Kernel Multivariate Analysis Framework for Supervised Subspace Learning: A Tutorial on Linear and Kernel Multivariate Methods

Feature extraction and dimensionality reduction are important tasks in many fields of science dealing with signal processing and analysis. The relevance of these techniques is increasing as current sensory devices are developed with ever higher resolution, and problems involving multimodal data sources become more common. A plethora of feature extraction methods are available in the literature collectively grouped under the field of Multivariate Analysis (MVA). This paper provides a uniform treatment of several methods: Principal Component Analysis (PCA), Partial Least Squares (PLS), Canonical Correlation Analysis (CCA) and Orthonormalized PLS (OPLS), as well as their non-linear extensions derived by means of the theory of reproducing kernel Hilbert spaces. We also review their connections to other methods for classification and statistical dependence estimation, and introduce some recent developments to deal with the extreme cases of large-scale and low-sized problems. To illustrate the wide applicability of these methods in both classification and regression problems, we analyze their performance in a benchmark of publicly available data sets, and pay special attention to specific real applications involving audio processing for music genre prediction and hyperspectral satellite images for Earth and climate monitoring

Sparse kernel orthonormalized PLS for feature extraction in large datasets

In this paper we are presenting a novel multivariate analysis method. Our scheme is based on a novel kernel orthonormalized partial least squares (PLS) variant for feature extraction, imposing sparsity constrains in the solution to improve scalability. The algorithm is tested on a benchmark of UCI data sets, and on the analysis of integrated short-time music features for genre prediction. The upshot is that the method has strong expressive power even with rather few features, is clearly outperforming the ordinary kernel PLS, and therefore is an appealing method for feature extraction of labelled data

Regularized multivariate analysis framework for interpretable high-dimensional variable selection

Multivariate Analysis (MVA) comprises a family of well-known methods for feature extraction which exploit correlations among input variables representing the data. One important property that is enjoyed by most such methods is uncorrelation among the extracted features. Recently, regularized versions of MVA methods have appeared in the literature, mainly with the goal to gain interpretability of the solution. In these cases, the solutions can no longer be obtained in a closed manner, and more complex optimization methods that rely on the iteration of two steps are frequently used. This paper recurs to an alternative approach to solve efficiently this iterative problem. The main novelty of this approach lies in preserving several properties of the original methods, most notably the uncorrelation of the extracted features. Under this framework, we propose a novel method that takes advantage of the,2,1 norm to perform variable selection during the feature extraction process. Experimental results over different problems corroborate the advantages of the proposed formulation in comparison to state of the art formulations.This work has been partly supported by MINECO projects TEC2013-48439-C4-1-R, TEC2014-52289-R and TEC2016-75161-C2-2-R, and Comunidad de Madrid projects PRICAM P2013/ICE-2933 and S2013/ICE-2933

Nonnegative OPLS for supervised design of filter banks: application to image and audio feature extraction

Audio or visual data analysis tasks usually have to deal with high-dimensional and nonnegative signals. However, most data analysis methods suffer from overfitting and numerical problems when data have more than a few dimensions needing a dimensionality reduction preprocessing. Moreover, interpretability about how and why filters work for audio or visual applications is a desired property, especially when energy or spectral signals are involved. In these cases, due to the nature of these signals, the nonnegativity of the filter weights is a desired property to better understand its working. Because of these two necessities, we propose different methods to reduce the dimensionality of data while the nonnegativity and interpretability of the solution are assured. In particular, we propose a generalized methodology to design filter banks in a supervised way for applications dealing with nonnegative data, and we explore different ways of solving the proposed objective function consisting of a nonnegative version of the orthonormalized partial least-squares method. We analyze the discriminative power of the features obtained with the proposed methods for two different and widely studied applications: texture and music genre classification. Furthermore, we compare the filter banks achieved by our methods with other state-of-the-art methods specifically designed for feature extraction.This work was supported in parts by the MINECO projects TEC2013-48439-C4-1-R, TEC2014-52289-R, TEC2016-75161-C2-1-R, TEC2016-75161-C2-2-R, TEC2016-81900-REDT/AEI, and PRICAM (S2013/ICE-2933)

Sparse and kernel OPLS feature extraction based on eigenvalue problem solving

Orthonormalized partial least squares (OPLS) is a popular multivariate analysis method to perform supervised feature extraction. Usually, in machine learning papers OPLS projections are obtained by solving a generalized eigenvalue problem. However, in statistical papers the method is typically formulated in terms of a reduced-rank regression problem, leading to a formulation based on a standard eigenvalue decomposition. A first contribution of this paper is to derive explicit expressions for matching the OPLS solutions derived under both approaches and discuss that the standard eigenvalue formulation is also normally more convenient for feature extraction in machine learning. More importantly, since optimization with respect to the projection vectors is carried out without constraints via a minimization problem, inclusion of penalty terms that favor sparsity is straightforward. In the paper, we exploit this fact to propose modified versions of OPLS. In particular, relying on the ℓ1 norm, we propose a sparse version of linear OPLS, as well as a non-linear kernel OPLS with pattern selection. We also incorporate a group-lasso penalty to derive an OPLS method with true feature selection. The discriminative power of the proposed methods is analyzed on a benchmark of classification problems. Furthermore, we compare the degree of sparsity achieved by our methods and compare them with other state-of-the-art methods for sparse feature extraction.This work was partly supported by MINECO projects TEC2011-22480 and PRIPIBIN-2011-1266.Publicad

Optimal filtering of dynamics in short-time features for music organization

There is an increasing interest in customizable methods for organizing music collections. Relevant music characterization can be obtained from short-time features, but it is not obvious how to combine them to get useful information. In this work, a novel method, denoted as the Positive Constrained Orthonormalized Partial Least Squares (POPLS), is proposed. Working on the periodograms of MFCCs time series, this supervised method finds optimal filters which pick up the most discriminative temporal information for any music organization task. Two examples are presented in the paper, the first being a simple proof-of-concept, where an altosax with and without vibrato is modelled. A more complex 11 music genre classification setup is also investigated to illustrate the robustness and validity of the proposed method on larger datasets. Both experiments showed the good properties of our method, as well as superior performance when compared to a fixed filter bank approach suggested previously in the MIR literature. We think that the proposed method is a natural step towards a customized MIR application that generalizes well to a wide range of different music organization tasks