About this Course
4.0
464 ratings
93 reviews
Specialization
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100% online

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Flexible deadlines

Flexible deadlines

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Intermediate Level

Intermediate Level

Hours to complete

Approx. 18 hours to complete

Suggested: 4 weeks of study, 4-5 hours/week...
Available languages

English

Subtitles: English

Skills you will gain

Python ProgrammingPrincipal Component Analysis (PCA)Projection MatrixMathematical Optimization
Specialization
100% online

100% online

Start instantly and learn at your own schedule.
Flexible deadlines

Flexible deadlines

Reset deadlines in accordance to your schedule.
Intermediate Level

Intermediate Level

Hours to complete

Approx. 18 hours to complete

Suggested: 4 weeks of study, 4-5 hours/week...
Available languages

English

Subtitles: English

Syllabus - What you will learn from this course

Week
1
Hours to complete
5 hours to complete

Statistics of Datasets

Principal Component Analysis (PCA) is one of the most important dimensionality reduction algorithms in machine learning. In this course, we lay the mathematical foundations to derive and understand PCA from a geometric point of view. In this module, we learn how to summarize datasets (e.g., images) using basic statistics, such as the mean and the variance. We also look at properties of the mean and the variance when we shift or scale the original data set. We will provide mathematical intuition as well as the skills to derive the results. We will also implement our results in code (jupyter notebooks), which will allow us to practice our mathematical understand to compute averages of image data sets....
Reading
8 videos (Total 27 min), 5 readings, 4 quizzes
Video8 videos
Welcome to module 141s
Mean of a dataset4m
Variance of one-dimensional datasets4m
Variance of higher-dimensional datasets5m
Effect on the mean4m
Effect on the (co)variance3m
See you next module!27s
Reading5 readings
About Imperial College & the team5m
How to be successful in this course5m
Grading policy5m
Additional readings & helpful references5m
Symmetric, positive definite matrices10m
Quiz3 practice exercises
Mean of datasets15m
Variance of 1D datasets15m
Covariance matrix of a two-dimensional dataset15m
Week
2
Hours to complete
4 hours to complete

Inner Products

Data can be interpreted as vectors. Vectors allow us to talk about geometric concepts, such as lengths, distances and angles to characterise similarity between vectors. This will become important later in the course when we discuss PCA. In this module, we will introduce and practice the concept of an inner product. Inner products allow us to talk about geometric concepts in vector spaces. More specifically, we will start with the dot product (which we may still know from school) as a special case of an inner product, and then move toward a more general concept of an inner product, which play an integral part in some areas of machine learning, such as kernel machines (this includes support vector machines and Gaussian processes). We have a lot of exercises in this module to practice and understand the concept of inner products....
Reading
8 videos (Total 36 min), 1 reading, 5 quizzes
Video8 videos
Dot product4m
Inner product: definition5m
Inner product: length of vectors7m
Inner product: distances between vectors3m
Inner product: angles and orthogonality5m
Inner products of functions and random variables (optional)7m
Heading for the next module!35s
Reading1 reading
Basis vectors20m
Quiz4 practice exercises
Dot product10m
Properties of inner products20m
General inner products: lengths and distances20m
Angles between vectors using a non-standard inner product20m
Week
3
Hours to complete
4 hours to complete

Orthogonal Projections

In this module, we will look at orthogonal projections of vectors, which live in a high-dimensional vector space, onto lower-dimensional subspaces. This will play an important role in the next module when we derive PCA. We will start off with a geometric motivation of what an orthogonal projection is and work our way through the corresponding derivation. We will end up with a single equation that allows us to project any vector onto a lower-dimensional subspace. However, we will also understand how this equation came about. As in the other modules, we will have both pen-and-paper practice and a small programming example with a jupyter notebook....
Reading
6 videos (Total 25 min), 1 reading, 3 quizzes
Video6 videos
Projection onto 1D subspaces7m
Example: projection onto 1D subspaces3m
Projections onto higher-dimensional subspaces8m
Example: projection onto a 2D subspace3m
This was module 3!32s
Reading1 reading
Full derivation of the projection20m
Quiz2 practice exercises
Projection onto a 1-dimensional subspace25m
Project 3D data onto a 2D subspace40m
Week
4
Hours to complete
5 hours to complete

Principal Component Analysis

We can think of dimensionality reduction as a way of compressing data with some loss, similar to jpg or mp3. Principal Component Analysis (PCA) is one of the most fundamental dimensionality reduction techniques that are used in machine learning. In this module, we use the results from the first three modules of this course and derive PCA from a geometric point of view. Within this course, this module is the most challenging one, and we will go through an explicit derivation of PCA plus some coding exercises that will make us a proficient user of PCA. ...
Reading
10 videos (Total 52 min), 5 readings, 2 quizzes
Video10 videos
Problem setting and PCA objective7m
Finding the coordinates of the projected data5m
Reformulation of the objective10m
Finding the basis vectors that span the principal subspace7m
Steps of PCA4m
PCA in high dimensions5m
Other interpretations of PCA (optional)7m
Summary of this module42s
This was the course on PCA56s
Reading5 readings
Vector spaces20m
Orthogonal complements10m
Multivariate chain rule10m
Lagrange multipliers10m
Did you like the course? Let us know!10m
Quiz1 practice exercise
Chain rule practice20m
4.0
93 ReviewsChevron Right

Top Reviews

By JSJul 17th 2018

This is one hell of an inspiring course that demystified the difficult concepts and math behind PCA. Excellent instructors in imparting the these knowledge with easy-to-understand illustrations.

By JVMay 1st 2018

This course was definitely a bit more complex, not so much in assignments but in the core concepts handled, than the others in the specialisation. Overall, it was fun to do this course!

Instructor

Avatar

Marc P. Deisenroth

Lecturer in Statistical Machine Learning
Department of Computing

About Imperial College London

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....

About the Mathematics for Machine Learning Specialization

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....
Mathematics for Machine Learning

Frequently Asked Questions

  • 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.

  • 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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