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We know from e.g. Koutis-Miller-Peng (based on work of Spielman & Teng), that we can very quickly solve linear systems $A x = b$ for matrices $A$ that are the graph Laplacian matrix for some sparse graph with non-negative edge weights.

Now (first question) consider using one of these graph Laplacian matrices $A$ as the covariance or (second question) inverse covariance matrix of a zero-mean multivariate normal distribution $\mathcal{N}(\boldsymbol{0}, A)$, or $\mathcal{N}(\boldsymbol{0}, A^{-1})$. For each of these cases, I have two questions:

A. How efficiently can we draw a sample from this distribution? (Typically to draw a sample, we compute the Cholesky decomposition $A = LL^T$, draw a standard normal $y \sim \mathcal{N}(\boldsymbol{0}, I)$, then compute a sample as $x = L^{-1} y$).

B. How efficiently can we compute the determinant of $A$?

Note that both of these could be solved easily given a Cholesky decomposition, but I don't immediately see how to extract $L$ more efficiently than just by using a standard sparse Cholesky algorithm, which wouldn't use the techniques presented in the above-referenced works, and which would have cubic complexity for sparse-but-high-treewidth graphs.

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  • $\begingroup$ I think it might help to be a little more specific on what you'd consider "efficient" in both cases. Is "efficient" the same as "not dependent on a Cholesky decomposition" ? $\endgroup$ – Suresh Venkat Aug 3 '12 at 19:52
  • $\begingroup$ Thanks for the suggestion. It's possible the answer to all the questions is "you need to compute a Cholesky decomposition, and there is no structure that can be leveraged beyond the sparseness of the matrix." I'd be interested to know if this were true (but I hope it's not). With respect to "efficiently" in the last paragraph, yes, I mostly mean more efficiently than standard sparse Cholesky algorithms. Although if there were a way to use the techniques of the above-referenced work to compute a Cholesky equally as fast as can be done via other means, that'd also be interesting. $\endgroup$ – dan_x Aug 7 '12 at 0:13
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There are two separate issues here.

  1. How to use efficient solvers for $Ax=b$ in order to apply $A^{1/2}b$.
  2. How to compute the determinant.

The short answers are 1) use rational matrix function approximations, and 2) you don't, but you don't need to anyways. I address both of these issues below.

Matrix square root approximations

The idea here is to convert a rational function approximation for scalar functions into a rational function approximation for matrix functions.

We know that there exist rational functions that can approximate the square root function extremely well, $$\sqrt{x} \approx r(x) := \frac{a_1}{x+b_1} + \frac{a_2}{x+b_2} + \dots + \frac{a_N}{x+b_N},$$ for positive $b_i$. Indeed, to get high accuracy on the interval $[m,M]$, you need $O(\log \frac{M}{m})$ terms in the series. To get the appropriate weights ($a_i$) and poles ($-b_i$), just look up rational function approximation online or in a book.

Now consider applying this rational function to your matrix: $$r(A) = a_1(A + b_1 I)^{-1} + a_2(A + b_2 I)^{-1} + \dots + a_N(A + b_N I)^{-1}.$$

Owing to the symmetry of $A$, we have \begin{align} ||A^{1/2} - r(A)||_2 &= ||U\left(\Sigma^{1/2} - r(\Sigma)\right)U^*||_2, \\ &= \max_i |\sqrt{\sigma_i} - r(\sigma_i)| \end{align} where $A = U \Sigma U^*$ is the singular value decomposition (SVD) of $A$. So, the quality of the rational matrix approximation is equivalent to the quality of the rational function approximation at the location of the eigenvalues.

Denoting the condition number of $A$ by $\kappa$, we can apply $A^{1/2}b$ to any desired tolerance by performing $O(\log \kappa)$ positively shifted graph Laplacian solutions of the form, $$(A + bI)x=b.$$

These solutions can be done with your favorite graph Laplacian solver—I prefer multigrid type techniques, but the one in the paper you cite should be fine, too. The extra $bI$ only helps the convergence of the solver.

For an excellent paper discussing this, as well as more general complex analysis techniques that apply to nonsymmetric matrices, see Computing $A^α$, $\log(A)$, and related matrix functions by contour integrals, by Hale, Higham, and Trefethen (2008).

Determinant "computation"

The determinant is harder to compute. As far as I know, the best way is to compute the Schur decomposition $A = Q U Q^*$ using the QR algorithm, then read off the eigenvalues from the diagonal of the upper-triangular matrix $U$. This takes $O(n^3)$ time, where $n$ is the number of nodes in the graph.

However, calculating determinants is an inherently ill-conditioned problem, so if you ever read a paper that relies on computing determinants of a large matrix, you should be very skeptical of the method.

Luckily, you probably don't actually need the determinant. For example,

  • To draw samples from a single Gaussian distribution $N(0,A^{-1})$, the normalization constant is the same at all points so you never need to compute it.
  • If your Laplacian matrix $A = A_x$ represents the inverse covariance of a local Gaussian approximation at point $x$ to a non-Gaussian distribution, then the determinant does indeed change from point to point. However, in every effective sampling scheme I know (including Markov chain Monte Carlo, importance sampling, etc.) what you really need is the determinant ratio, $$\det(A_{x_0}^{-1}A_{x_p}),$$ where $x_0$ is the current point, and $x_p$ is the proposed next sample.

We can view $A_{x_0}^{-1}A_{x_p}$ as a low-rank update to the identity, $$A_{x_0}^{-1}A_{x_p} = I + Q D Q^*,$$ where the effective numerical rank, $r$, of the low-rank update is a local measure of how non-Gaussian the true distribution is; typically this is much lower than the full rank of the matrix. Indeed, if $r$ is large, then the true distribution is locally so non-Gaussian that one ought to question the whole strategy of trying to sample this distribution using local Gaussian approximations.

The low-rank factors $Q$ and $D$ can be found with randomized SVD or Lanczos by applying the matrix $$A_{x_0}^{-1}A_{x_p} -I$$ to $O(r)$ different vectors, each application of which requires one graph Laplacian solution. Thus the overall work for getting these low rank factors is $O(r \max(n,E))$.

Knowing $D = \text{diag}(d_1,d_2,\dots,d_r)$, the determinant ratio is then $$\det(A_{x_0}^{-1}A_{x_p}) = \det(I + Q D Q^*) = \exp\left(\sum_{i=1}^r \log d_i\right).$$

These low rank determinant ration calculation techniques can be found in A Stochastic Newton MCMC Method for Large-Scale Statistical Inverse Problems with Application to Seismic Inversion, by Martin, et al. (2012). In this paper it is applied to continuum problems, so the "graph" is a grid in 3D space and the graph Laplacian is the actual Laplacian matrix. However, all the techniques apply to general graph Laplacians. There are probably other papers applying this technique to general graphs by now (the extension is trivial and basically what I just wrote).

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