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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 $M$? Can the SVD of $N$ be calculated using 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?

    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?

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

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 $M$? Can the SVD of $N$ be calculated using 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?

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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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 $M$? Can the SVD of $N$ be calculated using 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?

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 $M$? Can the SVD of $N$ be calculated using the SVD of $M$?

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 $M$? Can the SVD of $N$ be calculated using 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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Tensor matricizations and their decompositions

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 $M$? Can the SVD of $N$ be calculated using the SVD of $M$?