signal processing engineers

Abstract: We study over-the-air computation of functions, such as empirical variance, formed by combining two separately postprocessed sums of locally transformed observations. We consider continuous encoders subject to pointwise peak-power constraints over a single use of a complex multiple-access channel. With four or more transmitters, we identify conditions under which exact noiseless recovery forces every continuous encoder collection into a common affine structure that preserves both sums, reducing infinite-dimensional encoder design to selecting a finite set of coefficients. To quantify resilience against channel noise, we introduce a robustness score that measures the separation of noiseless received aggregates relative to variations in the desired function value. Within this characterized encoder family, we formulate robustness maximization as an exact convex semi-infinite program, eliminating rank constraints without a relaxation gap, and construct a decoder that attains the minimum uniform noise-amplification constant. For several function classes, including empirical variance, we derive closed-form robustness-optimal encoders and scores. Under mild regularity conditions, we also establish sharp limits on the Gaussian mean squared error at high signal-to-noise ratios and show how using the input distribution reduces average error at the expense of uniform robustness.

Abstract: Integrated Sensing and Communication (ISAC) systems rely on accurate Direction-of-Arrival (DoA) estimation for effective beamforming and spectral-efficiency enhancement. While numerous classical and deep-learning-based DoA estimation methods have been reported in the literature, their suitability for real-time ISAC operation remains insufficiently understood because practical deployment requires consideration of both estimation accuracy and computational latency. This paper presents a unified evaluation framework for systematically comparing classical and learning-based DoA estimators within an ISAC beamforming pipeline. Representative classical and learning-based DoA estimators, together with an ensemble architecture, are integrated with Singular Value Decomposition (SVD)-based beamforming and evaluated under identical operating conditions. The impact of DoA estimation accuracy, inference latency, beamforming-weight generation time, and resulting effective spectral efficiency is investigated. Results show that different estimators occupy distinct operating points in the accuracy-latency-performance design space. Although learning-based approaches substantially reduce estimation time compared with MUSIC, the overall processing latency increasingly becomes dominated by the computational cost of SVD-based beamforming. Among the evaluated methods, the Regression Convolutional Neural Network (RCNN) exhibits a favorable balance between estimation accuracy, computational efficiency, and beamforming performance, making it a promising candidate for real-time ISAC deployment. The findings further suggest that future low-latency ISAC systems may benefit from beamforming strategies that reduce or avoid computationally intensive SVD-based beamforming-weight generation.

Abstract: We study Boolean entropy and Fisher information in information inequalities and the Boolean central limit theorem. We prove a fractional Stam inequality for the Boolean Fisher information associated with arbitrary fractional partitions of subset sums, and derive an entropy power inequality for sums of Boolean independent random variables and its weighted version. We then establish a Boolean analogue of Barron's entropy convergence result along the Boolean central limit theorem: Boolean entropy converges to the entropy of the Rademacher law if and only if the initial entropy is finite. Furthermore, under a finite fourth moment, in the Boolean central limit theorem, we obtain \[ Γ(μ_n\mid\mathbf b)= \frac{m_4(μ)-1-m_3(μ)^2}{2}\frac{\log n}{n} +o\!\left(\frac{\log n}{n}\right), \] with an explicit example showing sharpness. For symmetric laws with finite fourth moment and finite Boolean Fisher information, \[ Ψ(μ_n\mid\mathbf b) =2\bigl(m_4(μ)-1\bigr)\frac{\log n}{n} +o\left(\frac{\log n}{n}\right). \] Our approach is based on the Nevanlinna representation of Cauchy transforms.

Abstract: Bilinear Compressive Security (BCS) is a lightweight encryption scheme that combines compressed sensing with a fresh random convolution filter to improve resistance against key-recovery attacks under key reuse. Previous work derived information-theoretic guarantees for BCS by relating key identifiability to phase retrieval under the assumption that an adversary has access to the full ciphertext distribution induced by the random filter. In practice, however, an adversary only observes finitely many encryptions, leaving the security of BCS in the finite-sample regime largely unexplored. In this paper, we study key recovery from only finitely many observations. We first develop a covariance-based attack that estimates the linear measurements from repeated encryptions of the same plaintext and subsequently recovers the secret measurement matrix. We analyze the estimation procedure and derive finite-sample guarantees together with corresponding sample-complexity bounds. We further investigate the complete attack experimentally and compare its observed recovery behavior with the previously established information-theoretic identifiability results. Our experiments show two qualitatively different recovery regimes on the structure of the measurement matrix. For dense matrices, including Gaussian matrices, the number of measurements required for successful recovery appears to grow linearly with the signal dimension. This differs from the worst-case behavior suggested by information theory. In contrast, for matrices with sparse rows, we observe a quadratic growth consistent with the theoretical bound. These results suggest that the sparsity structure of the measurement matrix plays an important role in the security of BCS and should be taken into account when selecting parameters and designing practical instances of the scheme.

Abstract: We consider the problem of learning structured linear dynamical systems over convex sets $\mathcal{K}$, where only a small subset of the observations are available at each time point. An estimator which minimizes a bias-corrected, potentially non-convex objective function is proposed. Non-asymptotic bounds are obtained for the statistical error, which depend on the local complexity of $\mathcal{K}$, the trajectory length $T$, and the sub-sampling probability $p$. Convergence of the projected gradient descent algorithm is also established. The general theory is applied to settings where (i) $\mathcal{K}$ is a subspace, (ii) $\mathcal{K}$ is the set of bi-isotonic matrices, and (iii) $\mathcal{K}$ is the set of matrices whose rows are formed by sampling Lipschitz functions. We show meaningful recovery of the transition matrix is possible for values of $T$ much smaller than what is required in the unconstrained case, and for $p = o(1)$.

Abstract: We study the sample complexity of noisy one-bit compressed sensing for signals drawn from a prior distribution. By characterizing the effective distributional complexity of the prior via its approximate covering number, we prove that posterior sampling achieves accurate recovery with high probability when the number of measurements scales with the logarithm of the approximate covering number, up to a one-bit separation gap factor. This upper bound is robust to learned prior mismatch. Specifically, we show that posterior sampling with an approximate prior remains reliable, provided that the learned prior distribution is sufficiently close to the true signal distribution in Wasserstein distance. In addition, we establish a sample complexity lower bound for any reliable method of noisy one-bit compressed sensing, showing that our upper bound is nearly matched in its main prior dependent term. To approximate the ideal posterior sampling process for real world scenarios, we instantiate posterior sampling through a plug-and-play algorithm with diffusion priors. Experiments on the FFHQ and ImageNet datasets demonstrate the effectiveness of our proposed approach.

Abstract: Zador's celebrated theorem is a cornerstone of optimal quantisation: it establishes both the weak limit of the empirical distribution of an optimal $n$-point quantiser in $R^d$ and the decay rate of the associated $L_s$-mean quantisation error. In large dimension, however, observing this asymptotic behaviour requires an astronomically large sample size. We prove that, for spherically symmetric target distributions, optimisation over all spherically symmetric distributions is a convex problem and derive an equivalence theorem that both characterises global optimality and yields a constructive algorithm. We show that, for moderate $n$, random quantisers uniformly distributed on a sphere of suitably chosen radius $R$ perform exceptionally well and, over a broad range of values of $n$, are numerically certified to be optimal among all random quantisers. Their expected distortion has an explicit integral representation that can be evaluated to arbitrary precision, and we prove concentration across random quantisers: the distortion variance tends to zero as $n\to\infty$ for fixed $d$. For $s=2$, both the optimal radius and the associated minimum expected distortion admit exact expressions. For general $s$, the optimal radius can be determined efficiently, and extreme-value theory provides useful approximations when $n$ grows with $d$. Depending on this growth rate, $R$ either converges to zero or approaches a positive limit that is independent of $s$.

Abstract: Physical reservoir computing exploits the nonlinear dynamics of physical systems to process time-dependent data with greater energy efficiency than conventional machine learning approaches. However, physical reservoirs have fixed intrinsic response timescales, whereas real-world signals combine deterministic and stochastic components across multiple timescales. Here we show, using a nanoporous niobium oxide reservoir, synthetic noisy signals and cryptocurrency-price volatility, that the relationship among noise correlation time, reservoir memory and forecast horizon determines whether correlated noise is filtered or predicted. Noise varying faster than the relevant reservoir memory and forecast horizon is averaged by the reservoir, whereas the temporal structure of slower-varying noise is sufficient for algorithmic forecasting. We introduce the reservoir memory horizon and forecasting regime index to distinguish these operating regimes. These contributions demonstrate that timescale matching can guide the encoding of input time series and development of physical reservoir architectures that filter, analyse and predict stochastic signal components across distinct temporal scales.

Abstract: A theoretical understanding of multi-head self-attention is fundamental to the study of modern neural networks. Using random matrix theory, we establish Gaussian equivalence for multi-head self-attention: replacing softmax attention with rescaled scores plus Gaussian noise preserves the limiting spectral law of the centered output. This equivalence also covers value and output projections that depend on the keys. The resulting laws separate the effects of head allocation and projection widths, and distinguish spectrum-preserving across-head sharing from within-head key--value dependence.

Abstract: Structured recovery phenomena, such as restricted isometry properties in compressed sensing, have shown that high-dimensional objects can often be reconstructed from remarkably low-dimensional linear measurements. This work develops an analogous recovery framework for low-rank signed measures on $\mathbb{R}^2$, defined here as measures that can be expressed as finite sums of product measures with one-dimensional factors. The framework is based on linear operators, termed "silhouette operators," that map a measure to a fixed finite collection of one-dimensional linear pushforwards. The main results show that a suitably chosen collection of $2k$ projected marginals suffices to identify every compactly supported rank-$\le k$ signed measure, that this number is optimal, and that the projection directions cannot be chosen arbitrarily. The framework is also extended to higher-dimensional sums of product measures by establishing sufficient conditions under which collections of pairwise marginals identify the full model. Building on this framework, a computationally efficient estimator, termed "silhouette mixture estimation" (SME), is introduced for constructing a low-rank empirical measure from data by matching its one-dimensional projected marginals to the corresponding empirical marginals in Wasserstein distance. When combined with one-dimensional density estimators, SME yields an efficient nonparametric density estimator that performs strongly relative to a range of parametric, nonparametric, and deep-learning baselines in settings of moderate dimension and sample size.

Abstract: This paper introduces an energy-adaptive noise scheduling and whitening strategy for transform-domain diffusion models. Existing spectral diffusion methods account for the non-uniform statistics of transform coefficients through coefficient scaling, normalization, or frequency prioritization, while the forward diffusion noise schedule remains largely independent of the underlying spectral-energy distribution. We investigate whether the temporal evolution of the forward diffusion process should also follow the spectral organization of natural images. The proposed formulation combines global spectral whitening with energy-conditioned noise allocation that jointly modulates the injected noise according to the energy of individual transform coefficients and an image-dependent energy path over diffusion time. The resulting forward process preserves Gaussian transitions with closed-form marginals and remains compatible with standard DDPM and DDIM procedures without modifying the diffusion architecture. Experiments on CIFAR-10 demonstrate the contribution of the proposed energy-conditioned noise schedule and spectral whitening, reducing Fréchet Inception Distance from 142.48 for a compact DCTdiff U-Net variant to 100.45.

Abstract: Preparing quantum states from classical data can cost more than the computation they serve; most loaders tailor a circuit to each input. Here we show that the signals of a real dataset share structure that can be learned once and reused. Our quantum-native loader learns a low-dimensional description of a dataset and prepares every signal with one fixed circuit set by a few numbers. Across seven views of five public datasets it meets the targets of the strongest structured loader at equal gate cost with several times fewer numbers per signal. These numbers can be inferred from a random subset: in a preregistered blind replication the subset needed to come within ten per cent of full-signal accuracy stayed constant within a prespecified margin as signals grew sixteenfold, whereas the structured loader needed ever more. It declines what it cannot represent, covering fewer cases than that baseline and no electrocardiogram.

Abstract: We study the average-case matrix discrepancy problem: given independent normalized $d\times d$ Gaussian orthogonal ensemble matrices $A_1,\dots,A_N$ and a fixed margin $κ>0$, find signs $σ_1,\dots,σ_N\in\{-1,1\}$ such that the operator norm of $\sum_{i=1}^N σ_i A_i$ is at most $κ\sqrt{N}$. Focusing on the proportional regime $N/d^2\to τ\in(0,\infty)$ as $d\to\infty$ followed by the small-margin limit $κ\downarrow 0$, we characterize the density required by stable offline algorithms and by online algorithms. In the offline setting, we construct a polynomial-time \emph{recenter-and-round} algorithm that is noise-stable and succeeds whenever $τ=Ω(\frac{1}{κ^2\log(1/κ)})$, along with a matching lower bound for all stable algorithms. In the online setting where each sign must be chosen irrevocably upon observing the corresponding matrix, we determine the exact limiting performance of the \emph{Frobenius-greedy} algorithm, establishing that it succeeds when $τ>τ_{\rm FG}(κ)\sim \fracπ{4κ^2}$, as well as a matching lower bound for all online algorithms by conditioning on a revealed prefix. At the core of our algorithms lies rotational symmetry, which enables us to transfer Frobenius norm control into operator norm guarantees. Together, our results identify the algorithmic phase transition points for random matrix discrepancy: $Θ(\frac{1}{κ^2\log(1/κ)})$ for stable offline algorithms and $Θ(\frac{1}{κ^2})$ for online algorithms. Both thresholds lie far above the satisfiability scale $Θ(\log(1/κ))$, as shown by Maillard~\cite{maillard2025}.

Abstract: Distinguishing periodic from chaotic dynamics in a time series is a fundamental challenge in both physics and engineering. Yet, end-to-end learned architectures must discover both a representation and a decision boundary from data, at substantial cost. We introduce TopTimeNet, which decouples these tasks: a fixed, non-learned stage extracts a $42$-dimensional geometric and topological descriptor from Takens delay embeddings and persistent homology, and a lightweight learnable stage performs classification. On a benchmark of $49$ nonlinear dynamical systems, a $1{,}638$-parameter configuration matches the mean accuracy of one with $33\times$ more trainable parameters. Additionally, this approach delivers mean accuracy comparable to convolutional neural networks and surpasses the average performance of converged Transformer models, while requiring three to four orders of magnitude fewer trainable parameters. Robustness also depends sharply on where noise is introduced: TopTimeNet degrades gracefully under perturbations to its precomputed features, but degrades sharply when noise is introduced into the raw signal and the full feature-extraction pipeline is recomputed, showing that robustness to perturbations of the precomputed features does not imply robustness of the complete raw-signal-to-prediction pipeline. These results show that decoupling fixed geometric and topological feature construction from a lightweight discriminative stage can achieve comparable classification accuracy with substantially fewer trainable parameters.

Abstract: We study the optimization landscape of low-tubal-rank tensor sensing through a balanced factorization. Under a tubal restricted isometry condition, we establish a quantitative strict-saddle landscape with no spurious local minima for arbitrary Fourier multi-rank profiles. We further show that the local geometry depends on the Fourier-slice ranks rather than the tubal rank alone. Uniform ranks yield quadratic growth transverse to the solution orbit, whereas nonuniform ranks produce quartically flat directions through hidden frequency-wise overparameterization, even when the factor width equals the exact tubal rank. Numerical experiments illustrate the global optimization behavior and the contrasting local geometries.

Abstract: We study the fundamental limits of transferability in algebraic signal processing through homomorphisms between algebraic signal models. Homomorphisms are linear maps between the signal spaces of two models that commute with filtering, so filtering a signal and transferring it across domains can be done in either order. The existence of such maps is governed entirely by coincidences among the filtered eigenvalues of the two models' shift operators, but existence alone is insufficient: the space of homomorphisms always contains trivial elements that destroy all information. We introduce the spectral transfer efficiency $η(θ)\in[0,1]$ to quantify information-preserving quality, prove that every homomorphism decomposes into unconstrained blocks over coincidence classes, derive the dimension of the homomorphism space, and characterize exactly when lossless transfer is achievable. Beyond normal shift operators, we quantify a departure-from-normality penalty and show how filter derivatives can repair spectral defectiveness. The theory yields concrete consequences in three settings: for sampling, eigenvalue interlacing converts transferability under subsampling into an explicit filter design constraint; for compressed sensing, $η(θ)$ controls the restricted isometry constant and the coherence of the resulting measurements, and recovery decouples across coincidence classes; and for machine learning, spectral aliasing emerges as the controlled symmetry breaking that makes transfer between mismatched domains possible at all.

Abstract: We study the computational complexity of hypothesis testing in the spiked Wigner model, a prototypical model for detecting low-rank structure in a large random matrix. Below the "BBP" eigenvalue transition, it is expected that strong detection --- with both type I and II errors vanishing --- requires exponential time. Assuming this as a conjecture, we determine the limits of polynomial-time weak detection, exactly characterizing the possible tradeoffs between type I and II errors. Specifically, the optimal tradeoff is achieved by a particular linear spectral statistic. Thus, the question of weak detection is entirely reduced to that of strong detection. The proof builds on ideas of Nagda-Raghavendra (2025) and Moitra-Wein (2025). The low-degree likelihood ratio (LDLR) plays a key role: any test that slightly beats the LDLR can be boosted to have an even higher success probability. This leads us to establish a computational analogue of the Neyman-Pearson lemma for a subclass of additive Gaussian models: for a given super-polynomial runtime, the best possible tradeoff between type I and II errors is either the one achieved by thresholding the LDLR, or the trivial tradeoff that results from strong detection.

Abstract: Range reduction plays a crucial role in the accuracy and performance of evaluating trigonometric functions, and is often the primary bottleneck for large floating-point inputs. While fast algorithms such as Cody--Waite work efficiently over narrow intervals, the Payne--Hanek algorithm remains the standard technique for accurate reduction across large floating-point inputs. However, existing implementations of Payne--Hanek suffer from high latency due to heavy branching, conversion overheads, and the use of multi-word integer arithmetic, which hinders SIMD vectorization. In this paper, we analyze and present a branch-free variation of the Payne--Hanek algorithm using only floating-point arithmetic. Our method operates directly over large double-precision inputs ($|x| \ge 2^{16}$) and is well suited to hardware with FMA instructions. We formulate the precision constraints in terms of a truncation error budget, construct a compact lookup table indexed by the input exponent, and prove that the reduced argument is accurate to within one ulp for every input. The same routine can serve both as the complete range reduction of a single-stage implementation and as the fast path of a correctly rounded one, and it improves both latency and throughput over existing implementations. The algorithm is currently implemented in the LLVM libc project.

Abstract: In this paper, we propose a representation of a digraph (directed graph) as a Hermitian matrix derived from its adjacency matrix. This representation characterizes both the connectivity and the edge orientation of the digraph. Based on the spectral decomposition of the Hermitian matrix, a digraph clustering algorithm with $k$-means is introduced to produce a partition on the graph. Applying this algorithm (bottom-up) recursively to a digraph with partially labeled vertices yields a spectral hierarchical digraph clustering (\myproj) algorithm that produces consistent nested partitions of the digraph, or equivalently, a tree structure. Furthermore, based on the in-degree and out-degree of each cluster in the digraph clustering, a pair of hierarchical interval partitions (filtrations) can be derived in a top-down manner to produce a pair of nested knot sequences. These knot sequences facilitate the construction of multilevel spline quasi-interpolants, enabling a noisy graph signal to be decomposed into a coarse approximation and inter-level details, followed by adaptive thresholding and reconstruction. Experiments on synthetic and real-world digraphs demonstrate the superiority of our {\myproj} algorithm for digraph clustering across diverse graph structural properties (homophily and heterophily) and supervision settings. Moreover, experiments on digraph signal processing using multilevel spline quasi-interpolants further demonstrate the effectiveness of signal recovery on digraphs in terms of RMSE and SNR.

Abstract: We present a deterministic CRT-based sparse Fourier architecture for exact recovery of noiseless, on-grid, at-most-\(k\)-sparse spectra with an engineered transform length. The transform length \(N\) is selected as an exact product of small pairwise-coprime stage primes, allowing each stage to use transforms whose size scales with the sparsity rather than the ambient dimension. Three co-prime-increment time shifts provide a one-sided singleton-consistency screen: every genuine singleton passes, while passing collision bins remain provisional candidates. Each candidate bin is mapped by a total label-emission rule to at most one global frequency label, and per-stage label sets are intersected, yielding a deterministic \(O(k)\) candidate bound for every input. Candidate construction uses \(O(k\log N)\) samples. In a comparison real-RAM model with exact arithmetic and exact phase evaluation but without unit-cost root-index decoding, it requires \(O(k\log^2 N/\log k)\) arithmetic operations; in a stronger root-index-oracle model the arithmetic cost is \(O(k\log N)\). Under all-stage genuine-singleton survival, the candidate set contains the true support. Exactness, however, does not depend on that survival condition: a consecutive-sample residual verifier accepts a sparse reconstruction only when it is exact, and all failures route to a dense FFT. The resulting hybrid algorithm therefore returns the exact spectrum for every input in the stated noiseless, on-grid, at-most-\(k\)-sparse model, with \(O(N\log N)\) worst-case runtime. The verified sparse path additionally incurs an \(O(k^2)\) verifier cost in the root-index-oracle model. Neither computational model is a bit-complexity model.

Abstract: Time series forecasting requires extrapolating the dynamics of an observed process beyond the last available measurement. Yet recurrent forecasting models typically treat observations as inputs that directly control their latent dynamics. It leads to a regime change when these observations become unavailable at prediction time. Following a state-estimation perspective, we introduce Observer State-Space Models (OSSMs), a class of recurrent models that interprets the observed input time series as measurements of an underlying autonomous dynamical system. OSSMs explicitly separate latent-state propagation from measurement assimilation: a single transition governs the dynamics across both context and forecasting intervals, while available observations correct the estimated state through an observer. This formulation naturally exposes classical control-theoretic properties, including observability and convergence of the state estimation error. We further show that conventional and recent SSMs can be recovered as particular instances of our OSSM framework, thereby providing a unified interpretation of their recurrent dynamics and revealing modeling inconsistencies. We perform experiments across several benchmarks showing that OSSM achieves substantial improvements while maintaining the same parameter count and training setup as the corresponding SSM baseline. These results support a simple principle for recurrent forecasting: observations should correct the estimated latent state, rather than control the dynamics used to propagate it.

Abstract: The presence of noise that depends on the latent variables poses a fundamental challenge to identifiability. Existing results rely on conditional independence among the observations given the latent variables. We study a more general \emph{misspecified structure}, where this conditional factorization does not hold, and establish both precise and approximate identifiability guarantees. We characterize structural misspecification as a perturbed factor analysis problem. For precise identifiability, we establish subspace identifiability under spectral separation and controlled perturbation, followed by component-wise identifiability under structural sparsity. When the precise condition is not guaranteed, we derive an approximate subspace-identifiability theorem. Based on these results, we develop an unsupervised variational estimator for recovering latent variables. Experiments demonstrate the effectiveness of the proposed framework.

Abstract: Bayesian optimization (BO) is a widely used framework for optimizing expensive black-box objectives, but standard BO methods often use Gaussian process (GP) surrogates whose Gaussian assumption is sensitive to outliers and heavy-tailed noise. We introduce q-ED-BO, a robust BO method whose surrogate follows a univariate q-exponential (q-ED) distribution, preserving GP-BO's closed-form posterior mean and variance while a shape parameter q controls the tail behavior, recovering the GP at q = 2 and growing heavier-tailed with wider confidence bounds as q decreases. This tractability yields a closed-form q-upper confidence bound (q-UCB) with sublinear regret, and an exact closed-form q-expected improvement (q-EI) that generalizes EI to the heavy-tailed predictive, recovering classical EI at q = 2. Experiments on beamformer and adaptive filter tuning with impulsive outliers show that q-ED-BO matches or exceeds existing baselines on clean data, and under corruption, improves the strongest baseline by approximately 0.7 dB in output SINR and 1.1 to 1.2 dB in misalignment reduction.

Abstract: We give a rank-23 algorithm for $3\times3$ matrix multiplication using 51 additions and subtractions in alternative bases. The input and output conversions require five further additions, giving 56 additions in ordinary coordinates. The construction combines linear-program reduction with a search over sparse basis changes, and the resulting programs are distributed as a machine-checkable certificate. We verify correctness by exact coefficient expansion and distinguish the kernel cost from the cost of a complete multiplication.

Abstract: We study how well gravitational-wave phase deviations can be identified using a neural network with classification and regression heads. The classification head distinguishes general relativity (GR) from six parametrized post-Einsteinian phase families with exponents $b\in\{-7,-5,-3,-1,+1,+2\}$; the regression head predicts the logarithm of the coupling magnitude, $\log_{10}|β|$. The network input is a response function quantifying the sensitivity of waveform mismatch to waveform deformations. We use stationary Gaussian noise with the Advanced LIGO design spectrum for 262 sources and a measured Hanford spectrum from the third observing run (O3) for 132 sources, and also test injections into recorded Hanford strain. On the two Gaussian datasets, the network identifies the exponent of deviations clearly above the noise with accuracies of $0.427$ and $0.329$. To interpret these results, we compare the network with an approximate classifier built from predicted waveform residuals for the six phase families and same sources. The two classifiers often confuse the same families. Examining residuals left by different phase corrections, we link these errors to similarities between the underlying deviations. On selected recorded Hanford strain, networks trained on simulated or recorded noise achieve overall classification accuracies close to those on simulated Hanford noise. Of thirteen neural-network variants tested on the Advanced LIGO design dataset, the recurrent network has the highest mean classification accuracy for loud deviations, although all variants remain below the approximate classifier. The network nearly matches that classifier's accuracy, indicating that identification is limited by similarities between neighboring phase families rather than by the network, for known intrinsic source parameters.

Abstract: Lipschitz continuity is a fundamental principle in the design of certifiably robust deep neural networks (DNNs), wherein adjusting the Lipschitz constant, which quantifies network robustness, is of central theoretical importance. A standard approach to enforcing Lipschitz continuity requires each layer of a DNN to be Lipschitz continuous, thereby guaranteeing overall Lipschitz continuity. However, this layer-wise approach typically imposes overly conservative restrictions by producing a loose estimate of the overall Lipschitz constant, which limits the expressive capacity of the DNN and degrades empirical performance at a prescribed level of robustness. To overcome this loose estimation, the recently proposed LipKernel transfers information across layers to yield a much tighter overall Lipschitz bound than conventional layer-wise construction. In this paper, we extend this concept to cascaded state-space models (SSMs) to construct Lipschitz-continuous DNNs capable of modeling longer-term dependencies. The proposed architecture, named LipSSM, is theoretically justified and empirically evaluated.

Abstract: In 2018, Soloveychik, Xiang and Tarokh considered a pseudo-random symmetric circulant matrix constructed from binary Golomb sequences of length $n=2^m-1$, and claimed a proof that its empirical spectral distribution converges almost surely to the semicircle law as $n$ grows to infinity by the method of moments. In this comment note we show that their argument contains several technical flaws and inappropriate applications of technical lemmas. Instead we demonstrate that the eigenvalues of the matrix are simply a normalized twisted Kloosterman sum varying over the multiplicative character, to which we apply Katz's result directly to establish the asymptotically semicircle spectral distribution deterministically.

Abstract: We investigate the expected reconstruction risk of trigonometric polynomial models under different sampling schemes. Through numerical experiments, we observe that when the sampling nodes $\{t_l\}_{l=1}^m$ are i.i.d. random variables uniformly distributed over $[0,1)$, the associated structured random matrix $\pmb{A} \in \mathbb{C}^{m \times N}$ with $A_{l,k} = e^{2π\mathrm{i} kt_l}, k \in Γ= \{-q, \dots, q\}, N = 2q+1$ frequently becomes nearly singular or severely ill-conditioned. As a consequence, the expected reconstruction risk exhibits divergent behavior. In contrast, when the sampling nodes $t_l$ are either equidistant points or small random perturbations of an equidistant grid, the expected reconstruction risk undergoes a sharp phase transition at the interpolation threshold $m=N$. To better understand the underlying mechanisms behind these different phenomena, we characterize the expected reconstruction risk through the spectral quantity $\sum_{i=1}^{r} \frac{1}{σ_i^2(\pmb{A})}$, where $σ_i(\pmb{A})$ denotes the singular values of the sampling matrix. Based on this spectral representation, we theoretically prove that the expected reconstruction risk diverges under uniformly distributed random sampling. Furthermore, we derive an explicit formula for the expected reconstruction risk in the equidistant sampling case and establish upper and lower bounds for the expected reconstruction risk under jittered sampling.

Abstract: Time-series explainability remains challenging because discriminative information is often encoded in latent frequency or time-frequency features rather than in the raw signal itself. Existing attribution methods typically operate either in the time domain or in a fixed transform domain, limiting their ability to capture salient information across different representations. We propose XACT, a general framework that learns sparse attribution masks over coefficients from arbitrary invertible time-frequency transforms. We evaluate the framework on the STFT, the continuous wavelet transform, and the discrete wavelet transform. In addition, we extend the virtual inspection layer approach from the STFT to both wavelet transforms, enabling LRP to generate explanations in these representations. On a synthetic dataset, XACT produces precise explanations and is less prone to highlighting spurious features than the tested baselines. Across two real-world datasets, XACT produces sparse and structured explanations, although no method performs best across all quantitative evaluation criteria. These results demonstrate that learning explanations directly in time-frequency representations offers a flexible approach to interpreting deep-learning models for time series data.

Abstract: Federated learning (FL) lets EEG decoders learn from recordings of several subjects without pooling them. We consider two light EEG decoders, the Riemannian SPDNet and the Euclidean EEGNet. Both split into a trunk, which builds a latent representation, and a head, which classifies it. Inter-subject variability, however, makes a single shared FL model a poor fit for each subject. Personalised FL addresses this: all subjects learn a common trunk, and each subject keeps its own head. We adapt it for SPDNet and study its effects against standard FL and centralised training, with EEGNet as a Euclidean baseline. Experiments cover three motor-imagery datasets that span diverse regimes in channels, subjects and classes. We observe that personalised SPDNet reaches higher accuracy than both standard FL and centralised training, while converging in fewer rounds and communicating fewer parameters than standard FL. It also outperforms every EEGNet configuration on two of the three datasets, although centralised EEGNet outperforms centralised SPDNet.

Abstract: Toeplitz matrices are matrices whose entries are constant along each diagonal. When only finitely many diagonals are nonzero, the determinants of successively larger matrices obey a fixed linear recurrence: each new determinant is a fixed linear combination of finitely many preceding ones. We ask whether the recurrence for a complicated band can be built from recurrences for simpler factors, and show that it can. Multiplying two finite banded Toeplitz matrices reproduces the expected product throughout the interior, with discrepancies only near two opposite corners. Shifting the factors relative to the main diagonal redistributes these boundary discrepancies, and the different shifts account exactly for the pieces from which the full determinant recurrence is assembled. For several factors, all allowed shifts are described by a finite system of linear inequalities, giving a systematic decomposition of the recurrence. This viewpoint also leads to a recursive construction that works directly with polynomial coefficients, without solving for their roots. When a factorization into bounded-degree pieces is supplied, a valid recurrence can be constructed using essentially a linear number of arithmetic operations in the number of coefficients that must be output. A five-diagonal example shows how a sixth-order recurrence is assembled from two tridiagonal Toeplitz factors together with two boundary contributions.

Abstract: These notes develop the mathematical foundations and construction of a Fourier-Bessel wavelet family inspired by the disk harmonics of Shaqfa et al.[9]. We begin with the relevant properties of Bessel and modified Bessel functions and introduce the wavelet properties required for the construction. We then derive the Fourier-Bessel disk harmonics as solutions to the Helmholtz equation on the unit disk subject to a Neumann boundary condition. Building on this basis, we construct a wavelet family by applying a Gaussian spatial envelope and introducing a zero-mean correction for the zeroth angular order. We derive the corresponding normalisation constants for $L^2$-based applications and discuss $L^1$-based normalisation for frequency-domain peak consistency. Finally, we derive a closed-form Fourier-domain representation of the resulting wavelets. The main motivation is the approximately linear spacing, which converges to $π$ between consecutive radial eigenvalues. Rather than replacing the conventional dyadic organisation of wavelet families, this construction lays out the foundation to explore whether a more uniform radial frequency allocation can be useful for applications in which broad and balanced frequency coverage is desirable.

Abstract: Principal component analysis (PCA) is a fundamental tool to reduce the dimensionality of the data in many applications. PCA finds a few signal directions that contain most of the variability of the data by computing the eigenvectors of the sample covariance matrix. In this work, we focus on the spiked covariance model, in which the data vectors are defined by a few orthogonal signals plus an isotropic Gaussian noise, and our goal is to estimate one or more of the leading signals. Our main theoretical finding is that the subspace spanned by several leading eigenvectors of the sample covariance matrix contains significant information about the desired signals long before the individual eigenvectors converge to the population principal components. To prove this, we derive a posteriori bounds for the angle between the subspace spanned by the desired population signals and the subspace obtained from the sample using perturbation theory for singular vectors. This leads to a new algorithm, SuperPCA (SUbsPace subsamplER PCA), which capitalizes on an approximate eigenspace of the sample covariance matrix to find the leading signals far more efficiently and accurately than classical PCA in the high-dimensional, multi-signal setting. SuperPCA exploits only a small number of subsampled coordinates of the data, which can lead to tremendous savings in data acquisition cost, especially when the signals are approximately sparse. For the same number of measurements, SuperPCA can offer a factor $10$ improvement in accuracy compared to the classical PCA method.

Abstract: We construct an explicit quaternary Legendre pair of length $64$. To the best of our knowledge, this is the first reported construction at this length.

Abstract: We consider the recovery of low-multilinear-rank tensors from linear measurements and propose an adaptive block-weighted modewise Riemannian gradient descent method. The method combines memory-efficient modewise measurements with a normalized adaptive weighting strategy for the core and factor components of the Riemannian gradient. The weighting improves convergence without increasing the multilinear-rank bound of the search direction or the size of the reduced core used for retraction. Under the tensor restricted isometry property and a suitable initialization, we establish local linear convergence and derive sampling guarantees for sub-Gaussian and subsampled orthogonal with random sign (SORS) measurements. Numerical experiments on synthetic low-Tucker-rank tensors show that the proposed method reduces iteration counts and computational time while maintaining reliable recovery performance, especially near the recovery threshold and for structured SORS measurements.

Abstract: Uncovering latent variables and their causal relations from observed data is a fundamental yet challenging problem. Existing methods often rely on restrictive assumptions, such as linear relations or invertible mixing functions. To better address this problem under general nonlinear mixing procedures, we propose a condition called the cross-Hessian Rank Constraint (HRC), which serves as a primitive rank-based tool for nonlinear latent causal discovery. In particular, we show that a rank-based property arises from the cross-Hessian of the observed-data log-density in the nonlinear case, revealing information about the latent variables, and reduces to the Tetrad constraints in the linear Gaussian case. More specifically, when two groups of observed variables are d-separated by a set of lower-dimensional latent variables, the rank of this cross-Hessian is equal to the dimension of the latent variables, under a mild affine derivative assumption on the conditional log-density derivatives. This assumption can be naturally satisfied when the noise level is low or the relevant nonlinearity is moderate. As a downstream application, we instantiate HRC in the pure one-factor measurement setting for locating latent variables and recovering their causal structure up to Markov equivalence. Experimental results on synthetic and real-world datasets support the theoretical claims.

Abstract: Dictionary learning seeks to recover an unknown dictionary $A$ from observations ${\bf y}_i = A{\bf x}_i$ with sparse coefficient vectors ${\bf x}_i$. We introduce the \emph{Iterative Atom Refinement} (IAR) algorithm, a simple procedure for recovering individual dictionary atoms. Starting from a random direction, IAR repeatedly selects the observations most strongly correlated with the current iterate and updates the direction by averaging the selected data. Our main contribution is a rigorous convergence theory of IAR. Using high-dimensional probabilistic estimates and a novel monotonicity principle for atom-selection probabilities, we show that a small initial advantage of one atom is amplified until that atom is isolated. Under our model assumptions, IAR identifies a generating atom after only three refinement steps. Numerical experiments support the theory and show that the resulting dynamics accurately capture the behavior observed in dictionary refinement.

Abstract: Inferring network topology from noisy node observations is a central problem in graph signal processing. In this paper, we consider Laplacian-constrained graph estimation for Gaussian Markov random fields, focusing on the underdetermined regime in which the number of samples is smaller than the number of graph nodes. Existing approaches often formulate the problem as a sparsity-regularized maximum-likelihood estimation problem. While effective, such methods typically require iterative optimization and are often computationally demanding, particularly under Laplacian constraints. Instead, we propose a non-iterative estimator of graph Laplacians that uses effective resistance for regularization, and evaluate the method using a simple sparsification procedure. Experiments show that with some trade-off in edge and weight recovery on the considered dataset, computational cost for moderately sized graphs can be substantially reduced.

Abstract: In this second paper on descent methods as proof mechanisms in harmonic analysis (the first treated Bourgain's $Λ(p)$ selection theorem), entropic mirror descent gives an improved restricted isometry bound, improving sampling and recovery bounds in the Fourier Ratio program. Let $H$ be a complex $n\times n$ matrix with $H^*H=nI$ and $|H_{ij}|=1$. For fixed $0<\varepsilon<\frac14$ and $A_0>0$, independent Bernoulli row selection with expected cardinality $C_{\varepsilon,A_0}r\log\frac{2en}{r}\log(2r)\leq m\leq n/2$ preserves, after normalization by $\sqrt m$, the squared norm of every complex vector $y$ with $\|y\|_1\leq\sqrt r\|y\|_2$ within a factor $1\pm\varepsilon$, with failure probability at most $Cn^{-A_0}$. This includes every $r$-sparse vector and also fully supported ones; support size enters only through this inequality. For $y=\widehat f$ the condition is exactly $\operatorname{FR}(f)^2\leq r$, where $\operatorname{FR}(f)=\|\widehat f\|_1/\|\widehat f\|_2$, giving uniform energy sampling and approximate recovery even for signals with full Fourier support. The same holds for a uniform subset of prescribed cardinality, and for discrete Fourier and real Hadamard matrices. At fixed accuracy the bound removes one sparsity logarithm from the earlier count and replaces $\log n$ by $\log\frac{2en}{r}$; for Walsh matrices it matches, up to constants, the lower bound of Błasiok et al.\ in their range. One relative-entropy potential controls the corrections of an amplitude predictor at every scale; counting them on the symmetric difference of two samples gives the uniform estimate. We also prove the sufficient bound $C_{A_0}\varepsilon^{-5}r\log\frac{2en}{r}\log\frac{2r}{\varepsilon}$, and stable sparse recovery from $Cs\log\frac{2en}{s}\log(2s)$ measurements, with error controlled by the noise and the best $s$-term approximation.

Abstract: We give an elementary proof of the Komlós conjecture by simplifying the recent proof of Guo, Fang, and Lu. We show that any vectors $v_1,\ldots,v_n\in\mathbb{R}^d$ with $\|v_i\|_2\le1$ admit signs $\varepsilon_i\in\{-1,1\}$ such that $\|\sum_{i=1}^n\varepsilon_i v_i\|_\infty\le36$. The proof uses only elementary combinatorial and probabilistic arguments and basic calculus.

Abstract: Marcus, Spielman, and Srivastava [MSS15] established the existence of Kadison--Singer partitions. We provide polynomial-time algorithms for the Kadison--Singer problem. For Hermitian matrices $A_1,\ldots,A_m\in\mathbb C^{n\times n}$ of rank at most one, we give two algorithms that find signs $σ\in\{\pm1\}^m$ satisfying $\|\sum_i σ_iA_i\|\le C\|\sum_i A_i^2\|^{1/2}$. The deterministic algorithm achieves $C=3.3443$ using $\widetilde O(mn^2+n^{56})$ arithmetic operations. The randomized algorithm achieves $C=4.8628$ using $\widetilde O(mn^2+n^{5.88})$ arithmetic operations in expectation. For vectors satisfying $\sum_i a_ia_i^*=I$ and $\|a_i\|^2\leα$, the algorithms yield partitions $[m]=I_1\cup I_2$ satisfying $\|\sum_{i\in I_j}a_ia_i^*-I/2\|\le (C/2)\sqrtα$ for $j=1,2$.

Abstract: We study machine-learning detection of controlled beyond-General-Relativity (beyond-GR) deviations in gravitational-wave signals, using both synthetic aLIGO-PSD noise and real LIGO H1 detector strain. Three deviation families are applied to General-Relativistic inspiral-merger-ringdown waveforms: amplitude modulation, phase modulation, and frequency modulation, each parameterized by a dimensionless strength coefficient $β$. A hybrid classifier combining a one-dimensional convolutional neural network with ten hand-crafted waveform statistics is trained on GR and modified waveforms and tested on a deviation type excluded from training. The central result is a quantitative detectability curve as a function of $β$. Using the real GW150914 strain as a template and real H1 detector noise, we find a detection threshold at $β\approx 0.25$, with accuracy rising smoothly from chance at $β\leq 0.2$ to perfect classification at $β\geq 0.5$. The threshold value is specific to the quadratic-in-time modulation form adopted here and should not be interpreted as a generic constraint on beyond-GR parameters. We nevertheless argue that the negative result at small $β$ is informative: it establishes a quantitative limit on machine-learning-only beyond-GR searches in real detector noise, in the absence of matched-filter signal extraction.

Abstract: Time-series language models provide a shared natural-language interface across temporal tasks, but plausible text does not guarantee reliable task outputs. Responses may appear reasonable while hallucinating the required object: numerical sequences can violate shape, scale, channel order, or temporal alignment, and textual decisions can fall outside the legal label space. We formulate reliable time-series language modeling, separating task-object reliability from predictive quality. We introduce ExecTS-QA, a contract-grounded benchmark spanning forecasting, imputation, classification, anomaly detection, and waveform analysis. We further propose WaveTLM, a unified compiler-executor model whose task compiler transforms user requests, visible arguments, and wave-grounded evidence into typed task states, while task-native executors construct numerical tensors, legal decisions, or structured records. On ExecTS-QA, a single WaveTLM checkpoint achieves 99.40% contract-valid coverage, compared with 37.83% for the strongest evaluated string-first baseline, while retaining balanced predictive performance across all five task families. Evaluations on SciTS, TSQA, IRTS-ToolBench, and ARFBench provide additional evidence of transfer. The code, construction scripts, and ExecTS-QA dataset will be publicly released upon publication. These results show that task compilation can convert plausible language generation into reliable time-series outputs.

Abstract: We study the expected number of zeros of the Short-Time Fourier Transform (STFT) with a Gaussian window for signals degraded by complex Gaussian colored noise. We provide an exact formula for this quantity and investigate its asymptotic behavior. From a computational perspective, we propose an algorithmic method to recover a smoothed power spectral density (PSD) directly from the spatial distribution of the transform's zeros. Our results formalize the commonly-held heuristic that detection algorithms based on spectrogram zeros, though designed for white noise, also perform adequately under moderately colored noise.

Abstract: We give a deterministic polynomial-time algorithm that, given rational Hermitian matrices $H_1,\dots,H_N$ of rank at most one, finds signs $s\in\{\pm1\}^N$ with $\|\sum_i s_i H_i\|\le 13\|\sum_i H_i^2\|^{1/2}$. As a corollary, for vectors $v_i$ with $\sum_i v_iv_i^*=I$ and $\|v_i\|^2\leδ$, the signs yield a partition $[N] = S_1 \cup S_2$ such that each part satisfies $\|\sum_{i \in S_j} v_i v_i^* - \frac{I}{2}\| \leq \frac{13}{2}\sqrtδ$ for $j = 1,2$. This gives a deterministic polynomial-time algorithm for the Kadison--Singer problem, in Weaver's equivalent discrepancy-theoretic $\mathsf{KS}_2$ formulation, with a universal constant.

Abstract: A finite structural characterization of the $U$-user multiuser channel-capacity region appears here. This region relies upon three concepts: (i) subset-based message atomization, (ii) synchronized receiver chain-rule/Fano reduction, and (iii) successive-decoding achievability via a finite-super-symbol closure. The resulting region is a finite union of order-indexed polytopes parameterized by the synchronized minimum mutual-information vector $\mathcal{I}_{\min}$. The framework removes auxiliary-random-variable proliferation and makes explicit the finite geometric structure underlying general multiuser converses, with the interference channel providing the primary illustration. For the linear matrix Gaussian multiuser-channel special case under per-user trace covariance constraints, Gaussian signaling is capacity-region-achieving within this framework. Also shown is that maximum rate sum for any Gaussian multiple-user-channel derives from simple iterative procedures.

Abstract: Regularization choice changes the recovery phase diagrams of annealing-based binary compressed sensing. We develop a regularization-selection method that combines systematic parameter search with random forest regression. Under noiseless Gaussian measurements with known sparsity, reference parameters are selected from a candidate grid by minimizing mean squared reconstruction error over repeated simulated annealing (SA) trials. The fitted model predicts these reference values from signal dimension, sampling ratio, and sparsity. With predicted regularization, the SA recovery transition broadly follows the asymptotic reference boundary for box-constrained $\ell_1$ recovery at the larger signal dimensions examined. Without retraining, the same predictor supplies identical regularization values to SA and a quantum--classical hybrid solver. On matched problem instances, the hybrid solver yields smaller mean squared reconstruction errors than SA in parts of the evaluated parameter space. The resulting rule reuses the searched information for subsequent reconstruction without repeating candidate searches at each setting. The results quantify empirical performance under the stated finite candidate grid and solver settings; they do not constitute a solver-independent recovery guarantee or a time-to-solution comparison.

Abstract: Warped Gaussian processes (GPs) handle non-Gaussian observations by mapping them into a latent standard GP via a parametric transformation called warping. Existing streaming variants, however, either optimize the warping parameters periodically or sacrifice analytical tractability for a higher model capacity. To bridge this gap, we show that the gradient of the instantaneous negative log-likelihood of a warped GP admits an exact recursive computation. Based on this result, we propose a novel online method for warped GPs that jointly updates the latent GP moments and optimizes the warping parameters.

Abstract: The Komlós conjecture is a classic problem in discrepancy theory; it asks whether an absolute constant $K$ exists such that given any $n$ vectors $a_1,\ldots,a_n$ inside the $m$-dimensional Euclidean ball, regardless of how large $m,n$ are, there is always a selection of signs $\varepsilon_1,\ldots,\varepsilon_n$ guaranteeing $$\|\varepsilon_1a_1+\ldots+\varepsilon_na_n\|_\infty \leq K.$$ We show that if the $\varepsilon_i$'s are allowed to take not just the values of $\pm 1$ but any unit modulus complex number, which we refer to as complex discrepancy, then the above inequality holds for a finite, explicit constant $K_{\mathbb{C}}$. Here, the $\ell^\infty$ norm of the resulting vector in $\mathbb{C}^m$ is the largest modulus of its entries, and thus the complex discrepancy of real vectors is equivalent to their rank-$2$ vector discrepancy. Therefore, our result resolves the Komlós problem for Gaussian discrepancy -- a discrepancy measure introduced by Chewi, Gerber, Rigollet and Turner. Our paper builds upon the recent work of Bansal and Jiang on the Beck-Fiala and Komlós conjectures, which we approach from the formalism of Burkholder and the Bellman function method from probability and harmonic analysis. Our work was in part motivated by the realization that the complex discrepancy of the columns of any unitary matrix is equal to 1, a fact that follows from a straightforward calculation based on Idel and Wolf's generalization of the Sinkhorn normal form for unitary matrices.

Abstract: Let $Y_s=\sqrt{s}X+Z$, where $Z$ is standard Gaussian and independent of the real random variable $X$. We prove that, under the square-exponential moment condition $\mathbb{E}e^{βX^2}<\infty$ for some $β>0$, the scalar minimum mean-square error $\operatorname{mmse}_X(s)$ is analytic at zero signal-to-noise ratio if and only if $X$ is Gaussian, with constant random variables included as degenerate Gaussians. The proof converts estimation in the Gaussian channel into a backward heat flow acting on the moment-generating function $M(z)=\mathbb{E}e^{zX}$. Under the stated tail condition, every non-Gaussian input forces $M$ to have a nonzero complex zero. We show that each zero cluster produces a finite singularity in its localized Borel transform at the action $ξ=z_0^2/2$. After removing the action scale, the Borel coefficients have a nonzero $n^{-1/2}$ prefactor for a simple zero. A zero of multiplicity $m\geq 2$ splits according to the roots of a Hermite polynomial and instead contributes a prefactor $n^{-m/2}e^{r_m\sqrt{2n}}$. A finite-disc localization and relative-cycle continuation argument then show that at least one such singularity survives in the full Borel transform. Thus, for every non-Gaussian input in the stated class, the formal zero-SNR expansion is Gevrey-1 but divergent. Rational-MMSE rigidity and the analogous analyticity criterion for mutual information follow as corollaries.

Abstract: The Fisher information matrix (FIM) determines the accuracy attainable in an estimation problem. For a parametric process with memory, it has a closed form only for narrow classes of models, and it cannot be computed when the model is available only as a simulator. This paper estimates the Fisher information rate of such a process, the per-observation limit of its FIM, from simulator output alone and without knowledge of the memory length. The construction uses the identity between the FIM and the local curvature of the Kullback--Leibler (KL) divergence. Directional divergence rates are estimated by context-tree weighting (CTW), and the matrix is recovered from them by least squares. We establish strong consistency in an iterated limit and a finite-sample bound, uniform over every working depth at least as large as the true memory length, which gives a mean error of order $n^{-1/3}$. The analysis also delivers a finite-sample rate for CTW-based estimation of the KL divergence rate. Experiments confirm the predicted exponents, show that the accuracy does not degrade as the working depth grows, and use the estimate to plan a sampling budget that attains a prescribed precision. The estimator stays accurate on a hidden-state source that has no finite memory length.

Abstract: Text-to-speech (TTS) models commonly address text--speech alignment by expanding phone-level encoder states to frame-level decoder inputs using predicted durations. While this length-regulation step resolves alignment structurally, this use of duration typically changes only where and how often latent states appear, not the values of the states themselves. This paper proposes a continuous-time mechanism for duration-aware acoustic modelling in TTS using neural controlled differential equations (CDEs). We formulate the phone representation as a temporally parameterised control path and use a neural acoustic vector field to produce a continuous-time hidden state whose values evolve with phonetic content and duration-derived timing. The resulting trajectory can be sampled at discrete points and integrated into a standard acoustic decoder pipeline. Objective results contrast CDEs and typical recurrent models. Subjective results suggest that CDE-based models evaluating one phone per step can improve rank-order agreement between synthesised and reference emotion intensity while maintaining comparable emotion-expression quality to a strong baseline. Additional experiments with half-phone step-sizes suggest that temporal resolution changes the trade-off between style tracking and absolute calibration. These results position CDEs as a promising design space for continuous-time and duration-aware style-sensitive TTS.

Abstract: Identifiability of tensor decompositions is often established through linear-algebraic conditions on the factor families. For nonnegative decompositions, however, positivity provides additional information that is not captured by dimension and independence alone: nonnegative terms cannot cancel, and their supports constrain competing decompositions. We introduce a positive scattering term that quantifies this additional source of identifiability and combine it with the dimension budget underlying the Lovitz--Petrov generalization of Kruskal's theorem. For every subset of components, we obtain two sufficient conditions: a threshold of $2|S|-2$ guarantees minimality and nonnegative rank, while the stronger threshold $2|S|-1$ guarantees uniqueness among nonnegative decompositions of the same length. The key result is a positive splitting inequality for irreducible exchanges of nonnegative rank-one tensors, which combines the dimension constraint with support-induced geometric rigidity. Although the scattering term is defined through an optimization over intermediate factor spaces, we show that its mode costs are exactly $0$, $1$, or $+\infty$, yielding an exact activation characterization in terms of graph connectivity. The resulting criterion can strictly certify sparse nonnegative tensor decompositions beyond the reach of Kruskal and Lovitz--Petrov conditions, including examples for which those conditions fail even after reshaping. In the matrix case, the two criteria reduce respectively to full-rank factorization and two-sided separability.

Abstract: The Ball-Nayar-Tkocz entropy concavity conjecture asserts that, if $X,Y$ are independent identically distributed real random variables with a common log-concave density, then the differential entropy of their weighted sum, \[ F(t)=h\bigl(\sqrt{1-t}\,X+\sqrt t\,Y\bigr),\qquad 0\le t\le1, \] is a concave function of the weight parameter $t$. We construct an asymmetric, strictly positive smooth probability density $f$ with mean zero, variance one, and $(\log f)''<-1/2$, for which the corresponding function satisfies $F''(t)>0$ throughout an endpoint neighborhood $0<t<δ$. This disproves the conjecture without an additional symmetry assumption. For a class of Gaussian perturbation densities, we first establish the endpoint expansion \[ F''(t)=-\frac{μ_3J_3}{16\sqrt t}+O(1),\qquad t\downarrow0. \] Here $μ_3=\int x^3f(x)\,dx$ is the third central moment. Writing $ρ=(\log f)'$ for the score function, its third moment is $J_3=\int f(x)ρ(x)^3\,dx$. A Hermite perturbation gives $μ_3>0$ and $J_3<0$, and explicit remainder estimates verify the constructed density and its endpoint curvature. The counterexample does not address the conjecture with an additional symmetry assumption.

Abstract: In the last decade, kilometre-scale interferometric gravitational-wave detectors have observed hundreds of compact binary mergers, the majority of which are binary black holes. However, the data are noise-dominated, and multiple independent search algorithms (pipelines) are used to enhance sensitivity and improve robustness. Rather than the standard approach of selecting the most significant pipeline output, we combine the outputs from all pipelines using a conformal prediction-based framework to provide statistically rigorous confidence estimates for candidate events. While combining pipelines improves sensitivity and ranking robustness, it requires a principled statistical framework that remains valid as data properties evolve across observing runs. A key challenge is distribution shifts between simulated datasets used for training and calibration and the real, unlabelled, observations used for testing, which can invalidate coverage guarantees and bias confidence estimates. In this work, we address this challenge by incorporating likelihood-ratio reweighting into our conformal prediction framework to account for covariate shift. Using mock datasets containing simulated signals, we demonstrate that weighted conformal prediction restores well-calibrated coverage under covariate shift and increases the confidence of events near the detection threshold, recovering true signals that would otherwise be missed.

Abstract: Minimum-norm confidence envelope strategies offer a nonparametric approach to constructing nonasymptotic, simultaneous confidence regions for band-limited functions, exploiting the theory of Reproducing Kernel Hilbert Spaces (RKHS). While the finite-sample coverage guarantees of these envelopes have been established, their consistency has not been analyzed so far. In this paper, we study this construction, here termed the Minimum-Norm Confidence Envelope (MiNCE) framework, and establish the strong uniform consistency of the resulting bands, both for noise-free and noisy observation models, under mild assumptions on the measurement noises. We further extend this formulation to the frequency domain, deriving nonasymptotic, simultaneous, strongly uniformly consistent confidence bands for the smoothed spectra. Numerical experiments in nonparametric regression and spectral estimation empirically confirm our theoretical results, illustrating the contraction of the confidence envelopes toward the target function as the sample size increases.

Abstract: Quantization schemes based on randomized rotations have recently received renewed attention, including the roles of MMSE and unbiased reconstruction scalings. In this note, we point out the connection to classical results in statistical signal processing and communication theory. Specifically, the two reconstruction scales used in the EDEN line of work admit a natural interpretation as finite-dimensional, realization-dependent counterparts of the Wiener and unbiased coefficients in the classical CDEF formulation. At finite blocklength, the CDEF +1 relation holds pointwise for each rotation realization as an exact geometric (Pythagorean) identity, but does not hold after averaging the distortions over the rotation. The classical SNR relation $\sf{SNR}_{\rm MMSE}=\sf{SNR}_{\rm MMSE,U}+1$ is recovered as $d\to\infty$: once the overall scale is handled separately, the empirical coordinate statistics of a randomly rotated vector approach their i.i.d. Gaussian counterparts, and the rotation-dependent quantities concentrate. Importantly, EDEN goes beyond this classical correspondence: for every finite $d$, its Haar-rotation formulation guarantees exact conditional unbiasedness, a stronger property than the second-order notion of unbiasedness in CDEF. We further comment on two distinct roles random rotations play in quantization: one is approximate Gaussianization of the coordinates; the other is decorrelation of reconstruction errors across quantization branches.

Abstract: The phase retrieval problem involves reconstructing a function or signal solely from the magnitude of linear measurements. Most theoretical analyses of phase retrieval algorithms rely on i.i.d. Gaussian random measurements or sub-Gaussian random measurements. In this paper, our focus is on the fusion frame phase retrieval problem, where the sampling matrices are i.i.d. rank-$r$ orthogonal projections drawn from the Haar measure. We present concentration inequalities for functions on the set of rank-$r$ orthogonal projection matrices. These inequalities are crucial for the theoretical analysis of the fusion frame phase retrieval problem. Based on these inequalities, we demonstrate that gradient descent, combined with a two-stage initialization, achieves linear convergence to the target signal up to a global phase with a measurement complexity of $O(d\log^2 d)$ when the rank $r = O(1)$. We verify this convergence through numerical results.

Abstract: We establish conditional central limit theorems in Fisher information under log-concavity in every fixed dimension. For conditionally centered normalized sums, after whitening by the averaged conditional covariance, the averaged conditional Fisher information converges to the dimension if and only if it is finite at one convolution level. The scalar criterion follows as the one-dimensional case; we also provide an independent scalar proof based on a second-order continuity theorem for Fisher production on Gaussian-smoothed, tail-controlled classes. For the original sums, the averaged Fisher information matrix converges in operator norm to the inverse averaged conditional covariance. Consequently, the conditional relative Fisher information with respect to the limiting Gaussian law vanishes, and the Gaussian logarithmic Sobolev inequality yields convergence in conditional relative entropy and conditional entropy. We give two operational consequences. For any fixed finite-constellation low-power input, the first-order conditional mutual-information slope converges to the Gaussian-noise benchmark. For Gaussian signaling at any fixed signal covariance, the mutual-information gap from that benchmark is bounded by the conditional relative Fisher deficit and hence vanishes asymptotically.

Abstract: In the recent work of Barbier, Ko, and the second present author on sublinear-rank symmetric matrix factorization [Math. Stat. Learn. 9 (2026), 1-68], a key result is that, in the Bayes-optimal setting, the large-size limit of the free entropy of the finite-rank spiked Wigner model is the same as in the rank-one case when the signal has centered i.i.d. entries. In this paper, we show that this rank-one equivalence result extends to the case of finite-rank, even-order, symmetric tensor factorization. Moreover, we give a natural reformulation of a hypothesis that was stated in the aforementioned work to be necessary for this result. As in the matrix case, we use information-theoretic identities and replica symmetry to reduce a known multi-dimensional variational formula for the limiting free entropy to its one-dimensional analog. The novelty stems from the fact that said formula involves a replica symmetric potential containing Hadamard (entrywise) powers, rather than squares, of the matrix-valued variational parameter, so the eigenvalue-based approach used in the matrix case must be adjusted.