In our work [1], we present a unnified theory for the pursuit of the block diagonal matrix structure for subspace clustering (this part is extended from our former work [2]). We also propose the first k-block diagonal regularizer for a direct pursuit of the block diagonal matrix. We then apply it for subpace clustering and name this method as Block Diagonal Representation (BDR).
Assume that is an affinity matrix, i.e.,
and
, the corresponding Laplacian matrix, denoted as
, is defined as
. Then the
-block diagonal regularizer is defined as the sum of the
smallest eigenvalues of
, i.e.,
It can be seen that is equivalent to the fact
that the affinity matrix
is
-block diagonal. So
can be regarded as the block diagonal matrix structure induced regularizer.
| [1] | Canyi Lu, Jiashi Feng, Tao Mei, Zhouchen Lin and Shuicheng Yan, Subspace Clustering by Block Diagonal Representation, IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI), 2019 |
| [2] | Can-Yi Lu, Hai Min, Zhong-Qiu Zhao, Lin Zhu, De-Shuang Huang and Shuicheng Yan, Robust and Efficient Subspace Segmentation via Least Squares Regression. European Conference on Computer Vision (ECCV), pp. 347-360, 2012 |