DKGE versus Partial Least Squares

This document provides a comparative analysis of Design-Kernel Group Embedding (DKGE) and the well-established partial least squares (PLS) framework for neuroimaging analysis. To establish a foundation for comparison, we first examine the classical PLS approach as described in McIntosh & Lobaugh (2004), following their established terminology and conceptual framework.

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What classical PLS does

Classical PLS neuroimaging analysis operates by constructing latent variables that capture the strongest relationships between brain activity and experimental or behavioral variables. The primary objective is to build latent variables (LVs) that maximise the covariance between an exogenous block (such as task design or behavior) and a brain-data block (elements × time).

The framework encompasses several methodological variants, each tailored to different research questions: task PLS analyzes condition differences, behavior PLS examines brain–behavior coupling, seed PLS investigates functional connectivity patterns, and spatiotemporal PLS treats space × time jointly for fMRI/ERP/MEG data.

The computational workflow follows a systematic sequence of steps: 1. Data are arranged as a single matrix $M \in \mathbb{R}^{(n k) \times (m t)}$ with observations nested inside conditions. 2. Cross-block covariance is formed with an orthonormal design matrix, followed by singular value decomposition $C^\top M = U S V^\top$. 3. Element/time saliences (singular images), design saliences, and singular values are extracted from the decomposition. 4. Brain scores $B = M U$ and design scores $D = C V$ are computed to characterize patterns in each domain. 5. For behavior PLS specifically, the design block is replaced with behavior matrices and correlated with $M$ before applying SVD.

Statistical inference in PLS relies on resampling approaches: permutation tests assess LV significance, bootstrap procedures evaluate voxel salience reliability, and Procrustes alignment stabilizes resampled LVs across iterations.

This methodology emphasizes whole-pattern effects and time-resolved couplings, leveraging linear algebra and resampling techniques to identify distributed, reliable neural patterns that relate to experimental or behavioral variables.

How DKGE relates

DKGE builds upon the fundamental concept of using a compact latent space to explain patterns in neuroimaging data, but extends this approach in ways specifically tailored for modern multi-subject neuroimaging analysis. While PLS operates on raw brain data to find latent variables, DKGE works with subject-wise GLM outputs and incorporates design-aware alignment and spatial transport mechanisms. The comprehensive comparison table below summarizes both the conceptual commonalities between these approaches and the key methodological advances that DKGE introduces.

Aspect	Partial Least Squares	DKGE
Latent-space construction	SVD on cross-block covariance between design/behaviour and brain data; columns of $U$ and $V$ are saliences.	Eigen-decomposition of a design-kernel-weighted covariance, producing orthonormal components $U$.
Design information	Implicit via orthonormal contrasts; no way to encode graded similarity between effects.	Explicit design kernel $K$ encodes factorial structure, smoothness, or prior correlations among effects.
Data normalisation	Conditions averaged or mean-centred before SVD; each voxel treated equally.	Row standardisation of subject betas using the pooled design Cholesky factor; optional spatial/reliability weights $\Omega_s$.
Cross-validation	Permutation for LV significance, bootstrap for salience stability (no LOSO cross-fitting).	LOSO / K-fold cross-fitting (dkge_contrast), analytic approximations, parametric or bootstrap inference with cached transports.
Transport / alignment	Outputs latent scores; spatial interpretation relies on the original voxel grid.	Provides barycentric kNN and Sinkhorn transports, anchor graphs, and voxel decoders for consistent spatial maps across parcellations.
Reliability weighting	All observations weighted equally; stability assessed post hoc via bootstrap ratios.	Subject- and cluster-level reliabilities enter directly (e.g. sizes, inverse variances), influencing fits and transport.
Spatiotemporal support	ST-PLS handles time by stacking features.	DKGE works on any GLM-derived beta blocks; temporal modelling is delegated to the design matrix and optional kernels.
Implementation focus	Exploratory LVs; complementary to other analyses.	Integrated workflow for group GLM analysis, transport, and inference tailored to fMRI/ERP pipelines.

Similarities worth noting

Despite their methodological differences, PLS and DKGE share several fundamental characteristics that reflect their common mathematical foundations. Both approaches rely on SVD or eigendecomposition techniques to obtain orthogonal latent patterns and their associated scores, ensuring that the derived components capture independent sources of variation in the data.

Resampling methodology plays a central role in both frameworks, though implemented differently: PLS employs permutation tests and bootstrap procedures for statistical inference, while DKGE provides a broader toolkit including analytic approximations, leave-one-subject-out (LOSO) cross-fitting, bootstrap procedures, and transport-aware resampling utilities.

Both methods require careful interpretation that involves examining latent loadings or saliences in conjunction with subject scores to properly understand how experimental conditions or behavioral variables relate to the underlying neural patterns.

Why DKGE can replace or complement PLS in modern pipelines

DKGE offers several methodological advantages that make it particularly well-suited for contemporary neuroimaging analysis pipelines, addressing limitations that can arise with classical PLS approaches:

1. Enhanced design control — The design kernel framework provides a principled mechanism for encoding factorial relations, smoothness constraints, or hierarchical structure among experimental conditions, capabilities that are difficult to achieve within the classical PLS framework.

2. Bias-aware statistical contrasts — Cross-fitting procedures systematically avoid the optimistic bias that can occur when estimating condition or behavior effects on the same data used for model fitting, while built-in inference tools provide rigorous statistical testing without requiring separate implementations.

3. Sophisticated spatial alignment — Transport operators enable mapping of component or contrast values onto consistent anchor or voxel grids, facilitating meaningful group-level interpretation even when working with heterogeneous parcellation schemes across subjects.

4. Principled reliability weighting — Subject-level and cluster-level reliability weights ensure that noisy measurements contribute proportionally less to the final estimates, improving overall stability compared to the uniform weighting typically employed in PLS analyses.

5. Seamless workflow integration — DKGE integrates directly into downstream analysis pipelines including rendering, bootstrapping, and component analysis without requiring reconstruction of transport operators or design alignment logic at each stage.

In summary, DKGE can be conceptualized as a design-aware, transport-enabled extension of fundamental PLS concepts. It preserves the interpretability and whole-pattern emphasis that make latent variable approaches valuable while providing the inference capabilities, reliability weighting, and spatial alignment mechanisms that modern multi-subject neuroimaging studies require.