Hexuan Liu

Anne Greenbaum, Hexuan Liu, Tyler Chen · uw

We consider three mathematically equivalent variants of the conjugate gradient (CG) algorithm and how they perform in finite precision arithmetic. It was shown in [{\em Behavior of slightly perturbed Lanczos and conjugate-gradient recurrences}, Lin.~Alg.~Appl., 113 (1989), pp.~7-63] that under certain conditions the convergence of a slightly perturbed CG computation is like that of exact CG for a matrix with many eigenvalues distributed throughout tiny intervals about the eigenvalues of the given matrix, the size of the intervals being determined by how closely these conditions are satisfied. We determine to what extent each of these variants satisfies the desired conditions, using a set of test problems and show that there is significant correlation between how well these conditions are satisfied and how well the finite precision computation converges before reaching its ultimately attainable accuracy. We show that for problems where the width of the intervals containing the eigenvalues of the associated exact CG matrix makes a significant difference in the behavior of exact CG, the different CG variants behave differently in finite precision arithmetic. For problems where the interval width makes little difference or where the convergence of exact CG is essentially governed by the upper bound based on the square root of the condition number of the matrix, the different CG variants converge similarly in finite precision arithmetic until the ultimate level of accuracy is achieved, although this ultimate level of accuracy may be different for the different variants. This points to the need for testing new CG variants on problems that are especially sensitive to rounding errors.

Hexuan Liu, Michael Wallner, Guan-Ru Yu

Tree-child networks are an important class of phylogenetic network used to model reticulate evolutionary processes. These networks have attracted increasing attention from researchers with interests in both combinatorics and algorithms. A fundamental open problem posed by Pons and Batle asks whether the number $TC_{n,k}$ of bicombining tree-child networks with $n$ leaves and $k$ reticulation nodes equals the number of certain constrained words, now called Pons-Batle words. In this paper, we confirm the conjecture for tree-child networks with a bounded number of reticulation nodes. Our approach is combinatorial and analytic. We introduce families of Young tableaux with walls and holes and construct explicit bijections with Pons-Batle words, yielding a direct combinatorial explanation of the identities. These tableaux encode structural features of the underlying networks, including the placement of reticulation nodes. By projecting them to decorated Dyck paths, we obtain algebraic generating functions with differential operators encoding step weights, leading to explicit recurrence relations and closed-form formulas for $TC_{n,k}$. Beyond finite verification for moderate $k$, the framework reveals an underlying probabilistic structure. For $k=1$, natural structural parameters, such as the position and value of distinguished cells, converge, after rescaling, to $\mathrm{Beta}(2,1)$, $\mathrm{Beta}(1,2)$, and Uniform (i.e., $\mathrm{Beta}(1,1)$) distributions. These limit laws arise from a coalescence of singularities at the dominant square-root singularity, producing a non-analytic transition in the local expansion. Overall, our results provide both combinatorial insight and a unified analytic perspective on the asymptotic behavior of tree-child networks, showing how algebraic generating functions with interacting singularities systematically produce Beta limit laws.

Hexuan Liu, Aleksandr Aravkin

A wide range of problems in computational science and engineering require estimation of sparse eigenvectors for high dimensional systems. Here, we propose two variants of the Truncated Orthogonal Iteration to compute multiple leading eigenvectors with sparsity constraints simultaneously. We establish numerical convergence results for the proposed algorithms using a perturbation framework, and extend our analysis to other existing alternatives for sparse eigenvector estimation. We then apply our algorithms to solve the sparse principle component analysis problem for a wide range of test datasets, from simple simulations to real-world datasets including MNIST, sea surface temperature and 20 newsgroups. In all these cases, we show that the new methods get state of the art results quickly and with minimal parameter tuning.