An improve face representation and recognition method based on graph regularized non-negative matrix factorization

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.

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

$$ Z\left(v,{v}^{\prime}\right)\ge F(v),Z\left(v,v\right)=F(v) $$

(17)

are satisfied, then F is non-increasing under the update

$$ {v}^{\left(t+1\right)}=\underset{v}{\arg \kern0.2em \min}\kern0.2em Z\left(v,{v}^{(t)}\right) $$

(18)

Proof

$$ F\left({v}^{\left(t+1\right)}\right)\le Z\left({v}^{\left(t+1\right)},{v}^{(t)}\right)\le Z\left({v}^{(t)},{v}^{(t)}\right)=F\left({v}^{(t)}\right) $$

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:

$$ {\displaystyle \begin{array}{c}\ell ={\left\Vert X-U{V}^T\right\Vert}_F^2+{\lambda}_1 tr\left({V}^T{L}^CV\right)-{\lambda}_2 tr\left({V}^T{L}^pV\right)\\ {}\begin{array}{l}=\sum \limits_{i=1}^M\sum \limits_{j=1}^N{\left({x}_{ij}-\sum \limits_{k=1}^K{u}_{ik}{v}_{jk}\right)}^2+{\lambda}_1\sum \limits_{k=1}^K\sum \limits_{j=1}^N\sum \limits_{l=1}^M{v}_{jk}{L}_{jl}^c{v}_{lk}\\ {}-{\lambda}_2\sum \limits_{k=1}^K\sum \limits_{j=1}^N\sum \limits_{l=1}^M{v}_{jk}{L}_{jl}^p{v}_{lk}\end{array}\end{array}} $$

(19)

Considering any element v_ab in V, we use F_ab to denote the part of ℓ which is only relevant to v_ab. It is easy to check that

$$ {\displaystyle \begin{array}{l}{F}_{ab}^{\prime }={\left(\frac{\partial {\ell}_1}{\partial v}\right)}_{ab}\\ {}\kern1.6em ={\left(-2{X}^TU+2V{U}^TU+2{\lambda}_1{L}^cV-2{\lambda}_2{L}^pV\right)}_{ab}\end{array}} $$

(20)

$$ {F}_{ab}^{{\prime\prime} }=2{\left({U}^TU\right)}_{bb}+2{\lambda}_1{L}_{aa}^c-2{\lambda}_2{L}_{aa}^p $$

(21)

Since our update is essentially element-wise, it is sufficient to show that each F_ab is non-increasing under the update step of Eq. (16).

Lemma 1: Function

$$ {\displaystyle \begin{array}{l}Z\left(v,{v}_{ab}^{(t)}\right)={F}_{ab}\left({v}_{ab}^{(t)}\right)+{F}_{ab}^{\prime}\left({v}_{ab}^{(t)}\right)\left(v-{v}_{ab}^{(t)}\right)\kern0.2em \\ {}\kern0.5em +\frac{{\left(V{U}^TU\right)}_{ab}+{\lambda}_1{\left({D}^cV\right)}_{ab}+{\lambda}_2{\left({W}^pV\right)}_{ab}}{v_{ab}^{(t)}}{\left(v-{v}_{ab}^{(t)}\right)}^2\end{array}} $$

(22)

is an auxiliary function for F_ab, and the part of ℓ which is only relevant to v_ab.

Proof

Since Z(v, v) = F_ab(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 F_ab(v).

$$ {\displaystyle \begin{array}{l}{F}_{ab}(v)={F}_{ab}\left({v}_{ab}^{(t)}\right)+{F}_{ab}^{\prime}\left({v}_{ab}^{(t)}\right)\left(v-{v}_{ab}^{(t)}\right)\kern0.2em \\ {}\kern2.899999em +\left[{\left({U}^TU\right)}_{bb}+{\lambda}_1{L}_{aa}^c-{\lambda}_2{L}_{aa}^p\right]{\left(v-{v}_{ab}^{(t)}\right)}^2\end{array}} $$

(23)

with Eq. (22) to find that $G\left(v,{v}_{ab}^{(t)}\right)\ge {F}_{ab}(v)$ is equivalent to

$$ {\displaystyle \begin{array}{l}\frac{{\left(V{U}^TU\right)}_{ab}+{\lambda}_1{\left({D}^cV\right)}_{ab}+{\lambda}_2{\left({W}^pV\right)}_{ab}}{v_{ab}^{(t)}}\\ {}\ge {\left({U}^TU\right)}_{bb}+{\lambda}_1{L}_{aa}^c-{\lambda}_2{L}_{aa}^p\end{array}} $$

(24)

We have

$$ {\left(V{U}^TU\right)}_{ab}=\sum \limits_{l=1}^k{v}_{al}^{(t)}{\left({U}^TU\right)}_{lb}\ge {v}_{ab}^{(t)}{\left({U}^TU\right)}_{bb} $$

(25)

and

$$ {\displaystyle \begin{array}{l}{\lambda}_1{\left({D}^cV\right)}_{ab}+{\lambda}_2{\left({W}^pV\right)}_{ab}\\ {}={\lambda}_1\sum \limits_{j=1}^M{D}_{aj}^c{v}_{jb}^{(t)}+{\lambda}_2\sum \limits_{j=1}^M{W}_{aj}^p{v}_{jb}^{(t)}\ge {\lambda}_1{D}_{aa}^c{v}_{ab}^{(t)}+{\lambda}_2{W}_{aa}^p{v}_{ab}^{(t)}\\ {}\ge {\lambda}_1{\left({D}^c-{W}^c\right)}_{aa}{v}_{ab}^{(t)}-{\lambda}_2{\left({D}^p-{W}^p\right)}_{aa}{v}_{ab}^{(t)}\\ {}={\lambda}_1{L}_{aa}^c{v}_{ab}^{(t)}-{\lambda}_2{L}_{aa}^p{v}_{ab}^{(t)}\end{array}} $$

(26)

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 u_ij and v_ij.

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:

$$ {\displaystyle \begin{array}{c}{v}_{ab}^{\left(t+1\right)}={v}_{ab}^{(t)}-{v}_{ab}^{(t)}\frac{F_{ab}^{\prime}\left({v}_{ab}^{(t)}\right)}{2{\left(V{U}^TU\right)}_{ab}+2{\lambda}_1{\left({D}^CV\right)}_{ab}+2{\lambda}_2{\left({W}^PV\right)}_{ab}}\\ {}={v}_{ab}^{(t)}\frac{{\left({X}^TU+{\lambda}_1{W}^CV+{\lambda}_2{D}^PV\right)}_{ab}}{{\left(V{U}^TU+{\lambda}_1{D}^CV+{\lambda}_2{W}^PV\right)}_{ab}}\end{array}} $$

(27)

Since Eq. (22) is an auxiliary function, F_ab is non-increasing under this update rules.