Local and Global Structure Preservation for Robust Unsupervised Spectral Feature Selection

A Robust Reduced Rank Graph Regression Method for Neuroimaging Genetic Analysis

“…More specifically, if $\sqrt{{false\Vert \mathrm{boldY}-{\mathbf{XUV}}^{\mathrm{T}}false\Vert }_F^2}$ is small (i.e., when XUV T is a good estimation of Y ), then ω 1 is assigned a large weight, and vice versa. On the other hand, it can be shown that the following equations hold (Zhu et al 2017a, 2018),…”

“…(15) converges. This implies that the following holds: $$\begin{gathered} \sqrt{{false\Vert \mathrm{boldY}-\mathrm{boldX}{\mathbf{U}}^{false(t+1false)}{\mathbf{V}}^{{\mathrm{normalT}}^{(t+1)}}false\Vert }_{2,1}}+\sqrt{{false\Vert \mathrm{boldX}-\mathrm{boldX}{\mathbf{S}}^{false(tfalse)}false\Vert }_{2,1}}+{\lambda }_2{\Vert {\mathrm{boldS}}^{(t)}\Vert }_1+{\lambda }_1{\Vert \left[{\mathrm{boldU}}^{(t+1)},{\mathrm{boldS}}^{(t)}\right]\Vert }_{2,1} \\ \le \sqrt{{false\Vert \mathrm{boldY}-\mathrm{boldX}{\mathbf{U}}^{false(t+1false)}{\mathbf{V}}^{{\mathrm{normalT}}^{(t)}}false\Vert }_{2,1}}+\sqrt{{false\Vert \mathrm{boldX}-\mathrm{boldX}{\mathbf{S}}^{false(tfalse)}false\Vert }_{2,1}}+{\lambda }_2{\Vert {\mathrm{boldS}}^{(t)}\Vert }_1+{\lambda }_1{\Vert \left[{\mathrm{boldU}}^{(t+1)},{\mathrm{boldS}}^{(t)}\right]\Vert }_{2,1}. \end{gathered}$$ According to conclusions in the literature (Hu et al 2017, Zhu et al 2017a, 2018), the framework of IRLS converges, so we have the following: $$\sqrt{{false\Vert \mathrm{boldY}-\mathrm{boldX}{\mathbf{U}}^{false(t+1false)}{\mathbf{V}}^{{\mathrm{normalT}}^{(t+1)}}false\Vert }_{2,1}}+\sqrt{{false\Vert \mathrm{boldX}-\mathrm{boldX}{\mathbf{S}}^{false(t+1false)}false\Vert }_{2,1}}+{\lambda }_2$$…”

“…According to conclusions in the literature (Hu et al 2017, Zhu et al 2017a, 2018), the framework of IRLS converges, so we have the following: $$\begin{gathered} \sqrt{{false\Vert \mathrm{boldY}-\mathrm{boldX}{\mathbf{U}}^{false(t+1false)}{\mathbf{V}}^{{\mathrm{normalT}}^{(t+1)}}false\Vert }_{2,1}}+\sqrt{{false\Vert \mathrm{boldX}-\mathrm{boldX}{\mathbf{S}}^{false(t+1false)}false\Vert }_{2,1}}+{\lambda }_2{\Vert {\mathrm{boldS}}^{(t+1)}\Vert }_1+{\lambda }_1{\Vert \left[{\mathrm{boldU}}^{(t+1)},{\mathrm{boldS}}^{(t+1)}\right]\Vert }_{2,1} \\ \le \sqrt{{false\Vert \mathrm{boldY}-\mathrm{boldX}{\mathbf{U}}^{false(t+1false)}{\mathbf{V}}^{{\mathrm{normalT}}^{(t+1)}}false\Vert }_{2,1}}+\sqrt{{false\Vert \mathrm{boldX}-\mathrm{boldX}{\mathbf{S}}^{false(tfalse)}false\Vert }_{2,1}}+{\lambda }_2{\Vert {\mathrm{boldS}}^{(t)}\Vert }_1+{\lambda }_1{\Vert \left[{\mathrm{boldU}}^{(t+1)},{\mathrm{boldS}}^{(t)}\right]\Vert }_{2,1}. \end{gathered}$$…”

“…However, high dimensionality of the neuroimaging and genetic data has been a major challenge, causing computational and statistical problems (Thompson et al 2013; Medland et al 2014). Firstly, due to resource limitations, neuroimaging genetic studies often have limited samples and the number of samples is much less than the dimensions of both neuroimaging and genetic data (Zhang et al 2011, 2012), leading to the so called “curse of dimensionality” problem (a, Peng and Fan 2017b; Zheng et al 2017; Zhu et al 2017a, 2018). Secondly, both noise and redundancy of the neuroimaging and genetic data could reduce the rank of the data, which may make it difficult to obtain robust machine learning models from such data (Vounou et al 2012; Greenlaw et al 2017; Lu et al 2017).…”

“…Anatomical atlases that define ROIs, such as Automated Anatomical Labeling (AAL) template (Tzourio-Mazoyer et al 2002), are widely adopted for computing regional imaging measures and for defining network nodes to compute brain connectivity measures in brain network analysis (Fan et al 2011). More recently, sparse models have been proposed to simultaneously deal with the issues of data noise and “curse of dimensionality” (Vounou et al 2012; Lu et al 2017; Zhu et al 2018). For example, Lu et al developed a Bayesian longitudinal low-rank regression model to jointly analyze correlations between high-dimensional longitudinal response variables and neuroimaging features (Lu et al 2017).…”