Mixing
Get global maximum slope
I have a function which computes values on images with the size of 32x32 pixels. Therefore, the function is applied on every single pixel $x_i$ but also depends on all 32x32 input pixels. The input is constrained to $-1 leq x_i leq 1$.
More formal: A function $f_i:mathbb [-1,1]^{1024} rightarrow mathbb R$ and an input vector $vec{x} in mathbb [-1,1]^{1024}$.
Now I am looking for the global maximum slope of $f_i$ (for any input) numerically. A local maximum slope is easy to find as one goes in the direction of the gradient. How to find the global maximum? I am not sure if brute force is an option because when the pixel values have a high precision (e.g. 1e-10) then the total number of input permutations is ridiculously high. The brute force method might work with a lower precision but this doesn't make sense in this scenario.
Can someone give me a hint? Is this even possible?
real-analysis numerical-methods
asked Nov 20 '18 at 17:55
Tobi
83
add a comment |
I have a function which computes values on images with the size of 32x32 pixels. Therefore, the function is applied on every single pixel $x_i$ but also depends on all 32x32 input pixels. The input is constrained to $-1 leq x_i leq 1$.
More formal: A function $f_i:mathbb [-1,1]^{1024} rightarrow mathbb R$ and an input vector $vec{x} in mathbb [-1,1]^{1024}$.
Now I am looking for the global maximum slope of $f_i$ (for any input) numerically. A local maximum slope is easy to find as one goes in the direction of the gradient. How to find the global maximum? I am not sure if brute force is an option because when the pixel values have a high precision (e.g. 1e-10) then the total number of input permutations is ridiculously high. The brute force method might work with a lower precision but this doesn't make sense in this scenario.
Can someone give me a hint? Is this even possible?
real-analysis numerical-methods
asked Nov 20 '18 at 17:55
Tobi
83
Does this answer your question?
– mm-crj
Nov 21 '18 at 18:59
add a comment |
I have a function which computes values on images with the size of 32x32 pixels. Therefore, the function is applied on every single pixel $x_i$ but also depends on all 32x32 input pixels. The input is constrained to $-1 leq x_i leq 1$.
More formal: A function $f_i:mathbb [-1,1]^{1024} rightarrow mathbb R$ and an input vector $vec{x} in mathbb [-1,1]^{1024}$.
Now I am looking for the global maximum slope of $f_i$ (for any input) numerically. A local maximum slope is easy to find as one goes in the direction of the gradient. How to find the global maximum? I am not sure if brute force is an option because when the pixel values have a high precision (e.g. 1e-10) then the total number of input permutations is ridiculously high. The brute force method might work with a lower precision but this doesn't make sense in this scenario.
Can someone give me a hint? Is this even possible?
real-analysis numerical-methods
asked Nov 20 '18 at 17:55
Tobi
83
I have a function which computes values on images with the size of 32x32 pixels. Therefore, the function is applied on every single pixel $x_i$ but also depends on all 32x32 input pixels. The input is constrained to $-1 leq x_i leq 1$.
More formal: A function $f_i:mathbb [-1,1]^{1024} rightarrow mathbb R$ and an input vector $vec{x} in mathbb [-1,1]^{1024}$.
Now I am looking for the global maximum slope of $f_i$ (for any input) numerically. A local maximum slope is easy to find as one goes in the direction of the gradient. How to find the global maximum? I am not sure if brute force is an option because when the pixel values have a high precision (e.g. 1e-10) then the total number of input permutations is ridiculously high. The brute force method might work with a lower precision but this doesn't make sense in this scenario.
Can someone give me a hint? Is this even possible?
real-analysis numerical-methods
real-analysis numerical-methods
asked Nov 20 '18 at 17:55
Tobi
83
asked Nov 20 '18 at 17:55
Tobi
83
asked Nov 20 '18 at 17:55
Tobi
83
asked Nov 20 '18 at 17:55
Tobi
83
asked Nov 20 '18 at 17:55
Tobi
83
83
Does this answer your question?
– mm-crj
Nov 21 '18 at 18:59
add a comment |
Does this answer your question?
– mm-crj
Nov 21 '18 at 18:59
Does this answer your question?
– mm-crj
Nov 21 '18 at 18:59
Does this answer your question?
– mm-crj
Nov 21 '18 at 18:59
add a comment |
1 Answer
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active
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This is a convex optimization problem. You can get a global minimum $implies$ that the local minimum from gradient descent if both your domain and the function that you are optimizing is convex.
In your case, you are maximizing $f'(x):[-1,1]rightarrow mathbb R$ (most probably as I don't know the domain of the derivative)in a $32times32$ discrete grid. As you can understand the domain is convex as it in an interval in $mathbb R$. Only need to check if the function is convex or not.
PS: And I think you can apply Newton's method or other faster algorithms depending upon the smoothness of the function.
answered Nov 20 '18 at 22:50
mm-crj
382212
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
This is a convex optimization problem. You can get a global minimum $implies$ that the local minimum from gradient descent if both your domain and the function that you are optimizing is convex.
In your case, you are maximizing $f'(x):[-1,1]rightarrow mathbb R$ (most probably as I don't know the domain of the derivative)in a $32times32$ discrete grid. As you can understand the domain is convex as it in an interval in $mathbb R$. Only need to check if the function is convex or not.
PS: And I think you can apply Newton's method or other faster algorithms depending upon the smoothness of the function.
answered Nov 20 '18 at 22:50
mm-crj
382212
add a comment |
This is a convex optimization problem. You can get a global minimum $implies$ that the local minimum from gradient descent if both your domain and the function that you are optimizing is convex.
In your case, you are maximizing $f'(x):[-1,1]rightarrow mathbb R$ (most probably as I don't know the domain of the derivative)in a $32times32$ discrete grid. As you can understand the domain is convex as it in an interval in $mathbb R$. Only need to check if the function is convex or not.
PS: And I think you can apply Newton's method or other faster algorithms depending upon the smoothness of the function.
answered Nov 20 '18 at 22:50
mm-crj
382212
add a comment |
This is a convex optimization problem. You can get a global minimum $implies$ that the local minimum from gradient descent if both your domain and the function that you are optimizing is convex.
In your case, you are maximizing $f'(x):[-1,1]rightarrow mathbb R$ (most probably as I don't know the domain of the derivative)in a $32times32$ discrete grid. As you can understand the domain is convex as it in an interval in $mathbb R$. Only need to check if the function is convex or not.
PS: And I think you can apply Newton's method or other faster algorithms depending upon the smoothness of the function.
answered Nov 20 '18 at 22:50
mm-crj
382212
This is a convex optimization problem. You can get a global minimum $implies$ that the local minimum from gradient descent if both your domain and the function that you are optimizing is convex.
In your case, you are maximizing $f'(x):[-1,1]rightarrow mathbb R$ (most probably as I don't know the domain of the derivative)in a $32times32$ discrete grid. As you can understand the domain is convex as it in an interval in $mathbb R$. Only need to check if the function is convex or not.
PS: And I think you can apply Newton's method or other faster algorithms depending upon the smoothness of the function.
answered Nov 20 '18 at 22:50
mm-crj
382212
answered Nov 20 '18 at 22:50
mm-crj
382212
answered Nov 20 '18 at 22:50
mm-crj
382212
answered Nov 20 '18 at 22:50
mm-crj
382212
382212
add a comment |
add a comment |
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Does this answer your question?
– mm-crj
Nov 21 '18 at 18:59