Towards a Unified Framework of Matrix Derivatives

Abstract

The need of processing and analyzing massive statistics simultaneously requires the derivatives of matrix-to-scalar functions (scalar-valued functions of matrices) or matrix-to-matrix functions (matrix-valued functions of matrices). Although derivatives of a matrix-to-scalar function have already been defined, the way to express it in algebraic expression, however, is not as clear as that of scalar-to-scalar functions (scalar-valued functions of scalars). Due to the fact that there does not exist a uniform way of applying “chain rule” on matrix derivation, we classify approaches utilized in existing schemes into two ways: the first relies on the index notation of several matrices, and they would be eliminated while being multiplied; the second relies on the vectorizing of matrices and thus they can be dealt with in the way we treat vector-to-vector functions (vector-valued functions of vectors), which has already been settled. On one hand, we find that the first approach holds a much lower time complexity than that of the second approach in general. On the other hand, until now though we know most typical functions that can be derived in the first approach, theoretically the second approach is more generally fit for any routine of ”chain rule.” The result of the second approach, nevertheless, can be also simplified to the same order of time complexity with the first approach under certain conditions. Therefore, it is important to establish these conditions. In this paper, we establish a sufficient condition under which not only the first approach can be applied but also the time complexity of results obtained from the second approach can be reduced. This condition is described in two equivalent individual conditions, each of which is a counterpart of an approach sequentially. In addition, we generalize the methods and use these two approaches to do the derivatives under the two conditions individually. This paper enables us to unify the framework of matrix derivatives, which would result in various applications in science and engineering.