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Assume an m-way tensor $\mathcal{Z}$.

$\mathcal{Z}_{p_1 p_2 ... p_m} = 0$ if any different indices match and $\mathcal{Z}_{p_1 p_2 ... p_m} = 1$ otherwise.

It is a symmetric tensor. Now if it is 2-way tensor, i.e., a matrix, I can decompose it by diagonalization (of a symmetric hollow matrix). For a general tensor, probably I can do a numerical tensor decomposition (e.g., a symmetric tensor decomposition).

But I was wondering, since it is such a simple tensor (elements are either 1 or 0), is there an analytical decomposed expression for this tensor?

I want to avoid storing the full tensor and then decompose it numerically.

I am a not a mathematician so I apologize if my terminologies are not correct.

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    $\begingroup$ You could store it as a function which associates to any $m$ indices the expressions you wrote down above. Then you don't need to store any entries; you just check a list of $m$ indices for duplicate indices. $\endgroup$
    – Ben McKay
    Jun 15, 2020 at 14:16
  • $\begingroup$ 1. What kind of decomposition are you looking for? 2. In the correspondence between symmetric tensors and polynomials, this corresponds to an elementary symmetric polynomial. $\endgroup$ Jun 15, 2020 at 14:47

1 Answer 1

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Thanks to Zach Teitler for the comment that this tensor is associated with elementary symmetric polynomial (ESP). I searched for the decomposed form of ESP and found a paper (Power Sum Decompositions of Elementary Symmetric Polynomials by Lee (2015)).

I followed that and all I had to do was to translate the algebraic expressions in the context of tensors. There might be small errors in my part but this is basically what I found.

The $N$-way, $M$-dimensional tensor $\mathcal{Z}$ is equivalent to the ESP $S_{N}^{M}$ -

\begin{align*} \mathcal{Z}_{p_1 ... p_N} &= \sum_{\mu (I) = 1}^{R} \: \alpha_{\mu} \: t_{p_1}^{\mu} ... t_{p_N}^{\mu} \\ S_{N}^{M} &= \sum_{1 \leq p_1 \leq ... \leq p_N \leq M} X_{p_1} ... X_{p_N} \quad (2 \leq N \leq M) \end{align*}

where R is the rank.

The vector $\alpha$ and matrix $t$ are defined as -

\begin{align*} t_{p}^{\mu (I)} &= - 1, \quad \mbox{if} \: p \in \{ I \} \\ &= + 1, \quad \mbox{otherwise} \end{align*}

\begin{align*} \alpha_{\mu (I)} &= (-1)^{n_I} \: \frac{1}{2^{N-1}} \: \begin{pmatrix} M - k - n_I - 1 \\ k - n_I \end{pmatrix}, \quad \mbox{if} \: k = \frac{N-1}{2} \: \mbox{(odd)} \\ &= (-1)^{n_I} \: \frac{1}{2^N \: (M-N)} \: \begin{pmatrix} M - k - n_I - 1 \\ k - n_I \end{pmatrix} \: (M - 2 \: n_I), \quad \mbox{if} \: k = \frac{N}{2} \: \mbox{(even)} \end{align*} where $ \{ I \} \subset \{ M \} $ and $ n_I = | I | \leq k $. All the subsets are ordered and assigned an integer which belongs to $ \{ \mu (I) \} $. R has an upper bound ($ R \leq \sum_{r=0}^{k} \: \begin{pmatrix} M \\ r \end{pmatrix}$).

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