Mathematics for Machine Learning: PCA

Go to Course: https://www.coursera.org/learn/pca-machine-learning

Introduction

### Course Review: Mathematics for Machine Learning: PCA on Coursera **Introduction to the Course** In the realm of machine learning and data science, dimensionality reduction techniques are crucial for processing and analyzing large datasets. One of the most significant methods in this arena is Principal Component Analysis (PCA). If you're interested in mastering PCA and the mathematical principles behind it, the "Mathematics for Machine Learning: PCA" course on Coursera is an excellent choice. This intermediate-level course not only provides a solid theoretical foundation but also integrates practical coding exercises to solidify understanding. **Overview of the Course** Designed for learners with a basic understanding of Python and numpy, this course takes you on a journey through the mathematics of PCA, starting with fundamental statistics and advancing through vectors and orthogonal projections. It culminates in deriving PCA itself, emphasizing both the geometric and algebraic intuition behind this essential technique. **Course Syllabus Breakdown** 1. **Statistics of Datasets**: The course begins with the basic statistical concepts necessary to summarize data sets, such as mean and variance. Here, students learn how to manipulate and understand the properties of datasets, particularly images. The emphasis on mathematical intuition lays a strong groundwork that enhances the upcoming coding exercises. 2. **Inner Products**: In this module, students delve into the concept of inner products, a pivotal concept for understanding geometric relationships in data. By starting with the familiar dot product and advancing to more generalized inner products, learners gain a robust framework that connects directly to machine learning practices, particularly with kernel methods. 3. **Orthogonal Projections**: This section introduces the concept of projecting high-dimensional vectors onto lower-dimensional subspaces. Students engage in both geometric reasoning and mathematical derivation, enabling them to comprehend how and why these projections work. The pen-and-paper exercises coupled with programming examples in Jupyter Notebooks enhance practical application. 4. **Principal Component Analysis**: The final module is the course’s crown jewel. Here, students utilize their accumulated knowledge to derive PCA from a geometric perspective. Although it presents the most challenge, it also offers the most reward, transforming theoretical concepts into functional skills that can be applied in real-world data analysis scenarios. **Course Experience** What sets this course apart is its balanced approach between theory and practice. The instructors encourage active learning through Jupyter Notebooks, where students implement their theoretical findings in code. This hands-on practice is especially beneficial for those who might find pure mathematical derivation challenging. Moreover, the course is structured to be progressively challenging, ensuring that students who persist through the early material are well-equipped to tackle the complexities of PCA in the final module. However, it's worth noting that a stronger background in Python and numpy is recommended from the outset to handle the programming assignments. **Recommendation** Overall, "Mathematics for Machine Learning: PCA" is highly recommended for data enthusiasts aiming to deepen their understanding of PCA and its mathematical underpinnings. Whether you're a budding data scientist or an experienced analyst looking to refresh your skills, this course will provide you with valuable insights and practical knowledge. **Final Thoughts** In an era where data is becoming increasingly complex, mastering dimensionality reduction techniques like PCA is not just beneficial—it's essential. This course serves as a comprehensive guide, merging mathematical rigor with practical application, thereby preparing students to tackle real-world data challenges with confidence. If you're ready to elevate your understanding of machine learning mathematics, this course on Coursera is a worthwhile investment in your education.

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.

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.

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.

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.