On the Real Part of Butterfly Factorizations, with Application to Fast Transforms
Fast transforms such as the Discrete Cosine Transform can be expressed as the real part of a product of complex-valued structured sparse matrices known as Kronecker-sparse factors. More generally, products of such structured sparse matrices appear in square dyadic butterfly and monarch matrices, which are known to represent many fast transforms efficiently. Yet, it has remained unclear whether taking the real or the imaginary part of such products preserves this structure. In this letter, we give a condition on the factors to ensure that the structure is preserved. This condition is satisfied in the case of the Discrete Cosine Transform, the Discrete Sine Transform and the Discrete Hartley Transform, allowing us to provide a new representation of these transforms as products of real-valued Kronecker-sparse factors, which opens new avenues for efficient GPU implementations using low-precision numerical format.