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Part of A Low Rank and Sparse Paradigm Free Mapping Algorithm for Deconvolution of fMRI Data

$\hat{\mathbf{L}}, \hat{\mathbf{S}} = \argmin_{\mathbf{L, S}} | \mathbf{Y} - \mathbf{HS} \mathbf{L} |_F^2 + \lambda_L |\mathbf{L}|_* + (1-\rho) | \mathbf{SD}_S |_{2,1} + \rho |\mathbf{SD}_S |_1$

$| \cdot |_F^2$ denotes the Frobenius norm.

The $\ell_{2,1} + \ell_1$ norm promotes temporal sparsity and spatial structure to the estimates of the activity-inducing signal.

$\rho = 0.8$ (empirically set) balances the trade-off between temporal sparsity and spatial structure.

$\mathbf{D}_S = diag(\lambda_{S_1}, ..., \lambda_{S_V})$ is a diagonal matrix with voxel-specific regularization parameters to promote the sparsity of $\mathbf{S}$ .

For each voxel, we set $\lambda_S$ to the median absolute deviation estimate of the noise standard deviation from the fine-scale wavelet coefficients of the voxel time series (Daubechies, order 3).

The nuclear norm $| \cdot |_*$ promotes the detection of low-rank, global components.

Referenced in

A Low Rank and Sparse Paradigm Free Mapping Algorithm for Deconvolution of fMRI Data