SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Graph convolutional networks. GCNs typically rely on an information aggregation procedure (referred to as graph convolutions) over a graph structure for data processing. Several implementation strategies for graph convolution operations have been proposed in the literature, including spectral convolutions [49], Chebyshev polynomials [50], ordinary polynomials [36], and diffusion based representations [51]. GCNs admit the properties of stability to topological perturbations and transferability across graphs of different sizes in various settings [38, 52-54], which makes them a well-motivated data analysis tool for graph-structured data.
In [31], coVariance neural networks (VNN) were proposed as GCNs with sample covariance matrices as graph and polynomial graph filters as convolution operation. Covariance matrices and principal component analysis (PCA) form the two cornerstones of non-parametric analyses in real world applications that have spatially distributed, multi-variate data acquisition protocols, including neuroimaging [19], computer vision [55, 56], weather modeling [57], traffic flow analysis [58], and cloud computing [59].
Explainability in GNNs. We refer the reader to [32, 60] for a detailed review on explainability in GNNs. Here, we adopt the taxonomy from [32], which categorizes the recent efforts to add explainability to GNN models into two categories: instance-level methods and model-level methods for explainability. Instance-level methods for explainability are extensions of the standard model-agnostic methods for wider categories of deep learning models to GNNs, and aim to identify features most important to the inference outcome. Examples of techniques used to determine the importance of features include gradient-based methods [61], perturbation-based methods [62], and surrogate methods [63]. Such methods are, in principle, input-dependent as they provide explanations based on the given instance of the dataset. The robustness of the conclusions drawn from such methods to perturbations in the input and variations in the model training algorithms is an active area of research.
Model-level explainability has previously been studied in terms of graph topology in [64]. Further, importance of subgraphs to inference outcomes using Shapley values was studied in [65]. Since graph topology informs convolution operations in GNNs, it can provide a more generic explanation to GNNs than those that focus on individual features (such as nodes or edges of the graph). Since the convolution operation in VNNs is equivalent to manipulating the input data according to the eigenvectors of the covariance matrix [31], the eigenvectors of the covariance matrix are instrumental to explaining the inference by VNNs. Hence, we argue that VNNs can offer model-level explainability, similar in spirit as discussed in [32], where the explainability hinges on the eigenvectors of the covariance matrix.
Interpretable brain age prediction. Limited focus has been on comparable studies that associate brain age gaps with regional profiles [20, 66]. The study in [20] adopts a convolutional neural network approach to infer brain age from MRI images directly and assigns importance to brain regions in evaluating the brain age. In principle, the interpretability offered by VNNs in the context of brain age is similar, as we infer a regional profile for  $\Delta$ -Age by isolating the brain regions that are contributors to the elevated  $\Delta$ -Age in neurodegeneration. In addition, the regional profile identified by VNNs is correlated with specific eigenvectors or the principal components of the anatomical covariance matrix. Hence, the  $\Delta$ -Age inferred by our framework is driven by the ability of a VNN to manipulate the input data according to certain principal components of the anatomical covariance matrix. Also, VNNs are significantly less complex deep learning models as compared to those studied in [20]. Our results demonstrate that the VNNs trained with less than 300 learnable parameters
exhibit regional interpretability in the context of brain age in AD. In general, the regional expressivity offered by VNNs is in stark contrast to a multitude of existing relevant studies that rely on less transparent statistical approaches and further use post-hoc analyses (such as ablation analysis [67-69] or exploring correlations with region-specific markers [25] and psychiatric symptoms [47, 70]) to assign interpretability to a scalar, elevated  $\Delta$ -Age effect.
Our study has proposed a methodologically transparent framework for brain age prediction using VNNs. In contrast to existing studies that primarily focus on the performance of the model in predicting the chronological age of a healthy population, we have focused on the properties that VNNs gained when trained for this task. In particular, the VNNs achieve learning by transforming the input data according to the principal components or the eigenvectors of the covariance matrix estimated from the data. Hence, training the VNNs to predict chronological age using cortical thickness features from a healthy population had enabled them to exploit the eigenvectors of the anatomical covariance matrix in a specific manner. Further, the anatomical interpretability associated with elevated  $\Delta$ -Age in Alzheimer's disease was derived by the statistical analyses of the features extracted at the final layer of the VNNs and projecting the results of these analyses on the brain surface. Thus, the anatomical interpretability offered by VNNs is an inherent feature of VNNs and fundamentally distinct from the state-of-the-art model agnostic explainability approaches in the context of brain age (such as SHAP, LIME, saliency maps etc., that provide importance weights to individual input features).
The lack of explainability in the machine learning models deployed in brain age prediction frameworks may be a fundamental reason behind exclusive focus on the performance-driven approaches in this domain thus far. VNNs are inherently explainable models, as their inference outcomes can be tied with the eigenvectors of the covariance matrix. Hence, the anatomical interpretability offered by VNNs to  $\Delta$ -Age could be explained by their ability to exploit specific eigenvectors of the anatomical covariance matrix. This observation is highly relevant, as the quality of prediction on the chronological prediction for healthy population by itself may not be a complete determinant of the quality of brain age prediction in neurodegeneration. Furthermore, the role of the age-bias correction step in the VNN-based brain age prediction framework was restricted to projecting the VNN outputs onto a space where one could infer accelerated biological aging with respect to the chronological age from a layman's perspective.
By associating  $\Delta$ -Age with a regional profile, VNNs also provide a feasible tool to distinguish pathologies if the distributions of  $\Delta$ -Age for them are overlapping. A larger focus is needed on principled statistical approaches for brain age prediction that can capture the factors that lead to accelerated aging. Locally interpretable and theoretically grounded deep learning models such as VNNs can provide a feasible, promising future direction to build statistically and conceptually legitimate brain age prediction models in broader contexts. Incorporating other modalities of neuroimaging or alternative metrics of aging other than chronological age (such as DNA methylation [71]) provide promising future directions that can help improve our understanding of aging.
Limitations. Existing studies, including this paper, fall short at concretely defining the notion of optimal brain age. From a broader perspective, quantifying biological age even for a healthy population is a complex task due to various factors that can contribute to accelerated aging in the absence of an adverse health condition [72-74]. Moreover, the impacts of the quality of MRI scans and brain atlases across datasets on  $\Delta$ -Age must also be explored.
Our analysis was limited to older individuals, and a dataset with more diverse age groups is expected to provide holistic information on brain age. Isolation of brain regions contributing more to  $\Delta$ -Age in AD than HC hinges on a binary group comparison. Such a comparison can be impacted by the composition of the dataset (for instance, a skewed dataset may not provide informative results).
Societal Concerns. We are grateful to the reviewers for suggesting the potential societal concerns that may arise due to brain age prediction algorithms in practice. These societal concerns are discussed next. While the adoption of brain age prediction algorithms in clinic can lead to better decision-making, it is also worth considering that inaccurate prediction of a brain age gap (given implications for neurodegenerative diseases) could potentially lead to unintended consequences (such as over-medication in the case where the brain age gap is over-estimated, as just one example). Furthermore,
any framework for predicting brain age in an anatomically interpretable manner could be abused in potentially unethical or dubious ways for commercial or sociopolitical reasons. For instance, using such predictions to curb immigration and asylum or increasing health insurance premiums can be two such examples.
D An Abstract Overview of VNN-based Brain Age Prediction
Figure 4 provides an abstract overview of the general procedure of evaluating brain age using machine learning (ML) models. From Fig. 4, we note that if the ML model is a black box, it may be infeasible to capture the contributors to elevated age-gap in Step 3. Furthermore, in this context, it is also unclear whether age-bias correction step (Step 2) influences final  $\Delta$ -Age prediction through some statistical artifact [23]. Hence, it can be desirable to minimize the role of age-bias correction in  $\Delta$ -Age evaluation by selecting an ML model that achieves a near perfect fit on chronological age of healthy controls in Step 1. However, there is no guarantee that achieving an 'perfect fit' on true age of healthy controls will enable the ML model to capture the impact of neurodegeneration in individuals with neurodegeneration.
Step 1. Train ML model to predict chronological age for healthy controls
Step 2. Linear regression-based age-bias correct for outputs of ML model
Step 3. Obtain Age-gap for a population of healthy controls and individuals with neurodegenerative condition.
Figure 4: A general overview of brain age evaluation using machine learning algorithms in the existing literature. Step 1 consists of training a machine learning (ML) model to predict chronological age (true age) for healthy controls. If the correlation between predicted age and true age is smaller than 1, an age-bias exists in ML model outputs as the age for older individuals tends to be under-estimated and that for younger individuals tends to be over-estimated. To correct for this bias, a linear regression based model is applied on the ML model outputs in Step 2. Under the hypothesis that ML model can capture accelerated aging in neurodegeneration, it is expected that  $\Delta$ -Age for individuals with neurodegeneration will be significantly higher than those of healthy controls (Step 3).
VNNs allow us to analyze the contribution of each feature (brain region) to the final output. Hence, by analyzing the elevations in contributions of different brain regions via studying group differences in regional residuals, we are able to characterize the brain regions that contribute to accelerated aging (or larger  $\Delta$ -Age) in individuals with neurodegeneration (Fig. 5). Thus, we can verify that VNNs captured neurodegeneration-driven effects that eventually led to elevated  $\Delta$ -Age for an individual. Our experiments show that VNNs do not obtain a perfect fit on chronological age of healthy individuals. Hence, age-bias correction is important to appropriately project the VNN model outputs via a linear model into an appropriate space such that a clinician can observe an elevated  $\Delta$ -Age effect in individuals with neurodegenerative condition (AD in this paper). Based on these observations, we remark that VNNs provide an interpretable framework for brain age prediction.
Figure 5: Interpretability offered by VNNs in brain age prediction. By analyzing the final layer outputs of VNNs, we can isolate brain regions that have larger regional residuals for individuals with AD with respect to healthy controls. Furthermore, the elevated regional residuals in these brain regions eventually contribute to elevated  $\Delta$ -Age after age-bias correction.
E VNN Architecture
a
CoVariance Filter
b
CoVariance Perceptron
C
3-Layer VNN
Figure 6: Basics of VNN architecture. Panel a illustrates that the coVariance filter  $\mathbf{H}(\mathbf{C})$  is a polynomial in  $\mathbf{C}$  and its application on input  $\mathbf{x}$ . Panel b shows the construction of a coVariance perceptron based on coVariance filter  $\mathbf{H}(\mathbf{C})$  and pointwise nonlinearity  $\sigma$ . coVariance perceptron specifies one layer of VNN. Panel c shows a basic multi-layer VNN architecture formed by stacking three coVariance perceptrons.
F VNN training
We randomly split the HC group into an approximately  $90/10$  train/test split. Thus, the test set consisted of 61 healthy individuals. The sample covariance matrix was evaluated using all samples in the training set ( $n = 550$ ). Furthermore, this covariance matrix was normalized such that its maximum eigenvalue was 1. Cortical thickness data was  $z$ -score normalized across the training set and this normalization was applied to the test set. Next, the training set was randomly split internally, such that, the VNN was trained with respect to the mean squared error loss between the predicted age and the true age in  $n = 489$  samples of the HC group. The loss was optimized using batch stochastic gradient descent with Adam optimizer available in PyTorch library [75] for up to 100 epochs. The batch size was 78 (determined via 'optuna' package [44]). The VNN model with the best minimum mean squared error performance on the remaining 61 samples in the training set (which acted as a validation set) was included in the set of nominal models for this permutation of the training set. For each dataset, we trained and validated the VNN models over 100 permutations of the complete training set of  $n = 550$  samples for the HC group, thus, leading to 100 trained VNN models (also referred to as nominal models) per dataset.
G VNN regression model outputs for HC group in OASIS-3 are correlated with the first eigenvector of anatomical covariance matrix
The study in [31] suggests that VNN based statistical inference draws conceptual similarities with PCA-driven analysis. Hence, we further investigated whether the regression performance achieved by VNNs in predicting the chronological age of HC group could be characterized by contributions of the eigenvectors of the anatomical covariance matrix. To avoid any selection bias, we report the results on the complete HC group, where the cortical thickness features were  $z$ -score normalized across the group such that the mean of cortical thickness for a brain region across the HC group was 0. Here, the notation  $\mathbf{C_H}$  denotes the anatomical covariance matrix derived from the complete HC group.
Recall that the final regression output by VNNs is formed by an unweighted average function as a readout function. Thus, we can equivalently represent the functionality of the readout as a simple aggregation of the contributions of different features or brain regions to the final estimate formed by the VNN (see (5)). Hence, for every individual, we evaluated the mean of the inner products (also equivalently referred to as dot product between vectors) between the vectors of contributions of every brain region with the eigenvector of the covariance matrix  $\mathbf{C_H}$  for all 100 VNN models. Note that a vector of regional contributions was of the same length as the number of cortical thickness features (i.e., 148 for OASIS-3) and therefore, each element of this vector was associated with a distinct brain region. We use the notation  $\mathbf{p}_{\mathrm{HC}}$  to represent the population of vectors obtained from the HC group. To evaluate the inner product, we used  $\bar{\mathbf{p}}_{\mathrm{HC}}$ , which was obtained from  $\mathbf{p}_{\mathrm{HC}}$  after normalization (norm  $= 1$ ). We denote the population of inner products across the HC group in OASIS-3 by  $|\langle \bar{\mathbf{p}}_{\mathrm{HC}}, \mathbf{v}_i \rangle|$  for an eigenvector  $\mathbf{v}_i$  of  $\mathbf{C_H}$ . Note that since all vectors in  $\bar{\mathbf{p}}_{\mathrm{HC}}$  were normalized to have norm 1 and the eigenvectors of  $\mathbf{C_H}$  were of length 1 by default,  $|\langle \bar{\mathbf{p}}_{\mathrm{HC}}, \mathbf{v}_i \rangle|$  represented the population of the cosine of the angles between the vectors in  $\mathbf{p}_{\mathrm{HC}}$  and eigenvectors  $\mathbf{v}_i$  of  $\mathbf{C_H}$  across the HC group in OASIS-3. Figure 7a plots the mean of the inner products observed across the HC group for the first 30 eigenvectors of  $\mathbf{C_H}$  for all 100 VNNs. Figure 7b illustrates the projection of  $\mathbf{v}_1$  on a brain template.
a
Inner product between outputs  $\bar{\mathbf{p}}_{\mathrm{HC}}$  from final layers of VNNs trained on OASIS-3 and eigenvectors of  $\mathbf{C_H}$
b
First eigenvector of  $\mathbf{C_H}$
Figure 7: Inner product between the normalized vector of regional contributions to the VNN outputs  $(\bar{\mathbf{p}}_{\mathsf{HC}})$  and eigenvectors of  $\mathbf{C_H}$  (anatomical covariance matrix for HC group in OASIS-3). Panel a illustrates a bar plot for  $|<\bar{\mathbf{p}}_{\mathsf{HC}},\mathbf{v}_i>|$  for  $i\in \{1,\dots ,30\}$ , where  $\mathbf{v}_i$  is the  $i$ -th eigenvector (principal component) of covariance matrix  $\mathbf{C_H}$  and associated with  $i$ -largest eigenvalue in terms of magnitude and the vectors of regional contributions,  $\bar{\mathbf{p}}_{\mathsf{HC}}$  were obtained by VNNs that were trained on OASIS-3 dataset. The inner product results for eigenvectors with coefficient of variation greater than  $30\%$  across the HC group of OASIS-3 were excluded (and hence, their respective entries set as 0). For every individual in HC group, the associations between their corresponding vector of regional contributions,  $\bar{\mathbf{p}}_{\mathsf{HC}}$  and eigenvectors of  $\mathbf{C_H}$  were evaluated over 100 nominal VNN models. The first eigenvector  $(\mathbf{v}_1)$  had the largest association. The eigenvector  $\mathbf{v}_1$  is plotted on a brain template in Panel b.
Figure 8: Results depicting the brain regions with significantly elevated regional residuals for AD+ group with respect to HC group in OASIS-3. The results here were derived by a VNN model that was trained as a regression model to predict chronological age from cortical thickness data for HC group in OASIS-3. Box plots depicting the distributions of regional residuals in the HC and AD+ groups are shown for a few representative regions.
H Illustration of regional residual analysis from VNN model outputs
In this section, we demonstrate the regional analysis described in Section 3.2 for a VNN model that was trained to predict chronological age for HC group in OASIS-3 dataset. All mathematical notations referred to in this section are borrowed from Section 3.2. Note that no further training was performed for this VNN model to evaluate brain age or regional residuals.
The covariance matrix in this VNN model was replaced with  $\mathbf{C}_{\mathrm{HA}}$  derived from the cortical thickness features from both HC and AD+ groups. Further, the cortical thickness features in the HC group were  $z$ -score normalized and this normalization was used to transform the cortical thickness features of the AD group.
The age prediction  $\hat{y}_i$  and a vector of residuals  $\mathbf{r}_i$  were obtained for an individual  $i\in \{1,\dots ,805\}$  in the dataset. The size of residual vector  $\mathbf{r}_i$  was  $148\times 1$  and hence, each element of  $\mathbf{r}_i$  corresponded to a distinct brain region as defined by the DKT brain atlas with 148 parcellations. By evaluating the vector of residuals  $\mathbf{r}_i$  for every individual in the combined dataset, a population of residual vectors from HC group (referred to as  $\mathbf{r}_{\mathrm{HC}}$ ) and AD+ group (referred to as  $\mathbf{r}_{\mathrm{AD + }}$ ) was constructed. The elements of these residual vectors are referred to as regional residuals throughout the paper.
Each dimension of these residual vectors was investigated for group differences between HC and  $\mathrm{AD + }$  groups via ANOVA as described in Section 3.2. Thus, for every VNN model, we eventually performed  $m = 148$  number of ANOVA tests and evaluated the brain regions for significance in group differences in their respective residuals. The significance of group differences between the distributions of regional residuals for HC and  $\mathrm{AD + }$  groups corresponding to a brain region was determined after correcting the  $p$ -values of ANOVA test for multiple comparisons via Bonferroni correction (Bonferroni corrected  $p$ -value  $< 0.05$ ). The group differences were additionally investigated for significance at an uncorrected level using ANCOVA with age and sex as covariates.
Figure 8 illustrates the results obtained via ANOVA in this context. The brain regions deemed significant according to the criteria provided in Section 3.2 have been shaded. The box plots for various brain regions show that the regional residuals were significantly elevated in  $\mathrm{AD + }$  group as compared to HC. The regional residuals that lead to the results in Fig. 8 are provided in the supplementary material (model ID 50) as part of the regional residuals extracted from all 100 VNN models that were trained on HC group in OASIS-3.
We had 100 trained VNN models for the OASIS-3 dataset and performed similar analyses for each of them. Further, we counted the number of models for which the above described analysis yielded a brain region to be significant. A brain region with robust group difference in its regional residual distribution in HC vs AD+ was expected to be more frequently labeled as significant by the VNN models. The results of this robustness analyses on the OASIS-3 dataset are shown in Fig. 2a.
I Cross-validation on ADNI-1 dataset
In this section, we provide results on the standardized 3.0 T ADNI1 dataset (see [34] for details), consisting of 47 controls (age = 75.06 ± 3.93 years, 29 females), 71 individuals with mild cognitive impairment (age = 74.03 ± 8.12 years, 26 females), and 33 individuals with dementia (age = 75.08 ± 8.07 years, 22 females) from the from the wider ADNI database. We chose this standardized dataset in the spirit of reproducibility and to avoid selection bias. The individuals with mild cognitive impairment and dementia diagnosis were combined to form the AD+ cohort, equivalent to that of the AD+ cohort for OASIS-3.
Data used from ADNI database were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database. The ADNI was launched in 2003 as a public-private partnership, led by Principal Investigator Michael W. Weiner, MD. The primary goal of ADNI has been to test whether serial magnetic resonance imaging (MRI), positron emission tomography (PET), other biological markers, and clinical and neuropsychological assessment can be combined to measure the progression of mild cognitive impairment (MCI) and early Alzheimer's disease (AD). For up-to-date information, see www.adni-info.org.
Data processing. The MRI images for the 3.0T standardized ADNI-1 dataset at the baseline visit were downloaded from https://adni.loni.usc.edu/. Cortical thickness features (curated according to DKT atlas) were derived using the open-access CAT12 pipeline [76] using their default options. We refer the reader to https://neuro-jena.github.io/cat12-help/ for detailed processing steps. All outputs were quality checked visually for errors in grey matter segmentation.