Algorithm for finding the minimum factorization of a tensor product expression

Question

I originally asked a question on the Mathematica stack exchange on a similar topic here.

But it seems like my question actually extends beyond Mathematica. The issue is the following. Let $a,b,d$ be elements of some vector space $V$ over $\mathbb{R}$, and let $c_1,c_2 \in \mathbb{R}$. I have some tensor product expression $a \otimes b$ that is non commutative, but is distributive over addition and linear and such: $$(c_1 a + c_2 b) \otimes d = c_1 a \otimes b + c_2 a \otimes d.$$

Given some long expression involving the sum of many tensor products. Something like:

$$a_1 \otimes b_1 + a_2 \otimes b_2 + a_1 \otimes d + a_2 \otimes d$$

I would like an algorithm that will find a factored expression with the minimum number of tensor products. For the above, that would be just two:

$$a_1 \otimes (b_1 + d) + a_2 \otimes (b_2 + d).$$

At the moment, my 'algorithm' is to:

1. Expand all tensor products so I do truly have a sum of single products
2. Recursively apply rules like $c_1 a\otimes b + c_2 a \otimes d \to a \otimes (c_1 b + c_2 d)$
3. Hope for the best.

This is working fine-ish, but is definitely not guaranteed to arrive at the factorization with the fewest terms. Because of the order that my program implements things it can get stuck on, for example:

$$a_1 \otimes b_1 + a_2 \otimes b_2 + (a_1+a_2) \otimes d$$

and doesn't realize this can be simplified further by instead grouping the $a$ terms together to get:

$$a_1 \otimes (b_1 + d) + a_2 \otimes (b_2 + d).$$

This is because I "group from the right" before I "group from the left" but regardless of which way I do it, there will be an analagous case to above where that runs into trouble.

This really feels like a problem that is common enough there should be a procedure for handling it. In fact, perhaps even one in Mathematica. I would be curious about a general thought as to a good algorithm, if I were to implement such a thing homebrew, or something specific to Mathematica.

In essence my problem is that my current algorithm is greedy, and not looking at the big picture

Answer 1

I think this could be solved with linear algebra and binary search. I don't know whether there is a better algorithm.

Given $k$, I will describe an algorithm to determine whether it is possible to rewrite the expression using at most $k$ tensor products. Then, you can use binary search on $k$ to find the smallest $k$ for which this is possible.

So, let's assume $k$ is fixed, and we want to know whether we can rewrite the expression to use at most $k$ tensor products. Let $S \subset V$ denote the set of vectors that appear in the original expression. Expand the original expression into the following form:

$$\sum_{s,t \in S} c_{s,t} \cdot (s \otimes t),$$

where the $c_{s,t} \in \mathbb{R}$ are known constants. I assume that any factored expression will have the form

$$\sum_{i=1}^k \left(\sum_{s \in S} d_{i,s} s\right) \otimes \left(\sum_{t \in S} d'_{i,t} t\right),$$

for some constants $d_{i,s},d'_{i,t} \in \mathbb{R}$. I believe this is a fully general form of your factored expressions. Now we want the original expression to be equal to the factored expression, i.e., we want to have

$$\sum_{s,t \in S} c_{s,t} \cdot (s \otimes t) = \sum_{i=1}^k \left(\sum_{s \in S} d_{i,s} s\right) \otimes \left(\sum_{t \in S} d'_{i,t} t\right).$$

This imposes some requirements on the constants $c_{s,t},d_{i,s},d'_{i,t}$. Let's work out what the requirements are. If we expand the factored expression, we find

$$\sum_{i=1}^k \left(\sum_{s \in S} d_{i,s} s\right) \otimes \left(\sum_{t \in S} d'_{i,t} t\right) = \sum_{s,t \in S} \left(\sum_{i=1}^k d_{i,s} d'_{i,t}\right) (s \otimes t).$$

We want the above to be equal to the original expression, i.e., we want

$$\sum_{s,t \in S} c_{s,t} \cdot (s \otimes t) = \sum_{s,t \in S} \left(\sum_{i=1}^k d_{i,s} d'_{i,t}\right) (s \otimes t).$$

This holds if for all $s,t \in S$, we have

$$c_{s,t} = \sum_{i=1}^k d_{i,s} d'_{i,t}.$$

Notice that this is a system of $|S|^2$ linear equations on the 2ドルk|S|$ unknowns $d_{i,s},d'_{i,t}$. Therefore, we can check whether there is any solution to this system of linear equations, e.g., using Gaussian elimination.

Finally, if there is a solution to this system of linear equations, then we have found a factored expression that is equivalent to the original expression and uses at most $k$ tensor products. If there is no solution to this system of linear equations, then there is no factored expression of the listed form that uses at most $k$ tensor products and is equivalent to the original expression. This gives us the claimed algorithm.

• Jack: This is a truly wonderful answer. I am very grateful. The final step of solving for the $d$ and $d'$ matrices can be done easily using the singular value decomposition. If it is of interest to future readers I describe such an implementation in this post: math.stackexchange.com/questions/4946689/…, where I also ask if anyone knows of a way to generalize the use of the singular value decomposition to the factorization of tensor product expressions involving more than one tensor product symbol.