A GENERAL FRAMEWORK FOR MANIFOLD RECONSTRUCTION FROM DIMENSIONALITY REDUCTION

Recently, many dimensionality reduction (DR) algorithms have been developed, which are successfully applied to feature extraction and representation in pattern classification. However, many applications need to re-project the features to the original space. Unfortunately, most DR algorithms cannot perform reconstruction. Based on the manifold assumption, this paper proposes a General Manifold Reconstruction Framework (GMRF) to perform the reconstruction of the original data from the low dimensional DR results. Comparing with the existing reconstruction algorithms, the framework has two significant advantages. First, the proposed framework is independent of DR algorithm. That is to say, no matter what DR algorithm is used, the framework can recover the structure of the original data from the DR results. Second, the framework is space saving, which means it does not need to store any training sample after training. The storage space GMRF needed for reconstruction is far less than that of the training samples. Experiments on different dataset demonstrate that the framework performs well in the reconstruction.

A note on immersing manifolds in euclidean spaces

Let M be a closed, connected smooth and 3-connected mod 2 (that is Hi(M;ℤ2) = 0, 0 < i ≤ 3) manifold of dimension n = 7 + 8k. Using a combination of cohomology operations on certain cohomology classes of M and on the Thom class of the stable normal bundle of M we show that under certain conditions M immerses in R2n−8. This extends previously known results for such a general manifold when the number of 1's in the dyadic expansion of n is less than 8.