Mathematics for Machine Learning: PCA from Coursera

This intermediate-level course introduces the mathematical foundations to derive Principal Component Analysis (PCA), a fundamental dimensionality reduction technique. We'll cover some basic statistics of data sets, such as mean values and variances, we'll compute distances and angles between vectors using inner products and derive orthogonal projections of data onto lower-dimensional subspaces. Using all these tools, we'll then derive PCA as a method that minimizes the average squared reconstruction error between data points and their reconstruction.

At the end of this course, you'll be familiar with important mathematical concepts and you can implement PCA all by yourself. If you’re struggling, you'll find a set of jupyter notebooks that will allow you to explore properties of the techniques and walk you through what you need to do to get on track. If you are already an expert, this course may refresh some of your knowledge.

The lectures, examples and exercises require:

1. Some ability of abstract thinking

2. Good background in linear algebra (e.g., matrix and vector algebra, linear independence, basis)

3. Basic background in multivariate calculus (e.g., partial derivatives, basic optimization)

4. Basic knowledge in python programming and numpy

Disclaimer: This course is substantially more abstract and requires more programming than the other two courses of the specialization. However, this type of abstract thinking, algebraic manipulation and programming is necessary if you want to understand and develop machine learning algorithms.

What's inside

Syllabus

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. Therefore, some python/numpy background will be necessary to get through this course. Note: If you have taken the other two courses of this specialization, this one will be harder (mostly because of the programming assignments). However, if you make it through the first week of this course, you will make it through the full course with high probability.

Inner Products

Data can be interpreted as vectors. Vectors allow us to talk about geometric concepts, such as lengths, distances and angles to characterize 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.

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.

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.

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Includes derivations of mathematical formulas to promote deep comprehension

Develops knowledge in probability, linear algebra, and optimization, core foundations for machine learning

Involves abstract thinking and mathematical operations, suitable for those comfortable with rigorous coursework

Taught by Marc Peter Deisenroth, a respected figure in machine learning research and education

Requires proficiency in Python programming and NumPy library for practical applications

Covers fundamental concepts in dimensionality reduction, a key technique in machine learning

Reviews summary

Mathematics for machine learning: pca

learners say this course is largely positive. It is heavily programming-focused and the math is not particularly difficult. However, the course is not very intuitive and many concepts are not well-explained in the lectures. Additionally, the programming assignments are often buggy and frustrating. Overall, learners say this is a good course for those with a strong math background and some programming experience, but it is not suitable for beginners.

The explanation of the model is very precise but there are unnecesary comments for PCA omit the comments related with std in the final assignment

"The explanation of the model is very precise but there are unnecesary comments for PCA omit the comments related with std in the final assignment"

Although the course was little out there and the instructor was trying his best to articulate a difficult topic, the overall experience is great.

"Although the course was little out there and the instructor was trying his best to articulate a difficult topic, the overall experience is great."

This has to be one of most nightmarish, ugly, courses I have taken on the Coursera platform. Lousy, boring instructor, assignments that are so full of bugs that even the staff cannot resolve the issues. Add to that very low participation from the mentors and teaching staff in responding to student concerns and questions.Hey Marc, teaching staff and Imperial College. Get your act together!!IOW This course sucks big time! Take it at your own risk

"This has to be one of most nightmarish, ugly, courses I have taken on the Coursera platform."

"Lousy, boring instructor, assignments that are so full of bugs that even the staff cannot resolve the issues."

"Add to that very low participation from the mentors and teaching staff in responding to student concerns and questions."

"Hey Marc, teaching staff and Imperial College. Get your act together!!IOW This course sucks big time! Take it at your own risk"

Great content however the programming part is too challenging for people without propre guidance in the subject. the videos aren't of much help.

"Great content however the programming part is too challenging for people without propre guidance in the subject."

"the videos aren't of much help."

difficult course but the instructor tried his best to deliver this course. Due to technical error I found it really frustrating at times and there is lack of examples and the study material wasn't enough and I found myself going through youtube many times through out this course. The discussion form is really helpful with out which I wouldn't have been able to complete this course

"difficult course but the instructor tried his best to deliver this course."

"Due to technical error I found it really frustrating at times and there is lack of examples and the study material wasn't enough and I found myself going through youtube many times through out this course."

"The discussion form is really helpful with out which I wouldn't have been able to complete this course"

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Mathematics for Machine Learning: PCA with these activities:

Organize Course Notes and Assignments

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Organizing your notes and assignments will help you consolidate your learning and prepare for assessments.

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Review your lecture notes and identify key concepts.
Summarize important points in your own words.
Compile all your notes and assignments into a central location.
Organize your materials using a logical structure.

Review Linear Algebra and Multivariate Calculus

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Reviewing Linear Algebra and Multivariate Calculus will help you strengthen your mathematical foundation and prepare you for the abstract concepts and derivations covered in this course.

View Linear Algebra Done Right (Undergraduate Texts... on Amazon

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Read the provided textbook chapters on linear algebra and multivariate calculus.
Work through the practice problems and exercises in the textbook.
Attend any optional review sessions or workshops offered by the course instructor.

Derive the Mean and Variance of a Dataset

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Practicing the derivation of the mean and variance will help you understand the underlying concepts and prepare you for more complex derivations later in the course.

Browse courses on Mean

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Review the definition of mean and variance.
Derive the mean of a dataset using the formula.
Derive the variance of a dataset using the formula.
Solve practice problems involving the calculation of mean and variance.

Five other activities

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Discuss PCA Applications with Classmates

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Engaging in discussions with classmates will help you exchange ideas, clarify concepts, and expand your understanding of PCA's applications.

Browse courses on Principal Component Analysis

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Join a study group or online forum for the course.
Participate in discussions on topics related to PCA.
Share your own insights and questions.
Collaborate on projects or assignments.

Develop Python Code to Compute Inner Products

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Creating Python code to compute inner products will reinforce your understanding of the concept and its implementation.

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Review the definition and properties of inner products.
Implement the dot product function in Python.
Implement a more general inner product function that takes a custom kernel function.
Test your code on various vectors and compare the results with theoretical calculations.

Explore PCA Implementations in Scikit-Learn

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Exploring PCA implementations in Scikit-Learn will expose you to practical applications of the technique and help you refine your understanding.

Browse courses on Principal Component Analysis

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Install the Scikit-Learn library.
Follow tutorials on how to use the PCA class in Scikit-Learn.
Apply PCA to a sample dataset and analyze the results.

Implement PCA from Scratch in Python

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Implementing PCA from scratch will provide you with a deep understanding of the algorithm and its mathematical foundations.

Browse courses on Principal Component Analysis

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Review the mathematical derivation of PCA.
Translate the mathematical steps into Python code.
Test your implementation on various datasets and compare the results with the output from Scikit-Learn's PCA.
Optimize your code for efficiency and scalability.

Assist Classmates with PCA Concepts

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Mentoring others will not only reinforce your understanding but also enhance your communication and teaching skills.

Browse courses on Principal Component Analysis

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Identify classmates who may need assistance.
Offer to help them understand PCA concepts.
Explain the concepts clearly and patiently.
Provide guidance and answer their questions.

Career center

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Data Scientist

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Machine Learning Engineer

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Statistician

The Statistics profession heavily relies on the mathematical foundations of Principal Component Analysis (PCA). With this course, you will strengthen your understanding and add to your competitive skill set. Statisticians use mathematical models to collect, analyze, interpret, and present data. As a Statistician, you will use your mathematical foundation to solve real-world problems in a variety of industries.

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Research Scientist

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Data Analyst

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Software Engineer

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Financial Analyst

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Actuary

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