Genome Biology ( IF 10.1 ) Pub Date : 2024-09-04 , DOI: 10.1186/s13059-024-03378-5 Kai Zhao 1 , Hon-Cheong So 2, 3, 4, 5, 6, 7 , Zhixiang Lin 1
Publisher Correction: Genome Biol 25, 223 (2024)
https://doi.org/10.1186/s13059-024-03345-0
Following publication of the original article [1], the authors identified a typesetting error in Eq. 3, 4 and 10, as well as in Algorithm 1 equation. An erroneous “ll” was typeset at the start of the equations.
The incorrect and corrected versions are published in this correction article.
Incorrect equation (3)
$$\left\{ \begin{array}{ll} ll\mathcal{L}(d, p, v, s, g) = & \frac{1}{2} \sum\nolimits_{i,m} \left( z_{i,m} - d_{j}^{\mathsf{T}} v_{m} - p_{t}^{\mathsf{T}} v_{m} - s_{i}^{\mathsf{T}} g_{m} \right)^{2} +\\ & \frac{1}{2} \lambda_{1} \left( \sum\nolimits_{j} \| d_{j} \|^{2}_{2} + \sum\nolimits_{t} \| d_{t} \|^{2}_{2} + \sum\nolimits_{m} \| v_{m} \|^{2}_{2}\right) +\\ & \lambda_{2} \left( \frac{1}{2} (1-\alpha) \sum\nolimits_{i} \|s_{i}\|_{2}^{2} + \alpha \sum\nolimits_{i}|s_{i}|_{1} \right),\\ \text{subject to} & \sum\nolimits_{m} g_{mk}^{2} \leq c, \forall k = 1, \ldots, K_{2}. \end{array}\right.$$(3)Correct equation (3)
$$\left\{ \begin{array}{ll}\mathcal{L}(d, p, v, s, g) = & \frac{1}{2} \sum\nolimits_{i,m} \left( z_{i,m} - d_{j}^{\mathsf{T}} v_{m} - p_{t}^{\mathsf{T}} v_{m} - s_{i}^{\mathsf{T}} g_{m} \right)^{2} +\\ & \frac{1}{2} \lambda_{1} \left( \sum\nolimits_{j} \| d_{j} \|^{2}_{2} + \sum\nolimits_{t} \| d_{t} \|^{2}_{2} + \sum\nolimits_{m} \| v_{m} \|^{2}_{2}\right) +\\ & \lambda_{2} \left( \frac{1}{2} (1-\alpha) \sum\nolimits_{i} \|s_{i}\|_{2}^{2} + \alpha \sum\nolimits_{i}|s_{i}|_{1} \right),\\ \text{subject to} & \sum\nolimits_{m} g_{mk}^{2} \leq c, \forall k = 1, \ldots, K_{2}. \end{array}\right.$$(3)Incorrect equation (4)
$$\left\{ \begin{array}{ll} ll \mathcal{L}(D, P, V, S, G) = & \frac{1}{2} \left\| Z - \left(X^{D} D + X^{P}P\right) V - SG\right\|_{\text{F}}^{2}+\\ & \frac{1}{2} \lambda_{1} \left( \|D\|^{2}_{\text{F}} + \|P\|^{2}_{\text{F}} + \|V\|^{2}_{\text{F}}\right) + \\ & \lambda_{2} \left[ \frac{1}{2} (1 - \alpha) \| S \|^{2}_{\text{F}} + \alpha\|S\|_{1}\right] \\ \text{subject to} & \left\| G_{2} \right\|_{2}^{2} \leq c, \forall k = 1, \ldots, K_{2}, \end{array}\right.$$(4)Correct equation (4)
$$\left\{ \begin{array}{ll}\mathcal{L}(D, P, V, S, G) = & \frac{1}{2} \left\| Z - \left(X^{D} D + X^{P}P\right) V - SG\right\|_{\text{F}}^{2}+\\ & \frac{1}{2} \lambda_{1} \left( \|D\|^{2}_{\text{F}} + \|P\|^{2}_{\text{F}} + \|V\|^{2}_{\text{F}}\right) + \\ & \lambda_{2} \left[ \frac{1}{2} (1 - \alpha) \| S \|^{2}_{\text{F}} + \alpha\|S\|_{1}\right] \\ \text{subject to} & \left\| G_{2} \right\|_{2}^{2} \leq c, \forall k = 1, \ldots, K_{2}, \end{array}\right.$$(4)Incorrect equation (10)
$$\left\{ \begin{array}{ll} ll\mathcal{L}(V, G) = & \frac{1}{2k} \sum\nolimits_{j=1}^{k} \left\| Z_{I_{j}} - \left( X_{I_{j}}^{D} D_{I_{j}} + X_{I_{j}}^{P} P_{I_{j}} \right) V - S_{I_{j}} G\right\|^{2}_{F} +\\ & \frac{1}{2} \lambda_{1} \left[ \frac{1}{k} \sum\nolimits_{j=1}^{k} \left(\left\| D_{I_{j}} \right\|^{2}_{\text{F}} + \left\| P_{I_{j}} \right\|^{2}_{F}\right) + \|V\|^{2}_{F}\right] + \\ & \frac{1}{k} \sum\nolimits_{j=1}^{k} \lambda_{2} \left[ \frac{1}{2} (1 - \alpha) \left\| S_{I_{j}} \right\|^{2}_{F} + \alpha \left\| S_{I_{j}} \right\|_{2} \right] , \\ \text{subject to} & \|G_{k}\|^{2}_{2} \leq c, \forall k = 1,\ldots, K_{2}.\end{array}\right.$$(10)Correct equation (10)
$$\left\{ \begin{array}{ll} \mathcal{L}(V, G) = & \frac{1}{2k} \sum\nolimits_{j=1}^{k} \left\| Z_{I_{j}} - \left( X_{I_{j}}^{D} D_{I_{j}} + X_{I_{j}}^{P} P_{I_{j}} \right) V - S_{I_{j}} G\right\|^{2}_{F} +\\ & \frac{1}{2} \lambda_{1} \left[ \frac{1}{k} \sum\nolimits_{j=1}^{k} \left(\left\| D_{I_{j}} \right\|^{2}_{\text{F}} + \left\| P_{I_{j}} \right\|^{2}_{F}\right) + \|V\|^{2}_{F}\right] + \\ & \frac{1}{k} \sum\nolimits_{j=1}^{k} \lambda_{2} \left[ \frac{1}{2} (1 - \alpha) \left\| S_{I_{j}} \right\|^{2}_{F} + \alpha \left\| S_{I_{j}} \right\|_{2} \right] , \\ \text{subject to} & \|G_{k}\|^{2}_{2} \leq c, \forall k = 1,\ldots, K_{2}.\end{array}\right.$$(10)Incorrect Algorithm 1
$$\left\{ \begin{array}{ll} ll A_{k} \leftarrow & A_{k-1} - \left( X_{I_{k}}^{D} D^{\prime}_{k} + X_{I_{k}}^{P} P^{\prime}_{k} \right)^{\mathsf{T}} \left( X_{I_{k}}^{D} D^{\prime}_{k} + X_{I_{k}}^{P} P^{\prime}_{k} \right) \\ B_{k} \leftarrow & B_{k-1} - \tilde{Z}^{\prime^{\mathsf{T}}}_{I_{k}} \left( X_{I_{k}}^{D} D^{\prime}_{k} + X_{I_{k}}^{P} P^{\prime}_{k} \right) \\ E_{k} \leftarrow & E_{k-1} - S^{\prime}_{I_{k}} {}^{\mathsf{T}} S^{\prime}_{I_{k}}\\ F_{k} \leftarrow & F_{k-1} - Z^{\prime}_{I_{k}} {}^{\mathsf{T}} S^{\prime}_{I_{k}}.\end{array}\right.$$
Correct Algorithm 1
$$\left\{ \begin{array}{ll}A_{k} \leftarrow & A_{k-1} - \left( X_{I_{k}}^{D} D^{\prime}_{k} + X_{I_{k}}^{P} P^{\prime}_{k} \right)^{\mathsf{T}} \left( X_{I_{k}}^{D} D^{\prime}_{k} + X_{I_{k}}^{P} P^{\prime}_{k} \right) \\ B_{k} \leftarrow & B_{k-1} - \tilde{Z}^{\prime^{\mathsf{T}}}_{I_{k}} \left( X_{I_{k}}^{D} D^{\prime}_{k} + X_{I_{k}}^{P} P^{\prime}_{k} \right) \\ E_{k} \leftarrow & E_{k-1} - S^{\prime}_{I_{k}} {}^{\mathsf{T}} S^{\prime}_{I_{k}}\\ F_{k} \leftarrow & F_{k-1} - Z^{\prime}_{I_{k}} {}^{\mathsf{T}} S^{\prime}_{I_{k}}.\end{array}\right.$$The original article [1] is corrected.
Zhao K, So HC, Lin Z. scParser: sparse representation learning for scalable single-cell RNA sequencing data analysis. Genome Biol. 2024;25:223. https://doi.org/10.1186/s13059-024-03345-0.
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Authors and Affiliations
Department of Statistics, The Chinese University of Hong Kong, Shatin, Hong Kong SAR, China
Kai Zhao & Zhixiang Lin
School of Biomedical Sciences, The Chinese University of Hong Kong, Shatin, Hong Kong SAR, China
Hon-Cheong So
KIZ-CUHK Joint Laboratory of Bioresources and Molecular Research of Common Diseases, Kunming Institute of Zoology and The Chinese University of Hong Kong, Shatin, Hong Kong SAR, China
Hon-Cheong So
Department of Psychiatry, The Chinese University of Hong Kong, Shatin, Hong Kong SAR, China
Hon-Cheong So
Margaret K.L. Cheung Research Centre for Management of Parkinsonism, The Chinese University of Hong Kong, Shatin, Hong Kong SAR, China
Hon-Cheong So
Brain and Mind Institute, The Chinese University of Hong Kong, Shatin, Hong Kong SAR, China
Hon-Cheong So
Hong Kong Branch of the Chinese Academy of Sciences Center for Excellence in Animal Evolution and Genetics, The Chinese University of Hong Kong, Shatin, Hong Kong SAR, China
Hon-Cheong So
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Correspondence to Hon-Cheong So or Zhixiang Lin.
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Zhao, K., So, HC. & Lin, Z. Publisher Correction: scParser: sparse representation learning for scalable single-cell RNA sequencing data analysis. Genome Biol 25, 238 (2024). https://doi.org/10.1186/s13059-024-03378-5
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