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How does the sparsity condition on an inverse covariance matrix affect the actual covariance matrix?

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  • $\begingroup$ Unfortunately, it does not $\endgroup$
    – Yair Daon
    Apr 12, 2016 at 13:02
  • $\begingroup$ There's got to be something. For instance identity matrix is sparse, and its inverse too. $\endgroup$
    – Aksakal
    Jun 23, 2016 at 18:17

1 Answer 1

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As already commented by Yair there is no specific sparsity condition of the inverse covariance matrix that affects the actual covariance matrix or vice versa. Anything other than trivial sparsity matrix patterns (ie. diagonal) have no guarantee that they will reflected on both a particular matrix and its inverse. Even tridiagonal matrices can easily have non sparse inverses.

For particular cases where the sparsity of the matrix occurs in blocks you might be able to derive some results stemming from the Block matrix pseudoinverse algorithm which states that:

$\left[\begin{array}{cc} A & B \\C & D \end{array}\right]^{-1} =\begin{bmatrix} (A - BD^{-1}C)^{-1} & -A^{-1}B(D - CA^{-1}B)^{-1} \\ -D^{-1}C(A - BD^{-1}C)^{-1} & (D - CA^{-1}B)^{-1} \end{bmatrix}$

but that's probably about it (purely anecdotally, I have tried to impose sparsity patterns through the Cholesky decomposition of a PSD matrix but I failed in my trial-and-error foray). You might also want to consider looking into the Cuthill–McKee algorithm (CM) if you expect some adjacency feature to be reflected in the covariance matrix. The CM algorithm permutes a sparse matrix that has a symmetric sparsity pattern into a band matrix form with a small bandwidth, this might help preserve some sparsity towards the off-diagonal entries of the inverse matrix but that is not guaranteed. (Applying CM -if reasonable- can be very helpful for particular applications (eg. in 2D smoothing routines) and may significantly speed-up your computations.)

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    $\begingroup$ (+1) Because the block-diagonal matrices of any given shape are a ring, their inverses (whenever they exist) have the same block-diagonal structure and so preserve that much of sparsity pattern. As an extreme example, diagonal matrices are block-diagonal, thereby exemplifying the case pointed out by @Aksakal. The furthest one can go in this direction is to conjugate block-diagonal matrices by permutation matrices (which obviously preserves all zero and non-zero entries, but just moves them around). $\endgroup$
    – whuber
    Jun 23, 2016 at 18:24
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    $\begingroup$ (+1) Thank you for this comment (short answer really). It is really insightful. I will definitely consider it in the future. $\endgroup$
    – usεr11852
    Jun 23, 2016 at 20:16

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