Spectral Uncertainty Overview

• Spectral uncertainty is the ambiguity in spectral representations, expressed via eigenvalue spectra, power spectral densities, and spectral risk measures.
• It applies across diverse domains such as graph signal processing, machine learning, and astrophysical spectroscopy, each employing tailored uncertainty constructs.
• Recent methods use entropic measures, Bayesian calibration, and marginalization techniques to propagate and certify uncertainty in spectral models.

Spectral uncertainty is an umbrella term for several technically distinct constructs in which uncertainty is expressed through a spectrum, a spectral measure, or a spectral-domain operator. In the cited literature, the relevant spectral object may be the eigenvalue spectrum of a kernel covariance operator, the Laplacian spectrum of a graph, a power spectral density, a reconstructed spectral function $\rho(\omega)$, a family of interval-valued eigenvalues, or the weights of a spectral risk measure. The corresponding notion of uncertainty likewise varies: entropy decomposition, localization bounds, diameter of an admissible set, empirical coverage of reported intervals, propagation of atomic-data errors, Bayesian posterior variance in eigenvalue problems, and distribution-free risk control all appear under the same broad label (Walha et al., 26 Sep 2025, Erb, 2019, Karlsson et al., 2012, Li, 10 Jun 2026).

1. Conceptual scope

The literature uses spectral uncertainty in several non-equivalent senses. In one family of works, uncertainty is a property of a spectral representation itself: for example, the spectrum of a density operator or kernel covariance operator is used to define entropy, and uncertainty is then a function of eigenvalues. In a second family, spectral uncertainty denotes a localization trade-off: a signal cannot be simultaneously concentrated in a graph vertex domain and in a graph spectral domain beyond a region determined by operator geometry. In a third family, the term refers to ambiguity of admissible spectra consistent with incomplete statistics or inverse data. In a fourth family, it denotes uncertainty attached to spectral models, such as PSDs, atomic spectra, or reconstructed transport spectra, and is assessed by marginalization, propagation, or calibration. In a fifth family, spectral representations are used to make uncertainty quantification tractable in machine learning, stochastic optimization, and reachability analysis.

Domain	Spectral object	Uncertainty construct
LLM uncertainty	density operator / kernel covariance spectrum	von Neumann entropy decomposition
Graph signal processing	Laplacian eigenbasis and localization operators	admissibility region / uncertainty curve
Spectral estimation	power spectra consistent with moments	diameter of an uncertainty set
Lattice QCD	reconstructed spectral functions	conditional adequacy test
Scientific ML and optimization	Hermitian matrices, PCE bases, Koopman spectra	posterior calibration, deterministic spectral counterpart, probabilistic reachability

A recurrent structural theme is that spectral uncertainty is rarely a single scalar notion detached from the underlying representation. It depends on what the spectrum is a spectrum of: a covariance operator, a Laplacian, a transfer kernel, a spectral density, or a parametric matrix model. This suggests that the term is best understood relationally rather than definitionally.

2. Entropic and evidential definitions in machine learning

The paper "Fine-Grained Uncertainty Decomposition in LLMs: A Spectral Approach" defines Spectral Uncertainty through the von Neumann entropy of a density operator constructed from RKHS embeddings of sampled model outputs. If $\rho$ is a density operator, then

$S(\rho) = - \mathrm{Tr}[ \rho \log \rho ].$

With $\Sigma$ denoting the density operator induced by the output distribution, the total predictive uncertainty is

$$H_{\text{total}} := H_{\mathrm{VN}}(\Sigma) = - \mathrm{Tr}[ \Sigma \log \Sigma ].$$

The core decomposition specializes a general concave-functional identity to $H_{\mathrm{VN}}$: $$H_{\text{total}} = E_W[H_{\mathrm{VN}}(P_{Y|W})] + \mathbb{H}(P_{Y|W}),$$ with the paper identifying

$$H_{\text{epi}} := E_W[H_{\mathrm{VN}}(P_{Y|W})], \qquad H_{\text{alea}} := \mathbb{H}(P_{Y|W}),$$

so that $$H_{\text{total}} = H_{\text{epi}} + H_{\text{alea}}$$. Practically, the method constructs inner Gram matrices for answers conditioned on each clarification and an outer Gram matrix over the flattened sample, then estimates the entropies from the spectra of the scaled Gram matrices. Matrix operations scale as $O((nm)^3)$ in the worst case, while $\rho$0 and $\rho$1 are kept to $\rho$2. On AmbigQA and AmbigInst, the method achieved up to $\rho$3 AUROC and $\rho$4 AUROC respectively; on TriviaQA with Phi-4, it reached $\rho$5 AUROC and $\rho$6 AUPR (Walha et al., 26 Sep 2025).

A different but related use appears in open-set hyperspectral domain generalization. There, spectral uncertainty is explicitly defined as the vacuity of a Dirichlet distribution produced from the spectral-only feature extractor. If the spectral pathway yields evidence $\rho$7, Dirichlet parameters $\rho$8, and total strength $\rho$9, then

$S(\rho) = - \mathrm{Tr}[ \rho \log \rho ].$0

This quantity is disentangled from spatial uncertainty and combined-pathway uncertainty by the Spectral-Spatial Uncertainty Disentanglement mechanism, which uses adaptive weighting and reliability-based decoupling. Reported unknown rejection rates are $S(\rho) = - \mathrm{Tr}[ \rho \log \rho ].$1 for PU $S(\rho) = - \mathrm{Tr}[ \rho \log \rho ].$2 PC, $S(\rho) = - \mathrm{Tr}[ \rho \log \rho ].$3 for HU13 $S(\rho) = - \mathrm{Tr}[ \rho \log \rho ].$4 HU18, and $S(\rho) = - \mathrm{Tr}[ \rho \log \rho ].$5 for Dioni $S(\rho) = - \mathrm{Tr}[ \rho \log \rho ].$6 Loukia; removing EDL causes the harmonic open-set score to drop by up to $S(\rho) = - \mathrm{Tr}[ \rho \log \rho ].$7 percentage points on Houston (Khoshbakht et al., 11 Jun 2025).

These two machine-learning uses share a spectral core but differ in semantics. In the LLM setting, uncertainty is the entropy of a spectrum. In the hyperspectral setting, uncertainty is the inverse total evidence of a pathway whose features are spectrally defined.

3. Localization trade-offs in graph and harmonic settings

In graph signal processing, spectral uncertainty usually denotes an uncertainty principle rather than probabilistic uncertainty quantification. One general framework introduces a spatial localization operator $S(\rho) = - \mathrm{Tr}[ \rho \log \rho ].$8 and a spectral localization operator $S(\rho) = - \mathrm{Tr}[ \rho \log \rho ].$9, with mean values

$\Sigma$0

and joint admissibility set

$\Sigma$1

For $\Sigma$2, this set is convex and coincides with the numerical range of $\Sigma$3. In the projection case, one recovers the Landau–Pollak curve

$\Sigma$4

with $\Sigma$5. The same framework gives polygonal approximations to the admissibility region by solving top-eigenvalue problems for rotated operators $\Sigma$6 (Erb, 2019).

The earlier graph uncertainty-curve formulation uses graph spread

$\Sigma$7

and spectral spread

$\Sigma$8

then defines the uncertainty curve

$\Sigma$9

Its lower boundary is achieved by eigenvectors of the affine family

$$H_{\text{total}} := H_{\mathrm{VN}}(\Sigma) = - \mathrm{Tr}[ \Sigma \log \Sigma ].$$0

The feasibility region is compact and convex, and the curve can be approximated to Hausdorff error $$H_{\text{total}} := H_{\mathrm{VN}}(\Sigma) = - \mathrm{Tr}[ \Sigma \log \Sigma ].$$1 with $$H_{\text{total}} := H_{\mathrm{VN}}(\Sigma) = - \mathrm{Tr}[ \Sigma \log \Sigma ].$$2 sparse eigenvalue evaluations. Closed-form curves are derived for complete and star graphs, while diffusion kernels $$H_{\text{total}} := H_{\mathrm{VN}}(\Sigma) = - \mathrm{Tr}[ \Sigma \log \Sigma ].$$3 closely trace, and in some cases exactly achieve, the uncertainty bound (Agaskar et al., 2012).

A 2026 extension replaces the graph Fourier transform by the graph fractional Fourier transform and constructs the sandwiched joint-localization operator

$$H_{\text{total}} := H_{\mathrm{VN}}(\Sigma) = - \mathrm{Tr}[ \Sigma \log \Sigma ].$$4

If $$H_{\text{total}} := H_{\mathrm{VN}}(\Sigma) = - \mathrm{Tr}[ \Sigma \log \Sigma ].$$5, then

$$H_{\text{total}} := H_{\mathrm{VN}}(\Sigma) = - \mathrm{Tr}[ \Sigma \log \Sigma ].$$6

or equivalently

$$H_{\text{total}} := H_{\mathrm{VN}}(\Sigma) = - \mathrm{Tr}[ \Sigma \log \Sigma ].$$7

The fractional order reshapes the vertex–fractional-spectral trade-off and can enlarge or shrink the uncertainty region relative to the standard graph Fourier case (Zhang et al., 2 Jul 2026).

An analogous non-graph harmonic-analysis line proves Ingham-type uncertainty principles for spectral projections on rank-one symmetric spaces. If $$H_{\text{total}} := H_{\mathrm{VN}}(\Sigma) = - \mathrm{Tr}[ \Sigma \log \Sigma ].$$8 and $$H_{\text{total}} := H_{\mathrm{VN}}(\Sigma) = - \mathrm{Tr}[ \Sigma \log \Sigma ].$$9, then a nonzero $H_{\mathrm{VN}}$0 vanishing on a nonempty open set cannot satisfy uniform decay of the form

$H_{\mathrm{VN}}$1

Similar results hold for spectral projections associated with the Dunkl Laplacian (Ganguly et al., 2020).

Across these works, spectral uncertainty denotes geometric admissibility and non-simultaneous localization. It is therefore structurally different from posterior variance or empirical coverage, even though the shared term can obscure that distinction.

4. Ambiguity sets, inverse problems, and adequacy of spectral claims

In spectral estimation from finite second-order statistics, spectral uncertainty is formalized as the uncertainty set of all spectra consistent with observed moments. For covariance estimates $H_{\mathrm{VN}}$2 with tolerance $H_{\mathrm{VN}}$3, the admissible set is

$H_{\mathrm{VN}}$4

Because the set may contain line spectra and discontinuities, distances are taken in the weak topology rather than by pointwise $H_{\mathrm{VN}}$5-type norms. A general weakly continuous metric has the integral form

$H_{\mathrm{VN}}$6

and the diameter

$H_{\mathrm{VN}}$7

acts as an upper bound on the distance between a nominal spectrum and the true one. For exact moments and Poisson-kernel metrics, explicit formulas are available, and a-priori bounds such as

$H_{\mathrm{VN}}$8

and

$H_{\mathrm{VN}}$9

connect uncertainty directly to lag depth or filter-bank design (Karlsson et al., 2012).

For inverse spectral problems, "Two-Spectra Theorem with Uncertainty" quantifies how interval-valued eigenvalue information interacts with partial knowledge of the potential. If Dirichlet and Neumann–Dirichlet eigenvalues are only known to lie in intervals $$H_{\text{total}} = E_W[H_{\mathrm{VN}}(P_{Y|W})] + \mathbb{H}(P_{Y|W}),$$0, the exact size of uncertainty is

$$H_{\text{total}} = E_W[H_{\mathrm{VN}}(P_{Y|W})] + \mathbb{H}(P_{Y|W}),$$1

with $$H_{\text{total}} = E_W[H_{\mathrm{VN}}(P_{Y|W})] + \mathbb{H}(P_{Y|W}),$$2 the interior Beurling–Malliavin density. Knowledge of the potential on $$H_{\text{total}} = E_W[H_{\mathrm{VN}}(P_{Y|W})] + \mathbb{H}(P_{Y|W}),$$3 yields uniqueness, while knowledge only on $$H_{\text{total}} = E_W[H_{\mathrm{VN}}(P_{Y|W})] + \mathbb{H}(P_{Y|W}),$$4 does not (Makarov et al., 2016).

For lattice QCD spectral reconstructions, the issue is not merely whether a spectrum fits the Euclidean correlator but whether a reported uncertainty law is adequate for a chosen target functional $$H_{\text{total}} = E_W[H_{\mathrm{VN}}(P_{Y|W})] + \mathbb{H}(P_{Y|W}),$$5. The proposed conditional model-adequacy test uses mock-data generation, a reconstruction adapter, empirical coverage

$$H_{\text{total}} = E_W[H_{\mathrm{VN}}(P_{Y|W})] + \mathbb{H}(P_{Y|W}),$$6

simulation-based calibration ranks, physical diagnostics, and stress tests. In the zero-temperature benchmark, $$H_{\text{total}} = E_W[H_{\mathrm{VN}}(P_{Y|W})] + \mathbb{H}(P_{Y|W}),$$7 is consistently the most calibratable summary, while $$H_{\text{total}} = E_W[H_{\mathrm{VN}}(P_{Y|W})] + \mathbb{H}(P_{Y|W}),$$8 is severely undercovered. In the finite-temperature shear application, a BG-style reconstruction family is Euclidean-compatible at $$H_{\text{total}} = E_W[H_{\mathrm{VN}}(P_{Y|W})] + \mathbb{H}(P_{Y|W}),$$9, yet pointwise peak-height intervals are not certified. The paper’s central conclusion is that Euclidean compatibility is necessary but not sufficient for spectral uncertainty claims (Li, 10 Jun 2026).

This family of works treats spectral uncertainty as a set-valued or target-wise statement: the problem is not only to infer a spectrum, but to determine how much of the spectrum is underdetermined and which claims are actually calibrated.

5. Propagation and marginalization in physical spectroscopy

In gravitational-wave parameter estimation, spectral uncertainty enters through uncertainty in the detector noise PSD. With data model $$H_{\text{epi}} := E_W[H_{\mathrm{VN}}(P_{Y|W})], \qquad H_{\text{alea}} := \mathbb{H}(P_{Y|W}),$$0, fixed-PSD likelihood

$$H_{\text{epi}} := E_W[H_{\mathrm{VN}}(P_{Y|W})], \qquad H_{\text{alea}} := \mathbb{H}(P_{Y|W}),$$1

and nuisance parameters $$H_{\text{epi}} := E_W[H_{\mathrm{VN}}(P_{Y|W})], \qquad H_{\text{alea}} := \mathbb{H}(P_{Y|W}),$$2 defining a flexible PSD model $$H_{\text{epi}} := E_W[H_{\mathrm{VN}}(P_{Y|W})], \qquad H_{\text{alea}} := \mathbb{H}(P_{Y|W}),$$3, the desired posterior is

$$H_{\text{epi}} := E_W[H_{\mathrm{VN}}(P_{Y|W})], \qquad H_{\text{alea}} := \mathbb{H}(P_{Y|W}),$$4

Using $$H_{\text{epi}} := E_W[H_{\mathrm{VN}}(P_{Y|W})], \qquad H_{\text{alea}} := \mathbb{H}(P_{Y|W}),$$5 BayesWave PSD draws and independent Bilby/dynesty runs, the reported widening of $$H_{\text{epi}} := E_W[H_{\mathrm{VN}}(P_{Y|W})], \qquad H_{\text{alea}} := \mathbb{H}(P_{Y|W}),$$6 credible intervals is typically a few percent: $$H_{\text{epi}} := E_W[H_{\mathrm{VN}}(P_{Y|W})], \qquad H_{\text{alea}} := \mathbb{H}(P_{Y|W}),$$7 for chirp mass, $$H_{\text{epi}} := E_W[H_{\mathrm{VN}}(P_{Y|W})], \qquad H_{\text{alea}} := \mathbb{H}(P_{Y|W}),$$8 for mass ratio, $$H_{\text{epi}} := E_W[H_{\mathrm{VN}}(P_{Y|W})], \qquad H_{\text{alea}} := \mathbb{H}(P_{Y|W}),$$9 for effective spin, and $$H_{\text{total}} = H_{\text{epi}} + H_{\text{alea}}$$0 for luminosity distance. For GW170817, the $$H_{\text{total}} = H_{\text{epi}} + H_{\text{alea}}$$1 sky area grew by $$H_{\text{total}} = H_{\text{epi}} + H_{\text{alea}}$$2 ($$H_{\text{total}} = H_{\text{epi}} + H_{\text{alea}}$$3); using the median PSD gives posteriors within $$H_{\text{total}} = H_{\text{epi}} + H_{\text{alea}}$$4 of the fully marginalized result (Biscoveanu et al., 2020).

In astrophysical spectral modeling, Bautista et al. derive analytic propagation equations for uncertainties in atomic rates. If level populations $$H_{\text{total}} = H_{\text{epi}} + H_{\text{alea}}$$5 solve a non-LTE balance system, the relative variances satisfy a coupled linear system

$$H_{\text{total}} = H_{\text{epi}} + H_{\text{alea}}$$6

This replaces brute-force Monte Carlo by a single algebraic solve. The paper recommends estimating $$H_{\text{total}} = H_{\text{epi}} + H_{\text{alea}}$$7 and $$H_{\text{total}} = H_{\text{epi}} + H_{\text{alea}}$$8 from the dispersion among independent calculations. For [Fe II], the mean lifetime uncertainty for $$H_{\text{total}} = H_{\text{epi}} + H_{\text{alea}}$$9 and $O((nm)^3)$0 levels is $O((nm)^3)$1, while the $O((nm)^3)$2 level has $O((nm)^3)$3 uncertainty (Bautista et al., 2013).

A complementary X-ray spectroscopy line implements Monte Carlo propagation in pyatomdb through the variableapec extension. Ionization and recombination rates are sampled from flat distributions within literature-based systematic uncertainties, line emissivities are recomputed, and uncertainty envelopes are produced for ionic fractions and diagnostic lines. For O ions near peak fraction, the $O((nm)^3)$4 error on O VI–O VIII fractions is $O((nm)^3)$5; for Fe L-shell ions, peak uncertainties are $O((nm)^3)$6, rising above $O((nm)^3)$7 in the wings. The paper recommends including atomic-data systematic uncertainties in high-resolution XRISM and Athena analyses (Heuer et al., 2020).

In $O((nm)^3)$8-ray spectrometry, hybrid spectral unmixing with deformed spectral signatures is treated in a Bayesian way. With

$O((nm)^3)$9

uniform priors on $\rho$00 and $\rho$01, and posterior mode $\rho$02, uncertainty intervals are computed either by Laplace approximation,

$\rho$03

or by NUTS-based MCMC. Repeated Monte Carlo experiments show that both methods are close to the expected $\rho$04 long-run success rate when constraints are inactive, but Laplace approximation degrades when constraints are active or background dominates, whereas MCMC remains robust (Phan et al., 22 Apr 2026).

A common misconception is that an apparently small residual or a convenient Gaussian approximation is sufficient. These papers instead show that propagation structure, constraints, and nuisance marginalization materially determine whether a reported spectral interval has the intended meaning.

6. Spectral representations for uncertainty quantification, optimization, and control

Several recent works use spectral representations not merely to analyze uncertainty, but to make uncertainty quantification computationally tractable. In causal inference with kernel methods, IMPspec starts from Mercer expansions

$\rho$05

and places Gaussian-process priors on spectral coefficients of RKHS conditional expectations. The resulting posteriors in coefficient space are Gaussian, which yields closed-form posterior moments for causal effects $\rho$06; the spectral basis is then tuned to optimize calibration of credible intervals. Reported experiments show lowest RMSE and best calibration in the ablation study, with coverage error $\rho$07 (Dance et al., 2024).

In scientific machine learning, Bayesian parametric matrix models extend Hermitian parametric matrix models by placing priors on parameters in

$\rho$08

and combining adaptive spectral perturbation bounds, manifold-aware variational inference, and finite-sample calibration guarantees. The framework reports $\rho$09 across matrix dimensions from $\rho$10 to $\rho$11, with graceful degradation under spectral ill-conditioning and reliable uncertainty estimates in near-degenerate regimes (Nooraiepour, 15 Sep 2025).

In stochastic optimization for virtual power plants, uncertainty is represented in the spectral domain by intrusive Polynomial Chaos Expansion. The stochastic program is reformulated as a low-dimensional deterministic spectral counterpart, with total-degree truncation

$\rho$12

Galerkin projection for equality constraints, and SOC moment-based approximations for chance constraints. In the reported case study, intrusive PCE achieves solution quality comparable to a scenario-based benchmark, with up to a $\rho$13 times reduction in computational effort (Zapparoli et al., 4 May 2026).

The term also appears in risk theory. A spectral risk measure has the form

$\rho$14

and one can quantify uncertainty in the scenario-dependent values $\rho$15 via deviation measures such as the full range, variance, standard deviation, or Gini deviation. In the NASDAQ case study with $\rho$16 three-month scenarios and Extended Gini Shortfall, the reported values are $\rho$17, $\rho$18, and worst-case spectral risk $\rho$19 (Berkhouch et al., 2019). A distribution-free control extension, Conformal Spectral Risk Control, uses weighted conformal risk control to guarantee that a chosen spectral risk of prediction-set loss stays below a user threshold $\rho$20, and then embeds the resulting tightened constraints into MPC. In dynamic obstacle avoidance, the reported violation rate drops from $\rho$21 for the baseline SAA-MPC to $\rho$22 for CSRC-MPC, while solve time decreases from $\rho$23 ms to $\rho$24 ms (Eom et al., 2 Jun 2026).

Finally, in dynamical systems reachability, Koopman spectral uncertainty is used to encode model uncertainty from finite data directly into Koopman spectral representation with quantifiable error bounds, and to determine time intervals within which trajectories from the initial set are guaranteed, with a prescribed probability, to reach the target set, without explicit computation of reachable sets (Ding et al., 28 Nov 2025). A separate dimensionality-reduction line introduces NGEU, which takes probability distributions as inputs, extends methods such as KPCA and MDA/KMFA, and analyzes generalization through Rademacher complexity (Laakom et al., 2022).

Taken together, these works indicate that spectral representations can make uncertainty both more structured and more actionable. A plausible implication is that the main divide in the field is no longer between “spectral” and “non-spectral” methods, but between methods that merely parameterize uncertainty in a spectral basis and methods that additionally calibrate, propagate, or certify that uncertainty for the target quantity of interest.