The central subspace of a pair of random variables $(y,x) \in \mathbb{R}^{p+1}$ is the minimal subspace $\mathcal{S}$ such that $y \perp \hspace{-2mm} \perp x\mid P_{\mathcal{S}}x$. In this paper, we consider the minimax rate of estimating the central space of the multiple index models $y=f(\beta_{1}^{\tau}x,\beta_{2}^{\tau}x,...,\beta_{d}^{\tau}x,\epsilon)$ with at most $s$ active predictors where $x \sim N(0,I_{p})$. We first introduce a large class of models depending on the smallest non-zero eigenvalue $\lambda$ of $var(\mathbb{E}[x|y])$, over which we show that an aggregated estimator based on the SIR procedure converges at rate $d\wedge((sd+s\log(ep/s))/(n\lambda))$. We then show that this rate is optimal in two scenarios: the single index models; and the multiple index models with fixed central dimension $d$ and fixed $\lambda$. By assuming a technical conjecture, we can show that this rate is also optimal for multiple index models with bounded dimension of the central space. We believe that these (conditional) optimal rate results bring us meaningful insights of general SDR problems in high dimensions.

The International Committee on Taxonomy of Viruses authorizes and organizes the taxonomic classification of viruses. Thus far, the detailed classifications for all viruses are neither complete nor free from dispute. For example, the current missing label rates in GenBank are 12.1% for family label and 30.0% for genus label. Using the proposed Natural Vector representation, all 2,044 single-segment referenced viral genomes in GenBank can be embedded in R^12. Unlike other approaches, this allows us to determine phylogenetic relations for all viruses at any level (e.g., Baltimore class, family, subfamily, genus, and species) in real time. Additionally, the proposed graphical representation for virus phylogeny provides a visualization of the distribution of viruses in R^12. Unlike the commonly used tree visualization methods which suffer from uniqueness and existence problems, our representation always exists and is unique. This approach is successfully used to predict and correct viral classification information, as well as to identify viral origins; e.g. a recent public health threat, the West Nile virus, is closer to the Japanese encephalitis antigenic complex based on our visualization. Based on cross validation results, the accuracy rates of our predictions are as high as 98.2% for Baltimore class labels, 96.6% for family labels, 99.7% for subfamily labels and 97.2% for genus labels.

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We prove that any “finite-type” component of a stability space of a triangulated category is contractible. The motivating example of such a component is the stability space of the Calabi–Yau–N category D(Gamma_N Q) associated to an ADE Dynkin quiver. In addition to showing that this is contractible we prove that the braid group Br(Q) acts freely upon it by spherical twists, in particular that the spherical twist group Br(Gamma_N Q) is isomorphic to Br(Q). This generalises the result of Brav–Thomas for the N=2 case. Other classes of triangulated categories with finite-type components in their stability spaces include locally finite triangulated categories with finite-rank Grothendieck group and discrete derived categories of finite global dimension.

We consider weighted ray-transforms PW (weighted Radon transforms along oriented straight lines) in Rd,d≥2, with strictly positive weights W. We construct an example of such a transform with non-trivial kernel in the space of infinitely smooth compactly supported functions on Rd. In addition, the constructed weight W is rotation-invariant continuous and is infinitely smooth almost everywhere on Rd×Sd−1. In particular, by this construction we give counterexamples to some well-known injectivity results for weighted ray transforms for the case when the regularity of W is slightly relaxed. We also give examples of continous strictly positive W such that dimkerPW≥n in the space of infinitely smooth compactly supported functions on Rd for arbitrary n∈N∪{∞}, where W are infinitely smooth for d=2 and infinitely smooth almost everywhere for d≥3.