Question

# a) Show that the smallest distance of a point (x∗,y∗) on the plane from a linear manifold (line) with equation

Ax+By+C = 0 is equal to |Ax∗+By∗+C|/√A2 + B2. b) Suppose now that you are given a collection of points {(x1,y1),...,(xn,yn)}. Deﬁne a cost function by considering the sum of the squares of the distances of each point from the manifold. By taking the derivative with respect to C, express C as a function of A and B and compute the new form of the cost. Put the resulting cost under a form which is suitable to apply the known result that computes the minimum value of the ratio minX X|ΩX X|X for a symmetric matrix Ω. c) Repeat questions a) and b) for a circular manifold with the circle deﬁned by the coordinates of its center (A,B) and its radius C > 0. For question c) you need to compute ﬁrst the shortest distance of a point from the circle as a function of A,B,C. Then deﬁne a cost by considering the sum of squares of the distances of the points from the circle. Next you must express C in terms of A,B by taking the derivative with respect to C and equating it to 0. Finally you must derive the new cost function that now depends only on A,B. d) Write the steepest descent algorithm for A,B that minimize this ﬁnal cost function.