A Statistically and Numerically Efficient Independence Test Based on Random Projections and Distance Covariance

Testing for independence plays a fundamental role in many statistical techniques. Among the nonparametric approaches, the distance-based methods (such as the distance correlation-based hypotheses testing for independence) have many advantages, compared with many other alternatives. A known limitation of the distance-based method is that its computational complexity can be high. In general, when the sample size is n, the order of computational complexity of a distance-based method, which typically requires computing of all pairwise distances, can be O(n2). Recent advances have discovered that in the univariate cases, a fast method with O(n log n) computational complexity and O(n) memory requirement exists. In this paper, we introduce a test of independence method based on random projection and distance correlation, which achieves nearly the same power as the state-of-the-art distance-based approach, works in the multivariate cases, and enjoys the O(nK log n) computational complexity and O( max{n, K}) memory requirement, where K is the number of random projections. Note that saving is achieved when K < n/ log n. We name our method a Randomly Projected Distance Covariance (RPDC). The statistical theoretical analysis takes advantage of some techniques on the random projection which are rooted in contemporary machine learning. Numerical experiments demonstrate the efficiency of the proposed method, relative to numerous competitors.

On the Generalization of the Random Projection Method for Problems of the Recovery of Object Signal Described by Models of Convolution Type

Introduction. In technical systems, there is a common situation when transformation input-output is described by the integral equation of convolution type. This situation accurses if the object signal is recovered by the results of remote measurements. For example, in spectrometric tasks, for an image deblurring, etc. Matrices of the discrete representation for the output signal and the kernel of convolution are known. We need to find a matrix of the discrete representation of a signal of the object. The well known approach for solving this problem includes the next steps. First, the kernel matrix has to be represented as the Kroneker product. Second, the input-output transformation has to be presented with the usage of Kroneker product matrices. Third, the matrix of the discrete representation of the object has to be found. The object signal matrix estimation obtained with the help of pseudo inverting of Kroneker decomposition matrices is unstable. The instability of the object signal estimation in the case of usage of Kroneker decomposition matrices is caused by their discrete ill posed matrix properties (condition number is big and the series of the singular numbers smoothly decrease to zero). To find solutions of discrete ill-posed problems we developed methods based on the random projection and the random projection with an averaging by the random matrices. These methods provide a stable solutions with a small computational complexity. We consider the problem of object signals recovering in the systems where an input-output transformation is described by the integral equation of a convolution. To find a solution for these problems we need to build a generalization for two-dimensional signals case of the random projection method. Purpose. To develop a stable method of the recovery of object signal for the case in which an input-output transformation is described by the integral equation of a convolution. Results and conclusions. We developed the method of a stable recovery of object signal for the case in which an input-output transformation is described by the integral equation of a convolution. The stable estimation of the object signal is provided by Kroneker decomposition of the kernel matrix of convolution, computation of random projections for Kroneker factorization matrices, and a selection of the optimal dimension of a projector matrix. The method is illustrated by its application in technical problems.

Recognition of Power Equipment Based on Multitask Sparse Representation

Image analysis of power equipment has important practical significance for power-line inspection and maintenance. This paper proposes an image recognition method for power equipment based on multitask sparse representation. In the feature extraction stage, based on the two-dimensional (2D) random projection algorithm, multiple projection matrices are constructed to obtain the multilevel features of the image. In the classification process, considering that the image acquisition process will inevitably be affected by factors such as light conditions and noise interference, the proposed method uses the multitask compressive sensing algorithm (MtCS) to jointly represent multiple feature vectors to improve the accuracy and robustness of reconstruction. In the experiment, the images of three types of typical power equipment of insulators, transformers, and circuit breakers are classified. The correct recognition rate of the proposed method reaches 94.32%. In addition, the proposed method can maintain strong robustness under the conditions of noise interference and partial occlusion, which further verifies its effectiveness.

Randomized Projection Learning Method for Dynamic Mode Decomposition

A data-driven analysis method known as dynamic mode decomposition (DMD) approximates the linear Koopman operator on a projected space. In the spirit of Johnson–Lindenstrauss lemma, we will use a random projection to estimate the DMD modes in a reduced dimensional space. In practical applications, snapshots are in a high-dimensional observable space and the DMD operator matrix is massive. Hence, computing DMD with the full spectrum is expensive, so our main computational goal is to estimate the eigenvalue and eigenvectors of the DMD operator in a projected domain. We generalize the current algorithm to estimate a projected DMD operator. We focus on a powerful and simple random projection algorithm that will reduce the computational and storage costs. While, clearly, a random projection simplifies the algorithmic complexity of a detailed optimal projection, as we will show, the results can generally be excellent, nonetheless, and the quality could be understood through a well-developed theory of random projections. We will demonstrate that modes could be calculated for a low cost by the projected data with sufficient dimension.

Dimensionality reduction for k-distance applied to persistent homology

Given a set P of n points and a constant k, we are interested in computing the persistent homology of the Čech filtration of P for the k-distance, and investigate the effectiveness of dimensionality reduction for this problem, answering an open question of Sheehy (The persistent homology of distance functions under random projection. In Cheng, Devillers (eds), 30th Annual Symposium on Computational Geometry, SOCG’14, Kyoto, Japan, June 08–11, p 328, ACM, 2014). We show that any linear transformation that preserves pairwise distances up to a $$(1\pm {\varepsilon })$$ ( 1 ± ε ) multiplicative factor, must preserve the persistent homology of the Čech filtration up to a factor of $$(1-{\varepsilon })^{-1}$$ ( 1 - ε ) - 1 . Our results also show that the Vietoris-Rips and Delaunay filtrations for the k-distance, as well as the Čech filtration for the approximate k-distance of Buchet et al. [J Comput Geom, 58:70–96, 2016] are preserved up to a $$(1\pm {\varepsilon })$$ ( 1 ± ε ) factor. We also prove extensions of our main theorem, for point sets (i) lying in a region of bounded Gaussian width or (ii) on a low-dimensional submanifold, obtaining embeddings having the dimension bounds of Lotz (Proc R Soc A Math Phys Eng Sci, 475(2230):20190081, 2019) and Clarkson (Tighter bounds for random projections of manifolds. In Teillaud (ed) Proceedings of the 24th ACM Symposium on Computational Geom- etry, College Park, MD, USA, June 9–11, pp 39–48, ACM, 2008) respectively. Our results also work in the terminal dimensionality reduction setting, where the distance of any point in the original ambient space, to any point in P, needs to be approximately preserved.

Fischer Linear Discrimination and Quadratic Discrimination Analysis–Based Data Mining Technique for Internet of Things Framework for Healthcare

The internet of reality or augmented reality has been considered a breakthrough and an outstanding critical mutation with an emphasis on data mining leading to dismantling of some of its assumptions among several of its stakeholders. In this work, we study the pillars of these technologies connected to web usage as the Internet of things (IoT) system's healthcare infrastructure. We used several data mining techniques to evaluate the online advertisement data set, which can be categorized as high dimensional with 1,553 attributes, and the imbalanced data set, which automatically simulates an IoT discrimination problem. The proposed methodology applies Fischer linear discrimination analysis (FLDA) and quadratic discrimination analysis (QDA) within random projection (RP) filters to compare our runtime and accuracy with support vector machine (SVM), K-nearest neighbor (KNN), and Multilayer perceptron (MLP) in IoT-based systems. Finally, the impact on number of projections was practically experimented, and the sensitivity of both FLDA and QDA with regard to precision and runtime was found to be challenging. The modeling results show not only improved accuracy, but also runtime improvements. When compared with SVM, KNN, and MLP in QDA and FLDA, runtime shortens by 20 times in our chosen data set simulated for a healthcare framework. The RP filtering in the preprocessing stage of the attribute selection, fulfilling the model's runtime, is a standpoint in the IoT industry.Index Terms: Data Mining, Random Projection, Fischer Linear Discriminant Analysis, Online Advertisement Dataset, Quadratic Discriminant Analysis, Feature Selection, Internet of Things.