Online PCA with Spectral Bounds

This paper revisits the online PCA problem. Given a stream of n vectors xt ∈ R (columns of X) the algorithm must output yt ∈ R (columns of Y ) before receiving xt+1. The goal of online PCA is to simultaneously minimize the target dimension ` and the error ‖X − (XY +)Y ‖. We describe two simple and deterministic algorithms. The first, receives a parameter ∆ and guarantees that ‖X − (XY +)Y ‖ is not significantly larger than ∆. It requires a target dimension of ` = O(k/ε) for any k, ε such that ∆ ≥ εσ 1 + σ k+1, with σi being the i’th singular value of X . The second receives k and ε and guarantees that ‖X−(XY +)Y ‖ ≤ εσ 1 +σ k+1. It requires a target dimension of O(k log n/ε). Different models and algorithms for Online PCA were considered in the past. This is the first that achieves a bound on the spectral norm of the residual matrix.