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Suppose we have $k$ $n\times n$ matrices $A,B,C,\ldots$. Is there a way to compute/approximate the trace of their product much faster than computing/approximating the full matrix product? IE, computing

$$\text{tr}(ABC\ldots)$$

A related problem of computing vector matrix product $v'ABC$ can be done in $O(n^2)$ time by skipping most of the computation needed for the full matrix product, so I'm wondering if there's a trick to speed up the trace as well

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  • $\begingroup$ By any chance, is the resulting matrix symmetric? Then could compute $v^\top ABC \ldots v$ for $v \sim \mathcal N(0^n, I_n))$, and your expectation should be the trace, unless I made a mistake. $\endgroup$
    – Dmitry
    Nov 15, 2021 at 11:32
  • $\begingroup$ Yes, that works (even for nonsymmetric), but it's not clear this is better than approximating full matrix product in a similar way, just put v's in the middle: AB...Cvv'DE.. is equal to the matrix product in expectation $\endgroup$ Nov 16, 2021 at 22:29
  • $\begingroup$ Should be the same, by the cyclic property of trace: $tr(AB...Cvv'DE..F)=tr(v'DE..FAB...Cv)$. But it already gives you the solution, no? (product of Gaussians should have decent concentration) $\endgroup$
    – Dmitry
    Nov 17, 2021 at 7:30
  • $\begingroup$ So here it seems that approximating trace has same cost as approximating the matrix product, since you can view this trick as taking trace of the matrix product approximation. I was curious if trace is fundamentally easier than full matrix product (like is the case with vector/matrix product), which seems like it isn't $\endgroup$ Nov 17, 2021 at 20:08

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For the worst-case time complexity, no algorithm faster than $\Theta(n^{\omega})$ time is known, even for the approximation version.

Indeed, there is a simple reduction from the triangle counting problem to the trace computation of a product matrix. Let $A$ be an adjacent matrix of an undirected graph, then $\mathrm{tr}(A^3)/2$ is the number of triangles of the graph. Note that this reduction preserves approximation factors.

As far as I know, this is the best-known way to count triangles in dense graphs. I don't have a proper reference but see this question on TCS.SE https://cstheory.stackexchange.com/questions/48539/fastest-approximate-triangle-counting-algorithms-in-dense-graphs for example.

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  • $\begingroup$ Thanks, connection to triangle counting is a useful to know $\endgroup$ Dec 15, 2021 at 18:52

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