Linear regression with dependent variables: express prediction with dot products
When dealing with Linear regressin with dependent variables, one can consider the optimization problem:
$$arg min_w 0.5lVert{w}rVert^2 \ s.t. Xw=y$$
Where $Xinmathbb{R}^{n,d}$ is the data matrix and $yinmathbb{R}^{n}$ is the labels vector.
Using Lagrangian multipliers I was able to show that $w^*$ can be written as $w^*=X^Talpha$ for $alpha in mathbb{R}^n$.
Now given a new $xinmathbb{R}^d$, consider $x^Tw^*$. How can this be expressed using dot products between $xinmathbb{R}^d$? The goal is to show that the kernel trick is working for that setup of linear regression.
Thanks.
linear-regression
asked Jan 4 at 16:14
galah92galah92
25418
add a comment |
When dealing with Linear regressin with dependent variables, one can consider the optimization problem:
$$arg min_w 0.5lVert{w}rVert^2 \ s.t. Xw=y$$
Where $Xinmathbb{R}^{n,d}$ is the data matrix and $yinmathbb{R}^{n}$ is the labels vector.
Using Lagrangian multipliers I was able to show that $w^*$ can be written as $w^*=X^Talpha$ for $alpha in mathbb{R}^n$.
Now given a new $xinmathbb{R}^d$, consider $x^Tw^*$. How can this be expressed using dot products between $xinmathbb{R}^d$? The goal is to show that the kernel trick is working for that setup of linear regression.
Thanks.
linear-regression
asked Jan 4 at 16:14
galah92galah92
25418
add a comment |
When dealing with Linear regressin with dependent variables, one can consider the optimization problem:
$$arg min_w 0.5lVert{w}rVert^2 \ s.t. Xw=y$$
Where $Xinmathbb{R}^{n,d}$ is the data matrix and $yinmathbb{R}^{n}$ is the labels vector.
Using Lagrangian multipliers I was able to show that $w^*$ can be written as $w^*=X^Talpha$ for $alpha in mathbb{R}^n$.
Now given a new $xinmathbb{R}^d$, consider $x^Tw^*$. How can this be expressed using dot products between $xinmathbb{R}^d$? The goal is to show that the kernel trick is working for that setup of linear regression.
Thanks.
linear-regression
asked Jan 4 at 16:14
galah92galah92
25418
When dealing with Linear regressin with dependent variables, one can consider the optimization problem:
$$arg min_w 0.5lVert{w}rVert^2 \ s.t. Xw=y$$
Where $Xinmathbb{R}^{n,d}$ is the data matrix and $yinmathbb{R}^{n}$ is the labels vector.
Using Lagrangian multipliers I was able to show that $w^*$ can be written as $w^*=X^Talpha$ for $alpha in mathbb{R}^n$.
Now given a new $xinmathbb{R}^d$, consider $x^Tw^*$. How can this be expressed using dot products between $xinmathbb{R}^d$? The goal is to show that the kernel trick is working for that setup of linear regression.
Thanks.
linear-regression
linear-regression
asked Jan 4 at 16:14
galah92galah92
25418
asked Jan 4 at 16:14
galah92galah92
25418
asked Jan 4 at 16:14
galah92galah92
25418
asked Jan 4 at 16:14
galah92galah92
25418
asked Jan 4 at 16:14
galah92galah92
25418
25418
add a comment |
add a comment |
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