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Sparse-spectrum Gaussian-process exact moment matching

Algorithms AIMech:SparseSpectrumGaussianProcessExactMomentMatching DRAFT

Aliases: SSGP-EMM

An uncertain-input prediction algorithm that analytically integrates sine/cosine feature moments of a fixed sparse-spectrum Gaussian-process posterior over an independently prescribed Gaussian input and represents the output by a Gaussian with the resulting exact mean and covariance.

Identity Notes & Concept Scope Compared all 442 pinned AIO IDs, labels and scoped synonyms and 543 maintained IDs, names and aliases: neither proposed identity nor its acronym matched. Full-text searches in the recovery, older discovery and original Documents checkouts included ignored and hidden files and returned 98, 0 and 0 lines; all 62 distinct matched strings were read. Hits concern pending sparse-spectrum leads, source reading notes and scope distinctions in existing GP records. Read all 39 semantic AIO rows and 107 native name/definition neighbors, seven complete annotated AIO CLASS nodes with incident edges and generated records, and four native context records. The fixed finite-feature posterior differs from the full squared-exponential kernel model; SSGP-EMM integrates its moments, whereas SSGP-Lin linearizes its mean and freezes its conditional variance. Neither identity denotes learning the underlying sparse-spectrum regression model, general Fourier feature construction, a complete Bayesian filter, or a model predictive controller. The earlier Girard Taylor record retains a conditional-variance Hessian term that SSGP-Lin omits. Absence is bounded to this inspected pin and corpus.

Claims & Grounded Evidence

5 Claims

Mechanism #fixed-feature-posterior

For fixed frequencies omega_i, the source uses a 2m-dimensional feature vector with cosine and sine entries scaled by sigma_k=sigma_f/sqrt(m). For n training input/response pairs, Phi=[phi(x_1),...,phi(x_n)] is the 2m by n feature-design matrix and y=[y_1,...,y_n]^T is the n-vector of scalar training responses. With Gaussian coefficient prior N(0,I) and scalar Gaussian observation noise of variance sigma_n^2, its trained coefficient posterior is N(alpha,S), where alpha=A^{-1}*Phi*y, S=sigma_n^2*A^{-1} and A=Phi*Phi^T+sigma_n^2*I. The fixed-input response has mean alpha^T*phi(x) and variance sigma_n^2+phi(x)^T*S*phi(x). Uncertain-input prediction integrates a prescribed X~N(mu,Sigma) independently of this fixed coefficient posterior, conditional on the training data, frequencies and hyperparameters. These prediction methods do not require one particular training solver once the posterior coefficient moments are supplied.

supports Source: Prediction under Uncertainty in Sparse Spectrum Gaussian Processes with Applications to Filtering and Control · Locator: Section 2, Equations 4, 6–7 and following remark, PDF pp. 2–3; Section 3 assumptions, PDF p. 3
supports Source: Appendix for Prediction under Uncertainty in Sparse Spectrum Gaussian Processes with Applications to Filtering and Control · Locator: Sections 2–3.1, Equations 1–5 and Gaussian-input assumptions, PDF pp. 1–2

Mechanism #exact-feature-moments

Let q=E[phi(X)] and Psi=E[phi(X)*phi(X)^T]. SSGP-EMM computes E[Y]=alpha^T*q and Var[Y]=sigma_n^2+tr(S*Psi)+alpha^T*(Psi-q*q^T)*alpha, then represents the output by a Gaussian with these moments. Gaussian characteristic-function identities give E[cos(omega^T*X)]=exp(-omega^T*Sigma*omega/2)*cos(omega^T*mu), with sine in place of cosine for E[sin]. Product-to-sum identities at frequency sums and differences give Psi. These are exact first two moments of the fixed finite-feature posterior under the stated independent Gaussian input, not exact moments of the original infinite-rank kernel model or integration over frequencies and hyperparameters.

supports Source: Prediction under Uncertainty in Sparse Spectrum Gaussian Processes with Applications to Filtering and Control · Locator: Section 3.1, Equations 8–9, Proposition 1 and feature-product blocks, PDF pp. 3–4
supports Source: Appendix for Prediction under Uncertainty in Sparse Spectrum Gaussian Processes with Applications to Filtering and Control · Locator: Sections 3.1–3.2 and 3.4, PDF pp. 3–5

Mechanism #cross-covariances

For a column-vector input, Cov(X,Y)=Cov(X,phi(X))*alpha. The cosine feature column is -sigma_k*Sigma*omega_i*E[sin(omega_i^T*X)] and the sine feature column is sigma_k*Sigma*omega_i*E[cos(omega_i^T*X)], with sigma_k the feature scale. For distinct conditionally independent scalar output models a and b with independent residual noises, Cov(Y_a,Y_b)=alpha_a^T*(Psi_ab-q_a*q_b^T)*alpha_b, where Psi_ab=E[phi_a(X)*phi_b(X)^T]. Shared input uncertainty can therefore correlate outputs even when their conditional distributions factorize. Conditional cross-output covariance or correlated noise would require an additional term.

supports Source: Prediction under Uncertainty in Sparse Spectrum Gaussian Processes with Applications to Filtering and Control · Locator: Section 3.1, Equations 10–11 and Proposition 2, PDF p. 4
supports Source: Appendix for Prediction under Uncertainty in Sparse Spectrum Gaussian Processes with Applications to Filtering and Control · Locator: Sections 3.1, 3.3–3.4, PDF pp. 3–5; Section 6.1, PDF p. 12

Limitation #gaussian-shape-boundary

Matching exact first two moments does not make the nonlinear predictive mixture Gaussian. As a curator boundary illustration, fixed coefficient Y=cos(X) with X~N(0,1) has mean exp(-1/2), variance (1+exp(-2))/2-exp(-1)>0 and support [-1,1]. Its moment-matched Gaussian has unbounded support. This uses deterministic coefficients and no observation noise as a limiting finite-feature example, not a fitted SSGP experiment. Direct deterministic quadrature of a separate nondegenerate two-feature Gaussian coefficient example checks moment and covariance formulas without establishing general empirical accuracy.

supports Source: Prediction under Uncertainty in Sparse Spectrum Gaussian Processes with Applications to Filtering and Control · Locator: Section 3, distinction between the predictive integral and Gaussian approximation, PDF p. 3; Proposition 1, PDF p. 3
supports Source: Appendix for Prediction under Uncertainty in Sparse Spectrum Gaussian Processes with Applications to Filtering and Control · Locator: Section 3 introductory assumptions and Section 3.4, PDF pp. 2 and 5

Limitation #sequential-use

The source uses these prediction transforms inside Bayesian filtering and future-state prediction for control. Gaussian moment propagation alone does not retain the full predictive shape or all dependence between a repeatedly used uncertain dynamics function and subsequent states. Hewing and colleagues explicitly analyze the successive-evaluation independence approximation in Pan-style trajectory prediction. This prediction transform is therefore not an exact shared-function trajectory inference algorithm or, by itself, a complete filter or controller.

supports Source: Prediction under Uncertainty in Sparse Spectrum Gaussian Processes with Applications to Filtering and Control · Locator: Section 4.1 prediction and filtering stages, PDF pp. 5–6; Section 4.2, PDF pp. 6–7
supports Source: On Simulation and Trajectory Prediction with Gaussian Process Dynamics · Locator: Introduction and Sections 2–3, PDF pp. 1–4; distinct trajectory constructions, PDF pp. 5–7

⚠️ Curated Limitations & Negative Results

Exactness is for first two moments of the fixed finite-feature posterior, not Gaussianity of the predictive distribution or exactness for the original stationary kernel.
The input Gaussian is prescribed independently of the fixed trained coefficient posterior. Frequencies, hyperparameters, training data and feature count are conditioned on rather than integrated out; finite-feature kernel approximation remains a separate source of error.
The displayed variance includes a future independent observation-noise term once. A latent-function prediction omits that additive future-noise term while retaining the trained posterior coefficient covariance.
Multivariate cross-output formulas assume conditionally independent scalar models and independent residual noises. Shared input covariance remains; correlated-output priors and correlated measurement noises require additional terms.
Gaussian representations and repeated independent-input propagation discard predictive shape and shared-function temporal dependence. Filtering updates and belief-space controller optimization are additional algorithms, not identities of this prediction transform.
Source notation inconsistencies were checked against the supplement and independent algebra. Table 1 spectral-density constants, controller derivative formulas, runtime claims and numerical comparative performance are not asserted here. No author implementation, training, paid provider or benchmark was run; analytic examples are curator illustrations. The record remains DRAFT.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:RegressionAnalysis` Regression Analysis	related_match	This procedure predicts a learned regression response while integrating or approximating uncertainty in its test input. The mapping expresses that regression-task relationship rather than algorithm identity or an asserted subclass.
`AIO:TimeSeriesForecasting` Time Series Forecasting	related_match	The primary source repeatedly applies the transform to predict future dynamical states in filtering and model predictive control. This is a demonstrated forecasting application, not a restriction of the generic prediction algorithm to time series.

Evidence Sources & Primary Literature

3 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`pan2017`	Prediction under Uncertainty in Sparse Spectrum Gaussian Processes with Applications to Filtering and Control Yunpeng Pan, Xinyan Yan, Evangelos A. Theodorou, Byron Boots	primary paper full_text	ID: https://proceedings.mlr.press/v70/pan17a.html sha256:f6419aa3e014…
`pan2017-supp`	Appendix for Prediction under Uncertainty in Sparse Spectrum Gaussian Processes with Applications to Filtering and Control Yunpeng Pan, Xinyan Yan, Evangelos A. Theodorou, Byron Boots	primary paper full_text	ID: https://proceedings.mlr.press/v70/pan17a.html sha256:4cdea13525c0…
`hewing2020`	On Simulation and Trajectory Prediction with Gaussian Process Dynamics Lukas Hewing, Elena Arcari, Lukas P. Fröhlich, Melanie N. Zeilinger	primary paper full_text	ID: https://proceedings.mlr.press/v120/hewing20a.html sha256:e03bc1f12364…

Curation Provenance

Curator: Codex

Method: ai_assisted

Model: GPT-6

Created: 2026-10-12T21:48:25Z

Updated: 2026-10-12T21:55:51Z

Source File: data/curated/SparseSpectrumGaussianProcessExactMomentMatching.yaml