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Fast linear transforms are ubiquitous in machine learning, including thediscrete Fourier transform, discrete cosine transform, and other structuredtransformations such as convolutions. All of these transforms can berepresented by dense matrix-vector multiplication, yet each has a specializedand highly efficient (subquadratic) algorithm. We ask to what extenthand-crafting these algorithms and implementations is necessary, whatstructural priors they encode, and how much knowledge is required toautomatically learn a fast algorithm for a provided structured transform.Motivated by a characterization of fast matrix-vector multiplication asproducts of sparse matrices, we introduce a parameterization ofdivide-and-conquer methods that is capable of representing a large class oftransforms. This generic formulation can automatically learn an efficientalgorithm for many important transforms; for example, it recovers the $O(N \logN)$ Cooley-Tukey FFT algorithm to machine precision, for dimensions $N$ up to$1024$. Furthermore, our method can be incorporated as a lightweightreplacement of generic matrices in machine learning pipelines to learnefficient and compressible transformations. On a standard task of compressing asingle hidden-layer network, our method exceeds the classification accuracy ofunconstrained matrices on CIFAR-10 by 3.9 points -- the first time a structuredapproach has done so -- with 4X faster inference speed and 40X fewerparameters.

Learning Fast Algorithms for Linear Transforms Using Butterfly Factorizations