In the last decade, there has been a continued effort to produce families of strong linearizations of a matrix polynomial P (λ), regular and singular, with good properties, such as, being companion forms, allowing the recovery of eigenvectors of a regular P (λ) in an easy way, allowing the computation of the minimal indices of a singular P (λ) in an easy way, etc. As a consequence of this research, families such as the family of Fiedler pencils, the family of generalized Fiedler pencils (GFP), the family of Fiedler pencils with repetition, and the family of generalized Fiedler pencils with repetition (GFPR) were constructed. In particular, one of the goals was to find in these families structured linearizations of structured matrix polynomials. For example, if a matrix polynomial P (λ) is symmetric (Hermitian), it is convenient to use linearizations of P (λ) that are also symmetric (Hermitian). Both the family of GFP and the family of GFPR contain block-symmetric linearizations of P (λ), which are symmetric (Hermitian) when P (λ) is. Now the objective is to determine which of those structured linearizations have the best numerical properties. The main obstacle for this study is the fact that these pencils are defined implicitly as products of so-called elementary matrices. Recent papers in the literature had as a goal to provide an explicit block-structure for the pencils belonging to the family of Fiedler pencils and any of its further generalizations to solve this problem. In particular, it was shown that all GFP and GFPR, after permuting some block-rows and block-columns, belong to the family of extended block Kronecker pencils, which are defined explicitly in terms of their block-structure. Unfortunately, those permutations that transform a GFP or a GFPR into an extended block Kronecker pencil do not preserve the block-symmetric structure. Thus, in this paper we consider the family of block-minimal bases pencils, which is closely related to the family of extended block Kronecker pencils, and whose pencils are also defined in terms of their block-structure, as a source of canonical forms for block-symmetric pencils. More precisely, we present four families of block-symmetric pencils which, under some generic nonsingularity conditions are block minimal bases pencils and strong linearizations of a matrix polynomial. We show that the block-symmetric GFP and GFPR, after some row and column permutations, belong to the union of these four families. Furthermore, we show that, when P (λ) is a complex matrix polynomial, any block-symmetric GFP and GFPR is permutationally congruent to a pencil in some of these four families. Hence, these four families of pencils provide an alternative but explicit approach to the block-symmetric Fiedler-like pencils existing in the literature.

The CP tensor decomposition is used in applications such as machine learning and signal processing to discover latent low‐rank structure in multidimensional data. Computing a CP decomposition via an alternating least squares (ALS) method reduces the problem to several linear least squares problems. The standard way to solve these linear least squares subproblems is to use the normal equations, which inherit special tensor structure that can be exploited for computational efficiency. However, the normal equations are sensitive to numerical ill‐conditioning, which can compromise the results of the decomposition. In this paper, we develop versions of the CP‐ALS algorithm using the QR decomposition and the singular value decomposition, which are more numerically stable than the normal equations, to solve the linear least squares problems. Our algorithms utilize the tensor structure of the CP‐ALS subproblems efficiently, have the same complexity as the standard CP‐ALS algorithm when the input is dense and the rank is small, and are shown via examples to produce more stable results when ill‐conditioning is present. Our MATLAB implementation achieves the same running time as the standard algorithm for small ranks, and we show that the new methods can obtain lower approximation error.

The CP tensor decomposition is used in applications such as machine learning and signal processing to discover latent low-rank structure in multidimensional data. Computing a CP decomposition via an alternating least squares (ALS) method reduces the problem to several linear least squares problems. The standard way to solve these linear least squares subproblems is to use the normal equations, which inherit special tensor structure that can be exploited for computational efficiency. However, the normal equations are sensitive to numerical ill-conditioning, which can compromise the results of the decomposition. In this paper, we develop versions of the CP-ALS algorithm using the QR decomposition and the singular value decomposition (SVD), which are more numerically stable than the normal equations, to solve the linear least squares problems. Our algorithms utilize the tensor structure of the CP-ALS subproblems efficiently, have the same complexity as the standard CP-ALS algorithm when the rank is small, and are shown via examples to produce more stable results when ill-conditioning is present. Our MATLAB implementation achieves the same running time as the standard algorithm for small ranks, and we show that the new methods can obtain lower approximation error and more reliably recover low-rank signals from data with known ground truth.

Introduction: Pediatric hypertension (HTN) is a growing concern with short and long-term adverse health effects. While children who were born preterm (<37 weeks’ gestation) likely have an increased HTN risk, it is unknown whether preterm birth is associated with more severe HTN once diagnosed. Objective: Determine whether youth referred for HTN disorders who were born preterm are more likely to have worse blood pressure (BP). Methods: This is a cross-sectional analysis of preliminary baseline data from The Study of the Epidemiology of Pediatric Hypertension (SUPERHERO) Registry, an ongoing multicenter retrospective cohort of youth referred to subspecialty clinics for HTN disorders. Inclusion criteria were <19 years of age, initial visit 1/01/2016-12/31/2021 (index date), and ICD-10 diagnostic codes for HTN disorders. Exclusion criteria were pregnancy, kidney failure on dialysis, or kidney transplantation by ICD-10 codes. We classified BP based on age, sex, and height per pediatric guidelines. We further defined high BP as elevated BP or any stage of HTN. Preterm birth was based on ICD-10 codes at the index date. We used unadjusted generalized linear models to estimate RR with 95% CL. Results: In the cohort, 939/3295 (29%) identified as Black/African American, 576/3295 (17%) Hispanic/Latino, 1216/3295 (37%) were female, and the median age was 14.2 years (IQR 10.5, 16.4);1703/3295 (52%) had obesity. Only 24/3295 (1%) had an ICD-10 code for preterm birth, and 1951/3228 (60%) had stage 1 or stage 2 HTN. Preterm birth ICD-10 codes were not associated with a higher risk of high BP (RR 0.78, 95% CL 0.57 to 1.06) or a higher risk of HTN (RR 0.84, 95% CL 0.56 to 1.27). Conclusion: Youth referred for HTN disorders who had ICD-10 codes for preterm birth did not have worse BP compared to those without these codes. It is possible that preterm birth is not accurately documented. Ongoing analyses include obtaining actual gestational age at birth and investigating the association with target organ damage.

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