Here is a simple way to accomplish non-linear dimensionality reduction: Input: High-dimensional dataset X = x1, . . . , xn R D, target dimension d Output: y1, . . . , yn R d as the low-dimensional mapping of the given dataset - Construct a k-nearest neighbor graph3 G on X - Let ij denote the shortest path between datapoints xi and xj according to G. - Select y1, . . . , yn R d according to the following minimization problem minimizey1,...,yn X i,j (yi yj ij )2 (ii) Is the optimization above convex with respect a fixed yi? Why or why not