Geometric medians in reconciliation spaces

In evolutionary biology, it is common to study how various entities evolve together, for example, how parasites coevolve with their host, or genes with their species. Coevolution is commonly modelled by considering certain maps or reconciliations from one evolutionary tree $P$ to another $H$, all of which induce the same map $\phi$ between the leaf-sets of $P$ and $H$ (corresponding to present-day associations). Recently, there has been much interest in studying spaces of reconciliations, which arise by defining some metric $d$ on the set $Rec(P,H,\phi)$ of all possible reconciliations between $P$ and $H$. In this paper, we study the following question: How do we compute a geometric median for a given subset $\Psi$ of $Rec(P,H,\phi)$ relative to $d$, i.e. an element $\psi_{med} \in Rec(P,H,\phi)$ such that $$\sum_{\psi' \in \Psi} d(\psi_{med},\psi') \le \sum_{\psi' \in \Psi} d(\psi,\psi')$$ holds for all $\psi \in Rec(P,H,\phi)$? For a model where so-called host-switches or transfers are not allowed, and for a commonly used metric $d$ called the edit-distance, we show that although the cardinality of $Rec(P,H,\phi)$ can be super-exponential, it is still possible to compute a geometric median for a set $\Psi$ in $Rec(P,H,\phi)$ in polynomial time. We expect that this result could be useful for computing a summary or consensus for a set of reconciliations (e.g. for a set of suboptimal reconciliations).Comment: 12 pages, 1 figur

Approximating the Distribution of the Median and other Robust Estimators on Uncertain Data

Robust estimators, like the median of a point set, are important for data analysis in the presence of outliers. We study robust estimators for locationally uncertain points with discrete distributions. That is, each point in a data set has a discrete probability distribution describing its location. The probabilistic nature of uncertain data makes it challenging to compute such estimators, since the true value of the estimator is now described by a distribution rather than a single point. We show how to construct and estimate the distribution of the median of a point set. Building the approximate support of the distribution takes near-linear time, and assigning probability to that support takes quadratic time. We also develop a general approximation technique for distributions of robust estimators with respect to ranges with bounded VC dimension. This includes the geometric median for high dimensions and the Siegel estimator for linear regression.Comment: Full version of a paper to appear at SoCG 201

Investigations on IMP2-2 and Kupffer cells in steatohepatitis

Alcoholic and non-alcoholic steatohepatitis (ASH and NASH) represent risk factors for the development of hepatocellular carcinoma. The prevalence of ASH and NASH is strongly increasing worldwide. Within this work, different mechanisms responsible for steatohepatitis disease progression were elucidated in murine models. The insulin-like growth factor 2 (IGF2) mRNA binding protein (IMP) p62/IMP2-2 was shown to promote progenitor or dedifferentiated cell populations in a model of NASH and thereby amplify fibrosis. In diet-induced steatohepatitis, epigenetic alterations strongly affected genetic regions playing a role in lipid metabolism and liver morphology. Depletion of Kupffer cells, the resident macrophages of the liver, induced liver damage and attenuated hepatic accumulation of storage lipids, while hepatotoxic lipids were incorporated. Taken together, this work provides evidence that p62 promotes the appearance of undifferentiated or dedifferentiated cells and thereby disease progression and furthermore that macrophages are crucial in hepatic lipid homeostasis and protection of lipotoxicity.Die alkoholische und nicht-alkoholische Steatohepatitis (ASH und NASH) stellen Risikofaktoren für die Entwicklung eines hepatozellulären Karzinoms dar. Die Verbreitung von ASH und NASH nimmt weltweit stark zu. Innerhalb dieser Arbeit wurden verschiedene Mechanismen in Mausmodellen aufgeklärt, die für das Fortschreiten der Steatohepatitis Erkrankung verantwortlich sind. Es wurde gezeigt, dass das Insulin-ähnliche Wachstumsfaktor 2 (IGF2) mRNA bindende Protein (IMP) p62/IMP2-2 das Auftreten hepatischer Progenitorzellen oder dedifferenzierter Zellpopulationen in einem NASH Modell fördert und dadurch eine Fibrose begünstigt. In Diät-induzierter Steatohepatitis beeinflussten epigenetische Veränderungen genetische Regionen, die eine Rolle im Lipidstoffwechsel und in der Morphologie der Leber spielen. Die Depletion von Kupffer-Zellen, den Makrophagen der Leber, rief Leberschäden hervor und verringerte die Menge an Speicherlipiden in der Leber, während hepatotoxische Lipide eingelagert wurden. Zusammengefasst bietet diese Arbeit Hinweise darauf, dass p62 das Auftreten von undifferenzierten oder dedifferenzierten Zellen und damit den Krankheitsverlauf fördert und dass Makrophagen von entscheidender Bedeutung in der Lipidhomöostase der Leber und im Schutz vor Lipotoxizität sind

Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent

We study first-order optimization algorithms for computing the barycenter of Gaussian distributions with respect to the optimal transport metric. Although the objective is geodesically non-convex, Riemannian GD empirically converges rapidly, in fact faster than off-the-shelf methods such as Euclidean GD and SDP solvers. This stands in stark contrast to the best-known theoretical results for Riemannian GD, which depend exponentially on the dimension. In this work, we prove new geodesic convexity results which provide stronger control of the iterates, yielding a dimension-free convergence rate. Our techniques also enable the analysis of two related notions of averaging, the entropically-regularized barycenter and the geometric median, providing the first convergence guarantees for Riemannian GD for these problems.Comment: 48 pages, 8 figure

Interior-point methods on manifolds: theory and applications

Interior-point methods offer a highly versatile framework for convex optimization that is effective in theory and practice. A key notion in their theory is that of a self-concordant barrier. We give a suitable generalization of self-concordance to Riemannian manifolds and show that it gives the same structural results and guarantees as in the Euclidean setting, in particular local quadratic convergence of Newton's method. We analyze a path-following method for optimizing compatible objectives over a convex domain for which one has a self-concordant barrier, and obtain the standard complexity guarantees as in the Euclidean setting. We provide general constructions of barriers, and show that on the space of positive-definite matrices and other symmetric spaces, the squared distance to a point is self-concordant. To demonstrate the versatility of our framework, we give algorithms with state-of-the-art complexity guarantees for the general class of scaling and non-commutative optimization problems, which have been of much recent interest, and we provide the first algorithms for efficiently finding high-precision solutions for computing minimal enclosing balls and geometric medians in nonpositive curvature.Comment: 85 pages. v2: Merged with independent work arXiv:2212.10981 by Hiroshi Hira

Discrete Approximation of Optimal Transport on Compact Spaces

We investigate the approximation of Monge--Kantorovich problems on general compact metric spaces, showing that optimal values, plans and maps can be effectively approximated via a fully discrete method. First we approximate optimal values and plans by solving finite dimensional discretizations of the corresponding Kantorovich problem. Then we approximate optimal maps by means of the usual barycentric projection or by an analogous procedure available in general spaces without a linear structure. We prove the convergence of all these approximants in full generality and show that our convergence results are sharp.Comment: 33 pages, 5 figure