The regression depth of a line is the minimum number of points it has to cross to take it from its initial position to vertical. The undirected depth of a point is the minimum number of lines a ray originating at the point will cross before escaping. If we use projective duality, then regression depth is same as undirected depth. I cannot see how that would be.
My attempt is as follows: By duality, points will map to lines, and lines will map to points. Suppose I am trying to compute the undirected depth of a point. So there is a point such that all rays emanating from it will meet n/3 points or more in each direction. In the dual plane, the point will coincide with a line. But what will rays correspond to?
EDIT: Quoting from paper by Nina Amenta, Marshall Bern et al, "Regression Depth and Center Points": Geometrically, the regression depth of a hyperplane is the minimum number of points intersected by the hyperplane as it undergoes any continuous motion taking it from its initial position to vertical. In the dual setting of hyperplane arrangements, the undirected depth of a point in an arrangement is the minimum number of hyperplanes touched by or parallel to a ray originating at the point. Standard techniques of projective duality transform any statement about regression depth to a mathematically equivalent statement about undirected depth and vice versa.
I was trying to come up with this duality but havent so far succeeded. In R_2, when we consider lines and points, I was considering a point that has high undirected depth, but am unable to make a transformation that would show exactly how the undirected depth of the point will map to the regression depth of a line in the dual plane.