Random Feature Mapping with Signed Circulant Matrix Projection

Random feature mappings have been successfully used for approximating non-linear kernels to scale up kernel methods. Some work aims at speeding up the feature mappings, but brings increasing variance of the approximation. In this paper, we propose a novel random feature mapping method that uses a signed Circulant Random Matrix (CRM) instead of an unstructured random matrix to project input data. The signed CRM has linear space complexity as the whole signed CRM can be recovered from one column of the CRM, and ensures loglinear time complexity to compute the feature mapping using the Fast Fourier Transform (FFT). Theoretically, we prove that approximating Gaussian kernel using our mapping method is unbiased and does not increase the variance. Experimentally, we demonstrate that our proposed mapping method is time and space efficient while retaining similar accuracies with state-of-the-art random feature mapping methods. Our proposed random feature mapping method can be implemented easily and make kernel methods scalable and practical for large scale training and predicting problems.