5,673 research outputs found
Randomized Robust Subspace Recovery for High Dimensional Data Matrices
This paper explores and analyzes two randomized designs for robust Principal
Component Analysis (PCA) employing low-dimensional data sketching. In one
design, a data sketch is constructed using random column sampling followed by
low dimensional embedding, while in the other, sketching is based on random
column and row sampling. Both designs are shown to bring about substantial
savings in complexity and memory requirements for robust subspace learning over
conventional approaches that use the full scale data. A characterization of the
sample and computational complexity of both designs is derived in the context
of two distinct outlier models, namely, sparse and independent outlier models.
The proposed randomized approach can provably recover the correct subspace with
computational and sample complexity that are almost independent of the size of
the data. The results of the mathematical analysis are confirmed through
numerical simulations using both synthetic and real data
Provable Self-Representation Based Outlier Detection in a Union of Subspaces
Many computer vision tasks involve processing large amounts of data
contaminated by outliers, which need to be detected and rejected. While outlier
detection methods based on robust statistics have existed for decades, only
recently have methods based on sparse and low-rank representation been
developed along with guarantees of correct outlier detection when the inliers
lie in one or more low-dimensional subspaces. This paper proposes a new outlier
detection method that combines tools from sparse representation with random
walks on a graph. By exploiting the property that data points can be expressed
as sparse linear combinations of each other, we obtain an asymmetric affinity
matrix among data points, which we use to construct a weighted directed graph.
By defining a suitable Markov Chain from this graph, we establish a connection
between inliers/outliers and essential/inessential states of the Markov chain,
which allows us to detect outliers by using random walks. We provide a
theoretical analysis that justifies the correctness of our method under
geometric and connectivity assumptions. Experimental results on image databases
demonstrate its superiority with respect to state-of-the-art sparse and
low-rank outlier detection methods.Comment: 16 pages. CVPR 2017 spotlight oral presentatio
Robust Subspace Learning: Robust PCA, Robust Subspace Tracking, and Robust Subspace Recovery
PCA is one of the most widely used dimension reduction techniques. A related
easier problem is "subspace learning" or "subspace estimation". Given
relatively clean data, both are easily solved via singular value decomposition
(SVD). The problem of subspace learning or PCA in the presence of outliers is
called robust subspace learning or robust PCA (RPCA). For long data sequences,
if one tries to use a single lower dimensional subspace to represent the data,
the required subspace dimension may end up being quite large. For such data, a
better model is to assume that it lies in a low-dimensional subspace that can
change over time, albeit gradually. The problem of tracking such data (and the
subspaces) while being robust to outliers is called robust subspace tracking
(RST). This article provides a magazine-style overview of the entire field of
robust subspace learning and tracking. In particular solutions for three
problems are discussed in detail: RPCA via sparse+low-rank matrix decomposition
(S+LR), RST via S+LR, and "robust subspace recovery (RSR)". RSR assumes that an
entire data vector is either an outlier or an inlier. The S+LR formulation
instead assumes that outliers occur on only a few data vector indices and hence
are well modeled as sparse corruptions.Comment: To appear, IEEE Signal Processing Magazine, July 201
Local Subspace-Based Outlier Detection using Global Neighbourhoods
Outlier detection in high-dimensional data is a challenging yet important
task, as it has applications in, e.g., fraud detection and quality control.
State-of-the-art density-based algorithms perform well because they 1) take the
local neighbourhoods of data points into account and 2) consider feature
subspaces. In highly complex and high-dimensional data, however, existing
methods are likely to overlook important outliers because they do not
explicitly take into account that the data is often a mixture distribution of
multiple components.
We therefore introduce GLOSS, an algorithm that performs local subspace
outlier detection using global neighbourhoods. Experiments on synthetic data
demonstrate that GLOSS more accurately detects local outliers in mixed data
than its competitors. Moreover, experiments on real-world data show that our
approach identifies relevant outliers overlooked by existing methods,
confirming that one should keep an eye on the global perspective even when
doing local outlier detection.Comment: Short version accepted at IEEE BigData 201
Robust Orthogonal Complement Principal Component Analysis
Recently, the robustification of principal component analysis has attracted
lots of attention from statisticians, engineers and computer scientists. In
this work we study the type of outliers that are not necessarily apparent in
the original observation space but can seriously affect the principal subspace
estimation. Based on a mathematical formulation of such transformed outliers, a
novel robust orthogonal complement principal component analysis (ROC-PCA) is
proposed. The framework combines the popular sparsity-enforcing and low rank
regularization techniques to deal with row-wise outliers as well as
element-wise outliers. A non-asymptotic oracle inequality guarantees the
accuracy and high breakdown performance of ROC-PCA in finite samples. To tackle
the computational challenges, an efficient algorithm is developed on the basis
of Stiefel manifold optimization and iterative thresholding. Furthermore, a
batch variant is proposed to significantly reduce the cost in ultra high
dimensions. The paper also points out a pitfall of a common practice of SVD
reduction in robust PCA. Experiments show the effectiveness and efficiency of
ROC-PCA in both synthetic and real data
- …