Mixing
On a nonlinear regression problem
$begingroup$
Consider the function $fcolon mathbb{R}^2to mathbb{R}$, $f(x_1,x_2)=x_1^2 +x_2$. Assume that I don't know the form of $f$ and I only have a set of $N$ independent "input-output" data ${(x_1^{(i)},x_2^{(i)}), f(x_1^{(i)},x_2^{(i)})}_{i=1}^N$.
I would like to estimate a "right inverse" of $f$. Specifically, I want to estimate from data a function $gcolon mathbb{R}to mathbb{R}^2$ such that $f(g(x))=x$, for all $xinmathbb{R}$.
In order to solve this problem I'm formulating the following nonlinear regression problem
$$tag{1} label{eq:1}
min_{ginmathcal{G}} |g(Y)-X|,
$$
where $mathcal{G}$ is the set of continuous functions from $mathbb{R}$ to $mathbb{R}^2$, $X:=begin{bmatrix}x_1^{(1)} & cdots & x_1^{(N)}\ x_2^{(1)} & cdots & x_2^{(N)}end{bmatrix}$, $Y:=begin{bmatrix}f(x_1^{(1)},x_2^{(1)}) & cdots & f(x_1^{(N)},x_2^{(N)})end{bmatrix}$, and $g(Y)$ stands for $g$ applied elementwise to vector $Y$.
Let $hat{g}$ be the minimizer of eqref{eq:1}.
My question: Does $hat{g}$ converge to a "right inverse" of $f$ for $N$ large enough? In other words, do we have $hat{g}(f(x))=x$, $xinmathbb{R}$, for $N$ large enough?
If $f$ is a linear function then eqref{eq:1} boils down to a linear regression problem and the answer to my question is in the affirmative. However I do not understand if this holds also for the nonlinear case (as in my simple example above). Numerical simulations suggest that this is not true, but I would like to understand why. (Note that, from a numerical viewpoint, I approximate $G(Y)$ using a finite set of basis functions.)
I'm sorry if my question is not very rigorous, but I've thought a lot about this problem with no luck. So, I would really appreciate any comment or feedback. Thanks!
analysis regression estimation regression-analysis inverse-problems
asked Jan 22 at 21:29
LudwigLudwig
813715
$endgroup$
add a comment |
$begingroup$
Consider the function $fcolon mathbb{R}^2to mathbb{R}$, $f(x_1,x_2)=x_1^2 +x_2$. Assume that I don't know the form of $f$ and I only have a set of $N$ independent "input-output" data ${(x_1^{(i)},x_2^{(i)}), f(x_1^{(i)},x_2^{(i)})}_{i=1}^N$.
I would like to estimate a "right inverse" of $f$. Specifically, I want to estimate from data a function $gcolon mathbb{R}to mathbb{R}^2$ such that $f(g(x))=x$, for all $xinmathbb{R}$.
In order to solve this problem I'm formulating the following nonlinear regression problem
$$tag{1} label{eq:1}
min_{ginmathcal{G}} |g(Y)-X|,
$$
where $mathcal{G}$ is the set of continuous functions from $mathbb{R}$ to $mathbb{R}^2$, $X:=begin{bmatrix}x_1^{(1)} & cdots & x_1^{(N)}\ x_2^{(1)} & cdots & x_2^{(N)}end{bmatrix}$, $Y:=begin{bmatrix}f(x_1^{(1)},x_2^{(1)}) & cdots & f(x_1^{(N)},x_2^{(N)})end{bmatrix}$, and $g(Y)$ stands for $g$ applied elementwise to vector $Y$.
Let $hat{g}$ be the minimizer of eqref{eq:1}.
My question: Does $hat{g}$ converge to a "right inverse" of $f$ for $N$ large enough? In other words, do we have $hat{g}(f(x))=x$, $xinmathbb{R}$, for $N$ large enough?
If $f$ is a linear function then eqref{eq:1} boils down to a linear regression problem and the answer to my question is in the affirmative. However I do not understand if this holds also for the nonlinear case (as in my simple example above). Numerical simulations suggest that this is not true, but I would like to understand why. (Note that, from a numerical viewpoint, I approximate $G(Y)$ using a finite set of basis functions.)
I'm sorry if my question is not very rigorous, but I've thought a lot about this problem with no luck. So, I would really appreciate any comment or feedback. Thanks!
analysis regression estimation regression-analysis inverse-problems
asked Jan 22 at 21:29
LudwigLudwig
813715
$endgroup$
add a comment |
$begingroup$
Consider the function $fcolon mathbb{R}^2to mathbb{R}$, $f(x_1,x_2)=x_1^2 +x_2$. Assume that I don't know the form of $f$ and I only have a set of $N$ independent "input-output" data ${(x_1^{(i)},x_2^{(i)}), f(x_1^{(i)},x_2^{(i)})}_{i=1}^N$.
I would like to estimate a "right inverse" of $f$. Specifically, I want to estimate from data a function $gcolon mathbb{R}to mathbb{R}^2$ such that $f(g(x))=x$, for all $xinmathbb{R}$.
In order to solve this problem I'm formulating the following nonlinear regression problem
$$tag{1} label{eq:1}
min_{ginmathcal{G}} |g(Y)-X|,
$$
where $mathcal{G}$ is the set of continuous functions from $mathbb{R}$ to $mathbb{R}^2$, $X:=begin{bmatrix}x_1^{(1)} & cdots & x_1^{(N)}\ x_2^{(1)} & cdots & x_2^{(N)}end{bmatrix}$, $Y:=begin{bmatrix}f(x_1^{(1)},x_2^{(1)}) & cdots & f(x_1^{(N)},x_2^{(N)})end{bmatrix}$, and $g(Y)$ stands for $g$ applied elementwise to vector $Y$.
Let $hat{g}$ be the minimizer of eqref{eq:1}.
My question: Does $hat{g}$ converge to a "right inverse" of $f$ for $N$ large enough? In other words, do we have $hat{g}(f(x))=x$, $xinmathbb{R}$, for $N$ large enough?
If $f$ is a linear function then eqref{eq:1} boils down to a linear regression problem and the answer to my question is in the affirmative. However I do not understand if this holds also for the nonlinear case (as in my simple example above). Numerical simulations suggest that this is not true, but I would like to understand why. (Note that, from a numerical viewpoint, I approximate $G(Y)$ using a finite set of basis functions.)
I'm sorry if my question is not very rigorous, but I've thought a lot about this problem with no luck. So, I would really appreciate any comment or feedback. Thanks!
analysis regression estimation regression-analysis inverse-problems
asked Jan 22 at 21:29
LudwigLudwig
813715
$endgroup$
Consider the function $fcolon mathbb{R}^2to mathbb{R}$, $f(x_1,x_2)=x_1^2 +x_2$. Assume that I don't know the form of $f$ and I only have a set of $N$ independent "input-output" data ${(x_1^{(i)},x_2^{(i)}), f(x_1^{(i)},x_2^{(i)})}_{i=1}^N$.
I would like to estimate a "right inverse" of $f$. Specifically, I want to estimate from data a function $gcolon mathbb{R}to mathbb{R}^2$ such that $f(g(x))=x$, for all $xinmathbb{R}$.
In order to solve this problem I'm formulating the following nonlinear regression problem
$$tag{1} label{eq:1}
min_{ginmathcal{G}} |g(Y)-X|,
$$
where $mathcal{G}$ is the set of continuous functions from $mathbb{R}$ to $mathbb{R}^2$, $X:=begin{bmatrix}x_1^{(1)} & cdots & x_1^{(N)}\ x_2^{(1)} & cdots & x_2^{(N)}end{bmatrix}$, $Y:=begin{bmatrix}f(x_1^{(1)},x_2^{(1)}) & cdots & f(x_1^{(N)},x_2^{(N)})end{bmatrix}$, and $g(Y)$ stands for $g$ applied elementwise to vector $Y$.
Let $hat{g}$ be the minimizer of eqref{eq:1}.
My question: Does $hat{g}$ converge to a "right inverse" of $f$ for $N$ large enough? In other words, do we have $hat{g}(f(x))=x$, $xinmathbb{R}$, for $N$ large enough?
If $f$ is a linear function then eqref{eq:1} boils down to a linear regression problem and the answer to my question is in the affirmative. However I do not understand if this holds also for the nonlinear case (as in my simple example above). Numerical simulations suggest that this is not true, but I would like to understand why. (Note that, from a numerical viewpoint, I approximate $G(Y)$ using a finite set of basis functions.)
I'm sorry if my question is not very rigorous, but I've thought a lot about this problem with no luck. So, I would really appreciate any comment or feedback. Thanks!
analysis regression estimation regression-analysis inverse-problems
analysis regression estimation regression-analysis inverse-problems
asked Jan 22 at 21:29
LudwigLudwig
813715
asked Jan 22 at 21:29
LudwigLudwig
813715
asked Jan 22 at 21:29
LudwigLudwig
813715
asked Jan 22 at 21:29
LudwigLudwig
813715
asked Jan 22 at 21:29
LudwigLudwig
813715
813715
add a comment |
add a comment |
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