【子空间聚类】Sparse Subspace Clustering(SSC) Algorithm
Symbol definition：
is n linear subspace of
.
is the dimension of
.
is N noise-free data points.
is a rank-
matrix, represent
(
>
) points that lie in
.
is a unknown permutation matrix.
SSC Algorithm:
Step 1: Solve the sparse optimization program
① Uncorrupted data
② Corrupted data
ps: E corresponds to a matrix of sparse outlying entries, and Z is a noise matrix.
Step 2: Normalize the columns of C as .
ps: max norm : .
Step 3: Form a similarity grahp with N nodes wegiths on the edges between the nodes by
.
ps:
Step: Use spectral clustering to the similarity graph.
Output: .
Subspace-Sparse Recovery Theory
Definition:
①
ps: is said to be independent.
②
③ Principle angle between and
Independent Subspace Model：
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其实，这篇paper主要是讲了SSC算法，在求稀疏解时的限定条件原本应该是0-范数最小，求最稀疏解，可是0-范数根本没法求，只具有实际意义，求解是个NP-hard问题，所以利用凸规划松弛方法，退而且其次，选择1-范数，使1-范数最小，得到稀疏解。Paper的后面证明了，利用1-范数最小解求解稀疏解，也可以得到理想的稀疏表示，可以使属于不同空间的点没有关联。也就是说Paper的精髓就在于证明1-范数最小值得到的稀疏解，和0-范数最小值得到的稀疏解的效果差不多。
