Energy-Aware Spectral Decomposition

• Energy-Aware Spectral Decomposition is a framework that integrates physical energy conservation or optimal energy partitioning within spectral or modal analysis.
• It employs diverse methods—from exact FFT-domain power identities and SPOD to graph spectral adaptations—to ensure that energy measures remain interpretable and balanced.
• Its applications span imaging, turbulent flows, control systems, and Hamiltonian discretizations, offering practical insights into energy‐optimal and physically consistent decompositions.

Energy-Aware Spectral Decomposition denotes a family of spectral or modal constructions in which energy, power, level spacing, or another physically meaningful budget is preserved, equalized, or made explicit in the decomposition itself. In the works considered here, that role is played by exact FFT-domain power identities, ensemble energy spectral densities on graphs, minimum-energy reachability through inverse controllability Gramians, Hamiltonian conservation in spectral discretizations, and energy-bin–resolved image models in spectral CT (Gokcesu et al., 2022, Behjat et al., 17 Feb 2025, Iskakov, 1 Apr 2026, Brugnano et al., 2014, Xie et al., 2019). This suggests that the term is not tied to a single algorithmic template; rather, it names a recurrent design principle in which spectral structure is constrained by an energy accounting that remains physically or statistically interpretable.

1. Conceptual scope

A recurring feature of these methods is that the decomposition is not introduced as a purely geometric change of basis. Instead, the spectral representation is tied to a conservation law, an equal-energy partition, an energy-optimal criterion, or a realization-specific spectral trend. In one line of work, the decomposition is required to satisfy an exact power split between extracted components and residual; in another, spectral kernels are warped so that each subband captures the same amount of ensemble energy; in another, the decomposed object is the controllability Gramian inverse that directly determines minimum-energy reachability; and in yet another, the space discretization of a Hamiltonian PDE is organized so that the truncated semi-discrete system remains Hamiltonian (Gokcesu et al., 2022, Behjat et al., 17 Feb 2025, Iskakov, 1 Apr 2026, Brugnano et al., 2014).

The same principle appears in modal data analysis. Spectral Proper Orthogonal Decomposition (SPOD) preserves the POD logic of energy ranking while imposing a temporal constraint through diagonal filtering of the snapshot correlation matrix, and multiscale POD decomposes data into energy-optimal modes within prescribed frequency bands (Sieber et al., 2015, Belda et al., 13 Apr 2026). In adaptive decomposition of nonstationary signals, the energy-preserving EMD construction shows that Parseval-type energy additivity can hold even when the components are not pairwise orthogonal, provided each component is orthogonal to the sum of those that follow (Singh et al., 2015). Taken together, these formulations suggest that “energy-aware” usually means that the decomposition is built around the quantity one ultimately wishes to interpret—variance, power, Hamiltonian, controllability energy, or spectral significance—rather than only around spectral localization.

2. Power-preserving and energy-ranked signal decompositions

In frequency-domain peak separation, robustness and energy accounting are made explicit. The method of "Robust, Nonparametric, Efficient Decomposition of Spectral Peaks under Distortion and Interference" models each extracted peak as a pseudo-symmetric function around a center bin and solves a magnitude-only isotonic regression under an alternating monotone ordering constraint. At iteration $r$, the residual is updated by

$W_k \leftarrow W_k - Z_k^{(r)},$

so that after $R$ iterations

$Y_k = \sum_{r=1}^{R} Z_k^{(r)} + W_k.$

The key energy-aware property is the exact power preserving equality

$$\sum_{r=1}^{R}\sum_{k=0}^K|Z_k^{(r)}|^2+\sum_{k=0}^K\left|Y_k-\sum_{r=1}^{R}Z_k^{(r)}\right|^2=\sum_{k=0}^{K}|Y_k|^2,$$

which gives each extracted spectral peak an interpretable share of the total observed spectral power (Gokcesu et al., 2022).

The Hilbert-spectrum literature addresses a related problem from the opposite direction: standard EMD reconstructs the signal exactly but does not, in general, preserve energy across IMFs because the components are not orthogonal. "The Hilbert spectrum and the Energy Preserving Empirical Mode Decomposition" introduces linearly independent, non orthogonal yet energy preserving decompositions and shows that if

$$\mathbf{x}=\sum_{i=1}^{n+1}\mathbf{x}_i,\qquad \mathbf{x}_i \perp \sum_{j=i+1}^{n+1}\mathbf{x}_j,$$

then

$$\|\mathbf{x}\|^2=\sum_{i=1}^{n+1}\|\mathbf{x}_i\|^2.$$

This yields EPIMFs and motivates reverse-order Gram–Schmidt orthogonalization, from the lowest-frequency IMF or residue to the highest-frequency IMF, to obtain orthogonal components that preserve energy while better retaining IMF character (Singh et al., 2015).

SPOD occupies an intermediate position between POD and Fourier decomposition. In SPOD, the snapshot correlation matrix $R$ is filtered along its diagonals,

$S_{i,j}=\sum_{k=-N_f}^{N_f} g_k\,R_{i+k,j+k},$

before eigendecomposition. The filter half-width $N_f$ continuously shifts the method from energetically optimal POD at $W_k \leftarrow W_k - Z_k^{(r)},$0 toward frequency-pure Fourier decomposition in the large-filter limit. The eigenvalues of the filtered operator still provide an energy ranking, but that ranking is now subject to a temporal coherence constraint (Sieber et al., 2015).

3. Adaptive decompositions of energy spectra and hidden energy scales

Energy-aware decomposition also appears in spectral statistics, where the issue is not power conservation but the removal of nonuniversal spectral trends without destroying fluctuation information. "Unfolding a composed ensemble of energy spectra using singular value decomposition" addresses the case where spectra come from a composed ensemble with strong realization-to-realization fluctuations, varying control parameters, varying matrix sizes, or different energy windows. Ordered eigenvalues are arranged into

$W_k \leftarrow W_k - Z_k^{(r)},$1

and decomposed as

$W_k \leftarrow W_k - Z_k^{(r)},$2

The right singular vectors encode common energy-dependent spectral trends, the left singular vectors encode realization-specific amplitudes, and dominant low-$W_k \leftarrow W_k - Z_k^{(r)},$3 modes define the unfolding trend. In the basic composed ensemble, the trend is reconstructed from the first two modes,

$W_k \leftarrow W_k - Z_k^{(r)},$4

and realization-specific local spacings are estimated by

$W_k \leftarrow W_k - Z_k^{(r)},$5

The scree-law observation

$W_k \leftarrow W_k - Z_k^{(r)},$6

is used to separate secular structure from GOE-like fluctuation content (Berkovits, 2023).

A different use of spectral decomposition appears in collider physics. "Spectral Decomposition of Missing Transverse Energy at Hadron Colliders" writes a visible recoil distribution as a superposition over invisible invariant masses. For a variable $W_k \leftarrow W_k - Z_k^{(r)},$7, especially MET, the hadron-level spectrum is expressed as

$W_k \leftarrow W_k - Z_k^{(r)},$8

and, after discretization,

$W_k \leftarrow W_k - Z_k^{(r)},$9

Here the basis functions are recoil spectra for virtual mediators of fixed invariant mass, while the coefficients encode the dark-matter invariant-mass distribution in a Källén–Lehmann-like manner (Bae et al., 2017). This suggests an energy-aware decomposition in which the latent variable is a hidden energy scale rather than a frequency.

4. Graph, network, operator, and control formulations

On graphs, the most explicit energy-aware construction is the use of an ensemble energy spectral density to shape the spectral partition itself. "Signal-adapted decomposition of graph signals" defines, for a normalized signal ensemble $R$0,

$R$1

and warps a prototype Parseval frame $R$2 into

$R$3

When the exact or approximate ensemble energy density is used, each subband captures equal ensemble energy: $R$4 The method therefore accounts at the same time for graph topology and signal features, and provides a meaningful interpretation of subbands in terms of coarse-to-fine representations (Behjat et al., 17 Feb 2025).

The same graph-spectral logic is used in time-series forecasting by xCPD. Channel-patches are treated as graph nodes, a shared graph Fourier basis is obtained from the average normalized Laplacian, and node-wise spectral energy responses are defined by

$R$5

Learnable low-, mid-, and high-frequency partitions aggregate these energies, and the dominant band routes each node through frequency-specific experts (Li et al., 14 Mar 2026). Here the decomposition is “energy-aware” because routing depends on how much of each channel-patch embedding lies in each graph-frequency range.

In low-inertia power networks, generalized spectral clustering is built from the linearized synchronization dynamics rather than from static topology. Around a synchronized operating point, the first-order reduced model is

$R$6

with dynamic edge weights

$R$7

The coherent partition is extracted from the generalized eigensystem

$R$8

This embeds buses using the slow synchronization modes of the operating-point-dependent network and balances dynamic cut strength against total damping within clusters (Ogbonna et al., 9 Jan 2026).

In discrete-time control, the decomposition target is the Gramian itself. "Spectral Decomposition of Discrete-Time Controllability Gramian and Its Inverse via System Eigenvalues" writes the infinite-horizon controllability Gramian as

$R$9

and gives an explicit decomposition of $Y_k = \sum_{r=1}^{R} Z_k^{(r)} + W_k.$0, the matrix appearing in

$Y_k = \sum_{r=1}^{R} Z_k^{(r)} + W_k.$1

The modal and pairwise terms expose which eigenmodes and modal interactions make target states cheap or expensive to reach (Iskakov, 1 Apr 2026).

A related operator-level viewpoint appears in spectral coarsening. "Chordal Decomposition for Spectral Coarsening" seeks a sparse PSD operator $Y_k = \sum_{r=1}^{R} Z_k^{(r)} + W_k.$2 that preserves low-frequency operator action through

$Y_k = \sum_{r=1}^{R} Z_k^{(r)} + W_k.$3

or, with additional low-frequency emphasis,

$Y_k = \sum_{r=1}^{R} Z_k^{(r)} + W_k.$4

Chordal decomposition replaces one global sparse PSD constraint by many clique-wise PSD constraints, making the energy-aware coarsening problem computationally tractable (Chen et al., 2020).

5. Fluids, turbulence, and Hamiltonian continuum systems

In Hamiltonian PDEs, energy-aware spectral decomposition begins with the spatial discretization itself. For the nonlinear wave equation,

$Y_k = \sum_{r=1}^{R} Z_k^{(r)} + W_k.$5

a Fourier-Galerkin expansion gives

$Y_k = \sum_{r=1}^{R} Z_k^{(r)} + W_k.$6

and leads to the infinite-dimensional Hamiltonian system

$Y_k = \sum_{r=1}^{R} Z_k^{(r)} + W_k.$7

After truncation, the semi-discrete Hamiltonian is

$Y_k = \sum_{r=1}^{R} Z_k^{(r)} + W_k.$8

The point is not only spectral accuracy in space, but preservation of the Hamiltonian structure of the projected system, with HBVMs used in time to preserve $Y_k = \sum_{r=1}^{R} Z_k^{(r)} + W_k.$9 exactly for suitable polynomial cases and practically exactly for smooth non-polynomial ones (Brugnano et al., 2014).

In wall turbulence, "Data-driven decomposition of the streamwise turbulence kinetic energy in boundary layers. Part 1. Energy spectra" uses spectral coherence as an energy filter. The measured streamwise spectrum is partitioned into

$$\sum_{r=1}^{R}\sum_{k=0}^K|Z_k^{(r)}|^2+\sum_{k=0}^K\left|Y_k-\sum_{r=1}^{R}Z_k^{(r)}\right|^2=\sum_{k=0}^{K}|Y_k|^2,$$0

with

$$\sum_{r=1}^{R}\sum_{k=0}^K|Z_k^{(r)}|^2+\sum_{k=0}^K\left|Y_k-\sum_{r=1}^{R}Z_k^{(r)}\right|^2=\sum_{k=0}^{K}|Y_k|^2,$$1

The threshold

$$\sum_{r=1}^{R}\sum_{k=0}^K|Z_k^{(r)}|^2+\sum_{k=0}^K\left|Y_k-\sum_{r=1}^{R}Z_k^{(r)}\right|^2=\sum_{k=0}^{K}|Y_k|^2,$$2

marks the onset of wall-coherent energy, and the decomposition is used to argue that a $$\sum_{r=1}^{R}\sum_{k=0}^K|Z_k^{(r)}|^2+\sum_{k=0}^K\left|Y_k-\sum_{r=1}^{R}Z_k^{(r)}\right|^2=\sum_{k=0}^{K}|Y_k|^2,$$3 region can only appear for $$\sum_{r=1}^{R}\sum_{k=0}^K|Z_k^{(r)}|^2+\sum_{k=0}^K\left|Y_k-\sum_{r=1}^{R}Z_k^{(r)}\right|^2=\sum_{k=0}^{K}|Y_k|^2,$$4 at $$\sum_{r=1}^{R}\sum_{k=0}^K|Z_k^{(r)}|^2+\sum_{k=0}^K\left|Y_k-\sum_{r=1}^{R}Z_k^{(r)}\right|^2=\sum_{k=0}^{K}|Y_k|^2,$$5 (Baars et al., 2018).

A related but climatological example is the zonal spectral decomposition of atmospheric meridional energy transport. Total transport is written as

$$\sum_{r=1}^{R}\sum_{k=0}^K|Z_k^{(r)}|^2+\sum_{k=0}^K\left|Y_k-\sum_{r=1}^{R}Z_k^{(r)}\right|^2=\sum_{k=0}^{K}|Y_k|^2,$$6

and decomposed into Fourier wavenumber contributions $$\sum_{r=1}^{R}\sum_{k=0}^K|Z_k^{(r)}|^2+\sum_{k=0}^K\left|Y_k-\sum_{r=1}^{R}Z_k^{(r)}\right|^2=\sum_{k=0}^{K}|Y_k|^2,$$7. Planetary waves ($$\sum_{r=1}^{R}\sum_{k=0}^K|Z_k^{(r)}|^2+\sum_{k=0}^K\left|Y_k-\sum_{r=1}^{R}Z_k^{(r)}\right|^2=\sum_{k=0}^{K}|Y_k|^2,$$8) and synoptic waves ($$\sum_{r=1}^{R}\sum_{k=0}^K|Z_k^{(r)}|^2+\sum_{k=0}^K\left|Y_k-\sum_{r=1}^{R}Z_k^{(r)}\right|^2=\sum_{k=0}^{K}|Y_k|^2,$$9) carry the bulk of poleward transport, winter and summer differ mainly because the planetary contribution contracts strongly in JJA, and extremes result from constructive or destructive interference among these scale bands (Lembo et al., 2019).

The multiscale POD literature makes the energy/frequency tradeoff explicit. In "A Fast Spectral Formulation of the Multiscale Proper Orthogonal Decomposition", the data matrix is written as

$$\mathbf{x}=\sum_{i=1}^{n+1}\mathbf{x}_i,\qquad \mathbf{x}_i \perp \sum_{j=i+1}^{n+1}\mathbf{x}_j,$$0

where each scale $$\mathbf{x}=\sum_{i=1}^{n+1}\mathbf{x}_i,\qquad \mathbf{x}_i \perp \sum_{j=i+1}^{n+1}\mathbf{x}_j,$$1 is associated with a prescribed frequency band, and the temporal modes satisfy

$$\mathbf{x}=\sum_{i=1}^{n+1}\mathbf{x}_i,\qquad \mathbf{x}_i \perp \sum_{j=i+1}^{n+1}\mathbf{x}_j,$$2

The fast spectral formulation replaces FIR filters with disjoint spectral masks, so each band yields a small independent eigenproblem whose size depends on the number of active frequencies in that band rather than on the full temporal dimension (Belda et al., 13 Apr 2026). This preserves energy-optimal decomposition within each band while enforcing exact spectral decoupling.

6. Spectral imaging and energy-bin decompositions

Spectral CT provides a direct instance where “energy-aware” refers literally to energy bins. FONT-SIR constructs, for each reference patch location, a fourth-order tensor

$$\mathbf{x}=\sum_{i=1}^{n+1}\mathbf{x}_i,\qquad \mathbf{x}_i \perp \sum_{j=i+1}^{n+1}\mathbf{x}_j,$$3

whose fourth mode is the spectral or energy-bin dimension. Each tensor is decomposed as

$$\mathbf{x}=\sum_{i=1}^{n+1}\mathbf{x}_i,\qquad \mathbf{x}_i \perp \sum_{j=i+1}^{n+1}\mathbf{x}_j,$$4

with a weighted nuclear norm on $$\mathbf{x}=\sum_{i=1}^{n+1}\mathbf{x}_i,\qquad \mathbf{x}_i \perp \sum_{j=i+1}^{n+1}\mathbf{x}_j,$$5 and a TV penalty on $$\mathbf{x}=\sum_{i=1}^{n+1}\mathbf{x}_i,\qquad \mathbf{x}_i \perp \sum_{j=i+1}^{n+1}\mathbf{x}_j,$$6,

$$\mathbf{x}=\sum_{i=1}^{n+1}\mathbf{x}_i,\qquad \mathbf{x}_i \perp \sum_{j=i+1}^{n+1}\mathbf{x}_j,$$7

The full reconstruction couples this decomposition with the forward model,

$$\mathbf{x}=\sum_{i=1}^{n+1}\mathbf{x}_i,\qquad \mathbf{x}_i \perp \sum_{j=i+1}^{n+1}\mathbf{x}_j,$$8

subject to $$\mathbf{x}=\sum_{i=1}^{n+1}\mathbf{x}_i,\qquad \mathbf{x}_i \perp \sum_{j=i+1}^{n+1}\mathbf{x}_j,$$9. On simulated five-bin data, the paper reports average RMSE $$\|\mathbf{x}\|^2=\sum_{i=1}^{n+1}\|\mathbf{x}_i\|^2.$$0 and average SSIM $$\|\mathbf{x}\|^2=\sum_{i=1}^{n+1}\|\mathbf{x}_i\|^2.$$1, better than TV, tPRISM, $$\|\mathbf{x}\|^2=\sum_{i=1}^{n+1}\|\mathbf{x}_i\|^2.$$2TDL, and ASSIST (Chen et al., 2020).

"ROI-Wise Material Decomposition in Spectral Photon-Counting CT" also treats the energy dimension explicitly, but through region-wise model adaptation. Reconstructed multi-energy images satisfy

$$\|\mathbf{x}\|^2=\sum_{i=1}^{n+1}\|\mathbf{x}_i\|^2.$$3

or, in matrix form,

$$\|\mathbf{x}\|^2=\sum_{i=1}^{n+1}\|\mathbf{x}_i\|^2.$$4

Spatio-energy segmentation combines a spectral kernel and a morphology-informed kernel,

$$\|\mathbf{x}\|^2=\sum_{i=1}^{n+1}\|\mathbf{x}_i\|^2.$$5

to define ROIs. A coarse sparse decomposition then determines which basis materials are plausible in each ROI, and the optimized decomposition matrix $$\|\mathbf{x}\|^2=\sum_{i=1}^{n+1}\|\mathbf{x}_i\|^2.$$6 retains only materials whose relative population exceeds a threshold $$\|\mathbf{x}\|^2=\sum_{i=1}^{n+1}\|\mathbf{x}_i\|^2.$$7. The final ROI-wise sparse decomposition reduces false positives substantially; on physical phantom data the paper reports a $$\|\mathbf{x}\|^2=\sum_{i=1}^{n+1}\|\mathbf{x}_i\|^2.$$8 FP improvement for iodine and a $$\|\mathbf{x}\|^2=\sum_{i=1}^{n+1}\|\mathbf{x}_i\|^2.$$9 FP improvement for gadolinium relative to the coarse baseline, although low-concentration iodine remains difficult (Xie et al., 2019).

7. Methodological tensions and limitations

Several tensions recur across these formulations. In SVD unfolding of composed energy spectra, mode truncation is essential but nontrivial: as ensemble diversity increases, more trend modes are required, and removing too many low-$R$0 modes can suppress genuine long-range fluctuations along with the trend (Berkovits, 2023). In signal-adapted graph decomposition, the adaptation depends on the representativeness of the signal set $R$1, on the chosen graph and Laplacian, and, in the large-graph setting, on the quality of polynomial approximation of both filters and spectral content (Behjat et al., 17 Feb 2025).

Methods that trade exact reconstruction for sharper spectral decoupling face a different difficulty. Fast spectral mPOD uses strictly disjoint masks, but this relaxes the partition-of-unity property and produces slight energy deficits near band boundaries (Belda et al., 13 Apr 2026). xCPD assumes that spectral energy is informative for dependency selection and uses only three bands, low, mid, and high; the paper explicitly notes that this is an interpretable but coarse spectral partition (Li et al., 14 Mar 2026). Low-inertia power-network clustering is derived from a first-order reduction, so explicit inertial energy does not remain in the final clustering operator, even though the decomposition is strongly tied to synchronization stiffness and damping (Ogbonna et al., 9 Jan 2026).

In imaging, robustness often trades against sensitivity. ROI-wise material decomposition reduces false positives, but can increase false negatives for low-concentration materials when the relative population threshold is too large (Xie et al., 2019). In tensor-based spectral CT reconstruction, the method is computationally heavy because it combines nonlocal patch search, PCA preprocessing, repeated tensor unfoldings and SVDs, Chambolle TV iterations, and conjugate-gradient image updates (Chen et al., 2020). More generally, these examples suggest that energy-aware spectral decomposition is most effective when the energy notion being preserved is well matched to the downstream interpretation, and least straightforward when spectral localization, structural sparsity, computational tractability, and exact energy accounting must all be enforced simultaneously.