À propos de ce cours : This 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. This 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
Les destinataires de ce cours : This is an intermediate level course. It is probably good to brush up your linear algebra and python programming before you start this course.
Créé par : Imperial College London
Enseigné par : Marc P. Deisenroth, Lecturer in Statistical Machine Learning
Department of Computing
| Informations de base | Cours 3 de 3 dans Mathematics for Machine Learning Specialization |
| Niveau | Intermediate |
| Engagement | 4 weeks of study, 3-5 hours/week |
| Langue | English |
| Comment réussir | Réussissez tous les devoirs notés pour terminer le cours. |
| Notes des utilisateurs |
Programme de cours
SEMAINE 1
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.
8 vidéos, 5 lectures, 2 quiz pour s'exercer
- Reading: About Imperial College & the team
- Reading: How to be successful in this course
- Reading: Grading policy
- Reading: Additional readings & helpful references
- Discussion Prompt: Nice to meet you!
- Ungraded Plugin: Survey
- Vidéo: Welcome to module 1
- Vidéo: Mean of a dataset
- Practice Quiz: Mean of datasets
- Vidéo: Variance of one-dimensional datasets
- Reading: Symmetric, positive definite matrices
- Vidéo: Variance of higher-dimensional datasets
- Practice Quiz: Covariance matrix of a two-dimensional dataset
- Vidéo: Effect on the mean
- Vidéo: Effect on the (co)variance
- Notebook: Mean/covariance of a dataset + effect of a linear transformation
- Vidéo: See you next module!
Noté: Variance of 1D datasets
Noté: Mean/covariance of a dataset + effect of a linear transformation
SEMAINE 2
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.
8 vidéos, 1 lecture, 2 quiz pour s'exercer
- Vidéo: Dot product
- Practice Quiz: Dot product
- Vidéo: Inner product: definition
- Vidéo: Inner product: length of vectors
- Vidéo: Inner product: distances between vectors
- Practice Quiz: General inner products: lengths and distances
- Reading: Basis vectors
- Vidéo: Inner product: angles and orthogonality
- Notebook: Inner products and angles
- Vidéo: Inner products of functions and random variables (optional)
- Vidéo: Heading for the next module!
Noté: Properties of inner products
Noté: Angles between vectors using a non-standard inner product
Noté: Inner products and angles
SEMAINE 3
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.
6 vidéos, 1 lecture, 1 quiz pour s'exercer
- Vidéo: Projection onto 1D subspaces
- Vidéo: Example: projection onto 1D subspaces
- Vidéo: Projections onto higher-dimensional subspaces
- Reading: Full derivation of the projection
- Vidéo: Example: projection onto a 2D subspace
- Practice Quiz: Project 3D data onto a 2D subspace
- Notebook: Orthogonal projections
- Vidéo: This was module 3!
Noté: Projection onto a 1-dimensional subspace
Noté: Orthogonal projections
SEMAINE 4
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.
10 vidéos, 5 lectures, 1 quiz pour s'exercer
- Reading: Vector spaces
- Reading: Orthogonal complements
- Vidéo: Problem setting and PCA objective
- Reading: Multivariate chain rule
- Practice Quiz: Chain rule practice
- Vidéo: Finding the coordinates of the projected data
- Vidéo: Reformulation of the objective
- Reading: Lagrange multipliers
- Vidéo: Finding the basis vectors that span the principal subspace
- Vidéo: Steps of PCA
- Vidéo: PCA in high dimensions
- Notebook: Principal Components Analysis (PCA)
- Vidéo: Other interpretations of PCA (optional)
- Vidéo: Summary of this module
- Vidéo: This was the course on PCA
- Reading: Did you like the course? Let us know!
- Ungraded Plugin: Post-Course Survey
Noté: PCA
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Imperial College London
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Notation et examens
Truly exceptional
Perfect course. It takes up more time and effort than the other two courses in the specialization. But what you learn by the end of it is totally worth the effort. Note that this is an Intermediate course compared to the other two which are beginner. So the extra rigor is expected.
This is a good course, you learn about the foundations of PCA.
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Very hard to follow, but you need to do it to understand machine learning very well.