How to derive a bound of distortion / error between two different tensor decompositions.
$begingroup$
Consider a tensor $mathcal{X}inmathbb{R}^{Itimes Jtimes K} $. It can be approximately decomposed/factored in multiple ways. Namely by using the TUCKER3 decomposition:
$mathcal{X}approx sum_{p=1}^Psum_{q=1}^Qsum_{r=1}^Rmathcal{G}_{pqr}mathbf{a}_pcircmathbf{b}_qcircmathbf{c}_r $,
where $mathcal{G}in mathbb{R}^{Ptimes Qtimes R}$ is a core mixing tensor, and $mathbf{a}_p,mathbf{b}_q,mathbf{c}_r$ are vectors from $mathbf{A}inmathbb{R}^{Itimes P},mathbf{B}inmathbb{R}^{Jtimes Q},mathbf{C}inmathbb{R}^{Ktimes R}$, and $circ$ denotes the outer product.
Now if the data tensor, $mathcal{X}$ can be assumed tri-linear in some $mathbf{A,B,C}$ components, a simpler decomposition is given by the CP-deomposition:
$mathcal{X}approx sum_{r=1}^R lambda_r cdot mathbf{a}_rcircmathbf{b}_rcircmathbf{c}_r $.
Now the question:
I know my data tensor $mathcal{X}$ is not trilinear in nature, but I wish to decompose it with the CP method anyway. After this decomp. I will use the $mathbf{C}$ matrix for a clustering problem. Is there anyway to derive some maths (e.g. bounds) for how much error/distortion I can expect if use elements of the $mathbf{C}$ from the CP decomposition, instead of the TUCKER3 which is theoretically "more suitable" for my tensor.
vector-spaces norm tensors tensor-rank
asked Jan 2 at 9:24
pche8701pche8701
135
$endgroup$
add a comment |
$begingroup$
Consider a tensor $mathcal{X}inmathbb{R}^{Itimes Jtimes K} $. It can be approximately decomposed/factored in multiple ways. Namely by using the TUCKER3 decomposition:
$mathcal{X}approx sum_{p=1}^Psum_{q=1}^Qsum_{r=1}^Rmathcal{G}_{pqr}mathbf{a}_pcircmathbf{b}_qcircmathbf{c}_r $,
where $mathcal{G}in mathbb{R}^{Ptimes Qtimes R}$ is a core mixing tensor, and $mathbf{a}_p,mathbf{b}_q,mathbf{c}_r$ are vectors from $mathbf{A}inmathbb{R}^{Itimes P},mathbf{B}inmathbb{R}^{Jtimes Q},mathbf{C}inmathbb{R}^{Ktimes R}$, and $circ$ denotes the outer product.
Now if the data tensor, $mathcal{X}$ can be assumed tri-linear in some $mathbf{A,B,C}$ components, a simpler decomposition is given by the CP-deomposition:
$mathcal{X}approx sum_{r=1}^R lambda_r cdot mathbf{a}_rcircmathbf{b}_rcircmathbf{c}_r $.
Now the question:
I know my data tensor $mathcal{X}$ is not trilinear in nature, but I wish to decompose it with the CP method anyway. After this decomp. I will use the $mathbf{C}$ matrix for a clustering problem. Is there anyway to derive some maths (e.g. bounds) for how much error/distortion I can expect if use elements of the $mathbf{C}$ from the CP decomposition, instead of the TUCKER3 which is theoretically "more suitable" for my tensor.
vector-spaces norm tensors tensor-rank
asked Jan 2 at 9:24
pche8701pche8701
135
$endgroup$
add a comment |
$begingroup$
Consider a tensor $mathcal{X}inmathbb{R}^{Itimes Jtimes K} $. It can be approximately decomposed/factored in multiple ways. Namely by using the TUCKER3 decomposition:
$mathcal{X}approx sum_{p=1}^Psum_{q=1}^Qsum_{r=1}^Rmathcal{G}_{pqr}mathbf{a}_pcircmathbf{b}_qcircmathbf{c}_r $,
where $mathcal{G}in mathbb{R}^{Ptimes Qtimes R}$ is a core mixing tensor, and $mathbf{a}_p,mathbf{b}_q,mathbf{c}_r$ are vectors from $mathbf{A}inmathbb{R}^{Itimes P},mathbf{B}inmathbb{R}^{Jtimes Q},mathbf{C}inmathbb{R}^{Ktimes R}$, and $circ$ denotes the outer product.
Now if the data tensor, $mathcal{X}$ can be assumed tri-linear in some $mathbf{A,B,C}$ components, a simpler decomposition is given by the CP-deomposition:
$mathcal{X}approx sum_{r=1}^R lambda_r cdot mathbf{a}_rcircmathbf{b}_rcircmathbf{c}_r $.
Now the question:
I know my data tensor $mathcal{X}$ is not trilinear in nature, but I wish to decompose it with the CP method anyway. After this decomp. I will use the $mathbf{C}$ matrix for a clustering problem. Is there anyway to derive some maths (e.g. bounds) for how much error/distortion I can expect if use elements of the $mathbf{C}$ from the CP decomposition, instead of the TUCKER3 which is theoretically "more suitable" for my tensor.
vector-spaces norm tensors tensor-rank
asked Jan 2 at 9:24
pche8701pche8701
135
$endgroup$
Consider a tensor $mathcal{X}inmathbb{R}^{Itimes Jtimes K} $. It can be approximately decomposed/factored in multiple ways. Namely by using the TUCKER3 decomposition:
$mathcal{X}approx sum_{p=1}^Psum_{q=1}^Qsum_{r=1}^Rmathcal{G}_{pqr}mathbf{a}_pcircmathbf{b}_qcircmathbf{c}_r $,
where $mathcal{G}in mathbb{R}^{Ptimes Qtimes R}$ is a core mixing tensor, and $mathbf{a}_p,mathbf{b}_q,mathbf{c}_r$ are vectors from $mathbf{A}inmathbb{R}^{Itimes P},mathbf{B}inmathbb{R}^{Jtimes Q},mathbf{C}inmathbb{R}^{Ktimes R}$, and $circ$ denotes the outer product.
Now if the data tensor, $mathcal{X}$ can be assumed tri-linear in some $mathbf{A,B,C}$ components, a simpler decomposition is given by the CP-deomposition:
$mathcal{X}approx sum_{r=1}^R lambda_r cdot mathbf{a}_rcircmathbf{b}_rcircmathbf{c}_r $.
Now the question:
I know my data tensor $mathcal{X}$ is not trilinear in nature, but I wish to decompose it with the CP method anyway. After this decomp. I will use the $mathbf{C}$ matrix for a clustering problem. Is there anyway to derive some maths (e.g. bounds) for how much error/distortion I can expect if use elements of the $mathbf{C}$ from the CP decomposition, instead of the TUCKER3 which is theoretically "more suitable" for my tensor.
vector-spaces norm tensors tensor-rank
vector-spaces norm tensors tensor-rank
asked Jan 2 at 9:24
pche8701pche8701
135
asked Jan 2 at 9:24
pche8701pche8701
135
asked Jan 2 at 9:24
pche8701pche8701
135
asked Jan 2 at 9:24
pche8701pche8701
135
asked Jan 2 at 9:24
pche8701pche8701
135
135
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