Related papers: An Analytical Formula for Spectrum Reconstruction
We give an efficient algorithm which can obtain a relative error approximation to the spectral norm of a matrix, combining the power iteration method with some techniques from matrix reconstruction which use random sampling.
Principal Component Analysis (PCA) is the workhorse tool for dimensionality reduction in this era of big data. While often overlooked, the purpose of PCA is not only to reduce data dimensionality, but also to yield features that are…
Optimization of convex functions subject to eigenvalue constraints is intriguing because of peculiar analytical properties of eigenvalues, and is of practical interest because of wide range of applications in fields such as structural…
Kernel Principal Component Analysis (KPCA) is a popular dimensionality reduction technique with a wide range of applications. However, it suffers from the problem of poor scalability. Various approximation methods have been proposed in the…
Understanding the singular value spectrum of a matrix $A \in \mathbb{R}^{n \times n}$ is a fundamental task in countless applications. In matrix multiplication time, it is possible to perform a full SVD and directly compute the singular…
Standard perturbation theory of eigenvalue problems consists of obtaining approximations of eigenmodes in the neighborhood of an operator where the corresponding eigenmode is known. Nevertheless, if the corresponding eigenmodes of several…
In datasets where the number of parameters is fixed and the number of samples is large, principal component analysis (PCA) is a powerful dimension reduction tool. However, in many contemporary datasets, when the number of parameters is…
Principal component analysis (PCA) is a widely used dimension reduction tool in the analysis of many kind of high-dimensional data. It is used in signal processing, mechanical engineering, psychometrics, and other fields under different…
Formulae for the value of a harmonic function at the center of a rectangle are found that involve boundary integrals. The central value of a harmonic function is shown to be well approximated by the mean value of the function on the…
Several fundamental tasks in data science rely on computing an extremal eigenspace of size $r \ll n$, where $n$ is the underlying problem dimension. For example, spectral clustering and PCA both require the computation of the leading…
Principal component analysis (PCA) is one of the most commonly used statistical procedures with a wide range of applications. Consider the points $X_1, X_2,..., X_n$ are vectors drawn i.i.d. from a distribution with mean zero and covariance…
Recent years have witnessed intense development of randomized methods for low-rank approximation. These methods target principal component analysis (PCA) and the calculation of truncated singular value decompositions (SVD). The present…
In this paper, we consider the sparse eigenvalue problem wherein the goal is to obtain a sparse solution to the generalized eigenvalue problem. We achieve this by constraining the cardinality of the solution to the generalized eigenvalue…
We derive high-probability bounds for the reconstruction error of PCA in infinite dimensions. We apply our bounds in the case that the eigenvalues of the covariance operator satisfy polynomial or exponential upper bounds.
We study sparse principal components analysis in the high-dimensional setting, where $p$ (the number of variables) can be much larger than $n$ (the number of observations). We prove optimal, non-asymptotic lower and upper bounds on the…
We precisely quantify the impact of statistical error in the quality of a numerical approximation to a random matrix eigendecomposition, and under mild conditions, we use this to introduce an optimal numerical tolerance for residual error…
The literature provides strong evidence that stock prices can be predicted from past price data. Principal component analysis (PCA) is a widely used mathematical technique for dimensionality reduction and analysis of data by identifying a…
Following the semiclassical formalism of Strutinsky et al., we have obtained the complete eigenvalue spectrum for a particle enclosed in an infinitely high spheroidal cavity. Our spheroidal trace formula also reproduces the results of a…
Our main objective is to develop a denoising strategy to increase the signal to noise ratio of individual spectral lines of stellar spectropolarimetric observations. We use a multivariate statistics technique called Principal Component…
Sparse Principal Component Analysis (SPCA) is a fundamental technique for dimensionality reduction, and is NP-hard. In this paper, we introduce a randomized approximation algorithm for SPCA, which is based on the basic SDP relaxation. Our…