Adaptive Shrinkage Estimation for Personalized Deep Kernel Regression in Modeling Brain Trajectories

Weaknesses

• We wish to clarify that our work does not aim to introduce for the first time the concept of personalized Gaussian Processes (GPs). Instead, we propose a novel method for personalized Deep Kernel Regression from high-dimensional multivariate data. Our method learns temporal functions of neuroimaging biomarker progression at any future time point and is able to adapt such temporal function leveraging any follow-up information. Also, we make no assumptions about biomarker trajectories. Our model neither requires temporally aligned data nor performs imputation of missing longitudinal observations in advance, thereby overcoming the limitations of predictive models—such as autoregressive ones—that rely on temporally aligned data. Through our composite framework, which comprises both the deep kernel regression for longitudinal prediction (p-DKGP and ss-DKGP models) and the adaptive shrinkage function for adaptation to follow-up observations, we are able to successfully perform the long-term prediction of longitudinal neuroimaging biomarkers from high-dimensional imaging and clinical data and adapt such predictions when subject's follow-ups become available.

• Furthermore, we want to highlight that our work differs significantly from Rudovic et al., 2019 in several key aspects:

1. Introduction of the Deep Kernel: Rudovic et al., 2019 approach employs multitask Gaussian Processes (GPs) with an RBF kernel to forecast up to two years ahead, predicting biomarker values at 6-month intervals (6, 12, 18, 24 months). However, this approach is not scalable to high-dimensional spaces, and the RBF kernel lacks the expressiveness needed to capture complex patterns in multivariate high-dimensional imaging data.

2. Long-term Predictions: Unlike Rudovic et al., 2019, which is limited to a fixed two-year forecasting horizon, our method learns and adapts biomarker progression functions over the long term without imposing such restrictions in the forecast window.

3. Handling Temporally Unaligned Data: Rudovic's method requires temporally aligned data with strict 6-month intervals, which significantly reduces the usable sample size of the dataset. In contrast, our approach does not impose such constraints, enabling broader applicability and higher data utilization.

4. Model Architecture: Rudovic et al., 2019 combines two versions of personalized GPs. Our model, by comparison, integrates a population-level predictor to capture general trends and a subject-specific predictor to model individual variations. These components are combined in a data-driven and explainable manner, enhancing interpretability and performance.

5. Scalability: Our model achieves scalability through dimensionality reduction applied to the input space, making it more effective in handling high-dimensional inputs by learning a mapping $\Phi$ that is informative of biomarker progression.

These distinctions emphasize the methodological advancements of our approach compared to Rudovic et al., 2019. We will ensure they are clarified in the manuscript.

• We have additional studies with longitudinal acquisitions through the iSTAGING consortium [1]. ADNI and BLSA studies that we have used for training the population model and the adaptive shrinkage function are the ones that are most longitudinally rich. We have already expanded our dataset with multiple studies, apart from ADNI and BLSA, for evaluating our model from a clinical perspective.

• We have added CIs in Figure 2, Figure 4, and Table 5 for consistent comparison.

• We report the mean AE with the number of observations. We initiate from the single scan where we employ the p-DKGP model and from the second scan where we employ the adaptive shrinkage function for the adaptation process. Based on the empirical evidence, the error indeed decreases as the follow-up observations increase.

  History	Mean AE	95% CI
  1	0.227297	0.002864
  2	0.233220	0.007580
  3	0.218854	0.008475
  4	0.153053	0.009802
  5	0.147692	0.010350

  Table: Mean Absolute Error (AE) and 95% Confidence Interval (CI) for the SPARE-AD biomarker across different histories.

Responses

• The pers-DKGP model has no minimum requirement for the number of data points per subject. For a single scan (baseline acquisition), the model employs the population predictor, while with multiple scans, it applies posterior correction through adaptive shrinkage.
• Thank you for pointing out the need to clarify Line 12 of Algorithm 1. To address this, we have explicitly detailed the step as follows:

Collect all $\hat{\alpha}_{s|h}$ for subject $s$ into a list.

• Thank you for your attention to detail! We have addressed all the minor comments in the manuscript and we also moved several equations inline in the method.

Weaknesses

• We respectfully disagree with the comment regarding the limited novelty due to Eleftheriadis et al., 2017. The key point of difference is the integration of the transformation $\Phi$ which is central to the success of our method as the Gaussian Processes alone are insufficient to capture meaningful relationships in high-dimensional multivariate inputs. Then, Adaptive Shrinkage is novel as an approach to combine population and individual trends in longitudinal progression prediction.
• We elaborate on why autoregressive models, such as LSTMs and Transformers did not work in the case of sparse temporal data thought we did implement them as baselines (Supplementary Section E.2)

Responses

• $\Phi$ captures relationships in the data that are predictive of progression of each biomarker, creating, population-informed feature space. By keeping these parameters fixed in the ss-DKGP, we ensure that each subject’s data is projected into the same representation space, maintaining alignment with population-level patterns and preventing overfitting to the limited, potentially noisy data available for an individual.

• We clarify that our method does not impose any structure in the latent space (clustering, subtyping etc). Thus, one should not expect the model to perform any latent clustering. Our method is designed to learn the function $\Phi$ that is the most informative for the progression prediction of each neuroimaging biomarker. Motivated by reviewer's comment, however, we decided to work on the interpreting and exploring the latent space. We performed two tasks: we interpreted the model to identify which imaging and clinical features are the most predictive for each task (biomarker progression prediction), and we used the latent features to classify subjects into progressors and non-progressors. A progressor is a subject that starts from Cognitive Normal (CN) and progresses to either Mild Cognitive Impairment (MCI) or Alzheimer's Disease (AD), and a non-progressor is a subject that remains CN.

Regarding the importance of imaging features for the SPARE-AD model, we observed several key findings. The bilateral hippocampus and left amygdala are highly important, aligning well with known AD pathology where these structures are often among the earliest and most affected. Then we performed the task of classifying a subject as a progressor or non-progressor based on its latent transformation. We showed that the transformation $\Phi$ effectively transforms the input data into a latent vector that is more informative for progression prediction. After balancing the dataset by undersampling the non-progressors, we were able to significantly improve the classification performance. The latent space classifier achieved an average accuracy of 62.26% across 5-fold cross-validation, with a precision of 70.83% and an ROC AUC of 71.99%, outperforming the input space classifier, which had an average accuracy of 50.15%, precision of 20.74%, and ROC AUC of 67.54%. We elaborate more on this analysis in Supplementary Section E.9

• We have included detailed analysis on the selection of XGBoost on the Supplementary Section D.2 We have also experimented with additional models for learning the Adaptive Shrinkage function such as Random Forest, DNN, Gradient Boosting Machine and through 5-fold cross validation across a 8 ROI biomarkers, the XGBoost appeared to have the best performance (SS D.2 Figure 6)

• We investigated the effect of varying constant $\alpha$ values on predictive accuracy, across multiple regions of interest (ROIs). The results reveal a U-shaped trend, where both very low and very high $\alpha$ values lead to higher prediction errors. This behavior reflects the need for posterior correction through the adaptive shrinkage parameter $\alpha$. Smaller $\alpha$ values place greater trust in subject-specific trajectories, which is advantageous when the observations are informative. However, neuroimaging data are noisy, making excessive reliance on subject-specific observations problematic. In contrast, higher $\alpha$ values rely more on population-level trajectories, which are robust to noise but may fail to capture deviations in individual progression. Regarding, the sensitivity and the trajectory shifts, we suggest the reviewer to look the qualitative examples we have attached on the Supplementary Section E.5, Figure 13 and 14.

• For simplicity, we opted for independence during the initial formulation. As detailed in Supplementary Section D.1.4 this assumption does not affect the estimation of the shrinkage parameter $\alpha$ or the combined predictive mean $y_c$ . The derivation of $\alpha$ minimizes the Mean Squared Error, relying solely on the means of the predictions, independent of error correlations. However, we recognize the limitations of this assumption when calculating the combined predictive variance $v_c$. We address this limitation in detail in Supplementary Section D.1.5.

Weakness

• We thank the reviewer for the valuable feedback. To address the concerns, we have expanded the literature review in the supplementary material, including recent works from the past two years, and situated our approach within the broader field of predictive modeling and Disease Progression Modeling (DPM). However, we want to clarify that our model is not specifically designed as a DPM, as it does not focus solely on homogeneous Alzheimer’s disease (AD) cohorts or simulate AD-related pathologies. Instead, it serves as a general predictive tool for modeling neuroimaging biomarker trajectories in the broader context of brain aging and dementia. Its design intentionally avoids strict assumptions about pathology or trajectory patterns, enabling it to handle heterogeneous populations and support real-world clinical applications. We hope these additions and clarifications address the reviewer’s concerns.
• We have enhanced the quantitative evaluations by incorporating CIs of the Mean AE (Figure 2 and Table 5), for clearer comparisons with the baselines. Additionally, we have included extensive qualitative results to demonstrate the pers-DKGP as well as its clinical utility for participant-level monitoring and trial design. (Supplementary E.6)

Responses

• While the ss-DKGP uses the same deep kernel transformation $\Phi$ as the p-DKGP, $\Phi$ serves as a population-informed feature extractor that patterns predictive of the progression of each biomarker, without imposing population-level trends on subject-specific trajectories. The ss-DKGP is trained exclusively on the subject’s own data, with its Gaussian Process component optimized solely for the subject’s observed trajectory. This ensures that ss-DKGP predictions reflect individual trends and remain independent of the population-level model. Combining p-DKGP and ss-DKGP predictions leverages the strengths of both models. The p-DKGP provides stable, general predictions when subject-specific data are sparse or noisy, while the ss-DKGP captures individual deviations with increasing accuracy as more observations are available. The shrinkage parameter $\alpha$ dynamically balances these contributions, favoring the p-DKGP when data are limited and the ss-DKGP as subject-specific evidence grows. This adaptive combination ensures robust, accurate, and personalized predictions across varying scenarios. Empirical evidence supporting this dynamic behavior is provided in Supplementary Section E.6

• We will include a dedicated section on the Supplementary Material (Section E.5) where we discuss the limitations of the baselines and existing methods.

• The OASIS, AIBL, and PreventAD studies were harmonized through the iSTAGING Consortium to minimize site-specific variability, as detailed in Supplementary Section B.1. This harmonization ensures consistency across datasets and enhances the reliability of our model's generalization results. By addressing site-related biases, this preprocessing step guarantees that the reported performance reflects true model robustness across diverse clinical populations.

• 1. Our approach leverages the entire imaging scan for deep kernel regression, utilizing all available ROI information. Since the information is accessible and relevant, we maximize the model’s predictive power by including the complete set of imaging data. We do not restrict the model to fewer ROIs or alternative biomarkers unless such information is unavailable or irrelevant.

  2. We do not impose constraints regarding monotonicity. The model is fully capable of learning non-monotonic and highly non-linear progression patterns if they are present in the observed data. This flexibility is primarily due to the RBF kernel, which captures complex trends in the data without enforcing a particular direction of progression. Therefore, our method is not limited to monotonic trends and can adapt to any progression pattern observed in the data. However, it is important to clarify that neuroimaging biomarkers, such as Tau PET, Amyloid, and atrophy markers, are monotonic due to the irreversible nature of aging and dementia-related pathologies.

• The p-DKGP model inherently handles missing data through its Gaussian Process (GP) framework, which models the temporal structure of sparse longitudinal data without requiring explicit imputation. During both training and inference, the GP provides principled posterior predictive distributions to reconstruct unobserved values. To evaluate this capability, we conducted an interpolation analysis (Supplementary Section E.8) where a data point was randomly removed from each subject's trajectory and reconstructed using the p-DKGP model. Compared to a Linear Interpolation baseline, p-DKGP achieved significantly lower reconstruction errors across multiple brain regions (SS E.8 Fig16). This demonstrates the model’s robustness in handling missing data and accurately reconstructing neuroimaging trajectories.

Weakness

• While hierarchical linear models (HLMs) effectively model individual trajectories based on population means, they face significant limitations in high-dimensional, multivariate settings like ours. Specifically, HLMs are prone to overfitting with a large number of fixed effects and assume linear relationships, which restricts their ability to capture the complex, nonlinear dynamics of brain changes over time. Additionaly, they are predominately used as inferential tools rather predictive models.
• Our method integrates p-DKGP and the ss-DKGP to deliver accurate and personalized predictions of biomarker trajectories. Both models share a deep representation, enabling the p-DKGP to capture population-wide patterns predictive of progression trends to the specific biomarker. The adaptive shrinkage parameter $\alpha$ dynamically balances the models, emphasizing population-level predictions when data is sparse and shifting to individual adjustments as more subject-specific data becomes available. This ensures that predictions evolve seamlessly from general to personalized, offering precise forecasting across diverse clinical scenarios.

Questions

• The optimization criterion in Eq. (17) learns the optimal value of $\alpha$ that results in the trajectory closest to the ground truth, thereby favoring the model with minimal error. These optimal $\alpha$ values are subsequently used to train the Adaptive Shrinkage function, which incorporates the predicted means and variances from the two models along with the time of observation ($T_{\text{obs}}$). Our explainability analysis (Supplementary Material D.3) reveals that the inferred shrinkage $\alpha$ values are strongly influenced by $T_{\text{obs}}$. Specifically, when evidence for a subject is limited and their trajectory deviates from the population, $\alpha$ approaches 1, placing greater trust in the population model. This is desirable, as the subject-specific model alone cannot reliably predict long-term biomarker trends with insufficient data. As more observations are gathered for a subject, $\alpha$ decreases, allowing greater reliance on the subject-specific model.

This adaptive behavior is further validated through correlation analysis between $\alpha$, $T_{\text{obs}}$, and the prediction deviation between p-DKGP and ss-DKGP ($\delta_y = |y_s - y_p|$) (see Table 5, Supplementary Section D.3). Key findings include:

• A consistent negative correlation between $T_{\text{obs}}$ and $\alpha$ for large deviations ($\delta_y$), with values as strong as $-0.64$ for biomarkers like SPARE-BA and $-0.55$ for Thalamus Proper. This indicates that as $T_{\text{obs}}$ increases, $\alpha$ decreases, shifting trust towards the subject-specific model.
• Strongest correlations are observed for biomarkers associated with neurodegenerative progression, SPARE-BA and SPARE-AD, highlighting the model’s capacity to adapt effectively in clinically relevant contexts.

These results confirm that $\alpha$ dynamically adjusts based on the subject’s data and deviation from population trends, rather than being biased toward a single model. Additionally, we provide qualitative examples in Supplementary Section E.6 to illustrate this adaptive shrinkage in real-world scenarios. Specifically, we refer the reviewer to check Figure 13 and Figure 14.

• This is a typo and we have fixed that. The correct is $Z_s$ and not $Z_p$.
• The 145 imaging ROIs were extracted using a multi-atlas segmentation algorithm. From these, we modeled the progression of six ROIs associated with aging and AD. Input features include the 145 baseline imaging ROIs, baseline diagnosis, age, sex, APOE4 alleles, education years, and the temporal variable Time. For the SPARE-AD and SPARE-BA models, baseline SPARE-AD and SPARE-BA values were also included. Detailed image and clinical data processing procedures are provided in Supplementary Material, Section B.1.
• Our model uses the 145 imaging ROIs and clinical covariates at baseline as input and is informed only by the baseline diagnosis status (CN, AD, or MCI), without any knowledge of progression status. Despite the overrepresentation of Healthy Controls in the dataset, we have demonstrated that the model performs robustly across all diagnosis categories. This is supported by empirical evidence obtained by stratifying population model errors based on progression status (e.g., Healthy Control [CN-CN], MCI Progressor [CN-MCI], AD Progressor [CN/MCI-AD], etc.) )(Manuscript, Figure 2). Additionally, during the adaptation process, the model is updated solely with new biomarker (target) values, ensuring no bias is introduced towards any specific progression status.
• On the Supplementary Material (Section E.6) we have included multiple qualitative examples of all the 6 ROIs as well as the SPARE-AD biomarker (Figures 11, 13 and 14).
• We have increased the font size of the Fig.3.

Dear Reviewers,

We want to inform you that we have updated the manuscript and the supplementary material.

For your convenience, we summarize, below all the changes and additions we have on the Manuscript and the Supplementary Material. The modifications are mentioned in the order that appear in the files.

Main Manuscript

• Introduction
  1. We replace the monotonic word with neuroimaging. Since we do not impose any monotonicity constraints in our model, the model, due to the flexibility of the RBF kernel of the DKGP is able to learn any trends are present in the data.
  2. We Elaborate how we differ from related works and specifically Rudovic et al., 2019
  3. Include in the contributions one additional point regarding our deep kernel regression models since are important part of the success of the method
• Method:
  1. Move some equations inline and implement minor changes that did not change the content
• Results:
  1. Figure 2 is updated with CIs for a more straightforward comparison as requested. Caption is also minimally modified to include the CIs.
  2. In Figure 3 we increase the font size of the legend as requested.
  3. Figure 4 is updated with CIs for a more straightforward comparison as requested. Caption is also minimally modified to include the CIs.
  4. Table 5 is also updated with the CIs reported in the parenthesis.
  5. Include a paragraph in the section Generalization to External Clinical Studies that describes the results with the CIs. The conclusion of course remains the same with our method to outperfom all the baselines in external studies.

Supplementary Material

1. Include the Section A: Extended Related Works in Predictive and Progression Modeling and Clinical Application
2. Include the Section E.3: Comparison with Baselines
   • We perform statistical significance tests on the performance metrics across stratified progression status for all the competing baselines
3. Include the Section E.4: Stratifying performance analysis by covariates: Sex, APOE4 Alleles and Education Years.
   • We present barplots with the mean AE and CIs and we show that our method outperforms all baselines across all covariates.
4. Include the Section E.5: Limitations of Baselines.
   • We elaborate more on the limitations of baselines.
5. Include the Section E.6: Qualitative Examples.
   • We present a plethora of predicted trajectories that highlight how our adaptation algorithm operates.
6. Included the Section E.8: p-DKGP learns smooth progression curves of observed trajectories
   • We elaborate and show how our DKGP model with the temporal variable Time, learns internally to impute the longitudinal biomarker values in time in the train population. We also show that our approach can perform better in interpolation of missing biomarker values in comparison with the Linear Interpolation.
7. Include the Section E.9: Latent Space Analysis
   • We show that the tranformation function $\Phi$ learns atrophy patterns related to AD and Brain Aging when modeling the progression of SPARE-AD and SPARE-BA biomarkers
   • We show that the the tranformation function $\Phi$ of SPARE-AD model tranforms the input data into a space that is more informative of progression. We show that the latent features are able to perform significantly better than the input features in the classification task Progressor vs Non-Progressor.

We sincerely appreciate the reviewers' insightful comments and have dedicated significant effort to enhance the presentation of our work. We hope these improvements meet your expectations and are always available for further discussion or clarification.

Best Regards,

Authors of Submission 5004

Discussion on the Potential of Adaptive Shrinkage Estimation for Personalized Deep Kernel Regression

Introduction to Adaptive Shrinkage

The Adaptive Shrinkage Function introduces a novel and versatile approach to predicting biomarker progression trajectories. By integrating two predictors that share a deep kernel, it projects inputs into a common feature space, achieving feature alignment across subjects. This shared representation informs both population-level trends and individual-specific trajectories, enabling our model to account for inter-subject variability while maintaining consistency with broader population dynamics.

Advancements Over Prior Models

Unlike prior models (e.g., Rudovic et al., 2019; Koval et al., 2021), which focus solely on predicting the next timepoint(s), our method uniquely addresses the entire trajectory from the baseline acquisition. It forecasts future biomarker values while also correcting noisy observations from baseline onward, producing robust and reliable estimates of progression rates (slopes). This dual functionality—integrating prediction and correction—marks a significant advancement over existing progression models. By providing comprehensive and accurate trajectory estimates, our approach enhances clinical decision-making in monitoring and managing neurodegenerative diseases.

Clinical Applications

Robust estimates of progression rates hold potential to impact several areas. They enable early diagnosis by detecting early signs of disease progression and support personalized treatment planning by facilitating tailored interventions. They improve clinical trial design and subject stratification by targeting specific subgroups, and they allow for more precise evaluation of therapeutic efficacy over time. These capabilities make our deep kernel framework with the Adaptive Shrinkage a valuable tool for both researchers and clinicians. (Maheux et al., 2023)

Flexibility and Minimal Assumptions

The Adaptive Shrinkage Function is highly flexible and operates without stringent assumptions about the backbone models or the transformation function $\Phi$. It requires only the predictive means, variances of the population DKGP and ss-DKGP, and the observation times, making it adaptable to diverse setups. This minimal set of requirements broadens its applicability and allows for seamless integration into other deep kernel regression frameworks, such as Stochastic Variational Deep Kernel Learning (SVDKL).

Versatility in Diverse Frameworks

This flexibility underscores the function's ability to integrate additional inductive biases, such as temporal smoothness, hierarchical modeling, or multi-modal data integration, to address more intricate modeling challenges. Its compatibility with diverse setups highlights its potential to enhance the scope and precision of deep kernel regression frameworks.

Future Directions

Future advancements aim to extend this framework by incorporating more expressive transformation functions $\Phi$, such as attention mechanisms, to capture complex relationships in the data. Additionally, moving to stochastic variational deep kernel techniques will help scale its application to more intricate scenarios, further unlocking the potential of Adaptive Shrinkage Estimation in personalized progression modeling. This remains an active area that we plan to explore.

Continues...

Summary of Changes and Additions

Main Manuscript

Introduction

Replaced "monotonic" with "neuroimaging" to better reflect the model's flexibility and clarify that no monotonicity constraints are imposed.
Expanded the discussion of differences from related methods, including Rudovic et al., 2019.
Enhanced the contributions section to emphasize the critical role of deep kernel regression models in our approach.

Methods

Reformatted by moving some equations inline for improved readability without altering the content.

Results

Figures and Tables

• Updated Figure 2 and Figure 4 with confidence intervals (CIs) for clearer comparisons. Captions were updated accordingly. (enPj)
• Increased the legend font size in Figure 3 for better readability. (7rm7)
• Updated Table 5 to include CIs in parentheses for clarity and consistency. (enPj)

Narrative Updates
Added a paragraph in the "Generalization to External Clinical Studies" section, describing results with CIs while maintaining our conclusion that the proposed method outperforms all baselines. (enPj)
Integrated supplementary insights into the main narrative, including:

• Quantitative comparisons against baselines, stratified errors, and qualitative results (Section 3.3).
• Error analysis based on the number of observations (Section 3.4). (enPj)
• Performance comparisons across external clinical studies (Section 3.5). (enPj)
• Illustrative examples of Adaptive Shrinkage in test samples (Section 3.6). (7rm7, Q2Nf)

Supplementary Material

We expanded the supplementary material significantly, adding 15 new pages to address reviewers' concerns comprehensively. The new sections include:

• Section A: Extended Related Works in Predictive and Progression Modeling and Clinical Applications. (Q2Nf)

• Section E.3: Statistical significance tests on performance metrics across progression statuses for all baselines for a more comprehensive presentation of the results. (Q2Nf)

• Section E.4: Stratified performance analysis by covariates (Sex, APOE4 Alleles, and Education Years) with barplots showing mean AE and CIs.

• Section E.5: Extended discussion on the limitations of baseline models. (Q2Nf)

• Section E.6: Qualitative examples highlighting how our adaptation algorithm operates, illustrated with predicted trajectories and elaboration on clinical applications. This improves the interpretation of our results and connects them to real clinical scenarios. (Q2Nf, 7rm7)

• Section E.7: By expanding the training datasets from 2 to 7 studies (3504 subjects in total), we developed p-DKGP model and Adaptive Shrinkage Function for the SPARE-AD biomarker. We conducted error analysis based on observation counts, presented SPARE-AD trajectory examples, and provided detailed result interpretations. (enPj)

• Section E.8: Demonstrated how p-DKGP effectively learns smooth progression curves without requiring imputation, handling missing longitudinal biomarkers internally. Quantitative results confirmed its superiority over Linear Interpolation for interpolation tasks. (Q2Nf)

• Section E.9: Latent Space Analysis revealed that the transformation function captures atrophy patterns linked to AD and brain aging, projecting input features into a space optimized for progression prediction.

We sincerely thank the reviewers for their valuable feedback, which has improved the clarity, presentation, and communication of the methodological and clinical significance of our work. During the discussion period, we had the opportunity to engage with only one of the four reviewers despite multiple follow-up requests for further feedback or responses to our rebuttal. We sincerely hope that this lack of engagement will not adversely affect or bias the decision-making process. We hope the updates, along with the expanded supplementary material, meet your expectations and contribute towards a favourable decision on our submission.

Regards,

Authors of Submission 5004