Robust Principal Component Analysis?

This paper is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a low-rank component and a sparse component. Can we recover each component individually? We prove that under some suitable assumptions, it is possible to recover both the low-rank and the sparse components exactly by solving a very convenient convex program called Principal Component Pursuit; among all feasible decompositions, simply minimize a weighted combination of the nuclear norm and of the L1 norm. This suggests the possibility of a principled approach to robust principal component analysis since our methodology and results assert that one can recover the principal components of a data matrix even though a positive fraction of its entries are arbitrarily corrupted. This extends to the situation where a fraction of the entries are missing as well. We discuss an algorithm for solving this optimization problem, and present applications in the area of video surveillance, where our methodology allows for the detection of objects in a cluttered background, and in the area of face recognition, where it offers a principled way of removing shadows and specularities in images of faces

Stable Principal Component Pursuit

In this paper, we study the problem of recovering a low-rank matrix (the principal components) from a high-dimensional data matrix despite both small entry-wise noise and gross sparse errors. Recently, it has been shown that a convex program, named Principal Component Pursuit (PCP), can recover the low-rank matrix when the data matrix is corrupted by gross sparse errors. We further prove that the solution to a related convex program (a relaxed PCP) gives an estimate of the low-rank matrix that is simultaneously stable to small entrywise noise and robust to gross sparse errors. More precisely, our result shows that the proposed convex program recovers the low-rank matrix even though a positive fraction of its entries are arbitrarily corrupted, with an error bound proportional to the noise level. We present simulation results to support our result and demonstrate that the new convex program accurately recovers the principal components (the low-rank matrix) under quite broad conditions. To our knowledge, this is the first result that shows the classical Principal Component Analysis (PCA), optimal for small i.i.d. noise, can be made robust to gross sparse errors; or the first that shows the newly proposed PCP can be made stable to small entry-wise perturbations.Comment: 5-page paper submitted to ISIT 201

CrC^r-Chain closing lemma for certain partially hyperbolic diffeomorphisms

For every $r\in\mathbb{N}_{\geq 2}\cup\{\infty\}$, we prove a $C^r$-orbit connecting lemma for dynamically coherent and plaque expansive partially hyperbolic diffeomorphisms with 1-dimensional orientation preserving center bundle. To be precise, for such a diffeomorphism $f$, if a point $y$ is chain attainable from $x$ through pseudo-orbits, then for any neighborhood $U$ of $x$ and any neighborhood $V$ of $y$, there exist true orbits from $U$ to $V$ by arbitrarily $C^r$-small perturbations. As a consequence, we prove that for $C^r$-generic diffeomorphisms in this class, periodic points are dense in the chain recurrent set, and chain transitivity implies transitivity

Dense Error Correction for Low-Rank Matrices via Principal Component Pursuit

We consider the problem of recovering a low-rank matrix when some of its entries, whose locations are not known a priori, are corrupted by errors of arbitrarily large magnitude. It has recently been shown that this problem can be solved efficiently and effectively by a convex program named Principal Component Pursuit (PCP), provided that the fraction of corrupted entries and the rank of the matrix are both sufficiently small. In this paper, we extend that result to show that the same convex program, with a slightly improved weighting parameter, exactly recovers the low-rank matrix even if "almost all" of its entries are arbitrarily corrupted, provided the signs of the errors are random. We corroborate our result with simulations on randomly generated matrices and errors.Comment: Submitted to ISIT 201

Numerical simulation of clouds and precipitation depending on different relationships between aerosol and cloud droplet spectral dispersion

The aerosol effects on clouds and precipitation in deep convective cloud systems are investigated using the Weather Research and Forecast (WRF) model with the Morrison two-moment bulk microphysics scheme. Considering positive or negative relationships between the cloud droplet number concentration (Nc) and spectral dispersion (ɛ), a suite of sensitivity experiments are performed using an initial sounding data of the deep convective cloud system on 31 March 2005 in Beijing under either a maritime (‘clean’) or continental (‘polluted’) background. Numerical experiments in this study indicate that the sign of the surface precipitation response induced by aerosols is dependent on the ɛ−Nc relationships, which can influence the autoconversion processes from cloud droplets to rain drops. When the spectral dispersion ɛ is an increasing function of Nc, the domain-average cumulative precipitation increases with aerosol concentrations from maritime to continental background. That may be because the existence of large-sized rain drops can increase precipitation at high aerosol concentration. However, the surface precipitation is reduced with increasing concentrations of aerosol particles when ɛ is a decreasing function of Nc. For the ɛ−Nc negative relationships, smaller spectral dispersion suppresses the autoconversion processes, reduces the rain water content and eventually decreases the surface precipitation under polluted conditions. Although differences in the surface precipitation between polluted and clean backgrounds are small for all the ɛ−Nc relationships, additional simulations show that our findings are robust to small perturbations in the initial thermal conditions. Keywords: aerosol indirect effects, cloud droplet spectral dispersion, autoconversion parameterization, deep convective systems, two-moment bulk microphysics schem

Smell machine -- Innovative city odor grid monitoring system based on MEMS sensing technology

As one of the important indicators of urban environmental quality, city odor has an important impact on residents’ quality of life and city image. Traditional odor monitoring methods have problems such as sampling diffi culty, time delay and low spatial resolution, which can not meet the needs of accurate and real-time monitoring of urban odor. Based on microelectromechanical system (MEMS) sensing technology, this paper proposes an innovative city odor grid monitoring system, the sniffi ng machine. The system uses the array of micro gas sensors, and arranges the sensors in diff erent areas of the city to form a grid layout, so as to realize the comprehensive monitoring of city odor. It provides a new idea and method for urban environmental management and improvement, and helps to promote the improvement of urban environmental quality and the improvement of residents’ quality of life

The first-order effect of Holocene Northern Peatlands on global carbon cycle dynamics

Given the fact that the estimated present-day carbon storage of Northern Peatlands (NP) is about 300–500 petagram (PgC, 1 petagram = 1015 gram), and the NP has been subject to a slow but persistent growth over the Holocene epoch, it is desirable to include the NP in studies of Holocene carbon cycle dynamics. Here we use an Earth system Model of Intermediate Complexity to study the first-order effect of NP on global carbon cycle dynamics in the Holocene. We prescribe the reconstructed NP growth based on data obtained from numerous sites (located in Western Siberia, North America, and Finland) where peat accumulation records have been developed. Using an inverse method, we demonstrate that the long-term debates over potential source and/or sink of terrestrial ecosystem in the Holocene are clarified by using an inverse method, and our results suggest that the primary carbon source for the changes (sinks) of atmospheric and terrestrial carbon is the ocean, presumably, due to the deep ocean sedimentation pump (the so-called alkalinity pump). Our paper here complements ref. 1 by sensitivity tests using modified boundary conditions