In the era of big data, reducing data dimensionality is critical in many areas of science. Widely used Principal Component Analysis (PCA) addresses this problem by computing a low dimensional data embedding that maximally explain variance of the data. However, PCA has two major weaknesses. Firstly, it only considers linear correlations among variables (features), and secondly it is not suitable for categorical data. We resolve these issues by proposing Maximally Correlated Principal Component Analysis (MCPCA). MCPCA computes transformations of variables whose covariance matrix has the largest Ky Fan norm. Variable transformations are unknown, can be nonlinear and are computed in an optimization. MCPCA can also be viewed as a multivariate extension of Maximal Correlation. For jointly Gaussian variables we show that the covariance matrix corresponding to the identity (or the negative of the identity) transformations majorizes covariance matrices of non-identity functions. Using this result we characterize global MCPCA optimizers for nonlinear functions of jointly Gaussian variables for every rank constraint. For categorical variables we characterize global MCPCA optimizers for the rank one constraint based on the leading eigenvector of a matrix computed using pairwise joint distributions. For a general rank constraint we propose a block coordinate descend algorithm and show its convergence to stationary points of the MCPCA optimization. We compare MCPCA with PCA and other state-of-the-art dimensionality reduction methods including Isomap, LLE, multilayer autoencoders (neural networks), kernel PCA, probabilistic PCA and diffusion maps on several synthetic and real datasets. We show that MCPCA consistently provides improved performance compared to other methods.

Direct contextual policy search methods learn to improve policy parameters and simultaneously generalize these parameters to different context or task variables. However, learning from high-dimensional context variables, such as camera images, is still a prominent problem in many real-world tasks. A naive application of unsupervised dimensionality reduction methods to the context variables, such as principal component analysis, is insufficient as task-relevant input may be ignored. In this paper, we propose a contextual policy search method in the model-based relative entropy stochastic search framework with integrated dimensionality reduction. We learn a model of the reward that is locally quadratic in both the policy parameters and the context variables. Furthermore, we perform supervised linear dimensionality reduction on the context variables by nuclear norm regularization. The experimental results show that the proposed method outperforms naive dimensionality reduction via principal component analysis and a state-of-the-art contextual policy search method.

I recently gave a free webinar on Principal Component Analysis. We had almost 300 researchers attend and didn't get through all the questions. This is part of a series of answers to those questions. If you missed it, you can get the webinar recording here. Principal Component Analysis specifically could be used with a training and test data set, but it doesn't make as much sense as doing so for Factor Analysis.

This review examines the applications of electronic noses and tongues in food analysis. A brief history of the development of sensors is included and this is illustrated by descriptions of the different types of sensors utilized in these devices. As pattern recognition techniques are widely used to analyse the data obtained from these multisensor arrays, a discussion of principal components analysis and artificial neural networks is essential. An introduction to the integration of electronic tongues and noses is also incorporated and the strengths and weaknesses of both are described. Applications described include identification and classification of flavour and aroma and other measurements of quality using the electronic nose.

Personalized treatment of patients based on tissue-specific cancer subtypes has strongly increased the efficacy of the chosen therapies. Even though the amount of data measured for cancer patients has increased over the last years, most cancer subtypes are still diagnosed based on individual data sources (e.g. gene expression data). We propose an unsupervised data integration method based on kernel principal component analysis. Principal component analysis is one of the most widely used techniques in data analysis. Unfortunately, the straight-forward multiple-kernel extension of this method leads to the use of only one of the input matrices, which does not fit the goal of gaining information from all data sources. Therefore, we present a scoring function to determine the impact of each input matrix. The approach enables visualizing the integrated data and subsequent clustering for cancer subtype identification. Due to the nature of the method, no free parameters have to be set. We apply the methodology to five different cancer data sets and demonstrate its advantages in terms of results and usability.

Given a matrix of observed data, Principal Components Analysis (PCA) computes a small number of orthogonal directions that contain most of its variability. Provably accurate solutions for PCA have been in use for a long time. However, to the best of our knowledge, all existing theoretical guarantees for it assume that the data and the corrupting noise are mutually independent, or at least uncorrelated. This is valid in practice often, but not always. In this paper, we study the PCA problem in the setting where the data and noise can be correlated. Such noise is often also referred to as ``data-dependent noise". We obtain a correctness result for the standard eigenvalue decomposition (EVD) based solution to PCA under simple assumptions on the data-noise correlation. We also develop and analyze a generalization of EVD, cluster-EVD, that improves upon EVD in certain regimes.

Direct contextual policy search methods learn to improve policy parameters and simultaneously generalize these parameters to different context or task variables. However, learning from high-dimensional context variables, such as camera images, is still a prominent problem in many real-world tasks. A naive application of unsupervised dimensionality reduction methods to the context variables, such as principal component analysis, is insufficient as task-relevant input may be ignored. In this paper, we propose a contextual policy search method in the model-based relative entropy stochastic search framework with integrated dimensionality reduction. We learn a model of the reward that is locally quadratic in both the policy parameters and the context variables. Furthermore, we perform supervised linear dimensionality reduction on the context variables by nuclear norm regularization. The experimental results show that the proposed method outperforms naive dimensionality reduction via principal component analysis and a state-of-the-art contextual policy search method.

Given a matrix of observed data, Principal Components Analysis (PCA) computes a small number of orthogonal directions that contain most of its variability. Provably accurate solutions for PCA have been in use for a long time. However, to the best of our knowledge, all existing theoretical guarantees for it assume that the data and the corrupting noise are mutually independent, or at least uncorrelated. This is valid in practice often, but not always. In this paper, we study the PCA problem in the setting where the data and noise can be correlated. Such noise is often also referred to as "data-dependent noise". We obtain a correctness result for the standard eigenvalue decomposition (EVD) based solution to PCA under simple assumptions on the data-noise correlation. We also develop and analyze a generalization of EVD, cluster-EVD, that improves upon EVD in certain regimes.

It was shown recently that the $K$ L1-norm principal components (L1-PCs) of a real-valued data matrix $\mathbf X \in \mathbb R^{D \times N}$ ($N$ data samples of $D$ dimensions) can be exactly calculated with cost $\mathcal{O}(2^{NK})$ or, when advantageous, $\mathcal{O}(N^{dK - K + 1})$ where $d=\mathrm{rank}(\mathbf X)$, $K

Principal component analysis (PCA) is often used to reduce the dimension of data by selecting a few orthonormal vectors that explain most of the variance structure of the data. L1 PCA uses the L1 norm to measure error, whereas the conventional PCA uses the L2 norm. For the L1 PCA problem minimizing the fitting error of the reconstructed data, we propose an exact reweighted and an approximate algorithm based on iteratively reweighted least squares. We provide convergence analyses, and compare their performance against benchmark algorithms in the literature. The computational experiment shows that the proposed algorithms consistently perform best.