Ad-nilpotent ideals and The Shi arrangement

We extend the Shi bijection from the Borel subalgebra case to parabolic subalgebras. In the process, the $I$-deleted Shi arrangement $\texttt{Shi}(I)$ naturally emerges. This arrangement interpolates between the Coxeter arrangement $\texttt{Cox}$ and the Shi arrangement $\texttt{Shi}$, and breaks the symmetry of $\texttt{Shi}$ in a certain symmetrical way. Among other things, we determine the characteristic polynomial $\chi(\texttt{Shi}(I), t)$ of $\texttt{Shi}(I)$ explicitly for $A_{n-1}$ and $C_n$. More generally, let $\texttt{Shi}(G)$ be an arbitrary arrangement between $\texttt{Cox}$ and $\texttt{Shi}$. Armstrong and Rhoades recently gave a formula for $\chi(\texttt{Shi}(G), t)$ for $A_{n-1}$. Inspired by their result, we obtain formulae for $\chi(\texttt{Shi}(G), t)$ for $B_n$, $C_n$ and $D_n$.Comment: The third version, quasi-antichains are shown to be in bijection with elements of L(Cox). arXiv admin note: text overlap with arXiv:1009.1655 by other author

Modulating Image Restoration with Continual Levels via Adaptive Feature Modification Layers

In image restoration tasks, like denoising and super resolution, continual modulation of restoration levels is of great importance for real-world applications, but has failed most of existing deep learning based image restoration methods. Learning from discrete and fixed restoration levels, deep models cannot be easily generalized to data of continuous and unseen levels. This topic is rarely touched in literature, due to the difficulty of modulating well-trained models with certain hyper-parameters. We make a step forward by proposing a unified CNN framework that consists of few additional parameters than a single-level model yet could handle arbitrary restoration levels between a start and an end level. The additional module, namely AdaFM layer, performs channel-wise feature modification, and can adapt a model to another restoration level with high accuracy. By simply tweaking an interpolation coefficient, the intermediate model - AdaFM-Net could generate smooth and continuous restoration effects without artifacts. Extensive experiments on three image restoration tasks demonstrate the effectiveness of both model training and modulation testing. Besides, we carefully investigate the properties of AdaFM layers, providing a detailed guidance on the usage of the proposed method.Comment: Accepted by CVPR 2019 (oral); code is available: https://github.com/hejingwenhejingwen/AdaF