L1-norm Regularized L1-norm Best-fit line problem

Background Conventional Principal Component Analysis (PCA) is a widely used technique to reduce data dimension. PCA finds linear combinations of the original features capturing maximal variance of data via Singular Value Decomposition (SVD). However, SVD is sensitive to outliers, and often leads to high dimensional results. To address the issues, we propose a new method to estimate best-fit one-dimensional subspace, called l1-norm Regularized l1-norm. Methods In this article, we describe a method to fit a lower-dimensional subspace by approximate a non-linear, non-convex, non-smooth optimization problem called l1 regularized l1-norm Best- Fit Line problem; minimize a combination of the l1 error and of the l1 regularization. The procedure can be simply performed using ratios and sorting. Also ,we present applications in the area of video surveillance, where our methodology allows for background subtraction with jitters, illumination changes, and clutters. Results We compared our performance with SVD on synthetic data. The numerical results showed our algorithm successfully found a better principal component from a grossly corrupted data than SVD in terms of discordance. Moreover, our algorithm provided a sparser principal component than SVD. However, we expect it to be faster on multi-node environment. Conclusions This paper proposes a new algorithm able to generate a sparse best-fit subspace robust to outliers. The projected subspaces sought on non-contaminated data, differ little from that of traditional PCA. When subspaces are projected from contaminated data, it attain arguably significant both smaller discordance and lower dimension than that of traditional PCA.https://scholarscompass.vcu.edu/gradposters/1074/thumbnail.jp

Modified mean curvature flow of star-shaped hypersurfaces in hyperbolic space

We define a new version of modified mean curvature flow (MMCF) in hyperbolic space $\mathbb{H}^{n+1}$, which interestingly turns out to be the natural negative $L^2$-gradient flow of the energy functional defined by De Silva and Spruck in \cite{DS09}. We show the existence, uniqueness and convergence of the MMCF of complete embedded star-shaped hypersurfaces with fixed prescribed asymptotic boundary at infinity. As an application, we recover the existence and uniqueness of smooth complete hypersurfaces of constant mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity, which was first shown by Guan and Spruck.Comment: 26 pages, 3 figure