signal processing developers

Abstract: We develop a unified framework for the instantaneous total level crossing rate (LCR) of products of independent positive nonstationary fading processes. Universal upper and lower bounds are first derived from an exact decomposition of the product derivative and signed crossing imbalance, requiring only factor-wise crossing statistics. We then introduce the drifting Gaussian-derivative (DGD) class, in which each conditional derivative is Gaussian with possibly nonzero drift. For products of DGD factors, an exact Gaussian-mixture representation yields bounds that jointly capture aggregate drift and derivative fluctuations. The lower bound reduces under stationarity to a known $\ell_2$ bound, whereas, to the best of the author's knowledge, the upper bound is new even in the stationary case. Furthermore, the bounds are asymptotically exact in drift-dominated regimes. The framework is specialized to nonstationary Double-Rayleigh fading and to quasi-identically distributed DGD factors, with closed forms for products of $α$-$μ$ envelopes. Numerical results show that the DGD bounds closely bracket the exact LCR.

Abstract: Temporal Predictive Coding (tPC) networks offer a layer-local alternative to backpropagation. For linear tPC, we show, by applying tools from cellular sheaf theory, that temporal credit assignment is a tug-of-war between spatial displacement and temporal errors, whose ratios are modulated by persistence and exposure operators $P$ and $Q$. Spatial inference therefore gives us the effective recurrent transition $F=PA$, whilst future credit is filtered by $P^\ast$. This creates a persistence--error tradeoff in spatial modes: under non-amplifying invariant recurrence, modes that preserve influence over long delays simultaneously fail to expose that influence strongly as temporal error. Recurrence can circumvent this via routing between persistent and exposed modes.

Abstract: Let $μ$ be a log-concave isotropic probability measure on $R^d$ , and let $μ^n$ denote the law of the normalized sum of $n$ independent random vectors with law $μ$. We prove that \[ H(μ^n|γ)\leq \frac{C}{n} \min\left\{1+\log(ed)^2H(μ|γ)\ ,\ d\right\}. \] Compared to the previous best known bound, our estimate replaces the polynomial dependence on the dimension by a polylogarithmic one. The proof relies on a new combination of the De Bruijn identity with Eldan's stochastic localization process.

Abstract: We construct a continuous function on $[-1,1]^2$ for which the infimum of the $L^2$-approximation error over the class of ReLU networks with two hidden layers and architecture $(2,2,1,1)$ is not attained. This counterexample shows that deep ReLU networks do not satisfy the best approximation property in general, even for continuous target functions. Our findings contrast with existing results for shallow neural networks, where ReLU is one of the few activation functions that guarantee the existence of global minima in typical regression tasks. The proof is based on the fact that the breaklines for neurons in the second hidden layer can bend, which breaks the symmetry arguments used in the one-hidden-layer case.

Abstract: Using the Hadamard matrix for dimension reduction can accelerate the computation due to the fast Hadamard transform. However, the previous target dimension bounds in terms of the Gaussian width depend on the ambient dimension. In this paper, we obtain the following ambient dimension free bound using three Hadamard-diagonal blocks. Let $H\in\mathbb R^{d\times d}$ be the normalized Walsh-Hadamard matrix, and let $D_1,D_2,D_3\in\mathbb R^{d\times d}$ be independent Rademacher diagonal matrices. We prove that for any symmetric set $T\subset \mathbb{S}^{d-1}$, \[ \mathbb P\left\{\sup_{\boldsymbol t\in T}\big|\frac dm\|P_ΩHD_3HD_2HD_1\boldsymbol t\|_2^2-1\big|\le\varepsilon\right\} \ge1-δ\] whenever \[ m\gtrsim\varepsilon^{-2}w(T)^2\log^2\frac eδ\,\log^4\!\left(e+\frac{w(T)}\varepsilon\log\frac eδ\right). \] Here $w(T)$ is the Gaussian width of $T$, and $Ω$ is a set of $m$ distinct coordinates chosen uniformly. This is the first ambient dimension free bound on the target dimension using Hadamard transforms, for an arbitrary symmetric set in terms of its Gaussian width. We also prove that it is impossible to obtain an ambient dimension free bound for an arbitrary set in terms of its Gaussian width using two Hadamard-diagonal blocks.

Abstract: Neural networks whose inputs are functions, as in operator learning, are usually shown to be universal by existence arguments, either by reduction to finite-dimensional networks or by Stone--Weierstrass-type density; these prove that approximants exist but say little about how the parameters depend on the target. We take a constructive route. We extend the ridgelet transform, an integral transform related to the Fourier, Radon, and wavelet transforms, to continuous maps on a separable infinite-dimensional Hilbert space, replacing the Lebesgue measure that is unavailable in infinite dimensions by Gaussian weights with a homogeneous mixture on the direction space. The resulting transform has a Plancherel identity and reconstruction formulas incorporating the frame operator, the correction induced by the weights, and it assigns to each target an explicit coefficient measure, one for every admissible analysis filter. From these representations we obtain uniform approximation on compact sets for all continuous non-polynomial activations of polynomial growth, including ReLU, and, by Monte Carlo sampling of the coefficient, finite-width networks with a dimension-free $O(N^{-1/2})$ error in the width $N$ for globally Lipschitz activations and coefficients with finite variation and second moments. Explicit examples include Gaussian targets and neural-operator layers. A companion Lean~4 formalization provides machine-checked proofs, with its correspondence to the manuscript documented at https://shosonoda.github.io/lean-operator-ridgelet/ .

Abstract: The Long-Range Ising Model benchmark (LRIM) guarantees long-range dependence by construction: each label is the node's spin times the spin field convolved with a power-law coupling, so any model's distance from the target is measurable. We prove that a readout whose filter is a polynomial in the adjacency, the linear core of isotropic message passing and spectral graph networks, leaves an error floor that no degree lowers, because the target varies along a level set of the spectrum on which every such filter is constant. Computed exactly, the floor and the best filter of each degree predict two spectral networks adapted from published architectures to within a tenth of a decade. A filter escapes the floor only if it weighs separately the offsets the adjacency lumps together, and must still resolve the coupling across the whole lattice. A strided filter of about fifty coefficients does both: fitted to the training labels, its test error is 52 times below the best published network's. A dilated convolutional network beats the best published result at 256 nodes, matches it at 1,024, and gives the first in-domain results at the three larger sizes, while a Fourier neural operator with its standard modes falls further behind as the lattice grows. At its standard budget the dilated network stays two to four orders of magnitude short of a larger strided filter fitted to the labels. The analysis carries over to the benchmark's second coupling.

Abstract: Neural networks often learn low-frequency components before high-frequency ones --- a phenomenon called spectral bias or termed F-principle. Many numerical studies have demonstrated this phenomenon, and theoretical studies have sought to explain it. However, several goals remain insufficiently addressed: (1) explaining spectral bias asymptotically, including its origin and why the preferred components are generally sinusoidal; (2) extending the explanation to general activation functions, such as sigmoid and ReLU; (3) making weak assumptions about the data distribution; (4) avoiding aliasing effects, such as those arising in finite-domain Fourier analysis; and (5) developing a theoretical framework applicable to arbitrary networks. We address these goals theoretically by analyzing the eigenvalue decay and eigenvector shapes of the dynamics matrix H, which arises when we expand the network output as a general Fourier series (GFS) in an orthogonal polynomial basis and express the gradient-descent dynamics of the GFS coefficients as ordinary differential equations. The graded structure of H enables an intuitive explanation of spectral bias, while orthogonal polynomial theory helps us analyze how learning bias depends on the activation function and, potentially, the network architecture.

Abstract: We present several advances on neural operators by viewing the action of operator layers as the minimizers of Bregman regularized optimization problems over Banach function spaces. The proposed framework allows interpreting the activation operators as Bregman proximity operators from dual to primal space. This novel viewpoint is general enough to recover classical neural operators as well as a new variant, coined Bregman neural operators, which includes the inverse activation operator and features the same expressivity of standard neural operators. Numerical experiments support the added benefits of the Bregman variant of Fourier neural operators for training deeper and more accurate models.

Abstract: The Zak-OTFS carrier waveform is a pulse in the delay-Doppler (DD) domain, and given the input-output (I/O) relation at one DD point in a frame, we can predict the I/O relation at all other points. While it is simple to acquire the effective wireless channel with Zak-OTFS, the cost of embracing inter-carrier interference is equalization complexity. The cost of linear minimum mean-square error (LMMSE) equalization grows cubically with the frame size. We describe how to use channel knowledge to reduce equalization complexity. For any pulse shaping filter, and for fractional delays and Dopplers, we show that the DD channel matrix is unitarily similar, through the discrete Zak transform, to a time-domain (TD) matrix that is a weighted sum of cyclic delay and Doppler operators, where the weights are the taps of the effective channel. We also obtain a dual frequency-domain (FD) form through an additional Fourier transform. The TD matrix is banded with a width set by the delay spread that is independent of the Doppler spread, whereas the FD matrix is banded with a width set only by the Doppler spread. We design a receiver that reads off the two widths from the channel estimate, selects the cheaper domain, and solves the banded system exactly, using a bordered Cholesky factorization that treats the quasi-periodic wrap-around without sacrificing degrees of freedom. Filters that decay slowly need a few preconditioned conjugate gradient steps. We show that the complexity of equalization becomes linear in the frame size. Simulations match direct DD-LMMSE error rates with three to four orders of magnitude fewer multiplications. A single receiver handles channels with very different delay and Doppler characteristics.

Abstract: We study the mean-squared error (MSE) rate-distortion function (RDF) of finite Gaussian mixture models (GMMs) in $\mathbb{R}^d$, for which neither the RDF nor an optimal test channel is known in closed form. It can be sandwiched between the genie-aided conditional RDF $R_{\mathbf{X} | T}(D)$, which can be characterized in closed-form for a GMM, and $R_{\mathbf{X} | T}(D)+I(\mathbf{X};T)$, where $T$ denotes the latent component label. We show that the upper bound can be achieved operationally by the proposed Gibbs-relabelling method (GRM), which conveys a label from the posterior $P_{T|\mathbf{X}}$ through \textit{channel simulation} at rate $I(\mathbf{X};T)$, followed by vector quantization of the resulting exactly Gaussian conditional sources by reverse water-filling. Using the Shannon lower bound (SLB), we prove that GRM attains the SLB, and hence the RDF, of a general finite mixture if and only if every component attains its own SLB. For a GMM this holds throughout the fully active interval $D/d<\min_{k,m}λ_{k,m}$, where $λ_{k,m}$ are the eigenvalues of the component covariances, and for scalar GMMs the condition is also necessary. Since GRM cannot operate below the label cost $I(\mathbf{X};T)$, we propose a complementary stochastic scalar quantization (SSQ) scheme for low rates, which simulates an $M$-ary reconstruction index. We show that at low rates SSQ with $M\geq 2$ matches the Shannon RDF to first order: $R(D_{\max}-δ)=δ/(2τ^2)+o(δ)$, where $τ^2$ is a directional sub-Gaussian proxy of the mixture, and quantify the price of a finite codebook through a strict rate gap at high rates and a distortion floor whose joint scaling in $M$ and number of mixture components is characterized. Numerical results illustrate both regimes and the finite-blocklength cost of channel simulation.

Abstract: Rydberg atomic quantum receivers (RAQR) exhibit exceptional sensitivity to radio-frequency (RF) signals, positioning them as promising candidates for next-generation wireless systems. Extending their capabilities to orthogonal frequency-division multiplexing (OFDM), however, requires atomic responses that go beyond broadband reception and are tailored to the structured interference of multicarrier signals. To bridge this gap, we develop an atomic-comb Rydberg receiver based on six-wave mixing (SWM) transduction. In particular, injecting a multi-tone local oscillator (LO) into the RAQR generates frequency-shifted replicas of the intrinsic SWM response. By subtly tuning the amplitude and phase of each LO tone, constructive and destructive interference can be selectively induced to preserve desired response components while attenuating harmful frequency offsets. Furthermore, we harness the response-reshaping capability of the proposed atomic comb to proactively mitigate OFDM interference. More explicitly, by exploiting the discrete difference-frequency structure of both carrier-frequency-offset (CFO)-induced and phase-noise (PN)-induced inter-carrier interference (ICI), we optimize the atomic-comb response to suppress dominant ICI components directly. We further reveal an intrinsic coupling between spectral shaping and the time-domain effective channel memory, motivating a joint optimization of both ICI and inter-symbol interference (ISI), which simultaneously suppresses inter-carrier leakage and post-cyclic-prefix (CP) tail energy. Extensive simulation results validate our atomic-comb design, unveiling that the proposed atomic comb can effectively reshape the receiver response over the OFDM, while mitigating ICI, shortening the effective channel memory, and substantially lowering CP overhead.

Abstract: Neural networks trained on modular addition tasks often develop Fourier-structured representations that support exact generalization. While prior work has identified these Fourier circuits, the mechanism by which gradient-based training selects them from the data distribution remains unclear. We address this question using probability signatures, which express leading gradient interactions through conditional statistics of the training distribution. For modular addition, these signatures are cyclic shift operators and are diagonalized by the discrete Fourier transform, yielding approximately decoupled Fourier-mode dynamics. This explains the emergence of Fourier sparsity, frequency matching, and phase alignment. The same framework resolves a puzzle under label noise: corrupted examples can show faster early loss decrease than clean examples, despite lacking a coherent generalization rule. We show that noise increases conditional label collisions, strengthening early shared-coordinate reinforcement. Finally, this method can be applied to other operators. Taking XOR as an example, we observed the predicted frequency in experiments.

Abstract: We introduce a novel second-order optimization framework for minimizing so-called variable projection functionals. We demonstrate that the proposed framework is especially usefulfor the training of variable projection based kernel methods. In particular, the problem of efficiently training variable projection support vector machines (VP-SVMs) is considered. We show the effectiveness of the proposed training methodology in a real-world application, namely we demonstrate how second-order trust region algorithms can be used to train VPSVM models to recognize road surface abnormalities based on 1D signals obtained from a tire sensor.

Abstract: State Space Models (SSMs) offer an efficient alternative to Transformers for sequence modeling, yet conditioning pre-trained SSMs for iterative generation typically operates outside the recurrent operator, through input injection or activation modulation. While such mechanisms expose the model to conditioning information, they leave the underlying temporal dynamics fixed. We introduce MaRK (Markov-adapted Recurrent Kernels), a dynamic operator-conditioning framework that maps context vectors directly into bounded modulations of a frozen SSM's recurrence ($A$), read-in ($B$), read-out ($C$), skip ($D$), and discretization ($Δ$) parameters. Viewed through the lens of LPV-SSM systems, MaRK induces a context-indexed family of Markov parameter sequences, allowing each diffusion timestep to reshape the model's input-output memory kernel. We instantiate MaRK on a frozen 111M-parameter Hydra SSM backbone and study three adapter geometries: Hypernet, Chebyshev polynomial, and Discrete Cosine Transform kernels. Since these adapters modify the Markov parameter sequence through low-rank auxiliary maps on the frozen backbone, parameter-efficient fine-tuning arises as a structural consequence of the adaptation mechanism itself, requiring only 6.3--11M trainable auxiliary parameters to transition from a bidirectional objective to an iterative diffusion regime. The bounded recurrence parameterization further yields an analytic Affine Quadratic Stability certificate for the modulated recurrence. Through synthetic LPV recovery experiments and Markov-operator diagnostics, we show that MaRK recovers coordinate-invariant temporal operators under matched assumptions and produces distinct, stable timestep-conditioned memory profiles. Empirically, the Chebyshev variant yields the strongest performance, achieving an average validation loss of 2.55, followed by the DCT (2.59) and Hypernet (3.77) geometries.

Abstract: We present an exposition of an argument, discovered by GPT-6 Pro, that gives an optimal bound for the polynomial Littlewood-Offord problem. Namely, let $F$ be a degree-$d$ multilinear polynomial that contains $r$ degree-$d$ monomials involving disjoint sets of variables. Then, for i.i.d. Rademacher random variables $ξ_1, \ldots, ξ_n$, we have $\mathbb{P}[F(ξ_1, \ldots, ξ_n) = 0] = O_d(r^{-1/2})$. This improves upon the previous bound of $(\log r)^{O_d(1)} r^{-1/2}$ due to Meka, O. Nguyen, and Vu, and resolves a conjecture attributed to H. Nguyen and Vu. The key part of the proof is an estimate for the total influence of bounded-degree rational functions, which resolves a recent conjecture of Kothari, Kovacs-Deak, Wang, and Yang.

Abstract: We present an efficient quantum algorithm for the non-uniform Chebyshev transform. It is defined as the projection of a function onto Chebyshev polynomials sampled at given nodes that are uniform in $x\in[-1,1]$, and hence non-uniform in the angle $θ=\arccos x$, a setting that QFT-based quantum Chebyshev transforms cannot handle. Our construction is based on an improvement of an existing Non-uniform Quantum Fourier Transform (NUQFT) whereby we remove the conditioning from non-uniform node sampling. Hence, error bounds are independent of the geometry-dependent parameter $κ$ of prior work. We use the fact that Chebyshev transform matrix is the average of two Type-II non-uniform DFTs, which we implement with a single controlled NUQFT circuit. We provide explicit oracle constructions, including the row-access oracle previously left as an assumption. The resulting $\varepsilon$-accurate block encoding has $O(1)$ normalization and uses $O(L)$ qubits and $\widetilde O(L^2)$ gates, where $L=\log N+\log(1/\varepsilon)$. We give an end-to-end implementation with complexity analysis, including the success probability and output-state error.

Abstract: Optimal Transport (OT) provides a principled framework for learning transformations between probability distributions from unpaired samples. In many applications, however, a single transformation must map several source distributions to a common target distribution. For example, image restoration might require handling different types of degradation without knowing the degradation of each input at inference time. Simple approaches of pooling the source distributions only encourage alignment with the target at the aggregate level and may leave individual sources misaligned. In our paper, we consider the simultaneous OT problem which formalizes the task of learning a shared transport map that minimizes the average transport cost while aligning each source distribution with a prescribed target. We propose a neural method for solving the simultaneous OT problem by learning a shared transport map that minimizes the average transport cost while aligning each source distribution with a prescribed target. We derive a max-min formulation for learning this map. We illustrate its application to image restoration, where a single model handles multiple degradation types using a common collection of clean target images.

Abstract: State space models (SSMs) achieve efficient sequence processing because their affine state updates are closed under composition and can therefore be evaluated with an associative parallel scan. Nonlinear recurrent models can provide richer, state-dependent dynamics, but generally lose this compositional structure: parallel evaluation then requires iterative methods that repeatedly linearize and scan the recurrence. We ask, what state-dependent nonlinear dynamics can be designed to remain exactly composable? We answer by introducing RiccatiSSM, a nonlinear SSM, in which each state dimension follows an input-conditioned Riccati differential equation. Its quadratic state dependence makes the local Jacobian explicitly state-dependent, while its exact per-step flow under piecewise-constant inputs is a Möbius transformation. Since Möbius maps are closed under composition and compose through $2\times 2$ matrix multiplication, the complete nonlinear state trajectory can be evaluated exactly with a single associative parallel scan, without iterative linearization. We further derive a constrained parameterization that ensures bounded, contractive dynamics, and avoids poles in the fractional-linear state update. Across long-sequence classification, regression, and forecasting tasks, RiccatiSSM achieves competitive predictive performance while reducing runtime by $22{-}33\%$ compared to the nonlinear LrcSSM under matched architectures. These results demonstrate that state-dependent nonlinear dynamics can retain exact composability and be evaluated efficiently within a single parallel scan.

Abstract: Interference is commonly treated as geometric overlap between learned features. We introduce effective interference, which combines feature geometry and code statistics to capture realized interactions, distinguishing constructive from destructive interference and frequent weak interactions from rare strong ones. Under local fixed-support assumptions, we characterize how architectural constraints shape interference through four mechanisms: feature orthogonalization, bias compensation, gain adaptation, and encoder-decoder separation. Experiments with sparse autoencoders show that constrained architectures selectively reduce overlap among co-active features, while bias, gain, and encoder freedom allow constructive cross-contributions to remain. Together, these results show that interference in learned representations depends not only on feature geometry, but also on how features are used and on the architecture that produces their codes.

Abstract: We give an exposition of the recent polynomial time algorithm of Guo, Fang, and Lu for the Komlos problem. We simplify various arguments, and highlight the key new spectral potential idea and how the algorithm follows naturally from it.

Abstract: Operator learning of periodic solutions requires deciding how simulation data should be recorded and represented. A natural choice is to integrate long enough for transients to decay and record a window wide enough to contain at least one full period of all trajectories. We find that these conservative choices can make the resulting trajectories difficult to learn, even when the underlying periodic orbits vary regularly with system parameters. Unaligned trajectories generalize poorly even within the training distribution. Phase alignment substantially improves in-distribution generalization, but models trained on a fixed physical-time window still have large errors on trajectories with periods outside the training range. We explain both failures through a common mechanism: frequency differences accumulate over time, so the target phase varies rapidly with the parameters. Predictors that cannot track this variation incur a population MSE floor in both settings; for fixed window prediction, we also derive a per-sample lower bound. We then study one of the simplest representations that escape these floors: learning an aligned, normalized waveform and its period separately. We establish regularity of the decoupled targets under ODE assumptions and show experimentally that this approach avoids both failures in ODE systems and a PDE case study.

Abstract: The design of optimal causal linear feedback schemes for additive white Gaussian noise (AWGN) channels with noisy output feedback has remained an open problem for over 60 years. Prior work has focused on restricted policy classes, especially passive (uncoded) noisy output feedback, where only the transmitter performs feedback coding. However, passive noisy output feedback fundamentally lacks the degrees of freedom required to attain the information-theoretic performance limit in general. In this paper, we consider the active (coded) noisy output feedback setting, where both the transmitter and the receiver perform feedback coding. We then develop a constructive KKT-optimal active linear feedback design that asymptotically attains the Elias-Butman SNR converse bound, thereby establishing MSE/SNR optimality over the entire class of causal linear schemes. Furthermore, we prove that the optimal passive feedback solution is recovered as a special case of the active design. This passive solution admits a Geometric Toeplitz (GT) structure with a Chance-Love (CL)-style one-shot polynomial characterization, and can be computed with O(log T) complexity. Thus, our results provide an affirmative answer to the long-standing optimality question for noisy output feedback under causal linear feedback coding, and our numerical results support the theoretical findings.

Abstract: We study minimal-norm interpolation and $\ell_2$-regularized logistic-loss minimization for binary classification by univariate two-layer ReLU networks. We give complete geometric characterizations of the optimal classifiers in function space, resolving how the solutions depend on whether hidden-layer biases are included in the parameter norm. When biases are unpenalized, the minimal-norm interpolators are exactly the continuous piecewise-affine functions that hug every label switch and have kinks of the appropriate convexity. When biases are penalized, the minimizer is unique in function space, has exactly one kink in each intermediate same-label segment, and is therefore a sparsest positive-margin classifier. We further show that adding a free affine skip connection leaves these function-space solutions unchanged but fundamentally improves the parameter-space landscape: every KKT point of the constrained problem becomes globally optimal, whereas suboptimal KKT points can occur without the skip connection. We establish analogous global-optimality and geometric results for sufficiently weak $\ell_2$-regularization of the logistic loss. In the unpenalized-bias case, we identify an additional sparsity-like restriction, implying that most minimal-norm interpolators cannot arise as small-regularization limits of margin-normalized logistic-loss minimizers. Numerical experiments across varying dataset complexity and network width support the predicted landscape and sparsity phenomena.

Abstract: Weighted sum-rate maximization in multicell multiple-input multiple-output (MIMO) networks is commonly addressed by the weighted minimum mean-square error (WMMSE) algorithm or fractional programming (FP), both of which, after fixing their auxiliary variables, solve a power-constrained quadratic transmit problem that requires costly dense operations for large arrays. Replacing the underlying curvature matrix with a scaled identity can avoid the matrix inverse operation and thereby reduce complexity, but it discards the curvature eigenvalue structure and yields a loose lower bound. We propose Spectral-NFP, which retains selected dominant curvature eigenpairs and uses a scaled identity matrix to bound the curvature on the remaining subspace from above. The retained rank thus traces a continuous path from NFP to the exact WMMSE transmit update. With the surrogate curvature fixed, Spectral-NFP can be interpreted as Euclidean projected-gradient ascent after a linear coordinate transformation, admitting Nesterov-type acceleration. We derive a lower bound on the one-step transmit-objective gain of Spectral-NFP relative to WMMSE, expressed in terms of the curvature eigenvalues. Under an idealized Wishart model, we analyze this bound in both finite dimensions and the large-system limit, obtaining an asymptotic rank-selection rule. Experimental results show that retaining at most 20% of the transmit dimension, and often less than 10%, achieves more than 99% of the WMMSE WSR. In large-array settings, the measured update time is below 20% of that required by WMMSE.

Abstract: Complex-valued neural networks (CVNNs) are increasingly adopted for complex-valued data; however, they are often trained with first-order optimizers inherited from the real-valued case. The efficiency of these methods depends largely on the step size, and their step-size rules ignore the angular information available in the complex plane. We address step-size adaptation in the complex domain by introducing AURA (Angular Update Rate Adaptation), a per-parameter step-size adaptation that can be added on top of any first-order optimizer, and removed from it, without altering its update direction. AURA measures the agreement between consecutive updates of each complex parameter, in length, alignment, and sense of rotation, and enlarges the step when they are consistent and reduces it when they are not. It requires no additional gradient evaluations and only inexpensive vector operations per step. We combine AURA with Adam and Muon and compare the resulting methods with well-known first-order optimizers on four test cases of increasing complexity, ranging from the approximation of scalar complex functions to physics-informed training. Fully connected neural networks are used throughout this work. All hyperparameters other than the step size are held fixed across test cases; for one case, we also tune the hyperparameters of each optimizer under the same budget. Our empirical tests show that AURA improves the convergence of its base optimizer in most cases with a small per-step overhead, and we identify the conditions under which it fails to do so.

Abstract: Signal processing techniques for wireless communications and sensing fundamentally differ between near-field and far-field propagation regimes. Accurately identifying the applicable propagation region is therefore essential for enabling efficient beamforming and channel estimation in ultra-massive MIMO (UM-MIMO) systems. This paper proposes a fully unsupervised learning framework to distinguish near-field from far-field propagation based solely on received signal measurements, before estimating the communication distance, and without relying on channel state information. The proposed approach exploits spatial signal power variations across subarrays of a UM array as a physics-inspired feature extraction stage, followed by the OPTICS clustering algorithm to infer the communication region. Simulation results under various system configurations and signal-to-noise ratio (SNR) levels demonstrate that the proposed method accurately identifies the near-field and far-field regions, showing agreement with theoretical boundaries.

Abstract: This paper provides a self-contained proof of the rank of the pathlifting Jacobian of a DAG ReLU network by performing an induction on the network's number of hidden nodes. In fact, the induction is elementary, and the key recipe is to consider the skeleton matrix of the network, a sparse matrix encoding the network paths, and transform the representation of one of its hidden neurons into an output node. The proof relies on intermediate propositions which link the pathlifting, its Jacobian, the network parameters, and its skeleton matrix, which, on top of permitting to conclude on the rank of the pathlifting Jacobian, also provide a way to compute it without backpropagation and whose computation cost is super efficient in practice compare to usual backpropagation. The paper is provided with a Python module that implements the different propositions of the paper for feed forward networks and is used to experimentally quantifies the computational gain of computing the pathlifting Jacobian with the proposed theory.

Abstract: In sub-terahertz (sub-THz) communications, the coupling of ultra-wide bandwidth and severe phase noise (PN) impairments renders conventional joint channel and PN estimation highly complex and computationally prohibitive. To address this, we propose CRFCAN, a complex-valued residual FFT convolutional attention network designed for joint channel and PN estimation. Unlike existing deep learning schemes that rely on cascaded networks or hybrid frameworks combining neural networks with conventional iterative estimators, CRFCAN performs joint recovery in a truly end-to-end fashion through a physics-inspired cross-domain structure. Specifically, Fast Fourier Transform (FFT) and inverse FFT modules are embedded within residual groups to enable iterative feature interaction across the time and frequency domains, thereby capturing both frequency-selective fading and time-varying phase distortions. In addition, two dedicated residual blocks are introduced for complex feature extraction and multiplicative phase-distortion modeling, respectively. A physics-aware PN output tail with soft normalization is further employed to improve estimation stability while preserving the physical characteristics of the effective PN process. Simulation results demonstrate that CRFCAN significantly outperforms conventional algorithms and state-of-the-art deep learning models in terms of normalized mean square error (NMSE) and bit error rate (BER). Notably, CRFCAN achieves superior performance with single-shot, fixed-complexity inference and generalizes well to unseen PN models without fine-tuning, highlighting its robustness and practicality for sub-THz receivers.

Abstract: Out-of-Distribution (OOD) data poses a significant threat to machine learning models, often leading to model failure during deployment. All existing OOD detection methods are post-hoc, relying on evaluation metrics such as accuracy and AUC-ROC during inference to indirectly assess the model's response to OOD data by measuring deviations. In contrast to existing approaches, the proposed work shifts the paradigm from OOD detection to OOD prediction by proposing a pre-hoc anticipatory framework called PLSP for OOD prediction. We make several key contributions: (a) a dataset-independent metric called the CREDibility Score (CREDS) is proposed for OOD prediction; (b) credibility curves are introduced to study the maximum credibility a model can attain; and (c) credibility heat maps (and volume under surface) are introduced to characterize pre-hoc model behavior across different datasets. This work provides a novel perspective on signal processing under distributional shifts. Experiments across multiple datasets demonstrate that the proposed metric serves as a valuable measure for improving the robustness of machine learning models toward OOD prediction.

Abstract: We consider channels with additive colored Gaussian noise and noiseless feedback. Kim's seminal work derived a stationary variational characterization of feedback capacity and further asserted that the capacity-achieving stationary input need not contain a feedback-independent Gaussian component. These results led to the construction of a simple coding scheme, based on the Schalkwijk--Kailath (SK) refinement principle, which was shown to be capacity-achieving. A recent note identified a gap in the proof of the feedback-independent component-removal assertion, thereby leaving the optimality of the SK scheme and subsequent results that rely on it incomplete. In this paper, we prove the component-removal assertion for channels with stationary Gaussian noise that has a rational power spectral density. Our proof uses a perturbation analysis of a convex optimization formulation of feedback capacity and, indeed, shows that every optimizer assigns zero power to the feedback-independent component. Using this stronger property, we construct from any optimizer an explicit SK coding scheme that achieves every rate below feedback capacity with doubly-exponentially decaying maximal error probability.

Abstract: We develop a time--frequency framework for lattice GKP codes in which ideal codewords are realized in the modulation space $M^\infty$ and identified, through a vector-valued Zak transform, with a finite logical fibre over the continuous syndrome torus. Multi-window Gabor analysis then represents the logical vector by a finite block of adjoint-lattice coefficients. We prove that the normalized block map is an isometry, obtain an explicit recovering projection, and derive stable logical reconstruction. We further construct normalizable GKP approximants as lattice-envelope Gabor multipliers and establish weak-$*$ convergence and asymptotically isometric encoding. Finally, we recover displacement syndromes from phase relations between translated coefficient blocks and quantify their stability under additive perturbations.

Abstract: Computing the optimal success probability for transmitting classical messages through a single use of a quantum channel is NP-hard, even for two messages. An existing semidefinite programming hierarchy based on symmetric extensions provides convergent upper bounds with an a priori error estimate that decays as the inverse square root of the extension level. In this work, we construct a Hermitian sum-of-squares hierarchy for an arbitrary number of messages and prove quadratic convergence in its level. The error bound is proportional to the advantage over random guessing. Our approach combines state-discrimination duality with positive polynomial kernels on products of spheres to construct feasible polynomial dual certificates. For binary messages, the resulting bounds give a multiplicative approximation from above of the trace-norm contraction coefficient.

Abstract: Neural networks acquire internal representations through learning. In this work, we formulate stochastic gradient descent (SGD) as a Markovian stochastic process and derive a Fisher-information flow speed limit that bounds the rate at which trainable parameters can acquire information about latent variables in the data-generating process. The resulting inequality decomposes the information flow into drift and noise contributions, thereby quantifying the roles of deterministic learning forces and SGD-induced fluctuations from an information-theoretic perspective. We verify the bound in analytically tractable basis-function linear regression, where the information budget predicted by the bound reproduces the ordering and characteristic time scales with which different latent variables are encoded in the learned parameters. These results establish Fisher-information speed limits as a quantitative framework for diagnosing when and how different aspects of the data-generating mechanism are acquired during stochastic learning.