Kernel Alignment Risk Estimator: Risk Prediction from Training Data

We study the risk (i.e. For this, we introduce two objects: the Signal Capture Threshold (SCT) and the Kernel Alignment Risk Estimator (KARE). The SCT \vartheta_{K,\lambda} is a function of the data distribution: it can be used to identify the components of the data that the KRR predictor captures, and to approximate the (expected) KRR risk. This then leads to a KRR risk approximation by the KARE \rho_{K, \lambda}, an explicit function of the training data, agnostic of the true data distribution. The key results then follow from a finite-size adaptation of the resolvent method for general Wishart random matrices.