Spectral Recoverability: Principles & Applications

• Spectral recoverability is the study of conditions under which unknown signals are reconstructed from spectral measurements with guarantees on identifiability, stability, and sample complexity.
• It spans diverse applications including phase retrieval, super-resolution, inverse spectral uniqueness, and graph recovery, each relying on tailored thresholds and preprocessing techniques.
• The framework leverages high-dimensional analysis, sparsity, and optimal estimator design to ensure robust recovery even under noise, missing data, and structural constraints.

Spectral recoverability denotes the conditions under which an unknown object can be reconstructed from spectral information, or recovered by a spectral estimator, with quantitative guarantees on identifiability, sample complexity, asymptotic overlap, or stability. In the literature summarized here, the term appears in several closely related senses: weak recovery by leading eigenvectors in high-dimensional generalized linear models and phase retrieval (Luo et al., 2018); exact or stable reconstruction from spectral measurements, such as line spectra, high-order spectra, or phaseless Fourier data (Mishra et al., 2014); and inverse-spectral uniqueness from eigenvalues, minors, or spectral measures (Maciążek et al., 2021). This suggests a common research program centered on thresholds, optimal preprocessing, structural priors, and robustness under missing data or noise.

1. Core meanings and formal criteria

In high-dimensional statistical estimation, spectral recoverability is typically defined through nontrivial correlation between a spectral estimator and the ground truth. For generalized linear estimation and phase retrieval, an estimator achieves “weak recovery’’ if it is positively correlated with the signal in the joint limit $n,d\to\infty$ with linear sample complexity (Mondelli et al., 2017). In the phase-retrieval formulation of Luo–Alghamdi–Lu, the central quantity is the normalized squared overlap

$$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$

where $x_1$ is the leading eigenvector of a weighted data matrix and $T$ is the preprocessing function (Luo et al., 2018).

In signal processing and super-resolution, the same phrase refers to exact or stable determination of a signal from structured spectral data. Examples include “off-the-grid’’ line-spectrum recovery with prior knowledge encoded through weighted, constrained, or conditional atomic norms (Mishra et al., 2014); recovery from a few linear measurements of the bispectrum or trispectrum (Bendory et al., 2021); and sparse phase retrieval from power spectral density or partial phaseless Fourier measurements (Jaganathan et al., 2013). In these settings, recoverability is expressed through uniqueness theorems, minimax error rates, or sample bounds such as $3s$, $O(k^2\log n)$, or $O(k\log n)$, depending on the model and the measurement design.

In inverse problems, spectral recoverability refers to unique reconstruction from eigenvalue data, spectral measures, or nested spectra. Real symmetric matrices can be recovered from the spectra of all leading principal minors together with sign indicators, and banded matrices admit generic recoverability with substantially fewer signs (Maciążek et al., 2021). Schrödinger operators on a finite interval can be uniquely determined from one full spectrum together with subsets of a second spectrum and point masses of the spectral measure (Hatinoğlu, 2019). A plausible implication is that “spectral recoverability” is best treated as a family of identifiability and stability statements rather than a single model-independent definition.

2. High-dimensional spectral initialization and weak recovery

For phase retrieval with Gaussian sensing vectors, the observation model is

$$y_i \sim p\!\bigl(y\mid|s_i|\bigr),\qquad s_i=\langle a_i,\xi\rangle,\;a_i\sim_{\text{i.i.d.}}\mathcal{CN}(0,I_n),$$

with $\|\xi\|=1$. Spectral initialization forms

$D=\frac1m\sum_{i=1}^m T(y_i)\,a_i\,a_i^*,$

and takes $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$0 to be the leading eigenvector. In the proportional limit $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$1 with $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$2, the asymptotic overlap is determined by the functions

$$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$3

and

$$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$4

If there is a unique solution $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$5 to $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$6 with $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$7, then

$$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$8

and otherwise $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$9 (Luo et al., 2018).

The optimal-design problem can be rewritten in a weighted $x_1$0 space. After the change of variables $x_1$1, the relaxed problem leads to a closed-form optimal spectral weight. Defining

$x_1$2

one obtains, for each $x_1$3, a unique $x_1$4 satisfying $x_1$5, the optimal performance curve

$x_1$6

and the optimal spectral weight

$x_1$7

The weak reconstruction threshold is

$x_1$8

Below $x_1$9, one has $T$0 for all $T$1; above it, positive overlap is possible. Under the mild technical condition $T$2, the same $T$3 is exactly optimal for every $T$4 (Luo et al., 2018).

A closely related threshold theory was developed for generalized linear models and phase retrieval with Gaussian measurements. In the vanishing-noise phase-retrieval regime, $T$5 implies that no estimator can do significantly better than random, while $T$6 allows a simple weighted spectral estimator to achieve positive correlation. In the noiseless phase-retrieval case, $T$7, and the optimal pre-processing converges to

$T$8

The same $T$9 also marks the “algorithmic’’ threshold for AMP (Mondelli et al., 2017).

The framework extends further. For right-unitarily invariant sensing matrices and arbitrary channel noise, the optimal spectral method can be derived from the linearization of message-passing algorithms and the Bethe Hessian. The optimal preprocessing is

$3s$0

and the weak-recovery transition occurs exactly at $3s$1 (Maillard et al., 2020). For multi-index models with fixed subspace dimension $3s$2, the top-$3s$3 eigenvalues of the spectral matrix exhibit an outlier phase transition, and the critical sampling ratio is

$3s$4

with matching optimal preprocessing $3s$5 in closed form (Kovačević et al., 3 Feb 2025).

3. Super-resolution, sparsity, and high-order spectral measurements

In grid-free line-spectrum estimation, prior knowledge can be built directly into convex optimization. For spectrally sparse signals

$3s$6

three classes of priors are considered: probabilistic priors through a weighted atomic norm, block priors through a constrained atomic norm, and known-pole priors through a conditional atomic norm. All three are expressed as SDP formulations via positive trigonometric polynomials and dual-polynomial constraints. The decisive recoverability statement is Theorem 4: if each $3s$7 lies in a very narrow band and one can enforce

$3s$8

then for any choice of $3s$9 frequencies, as long as $O(k^2\log n)$0 random samples are observed, with probability $O(k^2\log n)$1 the corresponding $O(k^2\log n)$2 system is non-degenerate, and perfect recovery is possible with only $O(k^2\log n)$3 samples (Mishra et al., 2014).

Stable recoverability in super-resolution also depends sharply on local geometry. For sparse measures

$O(k^2\log n)$4

with one cluster of $O(k^2\log n)$5 near-colliding nodes, the relevant parameter is $O(k^2\log n)$6. If

$O(k^2\log n)$7

then the minimax-optimal recovery rates separate clustered and non-clustered components. For $O(k^2\log n)$8 cluster,

$O(k^2\log n)$9

whereas for $O(k\log n)$0 cluster,

$O(k\log n)$1

The exponents in $O(k\log n)$2 and $O(k\log n)$3 are minimax-optimal, and Matrix Pencil achieves the predicted scaling numerically (Batenkov et al., 2019).

High-order spectra provide a different route to recoverability. For $O(k\log n)$4, the $O(k\log n)$5-th order spectrum

$O(k\log n)$6

is invariant under circular shifts. If the full map $O(k\log n)$7 is birational onto its image, then for a generic matrix $O(k\log n)$8 with $O(k\log n)$9, the composite $$y_i \sim p\!\bigl(y\mid|s_i|\bigr),\qquad s_i=\langle a_i,\xi\rangle,\;a_i\sim_{\text{i.i.d.}}\mathcal{CN}(0,I_n),$$0 is also birational onto its image. Equivalently, a generic signal can be recovered from only $$y_i \sim p\!\bigl(y\mid|s_i|\bigr),\qquad s_i=\langle a_i,\xi\rangle,\;a_i\sim_{\text{i.i.d.}}\mathcal{CN}(0,I_n),$$1 generic linear measurements of its $$y_i \sim p\!\bigl(y\mid|s_i|\bigr),\qquad s_i=\langle a_i,\xi\rangle,\;a_i\sim_{\text{i.i.d.}}\mathcal{CN}(0,I_n),$$2-th spectrum, up to circular shift (Bendory et al., 2021).

Sparse phase retrieval from Fourier magnitudes introduces a different limitation. Lifting-based SDP methods encounter a square–root bottleneck and typically recover correctly only when $$y_i \sim p\!\bigl(y\mid|s_i|\bigr),\qquad s_i=\langle a_i,\xi\rangle,\;a_i\sim_{\text{i.i.d.}}\mathcal{CN}(0,I_n),$$3. With partial phaseless Fourier measurements, the autocorrelation is at most $$y_i \sim p\!\bigl(y\mid|s_i|\bigr),\qquad s_i=\langle a_i,\xi\rangle,\;a_i\sim_{\text{i.i.d.}}\mathcal{CN}(0,I_n),$$4-sparse, so $$y_i \sim p\!\bigl(y\mid|s_i|\bigr),\qquad s_i=\langle a_i,\xi\rangle,\;a_i\sim_{\text{i.i.d.}}\mathcal{CN}(0,I_n),$$5 random Fourier rows suffice for exact recovery via $$y_i \sim p\!\bigl(y\mid|s_i|\bigr),\qquad s_i=\langle a_i,\xi\rangle,\;a_i\sim_{\text{i.i.d.}}\mathcal{CN}(0,I_n),$$6-minimization of the autocorrelation followed by phase retrieval from full autocorrelation. If the measurement matrices can be designed, then a $$y_i \sim p\!\bigl(y\mid|s_i|\bigr),\qquad s_i=\langle a_i,\xi\rangle,\;a_i\sim_{\text{i.i.d.}}\mathcal{CN}(0,I_n),$$7-sparse signal can be recovered using only $$y_i \sim p\!\bigl(y\mid|s_i|\bigr),\qquad s_i=\langle a_i,\xi\rangle,\;a_i\sim_{\text{i.i.d.}}\mathcal{CN}(0,I_n),$$8 phaseless measurements (Jaganathan et al., 2013).

4. Missing data, spectral gaps, and robust reconstruction

Recoverability from incomplete observations can be guaranteed by spectral degeneracy conditions. For discrete-time sequences $$y_i \sim p\!\bigl(y\mid|s_i|\bigr),\qquad s_i=\langle a_i,\xi\rangle,\;a_i\sim_{\text{i.i.d.}}\mathcal{CN}(0,I_n),$$9 with Z-transform $\|\xi\|=1$0, if $\|\xi\|=1$1 has a zero of effective order $\|\xi\|=1$2 at $\|\xi\|=1$3, then finitely many missing values indexed by $\|\xi\|=1$4 can be uniquely and stably recovered from $\|\xi\|=1$5. The explicit recovery filter is defined by a transfer function $\|\xi\|=1$6 that equals $\|\xi\|=1$7 on the pass-band and $\|\xi\|=1$8 near $\|\xi\|=1$9, with inverse Z-transform

$D=\frac1m\sum_{i=1}^m T(y_i)\,a_i\,a_i^*,$0

The zero-pattern $D=\frac1m\sum_{i=1}^m T(y_i)\,a_i\,a_i^*,$1 for $D=\frac1m\sum_{i=1}^m T(y_i)\,a_i\,a_i^*,$2 guarantees that unknown samples do not enter the recovery sum. Under additive $D=\frac1m\sum_{i=1}^m T(y_i)\,a_i\,a_i^*,$3-noise with $D=\frac1m\sum_{i=1}^m T(y_i)\,a_i\,a_i^*,$4, the reconstruction satisfies

$D=\frac1m\sum_{i=1}^m T(y_i)\,a_i\,a_i^*,$5

(Dokuchaev, 2018).

A different mechanism is provided by non-periodic spectrum gaps and $D=\frac1m\sum_{i=1}^m T(y_i)\,a_i\,a_i^*,$6-braided spectrum degeneracy. The classes $D=\frac1m\sum_{i=1}^m T(y_i)\,a_i\,a_i^*,$7 are uniformly recoverable from the single periodic subsequence

$D=\frac1m\sum_{i=1}^m T(y_i)\,a_i\,a_i^*,$8

and every $D=\frac1m\sum_{i=1}^m T(y_i)\,a_i\,a_i^*,$9 is uniquely determined by its full periodic subsequence $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$00, even after deletion of a finite central block. The same class is everywhere dense in $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$01, and the recovery remains uniformly accurate under additive $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$02-noise and truncation (Dokuchaev, 2018).

For dynamical sampling on a finite cyclic grid, one observes subsampled snapshots of $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$03, where $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$04 is a circular convolution operator on $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$05. After Fourier-domain aliasing, each channel yields a scalar sequence that is a mixture of at most $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$06 exponentials, and the associated Hankel matrix has low rank. In the presence of time-sparse corruptions, the recovery pipeline combines robust low-rank Hankel separation, low-rank completion, and Prony-type estimation. Under incoherence and conditioning assumptions, if

$$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$07

then with probability at least $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$08 the recovered Hankel blocks satisfy $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$09, and the Prony stage inherits $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$10-level root error (Cai et al., 10 Apr 2026).

Not all spectral reconstruction problems admit pointwise identification from finite noisy data. For Euclidean correlators of the form

$$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$11

the inverse Laplace problem is ill-posed. However, if one restricts to linear functionals

$$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$12

and imposes positivity $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$13, then a convex program and its Lagrange dual provide rigorous upper and lower bounds. These bounds are information-theoretically complete: for any value $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$14 inside the interval, there exists a nonnegative $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$15 consistent with the data and satisfying $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$16 (Lawrence, 2024).

5. Graphs, Fourier complexity, and community recovery

Graph recovery from partial information introduces a Fourier-analytic notion of recoverability. For a graph $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$17 on $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$18 vertices labeled by $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$19, the edge indicator $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$20 has discrete Fourier transform $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$21, and its Fourier ratio is

$$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$22

The isomorphism-invariant graph complexity is

$$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$23

A fundamental lower bound is

$$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$24

where $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$25 is graph energy and $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$26. If $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$27 satisfies $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$28 and $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$29, and each point of $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$30 is sampled independently with probability

$$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$31

then with probability at least $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$32, the solution $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$33 to Fourier-side $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$34-minimization satisfies

$$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$35

In particular, $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$36 random samples of adjacency entries suffice once a labeling with small Fourier ratio is available. The same framework applies to Laplacian spectral projectors, with

$$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$37

Natural labelings of cycles and circulants yield small Fourier complexity, whereas random labelings typically give large Fourier complexity (Gupta et al., 13 Jun 2026).

A different notion of spectral recoverability governs exact community detection in the Labeled Stochastic Block Model. In the logarithmic-degree regime, the information-theoretic threshold is

$$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$38

where $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$39. If $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$40 and, for each label $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$41, the matrix $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$42 has $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$43 distinct, nonzero eigenvalues, then a polynomial-time spectral algorithm exactly recovers the community labels with probability $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$44. The algorithm forms $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$45 indicator matrices $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$46, computes their top $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$47 eigenpairs, solves for weight vectors $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$48, enumerates the $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$49 sign choices, and outputs the candidate labeling with highest posterior probability (Gaudio et al., 2024).

These two graph-theoretic examples use “spectral” in different senses—Fourier compressibility of adjacency structure and eigenvector structure of random block models—but both tie recoverability to a threshold separating uninformative spectra from exploitable structure.

6. Inverse spectral uniqueness and multimodal recoverability

For real symmetric matrices, recoverability can be posed as a telescopic inverse problem. Let $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$50 denote the $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$51 leading principal minor, and let $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$52 be its spectrum. If $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$53 is regular—meaning that every $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$54 has simple spectrum and $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$55—then the data

$$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$56

uniquely determine $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$57. The reconstruction is inductive: one recovers the new diagonal entry from traces, solves the Cauchy system

$$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$58

obtains

$$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$59

and reconstructs $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$60. For symmetric $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$61-banded matrices with bandwidth $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$62, there is an open dense full-measure subset for which the spectra together with only the signs of $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$63 uniquely determine the matrix (Maciążek et al., 2021).

Inverse spectral uniqueness for Schrödinger operators takes an analogous form. For

$$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$64

one full spectrum $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$65, together with $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$66 from a second spectrum and $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$67 from the spectral measure, uniquely determine $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$68 almost everywhere on $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$69. The proof splits the Weyl $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$70-function as $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$71, uses a product representation and a Čebotarev-type representation

$$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$72

then recovers the missing zeros $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$73 and invokes the classical two-spectrum Borg–Levinson theorem (Hatinoğlu, 2019).

Recoverability results also arise in unregistered hyperspectral–multispectral fusion. Let $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$74 be the latent high-spectral MSI and $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$75 the latent high-spatial HSI. Under a shared-endmember linear mixture model and low-rank abundance maps, coupled LL1 tensor factorization yields exact MSI super-resolution: if the stated rank and dimension conditions hold, then any global optimizer recovers

$$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$76

and hence $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$77. For HSI super-resolution, a multimodal patch-generative model with continuous bijections and “Sufficiently Diverse Abundances” implies that the adversarial distribution-matching problem has unique minimizer $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$78, yielding exact recovery of $$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$79. If SDA is only approximately satisfied, then

$$\frac{|\langle\xi,x_1\rangle|^2}{\|\xi\|^2\,\|x_1\|^2}\xrightarrow{\;P\;}\rho(\alpha;T),$$80

(Song et al., 23 Mar 2026).

Taken together, these results show that spectral recoverability is governed less by a single universal mechanism than by a recurring set of structural principles: outlier emergence in random matrices, dual-certificate or convex-analytic identifiability, sparsity or low-rank structure, and inverse-spectral uniqueness from sufficiently rich auxiliary data. In each setting, the decisive question is not whether spectral information is present, but whether it crosses the relevant threshold for informative reconstruction.