Abstract
Based on recently proposed Non-negative Matrix Factorization (NMF) and Graph Embedded (GE) techniques with Discriminant Criterion (DC), we present in this paper a new algorithm of Face Representation and Recognition (FRR) called Discriminant Graph Regularized Non-negative Matrix Factorization (DGNMF) for dimensionality reduction (DR). Here, we firstly encode the geometrical class information by constructing an affinity graph using the DGNMF algorithm. After this, we determine a matrix factorization which adequately represents the graph structure. Finally, we conduct experiments to prove that DGNMF provides a better representation and achieves higher face recognition rates than previous approaches.
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Acknowledgements
This work is partially supported by National Key R&D Program Grant No. 2017YFC0804002, the National Science Foundation of China under Grant Nos. 61876213, 61462064, 6177227, 61861033, 61603192, the China Postdoctoral Science Foundation under Grant No. 2016 M600674, the Natural Science Fund of Jiangsu Province under Grants BK20161580, BK20171494 and China’s Jiangxi Province Natural Science Foundation (No. 20181BAB202022), and the Fund of China’s Jiangxi Provincial Department of Education (No. GJJ170599).
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Appendix
Appendix
1.1 The convergent analysis
Since the algorithm is an iterative method, we analyze its convergence in this section. Here we first introduce the definition of auxiliary function [19] to prove the convergence.
Definition 1. [19] Z(v, v ′) is an auxiliary function for F(v) if the conditions
are satisfied, then F is non-increasing under the update
Proof
Now we need to show that the updating step for V in Eq. (16) is exactly the update in Eq. (18) with a proper auxiliary function.
We can rewrite the objective function ℓ of DGNMF in Eq. (10) as follows:
Considering any element vab in V, we use Fab to denote the part of ℓ which is only relevant to vab. It is easy to check that
Since our update is essentially element-wise, it is sufficient to show that each Fab is non-increasing under the update step of Eq. (16).
Lemma 1: Function
is an auxiliary function for Fab, and the part of ℓ which is only relevant to vab.
Proof
Since Z(v, v) = Fab(v) is obvious, we need only show that \( Z\left(v,{v}_{ab}^{(t)}\right)\ge {F}_{ab}(v) \). To do this, we compare the Taylor series expansion of Fab(v).
with Eq. (22) to find that \( G\left(v,{v}_{ab}^{(t)}\right)\ge {F}_{ab}(v) \) is equivalent to
We have
and
Thus, Eq. (24) holds and \( Z\left(v,{v}_{ab}^{(t)}\right)\ge {F}_{ab}(v) \).
With the above preparations, we have the following theorem:
Theorem 1
The objective function ℓ is non-increasing under the updating rules uij and vij.
We can now demonstrate the convergence of Theorem 1.
Proof
Replacing \( Z\left(v,{v}_{ab}^{(t)}\right) \) in Eq. (18) by Eq. (22) results in the update rule:
Since Eq. (22) is an auxiliary function, Fab is non-increasing under this update rules.
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Wan, M., Lai, Z., Ming, Z. et al. An improve face representation and recognition method based on graph regularized non-negative matrix factorization. Multimed Tools Appl 78, 22109–22126 (2019). https://doi.org/10.1007/s11042-019-7454-2
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DOI: https://doi.org/10.1007/s11042-019-7454-2