Fast transforms correspond to factorizations of the form $\mathbf{Z} = \mathbf{X}^{(1)} \ldots \mathbf{X}^{(J)}$, where each factor $ \mathbf{X}^{(\ell)}$ is sparse and possibly structured. This paper investigates essential uniqueness of such factorizations, i.e., uniqueness up to unavoidable scaling ambiguities. Our main contribution is to prove that any $N \times N$ matrix having the so-called butterfly structure admits an essentially unique factorization into $J$ butterfly factors (where $N = 2^{J}$), and that the factors can be recovered by a hierarchical factorization method, which consists in recursively factorizing the considered matrix into two factors. This hierarchical identifiability property relies on a simple identifiability condition in the two-layer and fixed-support setting. This approach contrasts with existing ones that fit the product of butterfly factors to a given matrix via gradient descent. The proposed method can be applied in particular to retrieve the factorization of the Hadamard or the discrete Fourier transform matrices of size $N=2^J$. Computing such factorizations costs $\mathcal{O}(N^{2})$, which is of the order of dense matrix-vector multiplication, while the obtained factorizations enable fast $\mathcal{O}(N \log N)$ matrix-vector multiplications and have the potential to be applied to compress deep neural networks.

翻译：快速变换符合 $\ mathbf{ { { { { { { { { { { }\\ l}\ ldots\ mathbf{X{ { (J)} $ 的乘数, 其中每个乘数 $\ mathbf{X{ { { (ell)} 都稀少且可能结构化。 本文调查了这种乘数的基本独特性, 即独特性, 无法避免地缩放模糊性。 我们的主要贡献是证明任何 $N\ times N$ 矩阵中, 有所谓的蝴蝶结构允许基本上独特的乘数 $( $N = 2\ j} 美元) 的乘数, 以等级乘以等级化因数法, 将考虑的矩阵归为两个因数。 此等级的可识别性属性取决于两层和固定支持设置中简单的可识别性条件。 此方法与适合通过 梯度下降 度 矩阵中 的 的 。 提议的方法可以特别用于调取 $ncreal_ rental_ ral_ ral_ ral_ ral_ exal_ ormaxxxx 。