
Computability in Linear Algebra (2004) - ogogmad
https://www.sciencedirect.com/science/article/pii/S0304397504004086
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taliesinb
Interesting. Also, small world: One of the authors (Vasco) taught me
computability at the University of Cape Town.

What I find quite interesting to think about is how the complexity (as a
function of accuracy) of a “taken-for-granted” computational process works in
the context of another process. For example, the eigensystem of its adjacency
matrix provides powerful understanding of a graph — but how much accuracy do
you need in order to trust that understanding? If you need arbitrary precision
because of the pathological features of a particular family of graph (expander
graphs, say), and the scaling of the needed precision links directly to
aspects of that graph pathology, this is really telling you that the
eigenvector computation is not the most natural “computation basis” to
approach this phenomenon — and some explicitly discrete algorithm would make
this latent complexity relationship manifest. Probably individual instances of
this kind of phenomena are well known to those who study the relevant
structures, but I’m not sure if the general phenomena is described or modeled
anywhere.

In some weird sense you can link this to a kind of “computational curvature” —
ala some of Wolfram’s metamathematical musings. If one’s prior is to distrust
continuous mathematics and see only discrete computational processes as
“real”, it’s exactly the kind of phenomena one will be on the lookout for.
Anyway, the curvature analogy is that as you attempt to maintain a particular
computational basis for computing a phenomena of interest as its complexity
scales, other typically-uncorrelated complexity requirements that are normally
treated as constants will explode. Needs to be made more concrete though.

~~~
eru
> If you need arbitrary precision because of the pathological features of a
> particular family of graph (expander graphs, say), and the scaling of the
> needed precision links directly to aspects of that graph pathology, this is
> really telling you that the eigenvector computation is not the most natural
> “computation basis” to approach this phenomenon — and some explicitly
> discrete algorithm would make this latent complexity relationship manifest.

Or perhaps you just need to perform your eigenvector computation in a more
hospitable ring or field?

To give an example of the flavour: many graph computations are more natural
when performed on their adjacency matrix over tropical semi-rings. First and
foremost shortest path computations.

See eg
[http://r6.ca/blog/20110808T035622Z.html](http://r6.ca/blog/20110808T035622Z.html)
or
[http://stedolan.net/research/semirings.pdf](http://stedolan.net/research/semirings.pdf)

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jjgreen
Needs a [2004] (but great article nonetheless).

