why does lasso select at most n predictors?
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
asked 2 days ago
ErroriSalvoErroriSalvo
1336
add a comment |
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
asked 2 days ago
ErroriSalvoErroriSalvo
1336
add a comment |
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
asked 2 days ago
ErroriSalvoErroriSalvo
1336
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
asked 2 days ago
ErroriSalvoErroriSalvo
1336
asked 2 days ago
ErroriSalvoErroriSalvo
1336
asked 2 days ago
ErroriSalvoErroriSalvo
1336
asked 2 days ago
ErroriSalvoErroriSalvo
1336
asked 2 days ago
ErroriSalvoErroriSalvo
1336
1336
add a comment |
add a comment |
1 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.
answered 2 days ago
nopenope
3615
add a comment |
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1 Answer
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1 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.
answered 2 days ago
nopenope
3615
add a comment |
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.
answered 2 days ago
nopenope
3615
add a comment |
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.
answered 2 days ago
nopenope
3615
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.
answered 2 days ago
nopenope
3615
answered 2 days ago
nopenope
3615
answered 2 days ago
nopenope
3615
answered 2 days ago
nopenope
3615
3615
add a comment |
add a comment |
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