where Ωt is a selection operator that specifies the observable subset of entries at time t, at ∈ R r are the coefficients specifying the linear combination of the columns of Ut, and st ∈ R n is a sparse outlier vector. When the subspace Ut is stationary, we drop the subscript t from Ut and the problem reduces to robust matrix completion or robust principal component analysis where the task is t...

We consider the dimensionality-reduction problem (finding a subspace approximation of observed data) for contaminated data in the high dimensional regime, where the number of observations is of the same magnitude as the number of variables of each observation, and the data set contains some (arbitrarily) corrupted observations. We propose a High-dimensional Robust Principal Component Analysis (...

The outlier sensitivity of classical principal component analysis (PCA) has spurred the development of robust techniques. Existing robust PCA methods like ROBPCA work best if the non-outlying data have an approximately symmetric distribution. When the original variables are skewed, too many points tend to be flagged as outlying. A robust PCA method is developed which is also suitable for skewed...

We estimate the rate of convergence and sample complexity of a recent robust estimator for a generalized version of the inverse covariance matrix. This estimator is used in a convex algorithm for robust subspace recovery (i.e., robust PCA). Our model assumes a sub-Gaussian underlying distribution and an i.i.d. sample from it. Our main result shows with high probability that the norm of the diff...

We consider the problem of learning from noisy data in practical settings where the size of data is too large to store on a single machine. More challenging, the data coming from the wild may contain malicious outliers. To address the scalability and robustness issues, we present an online robust learning (ORL) approach. ORL is simple to implement and has provable robustness guarantee—in stark ...

Many real world datasets subsume a linear or non-linear low-rank structure in a very low-dimensional space. Unfortunately, one often has very little or no information about the geometry of the space, resulting in a highly under-determined recovery problem. Under certain circumstances, state-of-the-art algorithms provide an exact recovery for linear low-rank structures but at the expense of high...

In this paper we introduce a new method for robust principal component analysis. Classical PCA is based on the empirical covariance matrix of the data and hence it is highly sensitive to outlying observations. In the past, two robust approaches have been developed. The first is based on the eigenvectors of a robust scatter matrix such as the MCD or an S-estimator, and is limited to relatively l...

The Active Appearance Model (AAM) is a widely used method for model based vision showing excellent results. But one major drawback is that the method is not robust against occlusions. Thus, if parts of the image are occluded the method converges to local minima and the obtained results are unreliable. To overcome this problem we propose a robust AAM fitting strategy. The main idea is to apply a...