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arXiv2026

Trajectory-Wise Certification for Vector-Field Mirror Descent with Deterministic Finite Differences

We study mirror descent in which the objective gradient is replaced by a general vector field. Since such a field does not automatically relate the mirror update to the objective gap, we introduce a trajectory-wise generalized relative-smoothness condition and a generalized star-convexity interface. Together they yield a finite-horizon, a posteriori last-iterate certificate with an accumulated-stepsize term and an exceptional-region-dependent error term. We also give a pointwise sufficient condition for positive admissible stepsizes, displaying the effects of objective smoothness, mirror-map conditioning, and vector-field mismatch. We then construct a deterministic zeroth-order instance from coordinate central differences. The function values place the unknown gradient in an explicit uncertainty box, and verification of the interface reduces to robust conic dominance. We derive an explicit scaling of the central-difference vector that guarantees the required dominance over the gradient uncertainty set. Under uniform Hessian bounds, the resulting vector field satisfies the interface outside an explicit resolution-dependent neighborhood. The final guarantee combines a rate term governed by accumulated accepted stepsizes with a finite-resolution error floor.

math.OC↗

arXiv2026

Quantum Maximum Likelihood Prediction via Hilbert Space Embeddings

Maximum likelihood prediction (MLP) is a core task at the heart of modern large language models. Here, we study a quantum version of this task for a simplified data model consisting of independent and identically distributed samples, as a first step. The quantum maximum likelihood predictor (QMLP) is obtained by embedding of empirical probability distributions into quantum states and performing a minimization of quantum relative entropy over a given class of states. We derive non-asymptotic performance guarantees for QMLP in terms of convergence rates and concentration inequalities, both in trace norm and quantum relative entropy. Our approach provides a unified framework to handle MLP within both classical and quantum LLMs. We also consider the related problem of quantum information projection and generalize the quantum Pythagorean theorem to mixture families specified by possibly non-self-adjoint linear constraints. We further show that the Pythagorean inequality continues to hold in the infinite-dimensional setting whenever the convex information-projection problem attains a finite minimum.

cs.IT↗

arXiv2026

Optimal Transmitter Placement in Realistic Urban Environments

In a wireless network, transmitter locations strongly impact achievable rates. Cellular deployment is a difficult non-convex problem, typically addressed using simplified models and heuristics. We propose a mathematically rigorous framework incorporating detailed site-specific maps, material properties, and realistic attenuation. We introduce an aggregated network-quality functional scoring receiver-weighted signal quality, impose deployment costs via cardinality and budget constraints, and establish submodularity under practical conditions. To solve the optimization problem, we propose the Interference-Aware Submodular Placement Algorithm (IA-SPA) with a theoretical approximation guarantee relative to the optimum. IA-SPA incorporates existing base stations and prohibited areas, making it applicable to clean-slate and incremental deployments. We evaluate our approach using ray-tracing simulations on 3D maps of San Francisco and Florence, comparing against known deployments by AT and T, T-Mobile, and Iliad. Our strategy achieves significant increases in mean data rate (about 2x) and edge rate (2-8x) using the same number of transmitters. These gains persist under simultaneous exclusionary zones, small-scale fading, material/geometric perturbations, and incremental densification of existing networks. The pipeline has linear computational dependence on candidate sites, and wall-clock measurements demonstrate practical runtime at city scale.

cs.IT↗

arXiv2026

Joint Accuracy and Confidentiality in Semantic-Aware Secure Remote Reconstruction

In this paper, we consider remote reconstruction over wireless networks when simultaneous accuracy at the legitimate receiver and confidentiality against eavesdropping are required. These two objectives are often treated separately, even though they arise from the same update process and are marginals of a joint reconstruction event. This paper introduces confidential reconstruction accuracy (CRA), a metric to capture the joint event in which the legitimate receiver reconstructs correctly while the eavesdropper fails. Under randomized stationary policies, we develop a three-dimensional stationary analysis and derive closed-form expressions for the long-term average CRA and the optimal transmission probability. The results show that conventional marginal analysis can misidentify the optimal policy and misestimate the achievable simultaneous accuracy-confidentiality performance. They also reveal nontrivial behaviors: more frequent transmissions or better legitimate channels do not necessarily improve joint accurate and confidential reconstruction, and when the eavesdropping channel is strong, improving the legitimate channel alone may be insufficient. Finally, the framework induces the spatial safety boundary in a geofencing setting for secure remote reconstruction.

cs.IT↗

arXiv2026

Stabilizing Private LASSO under Heterogeneous Covariates via Anisotropic Objective Perturbation

We study high-dimensional LASSO under differential privacy via objective perturbation with heterogeneous covariate scales. In practical scenarios, covariates often exhibit diverse scales; however, standard preprocessing is problematic under privacy constraints, as it consumes additional privacy budget. This heterogeneity induces effective anisotropy in the objective perturbation via the inverse Gram matrix of covariates, which can degrade the stability and accuracy of algorithms. To address this, we propose a Gram-based anisotropic objective perturbation, a ``pre-distortion" strategy that counteracts the distortion from the covariate structure to restore isotropy in the estimation process. Using an Approximate Message Passing (AMP) framework and state evolution analysis, we demonstrate that our proposed perturbation significantly stabilizes convergence and improves both statistical efficiency and privacy performance compared to standard uniform noise injection. Our results provide theoretical insights into designing stable and efficient private estimators without relying on data-dependent preprocessing.

stat.ML↗

arXiv2026

Chase-like Decoding: Test-Pattern Design and Performance Analysis

Chase-like decoding algorithms are a popular choice for soft-input decoding of algebraic codes. We evaluate different test-pattern sets for Chase-like decoding. Structured sets, such as Chase-II patterns or patterns chosen by logistic weight, are analyzed using order statistics, while arbitrary sets are evaluated by calculating covered-space probabilities and by performing Monte Carlo simulation. We further propose an algorithm that designs test-pattern sets to cover likely error patterns, achieving comparable performance with half the number of test patterns compared with conventional sets for high-rate BCH codes.

cs.IT↗

arXiv2026

The Entropy of Floating-Point Numbers

Here we present an analytic approximation for the entropy of floating-point numbers, along with bounds on the error of this approximation. It is well-known that the differential entropy is tightly linked to the discrete entropy of a uniformly quantized random variable. Our approximation uncovers a different quantity that provides this link for floating-point quantization. Additionally, we prove that the entropy of a floating-point quantized random variable is approximately unchanged under scaling. Closed-form expressions for the floating-point entropy of common distributions are provided and compared to exact results.

cs.IT↗

arXiv2026

Covert Multi-bit LLM Watermarking: An Information Theory and Coding Approach

We study the problem of multi-bit watermarking for non-autoregressive large language models (LLMs). We introduce an information-theoretic model inspired by diffusion language models (DLM), in which the encoder has limited non-causal access to token distributions within each token block. This formulation enables an information-theoretic characterization of the non-causal watermarking capacity, in which knowledge of LLM cover statistics is leveraged to enable a multi-bit covert embedding. We study the information-theoretic limits of the model by combining Gelfand--Pinsker and channel synthesis coding techniques and obtain an exact characterization of the capacity. The embedding strategy is further optimized across blocks using a constrained Markov decision process (CMDP) and we develop an explicit algorithm based on polar codes following the information-theoretic principles. We simulate the error performance on LLaDA, and provide empirical total variation (TV) analysis as a function of key randomness.

cs.IT↗

arXiv2026

Divisibility of Trace Codes

A linear code is said to be $Δ$-divisible if the Hamming weights of all its codewords are divisible by $Δ$. The $p$-adic valuation of a code is defined as the greatest integer $t$ such that the code is $p^t$-divisible. In this paper, we establish a divisibility criterion for trace codes. Specifically, this criterion provides a systematic method to determine the $p$-adic valuation of the associated trace code, thereby extending Ward's classical divisibility criterion from standard generating sets (or matrices) to generalized generator matrices over an extension field. Furthermore, we present two applications of our framework. The first application provides a concise proof of the celebrated divisibility results on semisimple Abelian codes established by Delsarte and McEliece. The second application establishes several explicit lower bounds on the $p$-adic valuation of the number of solutions over $\mathbb{F}_{q^m}$ (where $q = p^e$) to the Artin--Schreier type equation $ f(x_1,\ldots,x_k)=y^q-y $. In particular, under the coprime condition $\left(d,\frac{q^m-1}{q-1}\right)=1$, we determine the exact minimum $p$-adic valuation of the number of solutions when $f$ is restricted to homogeneous polynomials of degree $d$.

math.CO↗

arXiv2026

Lossy Joint Source-Channel Coding over Unknown Channels

We analyze the performance of joint source-channel codes in an unknown-channel framework, where the true channel is unknown but the source distribution is known. We derive achievability bounds for a family of mismatched-design joint source-channel codes constructed for a design channel $Q_{Y|X}$ and operated over a possibly different true channel $P_{Y|X}$. Our one-shot achievability bound allows for standard Borel alphabets for the source, reproduction, channel input and channel output. The subsequent block coding result based on the normal approximation applies to stationary memoryless sources and memoryless, possibly nonstationary channels under regularity and moment conditions. The achievability bound is given in terms of the rate-distortion and rate-dispersion functions, as well as two channel-dependent quantities that we call the mismatched-design rate and mismatched-design rate-dispersion. We use a family of Gibbs posteriors parameterized by a single scalar as decoder-side kernels, and the envelope of the corresponding achievable rates recovers the generalized mutual information. In the stationary matched setting covered by our assumptions, our result recovers the achievability part of Kostina and Verdú's 2013 Gaussian approximation result and improves its third-order term. We also formalize a notion of a second-order universal family of source-channel codes under which there is no first- or second-order asymptotic penalty. We then construct two channel-blind families of source-channel codes: one that is second-order universal over a regular class of nonstationary block erasure channels and another that is second-order universal over stationary Gaussian channels. Our code construction uses Poisson functional representations of suitable conditional probability measures to produce the encoder and decoder outputs.

cs.IT↗

arXiv2026

Reinforcement Learning for Heterogeneous Sensor Selection in Maritime Surveillance

This paper presents an information-gain-guided reinforcement-learning sensor-selection framework for single-vessel tracking in heterogeneous maritime sensor networks. The proposed approach is motivated by information-theoretic sensor management: instead of activating all sensors or repeatedly performing computationally expensive online expected-information-gain evaluation, a learned policy selects one tracking-relevant sensor at each decision epoch. A Bayesian sequential Monte Carlo tracker estimates the vessel state from noisy measurements and provides a belief representation for scheduling under nonlinear and non-Gaussian conditions. A Proximal Policy Optimization agent selects one of five sensors in a georeferenced simulation of the CMMI Smart Marina testbed at Ayia Napa Marina, Cyprus. The policy is trained on the testbed's actual five-sensor configuration. The agent observes belief-state, detection-history, coverage, sensor-geometry, and realized-information-gain features. The reward is defined as a realized-information-gain term gated by an observability mask. Final-test simulations compare the proposed framework with random single-sensor selection, always-on sensing using all sensors simultaneously, and the expected-information-gain sensor-selection baseline proposed in our previous work. Results show that the learned policy achieves tracking performance close to always-on sensing while activating only one sensor per decision time step and avoiding the computationally expensive online entropy search required by expected-information-gain selection. Additional zero-shot evaluation without retraining on ten moderately perturbed versions of actual layout configuration showed broadly stable tracking, with any increase in positional tracking error remaining below 1 meter across all perturbations.

cs.AI↗

arXiv2026

The Rate-Distortion-Deception Tradeoff

The problem of finding the optimal compression rate for a given random variable has been traditionally studied under two main constraints: distortion and perception. The distortion constraint enforces the fidelity of our reconstruction with respect to the observed realization of the random variable, while the perception constraint ensures that the reconstruction is close to a sample from the distribution of the random variable of interest. In this work, we explore the possibility of reconstruction, such that the reconstructed sample is still within a desired fidelity level with our original realization of the random variable, but at the same time, it resembles a sample from a different target distribution. We term this criterion as the deception constraint and find the fundamental tradeoffs of rate-distortion and deception.

cs.IT↗

arXiv2026

New upper and lower bounds on covering codes K_q(n,R) for alphabets of size 5 <= q <= 21

Let K_q(n,R) denote the minimum cardinality of a q-ary code of length n with covering radius R. We improve the known bounds on K_q(n,R) in 83 cases (82 distinct cells). On the upper-bound side we give 25 improved bounds for 5<=q<=15 -- twenty-four found by search and one propagated by monotonicity -- using two complementary methods: an engineered focused local search seeded with structural constructions, and a large-neighbourhood search driven by exact full-space coverage transforms that evaluates every candidate codeword position simultaneously. These are, to our knowledge, the first improvements to any upper bound on K_q(n,R) with q >= 5 since the 2011 revision of Keri's tables; several bounds decrease by more than 20%, e.g. K_6(8,4)<=166 (previously 216) and K_8(10,5)<=1883 (previously 2461). On the lower-bound side we give 58 improved bounds for 6<=q<=21, obtained from the semidefinite programming hierarchy of Gijswijt and Polak, whose published results cover q<=5, by combining an exact-arithmetic reimplementation of the reduced program with a multiprecision solution pipeline. Every new lower bound is certified by a rational dual solution validated by a standalone exact-arithmetic checker; no floating-point computation is part of the trusted base. The same pipeline also gives strong numerical evidence of limits: on a dozen further cells the certified value of the relaxation, which the solver reports as optimal to within its working precision, lies below the best known bound, indicating that no improvement is available there at this level of the hierarchy. One cell is improved from both sides: 441<=K_6(10,4)<=2751, previously 417--2952. All codes and certificates are provided in machine-readable form together with standalone verifiers.

math.CO↗

arXiv2026

On cancellative pairs of families of subsets

A pair $(\mathcal{A}, \mathcal{B})$ of families of subsets of $[n]$ is cancellative if whenever $A, A' \in \mathcal{A}, B \in \mathcal{B}$ satisfy $A \cup B=A' \cup B$, then $A=A'$, and whenever $A \in \mathcal{A}, B, B' \in \mathcal{B}$ satisfy $A \cup B=A \cup B'$, then $B=B'$. We show that for every cancellative pair $(\mathcal{A}, \mathcal{B})$, the inequality $|\mathcal{A}||\mathcal{B}| \le 2.25^n$ holds, matching Tolhuizen's $(2.25-o(1))^n$ lower bound construction.

math.CO↗

arXiv2026

ISAC with Co-Prime Arrays: Virtual-Aperture Sensing and uplink downlink communications

Integrated sensing and communication (ISAC) enables simultaneous communication and environmental sensing in unmanned aerial vehicle (UAV) networks, but its performance is constrained by the physical antenna aperture and residual self-interference (SI) in full-duplex (FD) sensing. To address these issues, we propose a shared-aperture ISAC architecture in which a sparse co-prime array (CPA) is embedded in a uniform linear array (ULA) grid for FD sensing, while the remaining antenna positions support time-division duplexing (TDD) communication. We characterize the sensing performance through an order-wise Cramer-Rao bound (CRB) analysis, showing that the CPA achieves a stronger asymptotic sensing gain than the partitioned ULA benchmark in both single-target and nondegenerate multi-target scenarios. We further reveal a space-time sampling tradeoff under the same physical aperture. Based on the proposed architecture, we formulate a non-convex joint resource allocation problem that maximizes the weighted downlink-uplink sum rate by jointly designing the sensing transmit covariance, downlink precoder, and uplink receive beamformers under sensing accuracy, BS transmit-power, communication QoS, and residual SI constraints. An alternating-optimization-based algorithm is developed. Simulations demonstrate consistent performance gains over the considered baselines and confirm the complementary benefits of the CPA virtual aperture and sensing covariance optimization.

cs.IT↗

arXiv2026

Posterior Tempering Explains Variance Inflation in Linear and Generalized Linear Thompson Sampling

We study a variant of the Thompson Sampling (TS) algorithm, called $α$-TS, for solving stochastic generalized linear bandit problems. Existing analyses of TS require inflating the posterior variance to derive near-optimal regret guarantees. We formalize the idea of variance inflation by introducing $α$-TS that uses a fractional or $α$-posterior instead of the standard posterior. Our main contribution is to identify general regularity conditions on the prior and reward distributions that enable a regret analysis of $α$-TS without assuming any tractable approximation of the posterior distribution, unlike previous works. For a specific choice of $α\propto d^{-1}$, our general regret bound yields the best known regret bound of $O(d^{3/2}\sqrt{T}\log T)$ for both the exponential and sub-Gaussian families of reward distributions. We further provide an $α$-dependent lower bound showing that the regret constant depends on the product $αd$, and that when $α\propto d^{-1}$ the regret scales as $Ω(d^{3/2}\sqrt{T})$, explaining the origin of the $d^{3/2}$ factor in the upper bound. Our proof technique adapts and combines recent advancements in the analysis of linear bandit problems with first- and second-order posterior concentration theory from the Bayesian statistics literature.

stat.ML↗

arXiv2026

New binary optimal LCD codes using heuristic embedding

In this paper, we investigate the construction of binary optimal LCD codes through short LCD embeddings. For this purpose, we design heuristic frameworks based on a greedy algorithm. We explore the search spaces of LCD embeddings using the fact that an invertible matrix together with an arbitrary matrix yields an LCD embedding. We therefore use elementary row operations on the invertible block and single entry-flips on the arbitrary block as local moves in a greedy algorithm. Using this method, we have found $14$ optimal new LCD codes with dimensions 7 and 8 for lengths from 55 to 201.

cs.IT↗

arXiv2026

Instance Optimal Sparse Recovery from Nonlinear Observations: A Unified Framework

This paper develops a unified framework for instance optimal sparse recovery from nonlinear observations. The main ingredient is a signal-dependent restricted approximate invertibility condition (RAIC) of some gradient, which leads to the instance optimality of iterative hard thresholding. Under Gaussian designs, we apply the proposed framework to phaseless, one-bit, and ReLU measurements, which correspond to the problems of sparse phase retrieval, one-bit compressed sensing, and sparse ReLU regression, respectively. For sparse phase retrieval, we propose a variant of thresholded amplitude flow and show its instance optimality under $O(s^3)$ measurements (up to logarithmic factors), where $s$ is the sparsity level. To our best knowledge, this is the first instance optimal efficient algorithm for sparse phase retrieval and complements Gao, Wang and Xu (2016) that achieved this via a computationally intractable program. In one-bit compressed sensing, we establish the instance optimality of normalized binary iterative hard thresholding and strengthen the recent result of Matsumoto and Mazumdar (2024). In sparse ReLU regression, it is shown that a slight variant of the algorithm in Soltanolkotabi (2017) is instance optimal. Moreover, $(\ell_2,\ell_2)$ non-uniform instance optimal guarantees are obtained for these problems. The analysis is built upon a number of high-dimensional concentration bounds, including bounds on restricted eigenvalues and a novel instance-dependent hyperplane tessellation result.

cs.IT↗