Poor matrix reconstruction from eigenvectors in python vs R

Question

I am trying to learn python, and I have run into something confusing to me. Just as a sanity check, I wanted to make sure I could reconstruct a graph laplacian matrix from its eigenvectors and eigenvalues. In R this works as expected, but not in python. The python reconstructed matrix is pretty far off - the norm(Laplacian - estimate) ~ 0.99, while in R it is ~1e-16. I was hoping that someone could explain to me what is going on here. I've posted the code for both languages below:

In R:

library(igraph)
g <- watts.strogatz.game(1, 20, 3, 0, loops = FALSE, multiple = FALSE)
A <- as.matrix(as_adjacency_matrix(g, type = c("both"),
                                   attr = NULL, edges = FALSE, names = TRUE,
                                   sparse = FALSE))
A <- -A
diag(A) <- abs(rowSums(A))
D <- diag(diag(A)^-0.5, dmn[1])
Ln <- D %*% A %*% D
eL <- eigen(Ln)
rL <- eL$vectors %*% diag(eL$values) %*% t(eL$vectors)
print(norm(Ln - rL))

In Python:

import networkx as nx
import numpy as np

n=20
G = nx.watts_strogatz_graph(n, 3, 0) 
L = nx.normalized_laplacian_matrix(G).toarray()
evals, evecs = np.linalg.eig(L)
idx = evals.argsort()[::-1]   
evals = evals[idx]
F = evecs[:,idx]
D = np.diag(evals)
FDF = np.linalg.multi_dot([F, D, F.T])
rec = np.linalg.norm(L - FDF)
print(rec)

Thank you!

Paul

Igor Rivin · Accepted Answer · 2020-05-04T15:31:00.060

1

This does not really answer the question, but using np.linalg.svd instead of np.linalg.eig does seem to work fine.

edit actually, now I understand. numpy has linalg.eig which does not assume that the matrix is symmetric (or hermitian) and linalg.eigh (which does). Use the second, and all is well.

edited May 04 '20 at 15:31

answered May 04 '20 at 03:18

Igor Rivin

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Got it. I know that with R there are two functions for PCA, one of which uses eigen of the covariance matrix while the other just does SVD. The latter is preferred for numerical accuracy but still doesn't explain this kind of disparity. Thanks though! – user8271479 May 04 '20 at 05:45
@user8271479 I completely agree. When I try it I get a much lower (though still unacceptable, around 0.08) discrepancy with eig. – Igor Rivin May 04 '20 at 15:27
@user8271479 See the edit. R must check whether the matrix is symmetric, and then use the more stable/faster algorithm, but with python you have to do it by hand as in the answer. – Igor Rivin May 04 '20 at 15:32

Poor matrix reconstruction from eigenvectors in python vs R

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