Supervised Manifold Learning via Random Forest Geometry-Preserving Proximities

Manifold learning approaches seek the intrinsic, low-dimensional data structure within a high-dimensional space. Mainstream manifold learning algorithms, such as Isomap, UMAP, $t$-SNE, Diffusion Map, and Laplacian Eigenmaps do not use data labels and are thus considered unsupervised. Existing supervised extensions of these methods are limited to classification problems and fall short of uncovering meaningful embeddings due to their construction using order non-preserving, class-conditional distances. In this paper, we show the weaknesses of class-conditional manifold learning quantitatively and visually and propose an alternate choice of kernel for supervised dimensionality reduction using a data-geometry-preserving variant of random forest proximities as an initialization for manifold learning methods. We show that local structure preservation using these proximities is near universal across manifold learning approaches and global structure is properly maintained using diffusion-based algorithms.

Geometry- and Accuracy-Preserving Random Forest Proximities

Random forests are considered one of the best out-of-the-box classification and regression algorithms due to their high level of predictive performance with relatively little tuning. Pairwise proximities can be computed from a trained random forest which measure the similarity between data points relative to the supervised task. Random forest proximities have been used in many applications including the identification of variable importance, data imputation, outlier detection, and data visualization. However, existing definitions of random forest proximities do not accurately reflect the data geometry learned by the random forest. In this paper, we introduce a novel definition of random forest proximities called Random Forest-Geometry- and Accuracy-Preserving proximities (RF-GAP). We prove that the proximity-weighted sum (regression) or majority vote (classification) using RF-GAP exactly match the out-of-bag random forest prediction, thus capturing the data geometry learned by the random forest. We empirically show that this improved geometric representation outperforms traditional random forest proximities in tasks such as data imputation and provides outlier detection and visualization results consistent with the learned data geometry.

Supervised Visualization for Data Exploration

Dimensionality reduction is often used as an initial step in data exploration, either as preprocessing for classification or regression or for visualization. Most dimensionality reduction techniques to date are unsupervised; they do not take class labels into account (e.g., PCA, MDS, t-SNE, Isomap). Such methods require large amounts of data and are often sensitive to noise that may obfuscate important patterns in the data. Various attempts at supervised dimensionality reduction methods that take into account auxiliary annotations (e.g., class labels) have been successfully implemented with goals of increased classification accuracy or improved data visualization. Many of these supervised techniques incorporate labels in the loss function in the form of similarity or dissimilarity matrices, thereby creating over-emphasized separation between class clusters, which does not realistically represent the local and global relationships in the data. In addition, these approaches are often sensitive to parameter tuning, which may be difficult to configure without an explicit quantitative notion of visual superiority. In this paper, we describe a novel supervised visualization technique based on random forest proximities and diffusion-based dimensionality reduction. We show, both qualitatively and quantitatively, the advantages of our approach in retaining local and global structures in data, while emphasizing important variables in the low-dimensional embedding. Importantly, our approach is robust to noise and parameter tuning, thus making it simple to use while producing reliable visualizations for data exploration.