Gradient-MUSIC Algorithm

• Gradient-MUSIC is a nonconvex reformulation of the classical MUSIC technique that estimates frequencies and amplitudes from noisy time samples using a specially constructed landscape function.
• It uses coarse-grid thresholding and gradient descent in locally strongly convex regions to achieve exponential convergence and minimax optimal error rates even under ℓ^p-bounded noise.
• The algorithm significantly reduces computational complexity compared to classical MUSIC by limiting grid evaluations and leveraging efficient local search, making it robust for well-separated spectral signals.

The Gradient-MUSIC algorithm is a nonconvex optimization-based reformulation of the classical MUSIC (MUltiple SIgnal Classification) technique for estimating the frequencies and amplitudes of nonharmonic Fourier sums from noisy time samples. By optimizing a carefully constructed landscape function derived from the subspace method, Gradient-MUSIC achieves minimax optimal recovery rates with significant computational advantages over the classical approach in the presence of deterministic $\ell^p$-bounded perturbations, provided that the underlying frequencies are sufficiently ^^^^1^^^^ separated ($m\Delta\geq 8\pi$) (Fannjiang et al., 9 Apr 2025).

1. Mathematical Framework for Spectral Estimation

The signal model considered is

$h(\xi) = \sum_{j=1}^s a_j e^{ix_j\xi},$

where $\{x_j\}_{j=1}^s$ are unknown frequencies in $[0,2\pi)$ and $\{a_j\}_{j=1}^s$ are unknown complex amplitudes. The observations are noisy samples:

$y_k = h(k) + \eta_k,\quad k = -m+1, \ldots, m-1,$

with $\eta_k$ denoting noise, typically bounded in $\ell^p$. The goal is to recover both $\{x_j\}$ and $\{a_j\}$ from $y = (y_{-m+1},\ldots,y_{m-1})$.

The classical MUSIC algorithm proceeds by estimating a signal subspace via singular value decomposition (SVD) of a Hankel/Toeplitz matrix $T(y)$, then evaluates a trigonometric polynomial "landscape" function $q(t)$,

$q(t) = 1 - \|U^*\phi(t)\|_2^2,$

where $U\in\mathbb{C}^{m\times s}$ contains the estimated signal subspace, and $\phi(t)\in\mathbb{C}^m$ is the normalized steering vector. The minima of $q(t)$ approximate the true frequencies $\{x_j\}$.

2. Coarse-Grid Thresholding and Initialization

Gradient-MUSIC circumvents the computational cost of fine-grid search by evaluating $q(t)$ on a coarse grid $G\subset T$ with mesh size

$$\|(G)\| = \max_{t\in T} \min_{u\in G}|t-u| \le \frac{1}{2m}.$$

A threshold $\alpha\in(0,1)$ (with $\alpha = 0.529$ established as sufficient) is set. Any grid point $u\in G$ with $q(u)<\alpha$ is accepted; all others are rejected. Theoretical guarantees state that for sufficiently small subspace error $\vartheta$ and frequency separation $\Delta\ge 8\pi/m$, $q(t)\ge 0.529$ outside the union of intervals of width $8\pi/(3m)$ centered at each $x_j$, ensuring that each "well" (local minimum) contains an accepted coarse grid point. There is no advantage to a finer mesh for initialization purposes.

Accepted points $A=\{u\in G: q(u)<\alpha\}$ are algorithmically grouped into $s$ connected clusters; each cluster corresponds to the basin of attraction of a true frequency. A representative point $t_{j,0}$ from each cluster is initialized for subsequent gradient descent, with a guaranteed proximity of $|t_{j,0}-x_j|\le4\pi/(3m)$.

3. Gradient-Based Local Search and Geometry

Within each cluster, gradient descent is performed on the univariate $q(t)$ landscape. The function $q:T\to\mathbb{R}$ is $C^3$, with explicit formulas for its gradient and Hessian: $$q'(t) = -2\,\Re\left\langle \widetilde U^*\phi(t),\,\widetilde U^*\phi'(t) \right\rangle,$$

$$q''(t) = 2\|\widetilde U^*\phi'(t)\|_2^2 + 2\,\Re\langle \widetilde U^*\phi(t),\,\widetilde U^*\phi''(t) \rangle.$$

It is established that, in an interval of width $\pi/(3m)$ around each $x_j$, $q$ is strongly convex with $0.0271 m^2\leq q''(t)\leq 0.269 m^2$. This structure yields $L$-smoothness and strong convexity constants ($L = 0.269 m^2$, $\mu = 0.0271 m^2$) for efficient local optimization.

Gradient descent with step size $h \le 2/(L+\mu)\approx 6/m^2$ is guaranteed to converge exponentially fast to $x_j$ once the iterate falls within the convex core of $q$ around $x_j$. Entry into this core from the initialization region occurs in at most 31 iterations. The overall convergence satisfies

$$|t_{j,n} - x_j| \le \frac{7\vartheta}{m} + \frac{77\pi}{m}(0.839)^n,$$

where $(0.839)$ is the contraction factor per step in the strong convex region.

4. Minimax Optimality Under Deterministic $\ell^p$ Noise

For noise $\|\eta\|_p\le\epsilon$ and minimum amplitude $|a_j|\ge a_{\min}$, specialized Toeplitz-subspace perturbation and geometric landscape analysis yield minimax rates: $$\max_j|\widehat x_j - x_j| \le C\,\frac{\epsilon}{a_{\min} m^{1+1/p}}, \qquad \max_j|\widehat a_j - a_j| \le C\,\sqrt{s}\frac{\epsilon}{m^{1/p}}.$$ These error bounds are information-theoretically optimal up to constants: no algorithm, even those with unbounded computational resources, can guarantee better accuracy in this regime due to separate lower bounds on frequency and amplitude estimation.

Gradient-MUSIC thus achieves both minimax optimality and polynomial computational complexity for all $p\in[1,\infty]$, including the extreme cases of white noise and nonstationary independent Gaussian noise, up to logarithmic factors.

5. Computational Complexity and Comparison with Classical MUSIC

Classical MUSIC necessitates exhaustive search over a fine grid $G$ with mesh much smaller than $\vartheta/m$ to guarantee $O(\vartheta/m)$ frequency error. The number of required evaluations is $|G| \sim O(m\vartheta^{-1})$, yielding overall complexity $O(m^2s\vartheta^{-1})$ (each evaluation costs $O(ms)$). In contrast, Gradient-MUSIC evaluates $q(t)$ only on a much coarser grid ($|G|=O(m)$), performs $s$ clusters, and applies $n=O(\log(1/\vartheta))$ gradient steps per frequency. The resulting overall complexity is

$O(m^2 s + m s^2 \log(1/\vartheta)),$

which is significantly more efficient for small $\vartheta$, while achieving the same optimal estimation rates.

Comparative Summary Table

Algorithm	Grid Mesh	Complexity
Classical MUSIC	$\ll \vartheta/m$	$O(m^2 s \vartheta^{-1})$
Gradient-MUSIC	$1/(2m)$	$O(m^2 s + m s^2 \log(1/\vartheta))$

6. Algorithm Outline

The procedure for Gradient-MUSIC comprises:

1. Subspace Estimation:
   • Form Toeplitz matrix $T(y)$.
   • Estimate $s$ via singular value thresholding.
   • Compute leading $s$ singular vectors $\widetilde U \in \mathbb{C}^{m \times s}$.
2. Frequency Estimation:
   • Build a coarse grid $G$ with mesh $\leq 1/(2m)$.
   • Compute $q(t)$ on $G$.
   • Accept points where $q(u)<\alpha$, cluster into $s$ regions, select one representative per cluster.
   • For each, run gradient descent on $q$ with $h=6/m^2$ until convergence.
3. Amplitude Estimation:
   • Solve $\widehat a = [\Phi(\widehat x)]^\dagger y$, with $[\Phi(\widehat x)]_{kj} = e^{ik\widehat x_j}$.

The output $(\widehat x_j, \widehat a_j)_{j=1}^s$ achieves the minimax optimal error rates under $\ell^p$-bounded noise.

7. Landscape Structure and Guarantees

Under the frequency separation constraint $m\Delta\geq 8\pi$, the nonconvex objective $q(t)$ possesses exactly $s$ well-separated, locally strongly convex minima corresponding to the true frequencies. This structure is essential for both the initialization—via coarse grid thresholding—and the local exponential convergence of gradient descent. Theoretical analysis confirms the robust performance of the algorithm across the prescribed noise models and separation regimes. Both Gradient-MUSIC and classical MUSIC are thus the first provably optimal and computationally tractable spectral estimation methods for deterministic $\ell^p$ perturbations in the well-separated regime (Fannjiang et al., 9 Apr 2025).