Let ˆc be the optimal solution of minimizec 12kφ(y) φ(X)ck2 + λ2kck2, where φ is induced by Gaussian kernel and y is arbitrary. It is worth noting that Algorithm 1 can be easily implemented parallelly, which will reduce the time complexity to O(max(m,r)n2 +kmn). Denote vi = (vi1,...,vin) the i-th row of V and let vi = (vi1,...,vid), where d < n. Clustering the columns of X given by Definition C.1 according to the polynomials is actually a manifold clustering problem beyond the setting of subspaceclustering. The following theorem verifies the effectiveness of (15) followed by the truncation operation in manifolddetection. 2 TheoremC.3.

The performance of spectral clustering heavily relies on the quality of affinity matrix.