On semi-orthogonality and a special class of matrices

Groβ, Ju ̈rgen; Trenkler, Go ̈tz; Troschke, Sven-Oliver

doi:10.1016/s0024-3795(97)10002-7

Cited by 13 publications

(6 citation statements)

References 9 publications

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“…In practice, the Efficient ProMises approach projects the matrices into an n -lower-dimensional space using a specific semi-orthogonal transformation (Abadir & Magnus, 2005 ; Groß et al, 1999 ) , with dimensions , which preserve all the data’s information. It aligns, then, the reduced matrices by the perturbation or ProMises model.…”

Section: Efficient Promises Modelmentioning

confidence: 99%

Procrustes Analysis for High-Dimensional Data

Andreella

Finos

2022

Psychometrika

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The Procrustes-based perturbation model (Goodall in J R Stat Soc Ser B Methodol 53(2):285–321, 1991) allows minimization of the Frobenius distance between matrices by similarity transformation. However, it suffers from non-identifiability, critical interpretation of the transformed matrices, and inapplicability in high-dimensional data. We provide an extension of the perturbation model focused on the high-dimensional data framework, called the ProMises (Procrustes von Mises–Fisher) model. The ill-posed and interpretability problems are solved by imposing a proper prior distribution for the orthogonal matrix parameter (i.e., the von Mises–Fisher distribution) which is a conjugate prior, resulting in a fast estimation process. Furthermore, we present the Efficient ProMises model for the high-dimensional framework, useful in neuroimaging, where the problem has much more than three dimensions. We found a great improvement in functional magnetic resonance imaging connectivity analysis because the ProMises model permits incorporation of topological brain information in the alignment’s estimation process.

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Section: Efficient Promises Modelmentioning

confidence: 99%

Procrustes Analysis for High-Dimensional Data

Andreella

Finos

2022

Psychometrika

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“…To allow for whole-brain analysis, we propose the Efficient ProMises model, which allows for a faster functional alignment without loss of information. In practice, the Efficient ProMises model projects matrices X i into a t lower-dimensional space via specific semi-orthogonal transformations Q i ∈ R v×t (Abadir and Magnus, 2005;Groß et al, 1999) which preserve all of the information in the data. It aligns the reduced t × t matrices {X i Q i ∈ IR t×t } i=1,...,m , and back-projects the aligned matrices to the original t × v-size matrices {X i ∈ IR t×v } i=1,...,m using the transpose of these semi-orthogonal transformations…”

Section: Efficient Promises Modelmentioning

confidence: 99%

Enhanced hyperalignment via spatial prior information

Andreella¹,

Finos²,

Lindquist³

2022

Preprint

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Functional alignment between subjects is an important assumption of functional magnetic resonance imaging (fMRI) group-level analysis. However, it is often violated in practice, even after alignment to a standard anatomical template. Hyperalignment, based on sequential Procrustes orthogonal transformations, has been proposed as a method of aligning shared functional information into a common high-dimensional space and thereby improving inter-subject analysis. Though successful, current hyperalignment algorithms have a number of shortcomings, including difficulties interpreting the transformations, a lack of uniqueness of the procedure, and difficulties performing whole-brain analysis.To resolve these issues, we propose the ProMises (Procrustes von Mises-Fisher) model. We reformulate functional alignment as a statistical model and impose a prior distribution on the orthogonal parameters (the von Mises-Fisher distribution). This allows for the embedding of anatomical information into the estimation procedure by penalizing the contribution of spatially distant voxels when creating the shared functional high-dimensional space. Importantly, the transformations, aligned images, and related results are all unique. In addition, the proposed method allows for efficient whole-brain functional alignment. In simulations and application to data from four fMRI studies we find that ProMises improves inter-subject classification in terms of between-subject accuracy and interpretability compared to standard hyperalignment algorithms.

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Section: Efficient Promises Modelmentioning

confidence: 99%

Enhanced hyperalignment via spatial prior information

Andreella

Finos

Lindquist

2022

Human Brain Mapping

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Functional alignment between subjects is an important assumption of functional magnetic resonance imaging (fMRI) group-level analysis. However, it is often violated in practice, even after alignment to a standard anatomical template. Hyperalignment, based on sequential Procrustes orthogonal transformations, has been proposed as a method of aligning shared functional information into a common high-dimensional space and thereby improving inter-subject analysis. Though successful, current hyperalignment algorithms have a number of shortcomings, including difficulties interpreting the transformations, a lack of uniqueness of the procedure, and difficulties performing whole-brain analysis. To resolve these issues, we propose the ProMises (Procrustes von Mises-Fisher) model. We reformulate functional alignment as a statistical model and impose a prior distribution on the orthogonal parameters (the von Mises-Fisher distribution). This allows for the embedding of anatomical information into the estimation procedure by penalizing the contribution of spatially distant voxels when creating the shared functional high-dimensional space. Importantly, the transformations, aligned images, and related results are all unique. In addition, the proposed method allows for efficient whole-brain functional alignment. In simulations and application to data from four fMRI studies we find that ProMises improves inter-subject classification in terms of between-subject accuracy and interpretability compared to standard hyperalignment algorithms.

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On semi-orthogonality and a special class of matrices

Cited by 13 publications

References 9 publications

Procrustes Analysis for High-Dimensional Data

Procrustes Analysis for High-Dimensional Data

Enhanced hyperalignment via spatial prior information

Enhanced hyperalignment via spatial prior information

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