JEOS RP ISSN03 | Page 460

J. Eur. Opt. Society-Rapid Publ. 22, 45( 2026) 453

Appendix F: Implementation details of the Gaussian process Surrogate model

Rather than directly modeling the macroscopic set volume VðPÞ, the surrogate model approximates the underlying physical blackbox function f: X! R 3. Because the output coordinates( L, a, andb) are inherently correlated in the CIELAB color space, we employ a Multi-Output Gaussian Process based on the Linear Model of Coregionalization( LMC). Unlike independent, singletask GPs, the LMC captures cross-channel dependencies.

The multi-output surrogate model is defined by a vectorvalued mean function lðpÞ and a multi-output covariance kernel Kp ð; p 0 ÞR 2 33: fðpÞ GPðlðpÞ; Kp ð; p 0 ÞÞ

where the coregionalized kernel is expressed as the Kronecker product of a coregionalization matrix B( which models the cross-channel correlations) and a spatial kernel kp ð; p 0 Þ( which models the parameter dependencies):

Kp ð; p 0 Þ ¼ B kp ð; p 0 Þ

This architecture was implemented using the LMCVariationalStrategy provided by the GPyTorch library( https:// docs. gpytorch. ai / en / v1.12 / variational. html). Under the LMC framework, each output task f i ðpÞ is expressed as a linear combination of Q independent latent Gaussian processes:

f i ðpÞ ¼ XQ

a ðqÞ i g ðqÞ p q¼1

where i 2 f1; 2; 3g represents the specific output task( corresponding to the L, a, andb color channels, respectively), g ðqÞ ðpÞ is the q-th independent latent GP, and a iq ð Þ are the scalar weighting coefficients. In our formulation, the model was configured using Q ¼ 3 independent latent variables. ð Þ

Table F1. Comparison of mean / covariance choices and resulting color error.

Mean function Covariance function Dataset mean DE 76

These coefficients a iq ð Þ constitute the elements of a task projection matrix A. The coregionalization matrix B is AA T.

Finally, to assess the model performance under different structural assumptions, several combinations of mean and covariance functions were tested. Each model was trained on the same subset of data, and the kernel hyperparameters( e. g., length-scale, output scale, noise level) were optimized by minimizing the mean CIE qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DE 76, defined as DE 76 ¼ ðDL Þ 2 þ ðDa Þ 2 þ ðDb Þ 2, between the prediction and the ground truth. As shown in Table F1, the best model achieved a mean DE 76 of 4.5, which is exemplified in Figure F1. While this value exceeds the typical Just Noticeable Difference( JND 2:3) making the error visually perceptible, this level of precision is entirely acceptable for the purpose of the surrogate model. The GP is not used to precisely predict final colors for high-fidelity image printing, but rather to identify highprobability regions of gamut expansion to guide the physical exploration phase. The results shown in Table F1 correspond to predictions obtained from models trained on 20 measured colors( similar to the initial points) and used to predict, one at a time, other 10000 measurement points. The discrepancies between predictions and true values were averaged to quantify the model error. This process was repeated 10 times with different fixed random seeds to ensure the robustness and the relevance of the comparison.

Based on minimizing the mean color prediction error( CIE DE 76) on the initial measurement points, a constant mean function and a Radial Basis Function( RBF) spatial kernel k( p, p 0) were selected as they yielded the highest predictive accuracy for this physical system.

Figure

F. 1. Illustration of 4.5 dE 76 color difference.