dividing by variance to avoid trivial solutions in multidimensional Newton's method
$begingroup$
I'm using Newton's method as implemented by GSL (GNU Scientific Library) to try to find a cycle of a multi-variate function $F$ (reaction-diffusion simulation over two-channel 2D images collapsed to 1D vectors):
$$begin{aligned}F^{Q+P}(mathbf{v}) &= F^Q(mathbf{v})&mathbf{v} &in [0,1]^{S times S times 2}end{aligned}$$
Values are of the order $Q=1, P=1000,S=50$. The purpose of $Q$ is to mitigate the problem of $v_{ijk}$ being pushed outside the range $[0,1]$ by Newton iterations ($F$ will bring them into the range). To prevent convergence to a trivial solution (a featureless image), I am solving the system:
$$H(mathbf{v}) = frac{F^{Q+P}(mathbf{v}) - F^Q(mathbf{v})}{sigma(F^{Q+P}(mathbf{v}))} = mathbf{0}$$
where $sigma(mathbf{v})$ is the sum of the variances of each channel of the image:
$$sigma(mathbf{v}) = sum_{k=1}^2left(frac{S^2 left(sum_{i=1}^S sum_{j=1}^Sv_{ijk}^2right) - left(sum_{i=1}^Ssum_{j=1}^S v_{ijk}right)^2}{S^4}right)$$
For efficiency, I would like to use an FDF solver from GSL, which means I need to calculate the Jacobian derivative matrix for $H$. I get to:
$$J_H = frac{sigma(F^{Q+P}(mathbf{v})) times left(J_{F^{Q+P}}(mathbf{v}) - J_{F^Q}(mathbf{v})right) - left(F^{Q+P}(mathbf{v}) - F^Q(mathbf{v})right)times J_sigma(F^{Q+P}(mathbf{v}))} {sigma(F^{Q+P}(mathbf{v})^2)}$$
$J_F$ is not too hard to calculate. By a shape argument, I think $J_sigma(mathbf{v})$ must be a row vector that is combined with the $F$ column vectors using outer product, and concretely I get:
$$frac{partial g(mathbf{v})}{partial v_{ijk}} = frac{2 v_{ijk}}{S^2} - frac{2 sum_{l=1}^Ssum_{m=1}^S v_{lmk}}{S^4}$$
Questions:
- Is there a better way to solve my problem than Newton's method?
- Is my general approach of dividing by variance a good way to avoid trivial solutions?
- Is my expression for $J_H$ correct?
- Is my expression for $frac{partial g(mathbf{v})}{partial v_{ijk}}$ correct?
proof-verification numerical-methods variance newton-raphson
asked Dec 22 '18 at 20:50
ClaudeClaude
2,553523
$endgroup$
add a comment |
$begingroup$
I'm using Newton's method as implemented by GSL (GNU Scientific Library) to try to find a cycle of a multi-variate function $F$ (reaction-diffusion simulation over two-channel 2D images collapsed to 1D vectors):
$$begin{aligned}F^{Q+P}(mathbf{v}) &= F^Q(mathbf{v})&mathbf{v} &in [0,1]^{S times S times 2}end{aligned}$$
Values are of the order $Q=1, P=1000,S=50$. The purpose of $Q$ is to mitigate the problem of $v_{ijk}$ being pushed outside the range $[0,1]$ by Newton iterations ($F$ will bring them into the range). To prevent convergence to a trivial solution (a featureless image), I am solving the system:
$$H(mathbf{v}) = frac{F^{Q+P}(mathbf{v}) - F^Q(mathbf{v})}{sigma(F^{Q+P}(mathbf{v}))} = mathbf{0}$$
where $sigma(mathbf{v})$ is the sum of the variances of each channel of the image:
$$sigma(mathbf{v}) = sum_{k=1}^2left(frac{S^2 left(sum_{i=1}^S sum_{j=1}^Sv_{ijk}^2right) - left(sum_{i=1}^Ssum_{j=1}^S v_{ijk}right)^2}{S^4}right)$$
For efficiency, I would like to use an FDF solver from GSL, which means I need to calculate the Jacobian derivative matrix for $H$. I get to:
$$J_H = frac{sigma(F^{Q+P}(mathbf{v})) times left(J_{F^{Q+P}}(mathbf{v}) - J_{F^Q}(mathbf{v})right) - left(F^{Q+P}(mathbf{v}) - F^Q(mathbf{v})right)times J_sigma(F^{Q+P}(mathbf{v}))} {sigma(F^{Q+P}(mathbf{v})^2)}$$
$J_F$ is not too hard to calculate. By a shape argument, I think $J_sigma(mathbf{v})$ must be a row vector that is combined with the $F$ column vectors using outer product, and concretely I get:
$$frac{partial g(mathbf{v})}{partial v_{ijk}} = frac{2 v_{ijk}}{S^2} - frac{2 sum_{l=1}^Ssum_{m=1}^S v_{lmk}}{S^4}$$
Questions:
- Is there a better way to solve my problem than Newton's method?
- Is my general approach of dividing by variance a good way to avoid trivial solutions?
- Is my expression for $J_H$ correct?
- Is my expression for $frac{partial g(mathbf{v})}{partial v_{ijk}}$ correct?
proof-verification numerical-methods variance newton-raphson
asked Dec 22 '18 at 20:50
ClaudeClaude
2,553523
$endgroup$
add a comment |
$begingroup$
I'm using Newton's method as implemented by GSL (GNU Scientific Library) to try to find a cycle of a multi-variate function $F$ (reaction-diffusion simulation over two-channel 2D images collapsed to 1D vectors):
$$begin{aligned}F^{Q+P}(mathbf{v}) &= F^Q(mathbf{v})&mathbf{v} &in [0,1]^{S times S times 2}end{aligned}$$
Values are of the order $Q=1, P=1000,S=50$. The purpose of $Q$ is to mitigate the problem of $v_{ijk}$ being pushed outside the range $[0,1]$ by Newton iterations ($F$ will bring them into the range). To prevent convergence to a trivial solution (a featureless image), I am solving the system:
$$H(mathbf{v}) = frac{F^{Q+P}(mathbf{v}) - F^Q(mathbf{v})}{sigma(F^{Q+P}(mathbf{v}))} = mathbf{0}$$
where $sigma(mathbf{v})$ is the sum of the variances of each channel of the image:
$$sigma(mathbf{v}) = sum_{k=1}^2left(frac{S^2 left(sum_{i=1}^S sum_{j=1}^Sv_{ijk}^2right) - left(sum_{i=1}^Ssum_{j=1}^S v_{ijk}right)^2}{S^4}right)$$
For efficiency, I would like to use an FDF solver from GSL, which means I need to calculate the Jacobian derivative matrix for $H$. I get to:
$$J_H = frac{sigma(F^{Q+P}(mathbf{v})) times left(J_{F^{Q+P}}(mathbf{v}) - J_{F^Q}(mathbf{v})right) - left(F^{Q+P}(mathbf{v}) - F^Q(mathbf{v})right)times J_sigma(F^{Q+P}(mathbf{v}))} {sigma(F^{Q+P}(mathbf{v})^2)}$$
$J_F$ is not too hard to calculate. By a shape argument, I think $J_sigma(mathbf{v})$ must be a row vector that is combined with the $F$ column vectors using outer product, and concretely I get:
$$frac{partial g(mathbf{v})}{partial v_{ijk}} = frac{2 v_{ijk}}{S^2} - frac{2 sum_{l=1}^Ssum_{m=1}^S v_{lmk}}{S^4}$$
Questions:
- Is there a better way to solve my problem than Newton's method?
- Is my general approach of dividing by variance a good way to avoid trivial solutions?
- Is my expression for $J_H$ correct?
- Is my expression for $frac{partial g(mathbf{v})}{partial v_{ijk}}$ correct?
proof-verification numerical-methods variance newton-raphson
asked Dec 22 '18 at 20:50
ClaudeClaude
2,553523
$endgroup$
I'm using Newton's method as implemented by GSL (GNU Scientific Library) to try to find a cycle of a multi-variate function $F$ (reaction-diffusion simulation over two-channel 2D images collapsed to 1D vectors):
$$begin{aligned}F^{Q+P}(mathbf{v}) &= F^Q(mathbf{v})&mathbf{v} &in [0,1]^{S times S times 2}end{aligned}$$
Values are of the order $Q=1, P=1000,S=50$. The purpose of $Q$ is to mitigate the problem of $v_{ijk}$ being pushed outside the range $[0,1]$ by Newton iterations ($F$ will bring them into the range). To prevent convergence to a trivial solution (a featureless image), I am solving the system:
$$H(mathbf{v}) = frac{F^{Q+P}(mathbf{v}) - F^Q(mathbf{v})}{sigma(F^{Q+P}(mathbf{v}))} = mathbf{0}$$
where $sigma(mathbf{v})$ is the sum of the variances of each channel of the image:
$$sigma(mathbf{v}) = sum_{k=1}^2left(frac{S^2 left(sum_{i=1}^S sum_{j=1}^Sv_{ijk}^2right) - left(sum_{i=1}^Ssum_{j=1}^S v_{ijk}right)^2}{S^4}right)$$
For efficiency, I would like to use an FDF solver from GSL, which means I need to calculate the Jacobian derivative matrix for $H$. I get to:
$$J_H = frac{sigma(F^{Q+P}(mathbf{v})) times left(J_{F^{Q+P}}(mathbf{v}) - J_{F^Q}(mathbf{v})right) - left(F^{Q+P}(mathbf{v}) - F^Q(mathbf{v})right)times J_sigma(F^{Q+P}(mathbf{v}))} {sigma(F^{Q+P}(mathbf{v})^2)}$$
$J_F$ is not too hard to calculate. By a shape argument, I think $J_sigma(mathbf{v})$ must be a row vector that is combined with the $F$ column vectors using outer product, and concretely I get:
$$frac{partial g(mathbf{v})}{partial v_{ijk}} = frac{2 v_{ijk}}{S^2} - frac{2 sum_{l=1}^Ssum_{m=1}^S v_{lmk}}{S^4}$$
Questions:
- Is there a better way to solve my problem than Newton's method?
- Is my general approach of dividing by variance a good way to avoid trivial solutions?
- Is my expression for $J_H$ correct?
- Is my expression for $frac{partial g(mathbf{v})}{partial v_{ijk}}$ correct?
proof-verification numerical-methods variance newton-raphson
proof-verification numerical-methods variance newton-raphson
asked Dec 22 '18 at 20:50
ClaudeClaude
2,553523
asked Dec 22 '18 at 20:50
ClaudeClaude
2,553523
asked Dec 22 '18 at 20:50
ClaudeClaude
2,553523
asked Dec 22 '18 at 20:50
ClaudeClaude
2,553523
asked Dec 22 '18 at 20:50
ClaudeClaude
2,553523
2,553523
add a comment |
add a comment |
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