Randomised_CP function throws a Singular Matrix error

[hello-fri-end]

Randomised_CP function throws a Singular Matrix error

[chaoyihu]

Is there a bound on the value of rank for a given tensor?
Hi! Here I am offering you my findings on this problem: The parameter R you referred to is the number of components, i.e. rank-one tensors, you would like to have in the result of your CP decomposition. In the given reference of randomised_parafac(), (Casey Battaglino, Grey Ballard and Tamara G. Kolda, "A Practical Randomized CP Tensor Decomposition"), section 3.1, there was a discussion on the smallest number of rank-one tensors in an exact CP decomposition, i.e. the tensor rank. Here I am citing a few relevant points:

• There is no straightforward algorithm to determine the rank of a specific tensor. In fact, the problem is NP-hard.
• In practice, the rank of a tensor is determined numerically by fitting various rank-R CP models. (In section 3.4) Ideally, if the data is noise-free, and we have a procedure for calculating CP with a given number of components, then we can do that for R=1,2,3… and stop at the first value of R that gives a fit of 100%. However, there are many problems with this procedure. … When the data are noisy (as is frequently the case), then fit alone cannot determine the rank in any case; …
• ALS, today's workhorse method to compute CP at a given R, can take many iterations to converge, and is not guaranteed to converge to the global minimum or even a stationary point. The author also summarized typical ranks of three-way tensors over R space with or without frontal slice symmetry.

(Btw, I checked the arXiv paper again today and could not find the tensor rank section any more. Not sure why but I am trying to find the version I read a few days ago.)

Regarding the singular matrix error, specifically

factor = tl.transpose(tl.solve(pseudo_inverse, factor))

In numpy/linalg/linalg.py

def solve(a, b):
    """
    Solve a linear matrix equation, or system of linear scalar equations.

    Computes the "exact" solution, `x`, of the well-determined, i.e., full
    rank, linear matrix equation `ax = b`.
    …
    """

It seems that the singular matrix error occurs when a in ax=b is not invertible, which in this case means pseudo_inverse is not invertible. We confirm this by inspecting the determinant of pseudo_inverse:

############## START
R = 2
############## ITERATE
======== iteration 0
------------- mode 0
det of pseudo_inverse: 4.791666666666669
------------- mode 1
det of pseudo_inverse: 4.397817528932697e-29
------------- mode 2
det of pseudo_inverse: 0.0
Traceback (most recent call last):
...
numpy.linalg.LinAlgError: Singular matrix

Inspecting the calculation steps:

711             kr_prod, indices_list = sample_khatri_rao(                          
712                 factors, n_samples, skip_matrix=mode, random_state=rng          
713             )
…
724             pseudo_inverse = tl.dot(tl.transpose(kr_prod), kr_prod)

There can be multiple causes for det(pseudo_inverse) = 0, analytical or numerical.
Analytically, kr-prod is the sampled khatri-rao product of size (n_samples, R), and pseudo_inverse = tl.dot(tl.transpose(kr_prod), kr_prod) is of size (R, R). If we want pseudo_inverse to be invertible, kr_prod should have R non-zero eigenvalues.
Numerically, one possible reason is that column normalization of the updated factors is currently not implemented in randomised_parafac(). I'm not sure why this step is skipped.

[OsmanMalik]

chaoyihu is right about what the issues is here. I'll elaborate a bit further.

The issue is not with the choice of rank, but rather the choice of number of samples (n_samples) which is the 3rd parameter in randomised_parafac. In the ALS algorithm for CP decomposition, each step involves solving a highly overdetermined linear least squares problem, i.e., something of the form $\min_X || AX - B ||$ where $A$ is tall and skinny. What randomised_parafac does is draw a subset of the rows in this linear system uniformly at random. The number of rows that are drawn is determined by n_samples. When this is set to 1 as in your example, only a single row is drawn. When $R=1$, this doesn't break the code since you end up with a $1 \times 1$ linear system which is (typically) invertible. However, for $R > 1$, you'll end up with an underdetermined linear system.

To see why this is a problem, suppose $S$ is the sampling operator. Then, the sampled least squares system is $\min_X || SAX - SB||$. As implemented, randomised_parafac solves this via the normal equations, i.e., by reformulating it to the problem of solving $(SA)^\top (SA) X = (SA)^\top B$ for $X$. As chaoyihu points out, this system is solved by numpy's solve function which can only handle full-rank problems, i.e., when $(SA)^\top (SA)$ is of full rank. If n_samples < rank, this matrix will always be rank-deficient and solve won't work.

In practice, since the sampling is done uniformly at random, even if n_samples $\geq$ rank, there's a chance that the system $(SA)^\top (SA)$ is rank deficient. It is therefore a good idea to choose n_samples to be quite a bit bigger than rank.

As a side note, randomised_parafac doesn't actually implement any of the methods by Battaglino et al. That paper uses randomized trigonometric transforms to first mix the rows before sampling. The way randomised_parafac is implemented currently, it's easy to construct an adversarial example which breaks the method (which would give the kind of singular matrix error you noticed) with high likelihood even when a large number (say, half) of all rows are included in the linear least squares problems.