SVD-Based Low-Complexity Methods for Computing the Intersection of K≥2 Subspaces

Given the orthogonal basis (or the projections) of no less than two subspaces in finite dimensional spaces, we propose two novel algorithms for computing the intersection of those subspaces. By constructing two matrices using cumulative multiplication and cumulative sum of those projections, respectively, we prove that the intersection equals to the null spaces of the two matrices. Based on such a mathematical fact, we show that the orthogonal basis of the intersection can be efficiently computed by performing singular value decompositions on the two matrices with much lower complexity than most state-of-the-art methods including alternate projection method. Numerical simulations are conducted to verify the correctness and the effectiveness of the proposed methods.