Lecture

Linear Dimensionality Reduction

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In course

CS-233(b): Introduction to machine learning (BA4)

Machine learning and data analysis are becoming increasingly central in many sciences and applications. In this course, fundamental principles and methods of machine learning will be introduced, analyzed and practically implemented. ...

Description

This lecture covers the concept of linear dimensionality reduction, starting with Cover's Theorem and the importance of high-dimensional spaces. It then delves into Principal Component Analysis (PCA), focusing on variance maximization and optimal linear mapping. The instructor explains how PCA can be applied to real-world datasets like MNIST and the UCI Iris dataset, as well as medical applications such as cancer genome analysis. The lecture concludes with examples of PCA in 3D face modeling and eigenfaces for image reconstruction.

Instructor

Pascal Fua

Pascal Fua received an engineering degree from Ecole Polytechnique, Paris, in 1984 and the Ph.D. degree in Computer Science from the University of Orsay in 1989. He then worked at SRI International and INRIA Sophia-Antipolis as a Computer Scientist. He joined EPFL in 1996 where he is now a Professor in the School of Computer and Communication Science and heads the Computer Vision Laboratory. His research interests include shape modeling and motion recovery from images, analysis of microscopy images, and Augmented Reality. His research interests include shape modeling and motion recovery from images, analysis of microscopy images, and machine learning. He has (co)authored over 300 publications in refereed journals and conferences. He is an IEEE Fellow and has been an Associate Editor of IEEE journal Transactions for Pattern Analysis and Machine Intelligence. He often serves as program committee member, area chair, and program chair of major vision conferences and has cofounded three spinoff companies (Pix4D, PlayfulVision, and NeuralConcept).

Official source

Related concepts (32)

Dimensionality reduction

Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally close to its intrinsic dimension. Working in high-dimensional spaces can be undesirable for many reasons; raw data are often sparse as a consequence of the curse of dimensionality, and analyzing the data is usually computationally intractable (hard to control or deal with).

Nonlinear dimensionality reduction

Nonlinear dimensionality reduction, also known as manifold learning, refers to various related techniques that aim to project high-dimensional data onto lower-dimensional latent manifolds, with the goal of either visualizing the data in the low-dimensional space, or learning the mapping (either from the high-dimensional space to the low-dimensional embedding or vice versa) itself. The techniques described below can be understood as generalizations of linear decomposition methods used for dimensionality reduction, such as singular value decomposition and principal component analysis.

Covariance matrix

In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of a given random vector. Any covariance matrix is symmetric and positive semi-definite and its main diagonal contains variances (i.e., the covariance of each element with itself). Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions.

Curse of dimensionality

The curse of dimensionality refers to various phenomena that arise when analyzing and organizing data in high-dimensional spaces that do not occur in low-dimensional settings such as the three-dimensional physical space of everyday experience. The expression was coined by Richard E. Bellman when considering problems in dynamic programming. Dimensionally cursed phenomena occur in domains such as numerical analysis, sampling, combinatorics, machine learning, data mining and databases.

Principal component analysis

Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and enabling the visualization of multidimensional data. Formally, PCA is a statistical technique for reducing the dimensionality of a dataset. This is accomplished by linearly transforming the data into a new coordinate system where (most of) the variation in the data can be described with fewer dimensions than the initial data.

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