| ... | ... | @@ -234,7 +234,7 @@ Here, U and V are [unitary matrix]( https://en.wikipedia.org/wiki/Unitary_matrix |
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<img src="uploads/67baf97267820063eff52b30fb4d94ca/ica_2.png" width="600">
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We need to think about the last rotation: U. We need to rotate it back via $`U^*`$. How can we do that? We know that it is organized according to the variances in a hierarchical way. So I can find out on X in which direction the largest variances, i.e. the new coordinates. From the coordinates, we can find how much it was rotated ($`θ`$). The next question is; can I estimate how it may be stretched in the second step ($`Σ^{-1}`$). How was $`Σ^{-1}`$ operates in the first place? It was done according to singular values, variances. Since we can calculate the variances over the data, we can stretch it back via variances (i.e. moment). In the third step, we should again rotate back. Herein, we use another moment, kurtosis. Since we do not know how much to rotate, we will rotate in “a way that minimizes the kurtosis”. In short, we need to find the $`V`$ minimizing the kurtosis.
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We need to think about the last rotation: U. We need to rotate it back via $`U^*`$. How can we do that? We know that it is organized according to the variances in a hierarchical way. So I can find out on X in which direction the largest variances, i.e. the new coordinates. From the coordinates, we can find how much it was rotated ($`θ`$). The next question is; can I estimate how it may be stretched in the second step ($`Σ`$). How was $`Σ`$ operates in the first place? It was done according to singular values, variances. Since we can calculate the variances over the data, we can stretch it back ($`Σ{-1}`$) via these variances (i.e. moment). In the third step, we should rotate back ($`V`$) . Herein, we use another moment, kurtosis. Since we do not know how much to rotate, we approximate. We will rotate in “a way that minimizes the kurtosis”. In short, we need to find the $`V`$ minimizing the kurtosis.
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<img src="uploads/1b71bb1a15e8c1a2e1849e8ea14a9b55/ica_3.png" width="600">
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