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Variational sparse-spectrum Gaussian-process inference

Algorithms AIMech:VariationalSparseSpectrumGaussianProcessInference DRAFT

Aliases: VSSGP

The Gal–Turner finite spectral regression inference procedure that factorizes spectral parameters from output coefficients, analytically optimizes an unrestricted Gaussian coefficient factor, and optimizes the resulting evidence lower bound over the spectral variational distribution and model hyperparameters.

Identity Notes & Concept Scope Compared all 442 pinned AIO IDs, names and scoped synonyms and 546 maintained IDs, names and aliases; none matches the three proposed identities after normalization. Full-text searches of the recovery, older discovery and original Documents checkouts included hidden and ignored files, with 22, 0 and 0 matched lines. All 21 distinct matched contents were read: pending source leads and source-reading context, not existing records. Assessed 50 semantic AIO rows and 172 maintained names/definitions, ten full annotated AIO CLASS nodes with incident edges and generated YAML, and five complete native context records. Generic mean-field methods and stochastic gradient descent do not identify this finite spectral regression procedure. The paired-feature SSGP model and its uncertain-test-input transforms also differ: these methods infer uncertain spectral parameters from training data. The three source labels distinguish optimal unrestricted coefficient factors, explicit diagonal factors, and data-minibatch optimization. Absence is bounded to the searched pin and corpus.

Claims & Grounded Evidence

5 Claims

Mechanism #finite-feature-model

For one squared-exponential component, use K features phi_k(x)=sqrt(2*sigma^2/K)*cos(2*pi*w_k^T*L^{-1}*(x-z_k)+b_k), where L=diag(2*pi*l_j), l_j>0, and z_k are feature locations. Independent standardized frequencies have prior N(0,I); phases have uniform priors on [0,2*pi]. For N inputs, Phi has shape N by K. Each output column y_d has likelihood N(Phi*a_d,tau^{-1}*I), tau>0, with independent unit-Gaussian coefficient priors a_d. The variational family separates q(A) from q(omega), omega collecting frequencies and phases, and factors q(omega) across k with diagonal-Gaussian frequency factors and uniform phase intervals [alpha_k,beta_k] with 0<=alpha_k<beta_k<=2*pi. This is a finite-feature hierarchical regression model; integrating its random features does not generally yield the original full-kernel GP.

supports Source: Improving the Gaussian Process Sparse Spectrum Approximation by Representing Uncertainty in Frequency Inputs · Locator: Sections 2–4, feature construction, Equation 2 and variational family, PDF pp. 2–4
supports Source: Variational Inference for Sparse Spectrum Approximation in Gaussian Process Regression – Appendix · Locator: Propositions 1–2, PDF pp. 1–2

Mechanism #variational-objective

The evidence lower bound is the expected Gaussian log likelihood minus KL(q(A)||p(A)) and KL(q(omega)||p(omega)). For F=E_q[Phi], H=E_q[Phi^T*Phi], q(a_d)=N(m_d,S_d) and response column y_d, its output-d likelihood contribution is -N/2*log(2*pi/tau)-tau/2*[y_d^T*y_d-2*y_d^T*F*m_d+tr(H*(S_d+m_d*m_d^T))]. Gaussian trigonometric moments and uniform-phase integrals give F and H analytically. Distinct feature moments factor under the declared independent spectral factors; a feature’s second moment must be computed as E[phi_k^2], not (E[phi_k])^2.

supports Source: Improving the Gaussian Process Sparse Spectrum Approximation by Representing Uncertainty in Frequency Inputs · Locator: Section 4.1, Equation 6 and diagonal/off-diagonal moments, PDF p. 4
supports Source: Variational Inference for Sparse Spectrum Approximation in Gaussian Process Regression – Appendix · Locator: Identities 1–3, PDF p. 1

Mechanism #optimal-coefficient-factor

At fixed spectral distribution, tau and feature hyperparameters, define H=E_q[Phi^T*Phi], F=E_q[Phi], B=I+tau*H and g_d=F^T*y_d. Functional optimization over the coefficient factor gives S=B^{-1} and m_d=tau*S*g_d. S is generally full, shared across output columns. Substitution into the ELBO removes explicit coefficient variational parameters and yields the source’s Equation 7. This is optimal within the coefficient-versus-spectrum factorization; it does not integrate the true coupled frequency/coefficient posterior exactly or preserve the earlier diagonal coefficient restriction.

supports Source: Improving the Gaussian Process Sparse Spectrum Approximation by Representing Uncertainty in Frequency Inputs · Locator: Section 4.2 and Equation 7, PDF pp. 4–5; VSSGP label in Section 7, PDF p. 5
supports Source: Variational Inference for Sparse Spectrum Approximation in Gaussian Process Regression – Appendix · Locator: Proposition 3, PDF p. 2; diagonal versus non-diagonal discussion, PDF p. 4

Mechanism #predictive-moments

For a fixed query, let phi be its K-dimensional column feature vector, f=E_q[phi], R=E_q[phi*phi^T], and M=[m_1,...,m_D]. Under q(A)q(omega), the response mean is M^T*f and its covariance is tau^{-1}*I+diag_d[tr(R*S_d)]+M^T*(R-f*f^T)*M. Shared spectral uncertainty can therefore correlate different outputs even though their coefficient factors are independent. These are moments of the variational predictive mixture, which is not generally Gaussian; the latent-output covariance omits only the future-observation-noise term.

supports Source: Improving the Gaussian Process Sparse Spectrum Approximation by Representing Uncertainty in Frequency Inputs · Locator: Section 6, Equations 10–11, PDF p. 5
supports Source: Variational Inference for Sparse Spectrum Approximation in Gaussian Process Regression – Appendix · Locator: Propositions 4–5, PDF pp. 2–3

Scope #implementation-boundary

The inspected author model source computes Gaussian-frequency moments with the phase alpha fixed inside each cosine; its beta argument does not enter those moments and its objective contains frequency KL without a phase KL. Thus that revision implements a fixed-phase conditional variant, not integration over the paper’s positive-width uniform phase factors. Its second-moment diagonal uses the cosine-square expectation. The optimal-coefficient routine retains a full covariance and adds numerical diagonal regularization; source-code reading does not establish an executed result or exact correspondence to every published experiment.

supports Source: VSSGP author implementation: VSSGP_model.py · Locator: get_EPhi, lines 61–78; get_opt_A, lines 80–85; get_model and get_model_exact_A, lines 87–131
supports Source: Improving the Gaussian Process Sparse Spectrum Approximation by Representing Uncertainty in Frequency Inputs · Locator: Experimental setup, fixed randomized phases, Section 8.1, PDF p. 6

⚠️ Curated Limitations & Negative Results

The ELBO and predictive moments concern the specified finite-feature model and variational family. Spectral posterior factorization discards dependencies, and optimization can have local optima. Neither exact full-GP inference nor prevention of overfitting is guaranteed.
A single random-phase cosine per frequency differs from the earlier paired cosine/sine SSGP prior. Conditional finite-feature covariances need not be stationary. Frequency uncertainty here is not uncertainty in a supplied test-input distribution.
Paper phase integration and fixed phases in the inspected implementation are separate cases. A point mass against a continuous phase or frequency prior is not a finite-KL endpoint of the stated density family; no zero-width or zero-variance ELBO equivalence is asserted.
The supplement’s displayed covariance diagonal on its final page appears to substitute squared first moments for second moments. This record follows the main derivation, supplement Identity 3 and inspected implementation instead. Decay of expected features does not mean each random feature vanishes, and far-field variance need not equal the original GP prior variance.
All ten main and four supplementary pages were read. Author code was inspected statically only; no model fitting, author-code execution, paid provider or benchmark was run. Fixed finite arithmetic checks selected moment and objective identities, not convergence or empirical calibration. The record remains DRAFT.
The VSSGP acronym is used broadly in parts of the paper; this record follows its explicit Section 7 label for the analytically optimized coefficient variant. Explicit diagonal and stochastic variants retain separate records.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:RegressionAnalysis` Regression Analysis	related_match	This inference procedure fits a probabilistic mapping from covariates to response variables. The AIO target denotes the regression task; it does not identify this finite spectral model, variational family or optimizer. This is an algorithm-to-task relationship without an equivalence or subclass assertion.

Evidence Sources & Primary Literature

3 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`gal2015`	Improving the Gaussian Process Sparse Spectrum Approximation by Representing Uncertainty in Frequency Inputs Yarin Gal, Richard Turner	primary paper full_text	ID: https://proceedings.mlr.press/v37/galb15.html sha256:158bdd0ef09c…
`gal2015-supp`	Variational Inference for Sparse Spectrum Approximation in Gaussian Process Regression – Appendix Yarin Gal, Richard Turner	primary paper full_text	ID: https://proceedings.mlr.press/v37/galb15.html sha256:19b4948bcf25…
`author-implementation`	VSSGP author implementation: VSSGP_model.py Yarin Gal	official documentation documentation	ID: https://github.com/yaringal/VSSGP/blob/f0331a963c0779828a178f0c86f597f0c451dcaf/VSSGP_model.py sha256:1227bfd02ccd…

Curation Provenance

Curator: Codex

Method: ai_assisted

Model: GPT-6

Created: 2026-10-12T22:21:35Z

Updated: 2026-10-12T22:21:35Z

Source File: data/curated/VariationalSparseSpectrumGaussianProcessInference.yaml