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Introduction to JAX and Deep Learning

Vector Calculus - II

Explore the concept of Jacobian matrices as an extension of gradients for vector-valued functions. Understand how JAX implements forward and reverse mode auto-differentiation to efficiently calculate Jacobians, enhancing your skills in vector calculus for deep learning applications.

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Jacobian

The gradient is used for real-valued functions, f:Rn→Rf:R^n \to R.

The concept of gradients can be extended to vector-valued functions, f:Rn→Rmf: R^n \to R^m by Jacobian matrices.

A Jacobian matrix is defined as:

J=[∂f∂x1⋯∂f∂xn]=[∇Tf1⋮∇Tfm]=[∂f1∂x1⋯∂f1∂xn⋮⋱⋮∂fm∂x1⋯∂fm∂xn] J = \begin{bmatrix} \dfrac{\partial \mathbf{f}}{\partial x_1} & \cdots & \dfrac{\partial \mathbf{f}}{\partial x_n} \end{bmatrix} = \begin{bmatrix} \nabla^\mathsf{T} f_1 \\ \vdots \\ \nabla^\mathsf{T} f_m \end{bmatrix} = \begin{bmatrix} \dfrac{\partial f_1}{\partial x_1} & \cdots & \dfrac{\partial f_1}{\partial x_n}\\ \vdots & \ddots & \vdots\\ \dfrac{\partial f_m}{\partial x_1} & \cdots & \dfrac{\partial f_m}{\partial x_n} \end{bmatrix} ...