In this work, we propose a new invariant for 2D persistence modules called the compressed multiplicity and show that it generalizes the notions of the dimension vector and the rank invariant. In addition, we propose an "interval-decomposable approximation" δ * (M ) of a 2D persistence module M . In the case that M is interval-decomposable, we show that δ * (M ) = M . Furthermore, even for representations M not necessarily interval-decomposable, δ * (M ) preserves the dimension vector and the rank invariant of M .

In persistent homology [13] of filtrations, the indecomposable decompositions provide the persistence diagrams. In multidimensional persistence [9] it is known to be impossible to classify all indecomposable modules. One direction is to consider the subclass of intervaldecomposable persistence modules, which are direct sums of interval representations. We introduce the definition of pre-interval representations, a more algebraic definition, and study the relationships between pre-interval, interval, and indecomposable thin representations. We show that over the "equioriented" commutative 2D grid, these concepts are equivalent. Moreover, we provide an algorithm for determining whether or not an nD persistence module is interval/preinterval/thin-decomposable, under certain finiteness conditions and without explicitly computing decompositions.

Single-cell RNA sequencing (scRNA-seq) can determine gene expression in numerous individual cells simultaneously, promoting progress in the biomedical sciences. However, scRNA-seq data are high-dimensional with substantial technical noise, including dropouts. During analysis of scRNA-seq data, such noise engenders a statistical problem known as the curse of dimensionality (COD). Based on high-dimensional statistics, we herein formulate a noise reduction method, RECODE (resolution of the curse of dimensionality), for high-dimensional data with random sampling noise. We show that RECODE consistently resolves COD in relevant scRNA-seq data with unique molecular identifiers. RECODE does not involve dimension reduction and recovers expression values for all genes, including lowly expressed genes, realizing precise delineation of cell fate transitions and identification of rare cells with all gene information. Compared with representative imputation methods, RECODE employs different principles and exhibits superior overall performance in cell-clustering, expression value recovery, and single-cell–level analysis. The RECODE algorithm is parameter-free, data-driven, deterministic, and high-speed, and its applicability can be predicted based on the variance normalization performance. We propose RECODE as a powerful strategy for preprocessing noisy high-dimensional data.

This paper introduces a concept of dimension of a triangulated category with respect to a fixed full subcategory. For the bounded derived category of an abelian category, upper bounds of the dimension with respect to a contravariantly finite subcategory and a resolving subcategory are given. Our methods not only recover some known results on the dimensions of derived categories in the sense of Rouquier, but also apply to various commutative and non-commutative noetherian rings.

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