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I do recall looking up the definition of 'bijection', and seeing that it slightly resembles a graph. As a generalization: so an n-dimensional tensor can be view
by bhl 8y ago
I do recall looking up the definition of 'bijection', and seeing that it slightly resembles a graph. As a generalization: so an n-dimensional tensor can be viewed as a weighted n-partite graph?
- Twisol 8y agoAn n-partite graph has the problem that edges can still only connect two vertices at a time. I think you would also need to include hyper-edges (faces, solids, etc). The joint probabilities would be represented by these hyper-edges, and the bounding ("lower") edges represent progressively more marginal probabilities. Higher category theory studies these kinds of objects, so it wouldn't particularly surprise me. I wonder if simplicial complexes would work here? https://en.wikipedia.org/wiki/Simplicial_complex https://en.wikipedia.org/wiki/Simplicial_complex
- yorwba 8y agoNo, because the space of tensors of order n with k-dimensional arguments is k^n-dimensional, while a complete n-partite graph with partitions of size k has only nk(n-1)k/2 edges, which doesn't give you enough coefficients to uniquely identify a tensor if n > 2. You'd need a n-uniform n-partite hypergraph, where coefficients are assigned to n-tuples instead of pairs. So for a tensor of order 3, that would mean weighted triangles, which are somewhat hard to draw and everything gets less intuitive.