Isomap nonlinear dimensionality reduction and bimodality of Asian monsoon convection

It is known that the empirical orthogonal function method is unable to detect possible nonlinear structure in climate data. Here, isometric feature mapping (Isomap), as a tool for nonlinear dimensionality reduction, is applied to 1958–2001 ERA-40 sea-level pressure anomalies to study nonlinearity of the Asian summer monsoon intraseasonal variability. Using the leading two Isomap time series, the probability density function is shown to be bimodal. A two-dimensional bivariate Gaussian mixture model is then applied to identify the monsoon phases, the obtained regimes representing enhanced and suppressed phases, respectively. The relationship with the large-scale seasonal mean monsoon indicates that the frequency of monsoon regime occurrence is significantly perturbed in agreement with conceptual ideas, with preference for enhanced convection on intraseasonal time scales during large-scale strong monsoons. Trend analysis suggests a shift in concentration of monsoon convection, with less emphasis on South Asia and more on the East China Sea.

Gaussian processes autoencoder for dimensionality reduction

Learning low dimensional manifold from highly nonlinear data of high dimensionality has become increasingly important for discovering intrinsic representation that can be utilized for data visualization and preprocessing. The autoencoder is a powerful dimensionality reduction technique based on minimizing reconstruction error, and it has regained popularity because it has been efficiently used for greedy pretraining of deep neural networks. Compared to Neural Network (NN), the superiority of Gaussian Process (GP) has been shown in model inference, optimization and performance. GP has been successfully applied in nonlinear Dimensionality Reduction (DR) algorithms, such as Gaussian Process Latent Variable Model (GPLVM). In this paper we propose the Gaussian Processes Autoencoder Model (GPAM) for dimensionality reduction by extending the classic NN based autoencoder to GP based autoencoder. More interestingly, the novel model can also be viewed as back constrained GPLVM (BC-GPLVM) where the back constraint smooth function is represented by a GP. Experiments verify the performance of the newly proposed model.

A multidimensional collaborative filtering fusion approach with dimensionality reduction

Multidimensional data are getting increasing attention from researchers for creating better recommender systems in recent years. Additional metadata provides algorithms with more details for better understanding the interaction between users and items. While neighbourhood-based Collaborative Filtering (CF) approaches and latent factor models tackle this task in various ways effectively, they only utilize different partial structures of data. In this paper, we seek to delve into different types of relations in data and to understand the interaction between users and items more holistically. We propose a generic multidimensional CF fusion approach for top-N item recommendations. The proposed approach is capable of incorporating not only localized relations of user-user and item-item but also latent interaction between all dimensions of the data. Experimental results show significant improvements by the proposed approach in terms of recommendation accuracy.

Linear dimensionality reduction: Survey, insights, and generalizations

© 2015 John P. Cunningham and Zoubin Ghahramani. Linear dimensionality reduction methods are a cornerstone of analyzing high dimensional data, due to their simple geometric interpretations and typically attractive computational properties. These methods capture many data features of interest, such as covariance, dynamical structure, correlation between data sets, input-output relationships, and margin between data classes. Methods have been developed with a variety of names and motivations in many fields, and perhaps as a result the connections between all these methods have not been highlighted. Here we survey methods from this disparate literature as optimization programs over matrix manifolds. We discuss principal component analysis, factor analysis, linear multidimensional scaling, Fisher's linear discriminant analysis, canonical correlations analysis, maximum autocorrelation factors, slow feature analysis, sufficient dimensionality reduction, undercomplete independent component analysis, linear regression, distance metric learning, and more. This optimization framework gives insight to some rarely discussed shortcomings of well-known methods, such as the suboptimality of certain eigenvector solutions. Modern techniques for optimization over matrix manifolds enable a generic linear dimensionality reduction solver, which accepts as input data and an objective to be optimized, and returns, as output, an optimal low-dimensional projection of the data. This simple optimization framework further allows straightforward generalizations and novel variants of classical methods, which we demonstrate here by creating an orthogonal-projection canonical correlations analysis. More broadly, this survey and generic solver suggest that linear dimensionality reduction can move toward becoming a blackbox, objective-agnostic numerical technology.

Semisupervised Dimensionality Reduction and Classification Through Virtual Label Regression

Semisupervised dimensionality reduction has been attracting much attention as it not only utilizes both labeled and unlabeled data simultaneously, but also works well in the situation of out-of-sample. This paper proposes an effective approach of semisupervised dimensionality reduction through label propagation and label regression. Different from previous efforts, the new approach propagates the label information from labeled to unlabeled data with a well-designed mechanism of random walks, in which outliers are effectively detected and the obtained virtual labels of unlabeled data can be well encoded in a weighted regression model. These virtual labels are thereafter regressed with a linear model to calculate the projection matrix for dimensionality reduction. By this means, when the manifold or the clustering assumption of data is satisfied, the labels of labeled data can be correctly propagated to the unlabeled data; and thus, the proposed approach utilizes the labeled and the unlabeled data more effectively than previous work. Experimental results are carried out upon several databases, and the advantage of the new approach is well demonstrated.

Supervised Gaussian Process Latent Variable Model for Dimensionality Reduction

The Gaussian process latent variable model (GP-LVM) has been identified to be an effective probabilistic approach for dimensionality reduction because it can obtain a low-dimensional manifold of a data set in an unsupervised fashion. Consequently, the GP-LVM is insufficient for supervised learning tasks (e. g., classification and regression) because it ignores the class label information for dimensionality reduction. In this paper, a supervised GP-LVM is developed for supervised learning tasks, and the maximum a posteriori algorithm is introduced to estimate positions of all samples in the latent variable space. We present experimental evidences suggesting that the supervised GP-LVM is able to use the class label information effectively, and thus, it outperforms the GP-LVM and the discriminative extension of the GP-LVM consistently. The comparison with some supervised classification methods, such as Gaussian process classification and support vector machines, is also given to illustrate the advantage of the proposed method.

Semantics-Preserving Dimensionality Reduction: Rough and Fuzzy-Rough-Based Approaches

R. Jensen and Q. Shen. Semantics-Preserving Dimensionality Reduction: Rough and Fuzzy-Rough Based Approaches. IEEE Transactions on Knowledge and Data Engineering, 16(12): 1457-1471. 2004.

Temporal extension of Laplacian Eigenmaps for unsupervised dimensionality reduction of time series

A novel non-linear dimensionality reduction method, called Temporal Laplacian Eigenmaps, is introduced to process efficiently time series data. In this embedded-based approach, temporal information is intrinsic to the objective function, which produces description of low dimensional spaces with time coherence between data points. Since the proposed scheme also includes bidirectional mapping between data and embedded spaces and automatic tuning of key parameters, it offers the same benefits as mapping-based approaches. Experiments on a couple of computer vision applications demonstrate the superiority of the new approach to other dimensionality reduction method in term of accuracy. Moreover, its lower computational cost and generalisation abilities suggest it is scalable to larger datasets. © 2010 IEEE.

Using Recurrent Networks for Dimensionality Reduction

This report explores how recurrent neural networks can be exploited for learning high-dimensional mappings. Since recurrent networks are as powerful as Turing machines, an interesting question is how recurrent networks can be used to simplify the problem of learning from examples. The main problem with learning high-dimensional functions is the curse of dimensionality which roughly states that the number of examples needed to learn a function increases exponentially with input dimension. This thesis proposes a way of avoiding this problem by using a recurrent network to decompose a high-dimensional function into many lower dimensional functions connected in a feedback loop.

Understanding Subsystems in Biology through Dimensionality Reduction, Graph Partitioning and Analytical Modeling

Biological systems exhibit rich and complex behavior through the orchestrated interplay of a large array of components. It is hypothesized that separable subsystems with some degree of functional autonomy exist; deciphering their independent behavior and functionality would greatly facilitate understanding the system as a whole. Discovering and analyzing such subsystems are hence pivotal problems in the quest to gain a quantitative understanding of complex biological systems. In this work, using approaches from machine learning, physics and graph theory, methods for the identification and analysis of such subsystems were developed. A novel methodology, based on a recent machine learning algorithm known as non-negative matrix factorization (NMF), was developed to discover such subsystems in a set of large-scale gene expression data. This set of subsystems was then used to predict functional relationships between genes, and this approach was shown to score significantly higher than conventional methods when benchmarking them against existing databases. Moreover, a mathematical treatment was developed to treat simple network subsystems based only on their topology (independent of particular parameter values). Application to a problem of experimental interest demonstrated the need for extentions to the conventional model to fully explain the experimental data. Finally, the notion of a subsystem was evaluated from a topological perspective. A number of different protein networks were examined to analyze their topological properties with respect to separability, seeking to find separable subsystems. These networks were shown to exhibit separability in a nonintuitive fashion, while the separable subsystems were of strong biological significance. It was demonstrated that the separability property found was not due to incomplete or biased data, but is likely to reflect biological structure.

Comparing methods for dimensionality reduction when data are density functions

Functional Data Analysis (FDA) deals with samples where a whole function is observed for each individual. A particular case of FDA is when the observed functions are density functions, that are also an example of infinite dimensional compositional data. In this work we compare several methods for dimensionality reduction for this particular type of data: functional principal components analysis (PCA) with or without a previous data transformation and multidimensional scaling (MDS) for diferent inter-densities distances, one of them taking into account the compositional nature of density functions. The difeerent methods are applied to both artificial and real data (households income distributions)

Dimensionality reduction assisted tensor clustering

This paper is concerned with tensor clustering with the assistance of dimensionality reduction approaches. A class of formulation for tensor clustering is introduced based on tensor Tucker decomposition models. In this formulation, an extra tensor mode is formed by a collection of tensors of the same dimensions and then used to assist a Tucker decomposition in order to achieve data dimensionality reduction. We design two types of clustering models for the tensors: PCA Tensor Clustering model and Non-negative Tensor Clustering model, by utilizing different regularizations. The tensor clustering can thus be solved by the optimization method based on the alternative coordinate scheme. Interestingly, our experiments show that the proposed models yield comparable or even better performance compared to most recent clustering algorithms based on matrix factorization.

Semi-supervised dimensionality reduction based on partial least squares for visual analysis of high dimensional data

Dimensionality reduction is employed for visual data analysis as a way to obtaining reduced spaces for high dimensional data or to mapping data directly into 2D or 3D spaces. Although techniques have evolved to improve data segregation on reduced or visual spaces, they have limited capabilities for adjusting the results according to user's knowledge. In this paper, we propose a novel approach to handling both dimensionality reduction and visualization of high dimensional data, taking into account user's input. It employs Partial Least Squares (PLS), a statistical tool to perform retrieval of latent spaces focusing on the discriminability of the data. The method employs a training set for building a highly precise model that can then be applied to a much larger data set very effectively. The reduced data set can be exhibited using various existing visualization techniques. The training data is important to code user's knowledge into the loop. However, this work also devises a strategy for calculating PLS reduced spaces when no training data is available. The approach produces increasingly precise visual mappings as the user feeds back his or her knowledge and is capable of working with small and unbalanced training sets.