Statistical Embedding: Beyond Principal Components

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doi:10.1214/22-STS881

Statistical Embedding: Beyond Principal Components

2023-08-01

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DOI：10.1214/22-STS881

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There has been an intense recent activity in embedding of very high-dimensional and nonlinear data structures, much of it in the data science and machine learning literature. We survey this activity in four parts. In the first part, we cover nonlinear methods such as principal curves, multidimen-sional scaling, local linear methods, ISOMAP, graph-based methods and dif-fusion mapping, kernel based methods and random projections. The second part is concerned with topological embedding methods, in particular mapping topological properties into persistence diagrams and the Mapper algorithm. Another type of data sets with a tremendous growth is very high-dimensional network data. The task considered in part three is how to embed such data in a vector space of moderate dimension to make the data amenable to traditional techniques such as cluster and classification techniques. Arguably, this is the part where the contrast between algorithmic machine learning methods and statistical modeling, represented by the so-called stochastic block model, is at its greatest. In the paper, we discuss the pros and cons for the two approaches. The final part of the survey deals with embedding in R2, that is, visualization. Three methods are presented: t-SNE, UMAP and LargeVis based on meth-ods in parts one, two and three, respectively. The methods are illustrated and compared on two simulated data sets; one consisting of a triplet of noisy Ra-nunculoid curves, and one consisting of networks of increasing complexity generated with stochastic block models and with two types of nodes.

Keyword:

Statistical embedding

principal component

non-linear principal component

multidimensional scaling

local linear method

ISOMAP

graph spectral theory

diffusion mapping

reproducing kernel Hilbert space

random projection

topological data analysis and embedding

persistent homology

persistence diagram

the Mapper

network embedding

spectral embedding

stochastic block modeling

Skip-Gram

neighborhood sampling strategies

visualization

t-SNE

LargeVis

UMAP

Statistical Embedding: Beyond Principal Components

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