Why Is the Measure of a Nonconvex Hessian Matrix Convex?

[mertcan]

Hi, initially I would like to share this link: https://books.google.com.tr/books?id=gWeVPoBmBZ8C&pg=PA25&lpg=PA25&dq=matrix+measure+properties&source=bl&ots=N1unizFvG6&sig=kxijoOVlPAacZDEdyyCwam4RQnQ&hl=en&sa=X&ved=2ahUKEwjd7o-Ap53dAhWJGuwKHdRbAO04ChDoATABegQICBAB#v=onepage&q=matrix measure properties&f=false. Here, you are allowed to view the pages between 22-26 and they are about MEASURE MATRİX.

According to those pages, matrix measure u(A) is a convex function, and A is a matrix form. So no matter which matrix we put into "u()" function, it always ensures convexity. For instance as you can see on page 26, $$u_1(A)= max_j(a_{jj}+\sum_{i=!j} |a_{ij}|)$$ where a_ij are elements of matrix A. And let^s say that A is Hessian matrix which is derived from very complicated nonconvex nonlinear function.

MY QUESTION is: Although our Hessian matrix's elements are nonlinear and nonconvex, HOW is it POSSIBLE that the measure of Hessian matrix is convex? I can not believe measure of matrix is convex because for previous Hessian matrix, all elements of it which also take place in "u()" function are nonconvex nonlinear and nonconvex. Could you explain this situation to me?

 

[StoneTemplePython]

Check the definition of convexity. Then look at properties (M2) and (M5) on page 22 -- this meets the definition of convexity. In general homogeneity under (positive) rescaling and sub-additivity will do it. You need to go through these details slowly and verify this for yourself.

edit:
what worries me is this:

MY QUESTION is: Although our Hessian matrix's elements are nonlinear and nonconvex, HOW is it POSSIBLE that the measure of Hessian matrix is convex?

which suggests you don't know what convex means or aren't grasping that it is referring to the the 'measure' which is convex by design.

Your post title mentions norms and measure. If it were me, I'd look at the norms in detail and explore why they are convex. (Hint: M2 and M5 still apply.) Induced Norms (and Schatten Norms) are quite common and are what these 'matrix measures' are being built on...

 

[mertcan]

Check the definition of convexity. Then look at properties (M2) and (M5) on page 22 -- this meets the definition of convexity. In general homogeneity under (positive) rescaling and sub-additivity will do it. You need to go through these details slowly and verify this for yourself.

edit:
what worries me is this:
which suggests you don't know what convex means or aren't grasping that it is referring to the the 'measure' which is convex by design.

Your post title mentions norms and measure. If it were me, I'd look at the norms in detail and explore why they are convex. (Hint: M2 and M5 still apply.) Induced Norms (and Schatten Norms) are quite common and are what these 'matrix measures' are being built on...

Thanks for return @StoneTemplePython , I would like to ask another question as a reply to my post 1. Let's say we have 2 dimensions (x,y) for the elements of Hessian matrix in total, and concentrating on 5<=x<=10 and -3<=y<=2. Also we know that measure function is convex a function which means u(A) is convex. Besides, we can imagine (u,x,y) space and u is a function of x,y(because elements of Hessian consists of x,y). Here, can we say that in order to obtain MAXIMUM value of measure function for the intervals 5<=x<=10 and -3<=y<=2 we can just look at bound points? (for instance for a basic parabola function(z=x'^2+y^2), maximum value of parabola for intervals 5<=x<=10 and -3<=y<=2 would be at bounds...)

 

[StoneTemplePython]

Thanks for return @StoneTemplePython , I would like to ask another question as a reply to my post 1. Let's say we have 2 dimensions (x,y) for the elements of Hessian matrix in total, and concentrating on 5<=x<=10 and -3<=y<=2. Also we know that measure function is convex a function which means u(A) is convex. Besides, we can imagine (u,x,y) space and u is a function of x,y(because elements of Hessian consists of x,y). Here, can we say that in order to obtain MAXIMUM value of measure function for the intervals 5<=x<=10 and -3<=y<=2 we can just look at bound points? (for instance for a basic parabola function(z=x'^2+y^2), maximum value of parabola for intervals 5<=x<=10 and -3<=y<=2 would be at bounds...)

Again I'd suggest working with norms first... these 'bound points' you talk about are (x,y) coordinates that get mapped via a function (and really its second derivative) to the Hessian and then you apply a convex norm to it. I still think you need to write out what exactly is convex here and why. I.e. a careful mathematical writeup and proof of the convexity.

The domain for your convex function is given by the coordinates of the Hessian, not x and y. Alternatively you can have a composition of functions applied to these x and y coordinates, but you don't have per se reason to believe that the composition is convex.