Papers

Collecting multidimensional user data is essential for extracting rich insights across various applications. Local Differential Privacy (LDP) has emerged as a de facto standard for mitigating privacy risks in such scenarios. A key challenge in privacy-preserving multidimensional data collection lies in inter-attribute dependencies, as they can inadvertently reveal correlated information and increase privacy vulnerabilities. Therefore, accurately measuring correlation-induced privacy leakage (CPL) is essential for privacy analysis and privacy-utility trade-off. However, existing CPL analysis solutions either require accurate prior knowledge or face scalability challenges for large numbers of attributes and high-cardinality attributes. These limit their practical applicability in real data. To address this research gap, we propose a novel metric, “Dependency Triad” (DT), which summarizes the pairwise dependency information relevant to CPL using three parameters and yields a constant-time conservative estimator of pairwise CPL. DT explicitly models uncertainty in prior distributional knowledge through its parameters, delivering robust leakage estimates. Moreover, its robustness to sparse distributions makes it particularly suitable for high-cardinality attributes, while the pairwise formulation serves as a tractable building block for assessing total leakage in multidimensional settings. Extensive experiments on both synthetic and real datasets demonstrate that DT consistently estimates CPL across diverse dependency regimes and prior uncertainties.

In this work, we investigate permutation decoding for algebraic geometry (AG) codes arising from algebraic curves defined by separated polynomials. Using automorphisms of the underlying curves, we construct permutation automorphisms of the associated algebraic geometry codes and exploit the resulting orbit structure to determine information and check positions. We introduce a class of curves, called SAP curves (Separated Additive Polynomial curves), and investigate one-point AG codes defined on them. For these codes, we obtain permutation decoding sets that correct burst errors supported on coordinates associated with rational points sharing a common coordinate. We further identify a subclass of special SAP curves, including Hermitian curves, generalized Hermitian curves, and certain maximal curves, for which additional automorphisms yield more powerful decoding sets.

This paper investigates the performance of faster-than-Nyquist (FTN) signaling within the context of hyper-reliable low-latency communications (HRLLC), specifically focusing on the challenges imposed by the short-packet regime. While traditional Nyquist-based systems maintain symbol orthogonality to prevent inter-symbol interference (ISI), FTN intentionally introduces ISI to achieve higher transmission rates. While many existing FTN studies assume perfect channel state information, this assumption is often impractical for mission-critical HRLLC. In such scenarios, a portion of the limited packet length must be reserved for pilot symbols to ensure reliable estimation. To characterize the achievable error probability while accounting for imperfect channel estimation in the short-blocklength regime, we derive the random coding union bound with parameter $s$ (RCUs) under mismatched decoding for FTN systems. The numerical results demonstrate that FTN provides up to a 2 \dB SNR gain over Nyquist signaling, provided that power allocation and pilot overhead are optimized. These findings highlight the necessity of non-asymptotic analysis for designing efficient, next-generation HRLLC-FTN systems.

Enhanced Gabidulin Matrix Codes (EGMC), introduced by Aragon, Couvreur, Dyseryn, Gaborit, and Vincotte at Asiacrypt 2024, were designed to hide the algebraic structure of Gabidulin matrix codes while enabling very compact McEliece- and Niederreiter-type encryption schemes, with ciphertexts as small as 65 bytes at the claimed 128-bit security level. Their security relies on the assumption that a masked EGMC code is hard to distinguish from a random matrix code. We show that this enhanced construction leaves enough structure for an equivalent code of the secret key to be recovered. Unlike previous cryptanalysis, our attack combines combinatorial and algebraic techniques to recover a Gabidulin-equivalent compressed code. This code can then be extended to a full-length equivalent secret key in polynomial time. As a result, the attack provides both a distinguisher and a key-recovery attack against the EGMC encryption schemes. The attack breaks all 16 proposed EGMC parameter sets by large margins. For example, for the claimed 128-bit parameter set (2,17,37,4,0), it reduces the security level from 186 bits to 35 bits and in our implementation, the equivalent secret key of this parameter set is recovered in less than 10 minutes.

To support ultra-dense connectivity in terahertz (THz) communications, this paper proposes a dynamic frequency-position-fluid antenna (D-FPFA) architecture. Frequency-tunable local oscillators (LOs) are integrated into the RF chains to access different sub-bands, thereby expanding the total bandwidth of the system and providing frequency-domain diversity. To exploit spatial diversity, the base station employs movable subarrays, and each user is equipped with a movable antenna. We first develop a two-phase beam-split-aware frequency allocation strategy. In the first phase, we divide users into disjoint sub-bands according to their channel correlation coefficients to mitigate the interference. In the second phase, we investigate the wideband near-field beam-split effect for planar arrays and reveal an astigmatism phenomenon, in which the beam at a non-central subcarrier cannot be perfectly refocused at a single spatial point. Then, we establish a beam split multiplexing strategy, where we formulate the user grouping task as a minimum dominating set problem. To maximize the sum rate, we introduce a switch network along with a distance-based antenna selection strategy to account for the near-field channel gain variations, followed by a particle swarm optimization-based algorithm that jointly optimizes the antenna positions and precoders. Numerical results show that, the proposed D-FPFA achieves approximately 2.3 times the sum rate of a conventional phase-shifter (PS)-based array-of-subarrays (AoSA) architecture. It also attains 95% of the sum rate of its TTD counterpart while providing approximately 2.8 times its energy efficiency (EE). Moreover, the fully connected variant of D-FPFA, i.e., FPFA, achieves the highest EE among all considered architectures.

In this paper, we study the asymptotic behavior of deterministic identification (DID) over binary symmetric channels (BSCs) under vanishing error constraints. By introducing a minimum error parameter, we characterize how different error-decay regimes affect the achievable DID rate. General achievability and converse bounds are derived, with explicit asymptotic characterizations in the large-deviation, moderate-deviation, and central-limit regimes. The achievability analysis combines coding-theoretic constructions with probabilistic concentration techniques, while the converse links statistical distinguishability to the minimum-distance structure of DID codes via total variation and Hamming-type bounds. Our results show that the asymptotic behavior of DID over BSCs is governed by a Hamming-shell concentration geometry of channel outputs, offering insights into the finite-blocklength behavior of deterministic identification over discrete-output channels.

This paper investigates the minimum field size required for network maximum distance separable (MDS) codes, a critical parameter affecting computational complexity at network nodes. Focusing on generalized combination networks and Zosin Khuller networks, we develop a systematic framework for both scalar and vector network MDS codes. For scalar codes on generalized combination networks, we establish an equivalence between the minimum distance of network codes and the minimum Hamming distance of classical linear codes, converting network-level MDS constraints into coding theory conditions. This yields necessary and sufficient existence conditions linked to classical MDS codes and covering Grassmannian codes. Using refined greedy constructions and MRD code designs, we obtain improved bounds on the minimum field size, outperforming the prior universal bound. For vector network codes, we develop an analogous distance equivalence and characterize existence via covering Grassmannian codes, yielding bounds on the minimum effective field size. Notably, the gap between optimal scalar and vector MDS codes vanishes for several parameter regimes; we explicitly identify a family of such networks where vector coding offers no field size advantage over scalar coding. For Zosin Khuller networks, we derive lower bounds on the minimal effective field size of vector MDS codes using hypergraph homomorphisms and subset intersection arguments, strictly improving prior scalar bounds. We further provide a necessary and sufficient condition built upon hypergraph homomorphisms for vector MDS construction, yielding an upper bound on the minimum field size. Finally, we bound the MDS gap between optimal scalar and vector solutions.

We give an alternative proof of Yu's Thinning Lemma and of the Law of Thin Numbers using a stochastic variational formula for the relative entropy. We show that this approach yields new convergence rates for the Law of Thin Numbers, complementing and extending previously known results.

The containment of the code of the meet $G\wedge A$ in the hull and the identity $G\vee A-D=K-G\wedge A$ exchanging meet and join are known; imposing that $G\wedge A$ be principal constructs algebraic geometry codes with one-dimensional hull. We turn that construction into a measurement. For arbitrary divisors $G$ and $A$ we compute $C_L(D,G)\cap C_L(D,A)$ exactly: it is the code of the meet together with an excess $\varepsilon(G,A)$, canonically their quotient and a subquotient of $H^1(\mathcal O(G\wedge A))$. So $\varepsilon$ vanishes exactly when the meet is non-special; otherwise it certifies that $K-G\wedge A$ is linearly equivalent to an effective divisor, at degree zero the vanishing of a single class in the Picard group: the hull detects a linear equivalence rather than being built from one. For superelliptic curves $y^n=f(x)$ both sides can be computed: their weighted plane models in $\mathbb P^2_{(1,n/c,d/c)}$, $c=\gcd(n,d)$, identify codes $C_s$ of weighted forms of degree $s$ with those of $sD_\infty$ and turn hulls into lattice counts. The range on which $\varepsilon$ is blind is an explicit interval of degrees, where $\dim\operatorname{Hull}(C_s)=c\mu(s)-n\delta+1-g_X$, $\mu(s)=\min\{s,M-s\}$, depends only on its affine-point count. Outside it the meet and join are invariant under $s\mapsto M-s$ while $\varepsilon$ is not, so every asymmetry of the hull profile is excess and the threshold in $s$ refines the divisor class: two totally split curves of genus two, over $\mathbb F_7$ and over $\mathbb F_{11}$, present the same class at the same pair of degrees and are separated by the profile alone. If $0\leq\deg(G\wedge A)\leq2g_X-2$ the hull is at most $g_X+1$, so it is large only where it is blind, and over a prime field, under an explicit inequality on $(n,d,q)$, its maximum over the family is $\ell(\lfloor M/2\rfloor D_\infty)$, attained exactly on the totally split locus.

We study distributed testing of $\mathrm{Ber}(\alpha)$ versus $\mathrm{Ber}(\beta)$ in the broadcast, or shared-blackboard, model. For protocols with constant advantage, we characterise up to universal constant factors the information complexity under either hypothesis for every pair $\beta<\alpha$. The characterisation shows that the two information costs can be quite different and identifies three parameter regimes, with optimal protocols based respectively on clean samples, a noisy binary symmetric channel, and an asymmetric $Z$-channel. The lower bounds rely on a novel mixed Hellinger–Jensen–Shannon inequality that may be of independent interest. We also characterise the constant-advantage information complexity of testing arbitrary discrete distributions via an optimisation problem over channels, and show that binary-output channels suffice. We obtain bounds for bounded likelihood-ratio distributions, and give general upper bounds in terms of $\chi^2$ divergence. As applications, we recover the broadcast-model set-disjointness lower bound, and derive stronger lower bounds in the multi-pass streaming setting for some problems considered in prior work.

Polar codes achieve channel capacity as block length increases, but this asymptotic advantage comes at the cost of decoding speed: the conventional successive cancellation (SC) algorithm is inherently sequential, which limits its practicality for high-throughput applications. This work addresses that limitation by introducing the Speculative Successive-Cancellation (Spec-SC) framework, which incorporates speculative execution into the SC decoding process. While node-based decoders with special nodes enable partial parallelism for specific, predefined structures, Spec-SC extends this capability by permitting parallel execution in general nodes that do not match any special-node pattern. A built-in verification mechanism ensures that this speculation does not compromise decoding correctness. Simulation results show that Spec-SC achieves identical BER and FER performance to standard SC decoding, while reducing the average number of sequentially traversed nodes by up to $69\%$ for a polar code of length $N=4096$. These results demonstrate that Spec-SC enables additional algorithm-level parallelism beyond what special-node approaches alone can achieve, paving the way for speculative hardware decoder implementations.

We develop a unified framework for analyzing the complexity of quantum property testing through functionals of the form $\mathcal{L}_{\phi}(\rho) = \operatorname{tr}(\phi(d\rho))/d$, where $\rho$ is an unknown $d$-dimensional quantum state and $\phi$ is a given function. A master theorem is established that derives sample complexity lower bounds for estimating $\mathcal{L}_\phi(\rho)$ from properties of $\phi$, combining Haar-random moment encoding with moment matching and best polynomial approximation. Corresponding query complexity lower bounds follow from quantum sample-to-query lifting. The framework yields nearly tight bounds for a broad class of problems, including entropy estimation (von Neumann, Rényi, and Tsallis), closeness estimation (trace distance and Uhlmann fidelity), spectrum estimation, rank testing (operator rank, Schmidt rank, and matrix product states). Combined with known upper bounds, these results resolve several open problems and establish the optimality of 31 quantum algorithms since 2015, up to polylogarithmic factors.

Multi-level apportionment allocates integer seats through a hierarchy of groups. Schmidt-Kraepelin, Suksompong, and Wijaya proved that, at every fixed house size, lower and upper quota can be satisfied simultaneously; they also constructed house-monotone rules satisfying either quota separately. They left open whether one rule can satisfy lower quota, upper quota, and house monotonicity together, even when quota is required only relative to the root. We give a negative answer. For a full binary comb with $D$ equally entitled leaves, every house-monotone allocation sequence induces a sequence of seat recipients. Quota for the nested comb groups would force every grid-aligned prefix discrepancy to be below one. A midpoint embedding then bounds the full interval discrepancy by this quantity plus $1/2$, contradicting Schmidt's logarithmic lower bound. Conversely, a binary van der Corput seat schedule has comb-prefix error at most $\log D/(3\log 2)+1$. Thus the optimal worst-case error on the comb is $\Theta(\log D)$, and for sufficiently large finite $D$ no house-monotone quota rule exists. The proof isolates a static–dynamic gap: each house size admits a quota-feasible allocation, but the feasible allocations cannot be embedded into one monotone path. In quantization language, the result characterizes the order of the embedded-quantization penalty for progressive one-hot rounding on the comb.

This paper is concerned with one-bit mean estimation, where each independent sample is represented by a single binary message. We consider distributions on $\mathbb{R}$ with mean in $[-\lambda,\lambda]$ and absolute $k$-th central moment at most $\sigma^k$, where $k>1$ is fixed. For this class, previous work attained the optimal sample complexity for general queries using a two-stage protocol. The first stage localizes the mean. The second-stage queries are chosen after localization and refine the estimate around the decoded center. We show that this interaction can be avoided by constructing a randomized fully non-adaptive protocol that fixes all queries before observing the data and matches the optimal adaptive sample complexity. For target accuracy $\epsilon$ and confidence $1-\delta$, its sample complexity scales as \[ \log\frac{\lambda}{\sigma} + \begin{cases} (\sigma/\epsilon)^2\log(1/\delta), & k>2,\\ (\sigma/\epsilon)^2\log(\sigma/\epsilon)\log(1/\delta), & k=2,\\ (\sigma/\epsilon)^{k/(k-1)}\log(1/\delta), & 1<k<2, \end{cases} \] up to constants depending only on $k$. In the range covered by the known lower bound, this rate is minimax optimal even among fully adaptive protocols. This gives a negative answer to the COLT 2026 open problem asking whether interaction is necessary for order-optimal one-bit mean estimation with general queries \citep[Open Problem~1]{lau2026open}.

We characterize the finite sample behavior of the log-likelihood ratio statistic in binary logistic regression, uniformly over both the design and the target parameter. For $n\geq d\geq 3$, we determine, up to universal constants, its worst case $(1-\delta)$ quantile over all fixed collections of design vectors and all target parameters: \[ d\log\left(\frac{e n}{d}\right)+\log\left(\frac{1}{\delta}\right). \] This is a nonasymptotic analogue of the Wilks $\chi^2_d$ phenomenon and requires no regularity assumptions on the design. The low dimensional cases exhibit unusual behavior. The worst case quantile in dimension $d=2$ is sharply of order \[ \log\log\log n+\log\left(\frac{1}{\delta}\right). \] The worst case quantile in dimension $d=1$ is of order $\log(1/\delta)$, with no dependence on $n$. Finally, i.i.d. Gaussian design vectors recover the classical Wilks scale. In the regime $n\gtrsim d+\log(1/\delta)$, we prove the sharp bound \[ d+\log\left(\frac{1}{\delta}\right). \] Unlike existing asymptotic results, our bounds are uniform over the target parameter, which may depend on $n$, $d$, and $\delta$.

Recovering a complex-valued signal from coded diffraction patterns, namely the Fourier intensities obtained after modulating the signal with a collection of masks, is a fundamental structured phase retrieval problem arising in diffraction imaging and related applications. Despite its practical importance, the theoretical analysis of this structured framework remains scarce. In the standard random mask model, the optimal sampling rate achievable by computationally tractable recovery methods has remained open. In this paper, we establish the optimal sampling rate for the PhaseLift feasibility program. More precisely, PhaseLift achieves exact recovery of an unknown signal $\pmb{x}_0\in\mathbb{C}^n$, up to a global phase, from $\mathcal{O}(\log n)$ random masks, with polynomially decaying failure probability. Since $\Omega(\log n)$ masks are necessary to identify certain signals under the erasure mask ensemble, our result thereby achieves the optimal mask complexity. Equivalently, PhaseLift attains the optimal total sampling rate of $m=\mathcal{O}( n\log n)$ scalar intensity measurements. The proof is based on an approximate dual certificate construction via a refined golfing scheme that combines adaptive mask allocation with a dimension-independent truncation threshold.

Goppa codes are a well-known class of linear codes with important applications in cryptography. Determining the minimum distance of Goppa codes and constructing Goppa codes with prescribed automorphism groups are both meaningful and challenging problems in coding theory. In this paper, we first study the minimum distance of binary separable Goppa codes. For the two classes $g(X)=f(X^t)$ and $g(X)=A(X)h(\phi(X))$, we give criteria for attaining the designed distance and derive several infinite families whose minimum distances are determined. We then construct binary Goppa codes and their related codes with $A_4$ or $A_5$ automorphism groups. These constructions also naturally yield binary quasi-cyclic Goppa codes and their related codes. Moreover, by applying the minimum-distance criteria developed above, we determine the parameters of one class of the constructed $A_4$-invariant Goppa codes.

A common assumption in theoretical models of Bayesian inference is that the signal has i.i.d. components. To study the effect of correlations in the signal prior, we consider a minimal model: the planted spin glass on random regular graphs, where the signal is sampled from an Ising model with coupling $\kappa$. Depending on the phase of the prior, we find that adding structure in the signal can either help or hinder inference. In the paramagnetic regime, correlations in the signal lower the reconstruction threshold, so that weaker signal strength is sufficient for recovery. In the ferromagnetic regime, the prior alone already enables partial recovery, and we identify the threshold above which the observations provide additional information. When the prior itself is in a replica symmetry breaking (RSB) phase, we detect a static RSB transition in the posterior under Nishimori conditions. This provides an example where a non-separable, correlated prior leads to static RSB in a Bayes-optimal inference problem. We discuss the consequences of this glassy phase for algorithmic performance, in particular for Belief Propagation.

Massive connectivity, a key building block of 5G, is expected to play an important role in the next generation of wireless systems, and its expected requirements are being revolutionized through the modeling of the information dynamics related to the vast numbers of Internet of things (IoT) devices. Motivated by this, the present paper introduces a model that captures the spatio-temporal nature of freshness of information sent via random access channel policies from an extremely large set of IoT devices via simple scalar parameters, i.e., the probability of success and accuracy of received updates. There are many information freshness metrics, starting from the age of information (AoI), all of which are proxies for the actual application performance, characterized over the temporal dimension. Our model adds the spatial dimension to this picture, observing that sensors distributed over the same area may have a strong correlation, and information from multiple close-by sensors may improve the overall accuracy of the receiver. We focus on characterizing the uncertainty of the receiver, expressed through the conditional entropy, considering a network of partially reliable, spatially distributed sensors observing the same process and reporting their measurements over a slotted ALOHA channel. We consider a simple forgetful receiver and a more complete model which accounts for the full history of past observations, deriving their performance, and optimizing the transmission probability of nodes to minimize the expected uncertainty.

In this paper, we consider a general status update system, consisting of a source at which a physical process is monitored and a receiver. For transmission of status updates, a discrete set of blocklength error probability tuples is available. A typical example is a communication system with a fixed set of modulation and coding schemes. We assume that no feedback channel for acknowledgments exists and thus investigate the problem of constructing deterministic schedules for the status updating system. The average Age of Information (AoI) and average exponential AoI are appropriate performance metrics to capture the freshness of information. Interestingly, a static blocklength is shown to be suboptimal in general. While for the average AoI the gains that can be achieved by using a non-trivial periodic schedule are relatively small, they are significant for the average exponential AoI. Motivated by this, we prove conditions for single-blocklength optimality when two blocklengths are available. To achieve this, we use a Dinkelbach-like approach to reformulate the problem as an infinite-state transient average-cost Markov Decision Process and apply careful bounding techniques to derive the optimality conditions. As a converse, we analyze stationary randomized policies to certify the parameter regions where a non-trivial deterministic schedule outperforms the best constant schedule. Extensions to the case with more than two transmission lengths are provided. Numerical results show that the derived conditions cover a large portion of the parameter space and suggest that the optimality conditions for a constant schedule with the shorter blocklength, while sufficient, may also be necessary. As an example, we apply the results to the problem of age-optimal transmission with a small LDPC codebook, which illustrates the practical applicability.

Neural receivers have been recognized as a promising paradigm for the next-generation (NextG) communications. However, due to the reliance on a static network optimized for specific channel conditions, their generalization capability across diverse scenarios remains a significant challenge. To address this issue, this paper proposes a novel channel-adaptive neural receiver network (CARNet) based on the mixture-of-experts (MoE) framework. The proposed architecture employs multiple expert networks together with an efficient routing mechanism to enable signal detection in various scenarios. The experts are constructed via stacked ResNet blocks and specialize in robust signal detection within specific channel conditions, while the routing mechanism incorporates a lightweight representation learning module, which projects the coarse channel estimate into a low-dimensional latent embedding. The learned embedding characterizes task-relevant channel conditions and provides efficient guidance for accurate expert selection. Link-level simulation experiments demonstrate that the proposed CARNet achieves superior performance across diverse channel conditions.

This letter investigates an integrated sensing and communication (ISAC) system aided by movable subarrays (MSAs) using a hybrid near-far field channel model. The sensing target and communication users are assumed to lie in the near field of the overall MSA aperture but in the far-field region of each subarray. Accordingly, a hybrid near-far field channel model is established, and the equivalent Fisher information matrix and Cramér-Rao bound (CRB) for joint range, elevation, and azimuth estimation are derived. The transmit beamforming matrix and subarray positions are jointly optimized to minimize the trace CRB subject to minimum communication signal-to-interference-plus-noise ratio (SINR), maximum transmit power and subarray movement constraints. An alternating optimization algorithm is developed combining iterative rank-one-penalized semidefinite relaxation with projected finite-difference block descent and backtracking. Numerical results show that the hybrid-field model closely matches the spherical-wave model, while MSAs substantially reduce the CRB.

Joint frequency and chirp-rate estimation for a noisy chirp signal arises in radar, sonar, and burst satellite communications. Conventional estimators combine a coarse grid search with fine interpolation; accuracy degrades at the edges of the residual cell (the edge effect) and below the breakdown SNR (the threshold effect). We present a deterministic two-stage estimator that controls both failure modes uniformly over the residual cell. The estimator combines a time-centered, zero-padded dechirp-FFT acquisition bank with alternating selectable-$p$ amplitude-interpolation refinements on DTFT samples at fractional bins; in the centered frame, the frequency-chirp-rate cross-term of the Fisher information vanishes. The paper derives a mean-squared-error and threshold characterization over the full SNR range, in closed form except for one calibrated scalar (an effective cell count), to our knowledge the first for the joint problem: the breakdown threshold is governed by the cell count, and its cell-position dependence is dominated by the scalloping loss of the coarse FFT, which the padding bounds at 0.4 dB. An asymptotic uniformity analysis over the cell, including its corners, gives fixed-point variance ratios of $1.003$ and $0.998$, analytically free of the residual. A closed-form bias analysis under a cubic phase mismatch shows the centered chirp-rate estimate is insensitive to first order. Monte Carlo experiments at $N=256$ (validated at $N=32$-$512$) measure frequency- and chirp-rate-axis efficiencies with median $1.03$ and worst case $1.07$ over $144$ cell positions at $-5$ dB. Threshold predictions hold within $1.0$ dB on four configurations not used in the calibration. The dechirp-FFT bank is fully parallel, and each of the four refinement iterations evaluates three DTFT samples per axis; under fixed operating conditions, per-estimate latency is constant at $O(N\log N)$ cost.

We study vector-linear function computation over three-layer networks with a fixed target function and a fixed source-access pattern. We develop a support-constrained row-space framework that represents a linear computing code by a global row space. This space must contain the target row space and be generated by rows satisfying the local support-constraints of the network. We prove that this representation is equivalent to the existence of a linear computing code. For any prescribed global row space, we give a necessary and sufficient condition for its realization and determine the minimum uniform communication load at the middle nodes. The condition is expressed in terms of the ranks of the local subspaces supported on the source-access sets. It separates the exact local realization problem from the outer problem of designing the global row space and yields a variational characterization of the linear computing capacity. We then apply the framework to MDS targets over cyclic networks. We identify when the target row space alone is sufficient and when auxiliary rows are required. We determine the capacity in the dense regime and in the sparse regime whenever the cut-set bound is integral. For the remaining sparse parameters, we give a general linear construction whose achievable rate equals the integer part of the cut-set bound.

Drifting objectives compare a target and model distribution through a vector field observed noisily at finitely many locations. We ask what distributional conclusion such a frozen measurement system warrants. For integrable antisymmetric interactions and absolutely continuous laws in a declared finite density basis, the unnormalized sampled numerator satisfies $\operatorname{vec}(V_X)=Mc$, where $c$ is an antisymmetric mismatch and $M$ is probe-dependent. This identity yields an a posteriori total-variation (TV) upper confidence bound accounting for held-out field noise, estimated-operator error, and externally validated $L^1$ residual radii around normalized density approximants in the span; a nonpositive observability margin returns the trivial TV bound and abstains. The audit recomputes this numerator from held-out samples; a normalized drift statistic requires a separate joint numerator–denominator analysis. For Gaussian-RBF interactions, a global envelope supports distribution-free and empirical-Bernstein radii without truncation, with companion bounds for the Laplace similarity in the original drifting objective. We characterize random-probe observability by a population Gram matrix, identify rank and symmetry degeneracies, and prove large-bandwidth collapse toward mean matching. Synthetic studies exercise Gaussian and Laplace numerators, separately prespecified bounded-vector and variance-adaptive radii, Monte Carlo-calibrated operators, nonzero residual radii around normalized finite-basis approximants, outward-rounded observability bounds, and designed abstention. A joint basis-size/dimension stress path extends evaluation through $m=8$. The result is a conditional diagnostic for a finite density class, or for normalized finite-basis density approximants with external residual radii, not a universal guarantee from small training drift.

Sticky channels, which never destroy nor create runs, are some of the simplest types of channels with synchronization errors (such as deletions, insertions, and replications). Despite their simplicity, we know little about the capacity of the sticky channels considered in the literature so far beyond what can be gleaned from purely numerical methods. Towards a broader and systematic exploration of these channels, we initiate the study of additive-noise sticky channels, a basic family of sticky channels that extend each input run by an amount independently sampled from a fixed noise distribution. The capacity of these channels corresponds to the capacity per unit cost of additive-noise memoryless channels over the non-negative integers, and they capture some aspects of the loss of synchronization caused by homopolymer length miscalls in DNA sequencing. We first focus on the setting of Bernoulli additive noise with parameter $p$, and uncover curious behavior of the capacity beyond what numerical methods can tell us. For example, the capacity is constant when $p\in [1/\varphi^2,1/2]$ with $\varphi\approx 1.618$ the golden ratio, achieved by zero-error runlength-constrained coding, and behaves differently when $p\approx 0$ vs. when $p\approx 1$. More generally, we determine the capacity exactly for all $p\leq 1/2$, and when $1/2<p<1$ we give analytical bounds that allow us to characterize the asymptotic behavior of the capacity as $p\to 1$. We also extend our analysis to the setting where input strings have bounded runlengths, motivated by DNA-based data storage. Then, we study additive-noise sticky channels beyond Bernoulli noise. We derive capacity lower bounds for all additive-noise sticky channels whose noise distributions have a given mean and support. These lower bounds allow us to characterize the regime where zero-error runlength-constrained coding is never optimal.

The classical analysis of the Gaussian mechanism in differential privacy reduces privacy loss for a pair of neighboring datasets to a scalar random variable. While this scalar characterization is sufficient for privacy accounting, each perturbation instance also induces a high-dimensional random triangle formed by the sensitivity vector and the two corresponding noise vectors. In this work, we develop two complementary geometric representations of these random triangles. The first representation maps the normalized squared edge lengths to a simplex. We derive its exact joint density, characterize its elliptical support, and reconstruct the classical privacy loss random variable from the simplex coordinates. The second representation maps the spectral shape of each normalized triangle to a hemisphere. We derive the corresponding density and coordinate mappings, recover the same privacy loss, and characterize an equatorial drift together with band concentration as the dimension increases. These results provide two exact geometric coordinate systems that complement the scalar privacy loss and connect differential privacy with the probabilistic analysis of random shapes.

Masked prediction learns to infer missing variables from visible context selected by a mask schedule. When does small excess risk guarantee recovery of the true joint distribution? We study this question on finite product spaces under masked-block log loss, with conditionals induced by a single joint distribution. To quantify recovery, we introduce an $\varepsilon$-identifiability modulus measuring the worst-case error among joint distributions with population excess risk at most $\varepsilon$. For data with separated modes pinned down by sufficiently large visible contexts, schedules that always retain such contexts can permit nonvanishing mode-weight errors while incurring exponentially small excess risk. An information decomposition explains why: the loss captures mode-weight mismatch only where the visible context leaves the mode uncertain. We prove two-sided bounds showing that, over a fixed range of mode weights, sensitivity to mode reweighting is governed by residual mode uncertainty averaged over the mask schedule. Assigning schedule mass to low-visibility masks that retain this uncertainty yields recovery bounds within the reweighting family. Beyond this family, positive full-mask probability characterizes uniform control of joint KL divergence by excess risk. Exact calculations and controlled gradient optimization validate these predictions.

The pseudo-quantum representation re-encodes a finite, irreducible, reversible continuous-time Markov chain with $M$ states as a complex-orthogonal flow on a doubled space of dimension $2M$. After uniformisation, a square-root gauge, a doubling of the state space and a diagonal unitary twist, the chain generates an entire one-parameter group $W_z = e^{zK}$ with $z$ complex, and this single group is the object of the paper. Its real slice, $z$ real, reproduces the stochastic semigroup exactly through an explicit decoding whose probabilistic meaning is the parity of the number of ticks of the uniformised chain. Its imaginary slice, $z = i\theta$, is a finite-dimensional unitary quantum system. The chain and the quantum system are therefore not two analogous models but two restrictions of one entire representation. The setup also produces, with no further input, a pseudo-Schroedinger equation, a bilinear von Neumann equation for a complex-symmetric pseudo-density, and a pseudo-Bloch vector equation on a non-compact quadric. We develop the construction from first principles and work out one model completely, the usual symmetric Ehrenfest urn. Its gauged generator equals $(2/n) J_x$, the spin-$n/2$ operator, its relaxation modes are Lorentz boosts, and its imaginary slice is a depth-one quantum circuit whose Born distribution is the classical urn law under the clock $\theta(t) = \arccos(e^{-2t})$. The same clock identifies the Fisher-Rao lift of the urn trajectory with a rigid spin-coherent-state orbit. This is a preliminary simplified version of a longer paper. It treats only finite state spaces and only the symmetric Ehrenfest urn, and it previews without proofs the asymmetric urn, two queues, the symmetric simple exclusion process and the stochastic Ising model. Appendix primers make the quantum, geometric and information-theoretic language self-contained.

Let \(g\) be a monic polynomial of degree \(r<n\), and let \(C_g(n)\) be the coefficient-vector code formed by multiples \(ug\) with \(\deg(ug)<n\). We study the coefficient-space MDS locus \(M_{n,r}\). The companion construction identifies coefficient space with the moduli of cyclic matrix-vector pairs, and the remainder-orbit map embeds it as a smooth complete intersection in the standard big cell of \(\operatorname{Gr}(r,n)\). We prove that every normalized maximal Plucker coordinate pulls back, up to sign, to a power of the constant coefficient \(A_0\) times a Schur polynomial \(S_\kappa(g)=s_\kappa(\Lambda_g)\), where \(\kappa\subseteq (n-r)^{r-1}\). Hence the universal MDS polynomial is \[D_{n,r}=A_0\prod_{\kappa\subseteq (n-r)^{r-1}}S_\kappa.\] This description yields a flat non-MDS boundary over \(\mathbb{Z}\), explicit degree and finite-field estimates, and a length filtration governed by sparse multiples. It also gives bad-characteristic criteria on root-multiplicity strata and density-one results on the irreducible stratum. Finally, for \(r\ge 3\) and \(N\ge r+3\), every nonempty first-failure layer over an algebraically closed field has a dense open non-GRS locus.