we have $Ax=\lambda x$ (1) differentiate both side we have $\Delta Ax A\Delta x= \Delta\lambda x \lambda\Delta x$.

multiply both side with $x^T$ and we have $x^T\Delta Ax x^TA\Delta x= x^T\Delta\lambda x x^T\lambda\Delta x$ (2) from (1), by transposing both side we have $x^TA^T=\lambda x^T$.

I was also a member of the Image Formation and Processing Group(IFP) at the Beckman Institute for Advanced Science and Technologies at UIUC.

Say I used spectral clustering to cluster a data-set $D$ of points $X_0 - X_n$ into a number $C$ of clusters.

We describe different graph partition criteria, the definition of spectral clustering, and clustering steps, etc.

Finally, in order to solve the disadvantage of spectral clustering, some improvements are introduced briefly.

Spectral clustering algorithm is a kind of clustering algorithm based on spectral graph theory.

As spectral clustering has deep theoretical foundation as well as the advantage in dealing with non-convex distribution, it has received much attention in machine learning and data mining areas.Experimental results show that 1) as the number of data points increases, the mapping results of the proposed approach become closer and closer to that of batch-style approaches, including LTSA and LE, and 2) the proposed approach outperforms the incremental ISOMAP (IISOMAP, a typical incremental manifold learning algorithm) in mapping accuracy. We argue that the new algorithm is suitable for incremental learning of large-scale data streams. (2012) A New Manifold Learning Algorithm Based on Incremental Spectral Decomposition. Copyright (c) 2012, Ingo Bürk Copyright (c) 2003, Jochen Lenz Copyright (c) 2015, Oliver Woodford All rights reserved.Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: * Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.The connectivity of a graph is an important measure of its robustness as a network.

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