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Erschienen in:
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2016 | OriginalPaper | Buchkapitel

2. Principal Component Analysis

verfasst von : René Vidal, Yi Ma, S. Shankar Sastry

Erschienen in: Generalized Principal Component Analysis

Verlag: Springer New York

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Abstract

Principal component analysis (PCA) is the problem of fitting a low-dimensional affine subspace to a set of data points in a high-dimensional space. PCA is, by now, well established in the literature, and has become one of the most useful tools for data modeling, compression, and visualization.

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Vorheriges Kapitel Introduction
Nächstes Kapitel Robust Principal Component Analysis
Fußnoten
1
The reason for this is that both \(\boldsymbol{u}_{1}\) and its orthogonal complement \(\boldsymbol{u}_{1}^{\perp }\) are invariant subspaces of \(\Sigma _{\boldsymbol{x}}\).
 
2
In the statistical setting, \(\boldsymbol{x}_{j}\) and \(\boldsymbol{y}_{j}\) will be samples of two random variables \(\boldsymbol{x}\) and \(\boldsymbol{y}\), respectively. Then this constraint is equivalent to setting their means to zero.
 
3
From a statistical standpoint, the column vectors of U give the directions in which the data X has the largest variance, whence the name “principal components.”
 
4
In Section 1.​2.​1, we have seen an example in which a similar process can be applied to an ensemble of face images from multiple subspaces, where the first d = 3 principal components are calculated and visualized.
 
5
We leave as an exercise to the reader to calculate the number of parameters needed to specify a d-dimensional subspace in \(\mathbb{R}^{D}\) and then the additional parameters needed to specify a Gaussian distribution inside the subspace.
 
6
Even if one chooses to compare models by their algorithmic complexity, such as the minimum message length (MML) criterion (Wallace and Boulton 1968) (an extension of the Kolmogrov complexity to model selection), a strong connection with the above information-theoretic criteria, such as minimum description length (MDL), can be readily established via Shannon’s optimal coding theory (see (Wallace and Dowe 1999)).
 
7
It can be shown that the nuclear norm is a convex envelope of the rank function for matrices.
 
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Metadaten
Titel
Principal Component Analysis
verfasst von
René Vidal
Yi Ma
S. Shankar Sastry
Copyright-Jahr
2016
Verlag
Springer New York
Buch
Generalized Principal Component Analysis

Print ISBN: 978-0-387-87810-2

Electronic ISBN: 978-0-387-87811-9

Copyright-Jahr: 2016

DOI
https://doi.org/10.1007/978-0-387-87811-9_2

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