The idea used in this paper is very simple. Imagine that the feature points are scattered on the manifolds (locally linear patch), that is, a feature point can be described (linear combination) by its neiborhoods. Under this assumption, the low-dimensional description (weights) of each feature point is firstly obtained by searching its kNN and minimizing an error function. Given those weights, choose y (less than d) d-dimensional coordinates to minimize the sum of the reconstruction cost for every feature point.
My idea:
I think a major trick used in LLE is that the (rough) low-dimensional descriptions are found first. By this way, some information (local relationship in LLE) are preserved when the embedding coordinates are computed. My point is: If we obtain the weights by other methods (e.g., choose the neighbors that belong to the same class instead of using kNN), something other than local relationship (e.g., in-class relationship) can be preserved after the dimension reduction.
Reference:
S. T. Roweis and L. K. Saul, "Nonlinear Dimensionality Reduction by Locally Linear Embedding." Science, Vol. 290, pp. 2323-2326, 2000.
I think a major trick used in LLE is that the (rough) low-dimensional descriptions are found first. By this way, some information (local relationship in LLE) are preserved when the embedding coordinates are computed. My point is: If we obtain the weights by other methods (e.g., choose the neighbors that belong to the same class instead of using kNN), something other than local relationship (e.g., in-class relationship) can be preserved after the dimension reduction.
Reference:
S. T. Roweis and L. K. Saul, "Nonlinear Dimensionality Reduction by Locally Linear Embedding." Science, Vol. 290, pp. 2323-2326, 2000.
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