  The Matrix-based Renyi's entropy enables us to directly measure information
quantities from given data without the costly probability density estimation of
underlying distributions, thus has been widely adopted in numerous statistical
learning and inference tasks. However, exactly calculating this new information
quantity requires access to the eigenspectrum of a semi-positive definite (SPD)
matrix $A$ which grows linearly with the number of samples $n$, resulting in a
$O(n^3)$ time complexity that is prohibitive for large-scale applications. To
address this issue, this paper takes advantage of stochastic trace
approximations for matrix-based Renyi's entropy with arbitrary $\alpha \in R^+$
orders, lowering the complexity by converting the entropy approximation to a
matrix-vector multiplication problem. Specifically, we develop random
approximations for integer order $\alpha$ cases and polynomial series
approximations (Taylor and Chebyshev) for non-integer $\alpha$ cases, leading
to a $O(n^2sm)$ overall time complexity, where $s,m \ll n$ denote the number of
vector queries and the polynomial order respectively. We theoretically
establish statistical guarantees for all approximation algorithms and give
explicit order of s and m with respect to the approximation error
$\varepsilon$, showing optimal convergence rate for both parameters up to a
logarithmic factor. Large-scale simulations and real-world applications
validate the effectiveness of the developed approximations, demonstrating
remarkable speedup with negligible loss in accuracy.
