Minimazation problem with norm and matrix
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In the context of principal component analysis, I got to the minimazation problem:
$$min_{A, (v_{i})} sum_{i=0}^n leftlVert X_{i}-Av_{i}rightrVert^2$$
for $X_1,...,X_nin mathbb{R}^p$
For the mean $bar{X}=0$ it has the solutions:
$$hat{v_{i}}=hat{A^T}X_{i}$$ $$hat{A}=(w_1,...,w_q)$$
where $W=(w_1,...,w_p)in mathbb{R^{ptimes p}}$, and $W*X*W^T$ the eigenvaluedecomposition.
Its a long time I didnt do calculus and optimazation. I tried to compute the derivative and got (with f being the function we want to minimize):
$$frac{partial{f}}{partial{A}}= sum_{i=0}^n 2*leftlVert X_{i}-Av_{i}rightrVert * v_k^T$$
and
$$frac{partial{f}}{partial{v_k}} = 2*leftlVert X_{k}-Av_{k}rightrVert * A$$
Setting this zero, I really have no clue how to get to the seolution.
optimization norm matrix-calculus
asked Dec 10 '18 at 11:13
LosyresLosyres
354
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add a comment |
$begingroup$
In the context of principal component analysis, I got to the minimazation problem:
$$min_{A, (v_{i})} sum_{i=0}^n leftlVert X_{i}-Av_{i}rightrVert^2$$
for $X_1,...,X_nin mathbb{R}^p$
For the mean $bar{X}=0$ it has the solutions:
$$hat{v_{i}}=hat{A^T}X_{i}$$ $$hat{A}=(w_1,...,w_q)$$
where $W=(w_1,...,w_p)in mathbb{R^{ptimes p}}$, and $W*X*W^T$ the eigenvaluedecomposition.
Its a long time I didnt do calculus and optimazation. I tried to compute the derivative and got (with f being the function we want to minimize):
$$frac{partial{f}}{partial{A}}= sum_{i=0}^n 2*leftlVert X_{i}-Av_{i}rightrVert * v_k^T$$
and
$$frac{partial{f}}{partial{v_k}} = 2*leftlVert X_{k}-Av_{k}rightrVert * A$$
Setting this zero, I really have no clue how to get to the seolution.
optimization norm matrix-calculus
asked Dec 10 '18 at 11:13
LosyresLosyres
354
$endgroup$
$begingroup$
see here: stats.stackexchange.com/a/10260/55946
$endgroup$
– dimebucker
Dec 10 '18 at 11:22
$begingroup$
@dimebucker Thanks, that helps!
$endgroup$
– Losyres
Dec 11 '18 at 14:54
add a comment |
$begingroup$
In the context of principal component analysis, I got to the minimazation problem:
$$min_{A, (v_{i})} sum_{i=0}^n leftlVert X_{i}-Av_{i}rightrVert^2$$
for $X_1,...,X_nin mathbb{R}^p$
For the mean $bar{X}=0$ it has the solutions:
$$hat{v_{i}}=hat{A^T}X_{i}$$ $$hat{A}=(w_1,...,w_q)$$
where $W=(w_1,...,w_p)in mathbb{R^{ptimes p}}$, and $W*X*W^T$ the eigenvaluedecomposition.
Its a long time I didnt do calculus and optimazation. I tried to compute the derivative and got (with f being the function we want to minimize):
$$frac{partial{f}}{partial{A}}= sum_{i=0}^n 2*leftlVert X_{i}-Av_{i}rightrVert * v_k^T$$
and
$$frac{partial{f}}{partial{v_k}} = 2*leftlVert X_{k}-Av_{k}rightrVert * A$$
Setting this zero, I really have no clue how to get to the seolution.
optimization norm matrix-calculus
asked Dec 10 '18 at 11:13
LosyresLosyres
354
$endgroup$
In the context of principal component analysis, I got to the minimazation problem:
$$min_{A, (v_{i})} sum_{i=0}^n leftlVert X_{i}-Av_{i}rightrVert^2$$
for $X_1,...,X_nin mathbb{R}^p$
For the mean $bar{X}=0$ it has the solutions:
$$hat{v_{i}}=hat{A^T}X_{i}$$ $$hat{A}=(w_1,...,w_q)$$
where $W=(w_1,...,w_p)in mathbb{R^{ptimes p}}$, and $W*X*W^T$ the eigenvaluedecomposition.
Its a long time I didnt do calculus and optimazation. I tried to compute the derivative and got (with f being the function we want to minimize):
$$frac{partial{f}}{partial{A}}= sum_{i=0}^n 2*leftlVert X_{i}-Av_{i}rightrVert * v_k^T$$
and
$$frac{partial{f}}{partial{v_k}} = 2*leftlVert X_{k}-Av_{k}rightrVert * A$$
Setting this zero, I really have no clue how to get to the seolution.
optimization norm matrix-calculus
optimization norm matrix-calculus
asked Dec 10 '18 at 11:13
LosyresLosyres
354
asked Dec 10 '18 at 11:13
LosyresLosyres
354
asked Dec 10 '18 at 11:13
LosyresLosyres
354
asked Dec 10 '18 at 11:13
LosyresLosyres
354
asked Dec 10 '18 at 11:13
LosyresLosyres
354
354
$begingroup$
see here: stats.stackexchange.com/a/10260/55946
$endgroup$
– dimebucker
Dec 10 '18 at 11:22
$begingroup$
@dimebucker Thanks, that helps!
$endgroup$
– Losyres
Dec 11 '18 at 14:54
add a comment |
$begingroup$
see here: stats.stackexchange.com/a/10260/55946
$endgroup$
– dimebucker
Dec 10 '18 at 11:22
$begingroup$
@dimebucker Thanks, that helps!
$endgroup$
– Losyres
Dec 11 '18 at 14:54
$begingroup$
see here: stats.stackexchange.com/a/10260/55946
$endgroup$
– dimebucker
Dec 10 '18 at 11:22
$begingroup$
see here: stats.stackexchange.com/a/10260/55946
$endgroup$
– dimebucker
Dec 10 '18 at 11:22
$begingroup$
@dimebucker Thanks, that helps!
$endgroup$
– Losyres
Dec 11 '18 at 14:54
$begingroup$
@dimebucker Thanks, that helps!
$endgroup$
– Losyres
Dec 11 '18 at 14:54
add a comment |
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$begingroup$
see here: stats.stackexchange.com/a/10260/55946
$endgroup$
– dimebucker
Dec 10 '18 at 11:22
$begingroup$
@dimebucker Thanks, that helps!
$endgroup$
– Losyres
Dec 11 '18 at 14:54