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

Authors: Vasiliki Tassopoulou, Haochang Shou, Christos Davatzikos
Primary area: applications to neuroscience & cognitive science
Keywords: Deep Kernel Regression, Personalization, Posterior Correction, Longitudinal Biomarker Prediction

Abstract:
Longitudinal biomedical studies monitor individuals over time to capture dynamics in brain development, disease progression, and treatment effects. However, estimating trajectories of brain biomarkers is challenging due to biological variability, inconsistencies in measurement protocols (e.g., differences in MRI scanners) as well as scarcity and irregularity in longitudinal measurements. Herein,
we introduce a novel personalized deep kernel regression framework for forecasting brain biomarkers, with application to regional volumetric measurements. Our approach integrates two key components: a population model that captures brain trajectories from a large and diverse cohort, and a subject-specific model that captures individual trajectories. To optimally combine these, we propose Adaptive Shrinkage Estimation, which effectively balances population and subject-specific models. We assess our model’s performance through predictive accuracy metrics, uncertainty quantification, and validation against external clinical studies. Benchmarking against state-of-the-art statistical and machine learning models—including linear mixed effects models, generalized additive models, and deep learning methods—demonstrates the superior predictive performance of our approach. Additionally, we apply our method to predict trajectories of composite neuroimaging biomarkers, which highlights the versatility of our approach in modeling the progression of longitudinal neuroimaging biomarkers. Furthermore, validation on three external neuroimaging studies confirms the robustness of our method across different clinical contexts. We make the code available at https://github.com/vatass/AdaptiveShrinkageDKGP.

Decision: accept (poster)
Reviewer scores: 6, 6, 5, 6

Reviews and discussion:
<thread>
creation date:
1730104429184

thread_type:
official_review

<thread_object>
creator: reviewer_enpj

<content>
summary:
The authors have developed a deep kernel regression model to predict longitudinal brain volume changes. The model is made for personalised predictions and has two parts, one population model for global prediction and one subject-specific model for personalised predictions. The model is compared to several commonly used models on two open data sets. The resulting models are further evaluated on three external test sets. The results are convincing, and the method appears to work well.

strengths:
The paper is well-written and in general easy to follow. The method is straight-forward, clear, and appears to work well, and is applied with convincing results to real-world datasets. I really like that you evaluated on multiple external data; this strengthens the confidence in and validity of the results.

weaknesses:
It is not entirely clear where the methodological novelty lies. Personalised GPs have been proposed before (as mentioned in the introduction) and weighting schemes have also been proposed before (also correctly mentioned in the introduction). The adaptive shrinkage is interesting, and appears to work well, but must also be considered a small contribution.

It is great that the method is evaluated on no less than three external test sets, but it would have been good/more convincing if it was also trained on more than just two training sets.

There should be en ablation for the personalisation (or, equivalently, presenting results with \alpha=1), to see if it actually improves the results over just having the population level model.

There are not confidence intervals in the bar plots, which means it is not possible to compare those results.

questions:
Comments/questions/required changes:
 - How much data from individuals do you need for the personalised GP to be useful/to help in the predictions?
 - The step on Line 12 of Algorithm 1 is not atomic. Split this up or explain clearly somewhere what is being done.
 - It says the values in parentheses in Table 1 are standard deviations. Present the standard errors (standard deviations of the means) instead, so that the results can actually be compared properly.

Minor comments:
 - Do use \citet rather than \cite or \citep when you mention a reference in the body of the text, so that the names are not in parentheses in those cases.
 - Some more LaTeX-related issues: Remove blank lines after equations to avoid indented sentences after. If an equation ends a sentence, put a full stop after it (or, as in Equation 7, put the comma after the equation, not at the beginning of the next sentence). In general, make equations parts of sentences. There is a lot of space between Equations 6 and 7 and between Equations 11 and 12, probably because of an empty line between them (but do make these equations part of the sentence instead!).
 - Equation 2: The \Phi has not been defined.
 - Lines 82-83: These should rather be tuples, and not sets, since the order matters.
 - Line 163: A \cdot is missing as a wildcard for the first argument.
 - Line 417: A space missing after the colon.
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</thread>
<thread>
creation date:
1730594282503

thread_type:
official_review

<thread_object>
creator: reviewer_yf4s

<content>
summary:
This paper proposes a framework for personalized prediction to forecast longitudinal regional brain volumetric changes. Specifically, the authors provide a pipeline for integrating the population Deep Kernel Gaussian Process (DKGP) and the personalized DKGP by applying the Adaptive Posterior Shrinkage Estimation. The central innovation of this paper is the adaptive shrinkage mechanism, which could dynamically balance predictions from the population (p-DKGP) with the specific models (ss-DKGP) of the subject. Specifically, the authors propose an approach to learn the shrinkage parameter $\alpha$ to weigh the two models (as a linear combination). By combining these two models, this framework could make accurate predictions of brain volumes and could be stably generalized to diverse clinical populations. To sum up, the paper provides a flexible approach to modeling brain volume trajectories, which could offer a lot of potential applications in clinical settings.

strengths:
1. The logic of the paper is very clear and easy to follow. The paper's structure is well organized. 
2. The combination of population and subject-specific models and its application on data generalization for brain-related disease (AD, brain aging patterns) is a valuable contribution to the existing literature in this field.

weaknesses:
1. The contributions only have limited novelty. For example, a similar idea has already been proposed by Eleftheriadis et al., 2017. This paper already has a detailed demonstration of how the Gaussian Process can incorporate domain or individual-specific trends in a multi-task setup, which weakens the novelty of the proposed approach. Also, applying the machine learning techniques (like XGBoost) for weight estimation is also a common practice. The learning of the shrinkage parameters $\alpha$ might be a good application in the engineering field but is not a very novel way in ML research. A Bayesian Neural Network or probabilistic models that could provide more principled approaches for weight determination might be a better approach, such as *Weight Uncertainty in Neural Networks* by Blundell et al., 2015. 

2. The usage of latent space is very limited. In the current setup, the latent space serves primarily as a way to reduce the dimensionality of the input data and capture basic non-linear interactions through the MLP. However, it doesn’t appear that the authors further analyze or interpret the latent space itself. It would be very interesting if the authors could explore the temporal dynamics and conduct trajectory analysis in the latent space. 

3. In the experiment parts, the authors only compared some very traditional models; it would be much better if the authors could compare their approach with some more advanced deep learning based techniques. 

Reference:
Eleftheriadis, S., Rudovic, O., Deisenroth, M. P., & Pantic, M. (2017). Gaussian Process Domain Experts for Modeling of Facial Affect. IEEE transactions on image processing : a publication of the IEEE Signal Processing Society, 26(10), 4697–4711. https://doi.org/10.1109/TIP.2017.2721114

Blundell, C., Cornebise, J., Kavukcuoglu, K. &amp; Wierstra, D.. (2015). Weight Uncertainty in Neural Network. <i>Proceedings of the 32nd International Conference on Machine Learning</i>, in <i>Proceedings of Machine Learning Research</i> 37:1613-1622 Available from https://proceedings.mlr.press/v37/blundell15.html.

questions:
1. Could you elaborate on the rationale for freezing the latent transformation (MLP) parameters in the subject-specific model?
2. Did you explore the latent space and do some related analysis, such as subpopulation analysis or clustering of subjects with similar disease/ brain-trajectory patterns? 
3. What is the rationale for the choice of XGBoost to learn the shrinkage parameter $\alpha$? Did you try with other models?
4. How sensitive is the model's performance with different $\alpha$?  Can you provide some exploration of the impact of $\alpha$ on the prediction accuracy? 
4. The proposed model combines the variance of the GPs under the assumption of independence. How do you justify this assumption?
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</thread>
<thread>
creation date:
1730704333766

thread_type:
official_review

<thread_object>
creator: reviewer_q2nf

<content>
summary:
The authors proposed a method to model brain-changing trajectories. The method can capture brain volume trajectories from a large and diverse cohort. They introduced a personalization step that generates subject-specific models for individual trajectories. The paper was well written and easy to follow, although the method and the results were interesting. However, several points are unclear in the manuscript.

strengths:
The performance of the proposed method was evaluated on different neuroimaging data, which confirmed the generalizability and applicability of the method across different clinical contexts.

weaknesses:
1. The paper lacks of providing a comprehensive discussion of the wider literature on disease progression modeling. The interpretation of the results from the experiments is inadequate, raising questions about the model’s practical implications. 
2. Adding a clear comparison of the simulation data set will help readers better understand the advantages of the approach. 
3. Interpretation and Presentation of Results: The authors should provide a thorough interpretation of these results and a clearer discussion of the model’s real-world implications and its utility. In particular, the authors should clarify the implications of the findings from the experiment and how the modeling informs participant-level monitoring of disease progression.
4. Context and Literature Review: The authors should expand the context and literature review to properly situate their work within the broader field of disease progression modeling. The discussion should include a comparative analysis with other relevant works.
5. The references in this paper are relatively old. Please add the literature review of the past two years.

questions:
1. The subject-specific model ss-DKGP also used parameters of the population-level p-DKGP, how to guarantee the correctness of the predicted subject-specific results? If the answer is YES, why do we need to combine the prediction resultes of p-DKGP and ss-DKGP?
2. Expanding on the limitations of existing techniques would strengthen the positioning of your contribution.
3. The description of how external clinical studies were handled in the validation process could be clearer. Were there any preprocessing steps specific to the external datasets, and if so, how might these affect the generalization results?
4. The results demonstrate strong performance, but additional ablation studies would further strengthen the claims of the paper. For instance, how does the model perform when fewer ROIs are considered, or when different biomarkers are used? Additionally, could the method handle non-monotonic biomarkers, or is it limited to monotonic progressions?
5. The evaluation of external datasets is a strong point of the paper. However, it would be useful to see how the model handles missing data in longitudinal studies. What imputation strategies were employed, if any, and how did they impact the results?
</content>
</thread_object>

</thread>
<thread>
creation date:
1730758629707

thread_type:
official_review

<thread_object>
creator: reviewer_7rm7

<content>
summary:
This paper introduces a new personalized deep kernel regression framework to predict brain volume changes over time. The framework combines a population model, which captures population-level trends in brain volume changes, with a subject-specific model that tailors predictions to each individual subject’s trajectories. The optimal combination coefficient is determined using a shrinkage estimator to achieve the best personalized brain trajectory predictions. The framework is validated through metrics of predictive accuracy, uncertainty quantification, and comparisons with existing statistical and deep learning models, demonstrating superior performance. Additionally, the authors validated the framework's generalizability on brain datasets not seen during training.

strengths:
- This paper is overall well written and organized; hence it is easy for readers to follow.

- The motivation to develop a more precise predictive tool for diagnosing Alzheimer’s disease (AD) has strong clinical value.

- The proposed method is new in its approach to modeling both population-level and subject-specific brain trajectories. By combining these models to predict personalized brain volume changes over time through adaptive shrinkage estimation in a deep kernel regression framework, the author show a superior performance of the model compared with existing statistical and deep learning approaches.

weaknesses:
- While the approach of modeling population-level and individual-level trajectories for brain changes is promising, the current design, which trains these models separately, may be suboptimal. There is substantial research in computational anatomy [e.g., 1-3] demonstrating that individual trajectories should ideally be modeled as a hierarchical structure built upon the population-level mean. This approach naturally captures the hierarchical relationship between fixed group-level information and individual variability.

   [1] Hierarchical linear modeling (HLM) of longitudinal brain structural and cognitive changes in alcohol-dependent individuals during sobriety, 2007.

   [2] A hierarchical geodesic model for diffeomorphic longitudinal shape analysis, 2013.

   [3] Hierarchical multi-geodesic model for longitudinal analysis of temporal trajectories of anatomical shape and covariates, 2019.

- The connection between population-level trends and personalized trajectories in the proposed model is not clearly established.

questions:
This reviewer is happy to adjust the score if the questions and concerns are well addressed. 

- If an individual ROI differs significantly from the population mean, would it be possible that the optimization of Eq. (17) become biased towards minimizing the error between $y_t$ and $(1-\alpha) y_{st} $, as $y_{st}$ is learned to be fitted to a personalized brain trajectory. In this scenario, the value of $\alpha$ would likely be low if the changes in a particular ROI are subtle or difficult for the network to detect, and vice versa. It would be good if the authors can provide either empirical evidence or theoretical justification on how the model balances between population and individual trends in such cases.

- In Eq. (11), the expectation term appears to be conditioned on the learned latent representation $z_p$ from the population level. However, this condition seems to be missing in the final formulation. Please clarify if this is a typo. 

- Could the authors clarify how the six brain ROIs were selected and extracted from the original images? Additionally, please specify what information about the ROIs is input into the model (e.g., volume, size, or full ROI data).

- It is challenging to well justify the results in Fig. 3. The differences between the population and personalized trajectories may depend heavily on the distribution of training data and examine whether it is biased toward cognitively normal individuals / mild cognitive impairment (MCI) / AD. Each experiment should ideally account for and justify this factor.

- To enhance clarity, it would be helpful if the authors could summarize the other personalized trajectories for each of the six different ROIs in Fig. 3.

- It would be helpful to increase the font size of the legend in Fig. 3.
</content>
</thread_object>

</thread>
<thread>
creation date:
1732223685548

thread_type:
official_comment

<thread_object>
creator: authors

<content>
title:
Rebuttal and Response to Reviewers

comment:
We sincerely thank the reviewers for their thoughtful and constructive feedback. We are delighted to see that the reviewers recognized the value of our work! Through the extensive experimentation presented in the manuscript, we demonstrated and effectively communicated the *superior predictive performance* of our method, as well as its *generalizability* to external clinical studies. 
Some highlights from the reviews are the following:  
- **Methodology**:  “The proposed method is new in its approach to modeling both population-level and subject-specific brain trajectories” (7rm7), “The method is straight-forward, clear, and appears to work well” (enPj)  
- **Impact** : “The combination of population and subject-specific models and its application on data generalization for brain-related disease (AD, brain aging patterns) is a valuable contribution to the existing literature in this field.” (yf4s) 
- **Experiments and Evaluation**: “The performance of the proposed method was evaluated on different neuroimaging data, which confirmed the generalizability and applicability of the method across different clinical contexts” (Q2Nf). “I really like that you evaluated on multiple external data; this strengthens the confidence in and validity of the results”, “The method ... is applied with convincing results to real world datasets (enPj) 
- **Writting and Presentation**: “The logic of the paper is very clear and easy to follow. The paper's structure is well organized”  (yf4S) 

### **Contributions** 
Based on the reviewers' comments, we realized that our contributions were not emphasized clearly throughout the manuscript. To address this, we have revisited and explicitly outlined the key contributions of our work:

1. We present a novel deep kernel regression framework designed specifically for sparse longitudinal data. This framework employs a transformation function that maps multimodal and high-dimensional imaging and clinical features into a lower-dimensional space that is highly informative of biomarker progression.

2. We introduce a novel adaptation technique called Adaptive Shrinkage, which balances population-level trends with individual-specific trajectories. This innovative approach enables effective personalization within the deep kernel regression of longitudinal biomarkers.
3. We demonstrate that both the population models and the Adaptive Shrinkage function generalize well to external clinical studies, showcasing their robustness and applicability across diverse datasets.
4. We show that the transformation function, $\Phi$, is interpretable and captures meaningful patterns related to the modeled biomarker, 
progression 
5. We highlight that Adaptive Shrinkage is explainable, making it more trustworthy and practical for clinical applications where transparency is crucial.
6. We commit to making the code publicly available, enabling broader application and validation of our method.

Following, taking into account the valuable comments and questions we received from the reviewers, we put a lot of effort into implementing additional experiments/additions:
- Interpretability of the function $\Phi$ and Latent Space Exploration  (Supplementary Section E.9) 
   - We show that transformation function $\Phi$ captures atrophy patterns related to AD and Aging when predicting the biomarkers of SPARE-AD and SPARE-BA. 
   - We show that the transformation $\Phi$ gives a latent space more informative of progression in comparison with the input space
- Improvement of the presentation of the quantitative results and qualitative examples (Main Manuscript and Supplementary Section E.3, E.4 and E.6, respectively) 
    - We have included CIs along with the Mean AEs for a more straightforward comparison of the methods. 
    - We have included qualitative results that demostrate the behaviour our the Adaptive Shrinkage function 
- Handling the Missing Longitudinal Variables and Analysis of Interpolation abilities of our framework (Supplementary Section E.8) 
    - We elaborate on how our Deep Kernel GP framework handles internally the missing longitudinal observations without imputation in preprocessing.
    - We show that our model can effectively impute missing longitudinal biomarker values in time, in the observed population
- Extended related works and clinical applications of our method (Supplementary Section A) 

All the additions and modifications in the manuscript are colored blue and the revised manuscript will be updated shortly.

We will address all the comments and concerns of each reviewer, point by point in the individual responses below. 

All in all, we thank the reviewers for their valuable comments. We believe that the soundness of our work is improved with the changes on the manuscript as well as the additional results on the supplementary material.  
We hope our efforts to be recognized and we are always available for discussion. 

Sincerely, 

Authors of Submission 5004
</content>
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</thread>
<thread>
creation date:
1733259235747

thread_type:
official_comment

<thread_object>
creator: authors

<content>
title:
Final Remarks

comment:
---

## **Dear Area Chairs and Reviewers,**

We appreciate your thoughtful feedback and encouraging comments. Below, we summarize the strengths of our work as highlighted by the reviewers, outline the contributions of our work, and elaborate on the changes and additions we have made in response to reviewers’ feedback.

---
### **Acknowledgment of Positive Reviewer Feedback**


#### **Clinical Relevance**
- *"The motivation to develop a more precise predictive tool for diagnosing Alzheimer’s disease (AD) has strong clinical value."* (7rm7)

#### **Novelty and Methodological Strength**
- *"The proposed method is new in its approach to modeling both population-level and subject-specific brain trajectories. By combining these models to predict personalized brain volume changes over time through adaptive shrinkage estimation in a deep kernel regression framework, the authors show a superior performance of the model compared with existing statistical and deep learning approaches."* (7rm7)

#### **Robust Validation and Generalizability**
- *"The performance of the proposed method was evaluated on different neuroimaging data, which confirmed the generalizability and applicability of the method across different clinical contexts."* (Q2Nf)  
- *"I really like that you evaluated on multiple external data; this strengthens the confidence in and validity of the results."* (enPj)

#### **Scientific Contribution**
- *"The combination of population and subject-specific models and its application on data generalization for brain-related diseases (AD, brain aging patterns) is a valuable contribution to the existing literature in this field."* (yf4S)

#### **Clarity and Organization**
- *"This paper is overall well-written and organized; hence it is easy for readers to follow."* (7rm7)  
- *"The logic of the paper is very clear and easy to follow. The paper's structure is well organized."* (y4fs)  
- *"The paper is well-written and in general easy to follow."* (enPj)  

---

### **Contributions of Our Work**

1. **A Novel Framework for Sparse Longitudinal Data**  
   We present a deep kernel regression framework uniquely tailored for sparse longitudinal data, addressing key challenges in progression modeling. This framework leverages a transformation function to project multimodal, high-dimensional imaging, and clinical features into a compact, lower-dimensional space that captures the most informative aspects of biomarker progression.  
   Unlike traditional methods, our approach eliminates the need for imputation or temporal alignment across subjects, making it robust to irregular sampling and diverse observation patterns. Designed to predict long-term biomarker trajectories spanning several years, this framework offers a powerful and scalable solution for modeling complex, heterogeneous longitudinal datasets in neurodegenerative and other progressive diseases.

2. **Introduction of Adaptive Shrinkage**  
   We introduce a novel adaptation technique, **Adaptive Shrinkage**, which balances population-level trends with individual-specific trajectories. Population and subject-specific models share a common deep kernel that aligns features across subjects in the same latent space, providing a unified feature space for both population and subject-specific dynamics.  
   By leveraging this common representation, the Adaptive Shrinkage method enables effective personalization within the deep kernel regression framework for longitudinal biomarkers. This dual focus—capturing global trends while refining individual trajectories—provides a robust foundation for modeling heterogeneous progression patterns and enhances the interpretability and adaptability of the framework to diverse biomarker dynamics.

3. **Robust Generalizability**  
   We demonstrate that both the population models and the Adaptive Shrinkage function generalize well to external clinical studies, showcasing their robustness across diverse contexts.  
   - **AIBL**: Includes an older cohort primarily composed of AD patients.  
   - **OASIS**: Focuses on a younger cohort with a majority of healthy controls.  
   - **PreventAD**: Targets a healthier, younger population for early pre-symptomatic detection of AD.  
   These differences highlight the adaptability of our framework to varying age groups, disease stages, and clinical goals.

4. **Interpretable Transformation Function**  
   The transformation function captures meaningful neuroimaging patterns associated with biomarker progression, offering insights into the underlying processes driving disease dynamics.

5. **Explainable Adaptive Shrinkage**  
   Adaptive Shrinkage is inherently explainable (Section 3.5), ensuring transparency in its operation and enhancing trustworthiness for clinical applications where informed decision-making is paramount.

6. **Reproducibility**  
   To promote reproducibility and validation, we provide publicly available code and validation on open datasets as ADNI.

*Continues...*

---
</content>
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</thread>


Metareview:
metareview:
The proposed work provides a framework for modelling individual patient trajectories (e.g., in disease progression) across longitudinal data, using (deep) kernel regression. Their proposed regularization is adaptively set, and the authors provide a family of derived variants for subject-specific fitting.

Overall, the response from reviewers has been relatively mild but generally positive. Reviewers noted that the paper is well written and generally appears to have good results (modulo language noted by `enPj`), but that the contribution is difficult to define in light of previous GP work and previous disease progression literature.

By scores alone, this is definitionally borderline. However, after discussion, I find that this paper has value to both the methodological community and the disease modelling/longitudinal imaging community. This, combined with a clear description of their own method as well as deep kernel regression and its regularization, leads me to recommend acceptance as a poster.