This is the course page for an 18.S096 Special Subject in Mathematics at MIT taught in January 2024 (IAP) by Professors Alan Edelman and Steven G. Johnson.
- For the previous version of this course, see Matrix Calculus in IAP 2023 (OCW) on OpenCourseWare (also on github, with videos on YouTube). See also Matrix Calculus in IAP 2022 (OCW) (also on github).
Lectures: MWF time 11am–1pm, Jan 16 Jan 17–Feb 2 in room 2-131. 3 units, 2 problem sets due Jan 26 and Feb 2 — submitted electronically via Canvas, no exams. TA/grader: Fedir Yudin.
Piazza forum: Online discussions at this piazza link.
Description:
We all know that calculus courses such as 18.01 and 18.02 are univariate and vector calculus, respectively. Modern applications such as machine learning and large-scale optimization require the next big step, "matrix calculus" and calculus on arbitrary vector spaces.
This class covers a coherent approach to matrix calculus showing techniques that allow you to think of a matrix holistically (not just as an array of scalars), generalize and compute derivatives of important matrix factorizations and many other complicated-looking operations, and understand how differentiation formulas must be re-imagined in large-scale computing. We will discuss reverse/adjoint/backpropagation differentiation, custom vector-Jacobian products, and how modern automatic differentiation is more computer science than calculus (it is neither symbolic formulas nor finite differences).
Prerequisites: Linear Algebra such as 18.06 and multivariate calculus such as 18.02.
Course will involve simple numerical computations using the Julia language. Ideally install it on your own computer following these instructions, but as a fallback you can run it in the cloud here:
Topics:
Here are some of the planned topics:
- Derivatives as linear operators and linear approximation on arbitrary vector spaces: beyond gradients and Jacobians.
- Derivatives of functions with matrix inputs and/or outputs (e.g. matrix inverses and determinants). Kronecker products and matrix "vectorization".
- Derivatives of matrix factorizations (e.g. eigenvalues/SVD) and derivatives with constraints (e.g. orthogonal matrices).
- Multidimensional chain rules, and the significance of right-to-left ("forward") vs. left-to-right ("reverse") composition. Chain rules on computational graphs (e.g. neural networks).
- Forward- and reverse-mode manual and automatic multivariate differentiation.
- Adjoint methods (vJp/pullback rules) for derivatives of solutions of linear, nonlinear, and differential equations.
- Application to nonlinear root-finding and optimization. Multidimensional Newton and steepest–descent methods.
- Applications in engineering/scientific optimization and machine learning.
- Second derivatives, Hessian matrices, quadratic approximations, and quasi-Newton methods.
- part 1: overview (slides)
- part 2: derivatives as linear operators