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In many physical, statistical, biological and other investigations it is desirable to approximate a system of points by objects of lower dimension and/or complexity. For this purpose, Karl Pearson invented principal component analysis in 1901 and found lines and planes of closest fit to system of points. The famous k-means algorithm solves the approximation problem too, but by finite sets instead of lines and planes. This chapter gives a brief practical introduction into the methods of construction of general principal objects, i.e. objects embedded in the middle of the multidimensional data set. As a basis, the unifying framework of mean squared distance approximation of finite datasets is selected. Principal graphs and manifolds are constructed as generalisations of principal components and k-means principal points. For this purpose, the family of expectation/maximisation algorithms with nearest generalisations is presented. Construction of principal graphs with controlled complexity is based on the graph grammar approach.
We introduce a convolutional neural network that operates directly on graphs. These networks allow end-to-end learning of prediction pipelines whose inputs are graphs of arbitrary size and shape. The architecture we present generalizes standard molec
The recently proposed self-ensembling methods have achieved promising results in deep semi-supervised learning, which penalize inconsistent predictions of unlabeled data under different perturbations. However, they only consider adding perturbations
There has been a surge of recent interest in learning representations for graph-structured data. Graph representation learning methods have generally fallen into three main categories, based on the availability of labeled data. The first, network emb
Principal Component Analysis (PCA) is one of the most important methods to handle high dimensional data. However, most of the studies on PCA aim to minimize the loss after projection, which usually measures the Euclidean distance, though in some fiel
Dimensionality reduction on Riemannian manifolds is challenging due to the complex nonlinear data structures. While probabilistic principal geodesic analysis~(PPGA) has been proposed to generalize conventional principal component analysis (PCA) onto