基于截断L1范数的稀疏共典型关联分析：在脑影像遗传学中的应用
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本文主要探讨了"稀疏规范性共线性分析（Sparse Canonical Correlation Analysis, SCCA）"在脑成像遗传学中的应用，特别是在基因标记与神经影像定量特征之间的多变量关联挖掘。SCCA作为一种流行的方法，因其同时具备识别多变量关系和特征选择的强大能力而备受关注。然而，现有的SCCA方法通常依赖于L1范数或其变种，这可能限制了结果的稀疏性和模型的解释性。
传统的L1范数方法虽然能够引入稀疏性，但L0范数被认为更理想，因为它能更好地捕获非零关联，因为L0-norm最小化问题被证明是NP-hard，即在多项式时间内难以求解。因此，这篇2016年发表在IEEE International Conference on Bioinformatics and Biomedicine (BIBM)上的研究提出了一种创新的方法，即通过"截断L1范数"来解决这个问题。
作者们，来自西北工业大学自动化学院和印第安纳大学医学院的研究者，提出了一个新算法，旨在利用L1范数的特性同时结合截断操作，以克服L0范数求解的复杂性。这种方法允许在保持模型简洁的同时，发掘潜在的基因-影像特征间的稀疏相关性。他们将这一技术应用于阿尔茨海默病神经影像遗传学（Alzheimer's Disease Neuroimaging Initiative, ADNI）的数据集上，展示了其在实际应用中的可行性与有效性。
通过实施这种新颖的SCCA方法，研究人员不仅能够提高关联发现的效率，还能减少冗余特征的影响，使得结果更易于解读，这对于理解大脑功能与基因表达之间的复杂交互具有重要意义。该研究的贡献在于提供了一个有效且实用的工具，有望推动脑成像遗传学领域的进一步研究和发展。
2016 IEEE International Conference on Bioinformatics and Biomedicine (BIBM)
978-1-5090-1610-5/16/$31.00 ©2016 IEEE 707
Sparse Canonical Correlation Analysis via Truncated
1
-norm with Application to
Brain Imaging Genetics
Lei Du
∗
, Tuo Zhang
∗
, Kefei Liu
†
, Xiaohui Yao
†
, Jingwen Yan
†
,
Shannon L. Risacher
†
, Lei Guo
∗
, Andrew J. Saykin
†
and Li Shen
†§
for the ADNI
∗
School of Automation
Northwestern Polytechnical University, Xi’an, China 710072
Email: dulei@nwpu.edu.cn
†
Indiana University School of Medicine, Indianapolis, USA 46202
§
Corresponding to: Email: shenli@iu.edu
Abstract—Discovering bi-multivariate associations between
genetic markers and neuroimaging quantitative traits is a
major task in brain imaging genetics. Sparse Canonical
Correlation Analysis (SCCA) is a popular technique in this
area for its powerful capability in identifying bi-multivariate
relationships coupled with feature selection. The existing SCCA
methods impose either the
1
-norm or its variants. The
0
-
norm is more desirable, which however remains unexplored
since the
0
-norm minimization is NP-hard. In this paper, we
impose the truncated
1
-norm to improve the performance of
the
1
-norm based SCCA methods. Besides, we propose two
efficient optimization algorithms and prove their convergence.
The experimental results, compared with two benchmark meth-
ods, show that our method identifies better and meaningful
canonical loading patterns in both simulated and real imaging
genetic analyse.
Keywords-Sparse Canonical Correlation Analysis, Truncated
1
-norm, Brain Imaging Genetics
I. INTRODUCTION
Brain imaging genetics has gained more and more atten-
tions recently [1], [2]. A major task of imaging genetics
is to identify bi-multivariate associations between single
nucleotide polymorphisms (SNPs) and imaging quantitative
traits (QTs). Sparse canonical correlation analysis (SCCA),
which is powerful in bi-multivariate relationship discovery
coupled with feature selection, has become a popular tech-
nique in imaging genetic studies [3], [4], [5], [6], [7].
Witten et al. [3] introduced the
1
-norm (Lasso) to assure
sparsity which only selects a small proportion of the features.
Since then, many SCCA methods using the
1
-norm or its
variants are proposed [8]. There are two major concerns
regarding them. First, the
0
-norm, which only penalizes
those nonzero features, is the most ideal constraint. But it
is neither non-convex nor discontinuous [9]. Second, the
1
-
norm constraint is not a stable feature selector and thus could
incur estimation bias [10].
To overcome the problem above, the truncated
1
-norm
penalty (TLP) [10], [11] is proposed. The TLP is defined
as J
τ
(|x|)=min(
|x|
τ
, 1) with τ being a positive tuning
parameter. It approximates
0
-norm and permits desirable
sparsity. In addition, TLP can be equivalently transferred to
a piecewise linear function, and thus is easy to handle.
In this paper, we propose the TLP based SCCA (TLP-
SCCA) which embraces the TLP into the CCA model.
The TLP-SCCA has the following advantages [10]. First,
the TLP performs as a tradeoff between the
0
and
1
functions. This means that it not only has improved feature
selection, but also can be solved effectively. Second, it is an
adaptive shrinkage method if τ is tuned appropriately. We
propose two effective optimization algorithms, both using
the alternating direction method of multipliers (ADMM)
technique [12], and they are guaranteed to converge. The
experimental results, compared with two popular
1
-norm
based SCCA [3], [6], show that both TLP-SCCA exhibit
cleaner canonical loading patterns than the
1
-SCCA.
II. T
HE TRUNCATED
1
-NORM PENALTY
In this paper, a boldface lowercase letter denotes a vector,
and a boldface uppercase letter denotes a matrix. X ∈ R
n×p
denotes the SNP data, and Y ∈ R
n×q
is the QT data.
The truncated
1
-norm is defined as follows [13]:
P
TLP
(u)=
i
J
τ
(|u
i
|), where J
τ
(|u
i
|)=min(
|u
i
|
τ
, 1).
(1)
X
X
X
X
´
´
X
X
Figure 1. Visualization of the
0
-norm ball (left), TLP ball with τ =
1
4
and τ =
1
6
(middle), and
1
-norm ball (right).
The parameter τ is a threshold. Given an appropriate τ,
TLP balances between the
0
-norm and
1
-norm according
to the magnitude of the coefficients. Fig. 1 presents the norm
ball of
0
-norm,
1
-norm, and TLP with different τ ’s. The
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