Optimization algorithms for Kronecker approximation of high-dimensional covariance matrices

Question

I'm working with a high-dimensional covariance matrix and exploring Kronecker product approximations to make it computationally manageable.

Here's the setup:

1. I have a graph $G$ represented by a $D\times D$ matrix.
2. The covariance matrix S of this graph has dimensions $(D\times D)\times(D\times D)$
3. My goal is to approximate $S$ as a Kronecker product, $S \approx \hat U\otimes \hat V$
4. Ideally, I'd like to use this approximation in a matrix normal distribution setting, where $S$'s Kronecker structure can simplify computations significantly.

Given that $D$ could be as large as $60$ or more, are there specific optimization algorithms or approaches for finding $\hat U$ and $\hat V$ that minimize the approximation error between $S$ and $\hat U\otimes \hat V$?

I'm looking for algorithms that are efficient in high-dimensional settings, where S is too large to handle directly. Any insights into iterative methods, approximate solutions, or best practices would be greatly appreciated.

I also assume this algorithms should be written in C++ as it involves a lot matrices computation, is it?

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Edit ver

Given a large matrix S (e.g., dimensions 1M x 1M or 10,000 x 10,000), is it possible to factorize it exactly into matrices U and V in a Kronecker product setting, such that S=U⊗V? Or can we only achieve an approximation, where S≈U⊗V?

Below is an algorithm where it approximates a matrix into U,V using alternating least square. For example, given 10 nodes (D) of graph, you can depict this as a 10 x 10 matrix. The covariance (S) between each edge can be depicted as (10 x 10) x (10 x 10).

def kronecker_als_efficient(S, D, max_iter=100, tol=1e-6):
    U = np.random.rand(D, D)
    V = np.random.rand(D, D)
    residuals = []
    total_norm = norm(S, 'fro')

    for it in range(max_iter):
        S_tensor = S.reshape(D, D, D, D)  
        VVT = V @ V.T
        for i in range(D):
            for j in range(D):
                numerator = np.sum(S_tensor[i, :, j, :] * VVT)
                denominator = np.sum(VVT * VVT)
                U[i, j] = numerator / (denominator + 1e-8)
        
        UUT = U @ U.T
        for i in range(D):
            for j in range(D):
                numerator = np.sum(S_tensor[:, i, :, j] * UUT)
                denominator = np.sum(UUT * UUT)
                V[i, j] = numerator / (denominator + 1e-8)
        
        S_approx = np.kron(U, V)
        residual = norm(S - S_approx, 'fro')
        residuals.append(residual)
        print(f"Iteration {it+1}, Residual: {residual:.6f}")
        if residual / total_norm < tol:
            break
    return U, V, residuals

However, I think there should be a better way than this and find an exact solution.

Answer 1

A Kronecker product is essentially a rank-1 matrix, once you change indices: if $A_{ijkl} = U_{ik}V_{jl}$ (where all indices vary from $0$ to $n-1$, to keep the notation simpler), then the matrix defined by $M_{i+nj,k+nl} = A_{ijkl}$ has Kronecker product structure $M = U \otimes V$, but the matrix defined by $M_{i+nk,j+nl} = A_{ijkl}$ has rank 1.

So you can use the same techniques you would use for rank-1 approximation of a large matrix: sparse SVD, or the power method.

Answer 2

Answering to the edit:

• I still have no idea how $S$ is defined in terms of $G$. But that's fine, it can just be a $D^2\times D^2$ matrix for now.
• Not all matrices can be written as $S = U \otimes V$. If you rearrange entries as suggested in my other answer, that would mean that the permuted version $\tilde{S}$ of $S$ is rank-1, which clearly isn't true for all matrices. But you can approximate.
• ALS seems a reasonable algorithm. You can also try computing a sparse SVD of $\tilde{S}$ with scipy.sparse.linalg.svds, this might be competitive. In any case, if you don't have properties such as sparsity in $S$, or some additional structure, in most iterative algorithms each step is going to cost $O(D^4)$ operations.