arXiv subjects

Zane Smith

Publications and source records attributed to Zane Smith.

5 recordsLinked to original sources

arXiv2025

pHapCompass: Probabilistic Assembly and Uncertainty Quantification of Polyploid Haplotype Phase

Computing haplotypes from sequencing data, i.e. haplotype assembly, is an important component of molecular and population genetics problems, including interpreting the effects of genetic variation on complex traits and reconstructing genealogical relationships. Assembling the haplotypes of polyploid genomes remains a significant challenge due to the exponential search space of haplotype phasings and read assignment ambiguity; the latter challenge is particularly difficult for haplotype assemblers since the information contained within the observed sequence reads is often insufficient for unambiguous haplotype assignment in polyploid genomes. We present pHapCompass, probabilistic haplotype assembly algorithms for diploid and polyploid genomes that explicitly model and propagate read assignment ambiguity to compute a distribution over polyploid haplotype phasings. We develop graph theoretic algorithms to enable statistical inference and uncertainty quantification despite an exponential space of possible phasings. Since prior work evaluates polyploid haplotype assembly on synthetic genomes that do not reflect the realistic genomic complexity of polyploidy organisms, we develop a computational workflow for simulating genomes and DNA-seq for auto- and allopolyploids. Additionally, we generalize the vector error rate and minimum error correction evaluation criteria for partially phased haplotypes. Benchmarking of pHapCompass and several existing polyploid haplotype assemblers shows that pHapCompass yields competitive performance across varying genomic complexities and polyploid structures while retaining an accurate quantification of phase uncertainty. The source code for pHapCompass, simulation scripts, and datasets are freely available at https://github.com/bayesomicslab/pHapCompass.

q-bio.GN↗

arXiv2021

The Gromov-Hausdorff distance between ultrametric spaces: its structure and computation

The Gromov-Hausdorff distance ($d_\mathrm{GH}$) provides a natural way of quantifying the dissimilarity between two given metric spaces. It is known that computing $d_\mathrm{GH}$ between two finite metric spaces is NP-hard, even in the case of finite ultrametric spaces which are highly structured metric spaces in the sense that they satisfy the so-called \emph{strong triangle inequality}. Ultrametric spaces naturally arise in many applications such as hierarchical clustering, phylogenetics, genomics, and even linguistics. By exploiting the special structures of ultrametric spaces, (1) we identify a one parameter family $\{d_\mathrm{GH}^{(p)}\}_{p\in[1,\infty]}$ of distances defined in a flavor similar to the Gromov-Hausdorff distance on the collection of finite ultrametric spaces, and in particular $d_\mathrm{GH}^{(1)} =d_\mathrm{GH}$. The extreme case when $p=\infty$, which we also denote by $u_\mathrm{GH}$, turns out to be an ultrametric on the collection of ultrametric spaces. Whereas for all $p\in[1,\infty)$, $d_\mathrm{GH}^{(p)}$ yields NP-hard problems, we prove that surprisingly $u_\mathrm{GH}$ can be computed in polynomial time. The proof is based on a structural theorem for $u_\mathrm{GH}$ established in this paper; (2) inspired by the structural theorem for $u_\mathrm{GH}$, and by carefully leveraging properties of ultrametric spaces, we also establish a structural theorem for $d_\mathrm{GH}$ when restricted to ultrametric spaces. This structural theorem allows us to identify special families of ultrametric spaces on which $d_\mathrm{GH}$ is computationally tractable. These families are determined by properties related to the doubling constant of metric space. Based on these families, we devise a fixed-parameter tractable (FPT) algorithm for computing the exact value of $d_\mathrm{GH}$ between ultrametric spaces. We believe ours is the first such algorithm to be identified.

math.MG↗

arXiv2021

The Gromov-Hausdorff distance between spheres

We provide general upper and lower bounds for the Gromov-Hausdorff distance $d_{\mathrm{GH}}(\mathbb{S}^m,\mathbb{S}^n)$ between spheres $\mathbb{S}^m$ and $\mathbb{S}^n$ (endowed with the round metric) for $0\leq m< n\leq \infty$. Some of these lower bounds are based on certain topological ideas related to the Borsuk-Ulam theorem. Via explicit constructions of (optimal) correspondences we prove that our lower bounds are tight in the cases of $d_{\mathrm{GH}}(\mathbb{S}^0,\mathbb{S}^n)$, $d_{\mathrm{GH}}(\mathbb{S}^m,\mathbb{S}^\infty)$, $d_{\mathrm{GH}}(\mathbb{S}^1,\mathbb{S}^2)$, $d_{\mathrm{GH}}(\mathbb{S}^1,\mathbb{S}^3)$ and $d_{\mathrm{GH}}(\mathbb{S}^2,\mathbb{S}^3)$. We also formulate a number of open questions.

math.MG↗

arXiv2018

The Wasserstein transform

We introduce the Wasserstein Transform (WT), a general unsupervised framework for updating distance structures on given data sets with the purpose of enhancing features and denoising. Our framework represents each data point by a probability measure reflecting the neighborhood structure of the point, and then updates the distance by computing the Wasserstein distance between these probability measures. The Wasserstein Transform is a general method which extends the mean shift family of algorithms. We study several instances of WT, and in particular, in one of the instances which we call the Gaussian Transform (GT), we utilize Gaussian measures to model neighborhood structures of individual data points. GT is computationally cheaper than other instances of WT since there exists closed form solution for the $\ell^2$-Wasserstein distance between Gaussian measures. We study the relationship between different instances of WT and prove that each of the instances is stable under perturbations. We devise iterative algorithms for performing the above-mentioned WT and propose several strategies to accelerate GT, such as an observation from linear algebra for reducing the number of matrix square root computations. We examine the performance of the Wasserstein Transform method in many tasks, such as denoising, clustering, image segmentation and word embeddings.

cs.LG↗

arXiv2015

Symmetric Chromatic Polynomial of Trees

In a 1995 paper Richard Stanley defined $X_G$, the symmetric chromatic polynomial of a Graph $G=(V,E)$. He then conjectured that $X_G$ distinguishes trees; a conjecture which still remains open. $X_G$ can be represented as a certain collection of integer partitions of $|V|$ induced by each $S\subseteq E$, which is very approachable with the aid of a computer. Our research involved writing a computer program for efficient verification of this conjecture for trees up to 23 vertices. In this process, we also gather trees with matching collections of integer partitions of a fixed number of parts. For each $k=2, 3, 4, 5$, we provide the smallest pair of trees whose partitions of $k$ parts agree. In 2013, Orellana and Scott give a proof of a weaker version of Stanely's conjecture for trees with one centroid. We prove a similar result for arbitrary trees, and provide examples to show that this result, combined with that of Orellana and Scott, is optimal.

math.CO↗