Subspace clustering by (k,k)-sparse matrix factorization

Haixia Liu; Jian-Feng Cai; Yang Wang

doi:10.3934/ipi.2017025

Article Contents

2017, Volume 11, Issue 3: 539-551. Doi: 10.3934/ipi.2017025

This issue Previous Article Recovering the boundary corrosion from electrical potential distribution using partial boundary data Next Article Probabilistic interpretation of the Calderón problem

Subspace clustering by (k,k)-sparse matrix factorization

Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

* Corresponding author
* Corresponding author

Received: March 31, 2016

Revised: January 31, 2017

Published: August 2017

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Abstract

High-dimensional data often lie in low-dimensional subspaces instead of the whole space. Subspace clustering is a problem to analyze data that are from multiple low-dimensional subspaces and cluster them into the corresponding subspaces. In this work, we propose a $(k,k)$-sparse matrix factorization method for subspace clustering. In this method, data itself is considered as the "dictionary", and each data point is represented as a linear combination of the basis of its cluster in the dictionary. Thus, the coefficient matrix is low-rank and sparse. With an appropriate permutation, it is also blockwise with each block corresponding to a cluster. With an assumption that each block is no more than $k$-by-$k$ in matrix recovery, we seek a low-rank and $(k,k)$-sparse coefficient matrix, which will be used for the construction of affinity matrix in spectral clustering. The advantage of our proposed method is that we recover a coefficient matrix with $(k,k)$-sparse and low-rank simultaneously, which is better fit for subspace clustering. Numerical results illustrate the effectiveness that it is better than SSC and LRR in real-world classification problems such as face clustering and motion segmentation.

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Mathematics Subject Classification: Primary: 68T04; Secondary: 65F04.

Citation:

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Figure 1. The plot of RSS vs. the reduced dimension $s$

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Figure 2. The ExtendedYale data B

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Table Algorithm 1. ADMM Algorithm to solve Model (11)

Data: Initialize $E=0$, $U=0$, $V=0$, $\lambda_1=0$, $\lambda_2=0$.
1 while not convergence do
2 │ Update $Z$ with Equation (14);
3 │ Update $E$ with Equation (15) and $U$, $V$ with Equation (16), respectively;
4 │ Update $\lambda_1$ with Equation (17) and $\lambda_2$ with Equation (18);
5 end

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Table Algorithm 2. The algorithm for proximity operator of $\frac{1}{2\alpha}(\|\cdot\|_{sp}^{k})^2$ with input $\mathbf{h}$

Data: $\mathbf{h}\in\mathbb{R}^n$ and the parameter $\alpha$.
Result:

$\mathbf{p}=\underset{\mathbf{g}}{\arg\min}\left\{\frac{1}{2\alpha}(\|\mathbf{g}\|^{sp}_{k})^2+\frac{1}{2}\|\mathbf{g}-\mathbf{h}\|^2_2\right\}$

1 Let $\tilde{\mathbf{h}}=[\tilde{h}_1,\cdots,\tilde{h}_n]^T$, i.e., $\tilde{h}_i$ is the $i$-th largest element of $|\mathbf{h}|$. Let $\Pi$ be the permutation matrix such that $\tilde{\mathbf{h}}=\Pi|\mathbf{h}|$. For simplicity, define $\tilde{h}_0:=+\infty$, $\tilde{h}_{n+1}:=-\infty$ and $\gamma_{r,l}:=\sum^l_{i=k-r}\tilde{h}_i$
2 Find $r\in\{0,\cdots,k-1\}$ and $l\in\{k,\cdots,n\}$ such that
            $\frac{\tilde{h}_{k-r-1}}{\alpha+1}>\frac{\gamma_{r, l}}{l-k+(\alpha+1)(r+1)}\ge\frac{\tilde{h}_{k-r}}{\alpha+1}, $
            $\tilde{u}_l>\frac{\gamma_{r, l}}{l-k+(\alpha+1)(r+1)}\ge \tilde{g}_{l+1}.$
3 Define
            $ q_i = \left\{ \begin{array}{ll} \frac{\alpha}{\alpha+1}\tilde{h}_i & \textrm{if}\; i=1, \cdots, k-r-1\\ \tilde{h}_i-\frac{\gamma_{r, l}}{l-k+(\alpha+1)(r+1)}\\ & \textrm{if}\; i=k-r, \cdots, l\\ 0 & \textrm{if} \; i=l+1, \cdots, n \end{array} \right.$
4 Set $\mathbf{p}=[p_1,\ldots,p_n]^T$, where $p_i=\mathrm{sign}(h_i)(\Pi^{-1}\mathbf{q})_i$.

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Table 1. The error rate (mean % and median %) for face clustering on Extended Yale dataset B

# Classes	mean/median	SSC	LRR	(3, 3)-SMF		(4, 4)-SMF
# Classes	mean/median	SSC	LRR	error	s	error	s
2	mean	15.83	6.37	3.38	18	3.53	18
2	median	15.63	6.25	2.34	18	2.34	18
3	mean	28.13	9.57	6.19	25	6.06	25
3	median	28.65	8.85	5.73	25	5.73	25
5	mean	37.90	14.86	11.06	35	10.04	35
5	median	38.44	14.38	9.38	35	9.06	35
8	mean	44.25	23.27	23.08	50	22.51	50
8	median	44.82	21.29	27.54	50	26.06	50
10	mean	50.78	29.38	25.36	65	23.91	65
10	median	49.06	32.97	27.19	65	27.34	65

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Table 2. The error rate (mean %/median %) for motion segmentation on Hopkins155 dataset

	SSC	LRR	(3, 3)-SMF	(4, 4)-SMF
Mean	9.28	8.43	6.61	7.16
Median	0.24	1.54	1.20	1.32

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