arXiv subjects

Weilin Li

Publications and source records attributed to Weilin Li.

At least 19 recordsLinked to original sources

arXiv2026

A sharp analysis of Root-MUSIC: locations of correct and extraneous roots

Root-MUSIC is a spectral estimation algorithm that approximates the unknown signal frequencies by constructing a high-degree polynomial and finding a subset of roots which are closest to the complex unit circle. Previous works found asymptotic expectation formulas for the performance of Root-MUSIC under the implicit assumption that the aforementioned root selection criterion does not select extraneous roots -- those which are unrelated to the correct parameters. This paper removes the need for this assumption by showing all extraneous roots lie outside an annulus of a certain thickness and therefore are not selected by the algorithm. This paper also provides sharp, non-asymptotic, and explicit error bounds for the correct roots in terms of fundamental model parameters. All results hold under a natural separation condition on the correct signal frequencies and are applicable in both the single- and multi-snapshot models. More specifically, in the multi-snapshot model, we prove that Root-MUSIC estimates the frequencies with error at most $O(\sigma /(m \sqrt n))$, where $\sigma^2$ is the noise variance, $m$ is the number of sensors, and $n$ is the number of snapshots. A novelty of this non-asymptotic bound is the explicit $1/m$ decay, which indicates that there is a significant advantage in utilizing additional sensors. Numerical simulations confirm our theory. The main mathematical insight of this paper is a geometric property of the Root-MUSIC polynomial: its correct roots are highly stable to noise while its extraneous roots must lie outside of an annulus.

eess.SP↗

arXiv2026

Multidimensional Gradient-MUSIC: A Global Nonconvex Optimization Framework for Optimal Resolution

We develop a multidimensional version of Gradient-MUSIC for estimating the frequencies of a nonharmonic signal from noisy samples. The guiding principle is that frequency recovery should be based only on the signal subspace determined by the data. From this viewpoint, the MUSIC functional is an economical nonconvex objective encoding the relevant information, and the problem becomes one of understanding the geometry of its perturbed landscape. Our main contribution is a general structural theory showing that, under explicit conditions on the measurement kernel and the perturbation of the signal subspace, the perturbed MUSIC function is an admissible optimization landscape: suitable initial points can be found efficiently by coarse thresholding, gradient descent converges to the relevant local minima, and these minima obey quantitative error bounds. Thus the theory is not merely existential; it provides a constructive global optimization framework for multidimensional optimal resolution. We verify the abstract conditions in detail for two canonical sampling geometries: discrete samples on a cube and continuous samples on a ball. In both cases we obtain uniform, nonasymptotic recovery guarantees under deterministic as well as stochastic noise. In particular, for lattice samples in a cube of side length $4m$, if the true frequencies are separated by at least $\beta_d/m$ and the noise has $\ell^\infty$ norm at most $\varepsilon$, then Gradient-MUSIC recovers the frequencies with error at most \[ C_d \frac{\varepsilon}{m}, \] where $C_d, \beta_d>0$ depend only on the dimension. This scaling is minimax optimal in $m$ and $\varepsilon$. Under stationary Gaussian noise, the error improves to \[ C_d\frac{\sigma\sqrt{\log(m)}}{m^{1+d/2}}. \] This is the noisy super-resolution scaling: (see paper for rest of abstract)

math.OC↗

arXiv2025

Optimal structured approximation of Fourier subspaces, Toeplitz matrices, and exponential sums

This paper studies three structured approximation problems: (1) Recovering the range of a Fourier matrix from a single observation, (2) Recovering a corrupted low-rank Toeplitz/Hankel matrix, and (3) Recovering a finite exponential sum from noisy samples. All three problems are computationally challenging because their structural constraints are difficult to enforce directly. We show that all three tasks can be solved efficiently and optimally by applying the Gradient-MUSIC algorithm for spectral estimation. To provide an example, for a rank-$r$ Toeplitz matrix $T\in {\mathbb C}^{n\times n}$ that satisfies a regularity assumption and is corrupted by an arbitrary $E\in {\mathbb C}^{n\times n}$ such that $\|E\|_2\leq \alpha n$, our algorithm outputs a Toeplitz matrix $T_\sharp$ of rank exactly $r$ such that $\|T-T_\sharp\|_2 \leq C \|E\|_2$, where $C,\alpha>0$ are absolute constants. This performance guarantee is minimax optimal in $n$, $r$, and $\|E\|_2$. For the other two structured approximation problems, we also provide algorithms that are minimax optimal in the number of samples, rank/sparsity, and noise level. At the heart of this paper is a quantitative transference principle which shows how to convert computational methods and theory for spectral estimation into corresponding methods and theory for the other three problems.

cs.IT↗

arXiv2025

Optimality of Gradient-MUSIC for spectral estimation

We introduce the Gradient-MUSIC algorithm for estimating the unknown frequencies and amplitudes of a nonharmonic signal from noisy time samples. While the classical MUSIC algorithm performs a computationally expensive search over a fine grid, Gradient-MUSIC is significantly more efficient and eliminates the need for discretization over a fine grid by using optimization techniques. It coarsely scans the 1D landscape to find initialization simultaneously for all frequencies followed by parallelizable local refinement via gradient descent. We also analyze its performance when the noise level is sufficiently small and the signal frequencies are separated by at least $8\pi/m$, where $\pi/m$ is the standard resolution of this problem. Even though the 1D landscape is nonconvex, we prove a global convergence result for Gradient-MUSIC: coarse scanning provably finds suitable initialization and gradient descent converges at a linear rate. In addition to convergence results, we also upper bound the error between the true signal frequencies and amplitudes with those found by Gradient-MUSIC. For example, if the noise has $\ell^\infty$ norm at most $\varepsilon$, then the frequencies and amplitudes are recovered up to error at most $C\varepsilon/m$ and $C\varepsilon$ respectively for a universal $C>0$, which are minimax optimal in $m$, $\varepsilon$, and number of frequencies. Our theory can also handle stochastic noise with performance guarantees under nonstationary independent Gaussian noise. Our main approach is a comprehensive geometric analysis of the landscape, a perspective that has not been explored before.

cs.IT↗

arXiv2025

Subspace and DOA estimation under coarse quantization

We study direction-of-arrival (DOA) estimation from coarsely quantized data. We focus on a two-step approach which first estimates the signal subspace via covariance estimation and then extracts DOA angles by the ESPRIT algorithm. In particular, we analyze two stochastic quantization schemes which use dithering: a one-bit quantizer combined with rectangular dither and a multi-bit quantizer with triangular dither. For each quantizer, we derive rigorous high probability bounds for the distances between the true and estimated signal subspaces and DOA angles. Using our analysis, we identify scenarios in which subspace and DOA estimation via triangular dithering qualitatively outperforms rectangular dithering. We verify in numerical simulations that our estimates are optimal in their dependence on the smallest non-zero eigenvalue of the target matrix. The resulting subspace estimation guarantees are equally applicable in the analysis of other spectral estimation algorithms and related problems.

cs.IT↗

arXiv2024

An explicit spectral decomposition of the ADRT

The approximate discrete Radon transform (ADRT) is a hierarchical multiscale approximation of the Radon transform. In this paper, we factor the ADRT into a product of linear transforms that resemble convolutions and derive an explicit spectral decomposition of each factor. We further show that this implies -- for data lying in the range of the ADRT -- that the transform of an $N \times N$ image can be formally inverted with complexity $\mathcal{O}(N^2 \log^2 N)$. We numerically test the accuracy of the inverse on images of moderate size and find that it is competitive with existing iterative algorithms in this special regime.

math.NA↗

arXiv2024

Nonharmonic multivariate Fourier transforms and matrices: condition numbers and hyperplane geometry

Consider an operator that takes the Fourier transform of a discrete measure supported in $\mathcal{X}\subset[-\frac 12,\frac 12)^d$ and restricts it to a compact $\Omega\subset\mathbb{R}^d$. We provide lower bounds for its smallest singular value when $\Omega$ is either a closed ball of radius $m$ or closed cube of side length $2m$, and under different types of geometric assumptions on $\mathcal{X}$. We first show that if distances between points in $\mathcal{X}$ are lower bounded by a $\delta$ that is allowed to be arbitrarily small, then the smallest singular value is at least $Cm^{d/2} (m\delta)^{\lambda-1}$, where $\lambda$ is the maximum number of elements in $\mathcal{X}$ contained within any ball or cube of an explicitly given radius. This estimate communicates a localization effect of the Fourier transform. While it is sharp, the smallest singular value behaves better than expected for many $\mathcal{X}$, including when we dilate a generic set by parameter $\delta$. We next show that if there is a $\eta$ such that, for each $x\in\mathcal{X}$, the set $\mathcal{X}\setminus\{x\}$ locally consists of at most $r$ hyperplanes whose distances to $x$ are at least $\eta$, then the smallest singular value is at least $C m^{d/2} (m\eta)^r$. For dilations of a generic set by $\delta$, the lower bound becomes $C m^{d/2} (m\delta)^{\lceil (\lambda-1)/d\rceil }$. The appearance of a $1/d$ factor in the exponent indicates that compared to worst case scenarios, the condition number of nonharmonic Fourier transforms is better than expected for typical sets and improve with higher dimensionality.

math.NA↗

arXiv2024

Concerning the stability of exponential systems and Fourier matrices

Fourier matrices naturally appear in many applications and their stability is closely tied to performance guarantees of algorithms. The starting point of this article is a result that characterizes properties of an exponential system on a union of cubes in $\mathbb{R}^d$ in terms of a general class of Fourier matrices and their extreme singular values. This relationship is flexible in the sense that it holds for any dimension $d$, for many types of exponential systems (Riesz bases, Riesz sequences, or frames) and for Fourier matrices with an arbitrary number of rows and columns. From there, we prove new stability results for Fourier matrices by exploiting this connection and using powerful stability theorems for exponential systems. This paper provides a systematic exploration of this connection and suggests some natural open questions.

math.CA↗

arXiv2023

Uniform Approximation by Polynomials with Integer Coefficients via the Bernstein Lattice

Let $\mathscr{C}_\mathbb{Z}([0,1])$ be the metric space of real-valued continuous functions on $[0,1]$ with integer values at $0$ and $1$, equipped with the uniform (supremum) metric $d_\infty$. It is a classical theorem in approximation theory that the ring $\mathbb{Z}[X]$ of polynomials with integer coefficients, when considered as a set of functions on $[0,1]$, is dense in $\mathscr{C}_\mathbb{Z}([0,1])$. In this paper, we offer a strengthening of this result by identifying a substantially small subset $\bigcup_n \mathscr{B}_n$ of $\mathbb{Z}[X]$ which is still dense in $\mathscr{C}_\mathbb{Z}([0,1])$. Here $\mathscr{B}_n$, which we call the ``Bernstein lattice,'' is the lattice generated by the polynomials $$p_{n,k}(x) := \binom{n}{k} x^k(1-x)^{n-k}, ~~k=0,\dots,n.$$ Quantitatively, we show that for any $f \in \mathscr{C}_\mathbb{Z}([0,1])$, $$d_\infty(f, \mathscr{B}_n) \leq \frac{9}{4} ω_f(n^{-1/3}) + 2 n^{-1/3}, ~~n \geq 1,$$ where $ω_f$ stands for the modulus of continuity of $f$. We also offer a more general bound which can be optimized to yield better decay of approximation error for specific classes of continuous functions.

math.NT↗

arXiv2023

Unmasking Role-Play Attack Strategies in Exploiting Decentralized Finance (DeFi) Systems

The rapid growth and adoption of decentralized finance (DeFi) systems have been accompanied by various threats, notably those emerging from vulnerabilities in their intricate design. In our work, we introduce and define an attack strategy termed as Role-Play Attack, in which the attacker acts as multiple roles concurrently to exploit the DeFi system and cause substantial financial losses. We provide a formal definition of this strategy and demonstrate its potential impacts by revealing the total loss of \$435.1M caused by 14 historical attacks with applying this pattern. Besides, we mathematically analyzed the attacks with top 2 losses and retrofitted the corresponding attack pattern by concrete execution, indicating that this strategy could increase the potential profit for original attacks by \$3.34M (51.4%) and \$3.76M (12.0%), respectively.

cs.CR↗

arXiv2023

Multiscale estimates for the condition number of non-harmonic Fourier matrices

This paper studies the extreme singular values of non-harmonic Fourier matrices. Such a matrix of size $m\times s$ can be written as $\Phi=[ e^{-2\pi i j x_k}]_{j=0,1,\dots,m-1, k=1,2,\dots,s}$ for some set $\mathcal{X}=\{x_k\}_{k=1}^s$. Its condition number controls the stability of inversion, which is of great importance to super-resolution and nonuniform Fourier transforms. Under the assumption $m\geq 6s$ and without any restrictions on $\mathcal{X}$, the main theorems provide explicit lower bounds for the smallest singular value $\sigma_s(\Phi)$ in terms of distances between elements in $\mathcal{X}$. More specifically, distances exceeding an appropriate scale $\tau$ have modest influence on $\sigma_s(\Phi)$, while the product of distances that are less than $\tau$ dominates the behavior of $\sigma_s(\Phi)$. These estimates reveal how the multiscale structure of $\mathcal{X}$ affects the condition number of Fourier matrices. Theoretical and numerical comparisons indicate that the main theorems significantly improve upon classical bounds and recover the same rate for special cases but with relaxed assumptions.

math.NA↗

arXiv2023

Enhancing Smart Contract Security Analysis with Execution Property Graphs

Smart contract vulnerabilities have led to significant financial losses, with their increasing complexity rendering outright prevention of hacks increasingly challenging. This trend highlights the crucial need for advanced forensic analysis and real-time intrusion detection, where dynamic analysis plays a key role in dissecting smart contract executions. Therefore, there is a pressing need for a unified and generic representation of smart contract executions, complemented by an efficient methodology that enables the modeling and identification of a broad spectrum of emerging attacks. We introduce Clue, a dynamic analysis framework specifically designed for the Ethereum virtual machine. Central to Clue is its ability to capture critical runtime information during contract executions, employing a novel graph-based representation, the Execution Property Graph. A key feature of Clue is its innovative graph traversal technique, which is adept at detecting complex attacks, including (read-only) reentrancy and price manipulation. Evaluation results reveal Clue's superior performance with high true positive rates and low false positive rates, outperforming state-of-the-art tools. Furthermore, Clue's efficiency positions it as a valuable tool for both forensic analysis and real-time intrusion detection.

cs.CR↗

arXiv2023

Approximation of functions with one-bit neural networks

The celebrated universal approximation theorems for neural networks roughly state that any reasonable function can be arbitrarily well-approximated by a network whose parameters are appropriately chosen real numbers. This paper examines the approximation capabilities of one-bit neural networks -- those whose nonzero parameters are $\pm a$ for some fixed $a\not=0$. One of our main theorems shows that for any $f\in C^s([0,1]^d)$ with $\|f\|_\infty<1$ and error $\varepsilon$, there is a $f_{NN}$ such that $|f(\boldsymbol{x})-f_{NN}(\boldsymbol{x})|\leq \varepsilon$ for all $\boldsymbol{x}$ away from the boundary of $[0,1]^d$, and $f_{NN}$ is either implementable by a $\{\pm 1\}$ quadratic network with $O(\varepsilon^{-2d/s})$ parameters or a $\{\pm \frac 1 2 \}$ ReLU network with $O(\varepsilon^{-2d/s}\log (1/\varepsilon))$ parameters, as $\varepsilon\to0$. We establish new approximation results for iterated multivariate Bernstein operators, error estimates for noise-shaping quantization on the Bernstein basis, and novel implementation of the Bernstein polynomials by one-bit quadratic and ReLU neural networks.

cs.LG↗

arXiv2022

Approximation with one-bit polynomials in Bernstein form

We prove various theorems on approximation using polynomials with integer coefficients in the Bernstein basis of any given order. In the extreme, we draw the coefficients from $\{ \pm 1\}$ only. A basic case of our results states that for any Lipschitz function $f:[0,1] \to [-1,1]$ and for any positive integer $n$, there are signs $σ_0,\dots,σ_n \in \{\pm 1\}$ such that $$\left |f(x) - \sum_{k=0}^n σ_k \, \binom{n}{k} x^k (1-x)^{n-k} \right | \leq \frac{C (1+|f|_{\mathrm{Lip}})}{1+\sqrt{nx(1-x)}} ~\mbox{ for all } x \in [0,1].$$ More generally, we show that higher accuracy is achievable for smoother functions: For any integer $s\geq 1$, if $f$ has a Lipschitz $(s{-}1)$st derivative, then approximation accuracy of order $O(n^{-s/2})$ is achievable with coefficients in $\{\pm 1\}$ provided $\|f \|_\infty < 1$, and of order $O(n^{-s})$ with unrestricted integer coefficients, both uniformly on closed subintervals of $(0,1)$ as above. Hence these polynomial approximations are not constrained by the saturation of classical Bernstein polynomials. Our approximations are constructive and can be implemented using feedforward neural networks whose weights are chosen from $\{\pm 1\}$ only.

cs.IT↗

arXiv2022

Stable super-resolution limit and smallest singular value of restricted Fourier matrices

We consider the inverse problem of recovering the locations and amplitudes of a collection of point sources represented as a discrete measure, given $M+1$ of its noisy low-frequency Fourier coefficients. Super-resolution refers to a stable recovery when the distance $Δ$ between the two closest point sources is less than $1/M$. We introduce a clumps model where the point sources are closely spaced within several clumps. Under this assumption, we derive a non-asymptotic lower bound for the minimum singular value of a Vandermonde matrix whose nodes are determined by the point sources. Our estimate is given as a weighted $\ell^2$ sum, where each term only depends on the configuration of each individual clump. The main novelty is that our lower bound obtains an exact dependence on the {\it Super-Resolution Factor} $SRF=(MΔ)^{-1}$. As noise level increases, the {\it sensitivity of the noise-space correlation function in the MUSIC algorithm} degrades according to a power law in $SRF$ where the exponent depends on the cardinality of the largest clump. Numerical experiments validate our theoretical bounds for the minimum singular value and the sensitivity of MUSIC. We also provide lower and upper bounds for a min-max error of super-resolution for the grid model, which in turn is closely related to the minimum singular value of Vandermonde matrices.

cs.IT↗

arXiv2022

Stability and Super-resolution of MUSIC and ESPRIT for Multi-snapshot Spectral Estimation

This paper studies the spectral estimation problem of estimating the locations of a fixed number of point sources given multiple snapshots of Fourier measurements collected by a uniform array of sensors. We prove novel stability bounds for MUSIC and ESPRIT as a function of the noise standard deviation, number of snapshots, source amplitudes, and support. Our most general result is a perturbation bound of the signal space in terms of the minimum singular value of Fourier matrices. When the point sources are located in several separated clumps, we provide an explicit upper bound of the noise-space correlation perturbation error in MUSIC and the support error in ESPRIT in terms of a Super-Resolution Factor (SRF). The upper bound for ESPRIT is then compared with a new Cramér-Rao lower bound for the clumps model. As a result, we show that ESPRIT is comparable to that of the optimal unbiased estimator(s) in terms of the dependence on noise, number of snapshots and SRF. As a byproduct of our analysis, we discover several fundamental differences between the single-snapshot and multi-snapshot problems. Our theory is validated by numerical experiments.

cs.IT↗

arXiv2022

A range characterization of the single-quadrant ADRT

This work characterizes the range of the single-quadrant approximate discrete Radon transform (ADRT) of square images. The characterization follows from a set of linear constraints on the codomain. We show that for data satisfying these constraints, the exact and fast inversion formula [Rim, Appl. Math. Lett. 102 106159, 2020] yields a square image in a stable manner. The range characterization is obtained by first showing that the ADRT is a bijection between images supported on infinite half-strips, then identifying the linear subspaces that stay finitely supported under the inversion formula.

math.NA↗

arXiv2022

Quantization for spectral super-resolution

We show that the method of distributed noise-shaping beta-quantization offers superior performance for the problem of spectral super-resolution with quantization whenever there is redundancy in the number of measurements. More precisely, we define the oversampling ratio $λ$ as the largest integer such that $\lfloor M/λ\rfloor - 1\geq 4/Δ$, where $M$ denotes the number of Fourier measurements and $Δ$ is the minimum separation distance associated with the atomic measure to be resolved. We prove that for any number $K\geq 2$ of quantization levels available for the real and imaginary parts of the measurements, our quantization method combined with either TV-min/BLASSO or ESPRIT guarantees reconstruction accuracy of order $O(M^{1/4}λ^{5/4} K^{- λ/2})$ and $O(M^{3/2} λ^{1/2} K^{- λ})$ respectively, where the implicit constants are independent of $M$, $K$ and $λ$. In contrast, naive rounding or memoryless scalar quantization for the same alphabet offers a guarantee of order $O(M^{-1}K^{-1})$ only, regardless of the reconstruction algorithm.

cs.IT↗