Imperial College London
Mathematics for Machine Learning: Multivariate Calculus
Imperial College London

Mathematics for Machine Learning: Multivariate Calculus

Samuel J. Cooper
David Dye
A. Freddie Page

Instructors: Samuel J. Cooper

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Gain insight into a topic and learn the fundamentals.
4.7

(5,643 reviews)

Beginner level
No prior experience required
Flexible schedule
Approx. 17 hours
Learn at your own pace
91%
Most learners liked this course
Gain insight into a topic and learn the fundamentals.
4.7

(5,643 reviews)

Beginner level
No prior experience required
Flexible schedule
Approx. 17 hours
Learn at your own pace
91%
Most learners liked this course

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Assessments

25 assignments

Taught in English

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There are 6 modules in this course

Understanding calculus is central to understanding machine learning! You can think of calculus as simply a set of tools for analysing the relationship between functions and their inputs. Often, in machine learning, we are trying to find the inputs which enable a function to best match the data. We start this module from the basics, by recalling what a function is and where we might encounter one. Following this, we talk about the how, when sketching a function on a graph, the slope describes the rate of change of the output with respect to an input. Using this visual intuition we next derive a robust mathematical definition of a derivative, which we then use to differentiate some interesting functions. Finally, by studying a few examples, we develop four handy time saving rules that enable us to speed up differentiation for many common scenarios.

What's included

10 videos4 readings6 assignments1 discussion prompt1 plugin

Building on the foundations of the previous module, we now generalise our calculus tools to handle multivariable systems. This means we can take a function with multiple inputs and determine the influence of each of them separately. It would not be unusual for a machine learning method to require the analysis of a function with thousands of inputs, so we will also introduce the linear algebra structures necessary for storing the results of our multivariate calculus analysis in an orderly fashion.

What's included

9 videos5 assignments2 ungraded labs

Having seen that multivariate calculus is really no more complicated than the univariate case, we now focus on applications of the chain rule. Neural networks are one of the most popular and successful conceptual structures in machine learning. They are build up from a connected web of neurons and inspired by the structure of biological brains. The behaviour of each neuron is influenced by a set of control parameters, each of which needs to be optimised to best fit the data. The multivariate chain rule can be used to calculate the influence of each parameter of the networks, allow them to be updated during training.

What's included

6 videos3 assignments1 programming assignment1 discussion prompt1 ungraded lab

The Taylor series is a method for re-expressing functions as polynomial series. This approach is the rational behind the use of simple linear approximations to complicated functions. In this module, we will derive the formal expression for the univariate Taylor series and discuss some important consequences of this result relevant to machine learning. Finally, we will discuss the multivariate case and see how the Jacobian and the Hessian come in to play.

What's included

9 videos5 assignments1 plugin

If we want to find the minimum and maximum points of a function then we can use multivariate calculus to do this, say to optimise the parameters (the space) of a function to fit some data. First we’ll do this in one dimension and use the gradient to give us estimates of where the zero points of that function are, and then iterate in the Newton-Raphson method. Then we’ll extend the idea to multiple dimensions by finding the gradient vector, Grad, which is the vector of the Jacobian. This will then let us find our way to the minima and maxima in what is called the gradient descent method. We’ll then take a moment to use Grad to find the minima and maxima along a constraint in the space, which is the Lagrange multipliers method.

What's included

4 videos4 assignments1 discussion prompt1 ungraded lab

In order to optimise the fitting parameters of a fitting function to the best fit for some data, we need a way to define how good our fit is. This goodness of fit is called chi-squared, which we’ll first apply to fitting a straight line - linear regression. Then we’ll look at how to optimise our fitting function using chi-squared in the general case using the gradient descent method. Finally, we’ll look at how to do this easily in Python in just a few lines of code, which will wrap up the course.

What's included

4 videos1 reading2 assignments1 programming assignment1 ungraded lab1 plugin

Instructors

Instructor ratings
4.7 (674 ratings)
Samuel J. Cooper
Imperial College London
2 Courses424,480 learners
David Dye
Imperial College London
2 Courses424,480 learners
A. Freddie Page
Imperial College London
2 Courses424,480 learners

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