Supervised Discriminative Group Sparse Representation for Mild Cognitive Impairment Diagnosis

Abstract

Research on an early detection of Mild Cognitive Impairment (MCI), a prodromal stage of Alzheimer’s Disease (AD), with resting-state functional Magnetic Resonance Imaging (rs-fMRI) has been of great interest for the last decade. Witnessed by recent studies, functional connectivity is a useful concept in extracting brain network features and finding biomarkers for brain disease diagnosis. However, it still remains challenging for the estimation of functional connectivity from rs-fMRI due to the inevitable high dimensional problem. In order to tackle this problem, we utilize a group sparse representation along with a structural equation model. Unlike the conventional group sparse representation method that does not explicitly consider class-label information, which can help enhance the diagnostic performance, in this paper, we propose a novel supervised discriminative group sparse representation method by penalizing a large within-class variance and a small between-class variance of connectivity coefficients. Thanks to the newly devised penalization terms, we can learn connectivity coefficients that are similar within the same class and distinct between classes, thus helping enhance the diagnostic accuracy. The proposed method also allows the learned common network structure to preserve the network specific and label-related characteristics. In our experiments on the rs-fMRI data of 37 subjects (12 MCI; 25 healthy normal control) with a cross-validation technique, we demonstrated the validity and effectiveness of the proposed method, showing the diagnostic accuracy of 89.19 % and the sensitivity of 0.9167.

Appendices

Appendix A: Derivation of the Definitive Matrices

$$\begin{array}{@{}rcl@{}} &&\frac{1}{N^{c}}\sum\limits_{n~\text{s.t.}~l(n)=c}\left( \mathbf{w}_{r,g}(n)-\hat{w}_{r,g}^{c}\right)^{2}\\ &&{\kern.5pc}=\frac{1}{N^{c}}\sum\limits_{n~\text{s.t.}~l(n)=c}\left( \mathbf{w}_{r,g}{\mathbf{e}_{n}^{c}}-\mathbf{w}_{r,g}\mathbf{m}^{c}\right)^{2} \\ &&{\kern.5pc}=\frac{1}{N^{c}}\sum\limits_{n~\text{s.t.}~l(n)=c}\left\{ \mathbf{w}_{r,g}\left( {\mathbf{e}_{n}^{c}}-\mathbf{m}^{c}\right)\right\}\left\{ \mathbf{w}_{r,g}\left( {\mathbf{e}_{n}^{c}}-\mathbf{m}^{c}\right)\right\}^{T}\\ &&{\kern.5pc}=\frac{1}{N^{c}}\sum\limits_{n~\text{s.t.}~l(n)=c}\mathbf{w}_{r,g}\left( {\mathbf{e}_{n}^{c}}-\mathbf{m}^{c}\right)\left( {\mathbf{e}_{n}^{c}}-\mathbf{m}^{c}\right)^{T}\mathbf{w}^{T}_{r,g}\\ &&{\kern.5pc}=\frac{1}{N^{c}}\mathbf{w}_{r,g}\!\left(\sum\limits_{n~\text{s.t.}~l(n)=c} {\mathbf{e}_{n}^{c}}\mathbf{e}_{n}^{cT}\,-\,{\mathbf{e}_{n}^{c}}\mathbf{m}^{cT} \,-\,\mathbf{m}^{c}\mathbf{e}_{n}^{cT}\,+\,\mathbf{m}^{c}\mathbf{m}^{cT}\right)\mathbf{w}^{T}_{r,g}\\ &&{\kern.5pc}=\mathbf{w}_{r,g}\frac{1}{N^{c}}\left(\mathbf{I}^{c}-\mathbf{I}^{c}\mathbf{M}^{cT}-\mathbf{M}^{c}\mathbf{I}^{cT} +\mathbf{M}^{c}\mathbf{M}^{cT}\right)\mathbf{w}^{T}_{r,g}\\ &&{\kern.5pc}=\mathbf{w}_{r,g}K^{c}\mathbf{w}^{T}_{r,g} \end{array} $$

(15)

where N ^c is the number of training samples of the class c∈{+,−}, w _r,g (n) denotes the n-th element of a vector w _r,g , $\hat {w}_{r,g}^{c}=\frac {1}{N^{c}}{\sum }_{n~\text {s.t.}~l(n)=c}\mathbf {w}_{r,g}(n)$, ${\mathbf {e}_{n}^{c}}$ is an N-dimensional unit vector with the n-th element 1 if l(n)=c, 0 otherwise, I is a square diagonal matrix with $\mathbf {I}_{nn}^{c}=1$ if l(n)=c, 0 otherwise, M ^c is a square matrix with the columns set to $\mathbf {m}^{c}=\left [ {m^{c}_{1}}, \cdots , {m^{c}_{n}}, \cdots , {m^{c}_{N}}\right ]^{T}$, and ${m^{c}_{n}}=\frac {1}{N^{c}}$ if l(n)=c, otherwise 0. $K^{c}=\frac {1}{N^{c}}\left (\mathbf {I}^{c}-\mathbf {I}^{c}\mathbf {M}^{cT}- \mathbf {M}^{c}\mathbf {I}^{cT}+\mathbf {M}^{c}\mathbf {M}^{cT}\right )$.

$$\begin{array}{@{}rcl@{}} f_{W}(\mathbf{w}_{r,g})&=&\frac{1}{N^{+}}\sum\limits_{n~\text{s.t.}~l(n)=\text{`}+\text{'}}\left( \mathbf{w}_{r,g}(n)-\hat{w}_{r,g}^{+}\right)^{2} \\ &&+ \frac{1}{N^{-}}\sum\limits_{n~\text{s.t.}~l(n)=\text{`}-\text{'}}\left( \mathbf{w}_{r,g}(n)-\hat{w}_{r,g}^{-}\right)^{2}\\ &=&\mathbf{w}_{r,g}K^{+}\mathbf{w}^{T}_{r,g}+\mathbf{w}_{r,g}K^{-}\mathbf{w}^{T}_{r,g}\\ &=&\mathbf{w}_{r,g}\left(K^{+}+K^{-}\right)\mathbf{w}^{T}_{r,g}\\ &=&\mathbf{w}_{r,g}\hat{K}\mathbf{w}^{T}_{r,g}\\ &=&\mathbf{w}_{r,g}D_{1}{D_{1}^{T}}\mathbf{w}^{T}_{r,g}~~~~~~~~~~\text{(matrix decomposition)}\\ &=&\left(\mathbf{w}_{r,g}D_{1}\right)\left(\mathbf{w}_{r,g}D_{1}\right)^{T}\\ &=&\left\| \mathbf{w}_{r,g}D_{1}\right\|_{2}^{2} \end{array} $$

(16)

$$\begin{array}{@{}rcl@{}} f_{B}(\mathbf{w}_{r,g})&=&\left( \hat{w}_{r,g}^{+}-\hat{w}_{r,g}^{-}\right)^{2} \\ &=& \left( \mathbf{w}_{r,g}\mathbf{m}^{+}-\mathbf{w}_{r,g}\mathbf{m}^{-}\right)^{2} \\ &=& \left\{ \mathbf{w}_{r,g}\left(\mathbf{m}^{+}-\mathbf{m}^{-} \right)\right\}\left\{ \mathbf{w}_{r,g}\left(\mathbf{m}^{+}-\mathbf{m}^{-} \right)\right\}^{T}\\ &=&\mathbf{w}_{r,g}\left(\mathbf{m}^{+}-\mathbf{m}^{-} \right)\left(\mathbf{m}^{+}-\mathbf{m}^{-} \right)^{T}\mathbf{w}_{r,g}^{T}\\ &=&\left(\mathbf{w}_{r,g}D_{2} \right)\left(\mathbf{w}_{r,g}D_{2} \right)^{T}\\ &=&\left\|\mathbf{w}_{r,g}D_{2} \right\|_{2}^{2} \end{array} $$

(17)

where $D_{2}=\left (\mathbf {m}^{+}-\mathbf {m}^{-} \right )$.

Appendix B: Proof of Two Stage Proximal Operator

Given a target proximal operator of

$$ \uppi(\mathbf{v})= \underset{\mathbf{w}} {\text{argmin}}\frac{1}{2} \left\| \mathbf-\mathbf{v}\right\|_{2}^{2} + {\uplambda}_{1}\left\|\mathbf{w} \right\|_{2} + {\uplambda}_{2}\left(\left\|\mathbf{w} D_{1}\right\|_{2}^{2} - \left\|\mathbf{w} D_{2}\right\|_{2}^{2}\right). $$

(18)

we can decompose it into two proximal operators as follows:

$$\begin{array}{@{}rcl@{}} {\uppi}_{1}(\mathbf{v})&=& \underset{\mathbf{w}} {\text{argmin}}\frac{1}{2} \left\| \mathbf{w}-\mathbf{v}\right\|_{2}^{2} + {\uplambda}_{1}\left\|\mathbf{w} \right\|_{2} \end{array} $$

(19)

$$\begin{array}{@{}rcl@{}} {\uppi}_{1}(\mathbf{v})&=& \underset{\mathbf{w}} {\text{argmin}}\frac{1}{2} \left\| \mathbf{w}\!-\!\mathbf{v}\right\|_{2}^{2} \!+\! \uplambda_{2}\left(\!\left\|\mathbf{w} D_{1}\right\|_{2}^{2} \!-\! \left\|\mathbf{w} D_{2}\right\|_{2}^{2}\!\right).\! \end{array} $$

(20)

Then it holds that

$$ \uppi(\mathbf{v})={\uppi}_{2}\left({\uppi}_{1}(\mathbf{v})\right). $$

(21)

The necessary and sufficient optimality conditions for Eqs. 18, 19 and 20 can be written as

$$\begin{array}{@{}rcl@{}} 0&\in& \uppi(\mathbf{v})-\mathbf{v}+{\uplambda}_{1}\partial g\left(\uppi(\mathbf{v})\right) + {\uplambda}_{2}\partial h\left(\uppi(\mathbf{v})\right) \end{array} $$

(22)

$$\begin{array}{@{}rcl@{}} 0&\in& {\uppi}_{1}(\mathbf{v})-\mathbf{v}+{\uplambda}_{1}\partial g\left(\uppi(\mathbf{v})\right) \end{array} $$

(23)

$$\begin{array}{@{}rcl@{}} 0&\in& {\uppi}_{2}\left({\uppi}_{1}(\mathbf{v})\right)-{\uppi}_{1}(\mathbf{v})+ {\uplambda}_{2}\partial h\left({\uppi}_{2}\left({\uppi}_{1}(\mathbf{v})\right)\right) \end{array} $$

(24)

where the partial derivatives are defined as

$$ \partial g(\mathbf{x})=\left\{ \begin{array}{c c} \frac{\mathbf{x}}{\|\mathbf{x}\|_{2}} & \text{if }\mathbf{x} \neq\mathbf{0}\\ \left\{\mathbf{y}:\|\mathbf{y}\|_{2}\leq1 \right\} & \text{otherwise} \end{array} \right. $$

(25)

and

$$ \partial h(\mathbf{x})=2\mathbf{x}\left(D_{1}{D_{1}^{T}}-D_{2}{D_{2}^{T}}\right). $$

(26)

It follows from Eqs. 24 and 26 that if π₁(v)=0 then ${\uppi }_{2}\left ({\uppi }_{1}(\mathbf {v})\right )=0$. That is, the group sparsity π₁(v) via the group lasso still holds for ${\uppi }_{2}\left ({\uppi }_{1}(\mathbf {v})\right )$. Therefore, we have

$$ 0\in {\uppi}_{2}\left({\uppi}_{1}(\mathbf{v})\right)-{\uppi}_{1}(\mathbf{v})+ {\uplambda}_{2}{\uppi}_{2}\left({\uppi}_{1}(\mathbf{v})\right)\left(D_{1}{D_{1}^{T}}-D_{2}{D_{2}^{T}}\right). $$

(27)

Since Eq. 18 has a unique solution, we can get Eq. 21 from Eqs. 22 and 27. Note that thanks to the matrix multiplication of $D_{1}{D_{1}^{T}}$ in the partial derivative of Eq. 26, there is no need to explicitly decompose the matrix $\hat {K}$ in Eq. 16.