Applications such as reconstructing cell lineage trees (represented as phylogenetic trees) from single-cell sequencing data require reconstructing a [Formula: see text]-matrix that has many errors and missing entries. We introduce Sempervirens, a very fast matrix reconstruction algorithm for noisy and incomplete matrix representations of phylogenetic trees. Sempervirens uses an iterative maximum-likelihood approach to determine the topology tree represented by the corrupted data. We show that Sempervirens is at least three orders of magnitude faster than other methods on thousand by thousand matrices, with the speed gap widening with larger matrices. We also show that Sempervirens matches state-of-the-art methods in reconstruction accuracy. The speed of Sempervirens enables it to be tractably applied to reconstructing much larger matrices than those that other methods can reconstruct. In addition to experimental results, we justify the algorithm with a mathematical treatment of its subprocedures. History: Accepted by J. Paul Brooks, Area Editor for Applications in Biology, Medicine, & Healthcare. Funding: This work was supported by the Office of Science of the Department of Energy [Contract DE-AC02-05CH11231]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0373 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0373 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

We introduce a matrix decomposition method and prove that multiplication in GF$(2^k)$with a Type 1 optimal normal basis for can be performed using$k^2-1$XOR gates irrespective of the choice of the irreducible polynomial generating the field. The previous results achieved this bound only with special irreducible polynomials. Furthermore, the decomposition method performs the multiplication operation using$1.5k(k-1)$XOR gates for Type 2a and 2b optimal normal bases, which matches previous bounds.

A novel blind equalization method, based on a subgradient search over a convex cost surface, is proposed. This is an alternative to the existing iterative blind equalization approaches such as the constant modulus algorithm (CMA) which mostly suffer from the convergence problems caused by their non-convex cost functions. The proposed method is an iterative algorithm, for both real and complex constellations, with a very simple update rule that minimizes the l/sub /spl infin// norm of the equalizer output under a linear constraint on the equalizer coefficients. The algorithm has a nice convergence behavior, attributed to the convex l/sub /spl infin// cost surface. Examples are provided to illustrate the algorithm's performance.