In this paper, we investigate the butterfly factorization problem, i.e., the problem of approximating a matrix by a product of sparse and structured factors. We propose a new formal mathematical description of such factors, that encompasses many different variations of butterfly factorization with different choices of the prescribed sparsity patterns. Among these supports, we identify those that ensure that the factorization problem admits an optimum, thanks to a new property called ``chainability". For those supports we propose a new butterfly algorithm that yields an approximate solution to the butterfly factorization problem and that is supported by stronger theoretical guarantees than existing factorization methods. Specifically, we show that the ratio of the approximation error by the minimum value is bounded by a constant, independent of the target matrix.