Source-linked AI summary
PathFinder: Joint Decompositions of Linked Multimodal Datasets
Ying-Qiu Zheng, Alex Fung, Stephen M Smith, Rogier B Mars, Saad Jbabdi
TL;DR
Existing joint decompositions require datasets to share a common dimension, limiting co-analysis across domains and incomplete modalities. PathFinder links matrices through shared dimensions and paths, recovering missing stimuli and revealing cortico-basal pathways in real data.
Problem
Existing joint decompositions require a common dimension across datasets, hindering co-analysis across species or modalities when dimensions are incompatible or data are missing.
Method
PathFinder jointly decomposes matrices when pairs or subgroups share dimensions and paths connect all datasets, enabling prediction of missing matrices.
Results
PathFinder recovered held-out stimuli from remaining stimuli and fMRI data and recovered three canonical basal ganglia pathways in diffusion MRI tractography.
Takeaways & Limitations
PathFinder supports joint analysis across different domains with missing subsets, providing a common subspace for species comparison and prediction testing.
Takeaways & Limitations
PathFinder requires datasets to be arranged as a 2D matrix, limiting applicability to configurations where factor matrices would need to overlap.
Abstract
from arXiv · showhide
Low-rank matrix decompositions can uncover patterns and structure in data and have a number of different applications across many disciplines. Extensions to "joint" low-rank decompositions have been proposed to link datasets from different modalities. While these methods enable the discovery of common patterns across modalities, they require that all the multimodal data share one or more dimensions. We propose a new analysis method, PathFinder, that enables co-analysis of datasets that do not necessarily all share a dimension. The key insight is that as long as pairs or subgroups of matrices do share some dimension, and that there are one or more paths that link across the data matrices, a global joint decomposition can be sought out. This enables the joint estimation of common patterns across different modalities, species, or scales, where a one-to-one mapping across all data along some dimension is not necessarily available. We show that PathFinder is a general umbrella under which many matrix decomposition methods fall as special cases. It can be used to discover common patterns across disparate datasets and to make predictions for missing data or modalities.
1 Introduction
PathFinder extends joint low-rank decomposition to linked multimodal datasets that lack a single shared dimension, provided shared dimensions create paths across all matrices. The method also unifies existing approaches, supports interpretable components and missing-data analysis, and is demonstrated in two neuroimaging applications.
- Motivation: Low-rank decompositions, especially ICA, reveal hidden structure in neuroimaging data and support brain-network mapping and artifact removal.Extensions including concatenation, tensor-ICA and linked-ICA enable simultaneous analysis of multiple subjects or modalities.
- Problem: Existing joint analyses cannot always compare datasets lacking a common dimension, including resting-state fMRI across species or datasets with missing data.Across species, neither spatial nor temporal dimensions are compatible, and nonlinear registration into standard space is not possible.
- Method: PathFinder requires shared row- or column-dimensions between pairs or subgroups of matrices and one or more paths linking all matrices.These paths allow joint outer-product decomposition despite the absence of a single common dimension or the presence of missing data.
- Contributions: The paper develops PathFinder’s principles and algorithms, combines it with ICA for more interpretable latent components, and places linked-ICA and dual regression under its umbrella.The concept generalizes across multiple domains and modalities, with some modalities potentially missing for some domains.
- Applications: PathFinder is applied to stimulus prediction in a retinotopic task and analysis of cortico-subcortical connectivity loops in diffusion MRI tractography.These are the paper’s two real-dataset demonstrations.
2 Methods
PathFinder models available domain–modality matrices with shared domain-specific left factors and modality-specific right factors, enabling joint decomposition even when matrices lack a universal shared dimension. It minimizes regularized reconstruction error using alternating least squares and resolves rotational ambiguity post-hoc with ICA.
- Data structure: Each dataset X_dm has rows shared within domain d and columns shared within modality m, while only existing domain–modality combinations are included.The available matrix index set is M = {(d, m) | X_dm exists}.
- Joint decomposition: PathFinder factorizes every available matrix as X_dm = A_dS_m^T, enforcing shared left factors across domains and shared right factors across modalities.For rank r, A_d ∈ R^(r_d×r) and S_m ∈ R^(c_m×r), representing domain-specific and modality-specific subspaces.
- Objective and identifiability: The loss minimizes reconstruction error with L2 regularization to reduce overfitting and support prediction of unseen data, although factor solutions remain non-unique.Rotational ambiguity is addressed post-hoc with independent component analysis, which selects a rotation maximizing non-Gaussianity for more interpretable factors.
- Joint decomposition: The joint optimization relates matrices with neither domain nor modality in common through intermediate matrices when connecting domain–modality pairs exist.Thus, factors for otherwise incompatible matrices can be linked through available intermediate datasets.
- Optimization: Alternating least squares updates all A_d factors with S_m fixed and all S_m factors with A_d fixed, using closed-form ridge-regression subproblems.Each subproblem is convex; implementations typically run for 25 iterations or until relative loss change falls below a threshold.
3 Results
Across simulations and neuroimaging applications, PathFinder recovered observed and missing data, supported cyclic decompositions, predicted held-out stimuli, reproduced dual-regression goals, and revealed canonical basal ganglia pathways.
- Simulation results: PathFinder accurately recovered observed and missing matrices, with R2 maximized at the true rank and declining under stronger regularization, especially for unseen matrices.Accuracy remained high for small α values, while over-regularization degraded unseen matrices faster than observed ones.
- Simulation results: PathFinder achieved higher R2 than independently applied SVD at higher noise levels (σ > 0.5), likely because pooling information enabled effective denoising.The comparison used observed matrices because independent SVD cannot reconstruct unobserved data.
- Cyclic decomposition: For cyclic data, PathFinder recovered the data but not uniquely determined factors; applying ICA to concatenated reconstructed factors recovered the ground-truth factors.The ambiguity arose because the factors were identifiable only up to a rotation, which regularization did not resolve.
- Retinotopy fMRI: In retinotopy fMRI, PathFinder recovered a held-out counter-clockwise rotating wedge stimulus from the remaining 11 datasets and fMRI data.Excluding one hemisphere broke the prediction.
- Dual regression: PathFinder can implement dual regression as a single joint decomposition, using concatenated group and subject-specific columns to estimate group-level and individual spatial factors.The group spatial factor is constrained by every subject’s data, while each subject receives an additional subject-specific column.
- Cyclic decomposition: Applied to cortico-basal connectivity loops and followed by ICA, PathFinder recovered the three canonical basal ganglia pathways: motor, associative, and limbic.Winner-takes-all voxel assignments linked regions sharing components to the same functional circuits.
4 Discussion
PathFinder is a general framework for joint matrix decomposition that is especially useful for linking data from different domains with missing subsets. The discussion also identifies extensions to tensors, nonlinear encoders and decoders, and broader factor arrangements, while noting current 2D and bipartite limitations.
- Framework and applications: PathFinder provides a simple, general umbrella for joint matrix decomposition, including joint ICA and dual regression, and supports cross-domain data with potentially missing subsets.The discussion highlights cross-species analysis as a motivating case where different modalities may be available for different species.
- Extensions: Joint SVD can extend PathFinder to potentially missing datasets while resolving uniqueness up to sign flip through orthogonality and separating singular-value weights from factors.The passage identifies separation of weights from factors as an additional advantage, although the supplied text does not complete its consequence.
- Extensions: PathFinder can be extended beyond 2D matrices to higher-order tensors and to nonlinear encoder-decoder architectures such as auto-encoders.These extensions are presented as generalizations of matrix decomposition and linear low-dimensional embeddings.
- Limitations: A major limitation is that current PathFinder requires data to be arranged as 2D matrices with distinct A- and S-matrix sets, restricting applicability.The discussion notes that an A matrix for one dataset cannot currently serve as an S matrix for another dataset.
- Availability: The implementation of PathFinder, along with experiment data and supplementary files, has been released through Zenodo and OSF.The supplied passage provides the release locations but not additional implementation details.
Supplemental Materials · 5 Mini-batch implementation of Joint Decomposition
The supplemental materials describe memory-efficient mini-batch algorithms for PathFinder’s joint decomposition and Joint SVD. These methods update factors incrementally from row or column batches, with Joint SVD using power iteration to avoid storing full projection matrices.
- 5 Mini-batch implementation of Joint Decomposition: Mini-batching addresses prohibitively large concatenated matrices by processing large datasets in batches during each least-squares iteration.This is motivated by datasets such as MRI images, whose matrices may have thousands of rows and columns.
- 5 Mini-batch implementation of Joint Decomposition: For updating A_d, matrices are partitioned into column batches, with corresponding rows of S_m partitioned consistently.The loss and factor update are expressed using these aligned batches across modalities.
- 5 Mini-batch implementation of Joint Decomposition: The A_d update accumulates cross products and Gram matrices batch by batch, avoiding storage of the full concatenated S̃ and X̃_d matrices.The numerator and denominator of the update are both sums over batches.
- 5 Mini-batch implementation of Joint Decomposition: The S_m update is symmetric: matrices are partitioned along rows, and batchwise cross products are accumulated over the domains where each modality is observed.The pseudo-code for these alternating mini-batch updates is summarized in Table S1.
- 5.1 Implementation of Joint Singular Value Decomposition: Joint SVD shares orthogonal bases U_d and V_m across domains and modalities while allowing each matrix its own diagonal scaling D_dm.Alternating updates enforce orthogonality constraints; D_dm is obtained by projecting data onto the current orthonormal bases.
- 5.1 Implementation of Joint Singular Value Decomposition: Joint SVD updates each component of U_d and V_m from stacked projection matrices, then orthogonalizes the resulting factors with QR decomposition.The update for V_m follows the same construction as the update for U_d with rows and columns interchanged.
- 5.2 Mini-batch Implementation of JointSVD: Mini-batch Joint SVD uses power iteration on M M^T to obtain leading singular vectors without holding all modality projections in memory.The error contracts geometrically by (σ_2/σ_1)^2 per step, and row or column chunks compute matrix-vector products incrementally.
- 5.2 Mini-batch Implementation of JointSVD: The mini-batch Joint SVD algorithm samples subsets of domains or modalities, performs T_pow power iterations on chunked products, and uses batch_size for sampling and chunking.The same strategy updates U_d and V_m with rows and columns swapped; the pseudocode sets T_pow to 2 in the reported implementation example.