arXiv subjects

José Luis Romero

Publications and source records attributed to José Luis Romero.

At least 19 recordsLinked to original sources

arXiv2026

Zeros of the Spectrogram of Colored Noise

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.

math.PR↗

arXiv2026

PhaseJumps: fast computation of zeros from planar grid samples

We consider complex-valued functions on the complex plane and the task of computing their zeros from samples taken along a finite grid. We introduce PhaseJumps, an algorithm based on comparing changes in the complex phase and local oscillations among neighboring grid points. The algorithm is applicable to possibly non-analytic input functions, and also computes the direction of phase winding around zeros. PhaseJumps provides a first effective means to compute the zeros of the short-time Fourier transform of an analog signal with respect to a general analysis window, and makes certain recent signal processing insights more widely applicable, overcoming previous constraints to analytic transformations. We study the performance of (a variant of) PhaseJumps under a stochastic input model motivated by signal processing applications and show that the input instances that may cause the algorithm to fail are fragile, in the sense that they are regularized by additive noise (smoothed analysis). Precisely, given samples of a function on a grid with spacing $δ$, we show that our algorithm computes zeros with accuracy $\sqrtδ$ in the Wasserstein metric with failure probability $O\big(\log^2(\tfrac{1}δ) δ\big)$, while numerical experiments suggest even better performance.

math.NA↗

arXiv2026

Hyperuniformity and non-hyperuniformity of zeros of Gaussian Weyl-Heisenberg Functions

We study zero sets of twisted stationary Gaussian random functions on the complex plane, i.e., Gaussian random functions that are stochastically invariant under the action of the Weyl-Heisenberg group. This model includes translation-invariant Gaussian entire functions (GEFs), and also many other non-analytic examples, in which case winding numbers around zeros can be either positive or negative. We investigate zero statistics both when zeros are weighted with their winding numbers (charged zero set) and when they are not (uncharged zero set). We show that the variance of the charged zero statistic always grows linearly with the radius of the observation disk (hyperuniformity). Importantly, this holds for functions with possibly non-zero means and without assuming additional symmetries such as radiality. With respect to uncharged zero statistics, we provide an example for which the variance grows with the area of the observation disk (non-hyperuniformity). This is used to show that, while the zeros of GEFs are hyperuniform, the set of their critical points fails to be so. Our work contributes to recent developments in statistical signal processing, where the time-frequency profile of a non-stationary signal embedded into noise is revealed by performing a statistical test on the zeros of its spectrogram ("silent points"). We show that empirical spectrogram zero counts enjoy moderate deviations from their ensemble averages over large observation windows (something that was previously known only for pure noise). In contrast, we also show that spectrogram maxima ("loud points") fail to enjoy a similar property. This gives the first formal evidence for the statistical superiority of silent points over the competing feature of loud points, a fact that has been noted by practitioners.

math.PR↗

arXiv2026

Spectral deviation of concentration operators on reproducing kernel Hilbert spaces

We study the eigenvalue profile of concentration operators (multiplication by an indicator function followed by projection) acting on reproducing kernel Hilbert spaces. The spectral profile of such operators provides a useful notion of local degrees of freedom. We formalize this idea by estimating the number of eigenvalues that lie away from 0 and 1, commonly referred to as the plunge region. Our main motivation is to treat discrete and continuous settings simultaneously and uniformly, and to be able to argue that approximations arising from discretization schemes reflect, in a non-asymptotic sense, the spectral profile of their continuous counterparts. As a case in point, we show that Gabor multipliers computed on sufficiently fine grids obey spectral deviation estimates similar to those available for the short-time Fourier transform (STFT) with bounds that are uniform in the discretization step. Concretely, this means that the theoretical localization properties of the STFT are observable in practice.

math.SP↗

arXiv2026

Stability in unlimited sampling

Folded sampling replaces clipping in analog-to-digital converters by reducing samples modulo a threshold, thereby avoiding saturation artifacts. We study the reconstruction of bandlimited functions from folded samples and show that, for equispaced sampling patterns, the recovery problem is inherently unstable. We then prove that imposing any a priori energy bound restores stability, and that this regularization effect extends to non-uniform sampling geometries. Our analysis recasts folded-sampling stability as an infinite-dimensional lattice shortest-vector problem, which we resolve via harmonic-analytic tools (the spectral profile of Fourier concentration matrices) and, alternatively, via bounds for integer Tschebyschev polynomials. Our work brings context to recent results on injectivity and encoding guarantees for folded sampling and further supports the empirical success of folded sampling under natural energy constraints.

math.FA↗

arXiv2026

Model agnostic signal encoding by leaky integrate and fire, performance and uncertainty

Integrate-and-fire is a resource efficient time-encoding mechanism that summarizes into a signed spike train those time intervals where a signal's charge exceeds a certain threshold. We analyze the IF encoder in terms of a very general notion of approximate bandwidth, which is shared by most commonly-used signal models. This complements results on exact encoding that may be overly adapted to a particular signal model. We take into account, possibly for the first time, the effect of uncertainty in the exact location of the spikes (as may arise by decimation), uncertainty of integration leakage (as may arise in realistic manufacturing), and boundary effects inherent to finite periods of exposure to the measurement device. The analysis is done by means of a concrete bandwidth-based Ansatz that can also be useful to initialize more sophisticated model specific reconstruction algorithms, and uses the earth mover's (Wasserstein) distance to measure spike discrepancy.

math.FA↗

arXiv2025

Compiling Gradual Types with Evidence

Efficiently supporting sound gradual typing in a language with structural types is challenging. To date, the Grift compiler is the only close-to-the-metal implementation of gradual typing in this setting, exploiting coercions for runtime checks, and further extended with monotonic references for efficient access to statically-typed data structures. On the language design and semantics side, the Abstracting Gradual Typing (AGT) methodology has proven fruitful to elucidate existing designs and to innovate by deriving gradualizations of a wide variety of typing disciplines and language features. Grounded in abstract interpretation, the Curry-Howard inspired runtime semantics of AGT is based on the notion of evidence for consistent judgments that evolve during reduction, monitoring the plausibility of well-typedness. While expressive and versatile, it is unclear whether such evidence-based semantics are a viable route to realize an efficient implementation of gradual typing. In this work, we explore this question by designing, implementing, and evaluating an evidence-based compiler, called GrEv. We explain how to bridge the gap between the formal semantics and the GrEv compiler implementation, and identify novel monotonic semantics. We empirically evaluate the performance of GrEv on the Grift benchmark suite. The results show that an evidence-based compiler can be competitive with, and even faster than, a coercion-based compiler, exhibiting more stability across configurations on the static-to-dynamic spectrum. In addition to enriching the space of gradual typing compilers, this work opens a direct door to exploring efficient implementations of the many advanced gradual typing disciplines formally derived with AGT in the literature.

cs.PL↗

arXiv2025

Local repulsion between zeros and critical points of the Gaussian Entire Function

We study the zeros and critical points of different indices of the standard Gaussian entire function on the complex plane (whose zero set is stationary). We provide asymptotics for the second order correlations of all the corresponding number statistics on small observation disks, showing various rates of local repulsion. The results have consequences for signal processing, as they show extremely strong repulsion between the local maxima and zeros of spectrograms of noise computed with respect to Gaussian windows.

math.PR↗

arXiv2024

Enhancing sharp augmented Lagrangian methods with smoothing techniques for nonlinear programming

This paper proposes a novel approach to solving nonlinear programming problems using a sharp augmented Lagrangian method with a smoothing technique. Traditional sharp augmented Lagrangian methods are known for their effectiveness but are often hindered by the need for global minimization of nonconvex, nondifferentiable functions at each iteration. To address this challenge, we introduce a smoothing function that approximates the sharp augmented Lagrangian, enabling the use of primal minimization strategies similar to those in Powell--Hestenes--Rockafellar (PHR) methods. Our approach retains the theoretical rigor of classical duality schemes while allowing for the use of stationary points in the primal optimization process. We present two algorithms based on this method--one utilizing standard descent and the other employing coordinate descent. Numerical experiments demonstrate that our smoothing--based method compares favorably with the PHR augmented Lagrangian approach, offering both robustness and practical efficiency. The proposed method is particularly advantageous in scenarios where exact minimization is computationally infeasible, providing a balance between theoretical precision and computational tractability.

math.OC↗

arXiv2024

Gaussian beta ensembles: the perfect freezing transition and its characterization in terms of Beurling-Landau densities

The Gaussian $β$-ensemble is a real $n$-point configuration $\{x_j\}_1^n$ picked randomly with respect to the Boltzmann factor $e^{-\fracβ2H_n}$, $H_n=\sum_{i\ne j}\log\frac 1{|x_i-x_j|}+n\sum_{i=1}^n\tfrac 12x_i^2.$ The point process $\{x_j\}_1^n$ tends to follow the semicircle law $σ(x)=\tfrac 1{2π}\sqrt{(4-x^2)_+}$ in certain average senses. A Fekete configuration (minimizer of $H_n$) is spread out in a much more uniform way in the interval $[-2,2]$ with respect to the regularization $σ_n(x)=\max\{σ(x),n^{-\frac 1 3}\}$ of the semicircle law. In particular, Fekete configurations are "equidistributed" with respect to $σ_n(x)$, in a certain technical sense of Beurling-Landau densities. We consider the problem of characterizing sequences $β_n$ of inverse temperatures, which guarantee almost sure equidistribution as $n\to\infty$. We find that a necessary and sufficient condition is that $β_n$ grows at least logarithmically in $n$: $$β_n\gtrsim \log n.$$ We call this growth rate the perfect freezing regime. We give several further results on the distribution of particles when $β_n\gtrsim\log n$, for example on minimal spacing, discrepancies, and sampling and interpolation for weighted polynomials. The condition $β_n\gtrsim\log n$ was introduced by some of the authors in the context of two-dimensional Coulomb gas ensembles, where it is shown to be sufficient for equidistribution. Although the technical implementation requires some considerable modifications, the strategy from dimension two adapts well to prove sufficiency also for one-dimensional Gaussian ensembles. On a technical level, we use estimates for weighted polynomials due to Levin, Lubinsky, Gustavsson and others. The other direction (necessity) involves estimates due to Ledoux and Rider on the distribution of particles which fall near or outside the boundary.

math.PR↗

arXiv2024

Hölder-Continuity of Extreme Spectral Values of Pseudodifferential Operators, Gabor Frame Bounds, and Saturation

We build on our recent results on the Lipschitz dependence of the extreme spectral values of one-parameter families of pseudodifferential operators with symbols in a weighted Sjöstrand class. We prove that larger symbol classes lead to Hölder continuity with respect to the parameter. This result is then used to investigate the behavior of frame bounds of families of Gabor systems $\mathcal{G}(g,αΛ)$ with respect to the parameter $α>0$, where $Λ$ is a set of non-uniform, relatively separated time-frequency shifts, and $g\in M^1_s(\mathbb{R}^d)$, $0\leq s\leq 2$. In particular, we show that the frame bounds depend continuously on $α$ if $g\in M^1(\mathbb{R}^d)$, and are Hölder continuous if $g\in M^1_s(\mathbb{R}^d)$, $0<s\leq 2$, with the Hölder exponent explicitly given.

math.FA↗

arXiv2024

Sampling in the shift-invariant space generated by the bivariate Gaussian function

We study the space spanned by the integer shifts of a bivariate Gaussian function and the problem of reconstructing any function in that space from samples scattered across the plane. We identify a large class of lattices, or more generally semi-regular sampling patterns spread along parallel lines, that lead to stable reconstruction while having densities close to the critical value given by Landau's limit. At the critical density, we construct examples of sampling patterns for which reconstruction fails. In the same vein, we also investigate continuous sampling along non-uniformly scattered families of parallel lines and identify the threshold density of line configurations at which reconstruction is possible. In a remarkable contrast with Paley-Wiener spaces, the results are completely different for lines with rational or irrational slopes. Finally, we apply the sampling results to Gabor systems with bivariate Gaussian windows. As a main contribution, we provide a large list of new examples of Gabor frames with non-complex lattices having volume close to 1.

math.FA↗

arXiv2024

Spectral deviation of concentration operators for the short-time Fourier transform

Time-frequency concentration operators restrict the integral analysis-synthesis formula for the short-time Fourier transform to a given compact domain. We estimate how much the corresponding eigenvalue counting function deviates from the Lebesgue measure of the time-frequency domain. For window functions in the Gelfand-Shilov class, the bounds almost match known asymptotics, with the advantage of being effective for concrete domains and spectral thresholds. As such our estimates allow for applications where the spectral threshold depends on the geometry of the time-frequency concentration domain. We also consider window functions that decay only polynomially in time and frequency.

math.SP↗

arXiv2024

Eigenvalue estimates for Fourier concentration operators on two domains

We study concentration operators associated with either the discrete or the continuous Fourier transform, that is, operators that incorporate a spatial cut-off and a subsequent frequency cut-off to the Fourier inversion formula. Their spectral profiles describe the number of prominent degrees of freedom in problems where functions are assumed to be supported on a certain domain and their Fourier transforms are known or measured on a second domain. We derive eigenvalue estimates that quantify the extent to which Fourier concentration operators deviate from orthogonal projectors, by bounding the number of eigenvalues that are away from 0 and 1 in terms of the geometry of the spatial and frequency domains, and a factor that grows at most poly-logarithmically on the inverse of the spectral margin. The estimates are non-asymptotic in the sense that they are applicable to concrete domains and spectral thresholds, and almost match asymptotic benchmarks. Our work covers for the first time non-convex and non-symmetric spatial and frequency concentration domains, as demanded by numerous applications that exploit the expected approximate low dimensionality of the modeled phenomena. The proofs build on Israel's work on one dimensional intervals [arXiv: 1502.04404v1]. The new ingredients are the use of redundant wave-packet expansions and a dyadic decomposition argument to obtain Schatten norm estimates for Hankel operators.

math.FA↗

arXiv2024

Estimation of binary time-frequency masks from ambient noise

We investigate the retrieval of a binary time-frequency mask from a few observations of filtered white ambient noise. Confirming household wisdom in acoustic modeling, we show that this is possible by inspecting the average spectrogram of ambient noise. Specifically, we show that the lower quantile of the average of $\mathcal{O}(\log(|Ω|/\varepsilon))$ masked spectrograms is enough to identify a rather general mask $Ω$ with confidence at least $\varepsilon$, up to shape details concentrated near the boundary of $Ω$. As an application, the expected measure of the estimation error is dominated by the perimeter of the time-frequency mask. The estimator requires no knowledge of the noise variance, and only a very qualitative profile of the filtering window, but no exact knowledge of it.

cs.SD↗

arXiv2024

Improved discrepancy for the planar Coulomb gas at low temperatures

We study the planar Coulomb gas in the regime where the inverse temperature $β_n$ grows at least logarithmically with respect to the number of particles $n$ (freezing regime, $β_n\gtrsim \log n$). We show that, almost surely for large $n$, the discrepancy between the number of particles in any microscopic region and their expected value (given with adequate precision by the equilibrium measure) is, up to log factors, of the order of the perimeter of the observation window. The estimates are valid throughout the whole droplet (the region where the particles accumulate), and are particularly interesting near the boundary, while in the bulk they offer technical improvements over known results. Our work builds on recent results on equidistribution at low temperatures and improves on them by providing refined spectral asymptotics for certain Toeplitz operators on the range of the erfc-kernel (sometimes called Faddeeva or plasma dispersion kernel).

math.PR↗

arXiv2023

Random periodic sampling patterns for shift-invariant spaces

We consider multi-variate signals spanned by the integer shifts of a set of generating functions with distinct frequency profiles and the problem of reconstructing them from samples taken on a random periodic set. We show that such a sampling strategy succeeds with high probability provided that the density of the sampling pattern exceeds the number of frequency profiles by a logarithmic factor. The signal model includes bandlimited functions with multi-band spectra. While in this well-studied setting delicate constructions provide sampling strategies that meet the information theoretic benchmark of Shannon and Landau, the sampling pattern that we consider provides, at the price of a logarithmic oversampling factor, a simple alternative that is accompanied by favorable a priori stability margins (snug frames). More generally, we also treat bandlimited functions with arbitrary compact spectra, and different measures of its complexity and approximation rates by integer tiles. At the technical level, we elaborate on recent work on relevant sampling, with the key difference that the reconstruction guarantees that we provide hold uniformly for all signals, rather than for a subset of well-concentrated ones. This is achieved by methods of concentration of measure formulated on the Zak domain.

math.FA↗

arXiv2023

Lipschitz Continuity of Spectra of Pseudodifferential Operators in a Weighted Sjöstrand Class and Gabor Frame Bounds

We study one-parameter families of pseudodifferential operators whose Weyl symbols are obtained by dilation and a smooth deformation of a symbol in a weighted Sjöstrand class. We show that their spectral edges are Lipschitz continuous functions of the dilation or deformation parameter. Suitably local estimates hold also for the edges of every spectral gap. These statements extend Bellissard's seminal results on the Lipschitz continuity of spectral edges for families of operators with periodic symbols to a large class of symbols with only mild regularity assumptions. The abstract results are used to prove that the frame bounds of a family of Gabor systems $\mathcal{G}(g,αΛ)$, where $Λ$ is a set of non-uniform time-frequency shifts, $α>0$, and $g\in M^1_2(\mathbb{R}^d)$, are Lipschitz continuous functions in $α$. This settles a question about the precise blow-up rate of the condition number of Gabor frames near the critical density.

math.FA↗