Cross Validated
2022-09-11 06:06 UTC
By Galen
AI-113-20220911-social-media-ae96e691
What does Cayley's hyperdeterminant of a 2x2x2 mixed-product moment tensor tell us about how two variables are related?
Suppose we have a collection of random variables $S = \{ X_0, X_1 \}$ encoded into the $2 \times 2 \times 2$ tensor $$\mathcal{C}[i, j, k] = \mathbb{E}[X_i X_j X_k]$$ where $X_i, X_j, X_k \in S$ and $i,j,k \in \{ 0,1 \}$ . Cayley's hyperdeterminant for $\mathcal{C}$ can be (verbosely) expanded to: \begin{align} \det (\mathcal{C}) &= (\mathbb{E}[X_0^3]^2\mathbb{E}[X_1^3]^2+ 3\mathbb{E}[X_0^2X_1]^2 \mathbb{E}[X_0X_1^2]^2) \\ &-2(3\mathbb{E}[X_0^3] \mathbb{E}[X_0^2X_1]\mathbb{E}[X_0X_1^2]\mathbb{E}[X_1^3] + 3\mathbb{E}[X_0^2X_1]^2\mathbb{E}[X_0X_1^2]^2) \\ &+4(\mathbb{E}[X_0^3]\mathbb{E}[X_0X_1^2]^3 + \mathbb{E}[X_0^2X_1]^3 \mathbb{E}[X_1^3]) \end{align} What does Cayley's hyperdeterminant of a 2x2x2 mixed-product moment tensor tell us about how two $X_0$ and $X_1$ are related?
Suppose we have a collection of random variables $S = \{ X_0, X_1 \}$ encoded into the $2 \times 2 \times 2$ tensor $$\mathcal{C}[i, j, k] = \mathbb{E}[X_i X_j X_k]$$ where $X_i, X_j, X_k \in S$ and $i,j,k \in \{ 0,1 \}$ . Cayley's hyperdeterminant for $\mathcal{C}$ can be (verbosely) expanded to: \begin{align} \det (\mathcal{C}) &= (\mathbb{E}[X_0^3]^2\mathbb{E}[X_1^3]^2+ 3\mathbb{E}[X_0^2X_1]^2 \mathbb{E}[X_0X_1^2]^2) \\ &-2(3\mathbb{E}[X_0^3] \mathbb{E}[X_0^2X_1]\mathbb{E}[X_0X_1^2]\mathbb{E}[X_1^3] + 3\mathbb{E}[X_0^2X_1]^2\mathbb{E}[X_0X_1^2]^2) \\ &+4(\mathbb{E}[X_0^3]\mathbb{E}[X_0X_1^2]^3 + \mathbb{E}[X_0^2X_1]^3 \mathbb{E}[X_1^3]) \end{align} What does Cayley's hyperdeterminant of a 2x2x2 mixed-product moment tensor tell us about how two $X_0$ and $X_1$ are related?
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Cross Validated
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