I didn't see this explicitly stated anywhere.. - Line 181: alternative -> alternating(?) - LIne 232: Should ||r||_2 be ||r||_2^2? Otherwise the expression may not be strongly convex.

Robust PCA methods are typically based on batch optimization and have to load all the samples into memory. This prevents them from efficiently processing big data. In this paper, we develop an Online Robust Principal Component Analysis (OR-PCA) that processes one sample per time instance and hence its memory cost is independent of the data size, significantly enhancing the computation and storage efficiency. The proposed method is based on stochastic optimization of an equivalent reformulation of the batch RPCA method. Indeed, we show that OR-PCA provides a sequence of subspace estimations converging to the optimum of its batch counterpart and hence is provably robust to sparse corruption. Moreover, OR-PCA can naturally be applied for tracking dynamic subspace. Comprehensive simulations on subspace recovering and tracking demonstrate the robustness and efficiency advantages of the OR-PCA over online PCA and batch RPCA methods.

The performance of the standard Online Robust Principal Component Analysis (OR-PCA) technique depends on the optimum tuning of the explicit regularizers and this tuning is dataset sensitive. We aim to remove the dependency on these tuning parameters by using implicit regularization. We propose to use the implicit regularization effect of various modified gradient descents to make OR-PCA tuning free. Our method incorporates three different versions of modified gradient descent that separately but naturally encourage sparsity and low-rank structures in the data. The proposed method performs comparable or better than the tuned OR-PCA for both simulated and real-world datasets. Tuning-free ORPCA makes it more scalable for large datasets since we do not require dataset-dependent parameter tuning.

Robust PCA methods are typically based on batch optimization and have to load all the samples into memory during optimization. This prevents them from efficiently processing big data. In this paper, we develop an Online Robust PCA (OR-PCA) that processes one sample per time instance and hence its memory cost is independent of the number of samples, significantly enhancing the computation and storage efficiency. The proposed OR-PCA is based on stochastic optimization of an equivalent reformulation of the batch RPCA. Indeed, we show that OR-PCA provides a sequence of subspace estimations converging to the optimum of its batch counterpart and hence is provably robust to sparse corruption. Moreover, OR-PCA can naturally be applied for tracking dynamic subspace. Comprehensive simulations on subspace recovering and tracking demonstrate the robustness and efficiency advantages of the OR-PCA over online PCA and batch RPCA methods.

Robust PCA methods are typically based on batch optimization and have to load all the samples into memory. This prevents them from efficiently processing big data. In this paper, we develop an Online Robust Principal Component Analysis (OR-PCA) that processes one sample per time instance and hence its memory cost is independent of the data size, significantly enhancing the computation and storage efficiency. The proposed method is based on stochastic optimization of an equivalent reformulation of the batch RPCA method. Indeed, we show that OR-PCA provides a sequence of subspace estimations converging to the optimum of its batch counterpart and hence is provably robust to sparse corruption.

Max-norm regularizer has been extensively studied in the last decade as it promotes an effective low-rank estimation for the underlying data. However, such max-norm regularized problems are typically formulated and solved in a batch manner, which prevents it from processing big data due to possible memory budget. In this paper, hence, we propose an online algorithm that is scalable to large-scale setting. Particularly, we consider the matrix decomposition problem as an example, although a simple variant of the algorithm and analysis can be adapted to other important problems such as matrix completion. The crucial technique in our implementation is to reformulating the max-norm to an equivalent matrix factorization form, where the factors consist of a (possibly overcomplete) basis component and a coefficients one. In this way, we may maintain the basis component in the memory and optimize over it and the coefficients for each sample alternatively. Since the memory footprint of the basis component is independent of the sample size, our algorithm is appealing when manipulating a large collection of samples. We prove that the sequence of the solutions (i.e., the basis component) produced by our algorithm converges to a stationary point of the expected loss function asymptotically. Numerical study demonstrates encouraging results for the efficacy and robustness of our algorithm compared to the widely used nuclear norm solvers.

Low rank matrix recovery has shown its importance as a theoretic foundation in many areas of information processing. Its solutions are usually obtained in batch mode that requires to load all the data into memory during processing, and thus are hardly applicable on large scale data. Moreover, a fraction of data may be severely contaminated by outliers, which makes accurate recovery significantly more challenging. This paper proposes a novel online robust low rank matrix recovery method to address these difficulties. In particular, we first introduce an online algorithm to solve the problem of low rank matrix completion. Then we move on to low rank matrix recovery from observations with intensive outliers. The outlier support is robustly estimated from a perspective of mixture model. Experiments on both synthetic and real data are conducted to demonstrate the efficacy of our method and show its superior performance over the state-of-the-arts.

Robust PCA methods are typically based on batch optimization and have to load all the samples into memory. This prevents them from efficiently processing big data. In this paper, we develop an Online Robust Principal Component Analysis (OR-PCA) that processes one sample per time instance and hence its memory cost is independent of the data size, significantly enhancing the computation and storage efficiency. The proposed method is based on stochastic optimization of an equivalent reformulation of the batch RPCA method. Indeed, we show that OR-PCA provides a sequence of subspace estimations converging to the optimum of its batch counterpart and hence is provably robust to sparse corruption. Moreover, OR-PCA can naturally be applied for tracking dynamic subspace. Comprehensive simulations on subspace recovering and tracking demonstrate the robustness and efficiency advantages of the OR-PCA over online PCA and batch RPCA methods.