Linear Dimensionality Reduction: Survey, Insights, and Generalizations
Cunningham, John P.
Journal of Machine Learning Research
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Cunningham, J. P., & Ghahramani, Z. (2015). Linear Dimensionality Reduction: Survey, Insights, and Generalizations. Journal of Machine Learning Research, 16 2859-2900. http://jmlr.org/papers/v16/cunningham15a.html
This is the author accepted manuscript. The final version is available from MIT Press via http://jmlr.org/papers/v16/cunningham15a.html
Linear dimensionality reduction methods are a cornerstone of analyzing high dimensional data, due to their simple geometric interpretations and typically attractive computational properties. These methods capture many data features of interest, such as covariance, dynamical structure, correlation between data sets, input-output relationships, and margin between data classes. Methods have been developed with a variety of names and motivations in many fields, and perhaps as a result the connections between all these methods have not been highlighted. Here we survey methods from this disparate literature as optimization programs over matrix manifolds. We discuss principal component analysis, factor analysis, linear multidimensional scaling, Fisher's linear discriminant analysis, canonical correlations analysis, maximum autocorrelation factors, slow feature analysis, sufficient dimensionality reduction, undercomplete independent component analysis, linear regression, distance metric learning, and more. This optimization framework gives insight to some rarely discussed shortcomings of well-known methods, such as the suboptimality of certain eigenvector solutions. Modern techniques for optimization over matrix manifolds suggest a generic linear dimensionality reduction solver, which accepts as input data and an objective to be optimized, and returns, as output, an optimal low-dimensional projection of the data. This simple optimization framework further allows straightforward generalizations and novel variants of classical methods, which we demonstrate here by creating an orthogonal-projection canonical correlations analysis. More broadly, this survey and generic solver suggest that linear dimensionality reduction can move toward becoming a blackbox, objective-agnostic numerical technology.
JPC and ZG received funding from the UK Engineering and Physical Sciences Research Council (EPSRC EP/H019472/1). JPC received funding from a Sloan Research Fellowship, the Simons Foundation (SCGB#325171 and SCGB#325233), the Grossman Center at Columbia University, and the Gatsby Charitable Trust.
External link: http://jmlr.org/papers/v16/cunningham15a.html
This record's URL: http://www.repository.cam.ac.uk/handle/1810/248027
Attribution-NonCommercial 2.0 UK: England & Wales
Licence URL: http://creativecommons.org/licenses/by-nc/2.0/uk/
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