Projection Pursuit is a classic exploratory technique for finding interesting projections of a dataset. We propose a method for recovering projections containing either Imbalanced Clusters or a Bernoulli-Rademacher distribution using a gradient-based technique to optimize the projection index. As sample complexity is a major limiting factor in Projection Pursuit, we analyze our algorithm's sample complexity within a Planted Vector setting where we can observe that Imbalanced Clusters can be recovered more easily than balanced ones. Additionally, we give a generalized result that works for a variety of data distributions and projection indices. We compare these results to computational lower bounds in the Low-Degree-Polynomial Framework. Finally, we experimentally evaluate our method's applicability to real-world data using FashionMNIST and the Human Activity Recognition Dataset, where our algorithm outperforms others when only a few samples are available.

We propose a projection pursuit (PP) algorithm based on Gaussian mixture models (GMMs). The negentropy obtained from a multivariate density estimated by GMMs is adopted as the PP index to be maximised. For a fixed dimension of the projection subspace, the GMM-based density estimation is projected onto that subspace, where an approximation of the negentropy for Gaussian mixtures is computed. Then, Genetic Algorithms (GAs) are used to find the optimal, orthogonal projection basis by maximising the former approximation. We show that this semi-parametric approach to PP is flexible and allows highly informative structures to be detected, by projecting multivariate datasets onto a subspace, where the data can be feasibly visualised. The performance of the proposed approach is shown on both artificial and real datasets.

In this paper, we explore the limitations of PCA as a dimension reduction technique and study its extension, projection pursuit (PP), which is a broad class of linear dimension reduction methods. PCA is a popular dimension reduction technique commonly applied to scRNA sequencing data. Despite of huge success in practice, we will illustrate three drawbacks of PCA. It is well known that the eigenvalues of sample covariance matrix is not consistent in high dimensional cases. Every principal component is uncorrelated with each other but not independent.

Prediction accuracy and model explainability are the two most important objectives when developing machine learning algorithms to solve real-world problems. The neural networks are known to possess good prediction performance, but lack of sufficient model explainability. In this paper, we propose to enhance the explainability of neural networks through the following architecture constraints: a) sparse additive subnetworks; b) orthogonal projection pursuit; and c) smooth function approximation. It leads to a sparse, orthogonal and smooth explainable neural network (SOSxNN). The multiple parameters in the SOSxNN model are simultaneously estimated by a modified mini-batch gradient descent algorithm based on the backpropagation technique for calculating the derivatives and the Cayley transform for preserving the projection orthogonality. The hyperparameters controlling the sparse and smooth constraints are optimized by the grid search. Through simulation studies, we compare the SOSxNN method to several benchmark methods including least absolute shrinkage and selection operator, support vector machine, random forest, and multi-layer perceptron. It is shown that proposed model keeps the flexibility of pursuing prediction accuracy while attaining the improved interpretability, which can be therefore used as a promising surrogate model for complex model approximation. Finally, the real data example from the Lending Club is employed as a showcase of the SOSxNN application.

Associating distinct groups of objects (clusters) with contiguous regions of high probability density (high-density clusters), is central to many statistical and machine learning approaches to the classification of unlabelled data. We propose a novel hyperplane classifier for clustering and semi-supervised classification which is motivated by this objective. The proposed minimum density hyperplane minimises the integral of the empirical probability density function along it, thereby avoiding intersection with high density clusters. We show that the minimum density and the maximum margin hyperplanes are asymptotically equivalent, thus linking this approach to maximum margin clustering and semi-supervised support vector classifiers. We propose a projection pursuit formulation of the associated optimisation problem which allows us to find minimum density hyperplanes efficiently in practice, and evaluate its performance on a range of benchmark data sets. The proposed approach is found to be very competitive with state of the art methods for clustering and semi-supervised classification.

Product models of low dimensional experts are a powerful way to avoid the curse of dimensionality. We present the ``under-complete product of experts' (UPoE), where each expert models a one dimensional projection of the data. The UPoE is fully tractable and may be interpreted as a parametric probabilistic model for projection pursuit. Its ML learning rules are identical to the approximate learning rules proposed before for under-complete ICA. We also derive an efficient sequential learning algorithm and discuss its relationship to projection pursuit density estimation and feature induction algorithms for additive random field models.

In this paper, we propose a new algorithm for exploratory projection pursuit. The basis of the algorithm is the insight that previous approaches used fairly narrow definitions of interestingness / non interestingness. We argue that allowing these definitions to depend on the problem / data at hand is a more natural approach in an exploratory technique. This also allows our technique much greater applicability than the approaches extant in the literature. Complementing this insight, we propose a class of projection indices based on the spatial distribution function that can make use of such information. Finally, with the help of real datasets, we demonstrate how a range of multivariate exploratory tasks can be addressed with our algorithm. The examples further demonstrate that the proposed indices are quite capable of focussing on the interesting structure in the data, even when this structure is otherwise hard to detect or arises from very subtle patterns.

Our efforts proceed in the context of a problem suggested by the operational needs of a particular electric utility to make daily forecasts of short-term load or demand. Forecasts are made at midday (1 p.m.) on a weekday t ( Monday - Thursday), for the next evening peak e(t) (occuring usually about 8 p.m. in the winter), the daily minimum d(t

Our efforts proceed in the context of a problem suggested by the operational needs of a particular electric utility to make daily forecasts of short-term load or demand. Forecasts are made at midday (1 p.m.) on a weekday t ( Monday - Thursday), for the next evening peak e(t) (occuring usually about 8 p.m. in the winter), the daily minimum d(t

Some improved generalization properties are demonstrat.ed