  Under the framework of graph-based learning, the key to robust subspace
clustering and subspace learning is to obtain a good similarity graph that
eliminates the effects of errors and retains only connections between the data
points from the same subspace (i.e., intra-subspace data points). Recent works
achieve good performance by modeling errors into their objective functions to
remove the errors from the inputs. However, these approaches face the
limitations that the structure of errors should be known prior and a complex
convex problem must be solved. In this paper, we present a novel method to
eliminate the effects of the errors from the projection space (representation)
rather than from the input space. We first prove that $\ell_1$-, $\ell_2$-,
$\ell_{\infty}$-, and nuclear-norm based linear projection spaces share the
property of Intra-subspace Projection Dominance (IPD), i.e., the coefficients
over intra-subspace data points are larger than those over inter-subspace data
points. Based on this property, we introduce a method to construct a sparse
similarity graph, called L2-Graph. The subspace clustering and subspace
learning algorithms are developed upon L2-Graph. Experiments show that L2-Graph
algorithms outperform the state-of-the-art methods for feature extraction,
image clustering, and motion segmentation in terms of accuracy, robustness, and
time efficiency.
