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Suppose we have a 4-index tensor $t_{ijkl}$ (all 4 dimensions are equal size). We can make a matrix out of it by taking first and last two indexes as new indexes: $t_{ijkl} \rightarrow M_{ij, kl}$. Suppose we have an SVD of this matrix $M = U S V^\dagger$. Now we take another matricization of $t$, such that $t_{ijkl} \rightarrow N_{ik, jl}$. Can we say anything about the SVD of $N$ from the SVD of $M$? More precise, here are two questions I would like to answer

  1. Assuming the rank of M is R1, how can the rank of N be estimated?

  2. Is there a map between singular vectors of M and N, and if so how can it be built?

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    $\begingroup$ Do you know the paper A multilinear singular value decomposition? It gives some nice insight about SVD of tensors unfoldings. $\endgroup$
    – Surb
    Commented Mar 24, 2016 at 9:00
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    $\begingroup$ @Surb I've read it some time ago, but thank you for bringing this point $\endgroup$
    – qbit-
    Commented Mar 24, 2016 at 9:27
  • $\begingroup$ Just following on the idea of using multilinear SVD (HOSVD). In HOSVD the tensor is decomposed to a "core" tensor and orthogonal factor matrices along each dimension. $t_{ijkl} = \sum_{abcd} \sigma_{abcd} U_{ai} V_{bj} W_{ck} Y_{dl}$. Permuting the original tensor would mean permuting the core tensor and the order of factor matrices. However, I don't see how the core tensor from HOSVD is connected to the singular value matrix $S$ of $M$. In fact, if we set the size of the dimensions to be $n$, $\sigma$ can have up to $n^4$ elements, whereas $S$ will have only $n^2$ at most $\endgroup$
    – qbit-
    Commented Mar 24, 2016 at 17:28
  • $\begingroup$ Unfortunately I'm not sure how to answer this question right now because it is so general. For example, yes, the SVD of $N$ can be calculated using the SVD of $M$: use the SVD of $M$ to reconstruct $M$ itself, which can be used to construct $N$, which can be used to get the SVD of $N$. Of course, this certainly isn't what you actually want, but I'm not really sure what you do want either. It seems like you're perhaps interested in how the realignment map from quantum entanglement affects the singular values of a matrix. Is this correct? $\endgroup$
    – Nathaniel Johnston
    Commented Mar 31, 2016 at 17:20
  • $\begingroup$ @Nathaniel Johnston, thanks for commenting. Indeed, I'm interested in this problem because it's relation to quantum entanglement. I've corrected the question to clarify it $\endgroup$
    – qbit-
    Commented Apr 1, 2016 at 18:54

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This is a late answer, but there has been some movement regarding this topic. I humbly recommend to look into the article https://arxiv.org/abs/1701.08437 and in particular the citations therein.

There is also a segment about the equivalence of the tensor feasibility and the quantum marginal problem, which might be interesting depending on ones background.

The connection between the two matricization given in your example however turns out to be a bit more difficult to categorize. On the other hand, regarding the connection of singular values of $M_{i,jk\ell}$, $M_{ij,k\ell}$ and $M_{ijk,\ell}$, the problem is basically completely solved.

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