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Sparse-spectrum Gaussian-process linearized uncertain-input prediction

Algorithms AIMech:SparseSpectrumGaussianProcessLinearizedPrediction DRAFT

Aliases: SSGP-Lin

A local uncertain-input prediction algorithm for a fixed sparse-spectrum Gaussian-process posterior that linearizes the posterior mean at the Gaussian input mean, freezes the conditional variance there, and uses the resulting approximate mean and covariance for a Gaussian output representation.

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 #local-mean-frozen-variance

SSGP-Lin linearizes m(x)=alpha^T*phi(x) at the input mean mu and evaluates the conditional response variance at mu. With J=Dphi(mu) of size 2m by d and column gradient g=J^T*alpha, the approximate mean is m(mu) and variance is sigma_n^2+phi(mu)^T*S*phi(mu)+g^T*Sigma*g. The cosine feature derivative is -sigma_k*sin(omega_i^T*mu)*omega_i^T and the sine derivative is sigma_k*cos(omega_i^T*mu)*omega_i^T. This formula omits posterior-mean curvature and input-dependent conditional-variance curvature. Nonnegative component variances give a nonnegative scalar variance, which does not establish approximation accuracy.

supports Source: Prediction under Uncertainty in Sparse Spectrum Gaussian Processes with Applications to Filtering and Control · Locator: Section 3.2, Equations 12–14, PDF p. 5
supports Source: Appendix for Prediction under Uncertainty in Sparse Spectrum Gaussian Processes with Applications to Filtering and Control · Locator: Section 3.1, definition of M=Dphi, PDF p. 2; Section 3.5, PDF pp. 5–6

Mechanism #local-cross-covariance

With column input X, the local input-response cross-covariance is Sigma*g, a d by 1 column. For two conditionally independent scalar output models with independent residual noise, the off-diagonal response covariance is g_a^T*Sigma*g_b. This retains covariance induced by the shared uncertain input while freezing each conditional variance. It does not model conditional cross-output covariance. The main paper’s M underbrace and final input-response covariance orientation are inconsistent; the supplemental definition M=Dphi and explicit matrix dimensions determine the formulas used here.

supports Source: Prediction under Uncertainty in Sparse Spectrum Gaussian Processes with Applications to Filtering and Control · Locator: Section 3.2, Equations 12 and 15–16, PDF p. 5
supports Source: Appendix for Prediction under Uncertainty in Sparse Spectrum Gaussian Processes with Applications to Filtering and Control · Locator: Section 3.1, PDF p. 2; Section 3.5, PDF pp. 5–6; Section 6.1, PDF p. 12

Limitation #distinct-variance-truncation

Freezing the conditional variance differs from the earlier Girard local GP approximation, which adds one half of the trace of its conditional-variance Hessian times Sigma. In a curator finite-feature illustration Y=W*cos(X), with independent W~N(0,1) and X~N(0,2), SSGP-Lin gives variance 1, exact integration gives (1+exp(-4))/2, and retaining the second-order conditional-variance term gives -1. This limiting rank-one coefficient example distinguishes the truncations; it is not a trained-model benchmark or a claim that the earlier method always fails. Both local approximations can differ substantially from exact moments away from a sufficiently local input regime.

supports Source: Prediction under Uncertainty in Sparse Spectrum Gaussian Processes with Applications to Filtering and Control · Locator: Section 3.2, Equation 14, PDF p. 5
supports Source: Appendix for Prediction under Uncertainty in Sparse Spectrum Gaussian Processes with Applications to Filtering and Control · Locator: Section 3.5, PDF pp. 5–6
supports Source: Gaussian Process priors with Uncertain Inputs: Multiple-Step-Ahead Prediction · Locator: Sections 3.3–3.4, Equations 14–22, PDF pp. 8–9

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

Mean linearization and frozen conditional variance can miss both mean and variance curvature. Positive covariance is not a guarantee of accurate moments, calibrated uncertainty or performance superiority.
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

4 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…
`girard-tr2002`	Gaussian Process priors with Uncertain Inputs: Multiple-Step-Ahead Prediction Agathe Girard, Carl Edward Rasmussen, Roderick Murray-Smith	author technical report full_text	ID: TR-2002-119 sha256:f2e78afd674d…

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/SparseSpectrumGaussianProcessLinearizedPrediction.yaml