why does lasso select at most n predictors?












1














From the seminal paper on elastic net regularization from Zou and Hastie 2005, I read



For this kind of

p>>n and grouped variables situation, the lasso

is not the ideal method, because it can only select at most

n variables out of p candidates (Efron et al., 2004).


However, in Efron et al., 2004 I can not fin the proof/demonstration?
Any hint?






lasso elastic-net penalized







share|cite|improve this question



























    1














    From the seminal paper on elastic net regularization from Zou and Hastie 2005, I read



    For this kind of
    
p>>n and grouped variables situation, the lasso
    
is not the ideal method, because it can only select at most
    
n variables out of p candidates (Efron et al., 2004).


    However, in Efron et al., 2004 I can not fin the proof/demonstration?
    Any hint?






    lasso elastic-net penalized







    share|cite|improve this question

























      1












      1








      1







      From the seminal paper on elastic net regularization from Zou and Hastie 2005, I read



      For this kind of
      
p>>n and grouped variables situation, the lasso
      
is not the ideal method, because it can only select at most
      
n variables out of p candidates (Efron et al., 2004).


      However, in Efron et al., 2004 I can not fin the proof/demonstration?
      Any hint?






      lasso elastic-net penalized







      share|cite|improve this question













      From the seminal paper on elastic net regularization from Zou and Hastie 2005, I read



      For this kind of
      
p>>n and grouped variables situation, the lasso
      
is not the ideal method, because it can only select at most
      
n variables out of p candidates (Efron et al., 2004).


      However, in Efron et al., 2004 I can not fin the proof/demonstration?
      Any hint?






      lasso elastic-net penalized




      lasso elastic-net penalized






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 2 days ago









      ErroriSalvoErroriSalvo

      1336




      1336






















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          Consider a linear model $Y = Xbeta + varepsilon$ with $p$ variables and $n$ observations, $p>n$. Assuming the variables are not linear dependent, i.e. the matrix $X$ has rank $n$, $Y$ can be perfectly predictet ($Y = hat{Y}$) using only $n$ variables. So LASSO will ideally choose the $n$ variables such that $lambda ||beta||_1$ is minimal. This solution should be unique (because of linear independence on the individual variable level) and threrefore, all perfect fits of the linear model that include more than $n$ variables will have a higher $lambda ||beta||_1$ and are therefore not optimal.






          share|cite|improve this answer





















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            Consider a linear model $Y = Xbeta + varepsilon$ with $p$ variables and $n$ observations, $p>n$. Assuming the variables are not linear dependent, i.e. the matrix $X$ has rank $n$, $Y$ can be perfectly predictet ($Y = hat{Y}$) using only $n$ variables. So LASSO will ideally choose the $n$ variables such that $lambda ||beta||_1$ is minimal. This solution should be unique (because of linear independence on the individual variable level) and threrefore, all perfect fits of the linear model that include more than $n$ variables will have a higher $lambda ||beta||_1$ and are therefore not optimal.






            share|cite|improve this answer


























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              Consider a linear model $Y = Xbeta + varepsilon$ with $p$ variables and $n$ observations, $p>n$. Assuming the variables are not linear dependent, i.e. the matrix $X$ has rank $n$, $Y$ can be perfectly predictet ($Y = hat{Y}$) using only $n$ variables. So LASSO will ideally choose the $n$ variables such that $lambda ||beta||_1$ is minimal. This solution should be unique (because of linear independence on the individual variable level) and threrefore, all perfect fits of the linear model that include more than $n$ variables will have a higher $lambda ||beta||_1$ and are therefore not optimal.






              share|cite|improve this answer
























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                3






                Consider a linear model $Y = Xbeta + varepsilon$ with $p$ variables and $n$ observations, $p>n$. Assuming the variables are not linear dependent, i.e. the matrix $X$ has rank $n$, $Y$ can be perfectly predictet ($Y = hat{Y}$) using only $n$ variables. So LASSO will ideally choose the $n$ variables such that $lambda ||beta||_1$ is minimal. This solution should be unique (because of linear independence on the individual variable level) and threrefore, all perfect fits of the linear model that include more than $n$ variables will have a higher $lambda ||beta||_1$ and are therefore not optimal.






                share|cite|improve this answer












                Consider a linear model $Y = Xbeta + varepsilon$ with $p$ variables and $n$ observations, $p>n$. Assuming the variables are not linear dependent, i.e. the matrix $X$ has rank $n$, $Y$ can be perfectly predictet ($Y = hat{Y}$) using only $n$ variables. So LASSO will ideally choose the $n$ variables such that $lambda ||beta||_1$ is minimal. This solution should be unique (because of linear independence on the individual variable level) and threrefore, all perfect fits of the linear model that include more than $n$ variables will have a higher $lambda ||beta||_1$ and are therefore not optimal.







                share|cite|improve this answer












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                share|cite|improve this answer










                answered 2 days ago









                nopenope

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