Data forms the foundation of any machine learning algorithm, without it, Data Science can not happen. Sometimes, it can contain a huge number of features, some of which are not even required. Such redundant information makes modeling complicated. Furthermore, interpreting and understanding the data by visualization gets difficult because of the high dimensionality. This is where dimensionality reduction comes into play. Dimensionality reduction is the task of reducing the number of features in a dataset. In machine learning tasks like regression or classification, there are often too many variables to work with. These variables are also called features.
This is a tutorial and survey paper on unification of spectral dimensionality reduction methods, kernel learning by Semidefinite Programming (SDP), Maximum Variance Unfolding (MVU) or Semidefinite Embedding (SDE), and its variants. We first explain how the spectral dimensionality reduction methods can be unified as kernel Principal Component Analysis (PCA) with different kernels. This unification can be interpreted as eigenfunction learning or representation of kernel in terms of distance matrix. Then, since the spectral methods are unified as kernel PCA, we say let us learn the best kernel for unfolding the manifold of data to its maximum variance. We first briefly introduce kernel learning by SDP for the transduction task. Then, we explain MVU in detail. Various versions of supervised MVU using nearest neighbors graph, by class-wise unfolding, by Fisher criterion, and by colored MVU are explained. We also explain out-of-sample extension of MVU using eigenfunctions and kernel mapping. Finally, we introduce other variants of MVU including action respecting embedding, relaxed MVU, and landmark MVU for big data.
An analysis of high dimensional data can offer a detailed description of a system but is often challenged by the curse of dimensionality. General dimensionality reduction techniques can alleviate such difficulty by extracting a few important features, but they are limited due to the lack of interpretability and connectivity to actual decision making associated with each physical variable. Important variable selection techniques, as an alternative, can maintain the interpretability, but they often involve a greedy search that is susceptible to failure in capturing important interactions. This research proposes a new method that produces subspaces, reduced-dimensional physical spaces, based on a randomized search and forms an ensemble of models for critical subspaces. When applied to high-dimensional data collected from a composite metal development process, the proposed method shows its superiority in prediction and important variable selection.
In this paper, we investigate performing joint dimensionality reduction and classification using a novel histogram neural network. Motivated by a popular dimensionality reduction approach, t-Distributed Stochastic Neighbor Embedding (t-SNE), our proposed method incorporates a classification loss computed on samples in a low-dimensional embedding space. We compare the learned sample embeddings against coordinates found by t-SNE in terms of classification accuracy and qualitative assessment. We also explore use of various divergence measures in the t-SNE objective. The proposed method has several advantages such as readily embedding out-of-sample points and reducing feature dimensionality while retaining class discriminability. Our results show that the proposed approach maintains and/or improves classification performance and reveals characteristics of features produced by neural networks that may be helpful for other applications.
Dimensionality reduction is an unsupervised learning technique. Nevertheless, it can be used as a data transform pre-processing step for machine learning algorithms on classification and regression predictive modeling datasets with supervised learning algorithms. There are many dimensionality reduction algorithms to choose from and no single best algorithm for all cases. Instead, it is a good idea to explore a range of dimensionality reduction algorithms and different configurations for each algorithm. In this tutorial, you will discover how to fit and evaluate top dimensionality reduction algorithms in Python.
Recent work has shown the tremendous vulnerability to adversarial samples that are nearly indistinguishable from benign data but are improperly classified by the deep learning model. Some of the latest findings suggest the existence of adversarial attacks may be an inherent weakness of these models as a direct result of its sensitivity to well-generalizing features in high dimensional data. We hypothesize that data transformations can influence this vulnerability since a change in the data manifold directly determines the adversary's ability to create these adversarial samples. To approach this problem, we study the effect of dimensionality reduction through the lens of adversarial robustness. This study raises awareness of the positive and negative impacts of five commonly used data transformation techniques on adversarial robustness. The evaluation shows how these techniques contribute to an overall increased vulnerability where accuracy is only improved when the dimensionality reduction technique approaches the data's optimal intrinsic dimension. The conclusions drawn from this work contribute to understanding and creating more resistant learning models.
This work develops an analytically solvable unsupervised learning scheme that extracts the most informative components for predicting future inputs, termed predictive principal component analysis (PredPCA). Our scheme can effectively remove unpredictable observation noise and globally minimize the test prediction error. Mathematical analyses demonstrate that, with sufficiently high-dimensional observations that are generated by a linear or nonlinear system, PredPCA can identify the optimal hidden state representation, true system parameters, and true hidden state dimensionality, with a global convergence guarantee. We demonstrate the performance of PredPCA by using sequential visual inputs comprising hand-digits, rotating 3D objects, and natural scenes. It reliably and accurately estimates distinct hidden states and predicts future outcomes of previously unseen test input data, even in the presence of considerable observation noise. The simple model structure and low computational cost of PredPCA make it highly desirable as a learning scheme for biological neural networks and neuromorphic chips. Prediction is essential for both biological organisms [1,2] and machine learning [3,4]. In particular, they need to predict the dynamics of newly encountered sensory input data (i.e., test data) based on and only on knowledge learned from a limited number of past experiences (i.e., training data). Generalization error is a standard measure of the generalization capability of predicting the future consequences of previously unseen input data, which is defined as the difference between the training and test prediction errors.
Let's starts with the WHY we need to perform Dimensionality Reduction before analyzing data and coming down to some inferences, it is often necessary to visualize the data set, in order to get an idea of it. But, nowadays data sets contain a lot of random variables (also called features) due to which it becomes difficult in visualizing the data set. Sometimes it is even impossible to visualize such high dimensional data as we humans fall astray after we reach a dimension higher than 3. Here is where we come across dimensionality reduction. The process of reducing the number of random variables of the data set under consideration, via obtaining a set of principal variables.
Over the past few decades, we have witnessed a large family of algorithms that have been designed to provide different solutions to the problem of dimensionality reduction (DR). The DR is an essential tool to excavate the important information from the high-dimensional data by mapping the data to a low-dimensional subspace. Furthermore, for the diversity of varied high-dimensional data, the multi-view features can be utilized for improving the learning performance. However, many DR methods fail to integrating multiple views. Although the features from different views are extracted by different manners, they are utilized to describe the same sample, which implies that they are highly related. Therefore, how to learn the subspace for high-dimensional features by utilizing the consistency and complementary properties of multi-view features is important in the present. In this paper, we propose an effective multi-view dimensionality reduction algorithm named Multi-view Smooth Preserve Projection. Firstly, we construct a single view DR method named Smooth Preserve Projection based on the Smooth Representation model. The proposed method aims to find a subspace for the high-dimensional data, in which the smooth reconstructive weights are preserved as much as possible. Then, we extend it to a multi-view version in which we exploits Hilbert-Schmidt Independence Criterion to jointly learn one common subspace for all views. A plenty of experiments on multi-view datasets show the excellent performance of the proposed method.
Dimensionality reduction is a common method for analyzing and visualizing high-dimensional data. However, reasoning dynamically about the results of a dimensionality reduction is difficult. Dimensionality-reduction algorithms use complex optimizations to reduce the number of dimensions of a dataset, but these new dimensions often lack a clear relation to the initial data dimensions, thus making them difficult to interpret. Here we propose a visual interaction framework to improve dimensionality-reduction based exploratory data analysis. We introduce two interaction techniques, forward projection and backward projection, for dynamically reasoning about dimensionally reduced data. We also contribute two visualization techniques, prolines and feasibility maps, to facilitate the effective use of the proposed interactions. We apply our framework to PCA and autoencoder-based dimensionality reductions. Through data-exploration examples, we demonstrate how our visual interactions can improve the use of dimensionality reduction in exploratory data analysis.