Auto-Differentiating Linear Algebra

Development systems for deep learning, such as Theano, Torch, TensorFlow, or MXNet, are easyto-use tools for creating complex neural network models. Since gradient computations are automatically baked in, and execution is mapped to high performance hardware, these models can be trained endto-end on large amounts of data. However, it is currently not easy to implement many basic machine learning primitives in these systems (such as Gaussian processes, least squares estimation, principal components analysis, Kalman smoothing), mainly because they lack efficient support of linear algebra primitives as differentiable operators. We detail how a number of matrix decompositions (Cholesky, LQ, symmetric eigen) can be implemented as differentiable operators. We have implemented these primitives in MXNet, running on CPU and GPU in single and double precision. We sketch use cases of these new operators, learning Gaussian process and Bayesian linear regression models. Our implementation is based on BLAS/LAPACK APIs, for which highly tuned implementations are available on all major CPUs and GPUs.