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Lecture# Dimensionality Reduction

Description

This lecture covers the concepts of Singular Value Decomposition (SVD) and Principal Component Analysis (PCA) for dimensionality reduction. It explains how to find low-dimensional representations of high-dimensional data, with applications in visualization, noise reduction, and efficiency. The lecture also delves into the spectral theorem, SVD existence, low-rank approximation, and best rank(r)-approximation. Additionally, it explores the interpretation of SVD, covariance vs correlation matrix in PCA, Multidimensional Scaling (MDS), non-linear embedding techniques like Isomap, and concludes with a summary of lessons learned.

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Related concepts (102)

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.

Device file

In Unix-like operating systems, a device file or special file is an interface to a device driver that appears in a as if it were an ordinary . There are also special files in DOS, OS/2, and Windows. These special files allow an application program to interact with a device by using its device driver via standard input/output system calls. Using standard system calls simplifies many programming tasks, and leads to consistent user-space I/O mechanisms regardless of device features and functions.

Singular value decomposition

In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any matrix. It is related to the polar decomposition. Specifically, the singular value decomposition of an complex matrix M is a factorization of the form where U is an complex unitary matrix, is an rectangular diagonal matrix with non-negative real numbers on the diagonal, V is an complex unitary matrix, and is the conjugate transpose of V.

Linear subspace

In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces. If V is a vector space over a field K and if W is a subset of V, then W is a linear subspace of V if under the operations of V, W is a vector space over K.

File descriptor

In Unix and Unix-like computer operating systems, a file descriptor (FD, less frequently fildes) is a process-unique identifier (handle) for a or other input/output resource, such as a pipe or network socket. File descriptors typically have non-negative integer values, with negative values being reserved to indicate "no value" or error conditions. File descriptors are a part of the POSIX API.

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