A Simple and Efficient Tensor Calculus for Machine Learning

Computing derivatives of tensor expressions, also known as tensor calculus, is a fundamental task in machine learning. A key concern is the efficiency of evaluating the expressions and their derivatives that hinges on the representation of these expressions. Recently, an algorithm for computing higher order derivatives of tensor expressions like Jacobians or Hessians has been introduced that is a few orders of magnitude faster than previous state-of-the-art approaches. Unfortunately, the approach is based on Ricci notation and hence cannot be incorporated into automatic differentiation frameworks from deep learning like TensorFlow, PyTorch, autograd, or JAX that use the simpler Einstein notation. This leaves two options, to either change the underlying tensor representation in these frameworks or to develop a new, provably correct algorithm based on Einstein notation. Obviously, the first option is impractical. Hence, we pursue the second option. Here, we show that using Ricci notation is not necessary for an efficient tensor calculus and develop an equally efficient method for the simpler Einstein notation. It turns out that turning to Einstein notation enables further improvements that lead to even better efficiency. The methods that are described in this paper have been implemented in the online tool www.MatrixCalculus.org for computing derivatives of matrix and tensor expressions. An extended abstract of this paper appeared as "A Simple and Efficient Tensor Calculus", AAAI 2020

Matrix moments of the diffusion tensor distribution

Purpose: To facilitate the implementation/validation of signal representations and models using parametric matrix-variate distributions to approximate the diffusion tensor distribution (DTD) $\mathcal{P}(\mathbf{D})$. Theory: We establish practical mathematical tools, the matrix moments of the DTD, enabling to compute the mean diffusion tensor and covariance tensor associated with any parametric matrix-variate DTD whose moment-generating function is known. As a proof of concept, we apply these tools to the non-central matrix-variate Gamma (nc-mv-Gamma) distribution, whose covariance tensor was so far unknown, and design a new signal representation capturing intra-voxel heterogeneity via a single nc-mv-Gamma distribution: the matrix-variate Gamma approximation. Methods: Furthering this proof of concept, we evaluate the matrix-variate Gamma approximation in silico and in vivo, in a human-brain 'tensor-valued' diffusion MRI dataset. Results: The matrix-variate Gamma approximation fails to capture the heterogeneity arising from orientation dispersion and from simultaneous variances in the trace (size) and anisotropy (shape) of the underlying diffusion tensors, which is explained by the structure of the covariance tensor associated with the nc-mv-Gamma distribution. Conclusion: The matrix moments promote a more widespread use of matrix-variate distributions as plausible approximations of the DTD by alleviating their intractability, thereby facilitating the design/validation of matrix-variate microstructural techniques.Comment: 17 pages, 6 figure