Estimation of matrix trace using machine learning
This work addresses a computational bottleneck in large-scale linear algebra for researchers and engineers, offering a more efficient method for trace estimation, though it is incremental as it builds on the Hutchison estimator.
The paper tackles the problem of estimating the trace of a matrix when only matrix-vector products are available, proposing a machine learning-based estimator that uses probing vectors instead of random noise vectors. The result shows that with about 10 probing vectors, precision similar to using 10,000 random vectors is achieved in numerical experiments with random matrices.
We present a new trace estimator of the matrix whose explicit form is not given but its matrix multiplication to a vector is available. The form of the estimator is similar to the Hutchison stochastic trace estimator, but instead of the random noise vectors in Hutchison estimator, we use small number of probing vectors determined by machine learning. Evaluation of the quality of estimates and bias correction are discussed. An unbiased estimator is proposed for the calculation of the expectation value of a function of traces. In the numerical experiments with random matrices, it is shown that the precision of trace estimates with $\mathcal{O}(10)$ probing vectors determined by the machine learning is similar to that with $\mathcal{O}(10000)$ random noise vectors.