Semi-supervised classifier with projection graph embedding for motor imagery electroencephalogram recognition

Abstract

Brain computer interface (BCI) based on motor imagery (MI) provides a communication channel between the brain and a computer or other communication devices. In BCI system, electroencephalogram (EEG) is the most widely used brain signals, which has the function of responding to the physiological and emotional information of the brain. The recognition of MI EEG signals is the core technology of MI-BCI system. In this study, semi-supervised learning strategy is adopted in model training, and semi-supervised classifier with projection graph embedding (SCPGE) is proposed for MI EEG recognition. SCPGE combines graph embedding projection and sparse constraint into semi-supervised least squares model to learn the discriminative structure of EEG data in projection space, and adaptively learns the similarity matrix and pseudo-label matrix based on manifold learning. Unlike the “two-step” strategy of constructing graph embedding and classifier, these two factors, together with subspace projection and pseudo-label estimation, are simultaneously considered in the objection function and optimized in the iteration process. In addition, SCPGE uses the class by class based ℓ_{2, 1}-norm inter class sparsity constraint to exploit the correlation between samples, so as to obtain the discriminative projection matrix and classifier. Experiments on the publicly available MI EEG datasets show that SCPGE classifier can significantly improve the recognition accuracy.

Appendices

Appendix A

Optimization of objective Eq. (9).

As each ith element in s_iis independent, it can be calculated separately for each i, and Eq. (11) can be simplified as,

$${\displaystyle \begin{array}{c}\underset{{\textbf{s}}_i}{{\min}}{\left\Vert {\textbf{s}}_i+\frac{1}{2\alpha }{\textbf{d}}_i\right\Vert}_2^2,\\ {}s.t.\sum\limits_{j=1}^n{s}_{ij}=1,{s}_{ij}\ge 0,\end{array}}$$

(21)

where the matrix D = [d₁, d₂, …, d_n] is defined as ${d}_{ij}={d}_{ij}^x+\mu {d}_{ij}^f$.

Through the Lagrange multipliers method, the closed-form solution of s_ican be obtained as,

$${\textbf{s}}_i={\left(\frac{1+\sum\limits_{j=1}^m{\hat{d}}_{i_j}}{m}\textbf{1}-{\textbf{d}}_i\right)}_{+},$$

(22)

where x₊ =  max (x, 0)₊. The elements of ${\hat{\textbf{d}}}_i$ are the same as those of d_i with the ascending order. m is the parameter to control the number of nearest neighbors.

According to [18], the trade-off parameter α can be represented as,

$$\alpha =\frac{1}{n}\sum\limits_{i=1}^n\left(\frac{m}{2}{\hat{d}}_{i_{m+1}}-\frac{1}{2}\sum\limits_{j=1}^m{\hat{d}}_{i_j}\right).$$

(23)

Appendix B

Optimization of objective Eq. (12).

Eq. (12) can be represented as,

$${\displaystyle \begin{array}{l}\underset{{\textbf{F}}_u}{{\min}} Tr\left({{\textbf{F}}_u}^T\left(\mu {\textbf{L}}_{s, uu}+\lambda \textbf{I}\right){\textbf{F}}_u\right)-2 Tr\left(\left(\lambda {{\textbf{Z}}_u}^T\textbf{PW}-\mu {{\textbf{F}}_L}^T{\textbf{L}}_{s, lu}\right){\textbf{F}}_u\right),\\ {}s.t.{\textbf{F}}_u\ge 0,{\textbf{F}}_u\textbf{1}=\textbf{1}\end{array}}$$

(24)

The Lagrange dual method is used to solve Eq. (24). Let A = μL_{s, uu} + λI, B = λZ_u^TPW − μF_L^TL_{s, lu}, Eq. (24) can be represented as,

$$\underset{{\textbf{F}}_u}{{\min}} Tr\left({{\textbf{F}}_u}^T{\textbf{AF}}_u\right)-2 Tr\left({\textbf{BF}}_u\right)-\eta Tr\left({{\textbf{F}}_u}^T{\textbf{1}}_{n\times c}-\textbf{I}\right)- Tr\left(\overline{\boldsymbol{\upbeta}}{{\textbf{F}}_u}^T\right),$$

(25)

where η > 0and $\overline{\boldsymbol{\upbeta}}\ge \textbf{0}$ are Lagrangian multipliers.

Thus, F_u has the closed-form solution as,

$${\textbf{F}}_u={\left(2\textbf{A}\right)}^{-1}\left(2{\textbf{B}}^T+\eta {\textbf{1}}_{n\times c}+\overline{\boldsymbol{\upbeta}}\right).$$

(26)

Appendix C

Optimization of objective Eq. (16).

Similarly, we denote the diagonal matrix E ∈ R^p × p， $\textbf{E}=\left[\begin{array}{c}\frac{1}{2{\left\Vert {p}_1\right\Vert}_2}\dots 0\\ {}\vdots \vdots \\ {}0\cdots \frac{1}{2{\left\Vert {p}_p\right\Vert}_2}\end{array}\right]$, Eq. (16) can be represented as,

$${\displaystyle \begin{array}{l}\underset{\bf{P}}{{\min}} Tr\left({bf{Z}}^T{\textbf{P}\bf{L}}_s{\bf{P}}^T\bf{Z}\right)+\theta tr\left({\bf{P}}^T\bf{EP}\right)+\lambda {\left\Vert {\bf{Z}}^T\bf{P}\bf{W}-\bf{F}\right\Vert}_F^2+\delta \sum\limits_{i=1}^c Tr\left(\left({bf{W}}^T{\bf{P}}^T{\bf{Z}}_i\right){\bf{D}}_i\left({\bf{Z}}_i^T\bf{P}\bf{W}\right)\right),\\ {}s.t.{\bf{P}}^T\bf{P}=\bf{I}.\end{array}}$$

(27)

Using the Lagrangian multipliers method, Eq. (27) can be represented as,

$$\underset{P}{{\min}} Tr\left({\textbf{Z}}^T{\textbf{P}\textbf{L}}_s{\textbf{P}}^T\textbf{Z}\right)+\theta Tr\left({\textbf{P}}^T\textbf{EP}\right)+\lambda {\left\Vert {\textbf{Z}}^T\textbf{PW}-\textbf{F}\right\Vert}_F^2+\delta \sum\limits_{i=1}^c Tr\left(\left({\textbf{W}}^T{\textbf{P}}^T{\textbf{Z}}_i\right){\textbf{D}}_i\left({\textbf{Z}}_i^T\textbf{PW}\right)\right)+\frac{\sigma }{2} Tr\left({\textbf{P}}^T\textbf{P}-\textbf{I}\right),$$

(28)

where σ is the Lagrangian multiplier.

It is difficult to obtain the closed solution of P through Eq.(28). According to the gradient descent algorithm, the solution expression of P is

$${\textbf{P}}_{k+1}={\textbf{P}}_k-\alpha \frac{\nabla \textbf{P}}{\left\Vert \nabla \textbf{P}\right\Vert }.$$

(29)

The first derivative ∇P of P is,

$$\nabla \textbf{P}=2\left({\textbf{ZZ}}^T{\textbf{PL}}_s+\theta \textbf{EP}+\lambda {\textbf{ZZ}}^T{\textbf{PWW}}^T-\lambda {\textbf{ZFW}}^T+\delta {\textbf{Z}}_i{\textbf{D}}_i{\textbf{Z}}_i^T{\textbf{PWW}}^T\right)+\sigma \textbf{P}.$$

(30)