Localized Sliced Inverse Regression

by Qiang Wu, Sanyan Mukherjee and Feng Liang

Sliced inverse regression (SIR) for classification is equivalent to FDC. That is to solve a generalized eigenvalue problem Γβ=λΣβ, where Γ is the between-class covariance matrix and Σ is the total covariance matrix. The largest eigenvalues of this problem correspond to the disired directions. Well, in some cases we use the in-class covariance matrix B for Σ.

This has been discussed in Jieping Ye's paper extensively. In a way Σ is more likely to be non-singular than B. Due to the fact Σ = Γ + B, there exists a non-singular matrix A such that they can be diagonalized simultaneously with a transform Az = x. If they are non-singular, it matters not whether we use B or Σ. When B or Σ is singular, we might use its principal subspace or its null space for approximation (well it's not approximation I guess). I wonder what we shall do for SIR. Maybe we do the same rubbish too.

Now we come to the LSIR. The authors proposes a localized Γ by choosing the kNN samples of the same class. The only difference between LSIR and LFDC, I think, is the denominator part (i.e. a global covariance class v.s. a localized in-class covariance). Maybe I have to explore the corresponding graph to see their difference more clearly.

Their extension for SSL is to put all unlabelled samples into to all slices. That might be natural in the graph embedding framework. Why should I read this paper then?