Cognitive Assessment Prediction in Alzheimer’s Disease by Multi-Layer Multi-Target Regression

Abstract

Accurate and automatic prediction of cognitive assessment from multiple neuroimaging biomarkers is crucial for early detection of Alzheimer’s disease. The major challenges arise from the nonlinear relationship between biomarkers and assessment scores and the inter-correlation among them, which have not yet been well addressed. In this paper, we propose multi-layer multi-target regression (MMR) which enables simultaneously modeling intrinsic inter-target correlations and nonlinear input-output relationships in a general compositional framework. Specifically, by kernelized dictionary learning, the MMR can effectively handle highly nonlinear relationship between biomarkers and assessment scores; by robust low-rank linear learning via matrix elastic nets, the MMR can explicitly encode inter-correlations among multiple assessment scores; moreover, the MMR is flexibly and allows to work with non-smooth ℓ_2,1-norm loss function, which enables calibration of multiple targets with disparate noise levels for more robust parameter estimation. The MMR can be efficiently solved by an alternating optimization algorithm via gradient descent with guaranteed convergence. The MMR has been evaluated by extensive experiments on the ADNI database with MRI data, and produced high accuracy surpassing previous regression models, which demonstrates its great effectiveness as a new multi-target regression model for clinical multivariate prediction.

Appendix

Proof

By the definition of the nuclear norm, we can re-write it in terms of traces as follows

$$\begin{array}{@{}rcl@{}} ||S||_{*} & =& tr(\sqrt{S^{\top} S}) = tr(\sqrt{(U{\Sigma} V^{\top})^{\top}(U{\Sigma} V^{\top})})\\ & =& tr(\sqrt{V{\Sigma}^{\top} U^{\top} U {\Sigma} V^{\top}} = tr(\sqrt{V{\Sigma}^{\top} {\Sigma} V^{\top}})\\ & =& tr(\sqrt{V{\Sigma}^{\top} {\Sigma} V^{\top}})\\ & =& tr(\sqrt{V{\Sigma} V^{\top} V{\Sigma} V^{\top}})\\ & =& tr(V {\Sigma} V^{\top})\\ & =& tr({\Sigma}) \end{array} $$

(20)

Therefore, the nuclear norm of S can be also defined as the sum of the singular value decomposition of S. From (13), we have

$$ \partial S=\partial U{\Sigma} V^{\top}+U\partial{\Sigma} V^{\top}+U{\Sigma}\partial V^{\top}, $$

(21)

which gives rise to

$$ U\partial{\Sigma} V^{\top}=\partial S-\partial U{\Sigma} V^{\top}-U{\Sigma}\partial V^{\top}. $$

(22)

Multiplying U^⊤ on both sides of (22), we have

$$ U^{\top} U\partial{\Sigma} V^{\top} V =U^{\top}\partial SV-U^{\top}\partial U{\Sigma} V^{\top} V -U^{\top} U{\Sigma}\partial V^{\top} V $$

(23)

Since U is also an orthogonal matrix, we achieve

$$ \partial{\Sigma} =U^{\top}\partial SV-U^{\top}\partial U{\Sigma} - {\Sigma}\partial V^{\top} V. $$

(24)

Note that we have the fact that

$$ 0 = \partial I = \partial (U^{\top} U) = \partial U^{\top} U + U^{\top} \partial U, $$

(25)

where I is an identity matrix, and therefore U^⊤∂U is an antisymmetric matrix. We have

$$\begin{array}{@{}rcl@{}} tr(U^{\top} \partial U {\Sigma}) & =& tr((U^{\top} \partial U {\Sigma})^{\top}) = tr({\Sigma}^{\top} \partial U^{\top} U)\\ &=& - tr({\Sigma} U^{\top} \partial U) = - tr(U^{\top} \partial U {\Sigma}) \end{array} $$

(26)

which indicates that tr(U^⊤∂UΣ) = 0. Similarly, we also have tr(Σ∂V ^⊤V ) = 0. Therefore, we achieve

$$ tr(\partial{\Sigma}) =tr(U^{\top}\partial SV) $$

(27)

By taking the derivative of ||S||_∗ w.r.t. S, we obtain

$$ \frac{\partial \|S\|_{*}}{\partial S} =\frac{ tr(\partial{\Sigma})}{\partial S}=\frac{ tr(U^{\top}\partial SV)}{\partial S}= U V^{\top} $$

(28)

which closes the proof. □