An important machine learning method for dimensionality reduction is called Principal Component Analysis.

We propose a novel method called robust kernel principal component analysis (RKPCA) to decompose a partially corrupted matrix as a sparse matrix plus a high or full-rank matrix whose columns are drawn from a nonlinear low-dimensional latent variable model. RKPCA can be applied to many problems such as noise removal and subspace clustering and is so far the only unsupervised nonlinear method robust to sparse noises. We also provide theoretical guarantees for RKPCA. The optimization of RKPCA is challenging because it involves nonconvex and indifferentiable problems simultaneously. We propose two nonconvex optimization algorithms for RKPCA: alternating direction method of multipliers with backtracking line search and proximal linearized minimization with adaptive step size. Comparative studies on synthetic data and nature images corroborate the effectiveness and superiority of RKPCA in noise removal and robust subspace clustering.

In this paper we propose a new algorithm for streaming principal component analysis. With limited memory, small devices cannot store all the samples in the high-dimensional regime. Streaming principal component analysis aims to find the $k$-dimensional subspace which can explain the most variation of the $d$-dimensional data points that come into memory sequentially. In order to deal with large $d$ and large $N$ (number of samples), most streaming PCA algorithms update the current model using only the incoming sample and then dump the information right away to save memory. However the information contained in previously streamed data could be useful. Motivated by this idea, we develop a new streaming PCA algorithm called History PCA that achieves this goal. By using $O(Bd)$ memory with $B\approx 10$ being the block size, our algorithm converges much faster than existing streaming PCA algorithms. By changing the number of inner iterations, the memory usage can be further reduced to $O(d)$ while maintaining a comparable convergence speed. We provide theoretical guarantees for the convergence of our algorithm along with the rate of convergence. We also demonstrate on synthetic and real world data sets that our algorithm compares favorably with other state-of-the-art streaming PCA methods in terms of the convergence speed and performance.

Dimensionality Reduction is a category of unsupervised machine learning techniques which is used to reduce the number of features or variables of columns in a dataset.

Though there is a growing body of literature on fairness for supervised learning, the problem of incorporating fairness into unsupervised learning has been less well-studied. This paper studies fairness in the context of principal component analysis (PCA). We first present a definition of fairness for dimensionality reduction, and our definition can be interpreted as saying that a reduction is fair if information about a protected class (e.g., race or gender) cannot be inferred from the dimensionality-reduced data points. Next, we develop convex optimization formulations that can improve the fairness (with respect to our definition) of PCA and kernel PCA. These formulations are semidefinite programs (SDP's), and we demonstrate the effectiveness of our formulations using several datasets. We conclude by showing how our approach can be used to perform a fair (with respect to age) clustering of health data that may be used to set health insurance rates.

We revisit the problem of robust principal component analysis with features acting as prior side information. To this aim, a novel, elegant, non-convex optimization approach is proposed to decompose a given observation matrix into a low-rank core and the corresponding sparse residual. Rigorous theoretical analysis of the proposed algorithm results in exact recovery guarantees with low computational complexity. Aptly designed synthetic experiments demonstrate that our method is the first to wholly harness the power of non-convexity over convexity in terms of both recoverability and speed. That is, the proposed non-convex approach is more accurate and faster compared to the best available algorithms for the problem under study. Two real-world applications, namely image classification and face denoising further exemplify the practical superiority of the proposed method.

The purpose of this post is to give the reader detailed understanding of Principal Component Analysis with the necessary mathematical proofs. We plot the data and find various patterns in it or use it to train some machine learning models. One way to think about dimensions is that suppose you have an data point x, if we consider this data point as a physical object then dimensions are merely a basis of view, like where is the data located when it is observed from horizontal axis or vertical axis. As the dimensions of data increases, the difficulty to visualize it and perform computations on it also increases. Variance: It is a measure of the variability or it simply measures how spread the data set is. Mathematically, it is the average squared deviation from the mean score.

Principal Component Analysis (PCA) is a technique used to find the core components that underlie different variables. It comes in very useful whenever doubts arise about the true origin of three or more variables. There are two main methods for performing a PCA: naive or less naive. In the naive method, you first check some conditions in your data which will determine the essentials of the analysis. In the less-naive method, you set the those yourself, based on whatever prior information or purposes you had. I will tackle the naive method, mainly by following the guidelines in Field, Miles, and Field (2012), with updated code where necessary.

We introduce a new method for sparse principal component analysis, based on the aggregation of eigenvector information from carefully-selected random projections of the sample covariance matrix. Unlike most alternative approaches, our algorithm is non-iterative, so is not vulnerable to a bad choice of initialisation. Our theory provides great detail on the statistical and computational trade-off in our procedure, revealing a subtle interplay between the effective sample size and the number of random projections that are required to achieve the minimax optimal rate. Numerical studies provide further insight into the procedure and confirm its highly competitive finite-sample performance.