Get global maximum slope












1














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







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  • Does this answer your question?
    – mm-crj
    Nov 21 '18 at 18:59
















1














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







share|cite|improve this question






















  • Does this answer your question?
    – mm-crj
    Nov 21 '18 at 18:59














1












1








1







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







share|cite|improve this question













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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 20 '18 at 17:55









Tobi

83




83












  • 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




Does this answer your question?
– mm-crj
Nov 21 '18 at 18:59










1 Answer
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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.






share|cite|improve this answer





















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    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.






    share|cite|improve this answer


























      0














      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.






      share|cite|improve this answer
























        0












        0








        0






        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.






        share|cite|improve this answer












        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.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 20 '18 at 22:50









        mm-crj

        382212




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