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In this work, we generalize the notion of character for 2-representations of finite 2-groups. The properties of 2-characters bear strong similarities to those classical characters of finite groups, including conjugation invariance, additivity, multiplicativity and orthogonality. With a careful analysis using homotopy fixed points and quotients for categories with 2-group actions, we prove that the category of class functors on a 2-group $\mathcal G$ is equivalent to the Drinfeld center of the 2-group algebra $\mathrm{Vec}_{\mathcal G}$, which categorifies the Fourier transform on finite abelian groups. After transferring the canonical nondegenerate braided monoidal structure from $\mathfrak Z_1(\mathrm{Vec}_{\mathcal G})$, we discover that irreducible 2-characters of $\mathcal G$ coincide with full centers of the corresponding 2-representations, which are in a one-to-one correspondence with Lagrangian algebras in the category of class functors on $\mathcal G$. In particular, the fusion rule of $2\mathrm{Rep}(\mathcal G)$ can be calculated from the pointwise product of Lagrangian algebras as class functors. From a topological quantum field theory (TQFT) point of view, the commutative Frobenius algebra structure on a 2-character is induced from a 2D topological sigma-model with target space $\lvert \mathrm{B} \mathcal G \rvert$., Comment: 30 pages. Comments are welcome

Let $\mathcal{G}$ be a finite 2-group. We show that the 2-category $2\mathrm{Rep}(\mathcal{G})$ of finite semisimple 2-representations is a symmetric fusion 2-category. We also relate the auto-equivalence 2-group of the symmetric monoidal forgetful 2-functor $\omega : 2\mathrm{Rep}(\mathcal{G}) \to 2\mathrm{Vec}$ to the auto-equivalence 2-group of the regular algebra and show that they are equivalent to $\mathcal{G}$. This result categorifies the usual Tannaka-Krein duality for finite groups., Comment: 15 pages. Comments are welcome

In our Brief Communication (DOI: 10.1038/s41592-018-0033-z), we presented the method SAVER for recovering true gene expression levels in noisy single cell RNA sequencing data. We evaluated the performance of SAVER, along with comparable methods MAGIC and scImpute, in an RNA FISH validation experiment and a data downsampling experiment. In a Comment [arXiv:1908.07084v1], Li & Li were concerned with the use of the downsampled datasets, specifically focusing on clustering results obtained from the Zeisel et al. data. Here, we will address these comments and, furthermore, amend the data downsampling experiment to demonstrate that the findings from the data downsampling experiment in our Brief Communication are valid.