Linear regression with dependent variables: express prediction with dot products












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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







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    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







    share|cite|improve this question

























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      0







      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







      share|cite|improve this question













      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






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      share|cite|improve this question











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      asked Jan 4 at 16:14









      galah92galah92

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