A Test Prolem January 17, 2020
We consider a dimensionality reduction problem. There are N high-dimensional data points {xi}Ni=1, where xi ∈ RD. We want to find a low-dimensional mapping foreachdatadatapointxi →yi,whereyi ∈Rd withd
Obviously this is an unconstrained optimization problem of a smooth function. We can solve it by a gradient-based method. Your tasks are:
1. Derive the gradient ∂J (Y ) ∂Y
2. Implement the following function (in any language or in pseudocode) which returns the objective function J (Y ) and the gradient ∂J (Y ) for a given input
∂Y
function [obj, grad] = compute obj grad(P, lambda, Y)
3. (Bonus)ImplementasolverwhichminimizesJ(Y)forthegivenP,λandan initial guess of Y (denoted as Y0):
function Y = minimize J(P, lambda, Y0)
4. (Bonus) Run the solver on the given DOLPHINS data set, using λ = 0.1 and
d = 2. The P matrix is given in DOLPHINS.txt. 1
P, λ and Y (for example in Matlab):