Course Meeting Times
Lectures: 3 sessions / week, 2 hours / session
Prerequisites
Courses in linear algebra (such as 18.06 Linear Algebra) and multivariate calculus (such as 18.02 Multivariable Calculus)
Course Description
We all know that calculus courses such as 18.01 Single Variable Calculus and 18.02 Multivariable Calculus cover 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 reimagined 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).
Topics
Here are some of the topics covered:
- 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.
Grading
The course grade is based on performance on the two homework assignments.
