This is a very important paper that provides the first efficient and simple method to non-linearly project the high-dimensional signals onto a low-dimensional space. The key concept of the non-linear projection is that the data in fact lay on a manifold in the high-dimension space, which may not be represented by any simple slice of the space (manifold could become folded on the slice).

However, the locally connected graph can approximate the shape of the manifold. As long as the manifold is locally linear, we can obtain a linear reconstruction of each sample from the near neighbors. An optimal weighting set for each sample can be found by solving a least-square problem.

After the weighting set is obtain, the low-dimension representation of the sample on the manifold can then be defined, as long as the same weighting set can still reconstruct the low-dimension representation. Finding the optimal embedding is casted as a eigenvalue problem.

The bold sentences above are the key assumptions of this paper. Unlike Isomap which measure the geodesic distances of all sample pairs, only local distance is preserved. The degree of this effect may be application-dependent. For example, in Action Synopsis, it is shown that non metric replicated multi-dimensional scaling is better than LLE because the global analysis is preserved.

Also another problem is the number of the nearest neighbors (k), which should depends the local sampling rate. Also finding exact k-NN in a high-dimensional space is costly. However, since a effective dimension reduction method is so important, this cost is acceptable.

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