Age estimation based on improved discriminative Gaussian process latent variable model

Abstract

Affected by various factors (genes, living habits and so on), different people present distinct aging patterns. To discover the underlying trend of aging patterns, we propose an effective age estimation method based on DGPLVM (Discriminative Gaussian Process Latent Variable Model). DGPLVM is a kind of discriminative latent variable method for manifold learning. It discovers the low-dimensional manifold by employing a discriminative prior distribution over the latent space. DGPLVM with KFDA (Kernel Fisher Discriminant Analysis) prior has been studied and successfully applied to face verification. Different with face verification which is a two-class problem, age estimation is a linearly inseparable multi-class problem. In this paper, DGPLVM with KFDA is reformulated to get the low-dimensional representations for age estimation. After low-dimensional representations are obtained, Gaussian process regression model is adopted to find the age regressor mapping low-dimensional representations to ages. Experimental results on two widely used databases FG-NET and MORPH show that reformulated DGPLVM with KFDA is a good application in age estimation and achieves comparable results to state-of-the arts.

Appendix A: Derivations of (18) and (21)

Given (14),

$$J(\boldsymbol{w} )=\frac{\boldsymbol{w}^{\mathrm{T}}S_{b}\boldsymbol{w}}{\boldsymbol{w}\left( S_{w}+\lambda I_{n}\right)\boldsymbol{w}} $$

According to fisher discriminative analysis, the best projection direction w ^∗ satisfies S _b w ^∗=α(S _w +λ I _n )w ^∗. Assuming the samples are centralized, i.e. μ=0, then

$$\begin{array}{@{}rcl@{}} S_{b}\boldsymbol{w}^{\ast}&=&\sum\limits_{c=1}^{C} \frac{n_{c}}{n}\left( \boldsymbol{\mu}_{c}-\boldsymbol{\mu}\right)\left( \boldsymbol{\mu}_{c}- \boldsymbol{\mu}\right)^{\mathrm{T}\boldsymbol{w}^{\ast}}\\ &=&\sum\limits_{c=1}^{C} \frac{n_{c}}{n}\boldsymbol{\mu}_{c}\boldsymbol{\mu}_{c}^{\mathrm{T}\boldsymbol{w}^{\ast}} \end{array} $$

(32)

Let $U_{c} = \left [\phi \left (\boldsymbol {z}_{1}^{(c)}\right ),\cdots ,\phi \left (\boldsymbol {z}_{n_{c}}^{(c)}\right )\right ]$, U=[U ₁,⋯ ,U _C ], then

$$\begin{array}{@{}rcl@{}} \boldsymbol{w}^{\ast} &=& \left( S_{w}+\lambda I_{n}\right)^{-1}S_{b}\boldsymbol{w}^{\ast}\\ &=& \left( S_{w}+\lambda I_{n}\right)^{-1}\sum\limits_{c=1}^{C}{U_{c}}\frac{1}{n_{c}}\mathbf{1}_{n_{c}}\\ &=& \left( S_{w}+\lambda I_{n}\right)^{-1}U \boldsymbol{a} \end{array} $$

(33)

where $\boldsymbol {a} = \left [\frac {1}{n_{1}}\mathbf {1}_{n_{1}}^{\mathrm {T}},\cdots ,\frac {1}{n_{C}} \mathbf {1}_{n_{C}}^{\mathrm {T}}\right ]^{\mathrm {T}}$. Considering $S_{w} = {\sum }_{c=1}^{C} P_{c}\varSigma _{c}, P_{c} = \frac {n_{c}}{n}$, then

$$\begin{array}{@{}rcl@{}} \varSigma_{c} &=& \frac{1}{n_{c}}\sum\limits_{i=1}^{n_{c}}\left( \phi\left( \boldsymbol{z}_{i}^{(c)}\right)-\boldsymbol{\mu}_{c}\right)\left( \phi\left( \boldsymbol{z}_{i}^{(c)}\right)-\mu_{c}\right)^{\mathrm{T}}\\ &=& \frac{1}{n_{c}}\sum\limits_{i=1}^{n_{c}}\phi\left( \boldsymbol{z}_{i}^{(c)}\right)\phi\left( \boldsymbol{z}_{i}^{(c)}\right)^{\mathrm{T}}-\boldsymbol{\mu}_{c}\boldsymbol{\mu}_{c}^{\mathrm{T}}\\ &=&\frac{1}{n_{c}}U_{c}U_{c}^{\mathrm{T}}-\frac{1}{n_{c}^{2}}U_{c}\mathbf{1}_{n_{c}}\mathbf{1}_{n_{c}}^{\mathrm{T}}U_{c}^{\mathrm{T}}\\ &=& U_{c}J_{c}J_{c}U_{c}^{\mathrm{T}} \end{array} $$

(34)

where $J_{c} = \frac {1}{\sqrt {n_{c}}}\left (I_{n_{c}}-\frac {1}{n_{c}}\mathbf {1}_{n_{c}}\mathbf {1}_{n_{c}}^{\mathrm {T}}\right )$. Let

$$\begin{array}{@{}rcl@{}} A_{c} = \sqrt{P_{c}}J_{c} &=& \frac{\sqrt{n_{c}}}{\sqrt{n}}J_{c} = \frac{1}{\sqrt{n}}\left( I_{n_{c}}-\mathbf{1}_{n_{c}}\mathbf{1}_{n_{c}}^{\mathrm{T}}\right)\\ A &=& \left( \begin{array}{lll} A_{1} & &\\ &\ddots&\\ & &A_{C} \end{array}\right) \end{array} $$

(35)

then

$$\begin{array}{@{}rcl@{}} S_{w} &=& \sum\limits_{c=1}^{C}P_{c}\varSigma_{c}\\ &=&\sum\limits_{c=1}^{C}P_{c}U_{c}J_{c}J_{c}U_{c}^{\mathrm{T}}\\ &=&\sum\limits_{c=1}^{C}U_{c}\sqrt{P_{c}}J_{c}\sqrt{P_{c}}J_{c}U_{c}^{\mathrm{T}}\\ &=&\sum\limits_{c=1}^{C}U_{c}A_{c}A_{c}U_{c}^{\mathrm{T}}\\ &=&UAAU^{\mathrm{T}} \end{array} $$

(36)

According to Woodbury identity,

$$\begin{array}{@{}rcl@{}} \left( S_{W}+\lambda I_{n}\right)^{-1}&=&\left( \lambda I_{n}+UAAU^{\mathrm{T}}\right)^{-1}\\ &=&\frac{1}{\lambda}\left[I_{n}-UA\left( \lambda I_{n}+AU^{\mathrm{T}}UA\right)^{-1}AU^{\mathrm{T}}\right] \end{array} $$

(37)

Combining with equation (A.3), we have

$$\begin{array}{@{}rcl@{}} \boldsymbol{w}^{\ast}&=& \left( S_{w}+\lambda I_{n}\right)^{-1}Ua\\ &=& \frac{1}{\lambda}\left[I_{n}-UA\left( \lambda I_{n}+AU^{\mathrm{T}}UA\right)^{-1}AU^{\mathrm{T}}\right]\\ &=& U\frac{1}{\lambda}\left[I_{n}-A\left( \lambda I_{n}+AGA\right)^{-1}AG\right]\boldsymbol{a} \end{array} $$

(38)

Considering both (38) and (16), a ^∗ (18) can be obtained

$$ \boldsymbol{a}^{\ast} = \frac{1}{\lambda}\left[I_{n}-A\left( \lambda I_{n}+AGA\right)^{-1}AG\right]\boldsymbol{a} $$

(39)

For S _b ,

$$\begin{array}{@{}rcl@{}} S_{b} &=& \sum\limits_{c=1}^{C}\frac{n_{c}}{n}\boldsymbol{\mu}_{c}\boldsymbol{\mu}_{c}^{\mathrm{T}}\\ &=& \sum\limits_{c=1}^{C}\frac{n_{c}}{n}U_{c}\frac{1}{n_{c}}\mathbf{1}_{n_{c}}\left( U_{c}\frac{1}{n_{c}}\mathbf{1}_{n_{c}}\right)^{\mathrm{T}}\\ & =& \sum\limits_{c=1}^{C} U_{c}\frac{1}{nn_{c}}\mathbf{1}_{n_{c}}\mathbf{1}_{n_{c}}^{\mathrm{T}}U_{c}^{\mathrm{T}}\\ & =& UWU^{\mathrm{T}} \end{array} $$

(40)

where

$$\begin{array}{@{}rcl@{}} W &=& \left( \begin{array}{lll} W_{1} & &\\ &\ddots&\\ & &W_{C} \end{array}\right)\\ W_{c} &=& \frac{1}{nn_{c}}\boldsymbol{1}_{n_{c}}\boldsymbol{1}_{n_{c}}^{\mathrm{T}} \end{array} $$

(41)

Plugging S _b ,S _w ,w ^∗ into (14), J ^∗ is obtained

$$\begin{array}{@{}rcl@{}} J^{\ast}&=&\frac{\boldsymbol{w}^{\ast\mathrm{T}}S_{b}\boldsymbol{w}^{\ast}}{\boldsymbol{w}^{\ast\mathrm{T}}\left( S_{w}+\lambda I_{n}\right)\boldsymbol{w}^{\ast}}\\ &=&\frac{\boldsymbol{a}^{\ast\mathrm{T}}GWG\boldsymbol{a}^{\ast}}{\boldsymbol{a}^{\ast\mathrm{T}}\left( GAAG+\lambda G\right)\boldsymbol{a}^{\ast}} \end{array} $$

(42)