Least-squares regularization matrix is not positive definite?
$begingroup$
I am fitting data $(x_i,y_i)$ to the following model:
$$
f(x) = sum_j a_j g_j(x) = a^T g(x)
$$
where $g_j(x)$ are well-conditioned basis functions (in my application they are B-splines, but I don't think that matters for this question).
I want to regularize the result by smoothing (damping) the function by minimizing the second derivative. This means my solution $a$ is given by
$$
a = textrm{argmin } ||y - X a||^2 + lambda^2 int_{-infty}^{infty} left| frac{d^2 f(x)}{dx^2} right|^2 dx
$$
where $X_{ij} = g_j(x_i)$ is the least-squares matrix, and the regularization term can be expressed as:
begin{align*}
int_{-infty}^{infty} left| frac{d^2 f(x)}{dx^2} right|^2 dx &= int_{-infty}^{infty} left( a^T g'' right)^2 dx \
&= int_{-infty}^{infty} a^T g'' g''^T a dx \
&= a^T left( int_{-infty}^{infty} g'' g''^T dx right) a \
&= a^T G a
end{align*}
where
$$
G_{ij} = int_{-infty}^{infty} frac{d^2 g_i(x)}{dx^2} frac{d^2 g_j(x)}{dx^2} dx
$$
So, now my solution is given by the standard form:
$$
a = textrm{argmin } ||y - X a||^2 + lambda^2 a^T G a
$$
I was trying to compute the Cholesky factor of $G = L L^T$ so that I could express this as
$$
a = textrm{argmin } ||y - X a||^2 + lambda^2 ||L a||^2
$$
so that I can use a generalized SVD or QR approach to solve the system. However, to my surprise, I found that $G$ was not positive definite - it had one zero eigenvalue.
My question is: shouldn't the matrix $G$ be positive definite as defined above, since the basis function $g_j(x)$ are positive and well-conditioned? Again, in my application the $g_j(x)$ are B-splines.
I am trying to determine if I made a mistake when computing the $G$ matrix.
Alternatively, does anyone see another way to compute the regularization matrix $L$, aside from a Cholesky factorization of $G$?
matrices least-squares regularization
asked Dec 11 '18 at 23:54
vibevibe
1648
$endgroup$
add a comment |
$begingroup$
I am fitting data $(x_i,y_i)$ to the following model:
$$
f(x) = sum_j a_j g_j(x) = a^T g(x)
$$
where $g_j(x)$ are well-conditioned basis functions (in my application they are B-splines, but I don't think that matters for this question).
I want to regularize the result by smoothing (damping) the function by minimizing the second derivative. This means my solution $a$ is given by
$$
a = textrm{argmin } ||y - X a||^2 + lambda^2 int_{-infty}^{infty} left| frac{d^2 f(x)}{dx^2} right|^2 dx
$$
where $X_{ij} = g_j(x_i)$ is the least-squares matrix, and the regularization term can be expressed as:
begin{align*}
int_{-infty}^{infty} left| frac{d^2 f(x)}{dx^2} right|^2 dx &= int_{-infty}^{infty} left( a^T g'' right)^2 dx \
&= int_{-infty}^{infty} a^T g'' g''^T a dx \
&= a^T left( int_{-infty}^{infty} g'' g''^T dx right) a \
&= a^T G a
end{align*}
where
$$
G_{ij} = int_{-infty}^{infty} frac{d^2 g_i(x)}{dx^2} frac{d^2 g_j(x)}{dx^2} dx
$$
So, now my solution is given by the standard form:
$$
a = textrm{argmin } ||y - X a||^2 + lambda^2 a^T G a
$$
I was trying to compute the Cholesky factor of $G = L L^T$ so that I could express this as
$$
a = textrm{argmin } ||y - X a||^2 + lambda^2 ||L a||^2
$$
so that I can use a generalized SVD or QR approach to solve the system. However, to my surprise, I found that $G$ was not positive definite - it had one zero eigenvalue.
My question is: shouldn't the matrix $G$ be positive definite as defined above, since the basis function $g_j(x)$ are positive and well-conditioned? Again, in my application the $g_j(x)$ are B-splines.
I am trying to determine if I made a mistake when computing the $G$ matrix.
Alternatively, does anyone see another way to compute the regularization matrix $L$, aside from a Cholesky factorization of $G$?
matrices least-squares regularization
asked Dec 11 '18 at 23:54
vibevibe
1648
$endgroup$
add a comment |
$begingroup$
I am fitting data $(x_i,y_i)$ to the following model:
$$
f(x) = sum_j a_j g_j(x) = a^T g(x)
$$
where $g_j(x)$ are well-conditioned basis functions (in my application they are B-splines, but I don't think that matters for this question).
I want to regularize the result by smoothing (damping) the function by minimizing the second derivative. This means my solution $a$ is given by
$$
a = textrm{argmin } ||y - X a||^2 + lambda^2 int_{-infty}^{infty} left| frac{d^2 f(x)}{dx^2} right|^2 dx
$$
where $X_{ij} = g_j(x_i)$ is the least-squares matrix, and the regularization term can be expressed as:
begin{align*}
int_{-infty}^{infty} left| frac{d^2 f(x)}{dx^2} right|^2 dx &= int_{-infty}^{infty} left( a^T g'' right)^2 dx \
&= int_{-infty}^{infty} a^T g'' g''^T a dx \
&= a^T left( int_{-infty}^{infty} g'' g''^T dx right) a \
&= a^T G a
end{align*}
where
$$
G_{ij} = int_{-infty}^{infty} frac{d^2 g_i(x)}{dx^2} frac{d^2 g_j(x)}{dx^2} dx
$$
So, now my solution is given by the standard form:
$$
a = textrm{argmin } ||y - X a||^2 + lambda^2 a^T G a
$$
I was trying to compute the Cholesky factor of $G = L L^T$ so that I could express this as
$$
a = textrm{argmin } ||y - X a||^2 + lambda^2 ||L a||^2
$$
so that I can use a generalized SVD or QR approach to solve the system. However, to my surprise, I found that $G$ was not positive definite - it had one zero eigenvalue.
My question is: shouldn't the matrix $G$ be positive definite as defined above, since the basis function $g_j(x)$ are positive and well-conditioned? Again, in my application the $g_j(x)$ are B-splines.
I am trying to determine if I made a mistake when computing the $G$ matrix.
Alternatively, does anyone see another way to compute the regularization matrix $L$, aside from a Cholesky factorization of $G$?
matrices least-squares regularization
asked Dec 11 '18 at 23:54
vibevibe
1648
$endgroup$
I am fitting data $(x_i,y_i)$ to the following model:
$$
f(x) = sum_j a_j g_j(x) = a^T g(x)
$$
where $g_j(x)$ are well-conditioned basis functions (in my application they are B-splines, but I don't think that matters for this question).
I want to regularize the result by smoothing (damping) the function by minimizing the second derivative. This means my solution $a$ is given by
$$
a = textrm{argmin } ||y - X a||^2 + lambda^2 int_{-infty}^{infty} left| frac{d^2 f(x)}{dx^2} right|^2 dx
$$
where $X_{ij} = g_j(x_i)$ is the least-squares matrix, and the regularization term can be expressed as:
begin{align*}
int_{-infty}^{infty} left| frac{d^2 f(x)}{dx^2} right|^2 dx &= int_{-infty}^{infty} left( a^T g'' right)^2 dx \
&= int_{-infty}^{infty} a^T g'' g''^T a dx \
&= a^T left( int_{-infty}^{infty} g'' g''^T dx right) a \
&= a^T G a
end{align*}
where
$$
G_{ij} = int_{-infty}^{infty} frac{d^2 g_i(x)}{dx^2} frac{d^2 g_j(x)}{dx^2} dx
$$
So, now my solution is given by the standard form:
$$
a = textrm{argmin } ||y - X a||^2 + lambda^2 a^T G a
$$
I was trying to compute the Cholesky factor of $G = L L^T$ so that I could express this as
$$
a = textrm{argmin } ||y - X a||^2 + lambda^2 ||L a||^2
$$
so that I can use a generalized SVD or QR approach to solve the system. However, to my surprise, I found that $G$ was not positive definite - it had one zero eigenvalue.
My question is: shouldn't the matrix $G$ be positive definite as defined above, since the basis function $g_j(x)$ are positive and well-conditioned? Again, in my application the $g_j(x)$ are B-splines.
I am trying to determine if I made a mistake when computing the $G$ matrix.
Alternatively, does anyone see another way to compute the regularization matrix $L$, aside from a Cholesky factorization of $G$?
matrices least-squares regularization
matrices least-squares regularization
asked Dec 11 '18 at 23:54
vibevibe
1648
asked Dec 11 '18 at 23:54
vibevibe
1648
asked Dec 11 '18 at 23:54
vibevibe
1648
asked Dec 11 '18 at 23:54
vibevibe
1648
asked Dec 11 '18 at 23:54
vibevibe
1648
1648
add a comment |
add a comment |
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