This course is part of the Mathematics for Machine Learning Specialization

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Mathematics for Machine Learning Specialization

Imperial College London

About this Course

4.7

1,431 ratings

•

211 reviews

This course offers a brief introduction to the multivariate calculus required to build many common machine learning techniques. We start at the very beginning with a refresher on the “rise over run” formulation of a slope, before converting this to the formal definition of the gradient of a function. We then start to build up a set of tools for making calculus easier and faster. Next, we learn how to calculate vectors that point up hill on multidimensional surfaces and even put this into action using an interactive game. We take a look at how we can use calculus to build approximations to functions, as well as helping us to quantify how accurate we should expect those approximations to be. We also spend some time talking about where calculus comes up in the training of neural networks, before finally showing you how it is applied in linear regression models. This course is intended to offer an intuitive understanding of calculus, as well as the language necessary to look concepts up yourselves when you get stuck. Hopefully, without going into too much detail, you’ll still come away with the confidence to dive into some more focused machine learning courses in future.

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Suggested: 6 weeks of study, 2-5 hours/week...

Subtitles: English, Greek, Spanish

Linear RegressionVector CalculusMultivariable CalculusGradient Descent

Start instantly and learn at your own schedule.

Reset deadlines in accordance to your schedule.

Suggested: 6 weeks of study, 2-5 hours/week...

Subtitles: English, Greek, Spanish

Week

1Understanding 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. ...

10 videos (Total 46 min), 4 readings, 6 quizzes

Welcome to Module 1!1m

Functions4m

Rise Over Run4m

Definition of a derivative10m

Differentiation examples & special cases7m

Product rule4m

Chain rule5m

Taming a beast5m

See you next module!39s

About Imperial College & the team5m

How to be successful in this course5m

Grading Policy5m

Additional Readings & Helpful References5m

Matching functions visually20m

Matching the graph of a function to the graph of its derivative20m

Let's differentiate some functions20m

Practicing the product rule20m

Practicing the chain rule20m

Unleashing the toolbox20m

Week

2Building 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. ...

9 videos (Total 41 min), 5 quizzes

Variables, constants & context7m

Differentiate with respect to anything4m

The Jacobian5m

Jacobian applied6m

The Sandpit4m

The Hessian5m

Reality is hard4m

See you next module!23s

Practicing partial differentiation20m

Calculating the Jacobian20m

Bigger Jacobians!20m

Calculating Hessians20m

Assessment: Jacobians and Hessians20m

Week

3Having 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. ...

6 videos (Total 19 min), 4 quizzes

Multivariate chain rule2m

More multivariate chain rule5m

Simple neural networks5m

More simple neural networks4m

See you next module!34s

Multivariate chain rule exercise20m

Simple Artificial Neural Networks20m

Training Neural Networks25m

Week

4The 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. ...

9 videos (Total 41 min), 5 quizzes

Building approximate functions3m

Power series3m

Power series derivation9m

Power series details6m

Examples5m

Linearisation5m

Multivariate Taylor6m

See you next module!28s

Matching functions and approximations20m

Applying the Taylor series15m

Taylor series - Special cases10m

2D Taylor series15m

Taylor Series Assessment20m

Week

5If 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....

4 videos (Total 28 min), 4 quizzes

Newton-Raphson in one dimension20m

Checking Newton-Raphson10m

Lagrange multipliers20m

Optimisation scenarios20m

Week

6In 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....

4 videos (Total 25 min), 1 reading, 3 quizzes

General non linear least squares7m

Doing least squares regression analysis in practice6m

Wrap up of this course48s

Did you like the course? Let us know!10m

Linear regression25m

Fitting a non-linear function15m

4.7

211 Reviewsstarted a new career after completing these courses

got a tangible career benefit from this course

By DP•Nov 26th 2018

Great course to develop some understanding and intuition about the basic concepts used in optimization. Last 2 weeks were a bit on a lower level of quality then the rest in my opinion but still great.

By JT•Nov 13th 2018

Excellent course. I completed this course with no prior knowledge of multivariate calculus and was successful nonetheless. It was challenging and extremely interesting, informative, and well designed.

Imperial College London is a world top ten university with an international reputation for excellence in science, engineering, medicine and business. located in the heart of London. Imperial is a multidisciplinary space for education, research, translation and commercialisation, harnessing science and innovation to tackle global challenges.
Imperial students benefit from a world-leading, inclusive educational experience, rooted in the College’s world-leading research. Our online courses are designed to promote interactivity, learning and the development of core skills, through the use of cutting-edge digital technology....

For a lot of higher level courses in Machine Learning and Data Science, you find you need to freshen up on the basics in mathematics - stuff you may have studied before in school or university, but which was taught in another context, or not very intuitively, such that you struggle to relate it to how it’s used in Computer Science. This specialization aims to bridge that gap, getting you up to speed in the underlying mathematics, building an intuitive understanding, and relating it to Machine Learning and Data Science.
In the first course on Linear Algebra we look at what linear algebra is and how it relates to data. Then we look through what vectors and matrices are and how to work with them.
The second course, Multivariate Calculus, builds on this to look at how to optimize fitting functions to get good fits to data. It starts from introductory calculus and then uses the matrices and vectors from the first course to look at data fitting.
The third course, Dimensionality Reduction with Principal Component Analysis, uses the mathematics from the first two courses to compress high-dimensional data. This course is of intermediate difficulty and will require basic Python and numpy knowledge.
At the end of this specialization you will have gained the prerequisite mathematical knowledge to continue your journey and take more advanced courses in machine learning....

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Once you enroll for a Certificate, you’ll have access to all videos, quizzes, and programming assignments (if applicable). Peer review assignments can only be submitted and reviewed once your session has begun. If you choose to explore the course without purchasing, you may not be able to access certain assignments.

What will I get if I subscribe to this Specialization?

When you enroll in the course, you get access to all of the courses in the Specialization, and you earn a certificate when you complete the work. Your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. If you only want to read and view the course content, you can audit the course for free.

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