Graph regularized discriminative non-negative matrix factorization for face recognition

Abstract

Non-negative matrix factorization (NMF) has been widely employed in computer vision and pattern recognition fields since the learned bases can be interpreted as a natural parts-based representation of the input space, which is consistent with the psychological intuition of combining parts to form a whole. In this paper, we propose a novel constrained nonnegative matrix factorization algorithm, called the graph regularized discriminative non-negative matrix factorization (GDNMF), to incorporate into the NMF model both intrinsic geometrical structure and discriminative information which have been essentially ignored in prior works. Specifically, both the graph Laplacian and supervised label information are jointly utilized to learn the projection matrix in the new model. Further we provide the corresponding multiplicative update solutions for the optimization framework, together with the convergence proof. A series of experiments are conducted over several benchmark face datasets to demonstrate the efficacy of our proposed GDNMF.

Appendix (Proof of Theorem)

In order to prove Theorem, we need to show that O ₁ is non-increasing under the updating steps in (19), (20) and (21). For the objective function O ₁, we need to fix H and A if we update W, so, the first term of O ₁ exists. Similarly, we need to fix W and H if we update A, the third term of O ₁ exists. Therefore, we have exactly the same update formula for W and A in GDNMF as in the original NMF. Thus, we can use the convergence proof of NMF to show that O ₁ is nonincreasing under the update step in (20) and (21). These details can be found in [14].

Hence, we only need to prove that O ₁ is non-increasing under the updating step in (19). We follow the similar process depicted in [14]. Our proof make use of an auxiliary function similar to that used in the Expectation-Maximization algorithm [8]. We first give the definition of the auxiliary function.

Definition

G(h, h ^′) is an auxiliary function of F(h) if the following conditions are satisfied.

$$ G(h,h^{'})\geq F(h),~~~~ G(h,h)=F(h) $$

(22)

The above auxiliary function is very important because of the following lemma.

Lemma 1

If G is an auxiliary function of F, then F is non-increasing under the update

$$ h^{(t+1)}=\arg\min\limits_{h} G(h, h^{(t)}) $$

(23)

Proof

F(h ^(t + 1)) ≤ G(h ^(t + 1), h ^(t)) ≤ G(h ^(t), h ^(t)) = F(h ^(t))

Now, we show that the update step for H in (19) is exactly the update in (23) with a proper auxiliary function.

Considering any element h _ab in H, we use F _ab to denote the part of O ₁ which is only relevant to h _ab . It is easy to obtain the following derivatives.

$$ F_{ab}^{'}=\left(\dfrac{\partial O_{1}}{\partial\mathbf{H}}\right)_{ab}=[2\mathbf{W}^{T}(\mathbf{W}\mathbf{H}-\mathbf{X})+2\lambda\mathbf{H}(\mathbf{B}-\mathbf{C})+2\gamma\mathbf{A}^{T}(\mathbf{A}\mathbf{H}-\mathbf{S}) ]_{ab} $$

(24)

$$ F_{ab}^{''}=2(\mathbf{W}^{T}\mathbf{W})_{aa}+2\lambda\mathbf{B}_{bb}-2\lambda\mathbf{C}_{bb}+2\gamma(\mathbf{A}^{T}\mathbf{A})_{aa} $$

(25)

It is enough to show that each F _ab is nonincreasing under the update step of (19) because our update is essentially element-wise. Consequently, we introduce the following lemma. □

Lemma 2

Function

$$\begin{array}{rll} G(h,h_{ab}^{(t)})&=&F_{ab}(h_{ab}^{(t)})+F_{ab}^{'}(h_{ab}^{(t)})(h-h_{ab}^{(t)})\\ &&+\,\frac{(\mathbf{W}^{T}\mathbf{W}\mathbf{H}+\gamma\mathbf{A}^{T}\mathbf{A}\mathbf{H}+\lambda\mathbf{H}\mathbf{B})_{ab}}{h_{ab}^{(t)}}(h-h_{ab}^{(t)})^2 \end{array}$$

(26)

is an auxiliary function of F _ab .

Proof

We only need to prove that $G(h,h_{ab}^{(t)})\geq F_{ab}(h)$ because G(h,h) = F _ab (h) is obvious. Therefore, we first consider the Taylor series expansion of F _ab (h).

$$\begin{array}{rll} F_{ab}(h)&=&F_{ab}(h_{ab}^{(t)})+F_{ab}^{'}(h_{ab}^{(t)})(h-h_{ab}^{(t)})\\ &&+\,[(\mathbf{W}^{T}\mathbf{W})_{aa}+\lambda\mathbf{B}_{bb}-\lambda\mathbf{C}_{bb}+\gamma(\mathbf{A}^{T}\mathbf{A})_{aa}](h-h_{ab}^{(t)})^{2} \end{array}$$

(27)

We compare the (27) with (26) to find that $G(h,h_{ab}^{(t)})\geq F_{ab}(h)$ is equivalent to

$$\dfrac{(\mathbf{W}^{T}\mathbf{W}\mathbf{H}+\gamma\mathbf{A}^{T}\mathbf{A}\mathbf{H}+\lambda\mathbf{H}\mathbf{B})_{ab}}{h_{ab}^{(t)}}\geq (\mathbf{W}^{T}\mathbf{W})_{aa}+\lambda\mathbf{B}_{bb}-\lambda\mathbf{C}_{bb}+\gamma(\mathbf{A}^{T}\mathbf{A})_{aa} $$

(28)

In fact, we have

$$\begin{array}{rll} (\mathbf{W}^{T}\mathbf{W}\mathbf{H}+\gamma\mathbf{A}^{T}\mathbf{A}\mathbf{H})_{ab}&=&\sum\limits_{q=1}^{r}(\mathbf{W}^{T}\mathbf{W})_{aq}h_{qb}^{(t)}+\gamma\sum\limits_{q=1}^{r}(\mathbf{A}^{T}\mathbf{A})_{aq}h_{qb}^{(t)} \\&\geq& (\mathbf{W}^{T}\mathbf{W})_{aa}h_{ab}^{(t)}+\gamma(\mathbf{A}^{T}\mathbf{A})_{aa}h_{ab}^{(t)} \end{array}$$

(29)

and

$$ (\lambda\mathbf{H}\mathbf{B})_{ab}=\lambda\sum\limits_{j=1}^{n}h_{aj}^{(t)}\mathbf{B}_{jb}\geq \lambda h_{ab}^{(t)}\mathbf{B}_{bb}\geq\lambda h_{ab}^{(t)}\mathbf{B}_{bb}-\lambda h_{ab}^{(t)}\mathbf{C}_{bb} $$

(30)

Thus, (28) holds and $G(h,h_{ab}^{(t)})\geq F_{ab}(h)$. We can now demonstrate the convergence of Theorem: □

Proof of Theorem

Replacing $G(h,h_{ab}^{(t)})$ in (23) by (26) results in the following update rule:

$$\begin{array}{rll} h_{ab}^{(t+1)}&=&h_{ab}^{(t)}-h_{ab}^{(t)}\frac{F_{ab}^{'}(h_{ab}^{(t)})}{2(\mathbf{W}^{T}\mathbf{W}\mathbf{H}+\gamma\mathbf{A}^{T}\mathbf{A}\mathbf{H}+\lambda\mathbf{H}\mathbf{B})_{ab}}\\ &=&h_{ab}^{(t)}\frac{(\gamma\mathbf{A}^{T}\mathbf{S}+\mathbf{W}^{T}\mathbf{X}+\lambda\mathbf{H}\mathbf{C})_{ab}}{(\mathbf{W}^{T}\mathbf{W}\mathbf{H}+\gamma\mathbf{A}^{T}\mathbf{A}\mathbf{H}+\lambda\mathbf{H}\mathbf{B})_{ab}} \end{array}$$

(31)

Since (26) is an auxiliary function and F _ab is nonincreasing under this update rule. □