Canonical Correlation Analysis on SPD(n) Manifolds

Abstract

Canonical correlation analysis (CCA) is a widely used statistical technique to capture correlations between two sets of multivariate random variables and has found a multitude of applications in computer vision, medical imaging, and machine learning. The classical formulation assumes that the data live in a pair of vector spaces which makes its use in certain important scientific domains problematic. For instance, the set of symmetric positive definite matrices (SPD), rotations, and probability distributions all belong to certain curved Riemannian manifolds where vector-space operations are in general not applicable. Analyzing the space of such data via the classical versions of inference models is suboptimal. Using the space of SPD matrices as a concrete example, we present a principled generalization of the well known CCA to the Riemannian setting. Our CCA algorithm operates on the product Riemannian manifold representing SPD matrix-valued fields to identify meaningful correlations. As a proof of principle, we present experimental results on a neuroimaging data set to show the applicability of these ideas.

Appendices

Appendix 1

The iterative method Algorithm 1 for Riemannian CCA with exact projection needs first and second derivatives of g in (4.13). We provide more details here.

1.1 First Derivative of g for SPD

Given SPD(n), the gradient of g with respect to t is obtained by the following proposition in [39].

Proposition 4.1.

Let F(t) be a real matrix-valued function of the real variable t. We assume that, for all t in its domain, F(t) is an invertible matrix which does not have eigenvalues on the closed negative real line. Then,

$$\displaystyle{ \begin{array}{rl} \frac{d} {dt}\text{tr}[\log ^{2}F(t)] = 2\text{tr}[\log F(t)F(t)^{-1} \frac{d} {dt}F(t)] \end{array} }$$

(4.26)

The derivation of $\frac{d} {dt_{i}}g(t_{i},\boldsymbol{w}_{x})$ proceeds as

$$\displaystyle{ \begin{array}{rl} \frac{d} {dt_{i}}g(t_{i},\boldsymbol{w}_{x})& = \frac{d} {dt_{i}}\|\text{Log}(\text{Exp}(\mu _{x},t_{i}W_{x}),X_{i})\|^{2} \\ & = \frac{d} {dt_{i}}\text{tr}[\log ^{2}(X_{i}^{-1}S(t_{i}))]\\ \end{array} }$$

(4.27)

where $S(t_{i}) = \text{Exp}(\mu _{x},t_{i}W_{x}) =\mu _{ x}^{1/2}\mathop{ \mathrm{exp}}\nolimits ^{t_{i}A}\mu _{x}^{1/2}$ and $A =\mu _{ x}^{-1/2}W_{x}\mu _{x}^{-1/2}$.

In our formulation, $F(t) = X_{i}^{-1}S(t_{i})$. Then, we have $F(t)^{-1} = S(t_{i})^{-1}X_{i}$ and $\frac{d} {dt}F(t) = X_{i}^{-1}\dot{S}(t_{ i})$. Hence, the derivative of g with respect to t _i is given by

$$\displaystyle{ \begin{array}{rl} \frac{d} {dt_{i}}g(t_{i},\boldsymbol{w}_{x})& = 2\text{tr}[\log (X_{i}^{-1}S(t_{i}))S(t_{i})^{-1}X_{i}X_{i}^{-1}\dot{S}(t_{i})]\mbox{, according to Proposition 4.1} \\ & = 2\text{tr}[\log (X_{i}^{-1}S(t_{i}))S(t_{i})^{-1}\dot{S}(t_{i})] \end{array} }$$

(4.28)

where $\dot{S}(t_{i}) =\mu _{ x}^{1/2}A\mathop{\mathrm{exp}}\nolimits ^{t_{i}A}\mu _{x}^{1/2}$.

1.2 Numerical Expression for the Second Derivative of g

Riemannian CCA with exact projection can be optimized by Algorithm 1. Observe that the objective function of the proposed augmented Lagrangian method $\mathcal{L}_{A}$ includes the term ∇g in (4.13). The gradient of $\mathcal{L}_{A}$ involves the second derivative of g. More precisely, we need $\frac{d^{2}} {dwdt}g$ and $\frac{d^{2}} {dt^{2}} g$. These can be estimated by a finite difference method

$$\displaystyle{ f^{'}(x) =\lim _{h\rightarrow 0}\frac{f(x + h) - f(x)} {h} }$$

(4.29)

Obviously, $\frac{d^{2}} {dt^{2}} g$ can be obtained by the expression above using the analytical first derivative $\frac{d} {dt}g$. For $\frac{d^{2}} {dwdt}g$, we use the orthonormal basis in $T_{\mu _{x}}\mathcal{M}$ to approximate the derivative. By definition of directional derivative, we have

$$\displaystyle{ \lim _{h\rightarrow 0}\sum _{i}^{d}\left (\frac{f(x + hu_{i}) - f(x)} {h} \right )u_{i} =\sum _{ i}^{d}\langle \nabla _{ x}f(x),u_{i}\rangle u_{i} = \nabla _{x}f(x) }$$

(4.30)

where $x \in \mathcal{X}$, d is dimension of $\mathcal{X}$, and {u _i } is orthonormal basis of $\mathcal{X}$. Hence, perturbation along the orthonormal basis enables us to approximate the gradient. For example, on SPD(n) manifolds, the orthonormal basis in arbitrary tangent space $T_{p}\mathcal{M}$ can be obtained by following three steps:

Step a) :

Pick an orthonormal basis {e _i } of R ^n(n+1)∕2,

Step b) :

Convert {e _i } into n-by-n symmetric matrices {u _i } in $T_{I}\mathcal{M}$, i.e., {u _i } = mat({e _i }),

Step c) :

Transform basis {u _i } from $T_{I}\mathcal{M}$ to $T_{p}\mathcal{M}$.

Appendix 2

MR Image Acquisition and Processing All the MRI data were acquired on a GE 3.0 Tesla scanner Discovery MR750 MRI system with an 8-channel head coil and parallel imaging (ASSET). The DWI data were acquired using a diffusion-weighted, spin-echo, single-shot, echo planar imaging radiofrequency (RF) pulse sequence with diffusion weighting in 40 noncollinear directions at b = 1300 s/mm² in addition to 8 b = 0 (nondiffusion-weighted or T2-weighted) images. The cerebrum was covered using contiguous 2.5-mm-thick axial slices, FOV = 24 cm, TR = 8000 ms, TE = 67. 8 ms, $\mathrm{matrix} = 96 \times 96$, resulting in isotropic 2.5 mm³ voxels. High-order shimming was performed prior to the DTI acquisition to optimize the homogeneity of the magnetic field across the brain and to minimize EPI distortions. The brain region was extracted using the first b = 0 image as input to the brain extraction tool (BET), also part of the FSL software.

Eddy current-related distortion and head motion of each data set were corrected using FSL software package [46]. The b-vectors were rotated using the rotation component of the transformation matrices obtained from the correction process. Geometric distortion from the inhomogeneous magnetic field applied was corrected with the b = 0 field map and PRELUDE (phase region expanding labeler for unwrapping discrete estimates) and FUGUE (FMRIBs utility for geometrically unwarping EPIs) from FSL. Twenty-nine subjects did not have field maps acquired during their imaging session. Because these participants did not differ on APOE4 genotype, sex, or age compared to the participants who had field map correction, they were included in order to enhance the final sample size. The diffusion tensors were then estimated from the corrected DWI data using nonlinear least squares estimation using the Camino library [13].

Individual maps were registered to a population-specific template constructed using diffusion tensor imaging toolkit (DTI-TK¹), which is an optimized DTI spatial normalization and atlas construction tool that has been shown to perform superior registration compared to scalar-based registration methods [55]. The template is constructed in an unbiased way that captures both the average diffusion features (e.g., diffusivities and anisotropy) and anatomical shape features (tract size) in the population. A subset of 80 diffusion tensor maps was used to create a common space template. All diffusion tensor maps were normalized to the template with first rigid followed by affine and then symmetric diffeomorphic transformations. The diffeomorphic coordinate deformations themselves are smooth and invertible, that is, neuroanatomical neighbors remain neighbors under the mapping. At the same time, the algorithms used to create these deformations are symmetric in that they are not biased towards the reference space chosen to compute the mappings. Moreover, these topology-preserving maps capture the large deformation necessary to aggregate populations of images in a common space. The spatially normalized data were interpolated to 2×2×2 mm voxels for the final CCA analysis.

Along with the DWI data the T1-weighted images were acquired using BRAVO pulse sequence which uses 3D inversion recovery (IR) prepared fast spoiled gradient recalled echo (FSPGR) acquisition to produce isotropic images at 1×1×1 mm resolution. We extract the brain regions again using BET. We compute an optimal template space, i.e., a population-specific, unbiased average shape and appearance image derived from our population [4]. We use the openly available advanced normalization tools (ANTS²) to develop our template space and also perform the registration of the individual subjects to that space [5]. ANTS encodes current best practice in image registration, optimal template construction, and segmentation [35]. Once we perform the registrations we extract the Jacobian matrices (J) per voxel per subject from these deformation fields and compute the Cauchy deformation tensors ($\sqrt{J^{T } J}$) for performing CCA analysis.

Additional Experiments As described in the main text (Sect. 4.6.3), we examine the following multimodal linear relations but this time by performing CCA on the entire gray and white matter structures rather than just the hippocampi and cinguli:

$$\displaystyle\begin{array}{rcl} Y _{\mathrm{DTI}}& =& \beta _{0} +\beta _{1}\mathrm{Gender} +\beta _{2}X_{\mathrm{T1W}} +\beta _{3}X_{\mathrm{T1W}} \cdot \mathrm{ Gender} +\varepsilon {}\\ Y _{\mathrm{DTI}}& =& \beta '_{0} +\beta '_{1}\mathrm{AgeGroup} +\beta '_{2}X_{\mathrm{T1W}} +\beta '_{3}X_{\mathrm{T1W}} \cdot \mathrm{ AgeGroup}+\varepsilon {}\\ \end{array}$$

To enable this analysis, the gray matter region was defined as follows: First, we performed a three tissue segmentation of each of the spatially normalized T1-weighted images into gray, white, and cerebral spinal fluid using FAST segmentation algorithm [56] implemented in FSL. Then, we take the average of the gray matter probabilities using all the subjects and threshold it to obtain the final binary mask resulting in $\sim$ 700,000 voxels. The white matter region is simply defined as the region with fractional anisotropy (FA) obtained from the diffusion tensors > 0. 2 which resulted in about 50,000 voxels. We would like to contrast this with the region-specific analyses where the number of voxels was much smaller. To address this in our CCA we imposed an L ₁-norm penalty to the weight vectors (similar to Euclidean CCA [51]) in our CCA objective function with a tuning parameter $\lambda$.

In addition to the above linear relationships, CCA can also facilitate testing the following relationships by using the weight-vectors as substructures of interest and taking the average mean diffusivity ($\overline{MD}$) and average volumetric change ($\overline{\log \vert J\vert }$) in those structures as the outcome measures:

$$\displaystyle\begin{array}{rcl} \overline{MD}& =& \beta _{0} +\beta _{1}\delta _{\mathrm{Female}} +\beta _{2}\delta _{\mathrm{Male}} +\varepsilon {}\\ \overline{\log \vert J\vert }& =& \beta '_{0} +\beta '_{1}\delta _{\mathrm{Female}} +\beta '_{2}\delta _{\mathrm{Male}}+\varepsilon {}\\ \end{array}$$

where the null hypotheses to be tested are β ₁ = β ₂ and $\beta _{1}' =\beta _{2}'$. Similarly, we can find statistically significant AgeGroup differences by examining the following two models:

$$\displaystyle\begin{array}{rcl} \overline{MD}& =& \beta _{0} +\beta _{1}\delta _{\mathrm{Middle}} +\beta _{2}\delta _{\mathrm{Old}} +\varepsilon {}\\ \overline{\log \vert J\vert }& =& \beta '_{0} +\beta '_{1}\delta _{\mathrm{Middle}} +\beta '_{2}\delta _{\mathrm{Old}}+\varepsilon {}\\ \end{array}$$

The scatter and bar plots for the testing linear relationships using both Riemannian and Euclidean CCA are shown in Figs. 4.7 and 4.8.

Fig. 4.7

CCA projections revealing statistically significant volume and diffusivity interactions with gender (left) and age group (right). Top row shows the results using Riemannian CCA and the bottom row using the Euclidean CCA with vectorization. We can clearly see improvements in the statistical confidence (smaller p-values) for rejecting the null hypotheses when using Riemannian CCA

Fig. 4.8

Effect of gender and age group on the average mean diffusivity (MD) and the average $\log$ Jacobian determinant. Top row shows the results using Riemannian CCA and the bottom row using Euclidean CCA with vectorization. Again our approach produces smaller or similar p-values in rejecting the null hypotheses

We now present a comprehensive set of montages of all the slices of the brain overlaid by the weight vectors obtained from CCA from Figs. 4.9, 4.10, 4.11, and 4.12.

Fig. 4.9

Weight vector (obtained by Riemannian (tensor) CCA) visualization of T1W axial slices

Fig. 4.10

Weight vector (obtained by Riemannian (tensor) CCA) visualization of DTI axial slices

Fig. 4.11

Weight vector [obtained by Euclidean CCA (tensors)] visualization of T1W axial slices

Fig. 4.12

Weight vector [obtained by Euclidean CCA (tensors)] visualization of DTI axial slices

We can observe that the voxels with nonzero weights (highlighted in red-boxes) are spatially complimentary in DTI and T1W. Even more interestingly, CCA finds the cingulum regions in the white matter for the DTI modality. In our experiments, we observe the same regions for various settings of a sparsity parameter (used heuristically to obtain more interpretable regions). Similarly, Figs. 4.11 and 4.12 show the results using the Euclidean CCA performed using the full tensor information [51]. We can observe that the results are similar to those in Figs. 4.9 and 4.10 but the regions are thinner in the Euclidean version.