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I have a tensor $C$ of size $a \times b \times c$, all with real values.

I need to compute for a vector $u$ of length $a$ and a vector $w$ of length $c$ the product:

$(C \times_{3} w) \times_{1} u$

i.e. multiply the tensor by two vectors on the first and third dimensions.

$a$ and $c$ are rather large, so I want to reduce $C$ to $C'$ with dimensions $a' \times b \times c'$ such that I can transform $u$ to $u'$ and $w$ to $w'$ and compute

$(C' \times_{3} w') \times_{1} u'$

much faster (because the dimensions are smaller).

Any ideas how to do that? I don't mind if there is an error here (because we reduce the dimensions), but it would be nice if it reduced as much as possible.

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Why do yu care about the performance of your calculation? Is it a programming issue? –  draks ... Apr 5 '12 at 22:42

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