This intermediate-level course, "Mathematics for Machine Learning: PCA," presented by Imperial College London, provides a comprehensive exploration of the mathematical underpinnings essential for understanding Principal Component Analysis (PCA), a pivotal technique for dimensionality reduction in machine learning.
Throughout the course, you will delve into fundamental statistics of datasets, including mean values and variances, compute distances and angles between vectors using inner products, and derive orthogonal projections of data onto lower-dimensional subspaces. The culmination of these foundational concepts leads to the derivation of PCA as a method to minimize the average squared reconstruction error between data points and their reconstruction.
By the end of the course, you will be adept at implementing mathematical concepts using real-world data, deriving PCA from a projection perspective, understanding how orthogonal projections work, and mastering PCA. Proficiency in abstract thinking, a strong background in linear algebra and multivariate calculus, as well as basic knowledge in Python programming and NumPy are recommended for successful completion of this course.
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This course comprises four modules that delve into fundamental statistics of datasets, inner products, orthogonal projections, and the derivation of Principal Component Analysis (PCA) for machine learning applications.
This module introduces the fundamental statistics of datasets, covering mean values, variances, and the effects of linear transformations on datasets. Additionally, it includes a NumPy tutorial and pre-course and post-course surveys for assessment.
Module 2 focuses on inner products, delving into the dot product, lengths and distances, angles and orthogonality, and properties of inner products. An optional segment covers the K-nearest Neighbors Algorithm.
The third module explores orthogonal projections, including projection onto 1D and higher-dimensional subspaces, and provides a full derivation of the projection. The module also covers orthogonal complements and the multivariate chain rule.
Module 4, the final module, delves into Principal Component Analysis (PCA), addressing the problem setting and PCA objective, finding the basis vectors that span the principal subspace, and additional interpretations of PCA. It also includes a post-course survey for feedback.
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