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# Pattern Discovery in Brain Imaging Genetics via SCCA Modeling with a Generic Non-convex Penalty

## Abstract

Brain imaging genetics intends to uncover associations between genetic markers and neuroimaging quantitative traits. Sparse canonical correlation analysis (SCCA) can discover bi-multivariate associations and select relevant features, and is becoming popular in imaging genetic studies. The L1-norm function is not only convex, but also singular at the origin, which is a necessary condition for sparsity. Thus most SCCA methods impose $${\ell }_{{\bf{1}}}$$-norm onto the individual feature or the structure level of features to pursuit corresponding sparsity. However, the $${\ell }_{{\bf{1}}}$$-norm penalty over-penalizes large coefficients and may incurs estimation bias. A number of non-convex penalties are proposed to reduce the estimation bias in regression tasks. But using them in SCCA remains largely unexplored. In this paper, we design a unified non-convex SCCA model, based on seven non-convex functions, for unbiased estimation and stable feature selection simultaneously. We also propose an efficient optimization algorithm. The proposed method obtains both higher correlation coefficients and better canonical loading patterns. Specifically, these SCCA methods with non-convex penalties discover a strong association between the APOE e4 rs429358 SNP and the hippocampus region of the brain. They both are Alzheimer’s disease related biomarkers, indicating the potential and power of the non-convex methods in brain imaging genetics.

## Introduction

By identifying the associations between genetic factors and brain imaging measurements, brain imaging genetics intends to model and understand how genetic factors influence the structure or function of human brain1,2,3,4,5,6,7,8,9,10,11,12,13,14. Both genetic biomarkers such as single nucleotide polymorphisms (SNPs), and brain imaging measurements such as imaging quantitative traits (QTs) are multivariate. To address this problem, bi-multivariate association models, such as multiple linear regression15, reduced rank regression16,17,18, parallel independent component analysis19, partial least squares regression20,21, canonical correlation analysis (CCA)22 and their sparsity-inducing variants23, have been widely used to uncover the joint effect of multiple SNPs on one or multiple QTs. Among them, SCCA (Sparse CCA), which can discover bi-multivariate relationships and extract relevant features, is becoming popular in brain imaging genetics.

The CCA technique has been introduced for several decades24. CCA can only perform well when the number of observations is larger than the combined feature number of the two views. Unfortunately, the problem usually is a large-p-small-n problem in the biomedical and biology studies. And it gets even worse because in CCA we are facing a large-(p + q)-small-n problem. In order to overcome this limitation, sparse CCA (SCCA)25,26,27,28,29,30,31,32,33,34,35,36 employs a sparsity inducing regularization term to select a small set of relevant features and has received increasing attention. The $${\ell }_{1}$$-norm based SCCA method25 has gained great success for its sparsity pursuing capability. After that, there are many SCCA variants based on the $${\ell }_{1}$$-norm. For examples, the fused lasso penalty imposes the $${\ell }_{1}$$-norm onto the ordered pairwise features25, and the group lasso penalty imposes the $${\ell }_{1}$$-norm onto the group of features29,32. Further, the graph lasso or the graph guided lasso can be viewed as imposing the $${\ell }_{1}$$-norm onto the pairwise features defined by an undirected graph29.

However, the $${\ell }_{1}$$-norm penalty shows the conflict of optimal prediction and consistent feature selection37. In penalized least squares modeling, Fan and Li38 showed that a good penalty function should meet three properties. First, the penalty function should be singular at the origin to produce sparse results. Second, it should produce continuous models for stable model selection, and third, the penalty function should not penalize large coefficients to avoid estimation bias. The $${\ell }_{1}$$-norm penalty is successful in feature selection because it is singular at the origin. On the contrary, the $${\ell }_{1}$$-norm penalty over-penalizes large coefficients, and thus it may be suboptimal with respect to the estimation risk39,40. The $${\ell }_{0}$$-norm function which only involves the number of nonzero features is an ideal sparsity-inducing penalty. However, it is neither convex nor continuous, and thus solving $${\ell }_{0}$$-norm constrained problem is NP-hard41.

A number of non-convex penalties are proposed as the surrogate of the $${\ell }_{0}$$-norm to handle this issue. These penalties includes the $${\ell }_{\gamma }$$-norm (0 < γ < 1) penalty42, the Geman penalty43, the Smoothly Clipped Absolute Deviation (SCAD) penalty38, the Laplace penalty44, the Minimax Concave Penalty (MCP)45, the Exponential-Type Penalty (ETP)46 and the Logarithm penalty47. These non-convex functions have attractive theoretical properties for they all are singular at the origin and leave those larger coefficients unpenalized. Though they have gained great success in generalized linear models (GLMs), it is an unexplored topic to apply them to the SCCA models for achieving sparsity and unbiased prediction simultaneously.

Therefore, it is essential and of great interest to investigate performances of various SCCA models based on these non-convex penalties. A major challenge of non-convex function is the computational complexity. The local quadratic approximation (LQA) technique is introduced to solve the SCAD penalizing problem38. LQA approximates the objective by a locally quadratic expression which can be solved like a ridge constrained problem. Inspired by this, in this paper, we propose a generic non-convex SCCA models with these non-convex penalties, and propose a unified optimization algorithm based on the LQA technique and the Alternate Convex Search (ACS) method48. Using both synthetic data and real imaging genetic data, the experimental results show that with appropriate parameters, the non-convex SCCA methods have better performance on both canonical loading patterns and correlation coefficients estimation than the $${\ell }_{1}$$-norm based SCCA methods.

## Methods

Throughout this paper, scalars are denoted as italic letters, column vectors as boldface lowercase letters, and matrices as boldface capitals. The $$\Vert {\bf{u}}\Vert$$ denotes the Euclidean norm of a vector u.

### Preliminaries

#### Sparse Canonical Correlation Analysis (SCCA)

Let $${\bf{X}}\in { {\mathcal R} }^{n\times p}$$ be a matrix representing the SNP biomarkers data, where n is the number of participants and p is the number of SNPs. Let $${\bf{Y}}\in { {\mathcal R} }^{n\times q}$$ be the QT data with q being the number of imaging measurements. A typical SCCA model is defined as

$$\mathop{{\rm{\min }}}\limits_{{\bf{u}},{\bf{v}}}-{{\bf{u}}}^{{\rm{{\rm T}}}}{{\bf{X}}}^{{\rm{{\rm T}}}}{\bf{Y}}{\bf{v}}$$
(1)
$$s\mathrm{.}t\mathrm{.\ \ }{\Vert {\bf{X}}{\bf{u}}\Vert }^{2}\le 1,{\Vert {\bf{Y}}{\bf{v}}\Vert }^{2}\le 1,{\rm{\Omega }}({\bf{u}})\le {c}_{1},{\rm{\Omega }}({\bf{v}})\le {c}_{2},$$

where Xu and Yv are the canonical variables, u and v are the corresponding canonical vectors we desire to estimate, and c 1, c 2 are the tuning parameters that control the sparsity level of the solution. The penalty function could be the $${\ell }_{1}$$-norm penalty, or its variants such as the fused lasso, group lasso and graph lasso25,27,29,32,34.

#### Non-convex Penalty Functions for SCCA

In this paper, we investigate seven non-convex surrogate penalties of $${\ell }_{0}$$-norm in the SCCA model. They are singular at the origin, which is essential to achieve sparsity in the solution. And they do not overly penalize large coefficients. In order to facilitate a unified description, we denote the non-convex penalty as

$${{\rm{\Omega }}}_{{\rm{nc}}}({\bf{u}})=\sum _{i=1}^{p}{P}_{\lambda ,\gamma }(|{u}_{i}|),$$
(2)

where λ and γ are nonnegative parameters, and P λ,γ (|u i |) is a non-convex function. We absorb λ into the penalty because it cannot be decoupled from several penalties, such as the SCAD function38. We here have seven penalties and they are described in Table 1 and visualized in Fig. 1, where for clarity we have dropped the subscript i in u i . There is a sharp point at the origin for each of them, indicating that they are singular at the origin. This is essential to achieve sparseness in the solution. Besides, these curves are concave in |u i | and monotonically decreasing on (−∞, 0], and monotonically increasing on [0, ∞). Therefore, though these penalties are not convex, they are piecewise continuously differentiable and their supergradients exist on both (−∞, 0] and [0, ∞)49. Table 1 also shows their supergradients P λ,γ (|u i |) with respect to |u i |.

### The Proposed Non-convex SCCA Model and Optimization Algorithm

Replacing the $${\ell }_{1}$$-norm constraints in the SCCA model, we define the unified non-convex SCCA model as follows

$$\mathop{{\rm{\min }}}\limits_{{\bf{u}},{\bf{v}}}-{{\bf{u}}}^{{\rm{{\rm T}}}}{{\bf{X}}}^{{\rm{{\rm T}}}}{\bf{Y}}{\bf{v}}+{{\rm{\Omega }}}_{{\rm{nc}}}({\bf{u}})+{{\rm{\Omega }}}_{{\rm{nc}}}({\bf{v}})$$
(3)
$$s\mathrm{.}t\mathrm{.\ \ }{\Vert {\bf{X}}{\bf{u}}\Vert }^{2}\le \mathrm{1,}\,{\Vert {\bf{Y}}{\bf{v}}\Vert }^{2}\le \mathrm{1,}$$

where Ωnc(u) and Ωnc(v) can be any of the non-convex functions listed in Table 1.

To solve the non-convex SCCA problem, we use the Lagrangian method,

$${\mathcal L} {\boldsymbol{(}}{\bf{u}}{\boldsymbol{,}}{\bf{v}}{\boldsymbol{)}}=-{{\bf{u}}}^{{\rm{{\rm T}}}}{{\bf{X}}}^{{\rm{{\rm T}}}}{\bf{Y}}{\bf{v}}+{{\rm{\Omega }}}_{{\rm{nc}}}({\bf{u}})+{{\rm{\Omega }}}_{{\rm{nc}}}({\bf{v}})+\frac{{\alpha }_{1}}{2}({\Vert {\bf{X}}{\bf{u}}\Vert }^{2}-1)+\frac{{\alpha }_{2}}{2}({\Vert {\bf{Y}}{\bf{v}}\Vert }^{2}-1),$$
(4)

which is equivalent to

$${\mathcal L} {\boldsymbol{(}}{\bf{u}}{\boldsymbol{,}}{\bf{v}}{\boldsymbol{)}}=-{{\bf{u}}}^{{\rm{{\rm T}}}}{{\bf{X}}}^{{\rm{{\rm T}}}}{\bf{Y}}{\bf{v}}+{{\rm{\Omega }}}_{{\rm{nc}}}({\bf{u}})+{{\rm{\Omega }}}_{{\rm{nc}}}({\bf{v}})+\frac{{\alpha }_{1}}{2}{\Vert {\bf{X}}{\bf{u}}\Vert }^{2}+\frac{{\alpha }_{2}}{2}{\Vert {\bf{Y}}{\bf{v}}\Vert }^{2}$$
(5)

from the point of view of optimization. α 1, α 2, λ 1, λ 2 and γ are nonnegative tuning parameters. Next we will show how to solve this non-convex problem.

The first term −u Τ X Τ Yv on the right of equation (5) is biconvex in u and v. $${\Vert {\bf{X}}{\bf{u}}\Vert }^{2}$$ is convex in u, and $${\Vert {\bf{Y}}{\bf{v}}\Vert }^{2}$$ is convex in v. It remains to approximate both Ωnc(u) and Ωnc(v) and transform them into convex ones.

The local quadratic approximation (LQA) technique was introduced to quadratically expresses the SCAD penalty38. Based on LQA, we here show how to represent these non-convex penalties in a unified way. First, we have the first-order Taylor expansion of $${P}_{{\lambda }_{1},\gamma }(\sqrt{\mu })$$ at μ 0 P λ,γ ((μ)1/2) at μ 0

$${P}_{\lambda ,\gamma }(\sqrt{\mu })\approx {P}_{\lambda ,\gamma }(\sqrt{{\mu }_{0}})+{P^{\prime} }_{\lambda ,\gamma }(\sqrt{{\mu }_{0}})\frac{1}{2\sqrt{{\mu }_{0}}}(\mu -{\mu }_{0}),$$
(6)

where μ 0 and μ are neighbors, e.g., the estimates at two successive iterations during optimization. Substituting $$\mu ={u}_{i}^{2}$$ and $${\mu }_{0}={({u}_{i}^{t})}^{2}$$ into (6), we have

$${P}_{\lambda ,\gamma }(|{u}_{i}|)\approx {P}_{\lambda ,\gamma }(|{u}_{i}^{t}|)+{P^{\prime} }_{\lambda ,\gamma }(|{u}_{i}^{t}|)\frac{1}{\mathrm{2|}{u}_{i}^{t}|}({u}_{i}^{2}-{({u}_{i}^{t})}^{2})$$
(7)

with $${P^{\prime} }_{\lambda ,\gamma }(|{u}_{i}^{t}|)$$ being the supergradient of $${P}_{\lambda ,\gamma }(|{u}_{i}^{t}|)$$ (as shown in Table 1) at $$|{u}_{i}^{t}|$$.

Then we obtain a quadratic approximation to Ωnc(u):

$${{\rm{\Omega }}}_{{\rm{nc}}}({\bf{u}})=\sum _{i\mathrm{=1}}^{p}{P}_{\lambda ,\gamma }(|{u}_{i}|)\approx \sum _{i\mathrm{=1}}^{p}\frac{{P^{\prime} }_{\lambda ,\gamma }(|{u}_{i}^{t}|)}{\mathrm{2|}{u}_{i}^{t}|}{u}_{i}^{2}+{C}_{{\bf{u}}},$$
(8)

where

$${C}_{{\bf{u}}}=\sum _{i\mathrm{=1}}^{p}[{P}_{\lambda ,\gamma }(|{u}_{i}^{t}|)-\frac{1}{2}{P^{\prime} }_{\lambda ,\gamma }(|{u}_{i}^{t}|)|{u}_{i}^{t}|]$$

is not a function of u and thus will not contribute to the optimization.

In a similar way, we can construct a quadratic approximation to Ωnc(v)

$${{\rm{\Omega }}}_{{\rm{nc}}}({\bf{v}})=\sum _{j\mathrm{=1}}^{q}{P}_{\lambda ,\gamma }(|{v}_{j}|)\approx \sum _{j\mathrm{=1}}^{q}\frac{{P^{\prime} }_{\lambda ,\gamma }(|{v}_{j}^{t}|)}{\mathrm{2|}{v}_{j}^{t}\mathrm{|}}{v}_{j}^{2}+{C}_{{\bf{v}}},$$
(9)

where

$${C}_{{\bf{v}}}=\sum _{j\mathrm{=1}}^{q}[{P}_{\lambda ,\gamma }(|{v}_{j}^{t}|)-\frac{1}{2}{P^{\prime} }_{\lambda ,\gamma }(|{v}_{j}^{t}|)|{v}_{j}^{t}|]$$

is not a function of v and makes no contribute towards the optimization.

Denote the estimates of u and v in the t-th iteration as u t and v t, respectively. To update the estimates of u and v in the (t + 1)-th iteration, we substitute the approximate functions of Ωnc(u) and Ωnc(v) in equations (8) and (9) into $${\mathcal L} {\boldsymbol{(}}{\bf{u}}{\boldsymbol{,}}{\bf{v}}{\boldsymbol{)}}$$ in 5, and solve the resultant approximate version of the original problem:

$$\begin{array}{rcl}{\rm{\arg }}\,{\rm{\min }}\, {\mathcal L} {\boldsymbol{(}}{\bf{u}}{\boldsymbol{,}}{\bf{v}}{\boldsymbol{)}} & = & {\rm{\arg }}\,{\rm{\min }}\,-{{\bf{u}}}^{{\rm{{\rm T}}}}{{\bf{X}}}^{{\rm{{\rm T}}}}{\bf{Y}}{\bf{v}}+\sum _{i\mathrm{=1}}^{p}\frac{{P^{\prime} }_{{\lambda }_{1},\gamma }(|{u}_{i}^{t}|)}{\mathrm{2|}{u}_{i}^{t}|}{u}_{i}^{2}\\ & & +\,\sum _{j\mathrm{=1}}^{q}\frac{{P^{\prime} }_{{\lambda }_{1},\gamma }(|{v}_{j}^{t}|)}{\mathrm{2|}{v}_{j}^{t}|}{v}_{j}^{2}+\frac{{\alpha }_{1}}{2}||{\bf{X}}{\bf{u}}{||}^{2}+\frac{{\alpha }_{2}}{2}||{\bf{Y}}{\bf{v}}{||}^{2}\end{array}$$
(10)

Obviously, the equation (10) is a quadratical expression, and is biconvex in u and v. This means it is convex in terms of u given v, and vice versa. Then according to the alternate convex search (ACS) method which is designed to solve biconvex problems48, the (t + 1)-th estimation of u and v can be calculated via

$$\begin{array}{rcl}{{\bf{u}}}^{t+1} & = & {\rm{\arg }}\,\mathop{{\rm{\min }}}\limits_{{\bf{u}}}-{{\bf{u}}}^{{\rm{{\rm T}}}}{{\bf{X}}}^{{\rm{{\rm T}}}}{\bf{Y}}{{\bf{v}}}^{t}+\sum _{i\mathrm{=1}}^{p}\frac{{P^{\prime} }_{{\lambda }_{1},\gamma }(|{u}_{i}^{t}|)}{\mathrm{2|}{u}_{i}^{t}|}{u}_{i}^{2}+\frac{{\alpha }_{1}}{2}||{\bf{X}}{\bf{u}}{||}^{2},\\ {{\bf{v}}}^{t+1} & = & {\rm{\arg }}\,\mathop{{\rm{\min }}}\limits_{{\bf{v}}}-{({{\bf{u}}}^{t+1})}^{{\rm{{\rm T}}}}{{\bf{X}}}^{{\rm{{\rm T}}}}{\bf{Y}}{\bf{v}}+\sum _{j\mathrm{=1}}^{q}\frac{{P^{\prime} }_{{\lambda }_{2},\gamma }(|{v}_{j}^{t}|)}{\mathrm{2|}{v}_{j}^{t}|}{v}_{j}^{2}+\frac{{\alpha }_{2}}{2}||{\bf{Y}}{\bf{v}}{||}^{2}\mathrm{.}\end{array}$$
(11)

Both equations above are quadratic, and thus their closed-form solutions exist. Taking the partial derivative of $${\mathcal L} {\boldsymbol{(}}{\bf{u}}{\boldsymbol{,}}{\bf{v}}{\boldsymbol{)}}$$ in (5) with respect to u and v and setting the results to zero, we have

$${\bf{0}}\in -{{\bf{X}}}^{{\rm{{\rm T}}}}{\bf{Y}}{\bf{v}}+({{\bf{D}}}_{1}^{t}+{\alpha }_{1}{{\bf{X}}}^{{\rm{{\rm T}}}}{\bf{X}}){\bf{u}},$$
(12)
$${\bf{0}}\in -{{\bf{Y}}}^{{\rm{{\rm T}}}}{\bf{X}}{\bf{u}}+({{\bf{D}}}_{2}^{t}+{\alpha }_{2}{{\bf{Y}}}^{{\rm{{\rm T}}}}{\bf{Y}}){\bf{v}},$$
(13)

where $${{\bf{D}}}_{1}^{t}$$ is a diagonal matrix with the i-th diagonal entry as $$\frac{{P^{\prime} }_{{\lambda }_{1},\gamma }(|{u}_{i}^{t}|)}{|{u}_{i}^{t}|}$$ (i[1, p]). It can be calculated by taking the partial derivative of equation (7) with respect to u i . $${{\bf{D}}}_{2}^{t}$$ is also a diagonal matrix with the j-th diagonal entry as $$\frac{{P^{\prime} }_{{\lambda }_{1},\gamma }(|{v}_{j}^{t}|)}{|{v}_{j}^{t}|}$$ (j[1, q]), and can be computed similarly. However, the i-th element of $${{\bf{D}}}_{1}^{t}$$ does not exist if $${u}_{i}^{t}=0$$. According to perturbed version of LQA50, we address this by adding a slightly perturbed term. Then the i-th element of $${{\bf{D}}}_{1}^{t}$$ is

$${{\bf{D}}}_{1}^{t}(i,i)=\frac{{P^{\prime} }_{{\lambda }_{1},\gamma }(|{u}_{i}|)}{|{u}_{i}|+\zeta }$$
(14)

where ζ is a tiny positive number. Hunter and Li50 showed that this modification guarantees optimizing the equation (11). Then we have the updating expressions at the (t + 1)-th iteration

$${{\bf{u}}}^{t+1}=({{\bf{D}}}_{1}^{t}+{\alpha }_{1}{{\bf{X}}}^{{\rm{{\rm T}}}}{\bf{X}}{)}^{-1}{{\bf{X}}}^{{\rm{{\rm T}}}}{\bf{Y}}{{\bf{v}}}^{t},$$
(15)
$${{\bf{v}}}^{t+1}=({{\bf{D}}}_{2}^{t}+{\alpha }_{2}{{\bf{Y}}}^{{\rm{{\rm T}}}}{\bf{Y}}{)}^{-1}{{\bf{Y}}}^{{\rm{{\rm T}}}}{\bf{X}}{{\bf{u}}}^{t+1}\mathrm{.}$$
(16)

We alternate between the above two equations to graduate refine the estimates for u and v until convergence. The pseudo code of the non-convex SCCA algorithm is described in Algorithm 1.

### Computational Analysis

In Algorithm 1, Step 3 and Step 6 are linear in the dimension of u and v, and are easy to compute. Step 4 and Step 7 are the critical steps of proposed algorithm. Since we have closed-form updating expressions, they can be calculated via solving a system of linear equations with quadratic complexity which avoids computing the matrix inverse with cubic complexity. Step 5 and 8 are the re-scale step and very easy to calculate. Therefore, the whole algorithm is efficient.

### Data Availability

The synthetic data sets generated in this work are available from the corresponding authors’ web sites, http://www.escience.cn/people/dulei/code.html and http://www.iu.edu/ shenlab/tools/ncscca/. The real data set is publicly available in the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database repository, http://adni.loni.usc.edu.

## Experiments and Datasets

### Data Description

#### Synthetic Dataset

There are four data sets with sparse true signals for both u and v, i.e., only a small subset of features are nonzero. The number of features of both u and v are larger than the observations to simulate a large-(p + q)-small-n task. The generating process is as follows. We first generate u and v with most feature being zero. After that, the latent variable z is constructed from Gaussian distribution N(0, I n × n ). Then we create the data X from $${{\bf{x}}}_{i}\sim N({z}_{i}{\bf{u}},{\sum }_{x})$$ and data $${{\bf{y}}}_{i}\sim N({z}_{i}{\bf{v}},{\sum }_{y})$$, where (∑ x ) jk = exp(−|u j  − u k |) and (∑ y ) jk = exp(−|v j  − v k |). The first three sets have 250 features for u and 600 ones for v, but they have different correlation coefficients. There are 500 features and 900 features in u and v respectively for the last data set. We show the true signal of every data set in Fig. 2 (top row).

#### Real Neuroimaging Genetics Dataset

The real neuroimaging genetics dataset were collected from 743 participants, and the details was presented in Table 2. There were 163 candidate SNP biomarkers from the AD-risk genes, e.g., APOE, in the genotyping data. The structural MRI scans were processed with voxel-based morphometry (VBM) in SPM851,52. Briefly, scans were aligned to a T1-weighted template image, segmented into gray matter (GM), white matter (WM) and cerebrospinal fluid (CSF) maps, normalized to MNI space, and smoothed with an 8mm FWHM kernel. We subsampled the whole brain and generated 465 voxels spanning the whole brain ROIs. The regression technique was employed to remove the effects of the baseline age, gender, education, and handedness for these VBM measures. The aim of this study is to evaluate the correlation between the SNPs and the VBM measures, and further identify which SNPs and ROIs are associated.

### Experimental Setup

#### Benchmarks

In this paper, we are mainly interested in whether these non-convex SCCA methods could enhance the performance of $${\ell }_{1}$$-SCCA method based on our motivation. It is reasonable to employ the $${\ell }_{1}$$-norm based methods in comparison. Therefore, the structure-aware SCCA methods such as28,29,32,34 are not contained here as benchmark. Based on different mathematical techniques, there are three different $${\ell }_{1}$$-SCCA algorithms. They are the singular value decomposition based method25, the primal-dual based method29 and the LQA based method32. Though the latter two are proposed for capturing group or network structure, they can be easily reformulated to the $${\ell }_{1}$$-norm constrained methods, such as setting the parameters associated with the structure penalty to zero29. Therefore, to make the comparison fair and convincing, we choose all of them as benchmarks. With a slight abuse of notation, we use the penalty name to refer a non-convex SCCA method, e.g. ETP for ETP based SCCA method. For the $${\ell }_{1}$$-norm based methods, we call them L1-SCCA25, L1-S2CCA32, and L1-NSCCA29.

#### Parameter Tuning

There are four parameters λ i (i = 1, 2) and α i (i = 1, 2) associated with the non-convex SCCA methods, and one pivotal parameter γ. According to their equations, these non-convex penalties can approximate the $${\ell }_{0}$$-norm by providing an appropriate γ. In this situation, the λ i and α i play a very weak role because theoretically the $${\ell }_{0}$$-norm penalized problem does not rely on the parameters. Based on this consideration, we here only tune the γ other than tuning λ i and α i by a grid search strategy. This reduces the time consumption dramatically but does not affect the performance significantly. Further, we observe that two γ's perform similarly if they are not significantly different. Thus the tuning range of γ is not continuous. Besides, we set γ = 3.7 for SCAD penalty since38 suggested that this is a very reasonable choice. The details of tuning range for each penalty are contained in Table 3. For λ i and α i , we simply set them to 1 in this study.

#### Termination Criterion

We use $${{\rm{\max }}}_{i}|{u}_{i}^{t+1}-{u}_{i}^{t}|\le \varepsilon$$ and $${{\rm{\max }}}_{j}|{v}_{j}^{t+1}-{v}_{j}^{t}|\le \varepsilon$$ as the termination condition for Algorithm 1, where ε is the user defined error bound. In this study, we set ε = 10−5 according to experiments. All methods use the same setup, i.e., the same partition of the five-fold cross-validation, running on the same platform.

### Results on Synthetic Data

Figure 2 shows the heat maps of canonical loadings estimated from all SCCA methods, where each row corresponds to an experimental method. We clearly observe that the non-convex SCCA methods and L1-SCCA correctly identify the identical signal positions to the ground truth across four data sets. Besides true signals, L1-SCCA introduces several undesired signals which makes it be inferior to our methods. As a contrast, L1-NSCCA finds out an incomplete proportion of the ground truth, and L1-S2CCA performs unstably as it fails on some folds. Moreover, we also prioritize these methods using the AUC (area under ROC) criterion in Table 4, where a higher value indicates a better performance. The results exhibit that the non-convex SCCA methods have the highest score at almost every case. L1-SCCA scores similarly to the proposed methods, but later we can see it pays the price at a reduced prediction ability. Table 5 presents the estimated correlation coefficients on both training and testing data, where the best values are shown in boldface. The proposed SCCA methods alternatively gain the best value, and the Log method wins out for the most times. This demonstrates that the non-convex methods outperform $${\ell }_{1}$$-norm based SCCA methods in terms of the prediction power. In summary, the proposed methods identify accurate and sparse canonical loading patterns and obtain high correlation coefficients simultaneously, while those $${\ell }_{1}$$-norm based SCCA methods cannot.

### Results on Real Neuroimaging Genetics Data

In this real data study, the genotyping data is denoted by X, and the imaging data is denoted by Y. The u is a vector of weights of all SNPs, and v is a vector of weights of all imaging markers.The canonical correlation coefficients are defined as Pearson correlation coefficient between Xu and Yv, i.e., $${({\bf{X}}{\bf{u}})}^{{\rm{{\rm T}}}}{\bf{Y}}{\bf{v}}/(\Vert {\bf{X}}{\bf{u}}\Vert \Vert {\bf{Y}}{\bf{v}}\Vert )$$.

Figure 3 presents the heat maps regrading the canonical loadings generated from the training set. In this figure, each row shows two weights of a SCCA method, where a larger weight stands for a more importance. The weight associated with the SNPs is on the left panel, and that associated with the voxels is on the right. The proposed non-convex SCCA methods obtain very clean and sparse weights for both u and v. The largest signal on the genetic side is the APOE e4 SNP rs429358, which has been previously reported to be related to AD53. On the right panel, the largest signal for all SCCA methods comes from the hippocampus region. This is one of the most notable biomarkers as an indicator of AD, since atrophy of hippocampus has been shown to be related to brain atrophy and neuron loss measured with MRI in AD cohort53. In addition, the L1-S2CCA and SCAD methods identify a weak signal from the parahippocampal gyrus, which is previously reported as an early biomarker of AD54. On some folds, the Log method also finds out the lingual region, parahippocampal gyrus, vermis region. Interestingly, all the three regions have shown to be correlated to AD, and could be further considered as an indicating biomarker that can be observed prior to a dementia diagnosis. For example, Sjöbeck and Englund reported that molecular layer gliosis and atrophy in the vermis are clearly severer in AD patients than in the health controls55. This is meaningful since the non-convex SCCA methods identify the correct clue for further investigation. On this account, both L1-SCCA and L1-NSCCA are not good choices since they identify too many signals, which may misguide subsequent investigation. The figure shows that L1-S2CCA could be an alternative choice for sparse imaging genetics analysis, but it performs unstably across the five folds. And, the non-convex methods is more consistent and stable than those $${\ell }_{1}$$-SCCA methods. To show the results more clearly, we map the canonical weights (averaged across 5 folds) regarding the imaging measurements from each SCCA method onto the brain in Fig. 4. The figure confirms that the L1-SCCA and L1-NSCCA find out many signals that are not sparse. The L1-S2CCA identifies fewer signals than both L1-SCCA and L1-NSCCA, but more than all these non-convex SCCA methods. All the non-convex SCCA only highlights a small region of the whole brain. This again reveals that the proposed methods have better canonical weights which reduces the effort of further investigation.

Besides, we include both training and testing correlation coefficients in Table 6, where their mean and standard deviation are shown. The training results of all methods are similar, with the Log method gains the highest value of 0.33 ± 0.03. As for the testing results, which is our primary interest, all the non-convex SCCA methods obtain better values than these $${\ell }_{1}$$-SCCA methods. Besides, the difference between the training and testing performance of the proposed methods is much smaller than that of three $${\ell }_{1}$$-SCCA methods. This means that the non-convex methods have better generalization performance as they are less likely to fall into overfitting issue. The result of this real imaging genetics data reveals that the proposed SCCA methods can extract more accurate and sparser canonical weights for both genetic and imaging biomarkers, and obtain higher correlation coefficients than those $${\ell }_{1}$$-SCCA methods.

## Conclusion

We have proposed a unified non-convex SCCA model and an efficient optimization algorithm using a family of non-convex penalty functions. These penalties are concave and piecewise continuous, and thus piecewise differentiable. We approximate these non-convex penalties by an $${\ell }_{2}$$ function via the local quadratic approximation (LQA)38. Therefore, the proposed algorithm is effective and runs fast.

We compare the non-convex methods with three state-of-the-art $${\ell }_{1}$$-SCCA methods using both simulation data and real imaging genetics data. The simulation data have different ground truth structures. The results on the simulation data show that the non-convex SCCA methods identify cleaner and better canonical loadings than the three $${\ell }_{1}$$-SCCA methods, i.e. L1-SCCA25, L1-S2CCA32, and L1-NSCCA29. These non-convex methods also recover higher correlation coefficients than $${\ell }_{1}$$-SCCA methods, demonstrating that $${\ell }_{1}$$-SCCA methods have suboptimal prediction capability as they may over penalize large coefficients. The results on the real data show that the proposed methods discover a pair of meaningful genetic and brain imaging biomarkers, while the $${\ell }_{1}$$-SCCA methods return too many irrelevant signals. The correlation coefficients show that the non-convex SCCA methods hold better testing values. This verifies our motivation that the non-convex penalty can improve the prediction ability, and thus has better generalization capability. Obviously, the parameter γ plays a key role in these non-convex penalties. In the future work, we will investigate how to choose a reasonable γ; and explore how to incorporate structure information into the model as structure information extraction is an important task for brain imaging genetics as well as biology studies.

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A comprehensive list of consortium members appears at the end of the paper

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## Acknowledgements

Data collection and sharing for this project was funded by the Alzheimer’s Disease Neuroimaging Initiative (ADNI) (National Institutes of Health Grant U01 AG024904) and DOD ADNI (Department of Defense award number W81XWH-12-2-0012). ADNI is funded by the National Institute on Aging, the National Institute of Biomedical Imaging and Bioengineering, and through generous contributions from the following: AbbVie, Alzheimer’s Association; Alzheimer’s Drug Discovery Foundation; Araclon Biotech; BioClinica, Inc.; Biogen; Bristol-Myers Squibb Company; CereSpir, Inc.; Cogstate; Eisai Inc.; Elan Pharmaceuticals, Inc.; Eli Lilly and Company; EuroImmun; F. Hoffmann-La Roche Ltd and its affiliated company Genentech, Inc.; Fujirebio; GE Healthcare; IXICO Ltd.; Janssen Alzheimer Immunotherapy Research & Development, LLC.; Johnson & Johnson Pharmaceutical Research & Development LLC.; Lumosity; Lundbeck; Merck & Co., Inc.; Meso Scale Diagnostics, LLC.; NeuroRx Research; Neurotrack Technologies; Novartis Pharmaceuticals Corporation; Pfizer Inc.; Piramal Imaging; Servier; Takeda Pharmaceutical Company; and Transition Therapeutics. The Canadian Institutes of Health Research is providing funds to support ADNI clinical sites in Canada. Private sector contributions are facilitated by the Foundation for the National Institutes of Health (www.fnih.org). The grantee organization is the Northern California Institute for Research and Education, and the study is coordinated by the Alzheimer’s Therapeutic Research Institute at the University of Southern California. ADNI data are disseminated by the Laboratory for Neuro Imaging at the University of Southern California. L. Du was supported by the National Natural Science Foundation of China (61602384); the Natural Science Basic Research Plan in Shaanxi Province of China (2017JQ6001); the China Postdoctoral Science Foundation (2017M613202); and the Fundamental Research Funds for the Central Universities (3102016OQD0065) at Northwestern Polytechnical University. This work was also supported by the National Institutes of Health R01 EB022574, R01 LM011360, U01 AG024904, P30 AG10133, R01 AG19771, UL1 TR001108, R01 AG 042437, R01 AG046171, R01 AG040770; the Department of Defense W81XWH-14-2-0151, W81XWH-13-1-0259, W81XWH-12-2-0012; the National Collegiate Athletic Association 14132004 at Indiana University.

## Author information

### Affiliations

1. #### School of Automation, Northwestern Polytechnical University, Xi’an, 710072, China

• Lei Du
• , Junwei Han
•  & Lei Guo
2. #### Radiology and Imaging Sciences, Indiana University School of Medicine, Indianapolis, IN, 46202, USA

• Kefei Liu
• , Xiaohui Yao
• , Jingwen Yan
• , Shannon L. Risacher
• , Andrew J. Saykin
• , Li Shen
• , Tatiana M. Foroud
• , Kelley Faber
• , Sungeun Kim
• , Kwangsik Nho
• , Martin R. Farlow
• , Ann Marie Hake
• , Brandy R. Matthews
• , Jared R. Brosch
•  & Scott Herring
3. #### University of California, San Francisco, USA

• Michael W. Weiner
• , Norbert Schuff
• , Howard J. Rosen
• , Bruce L. Miller
•  & David Perry
4. #### University of Southern California, Los Angeles, USA

• Paul Aisen
• , Arthur W. Toga
• , Karen Crawford
• , Scott Neu
• , Lon S. Schneider
• , Sonia Pawluczyk
• , Mauricio Beccera
• , Liberty Teodoro
•  & Bryan M. Spann
5. #### Mayo Clinic, Rochester, Minnesota, USA

• Ronald Petersen
• , Clifford R. Jack
• , Matthew Bernstein
• , Bret Borowski
• , Jeff Gunter
• , Matt Senjem
• , Prashanthi Vemuri
• , David Jones
• , Kejal Kantarci
• , Sara S. Mason
• , Colleen S. Albers
• , David Knopman
•  & Kris Johnson
6. #### University of California, Berkeley, Berkeley, USA

• William Jagust
•  & Susan Landau
7. #### University of Pennsylvania, Philadelphia, USA

• John Q. Trojanowki
• , Leslie M. Shaw
• , Virginia Lee
• , Magdalena Korecka
• , Michal Figurski
• , Steven E. Arnold
• , Jason H. Karlawish
• , David A. Wolk
•  & Christopher M. Clark
8. #### University of California, Davis, Davis, USA

• Laurel Beckett
• , Danielle Harvey
• , Charles DeCArli
• , Evan Fletcher
• , Pauline Maillard
• , John Olichney
• , Charles DeCarli
•  & Owen Carmichael
9. #### Brigham and Women’s Hospital/Harvard Medical School, Boston, USA

• Robert C. Green
• , Reisa A. Sperling
• , Keith A. Johnson
10. #### Washington University St. Louis, St. Louis, USA

• John Morris
• , Marc Raichle
• , David Holtzman
• , Nigel J. Cairns
• , Erin Franklin
• , Lisa Taylor-Reinwald
• , Beau Ances
• , Maria Carroll
• , Mary L. Creech
• , Erin Franklin
• , Mark A. Mintun
• , Stacy Schneider
•  & Angela Oliver
11. #### Prevent Alzheimer’s Disease, 2020, Rockville, USA

• Zaven Khachaturian
12. #### Siemens, Munich, Germany

• Greg Sorensen
13. #### Alzheimer’s Association, Illinois, USA

• Maria Carrillo
14. #### University of Pittsburgh, Pennsylvania, USA

• Lew Kuller
• , Chet Mathis
• , Oscar L. Lopez
• , Mary Ann Oakley
•  & Donna M. Simpson
15. #### Cornell University, New York, USA

• Steven Paul
• , Norman Relkin
• , Gloria Chaing
• , Michael Lin
•  & Lisa Ravdin
16. #### Albert Einstein College of Medicine of Yeshiva University, New York, USA

• Peter Davies
17. #### AD Drug Discovery Foundation, New York, USA

• Howard Fillit
18. #### Acumen Pharmaceuticals, California, USA

• Franz Hefti
19. #### Northwestern University, Illinois, USA

• M. Marcel Mesulam
• , Emily Rogalski
• , Kristine Lipowski
• , Sandra Weintraub
• , Borna Bonakdarpour
• , Diana Kerwin
• , Chuang-Kuo Wu
•  & Nancy Johnson
20. #### National Institute of Mental Health, Maryland, USA

• William Potter
21. #### Brown University, Rhode Island, USA

• Peter Snyder

23. #### University of Washington, Washington, USA

• Tom Montine
24. #### University of California, San Diego, California, USA

• Ronald G. Thomas
• , Michael Donohue
• , Sarah Walter
• , Devon Gessert
• , Tamie Sather
• , Gus Jiminez
• , Archana B. Balasubramanian
• , Jennifer Mason
• , Iris Sim
• , Leon Thal
• , James Brewer
• , Helen Vanderswag

• Nick Fox
26. #### University of California, Los Angeles, California, USA

• Paul Thompson
• , Liana Apostolova
• , Kathleen Tingus
• , Ellen Woo
• , Daniel H. S. Silverman
• , Po H. Lu
•  & George Bartzokis
27. #### University of Michigan, Michigan, USA

• Robert A. Koeppe
• , Judith L. Heidebrink
•  & Joanne L. Lord
28. #### University of Utah, Utah, USA

• Norm Foster
29. #### Banner Alzheimer’s Institute, Arizona, USA

• Eric M. Reiman
• , Kewei Chen
• , Pierre Tariot
• , Anna Burke
• , Ann Marie Milliken
•  & Stephanie Reeder
30. #### University of California, Irvine, California, USA

• Steven Potkin
•  & Dana Nguyen
31. #### National Institute on Aging, Maryland, USA

• Neil Buckholtz
•  & John Hsiao
32. #### Johns Hopkins University, Maryland, USA

• Marilyn Albert
• , Daniel D’Agostino
•  & Stephanie Kielb
33. #### Richard Frank Consulting, New Hampshire, USA

• Richard Frank
34. #### Oregon Health and Science University, Oregon, USA

• Jeffrey Kaye
• , Joseph Quinn
• , Lisa Silbert
• , Betty Lind
• , Raina Carter
•  & Sara Dolen
35. #### Baylor College of Medicine, Texas, USA

• Rachelle S. Doody
• , Javier Villanueva-Meyer
• , Valory Pavlik
• , Victoria Shibley
• , Munir Chowdhury
• , Susan Rountree
•  & Mimi Dang
36. #### Columbia University Medical Center, New York, USA

• Yaakov Stern
• , Lawrence S. Honig
•  & Karen L. Bell
37. #### University of Alabama-Birmingham, Alabama, USA

• Daniel Marson
• , David Geldmacher
• , Marissa Natelson Love
• , Randall Griffith
• , David Clark
• , John Brockington
•  & Erik Roberson
38. #### Mount Sinai School of Medicine, New York, USA

• Hillel Grossman
•  & Effie Mitsis
39. #### Rush University Medical Center, Rush University, Illinois, USA

• Raj C. Shah
•  & Leyla deToledo-Morrell
40. #### Wien Center, Florida, USA

• Ranjan Duara
• , Maria T. Greig-Custo
•  & Warren Barker
41. #### New York University, New York, USA

• , Mohammed O. Sheikh
• , Anaztasia Ulysse
42. #### Duke University Medical Center, North Carolina, USA

• P. Murali Doraiswamy
• , Jeffrey R. Petrella
• , Terence Z. Wong
•  & Edward Coleman
43. #### University of Kentucky, Kentucky, USA

• Charles D. Smith
• , Greg Jicha
• , Peter Hardy
• , Partha Sinha
• , Elizabeth Oates
44. #### University of Rochester Medical Center, New York, USA

• Anton P. Porsteinsson
• , Bonnie S. Goldstein
• , Kim Martin
• , Kelly M. Makino
• , M. Saleem Ismail
•  & Connie Brand
45. #### University of Texas Southwestern Medical School, Texas, USA

• Kyle Womack
• , Dana Mathews
•  & Mary Quiceno
46. #### Emory University, Georgia, USA

• Allan I. Levey
• , James J. Lah
•  & Janet S. Cellar
47. #### University of Kansas, Medical Center, Kansas, USA

• Jeffrey M. Burns
• , Russell H. Swerdlow
•  & William M. Brooks
48. #### Mayo Clinic, Jacksonville, Florida, USA

• , Francine Parfitt
•  & Kim Poki-Walker
49. #### Yale University School of Medicine, Connecticut, USA

• Christopher H. van Dyck
• , Richard E. Carson
• , Martha G. MacAvoy
50. #### McGill University, Montreal-Jewish General Hospital, Quebec, Canada

• Howard Chertkow
• , Howard Bergman
•  & Chris Hosein
51. #### Sunnybrook Health Sciences, Ontario, Canada

• Sandra Black
• , Bojana Stefanovic
•  & Curtis Caldwell
52. #### U.B.C. Clinic for AD & Related Disorders, British Columbia, Canada

• Ging-Yuek Robin Hsiung
• , Benita Mudge
• , Vesna Sossi
• , Howard Feldman
•  & Michele Assaly
53. #### Cognitive Neurology-St. Joseph’s, Ontario, Canada

• Elizabeth Finger
• , Stephen Pasternack
• , Irina Rachisky
• , John Rogers
• , Dick Trost
•  & Andrew Kertesz
54. #### Cleveland Clinic Lou Ruvo Center for Brain Health, Ohio, USA

• Charles Bernick
•  & Donna Munic
55. #### Premiere Research Inst (Palm Beach Neurology), Florida, USA

•  & Teresa Villena
56. #### Georgetown University Medical Center, Washington D.C, USA

• Raymond Scott Turner
• , Kathleen Johnson
•  & Brigid Reynolds
57. #### Stanford University, California, USA

• Jerome Yesavage
• , Joy L. Taylor
• , Barton Lane
• , Allyson Rosen
•  & Jared Tinklenberg
58. #### Banner Sun Health Research Institute, Arizona, USA

• Marwan N. Sabbagh
• , Christine M. Belden
• , Sandra A. Jacobson
•  & Sherye A. Sirrel
59. #### Boston University, Massachusetts, USA

• Neil Kowall
• , Ronald Killiany
• , Andrew E. Budson
• , Alexander Norbash
•  & Patricia Lynn Johnson
60. #### Howard University, Washington D.C, USA

• Thomas O. Obisesan
• , Saba Wolday
•  & Joanne Allard
61. #### Case Western Reserve University, Ohio, USA

• Alan Lerner
• , Paula Ogrocki
• , Curtis Tatsuoka
•  & Parianne Fatica
62. #### Neurological Care of CNY, New York, USA

• Smita Kittur
63. #### Parkwood Hospital, Pennsylvania, USA

• Michael Borrie
• , T.-Y. Lee
•  & Rob Bartha
64. #### University of Wisconsin, Wisconsin, USA

• Sterling Johnson
• , Sanjay Asthana
•  & Cynthia M. Carlsson
65. #### Dent Neurologic Institute, New York, USA

• Vernice Bates
• , Horacio Capote
•  & Michelle Rainka
66. #### Ohio State University, Ohio, USA

• Douglas W. Scharre
• , Maria Kataki
•  & Brendan Kelly
67. #### Albany Medical College, New York, USA

• Earl A. Zimmerman
• , Dzintra Celmins
•  & Alice D. Brown
68. #### Hartford Hospital, Olin Neuropsychiatry Research Center, Connecticut, USA

• Godfrey D. Pearlson
• , Karen Blank
•  & Karen Anderson
69. #### Dartmouth-Hitchcock Medical Center, New Hampshire, USA

• Laura A. Flashman
• , Marc Seltzer
• , Mary L. Hynes
•  & Robert B. Santulli
70. #### Wake Forest University Health Sciences, North Carolina, USA

• Kaycee M. Sink
• , Leslie Gordineer
• , Jeff D. Williamson
•  & Franklin Watkins
71. #### Rhode Island Hospital, Rhode Island, USA

• Brian R. Ott
• , Geoffrey Tremont
•  & Lori A. Daiello
72. #### Butler Hospital, Rhode Island, USA

• Stephen Salloway
• , Paul Malloy
•  & Stephen Correia
73. #### Medical University South Carolina, Carolina, USA

• Jacobo Mintzer
• , Kenneth Spicer
•  & David Bachman
74. #### Nathan Kline Institute, New York, USA

• Nunzio Pomara
• , Raymundo Hernando
•  & Antero Sarrael
75. #### University of Iowa College of Medicine, Iowa, USA

• Susan K. Schultz
• , Karen Ekstam Smith
• , Hristina Koleva
• , Ki Won Nam
•  & Hyungsub Shim
76. #### USF Health Byrd Alzheimer’s Institute, University of South Florida, Florida, USA

• Amanda Smith
• , Balebail Ashok Raj
•  & Kristin Fargher

### Contributions

L.D., L.G. and L.S. conceived and designed the research. L.D., K.L. and J.H. carried out the study analysis. X.Y., J.Y, S.L.R. and A.J.S. collected the data from ADNI database. L.D., K.L., L.S. and A.J.S. analyzed the results and wrote the paper. Data used in preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report.

### Competing Interests

The authors declare that they have no competing interests.

### Corresponding authors

Correspondence to Lei Du or Li Shen.