Uncertainty quantification for tensor network constrained kernel machines

Abstract

This research aims at quantifying the uncertainty in the predictions of tensor network constrained kernel machines, focusing on the Canonical Polyadic Decomposition (CPD) constrained kernel machine. Constraining the parameters in the kernel machine optimization problem to be a CPD results in a linear computational complexity in the number of features, whereas the original problem suffers heavily from the curse of dimensionality as the number of parameters scale exponentially. By employing a product feature map with polynomial features, the original data input is transformed to a higher-dimensional space.

Three different methods are investigated for quantifying the uncertainty of the predictions of the CPD constrained kernel machine. Firstly, the delta method is proposed which is a frequentist approach that linearizes a nonlinear parametric model around the estimated model. By estimating the covariance of the model parameters, the delta method can estimate the uncertainty in the model predictions based on the estimated parameter uncertainties. The delta method is compared to two other methods that are able to reflect the prediction uncertainty: the Bayesian method and Single Bayesian Core (SBC) method. The Bayesian method treats the parameters in the factor matrices of the CPD as probability distributions rather than single values and the SBC method incorporates both frequentist and Bayesian aspects. A comparison between the three different methods is performed based on an assessment on the correctness and informativeness of the uncertainty measures of prediction intervals and confidence Intervals.

It was found by regression and classification experiments that all three methods can provide valuable uncertainty quantification measures in terms of correctness and informativeness for the CPD constrained kernel machine. However, the Bayesian method provides in general more conservative uncertainty intervals compared to the delta and SBC method. A major drawback of the Bayesian method is its lack of scalability as the size of the mean and covariance, constructed by the unscented transform in the Bayesian method, scale exponentially. Furthermore, the delta and SBC method produce high quality uncertainty intervals and the methods provide remarkably similar uncertainty quantification on the prediction error variance.