Q1. For a tensor whose size is (N \times N \times N), what is the order of the number of parameters of its one-body approximation?
a. O(3N)
b. O(3^N)
c. O(N^3)
d. O(N!)
a. The Expectation Maximization (EM) algorithm always guarantees the globally optimal solution, regardless of the choice of the objective function.
b. The non-negative rank-1 approximation minimizing the KL divergence can be exactly solved within polynomial time for the tensor size.
c. The KL divergence minimization offers strong outlier robustness.
d. The KL divergence is not a Bregman divergence.
Q3. As discussed in the lecture, the one-body tensor is always a rank-1 tensor. How about a two-body tensor? If each element of the given tensor P was given as P_ijk = A_ij B_jk C_ki, the tensor P is called two-body tensor. Can we say that the two-body tensor P is always a rank-2 tensor?
a. Yes, a two-body tensor is always a rank-2 tensor.
b. No, a two-body tensor is not necessarily a rank-2 tensor.`
Exercise for Tensor Networks for Density Estimation Beyond KL Divergence
at PhD Summer School at DTU (2026), 02901 Advanced Topics in Machine Learning: Tensor Networks for Machine Learning
Please fill in the blank of the Jupyter notebook to implement the EEM algorithm using the closed-form update formula we discussed in the lecture. Change the value of q and rank, and observe the quality of reconstruction. How would the results change if you changed the noise level? I assume you implement CPD with a third-order tensor, but if you have a specific preference, you can implement other structures and/or dimensions.
You can find more details of the EEM algorithm here