Random features for Grassmannian kernel approximation with bounded rank-one projections

We propose a family of random feature maps for scalable kernel machines defined over low-dimensional subspaces in high dimensions, i.e. over the Grassmannian manifold. This is typically useful in a machine learning context when data classes or clusters are well represented by the span of a few data points. Classical Grassmannian kernels such as the projection or Binet–Cauchy kernels require constructing full Gram matrices for practical applications, leading to prohibitive computational and memory costs for large subspace datasets in high dimensions. We address this limitation by computing specific random features of subspaces. These combine random rank-one projections of the subspace projection matrices with bounded non-linear transforms—periodic or binary—to tame the resulting heavy-tailed distribution.

We show that, in the random feature space, inner products approximate well-defined, rotation-invariant Grassmannian kernels, i.e. depending only on the principal angles of the considered subspaces. Provided the number of random features is large compared to the subspace intrinsic dimension, we show that this approximation holds uniformly over all subspaces of fixed dimensions with high probability.

When the non-linear transform is periodic, the approximated kernel admits a closed-form expression with a tunable behaviour bridging inverse Binet–Cauchy and Gaussian-type regimes, while the binarised feature has no known closed-form kernel but lends itself to even more compactly represented one-bit subspace features.

Moreover, we show how structured rank-one projections, leveraging randomised fast Fourier transforms, further reduce the random feature computational complexity without sacrificing accuracy in practical experiments. We demonstrate the practicality of these techniques with synthetic experiments and classification tasks on the ETH-80 dataset representing visual object images from different viewpoints. The proposed random features recover Grassmannian geometry with high accuracy while reducing computation, memory, and storage requirements. This demonstrates that rank-one embeddings offer a practical and scalable alternative to classical Grassmannian kernels.