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There's an easily understandable intuitive understanding of the 1st eigenvector (for the eigenvalue == 1) of the pagerank matrix (loosely speaking: assuming you
by frig 17y ago
There's an easily understandable intuitive understanding of the 1st eigenvector (for the eigenvalue == 1) of the pagerank matrix (loosely speaking: assuming you browse a-la the pagerank matrix, the ith component of the (unit) eigenvector for the 1.0 eigenvalue is ~the same as lim n->infinity of #(of visits to i-th page on the internet)/n; some details are getting blurred (of course) but it's a helpful intuitive explanation).
Does anyone have any insight into intuitive interpretations of the eigenvectors fpr the eigenvalues < 1? I'm drawing blanks.
(Edited: to be clear, I understand what the linked algorithm is doing and I think it is very clever, and I am not surprised that it works reasonably well for finding similar topics on wikipedia. I'm just at a blank for what it means to say that 'the i-th component of the (unit) eigenvector for the k-th largest eigenvalue is X'.)
- diiq 17y agoI would reccomend looking at 'Proto Value Functions' ( http://www.machinelearning.org/.../070_ProtoValue_Mahadevan.pdf http://www.machinelearning.org/.../070_ProtoValue_Mahadevan.... )[1]; they use eigenvectors of adjaceny matrices for reinforcement learning --- but because gridworld has a simple adjacency martix, the resulting eigenvectors are somewhat human-readable, and the pictures help a lot. If you're really curious, I can probably bang up some python or MATLAB or something. The essential intuition is that higher numbered eigenvectors represent more local adjacency information --- so the first eigenvector deals with global connectivity (the relative value of i-th and j-th component says something about how interconnected nodes i and j are in terms of the whole graph). A higher eigenvalue's corresponding vector has a higher local gradient, and the direction of the gradient is less constant across the graph (there is not necessarily a monotonic path from node to node) so says less about global connectivity --- but has a higher granularity when examining local connectivity. 1[Edited to warn that Mahadevan's papers are fantastically dense --- and you don't need to understand the paper to gain intuition from the pretty pictures; read the text at your own risk]