Spectral Representation in Complex Systems

• Spectral Representation View is a framework that decomposes signals, stochastic processes, and operators into their underlying spectral components using Fourier, eigenfunction, or harmonic analysis.
• It underpins diverse applications from stochastic process theory and quantum many-body physics to multi-view clustering and neural scene modeling, facilitating both theoretical insights and practical computations.
• By leveraging spectral properties, this approach enables precise control over information content and system behavior, supporting effective simulation, dimension reduction, and efficient algorithm design.

The spectral representation view comprises the conceptual and analytical framework by which signals, stochastic processes, fields, operators, or multi-modal data are characterized, decomposed, or described in terms of their spectral components—typically meaning their behavior under a Fourier, eigenfunction, or harmonic analysis. This approach is foundational in probability, statistical physics, random field theory, operator theory, and representation learning, providing tools for both theoretical understanding and practical algorithmic manipulation of complex systems.

1. Spectral Representation in Stochastic Processes and Random Fields

In the theory of stationary random fields, the spectral representation expresses such fields as stochastic integrals over spectral measures. For a homogeneous scalar field $X(x)$ on $\mathbb{R}^n$, one writes

$$X(x) = \int_{\mathbb{R}^n} e^{i k \cdot x} F(k) \xi(k) \, dk$$

where $F(k) = \sqrt{S(k)}$ (with $S(k)$ the power spectrum) and $\xi(k)$ is a complex Gaussian white noise field, $E[\xi(k) \xi(k')^*] = \delta_{k, k'}$ (Chepurnov, 2021). This integral representation is statistically convergent (mean-square sense) rather than pointwise, providing a physical basis for simulation and analysis. For vector fields, spectral projectors $T_{ij}(k)$ encode polarization (e.g., solenoidal, compressible, Alfvénic modes), yielding

$$u_i(r) = \int_{\mathbb{R}^3} e^{i k \cdot r} F(k) T_{ij}(k) \xi_j(k) \, dk$$

which allows geometric interpretation and efficient computation of higher-order statistics and numerical realizations.

2. Spectral Representation of Operators and Hilbertian Time Series

The spectral representation (or spectral theorem) for closed, densely defined operators $A$ on a Hilbert space $H$ states that $A$ can be written as an operator-valued integral:

$A x = \int_{\mathbb{R}} \lambda \, dE(\lambda) x$

where $E(\cdot)$ is a projection-valued measure (Stone–von Neumann representation) (Gill et al., 2012). For arbitrary closed operators (possibly non-self-adjoint), polar decomposition yields the "deformed" spectral representation via a vector measure $F(\Delta) = U E(\Delta)$, with $U$ a partial isometry. These tools generalize functional calculus on Hilbert and Banach spaces, enabling rigorous analysis of PCA, covariance operators, and time series in separable Hilbert spaces (Horta et al., 2016).

In Hilbertian time series, the spectral (eigen-)decomposition of the covariance operator $C$ leads to the Karhunen–Loève expansion:

$$\xi(\omega) = \sum_{j=1}^\infty \langle \xi(\omega), \varphi_j \rangle \varphi_j$$

with $\varphi_j$ orthonormal eigenfunctions and $\lambda_j$ variances of the principal components, capturing all the probabilistic content for dimension reduction or PCA.

3. Spectral Representation in Quantum Many-Body Physics and Non-Equilibrium Dynamics

For quantum systems, the Lehmann (spectral) representation expresses thermal correlation functions in terms of energy eigenstates and universal kernel functions, decoupling frequency dependence from dynamical matrix elements (Halbinger et al., 2023). For Matsubara $n$-point functions,

$$G_{A_1\cdots A_n}(\omega_1,\dots,\omega_{n-1}) = \frac{1}{Z} \sum_{p \in S_n} \zeta(p) \sum_{a_1\cdots a_n} e^{-\beta E_{a_1}} (A_{p(1)})^{a_1a_2}\cdots K_n(\Omega_{p(1)}^{a_1a_2},\dots)$$

with $K_n$ an explicit universal kernel that encodes both regular and anomalous frequency structures. In non-equilibrium field theory, spectral decomposition of contour-ordered correlators—especially in the Schwinger-Keldysh formalism—organizes time/frequency-ordered observables into a basis generated by nested or double commutators, generalizing the retarded/advanced decomposition for OTO (out-of-time-ordered) correlators (Chaudhuri et al., 2018).

4. Spectral Representation in Multi-view and Tensor-based Representation Learning

Spectral representation views undergird algorithms for multi-view clustering and representation learning. Classical low-rank representation (LRR) (Wang et al., 2017, Wang et al., 2016) and tensor-based spectral clustering (Jia et al., 2020) rely on constructing graph Laplacians or affinity matrices whose spectral properties (low-rank, block-diagonal structure) encode consensus and local geometry across multiple data modalities.

For multi-view tensor affinity $W \in \mathbb{R}^{n \times n \times V}$, tailored norms impose nuclear-norm (low-rank) and sparse constraints on frontal (intra-view) slices and horizontal (inter-view) slices, with a convex optimization yielding a clean tensor $L$ whose slices correspond to optimal affinity graphs:

$$\min_{L, E} \|L\|_\oplus + \lambda \|E\|_F^2 \qquad \text{s.t.} \quad W = L + E,\, L(:,:,v) = L(:,:,v)^T$$

Block-diagonal affinities and their spectra directly correspond to perfect clustering embeddings, as supported by numerical experiments reporting accuracy near 1.0 on standard benchmarks (Jia et al., 2020).

5. Spectral Representation of Lévy-driven Processes and Infinite Variance Noise

For Lévy-driven processes, especially those with infinite variance (index $\alpha \in (1,2)$), the spectral representation generalizes $L^2$ stochastic integrals and CARMA (continuous-time autoregressive moving average) models (Fuchs et al., 2011). A regularly varying Lévy process $L_t$ is represented as a stochastic Fourier integral:

$$L_t = \mathrm{P\!\!-\!\!lim}_{\Lambda \to \infty} \int_{-\Lambda}^\Lambda \frac{e^{i t \mu} - 1}{i \mu} M(d\mu)$$

where $M$ is a set-indexed random content with additive properties and a Lévy–Khintchine triplet. For MCARMA($p,q$) processes,

$$Y_t = \mathrm{P\!\!-\!\!lim}_{\Lambda \to \infty} \int_{-\Lambda}^{\Lambda} e^{i t \mu} H(\mu) M(d\mu)$$

with $H(\mu) = P(i\mu)^{-1} Q(i\mu)$, which reduces under $L^2$ conditions to the classical orthogonal random measure case. Stationarity, continuity, and explicit moment and dependence structure are derived directly from the spectral properties of $M$ and the associated Lévy measure.

6. Spectral Representation in Neural and Visual Scene Models

Recent advances in scene representation leverage spectral principles for multi-spectral data synthesis and rendering. Neural implicit spectral reconstruction (Xu et al., 2021) queries spectral intensities as a continuous function over wavelength, using MLP architectures enriched by spectral profile interpolation and neural attention. In visual scene modeling, 3D Gaussian splatting frameworks encode multi-spectral information into unified neural representations, exploiting shared spatial-spectral correlations and enabling physically-based rendering, segmentation, and editing in real time (Meyer et al., 3 Jun 2025, Sinha et al., 2024).

In multi-spectral splatting, each scene primitive (splat) carries a joint feature embedding decoded by an MLP to yield spectral colors for novel views, outperforming prior approaches in PSNR and SSIM for multi-spectral rendering tasks, and supporting application-specific indices (e.g., NDVI for vegetation analysis).

7. Theoretical and Computational Significance

Spectral representation views unify harmonic analysis, stochastic calculus, operator theory, and deep representation learning. They enable precise control over the structure, regularity, and information content of complex data, and furnish rigorous and efficient strategies for inference, simulation, clustering, and prediction across a broad range of disciplines. Whether via random measure integrals, operator-functional calculi, graph spectra, or neural encoding, the spectral viewpoint imposes a canonical decomposition that exposes the essential features underlying system behavior.