodes with length $(r+δ-1)$ and minimum distance $δ$. We prove that such codes are $(r, t)_q$-SLRC ($t \geq δt_i+1$), which implies that they permit sequential recovery for up to $t$ erasures each one by $r$ other code symbols.