Robust and Scalable Learning of Complex Intrinsic Dataset Geometry via ElPiGraph

Multidimensional datapoint clouds representing large datasets are frequently characterized by non-trivial low-dimensional geometry and topology which can be recovered by unsupervised machine learning approaches, in particular, by principal graphs. Principal graphs approximate the multivariate data by a graph injected into the data space with some constraints imposed on the node mapping. Here we present ElPiGraph, a scalable and robust method for constructing principal graphs. ElPiGraph exploits and further develops the concept of elastic energy, the topological graph grammar approach, and a gradient descent-like optimization of the graph topology. The method is able to withstand high levels of noise and is capable of approximating data point clouds via principal graph ensembles. This strategy can be used to estimate the statistical significance of complex data features and to summarize them into a single consensus principal graph. ElPiGraph deals efficiently with large datasets in various fields such as biology, where it can be used for example with single-cell transcriptomic or epigenomic datasets to infer gene expression dynamics and recover differentiation landscapes.

Howe–Moore type theorems for quantum groups and rigid -tensor categories

We formulate and study Howe–Moore type properties in the setting of quantum groups and in the setting of rigid $C^{\ast }$-tensor categories. We say that a rigid $C^{\ast }$-tensor category ${\mathcal{C}}$ has the Howe–Moore property if every completely positive multiplier on ${\mathcal{C}}$ has a limit at infinity. We prove that the representation categories of $q$-deformations of connected compact simple Lie groups with trivial center satisfy the Howe–Moore property. As an immediate consequence, we deduce the Howe–Moore property for Temperley–Lieb–Jones standard invariants with principal graph $A_{\infty }$. These results form a special case of a more general result on the convergence of completely bounded multipliers on the aforementioned categories. This more general result also holds for the representation categories of the free orthogonal quantum groups and for the Kazhdan–Wenzl categories. Additionally, in the specific case of the quantum groups $\text{SU}_{q}(N)$, we are able, using a result of the first-named author, to give an explicit characterization of the central states on the quantum coordinate algebra of $\text{SU}_{q}(N)$, which coincide with the completely positive multipliers on the representation category of $\text{SU}_{q}(N)$.

A simple sufficient condition for triviality of obstructions in the orbifold construction for subfactors

We present a simple sufficient condition for triviality of obstructions in the orbifold construction. As an application, we can show the existence of subfactors with principal graph $D_{2n}$ without full use of Ocneanu's paragroup theory.

Reversed graph embedding resolves complex single-cell developmental trajectories

Organizing single cells along a developmental trajectory has emerged as a powerful tool for understanding how gene regulation governs cell fate decisions. However, learning the structure of complex single-cell trajectories with two or more branches remains a challenging computational problem. We present Monocle 2, which uses reversed graph embedding to reconstruct single-cell trajectories in a fully unsupervised manner. Monocle 2 learns an explicit principal graph to describe the data, greatly improving the robustness and accuracy of its trajectories compared to other algorithms. Monocle 2 uncovered a new, alternative cell fate in what we previously reported to be a linear trajectory for differentiating myoblasts. We also reconstruct branched trajectories for two studies of blood development, and show that loss of function mutations in key lineage transcription factors diverts cells to alternative branches on the a trajectory. Monocle 2 is thus a powerful tool for analyzing cell fate decisions with single-cell genomics.

Quotients of A2 * T2

We study unitary quotients of the free product unitary pivotal category A2 * T2. We show that such quotients are parametrized by an integer n ≥ 1 and an 2n–th root of unity ω. We show that for n = 1, 2, 3, there is exactly one quotient and ω = 1. For 4 ≤ n ≤ 10, we show that there are no such quotients. Our methods also apply to quotients of T2 * T2, where we have a similar result.The essence of our method is a consistency check on jellyfish relations. While we only treat the specific cases of A2 × T2 and T2 . T2, we anticipate that our technique can be extended to a general method for proving the nonexistence of planar algebras with a specified principal graph.During the preparation of this manuscript, we learnt of Liu's independent result on composites of A3 and A4 subfactor planar algebras (arxiv:1308.5691). In 1994, BischHaagerup showed that the principal graph of a composite of A3 and A4 must fit into a certain family, and Liu has classified all such subfactor planar algebras. We explain the connection between the quotient categories and the corresponding composite subfactor planar algebras. As a corollary of Liu's result, there are no such quotient categories for n ≥ 4.This is an abridged version of arxiv:1308.5723.