Large-Scale Learning with Fourier Features and Tensor Decompositions
This addresses scalability issues in kernel methods for high-dimensional data, though it is incremental as it builds on existing deterministic Fourier features.
The paper tackles the curse of dimensionality in deterministic Fourier features for kernel methods by using low-rank tensor decompositions, achieving linear complexity in sample size and dimensionality while matching nonparametric model performance and outperforming random Fourier features.
Random Fourier features provide a way to tackle large-scale machine learning problems with kernel methods. Their slow Monte Carlo convergence rate has motivated the research of deterministic Fourier features whose approximation error can decrease exponentially in the number of basis functions. However, due to their tensor product extension to multiple dimensions, these methods suffer heavily from the curse of dimensionality, limiting their applicability to one, two or three-dimensional scenarios. In our approach we overcome said curse of dimensionality by exploiting the tensor product structure of deterministic Fourier features, which enables us to represent the model parameters as a low-rank tensor decomposition. We derive a monotonically converging block coordinate descent algorithm with linear complexity in both the sample size and the dimensionality of the inputs for a regularized squared loss function, allowing to learn a parsimonious model in decomposed form using deterministic Fourier features. We demonstrate by means of numerical experiments how our low-rank tensor approach obtains the same performance of the corresponding nonparametric model, consistently outperforming random Fourier features.