Representing a function of two variables as a superposition over angles of ridge functions with a given shape via linear programming

Abstract

A ridge function with shape function g in the horizontal direction is a function of the form g ( x ) h ( y , 0 ) . Along each horizontal line it has the shape g ( x ) , multiplied by a function h ( y , 0 ) which depends on the y-value of the horizontal line. Similarly a ridge function with shape function g in the vertical direction has the form g ( y ) h ( x , π / 2 ) . For a given shape function g it may or may not be possible to represent an arbitrary function f ( x , y ) as a superposition over all angles of a ridge function with shape g in each direction, where h = h f = h f , g depends on the functions f and g and also on the direction, θ : h = h f , g ( · , θ ) . We show that if g is Gaussian centered at zero then this is always possible and we give the function h f , g for a given f ( x , y ) . For highpass or for odd shapes g, we show it is impossible to represent an arbitrary f ( x , y ) , i.e. in general there is no h f , g . Note that our problem is similar to tomography, where the problem is to invert the Radon transform, except that the use of the word inversion is here somewhat “inverted”: in tomography f ( x , y ) is unknown and we find it by inverting the projections of f; here, f ( x , y ) is known, g ( z ) is known, and h f ( · , θ ) = h f , g ( · , θ ) is the unknown.