Temporal Matrix Factorization

[Question A] For any partially observed data matrix with the observed index set , suppose is the orthogonal projection supported on , then the matrix factorization (i.e., are the rank- latent factor matrices) involving temporal modeling on latent temporal factors (i.e., ) can be formulated as follows,

where is the coefficient matrix. The symbol denotes Kronecker product.

If and are known, then how to optimize the variables ? Do these variables have any closed-form solutions?

Note that

where is the th column of , while is the th column of .

Is it possible to use alternating least squares (ALS) method to solve the optimization problem?

[Note] Let be the objective, the first-order partial derivatives are:

  • with respect to :

  • with respect to :

Let , then we have a vectorized closed-form solution to as

where , and we have

In this case, we also define a block matrix

with the following sub-matrix

  • with respect to :

where .

Here, we have a sequence of matrices:

[Question B] How do the following two models perform?


Which is better?


[Question C] What is the difference between


[Note] For , the first-order partial derivative is given by