Kernel-combination procedures used in classification often return only a combined kernel matrix on the training sample, rather than a kernel function that can be evaluated consistently at new points. This limitation is especially important for supervised or label-aware combinations, whose entries may depend on training labels and therefore have no immediate out-of-sample meaning. We study the problem of constructing an inductive, finite-rank kernel extension from such empirical matrices. The proposed framework makes the non-uniqueness of this extension explicit: it is determined by empirical coordinates, a positive-semidefinite coefficient matrix, and a continuation model for the coordinates. Experiments on vector, tabular, and relational classification problems give a deliberately diagnostic picture. Smooth direct combinations are stable: on Synthetic, the direct mean gives error $0.0793\pm 0.0227$, essentially matching the best individual RBF kernel ($0.0809\pm 0.0231$), and on Telco it remains close to the best individual polynomial kernel ($0.2061\pm 0.0154$ versus $0.2045\pm 0.0154$). In the controlled Synthetic oracle diagnostic, reconstructing a smooth sum/mean gives relative Frobenius error $4.13\times {10}^{-6}\pm 9.41\times {10}^{-6}$ and functional MSE at numerical scale. By contrast, abrupt label-aware matrix-only rules are less robust: the Synthetic percentile_inout_auto rule has error $0.1404\pm 0.1198$, Telco matrix-only supervised rules are around $0.307$–$0.326$ error, and the Chickenpieces pickout_auto rule fails under strict out-of-sample reconstruction ($0.3545\pm 0.2666$ error), whereas direct relational combinations match the best individual relational kernel within ${10}^{-3}$. Overall, the empirical evidence supports the method as a bridge from finite matrix-level information fusion to deployable kernels, while also identifying abrupt label-aware geometries as the main limitation for stable generalization.

Information about the paper titled "INTRODUCING SYSTEMS THINKING SKILLS TO IMPROVE THE LEARNING PROCESS ABOUT COMPLEXITY AND COMPLEX SYSTEMS" at IATED Digital Library