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I am looking for information on a specific type of tensor/matrix decomposition which is quite similar to the SVD for matrices but does not look like the HOSVD since the core tensor is only a vector. Given a rank-$p$ Tensor $A_{i_1,\ldots,i_p}\in\mathbb{F}^{n_1,\ldots,n_p}$ the decomposition I am looking for should be something like

$$ A_{i_1,\ldots,i_p} = \sum_{\alpha} U_{i_1,\ldots,i_q,\alpha}\ \Sigma_{\alpha}\ V^{*}_{\alpha,i_{q+1}\ldots,i_p}$$

for some $1\leq q\leq p$ and where $^*$ denotes the conjugate transpose. Basically, $A$ is factored into two tensors $U$ and $V$ and a "singular value" vector $\Sigma$ with $\Sigma_{\alpha}>0$. Most importantly, note that the dimensions of $A$ are "split" after $i_q$ with $U$ having $q+1$ dimensions and $V^*$ having $i-q+1$. The requirements on $U$ and $V$ should be similar to those from the regular SVD. For example,

$$ \sum_{\alpha} U_{i_1,\ldots,i_q,\alpha} U^{*}_{i_1',\ldots,i_q',\alpha} = \prod_{j}\delta_{i_j,i_j'}$$

but also, more generally

$$ \sum_{i_{\beta}} U_{i_1,\ldots,i_{\beta},\ldots,i_q} U^{*}_{i_1',\ldots,i_{\beta},\ldots,i_q',\alpha} = \prod_{j\neq \beta}\delta_{i_j,i_j'}$$

for all $1\leq \beta \leq q$ and analogous for $V$.

which would correspond to the unitarity of the matrices $U$ and $V$ for the regular SVD.

An interesting special case of such a decomposition would be the symmetric case $U=V$.

Several questions arise from this, such as the existence and uniqueness of such a decomposition. Does it have an established name? Is there more information you can give me on such a decomposition? I am also looking for helpful literature.

Update

From what I have found, this could be described as PARAFAC with orthogonal rank-$1$ vectors and positive $\lambda_r$. I have found something called WON-PARAFAC ("Weighted orthogonal non-negative PARAFAC") but I still need to do more research.

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  • $\begingroup$ Does such decomposition exist? $\endgroup$ Jun 4, 2021 at 21:59
  • $\begingroup$ @მამუკაჯიბლაძე I don't know in general. For the problem I am working on, it should exist. $\endgroup$
    – user141399
    Jun 4, 2021 at 22:00
  • $\begingroup$ Could you provide sufficient conditions that suit you? $\endgroup$ Jun 4, 2021 at 22:01
  • $\begingroup$ @მამუკაჯიბლაძე There are no real conditions I would know of. Everything that should be necessary is written in the text, such as the unitarity. $\endgroup$
    – user141399
    Jun 5, 2021 at 11:53

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