This paper introduces an adaptation of the sequential matrix diagonalization (SMD) method to high-dimensional functional magnetic resonance imaging (fMRI) data. SMD is currently the most efficient statistical method to perform polynomial eigenvalue decomposition. Unfortunately, with current implementations based on dense polynomial matrices, the algorithmic complexity of SMD is intractable and it cannot be applied as such to high-dimensional problems. However, for certain applications, these polynomial matrices are mostly filled with null values and we have consequently developed an efficient implementation of SMD based on a GPU-parallel representation of sparse polynomial matrices. We then apply our “sparse SMD” to fMRI data, i.e. the temporal evolution of a large grid of voxels (as many as 29,328 voxels). Because of the energy compaction property of SMD, practically all the information is concentrated by SMD on the first polynomial principal component. Brain regions that are activated over time are thus reconstructed with great fidelity through analysis-synthesis based on the first principal component only, itself in the form of a sequence of polynomial parameters.

中文翻译：

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顺序矩阵对角化算法在高维功能性MRI数据中的应用

本文介绍了顺序矩阵对角化（SMD）方法对高维功能磁共振成像（fMRI）数据的适应性。SMD是当前执行多项式特征值分解的最有效的统计方法。不幸的是，在当前基于密集多项式矩阵的实现中，SMD的算法复杂度难以解决，因此无法应用于高维问题。但是，对于某些应用程序，这些多项式矩阵大多填充有空值，因此我们基于稀疏多项式矩阵的GPU并行表示开发了SMD的有效实现。然后，我们将“稀疏SMD”应用于fMRI数据，即大体素网格（多达29,328个体素）的时间演化。由于SMD的能量压缩特性，实际上所有信息都通过SMD集中在第一个多项式主分量上。因此，仅基于第一主成分，其本身就以多项式参数序列的形式，通过分析-合成，可以高度保真地重建随时间激活的大脑区域。