DS-GA 1018: Homework 3
Due Friday October 27th at 5:00 pm
Problem 1 (17 points): Consider the latent space model we presented in
class defined by:
zt = Azt−1 + wt (1) xt = Czt + vt (2)
where the latent space z is has dimension d and the data x has dimension n. Our noise is being drawn from wt ∼ N(0,Q) and vt ∼ N(0,R). Assume thatn=dandthatwehaveA=I,C=I,andQ=0·I. Finally,assume that Σ0 → ∞. We make no specific assumption for R or μ0. We want to show that the smoothed mean μt|t+1 simplifies to the mean of the data
T1 PTi=1 xi. This problem will walk you through the steps:
i. (5 points): Show that μ1|1 = x1 and Σ1|1 = R. You will need to take the limit Σ0 → ∞. Hint: it will be useful to know that (I+M)−1 = P∞i=0(−1)iMi, where M is an arbitrary matrix.
ii. (2 points): Explain the intuition behind the result for μ1|1 and Σ1|1. iii. (4 points): Assume that μ = 1 Pt−1 x and that Σ =
t−1|t−1 t−1 i=1 i t−1|t−1
1 R (this is consistent with your previous result). Show that μt|t =
t−1
1t Pti=1 xi and that Σt|t = 1t R.
iv. (4 points): Finally, show that μt|t+1 = T1 PTi=0 xi for all t. v. (2 points): Explain the intuition behind the final result.
Problem 2 (20 points): In lecture we discussed the importance of the ini- tial parameter values for the results of the expectation-maximization algo- rithm. In this problem we will quantitatively demonstrate that importance.
Consider the EM algorithm for an LDS system like the one in class. Imagine that we have an initial guess for our parameters:
θ0 = {μ0,0,Σ0,0,A0,Q0,C0,R0}. (3)
We have also set our distribution q1(Z) = p(Z|X,θ0). In other words, q1 is the distributions set by the Kalman smoothing means and covariances (the standard choice).
i. (3 points): Consider the case where A0 = 0 · I and C0 = 0 · I. what will the value of μ0,1 be in terms of the data in X and the parameters in θ0? For all the subproblems, don’t forget to consider how our choice of θ0 affects terms like μt|t−1.
ii. (3 points): Still considering the case where A0 = 0 · I and C0 = 0 · I, what will the value of Σ0,1 be in terms of the data in X and the parameters in θ0?
iii. (4 points): Still considering the case where A0 = 0·I and C0 = 0·I, what will the value of A1 be in terms of the data in X and the parameters in θ0?
iv. (4 points): Still considering the case where A0 = 0·I and C0 = 0·I, what will the value of Q1 be in terms of the data in X and the parameters in θ0?
v. (4 points): Still considering the case where A0 = 0 · I and C0 = 0 · I, what will the value of C1 be in terms of the data in X and the parameters in θ0?
vi. (2 points): Given these results, what is the solution to μ0,n, Σ0,n, An, Cn, and Qn?
