Unitary Kernel Quadrature for Training Parameter Distributions

A shallow model commonly appears in machine learning and signal processing. Whether it is parametric or non-parametric, a shallow model is formulated as the integration of feature maps against an unknown \emph{parameter distribution}, or a complex-valued measure. As is often the case with neural networks, if a model is parameterized in vectors, the parameter dimension needs to be manually determined by model selection; and the gradient descent training often results in a non-convex optimization problem, even when the loss function is convex. On the other hand, if the model is re-parameterized in measures, the parameter dimension can be automatically determined; and the training is convex when the loss function is convex. Therefore, it is natural to consider training the parameter distribution. However, handling a measure is difficult in practice because (1) we need a finite sum of point masses, and (2) the parameterization is not always unique. In other words, two different parameter distributions may indicate the same function. For example, kernels on the input space, priors on the parameter space, and random feature methods are too weak to handle point masses, because they turn point masses into smooth functions. On the other hand, versatile topologies for measures such as the total variation and the Wasserstein distance are too strong when the parameterization is not unique. Namely, these topologies unnecessarily distinguish two measures that indicate the same function, which causes another non-convexity. To address these difficulties, we investigate the \emph{generalized kernel quadrature} for handling complex-valued point masses; and propose to employ \emph{unitary kernel embedding} for killing non-uniqueness. The proposed method converges in $L^2$-norm at Barron's theoretical fast ratio.