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.
Este curso forma parte de Programa especializado: Matemática aplicada al aprendizaje automático
ofrecido por
Acerca de este Curso
Qué aprenderás
Implement mathematical concepts using real-world data
Derive PCA from a projection perspective
Understand how orthogonal projections work
Master PCA
Habilidades que obtendrás
- Dimensionality Reduction
- Python Programming
- Linear Algebra
ofrecido por

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.
Programa - Qué aprenderás en este curso
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.
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.
Reseñas
- 5 stars51,01 %
- 4 stars22,61 %
- 3 stars12,83 %
- 2 stars6,73 %
- 1 star6,80 %
Principales reseñas sobre MATHEMATICS FOR MACHINE LEARNING: PCA
Rather difficult course and will probably reqire to watch additional video-explanations on YouTube as well as studing math notation, etc. Otherwise, helpfull and comprehensive.
Course content is interesting and well planned, Can be improved by making it Simpler for Students as it was more technical than the other 2 courses of the Specialization.
Relatively tougher than previous two courses in the specialization. I'd suggest giving more time and being patient in pursuit of completing this course and understanding the concepts involved.
This course is well worth the time. I have a better understanding of one of the most foundational and biologically plausible machine learning algorithms used today! Love it.
Acerca de Programa especializado: Matemática aplicada al aprendizaje automático
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.

Preguntas Frecuentes
¿Cuándo podré acceder a las lecciones y tareas?
¿Qué recibiré si me suscribo a este Programa especializado?
¿Hay ayuda económica disponible?
What level of programming is required to do this course?
How difficult is this course in comparison to the other two of this specialization?
¿Tienes más preguntas? Visita el Centro de Ayuda al Alumno.