Publisher Correction: scParser: sparse representation learning for scalable single-cell RNA sequencing data analysis

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.