Gravitational Dimensionality Reduction Using Newtonian Gravity and Einstein's General Relativity

Supervised dimensionality reduction strategies have been of great interest. However, current supervised dimensionality reduction approaches are difficult to scale for situations characterized by large datasets given the high computational complexities associated with such methods. While stochastic approximation strategies have been explored for unsupervised dimensionality reduction to tackle this challenge, such approaches are not well-suited for accelerating computational sp...

Often the relation between the variables constituting a multivariate data space might be characterized by one or more of the terms: ``nonlinear'', ``branched'', ``disconnected'', ``bended'', ``curved'', ``heterogeneous'', or, more general, ``complex''. In these cases, simple principal component analysis (PCA) as a tool for dimension reduction can fail badly. Of the many alternative approaches proposed so far, local approximations of PCA are among the most promising. This pape...

The central goal of this paper is to establish two commonly available dimensionality reduction (DR) methods i.e. t-distributed Stochastic Neighbor Embedding (t-SNE) and Multidimensional Scaling (MDS) in Matlab and to observe their application in several datasets. These DR techniques are applied to nine different datasets namely CNAE9, Segmentation, Seeds, Pima Indians diabetes, Parkinsons, Movement Libras, Mammographic Masses, Knowledge, and Ionosphere acquired from UCI machi...

We propose a novel algorithm for supervised dimensionality reduction named Manifold Partition Discriminant Analysis (MPDA). It aims to find a linear embedding space where the within-class similarity is achieved along the direction that is consistent with the local variation of the data manifold, while nearby data belonging to different classes are well separated. By partitioning the data manifold into a number of linear subspaces and utilizing the first-order Taylor expansion...

Dimensional reduction~(DR) maps high-dimensional data into a lower dimensions latent space with minimized defined optimization objectives. The DR method usually falls into feature selection~(FS) and feature projection~(FP). FS focuses on selecting a critical subset of dimensions but risks destroying the data distribution (structure). On the other hand, FP combines all the input features into lower dimensions space, aiming to maintain the data structure; but lacks interpretabi...

Visualizing high dimensional data by projecting them into two or three dimensional space is one of the most effective ways to intuitively understand the data's underlying characteristics, for example their class neighborhood structure. While data visualization in low dimensional space can be efficient for revealing the data's underlying characteristics, classifying a new sample in the reduced-dimensional space is not always beneficial because of the loss of information in exp...

This survey is written in summer, 2016. The purpose of this survey is to briefly introduce nonlinear dimensionality reduction (NLDR) in data reduction. The first two NLDR were respectively published in Science in 2000 in which they solve the similar reduction problem of high-dimensional data endowed with the intrinsic nonlinear structure. The intrinsic nonlinear structure is always interpreted as a concept in manifolds from geometry and topology in theoretical mathematics by ...

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 reduct...

Manifold learning-based encoders have been playing important roles in nonlinear dimensionality reduction (NLDR) for data exploration. However, existing methods can often fail to preserve geometric, topological and/or distributional structures of data. In this paper, we propose a deep manifold learning framework, called deep manifold transformation (DMT) for unsupervised NLDR and embedding learning. DMT enhances deep neural networks by using cross-layer local geometry-preservi...

Distance metric learning can be viewed as one of the fundamental interests in pattern recognition and machine learning, which plays a pivotal role in the performance of many learning methods. One of the effective methods in learning such a metric is to learn it from a set of labeled training samples. The issue of data imbalance is the most important challenge of recent methods. This research tries not only to preserve the local structures but also covers the issue of imbalanc...