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Principal Component Analysis


Understanding Principal Component Analysis - GreatLearning

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While working on different Machine Learning techniques for Data Analysis, we deal with hundreds or thousands of variables. Most of the variables are correlated with each other. Principal Component Analysis and Factor Analysis techniques are used to deal with such scenarios. Principal Component Analysis (PCA) is an unsupervised statistical technique algorithm. PCA is a "dimensionality reduction" method.


What is Principal Component Analysis in Machine Learning? Super Easy!

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Do you wanna know What is Principal Component Analysis?. If yes, then this blog is just for you. Here I will discuss What is Principal Component Analysis, its purpose, and How PCA works?. So, give your few minutes to this article in order to get all the details regarding Principal Component Analysis. Principal Component Analysis(PCA) is one of the best-unsupervised algorithms.


An overview of Principal Component Analysis

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This article will explain you what Principal Component Analysis (PCA) is, why we need it and how we use it. I will try to make it as simple as possible while avoiding hard examples or words which can cause a headache. A moment of honesty: to fully understand this article, a basic understanding of some linear algebra and statistics is essential. Let's say we have 10 variables in our dataset and let's assume that 3 variables capture 90% of the dataset, and 7 variables capture 10% of the dataset. Let's say we want to visualize 10 variables.


Distributed Estimation for Principal Component Analysis: a Gap-free Approach

arXiv.org Machine Learning

The growing size of modern data sets brings many challenges to the existing statistical estimation approaches, which calls for new distributed methodologies. This paper studies distributed estimation for a fundamental statistical machine learning problem, principal component analysis (PCA). Despite the massive literature on top eigenvector estimation, much less is presented for the top-$L$-dim ($L > 1$) eigenspace estimation, especially in a distributed manner. We propose a novel multi-round algorithm for constructing top-$L$-dim eigenspace for distributed data. Our algorithm takes advantage of shift-and-invert preconditioning and convex optimization. Our estimator is communication-efficient and achieves a fast convergence rate. In contrast to the existing divide-and-conquer algorithm, our approach has no restriction on the number of machines. Theoretically, we establish a gap-free error bound and abandon the assumption on the sharp eigengap between the $L$-th and the ($L+1$)-th eigenvalues. Our distributed algorithm can be applied to a wide range of statistical problems based on PCA. In particular, this paper illustrates two important applications, principal component regression and single index model, where our distributed algorithm can be extended. Finally, We provide simulation studies to demonstrate the performance of the proposed distributed estimator.


Machine Learning in Python: Principal Component Analysis (PCA) for Handling High-Dimensional Data

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Machine Learning in Python: Principal Component Analysis (PCA) for Handling High-Dimensional Data In this video, I will be showing you how to perform principal component analysis (PCA) in Python using the scikit-learn package. PCA represents a powerful learning approach that enables the analysis of high-dimensional data as well as reveal the contribution of descriptors in governing the distribution of data clusters. Particularly, we will be creating PCA scree plot, scores plot and loadings plot. This video is part of the [Python Data Science Project] series. If you're new here, it would mean the world to me if you would consider subscribing to this channel.


Manifold denoising by Nonlinear Robust Principal Component Analysis

Neural Information Processing Systems

This paper extends robust principal component analysis (RPCA) to nonlinear manifolds. Suppose that the observed data matrix is the sum of a sparse component and a component drawn from some low dimensional manifold. Is it possible to separate them by using similar ideas as RPCA? Is there any benefit in treating the manifold as a whole as opposed to treating each local region independently? We answer these two questions affirmatively by proposing and analyzing an optimization framework that separates the sparse component from the manifold under noisy data.


Robust Principal Component Analysis with Adaptive Neighbors

Neural Information Processing Systems

Suppose certain data points are overly contaminated, then the existing principal component analysis (PCA) methods are frequently incapable of filtering out and eliminating the excessively polluted ones, which potentially lead to the functional degeneration of the corresponding models. To tackle the issue, we propose a general framework namely robust weight learning with adaptive neighbors (RWL-AN), via which adaptive weight vector is automatically obtained with both robustness and sparse neighbors. More significantly, the degree of the sparsity is steerable such that only exact k well-fitting samples with least reconstruction errors are activated during the optimization, while the residual samples, i.e., the extreme noised ones are eliminated for the global robustness. Additionally, the framework is further applied to PCA problem to demonstrate the superiority and effectiveness of the proposed RWL-AN model. Papers published at the Neural Information Processing Systems Conference.


Interpret Principal Component Analysis (PCA)

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Data can tell us stories. That's what I've been told anyway. As a Data Scientist working for Fortune 300 clients, I deal with tons of data daily, I can tell you that data can tell us stories. You can apply a regression, classification or a clustering algorithm on the data, but feature selection and engineering can be a daunting task. A lot of times, I have seen data scientists take an automated approach to feature selection such as Recursive Feature Elimination (RFE) or leverage Feature Importance algorithms using Random Forest or XGBoost.


Fair Principal Component Analysis and Filter Design

arXiv.org Machine Learning

We consider Fair Principal Component Analysis (FPCA) and search for a low dimensional subspace that spans multiple target vectors in a fair manner. FPCA is defined as a non-concave maximization of the worst projected target norm within a given set. The problem arises in filter design in signal processing, and when incorporating fairness into dimensionality reduction schemes. The state of the art approach to FPCA is via semidefinite relaxation and involves a polynomial yet computationally expensive optimization. To allow scalability, we propose to address FPCA using naive sub-gradient descent. We analyze the landscape of the underlying optimization in the case of orthogonal targets. We prove that the landscape is benign and that all local minima are globally optimal. Interestingly, the SDR approach leads to sub-optimal solutions in this simple case. Finally, we discuss the equivalence between orthogonal FPCA and the design of normalized tight frames.


Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Matrices via Convex Optimization

Neural Information Processing Systems

Principal component analysis is a fundamental operation in computational data analysis, with myriad applications ranging from web search to bioinformatics to computer vision and image analysis. However, its performance and applicability in real scenarios are limited by a lack of robustness to outlying or corrupted observations. This paper considers the idealized "robust principal component analysis" problem of recovering a low rank matrix A from corrupted observations D A E. Here, the error entries E can be arbitrarily large (modeling grossly corrupted observations common in visual and bioinformatic data), but are assumed to be sparse. We prove that most matrices A can be efficiently and exactly recovered from most error sign-and-support patterns, by solving a simple convex program. Our result holds even when the rank of A grows nearly proportionally (up to a logarithmic factor) to the dimensionality of the observation space and the number of errors E grows in proportion to the total number of entries in the matrix.