We propose a high dimensional semiparametric scale-invariant principle component analysis, named TCA, by utilize the natural connection between the elliptical distribution family and the principal component analysis. Elliptical distribution family includes many well-known multivariate distributions like multivariate Gaussian, t and logistic and it is extended to the meta-elliptical by Fang et.a...

Measuring similarity is crucial to many learning tasks. To this end, metric learning has been the dominant paradigm. However, similarity is a richer and broader notion than what metrics entail. For example, similarity can arise from the process of aggregating the decisions of multiple latent components, where each latent component compares data in its own way by focusing on a different subset o...

Consider a situation in which we have a number of sources emitting signals which are interfering with one another. Familiar situations in which this occurs are a crowded room with many people speaking at the same time, interfering electromagnetic waves from mobile phones or crosstalk from brain waves originating from different areas of the brain. In each of these situations the mixed signals ar...

A. Two quite different forms of nonlinear principal component analysis have been proposed in the literature. The first one is associated with the names of Guttman, Burt, Hayashi, Benzécri, McDonald, De Leeuw, Hill, Nishisato. We call itmultiple correspondence analysis. The second form has been discussed by Kruskal, Shepard, Roskam, Takane, Young, De Leeuw, Winsberg, Ramsay. We call it no...

Principal Component Analysis (PCA) is a feature extraction approach directly based on a whole vector pattern and acquires a set of projections that can realize the best reconstruction for an original data in the mean squared error sense. In this paper, the progressive PCA (PrPCA) is proposed, which could progressively extract features from a set of given data with large dimensionality and the e...

The vanishing ideal of a set of points, S ⊂ R, is the set of all polynomials that attain the value of zero on all the points in S. Such ideals can be compactly represented using a small set of polynomials known as generators of the ideal. Here we describe and analyze an efficient procedure that constructs a set of generators of a vanishing ideal. Our procedure is numerically stable, and can be ...

The purpose of sufficient dimension reduction (SDR) is to find a low-dimensional expression of input features that is sufficient for predicting output values. In this paper, we propose a novel distribution-free SDR method called sufficient component analysis (SCA), which is computationally more efficient than existing methods. In our method, a solution is computed by iteratively performing depe...

The famous example to illustrate ICA method is so-called cocktail party. Imagine two people are talking in a party and two different microphones are recording. Then the two records X1(t), X2(t) from those microphones are both the mixtures of the speech signal S1(t), S2(t) from the two speakers. Let us assume that only additive mixed effect exists, at this time, we can use the following equation...

Classical methods such as Principal Component Analysis (PCA) and Canonical Correlation Analysis (CCA) are ubiquitous in statistics. However, these techniques are only able to reveal linear relationships in data. Although nonlinear variants of PCA and CCA have been proposed, these are computationally prohibitive in the large scale. In a separate strand of recent research, randomized methods have...