More Like this

1. Inference on the Primary Parameter of Interest with the Aid of Dimension Reduction Estimation

   https://doi.org/10.1111/j.1467-9868.2010.00759.x  

   Li, Lexin ; Zhu, Liping ; Zhu, Lixing ( October 2010 , Journal of the Royal Statistical Society Series B: Statistical Methodology)

   As high dimensional data become routinely available in applied sciences, sufficient dimension reduction has been widely employed and its research has received considerable attention. However, with the majority of sufficient dimension reduction methodology focusing on the dimension reduction step, complete analysis and inference after dimension reduction have yet to receive much attention. We couple the strategy of sufficient dimension reduction with a flexible semiparametric model. We concentrate on inference with respect to the primary variables of interest, and we employ sufficient dimension reduction to bring down the dimension of the regression effectively. Extensive simulations demonstrate the efficacy of the method proposed, and a real data analysis is presented for illustration.

    

   more » « less
2. A topological approach to inferring the intrinsic dimension of convex sensing data

   https://doi.org/10.1007/s41468-021-00081-3  

   Wu, Min-Chun ; Itskov, Vladimir ( November 2021 , Journal of Applied and Computational Topology)

   We consider a common measurement paradigm, where an unknown subset of an affine space is measured by unknown continuous quasi-convex functions. Given the measurement data, can one determine the dimension of this space? In this paper, we develop a method for inferring the intrinsic dimension of the data from measurements by quasi-convex functions, under natural assumptions. The dimension inference problem depends only on discrete data of the ordering of the measured points of space, induced by the sensor functions. We construct a filtration of Dowker complexes, associated to measurements by quasi-convex functions. Topological features of these complexes are then used to infer the intrinsic dimension. We prove convergence theorems that guarantee obtaining the correct intrinsic dimension in the limit of large data, under natural assumptions. We also illustrate the usability of this method in simulations.

    

   more » « less
3. The essential dimension of congruence covers

   https://doi.org/10.1112/S0010437X21007594  

   Farb, Benson ; Kisin, Mark ; Wolfson, Jesse ( November 2021 , Compositio Mathematica)

   Abstract Consider the algebraic function $\Phi _{g,n}$ that assigns to a general $g$ -dimensional abelian variety an $n$ -torsion point. A question first posed by Klein asks: What is the minimal $d$ such that, after a rational change of variables, the function $\Phi _{g,n}$ can be written as an algebraic function of $d$ variables? Using techniques from the deformation theory of $p$ -divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $p$ -dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $p$ -dimension of congruence covers of the moduli space of genus $g$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties. These results include cases where the locally symmetric variety $M$ is proper . As far as we know, these are the first examples of nontrivial lower bounds on the essential dimension of an unramified, nonabelian covering of a proper algebraic variety. 

   more » « less
4. Dynamic asymptotic dimension and Matui's HK conjecture

   https://doi.org/10.1112/plms.12510  

   Bönicke, Christian ; Dell'Aiera, Clément ; Gabe, James ; Willett, Rufus ( January 2023 , Proceedings of the London Mathematical Society)

   We prove that the homology groups of a principal ample groupoid vanish in dimensions greater than the dynamic asymptotic dimension of the groupoid (as a side‐effect of our methods, we also give a new model of groupoid homology in terms of the Tor groups of homological algebra, which might be of independent interest). As a consequence, the K‐theory of the ‐algebras associated with groupoids of finite dynamic asymptotic dimension can be computed from the homology of the underlying groupoid. In particular, principal ample groupoids with dynamic asymptotic dimension at most two and finitely generated second homology satisfy Matui's HK‐conjecture.

   We also construct explicit maps from the groupoid homology groups to the K‐theory groups of their ‐algebras in degrees zero and one, and investigate their properties.

    

   more » « less
5. Astrophysical constraints on decaying dark gravitons

   https://doi.org/10.1007/JHEP06(2024)047  

   Law-Smith, Jamie_A P ; Obied, Georges ; Prabhu, Anirudh ; Vafa, Cumrun ( June 2024 , Journal of High Energy Physics)

   In the dark dimension scenario, which predicts an extra dimension of micron scale, dark gravitons (KK modes) are a natural dark matter candidate. In this paper, we study observable features of this model. In particular, their decay to standard matter fields can distort the CMB and impact other astrophysical signals. Using this we place bounds on the parameters of this model. In particular we find that the natural range of parameters in this scenario is consistent with these constraints and leads to the prediction that the mean mass of the dark matter today is close to a few hundred keV and the effective size of the extra dimension is around 1–30 μm.

    

   more » « less