Failure Idea from Locality Preserving Projection (LPP)
Recently, because of the assignment of the course of "Computational Intelligence", I investigate the method of Locality Preserving Projection (LPP) in more detail. I suppose I found some defects of LPP. It only preserve the local geometric structure of the original data but loss the global geometric relationship. Although the authors of LPP argue that the local geometric structure is more useful than global structure, especially in the application of information retrieval (k nearest neighbors retrieval), the fact is that the global geometric structures reflect the clustering information and can be used to discriminate each class from others. So LPP is useful to reveal the structure of small difference like human pose, expression etc. but will mix the data from multiple classes. Compared with nonlinear method like ISOMap, it's objective function only consider large weight (local structure) and the adjacency graph contains only non-zero weight in local neighbors. LPP has less discriminant power than ISOMap which try to preserve all distance nonlinearly.
My first thought is that if we can find a linear projection that preserving all distances to approximate Multidimensional Scaling (MDS) given a graph constructed similar in ISOMap representing geometric structure in original space. Sooner, I lost in this direction. Two questions I need ask before I am sure I can go deeply in this idea.
- Is it possible to linearly achieve all distance preserving that approximate MDS?
- If possible, why need this linear projection given MDS exists?
So I FAIL and STOP in this idea. Other ways to extend this idea are like how to apply the idea of LPP with other subspace method like GPCA.
