MATCH-AD: Adaptive Transport Clustering for AD

• The paper introduces MATCH-AD, which fuses deep representation learning, graph-based label propagation, and optimal transport clustering to extract disease-relevant signals from heterogeneous neuroimaging data.
• It achieves robust quantification of Alzheimer’s progression with strong theoretical guarantees and near-perfect diagnostic reliability under limited label availability.
• Extensive evaluations demonstrate that MATCH-AD significantly outperforms baseline methods in accuracy and Cohen's kappa, enhancing clinical interpretability and decision-making.

Multi-view Adaptive Transport Clustering for Heterogeneous Alzheimer’s Disease (MATCH-AD) is a semi-supervised learning framework designed to address diagnostic challenges in Alzheimer’s disease (AD) using heterogeneous neuroimaging datasets with limited ground truth annotations. By integrating deep representation learning, graph-based label propagation, and optimal transport clustering, MATCH-AD enables the extraction of disease-relevant structure, label-efficient classification, and explicit quantification of disease progression, with strong theoretical guarantees and empirical superiority over existing methods (Moayedikia et al., 19 Dec 2025).

1. Overview of MATCH-AD Framework

MATCH-AD is constructed as a unified, alternating-minimization pipeline comprising three tightly coupled modules:

• Deep Representation Learning: Heterogeneous input data—including structural MRI features ($X_{MRI}\in\mathbb{R}^{n\times190}$), CSF biomarkers ($X_{CSF}\in\mathbb{R}^{n\times6}$), and clinical/demographic variables ($X_{demo}\in\mathbb{R}^{n\times23}$)—are preprocessed by kNN imputation (k=5) and robust scaling, then concatenated into a tensor $X\in\mathbb{R}^{n\times219}$. An autoencoder $f_\theta,g_\phi$ with encoder-decoder architecture learns a latent representation $Z=f_\theta(X)\in\mathbb{R}^{n\times32}$ designed to preserve disease-relevant manifold structure.
• Graph-based Label Propagation: A kNN graph is built over $Z$, constructing an affinity matrix $W$ from Gaussian kernels and normalizing to a similarity matrix $S=D^{-1/2} W D^{-1/2}$. This graph is used to propagate sparse clinical labels ($\sim$29% labeled) to the majority of the cohort.
• Optimal Transport Clustering: Using the propagated multi-class labels, samples are partitioned into disease stages. Entropically regularized Wasserstein distances are computed between empirical distributions at adjoining stages, providing a continuous, metric-based quantification of disease progression using the Sinkhorn algorithm.

All modules contribute to a joint training objective, with alternating updates for representations, label distributions, and transport plans.

2. Mathematical Foundation

2.1 Input Views and Shared Embedding

• Let $X_{CSF}\in\mathbb{R}^{n\times6}$0, $X_{CSF}\in\mathbb{R}^{n\times6}$1, $X_{CSF}\in\mathbb{R}^{n\times6}$2; $X_{CSF}\in\mathbb{R}^{n\times6}$3
• Encoder: $X_{CSF}\in\mathbb{R}^{n\times6}$4
• Decoder: $X_{CSF}\in\mathbb{R}^{n\times6}$5
• Latent representation: $X_{CSF}\in\mathbb{R}^{n\times6}$6

2.2 Graph Construction

• Pairwise Euclidean distance matrix in latent space: $X_{CSF}\in\mathbb{R}^{n\times6}$7
• For kNN graph, affinity weights:

$X_{CSF}\in\mathbb{R}^{n\times6}$8

with $X_{CSF}\in\mathbb{R}^{n\times6}$9 the mean distance to $X_{demo}\in\mathbb{R}^{n\times23}$0’s $X_{demo}\in\mathbb{R}^{n\times23}$1-th neighbor.

2.3 Label Propagation

• Label matrix $X_{demo}\in\mathbb{R}^{n\times23}$2: labeled rows one-hot, unlabeled rows uniform ($X_{demo}\in\mathbb{R}^{n\times23}$3)
• Propagation update:

$X_{demo}\in\mathbb{R}^{n\times23}$4

This admits a closed-form solution:

$X_{demo}\in\mathbb{R}^{n\times23}$5

2.4 Optimal Transport

• Successive disease stages $X_{demo}\in\mathbb{R}^{n\times23}$6, $X_{demo}\in\mathbb{R}^{n\times23}$7 induce empirical distributions over $X_{demo}\in\mathbb{R}^{n\times23}$8; 2-Wasserstein distance:

$X_{demo}\in\mathbb{R}^{n\times23}$9

where $X\in\mathbb{R}^{n\times219}$0 and $X\in\mathbb{R}^{n\times219}$1 is the entropy regularizer.

3. Training Objectives and Optimization

MATCH-AD is optimized by minimizing a composite loss:

$X\in\mathbb{R}^{n\times219}$2

• Autoencoder Loss $X\in\mathbb{R}^{n\times219}$3:

$X\in\mathbb{R}^{n\times219}$4

• Propagation Loss $X\in\mathbb{R}^{n\times219}$5:

$X\in\mathbb{R}^{n\times219}$6

• Optimal Transport Loss $X\in\mathbb{R}^{n\times219}$7: sum of Wasserstein distances
• Smoothness Loss $X\in\mathbb{R}^{n\times219}$8: promotes local consistency of soft label distributions

Algorithmic optimization proceeds by pretraining the autoencoder, followed by alternating updates of labels, transport plans, and representations until convergence.

4. Theoretical Analysis

MATCH-AD is accompanied by several theoretical guarantees:

• Convergence of Label Propagation: The label propagation update converges geometrically for $X\in\mathbb{R}^{n\times219}$9, with

$f_\theta,g_\phi$0

• Label Consistency Bound: Under appropriate manifold assumptions, the expected label error is bounded:

$f_\theta,g_\phi$1

where $f_\theta,g_\phi$2 is the intrinsic dimension and $f_\theta,g_\phi$3 the geodesic error.

• Stability of Wasserstein Distance: Empirical Wasserstein distances are stable to sampling, with

$f_\theta,g_\phi$4

• Global Convergence: Alternating minimization is guaranteed to reach a stationary point (under Lipschitz and boundedness assumptions).
• Sample Complexity: To achieve $f_\theta,g_\phi$5-accurate label propagation with probability $f_\theta,g_\phi$6 requires

$f_\theta,g_\phi$7

5. Experimental Evaluation

5.1 Data and Preprocessing

• Cohort: $f_\theta,g_\phi$8 subjects from National Alzheimer’s Coordinating Center; 219 total features per subject after integration
• Semi-supervised regime: 29.1% labeled for training/testing, 70.9% unlabeled
• Class distribution (labeled subset): Normal (63.1%), Impaired-not-MCI (3.7%), MCI (19.9%), Dementia (13.3%)
• Preprocessing: $f_\theta,g_\phi$950% missing—excluded; remaining imputed by kNN (k=5)
• Train/test split: 80/20 stratified on labeled data

5.2 Performance under Label Scarcity

Labeled Fraction	Accuracy (\%)	Cohen’s $Z=f_\theta(X)\in\mathbb{R}^{n\times32}$0
5\% ($Z=f_\theta(X)\in\mathbb{R}^{n\times32}$170)	59.1 ± 1.8	0.159
30\%	72.6 ± 0.6	0.430
50\%	81.1 ± 0.7	0.614
80\%	91.9 ± 0.3	0.842
100\% ($Z=f_\theta(X)\in\mathbb{R}^{n\times32}$2 test set)	98.4	0.970

• MATCH-AD retains clinically meaningful κ ($Z=f_\theta(X)\in\mathbb{R}^{n\times32}$30.4) with only 30% labels, reaching “almost perfect agreement” (κ$Z=f_\theta(X)\in\mathbb{R}^{n\times32}$40.8) at 80% (Moayedikia et al., 19 Dec 2025).

5.3 Baseline Comparisons and Ablations

• Best baseline (SelfTraining_SVM): 71.3% accuracy, κ=0.329
• MATCH-AD: 98.4% accuracy, κ=0.970, F1=0.976
• SelfTraining_SVM underperforms on minority classes, while MATCH-AD maintains F1$Z=f_\theta(X)\in\mathbb{R}^{n\times32}$50.86 across all classes (“Normal”: 98.9%, “Impaired-not-MCI”: 90.9%, “MCI”: 86.2%, “Dementia”: 97.4%)
• Removal of the autoencoder collapses κ to zero/negative; removal of optimal transport impairs modeling of progression but weakly affects classification performance

5.4 Hyperparameter Sensitivity

• Propagation $Z=f_\theta(X)\in\mathbb{R}^{n\times32}$6: peak performance at $Z=f_\theta(X)\in\mathbb{R}^{n\times32}$7–$Z=f_\theta(X)\in\mathbb{R}^{n\times32}$8
• Neighborhood $Z=f_\theta(X)\in\mathbb{R}^{n\times32}$9: plateau of robust performance for $Z$0 (default: $Z$1)
• Outer iterations $Z$2–20 for convergence

6. Clinical Relevance and Interpretability

• Accuracy alone understates performance under class imbalance; Cohen’s κ provides a stricter criterion
• MATCH-AD illustrates resource allocation tradeoffs: moderate agreement (κ$Z$30.4) attainable with only ~30% labeled subjects, substantial (κ$Z$40.6) at ~50–60%, and almost perfect (κ$Z$50.8) at ~80%
• Minority/early-stage classes (Impaired-not-MCI) benefit most from the integrated representation+propagation approach
• The explicit transport-based progression quantifies state transition distances, potentially offering interpretable “disease trajectories” in clinical settings

7. Summary and Availability

MATCH-AD introduces a mathematically principled, scalable, and label-efficient framework for mining disease structure in heterogeneous neuroimaging datasets. It achieves nearly perfect diagnostic reliability (κ up to 0.97), robust performance across extreme label scarcity (≥5% labeled), and continuous quantification of progression via Wasserstein distances, with strong theoretical guarantees at every stage (Moayedikia et al., 19 Dec 2025). All code and data splits are provided at https://github.com/amoayedikia/brain-network.git.