Abstract. We propose a superfast solver for Toeplitz linear systems based on rank structured matrix methods and randomized sampling. The solver uses displacement equations to transform a Toeplitz matrix T into a Cauchy-like matrix C, which is known to have low-numerical-rank offdiagonal blocks. Thus, we design a fast scheme for constructing a hierarchically semiseparable (HSS) matrix approximation to C, where the HSS generators have internal structures. Unlike classical HSS methods, our solver employs randomized sampling techniques together with fast Toeplitz matrixvector multiplications, and thus converts the direct compression of the off-diagonal blocks of C into the compression of much smaller blocks. A strong rank-revealing QR factorization method is used to generate/preserve certain special structures, and also to ensure stability. A fast ULV HSS factorization scheme is provided to take advantage of the special structures. We also propose a precomputation procedure for the HSS construction so as to further improve the efficiency. The complexity of these methods is significantly lower than some similar Toeplitz solvers for large matrix size n. Detailed flop counts are given, with the aid of a rank relaxation technique. The total cost of our methods includes O(n) flops for HSS operations and O(n log 2 n) flops for matrix multiplications via FFTs, where n is the order of T . Various numerical tests on classical examples, including ill-conditioned ones, demonstrate the efficiency, and also indicate that the methods are stable in practice. This work shows a practical way of using randomized sampling in the development of fast rank structured methods.

Esophageal squamous-cell carcinoma (ESCC), one of the most prevalent and lethal malignant disease, has a complex but unknown tumor ecosystem. Here, we investigate the composition of ESCC tumors based on 208,659 single-cell transcriptomes derived from 60 individuals. We identify 8 common expression programs from malignant epithelial cells and discover 42 cell types, including 26 immune cell and 16 nonimmune stromal cell subtypes in the tumor microenvironment (TME), and analyse the interactions between cancer cells and other cells and the interactions among different cell types in the TME. Moreover, we link the cancer cell transcriptomes to the somatic mutations and identify several markers significantly associated with patients’ survival, which may be relevant to precision care of ESCC patients. These results reveal the immunosuppressive status in the ESCC TME and further our understanding of ESCC.

Esophageal squamous cell carcinoma (ESCC) is prevalent in some geographical regions of the world. ESCC development presents a multistep pathogenic process from inflammation to invasive cancer; however, what is critical in these processes and how they evolve is largely unknown, obstructing early diagnosis and effective treatment. Here, we create a mouse model mimicking human ESCC development and construct a single-cell ESCC developmental atlas. We identify a set of key transitional signatures associated with oncogenic evolution of epithelial cells and depict the landmark dynamic tumorigenic trajectories. An early downregulation of CD8 + response against the initial tissue damage accompanied by the transition of immune response from type 1 to type 3 results in accumulation and activation of macrophages and neutrophils, which may create a chronic inflammatory environment that promotes carcinogen-transformed epithelial cell survival and proliferation. These findings shed light on how ESCC is initiated and developed.

We present some superfast (O((m + n) log 2 (m + n)) complexity) and stable structured direct solvers for m × n Toeplitz least squares problems. Based on the displacement equation, a Toeplitz matrix T is first transformed into a Cauchy-like matrix C, which can be shown to have small off-diagonal numerical ranks when the diagonal blocks are rectangular. We generalize standard hierarchically semiseparable (HSS) matrix representations to rectangular ones, and construct a rectangular HSS approximation to C in nearly linear complexity with randomized sampling and fast multiplications of C with vectors. A new URV HSS factorization and a URV HSS solution are designed for the least squares solution. We also present two structured normal equation methods. Systematic error and stability analysis for our HSS methods is given, which is also useful for studying other HSS and rank structured methods. We derive the growth factors and the backward error bounds in the HSS factorizations, and show that the stability results are generally much better than those in dense LU factorizations with partial pivoting. Such analysis has not been done before for HSS matrices. The solvers are tested on various classical Toeplitz examples ranging from well-conditioned to highly ill-conditioned ones. Comparisons with some recent fast and superfast solvers are given. Our new methods are generally much faster, and give better (or at least comparable) accuracies, especially for ill-conditioned problems.

Abstract. Polynomial filtering can provide a highly effective means of computing all eigenvalues of a real symmetric (or complex Hermitian) matrix that are located in a given interval, anywhere in the spectrum. This paper describes a technique for tackling this problem by combining a ThickRestart version of the Lanczos algorithm with deflation ('locking') and a new type of polynomial filters obtained from a least-squares technique. The resulting algorithm can be utilized in a 'spectrumslicing' approach whereby a very large number of eigenvalues and associated eigenvectors of the matrix are computed by extracting eigenpairs located in different sub-intervals independently from one another.