| Maybe someone else can summarize more accurately or do a better job but I'll take a shot: The Jacobian often appears in the final segment of a three part calculus series when exploring chain rules and variable transformations. Look up the Jacobian used in converting between x,y,z and spherical coordinates ρ,φ,θ and note its matrix structure. Skimming your Medium Lobosi article it seems it emphasizes this aspect. The Jacobian also serves another purpose. As stated in the OP's article "The Jacobian operator Df:x⟼Df(x) is a linear map which provides the best linear approximation of f around a given point x." We like approximations, we can make a speed vs accuracy/memory trade off, you only have so much space in a register or memory cell, and trying to get more accuracy past a certain point takes more memory/computations/time. The article then notes that many computations involve Jacobians with sparse matrices meaning some matrix elements can be ignored so we don't have to waste our time on them if handled cleverly. Subsequent sections cover methods to identify and label sparsity patterns. The article explains how applying their proposed coloring techniques to large matrices common in machine learning yields significant efficiency gains. As far as the mathematical heritage, I don't know the family tree, but I suspect it stems from courses blending matrix theory linear algebra and algorithms so you'd want the computer science version of such math. Functional approximation ties to numerical methods though I am uncertain if introductory texts cover Jacobians. Check out Newton's method to grasp its mechanics and understand how that works then explore its Jacobian extension. For the coloring aspect graph theory is where to turn. You can learn its basics with minimal prerequisites by studying the seven bridges problem or the map coloring problem, do the five color version. Many of these concepts can be simplified into small programming projects. They will not rival Matlab but they will solidify your understanding. |