What is the (sub-)gradient of $|W^TW-I|_*$ w.r.t. the matrix variable $W$?
$begingroup$
For $|W^TW-I|_*$ , $Win R^{mtimes n}$ with $nleq m$ and $Iin R^{ntimes n}$ is an identity matrix, $||_*$ is nuclear norm, also named as trace norm.
Q1: $|W^TW-I|_*$ is usually used in machine learning algorithms as a regularization term. Minimizing such term can be done by calculating its (sub-)gradient with regard to the matrix variable $W$. Then how to do? I only know the (sub-)gradient of $|W|_*$ as answered in Derivative of the nuclear norm with respect to its argument.
Q2: Maybe Q2 will be much complicated. The (sub-)gradient of $|W|_*$ involves Singular Value Decomposition, and Is there a nuclear norm approximation for stochastic gradient descent optimization? provides some way to make it efficient. I am afraid the solution of Q1 will also be computationally expensive. Then any method to make it efficient?
calculus linear-algebra matrices machine-learning
asked Dec 26 '18 at 16:37
oliviaolivia
791617
$endgroup$
add a comment |
$begingroup$
For $|W^TW-I|_*$ , $Win R^{mtimes n}$ with $nleq m$ and $Iin R^{ntimes n}$ is an identity matrix, $||_*$ is nuclear norm, also named as trace norm.
Q1: $|W^TW-I|_*$ is usually used in machine learning algorithms as a regularization term. Minimizing such term can be done by calculating its (sub-)gradient with regard to the matrix variable $W$. Then how to do? I only know the (sub-)gradient of $|W|_*$ as answered in Derivative of the nuclear norm with respect to its argument.
Q2: Maybe Q2 will be much complicated. The (sub-)gradient of $|W|_*$ involves Singular Value Decomposition, and Is there a nuclear norm approximation for stochastic gradient descent optimization? provides some way to make it efficient. I am afraid the solution of Q1 will also be computationally expensive. Then any method to make it efficient?
calculus linear-algebra matrices machine-learning
asked Dec 26 '18 at 16:37
oliviaolivia
791617
$endgroup$
add a comment |
$begingroup$
For $|W^TW-I|_*$ , $Win R^{mtimes n}$ with $nleq m$ and $Iin R^{ntimes n}$ is an identity matrix, $||_*$ is nuclear norm, also named as trace norm.
Q1: $|W^TW-I|_*$ is usually used in machine learning algorithms as a regularization term. Minimizing such term can be done by calculating its (sub-)gradient with regard to the matrix variable $W$. Then how to do? I only know the (sub-)gradient of $|W|_*$ as answered in Derivative of the nuclear norm with respect to its argument.
Q2: Maybe Q2 will be much complicated. The (sub-)gradient of $|W|_*$ involves Singular Value Decomposition, and Is there a nuclear norm approximation for stochastic gradient descent optimization? provides some way to make it efficient. I am afraid the solution of Q1 will also be computationally expensive. Then any method to make it efficient?
calculus linear-algebra matrices machine-learning
asked Dec 26 '18 at 16:37
oliviaolivia
791617
$endgroup$
For $|W^TW-I|_*$ , $Win R^{mtimes n}$ with $nleq m$ and $Iin R^{ntimes n}$ is an identity matrix, $||_*$ is nuclear norm, also named as trace norm.
Q1: $|W^TW-I|_*$ is usually used in machine learning algorithms as a regularization term. Minimizing such term can be done by calculating its (sub-)gradient with regard to the matrix variable $W$. Then how to do? I only know the (sub-)gradient of $|W|_*$ as answered in Derivative of the nuclear norm with respect to its argument.
Q2: Maybe Q2 will be much complicated. The (sub-)gradient of $|W|_*$ involves Singular Value Decomposition, and Is there a nuclear norm approximation for stochastic gradient descent optimization? provides some way to make it efficient. I am afraid the solution of Q1 will also be computationally expensive. Then any method to make it efficient?
calculus linear-algebra matrices machine-learning
calculus linear-algebra matrices machine-learning
asked Dec 26 '18 at 16:37
oliviaolivia
791617
asked Dec 26 '18 at 16:37
oliviaolivia
791617
asked Dec 26 '18 at 16:37
oliviaolivia
791617
asked Dec 26 '18 at 16:37
oliviaolivia
791617
asked Dec 26 '18 at 16:37
oliviaolivia
791617
791617
add a comment |
add a comment |
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