Detecting outlying subspaces for high-dimensional data: the new task, algorithms and performance

[Abstract]: In this paper, we identify a new task for studying the outlying degree (OD) of high-dimensional data, i.e. finding the subspaces (subsets of features) in which the given points are outliers, which are called their outlying subspaces. Since the state-of-the-art outlier detection techniques fail to handle this new problem, we propose a novel detection algorithm, called High-Dimension Outlying subspace Detection (HighDOD), to detect the outlying subspaces of high-dimensional data efficiently. The intuitive idea of HighDOD is that we measure the OD of the point using the sum of distances between this point and its k nearest neighbors. Two heuristic pruning strategies are proposed to realize fast pruning in the subspace search and an efficient dynamic subspace search method with a sample-based learning process has been implemented. Experimental results show that HighDOD is efficient and outperforms other searching alternatives such as the naive top–down, bottom–up and random search methods, and the existing outlier detection methods cannot fulfill this new task effectively

Robust Recovery of Subspace Structures by Low-Rank Representation

In this work we address the subspace recovery problem. Given a set of data samples (vectors) approximately drawn from a union of multiple subspaces, our goal is to segment the samples into their respective subspaces and correct the possible errors as well. To this end, we propose a novel method termed Low-Rank Representation (LRR), which seeks the lowest-rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary. It is shown that LRR well solves the subspace recovery problem: when the data is clean, we prove that LRR exactly captures the true subspace structures; for the data contaminated by outliers, we prove that under certain conditions LRR can exactly recover the row space of the original data and detect the outlier as well; for the data corrupted by arbitrary errors, LRR can also approximately recover the row space with theoretical guarantees. Since the subspace membership is provably determined by the row space, these further imply that LRR can perform robust subspace segmentation and error correction, in an efficient way.Comment: IEEE Trans. Pattern Analysis and Machine Intelligenc

Spatial Random Sampling: A Structure-Preserving Data Sketching Tool

Random column sampling is not guaranteed to yield data sketches that preserve the underlying structures of the data and may not sample sufficiently from less-populated data clusters. Also, adaptive sampling can often provide accurate low rank approximations, yet may fall short of producing descriptive data sketches, especially when the cluster centers are linearly dependent. Motivated by that, this paper introduces a novel randomized column sampling tool dubbed Spatial Random Sampling (SRS), in which data points are sampled based on their proximity to randomly sampled points on the unit sphere. The most compelling feature of SRS is that the corresponding probability of sampling from a given data cluster is proportional to the surface area the cluster occupies on the unit sphere, independently from the size of the cluster population. Although it is fully randomized, SRS is shown to provide descriptive and balanced data representations. The proposed idea addresses a pressing need in data science and holds potential to inspire many novel approaches for analysis of big data