Incremental Laplacian Eigenmaps

The incremental Laplacian eigenmaps algorithm [32] seeks to incrementally incorporate new data points by adjusting the local sub-manifold of the new data point s neighbourhood. The three steps followed by incremental Laplacian eigenmaps are update the adjacency matrix project the new data point update the local sub-manifold affected by the insertion of the new data point. 

The adjacency matrix, F, is initially updated in light of the addition of the new data point X. This is done by first extending the matrix so that it is now of size (n -I-1) x 

The low-dimensional representation, y, of x is then found by using an alternative formulation of the linear incremental method or the sub-manifold analysis method. Both seek to find y in terms of either the entire weight matrix (linear incremental) or a subset of the weights (sub-manifold analysis method). 

The incremental Laplacian eigenmaps method is fast to compute due to its simplicity. It is however dependent on whether the sub-manifold or linear incremental method is used to obtain the low-dimensional representation. The sub-manifold method does provide improved results over the linear incremental method but at an increased computational cost [32]. 

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Jia, R, Yin, J., Huang, X., Hu, D. Incremental Laplacian Eigenmaps by preserving adjacent information between data points. Pattern Recognition Letters 30, 1457-1463 (2009)... 

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