8,358 research outputs found
Convex Subspace Clustering by Adaptive Block Diagonal Representation
Subspace clustering is a class of extensively studied clustering methods and
the spectral-type approaches are its important subclass whose key first step is
to learn a coefficient matrix with block diagonal structure. To realize this
step, sparse subspace clustering (SSC), low rank representation (LRR) and block
diagonal representation (BDR) were successively proposed and have become the
state-of-the-arts (SOTAs). Among them, the former two minimize their convex
objectives by imposing sparsity and low rankness on the coefficient matrix
respectively, but so-desired block diagonality cannot neccesarily be guaranteed
practically while the latter designs a block diagonal matrix induced
regularizer but sacrifices convexity. For solving this dilemma, inspired by
Convex Biclustering, in this paper, we propose a simple yet efficient
spectral-type subspace clustering method named Adaptive Block Diagonal
Representation (ABDR) which strives to pursue so-desired block diagonality as
BDR by coercively fusing the columns/rows of the coefficient matrix via a
specially designed convex regularizer, consequently, ABDR naturally enjoys
their merits and can adaptively form more desired block diagonality than the
SOTAs without needing to prefix the number of blocks as done in BDR. Finally,
experimental results on synthetic and real benchmarks demonstrate the
superiority of ABDR.Comment: 13 pages, 11 figures, 8 table
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
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