For a non-negative integer $sle |V(G)|-3$, a graph $G$ is $s$-Hamiltonian if the removal of any $kle s$ vertices results in a Hamiltonian graph. Given a connected simple graph $G$ that is not isomorphic to a path, a cycle, or a $K_{1,3}$, let $delta(
G)$ denote the minimum degree of $G$, let $h_s(G)$ denote the smallest integer $i$ such that the iterated line graph $L^{i}(G)$ is $s$-Hamiltonian, and let $ell(G)$ denote the length of the longest non-closed path $P$ in which all internal vertices have degree 2 such that $P$ is not both of length 2 and in a $K_3$. For a simple graph $G$, we establish better upper bounds for $h_s(G)$ as follows. begin{equation*} h_s(G)le left{ begin{aligned} & ell(G)+1, &&mbox{ if }delta(G)le 2 mbox{ and }s=0; & widetilde d(G)+2+lceil lg (s+1)rceil, &&mbox{ if }delta(G)le 2 mbox{ and }sge 1; & 2+leftlceillgfrac{s+1}{delta(G)-2}rightrceil, && mbox{ if } 3ledelta(G)le s+2; & 2, &&{rm otherwise}, end{aligned} right. end{equation*} where $widetilde d(G)$ is the smallest integer $i$ such that $delta(L^i(G))ge 3$. Consequently, when $s ge 5$, this new upper bound for the $s$-hamiltonian index implies that $h_s(G) = o(ell(G)+s+1)$ as $s to infty$. This sharpens the result, $h_s(G)leell(G)+s+1$, obtained by Zhang et al. in [Discrete Math., 308 (2008) 4779-4785].
This is the third in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field $K$, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the s
pecial linear group. This is more subtle than the results for the general linear and symplectic groups obtained in the first two papers because the arc space of the corresponding affine quotients can be nonreduced.
In an autonomous driving system, it is essential to recognize vehicles, pedestrians and cyclists from images. Besides the high accuracy of the prediction, the requirement of real-time running brings new challenges for convolutional network models. In
this report, we introduce a real-time method to detect the 2D objects from images. We aggregate several popular one-stage object detectors and train the models of variety input strategies independently, to yield better performance for accurate multi-scale detection of each category, especially for small objects. For model acceleration, we leverage TensorRT to optimize the inference time of our detection pipeline. As shown in the leaderboard, our proposed detection framework ranks the 2nd place with 75.00% L1 mAP and 69.72% L2 mAP in the real-time 2D detection track of the Waymo Open Dataset Challenges, while our framework achieves the latency of 45.8ms/frame on an Nvidia Tesla V100 GPU.
Photoacoustic imaging is an emerging technology based on the photoacoustic effect that has developed rapidly in recent years. It combines the high contrast of optical imaging and the high penetration and high resolution of acoustic imaging. As a non-
destructive biological tissue imaging technology, photoacoustic imaging has important application value in the field of biomedicine. With its high efficiency bi-oimaging capabilities and excellent biosafety performance, it has been favored by researchers. The visualization of photoacoustic imaging has great research signifi-cance in the early diagnosis of some diseases, especially tumors. In photoacoustic imaging, light transmission and thermal effects are important processes. This article is based on COMSOL software and uses finite element analysis to construct a physi-cal model for simulation. Through laser pulses into the stomach tissue containing tumor, the physical process of light transmission and biological heat transfer was studied, and a photothermal model composed of two physical fields was built, and finally a series of visualization graphics were obtained. This work has certain theo-retical guiding significance for further promoting the application of photoacoustic imaging in the field of biomedicine.
The Feature Pyramid Network (FPN) presents a remarkable approach to alleviate the scale variance in object representation by performing instance-level assignments. Nevertheless, this strategy ignores the distinct characteristics of different sub-regi
ons in an instance. To this end, we propose a fine-grained dynamic head to conditionally select a pixel-level combination of FPN features from different scales for each instance, which further releases the ability of multi-scale feature representation. Moreover, we design a spatial gate with the new activation function to reduce computational complexity dramatically through spatially sparse convolutions. Extensive experiments demonstrate the effectiveness and efficiency of the proposed method on several state-of-the-art detection benchmarks. Code is available at https://github.com/StevenGrove/DynamicHead.
The Learnable Tree Filter presents a remarkable approach to model structure-preserving relations for semantic segmentation. Nevertheless, the intrinsic geometric constraint forces it to focus on the regions with close spatial distance, hindering the
effective long-range interactions. To relax the geometric constraint, we give the analysis by reformulating it as a Markov Random Field and introduce a learnable unary term. Besides, we propose a learnable spanning tree algorithm to replace the original non-differentiable one, which further improves the flexibility and robustness. With the above improvements, our method can better capture long-range dependencies and preserve structural details with linear complexity, which is extended to several vision tasks for more generic feature transform. Extensive experiments on object detection/instance segmentation demonstrate the consistent improvements over the original version. For semantic segmentation, we achieve leading performance (82.1% mIoU) on the Cityscapes benchmark without bells-and-whistles. Code is available at https://github.com/StevenGrove/LearnableTreeFilterV2.
This paper studies a strategy for data-driven algorithm design for large-scale combinatorial optimization problems that can leverage existing state-of-the-art solvers in general purpose ways. The goal is to arrive at new approaches that can reliably
outperform existing solvers in wall-clock time. We focus on solving integer programs, and ground our approach in the large neighborhood search (LNS) paradigm, which iteratively chooses a subset of variables to optimize while leaving the remainder fixed. The appeal of LNS is that it can easily use any existing solver as a subroutine, and thus can inherit the benefits of carefully engineered heuristic or complete approaches and their software implementations. We show that one can learn a good neighborhood selector using imitation and reinforcement learning techniques. Through an extensive empirical validation in bounded-time optimization, we demonstrate that our LNS framework can significantly outperform compared to state-of-the-art commercial solvers such as Gurobi.
Learning discriminative global features plays a vital role in semantic segmentation. And most of the existing methods adopt stacks of local convolutions or non-local blocks to capture long-range context. However, due to the absence of spatial structu
re preservation, these operators ignore the object details when enlarging receptive fields. In this paper, we propose the learnable tree filter to form a generic tree filtering module that leverages the structural property of minimal spanning tree to model long-range dependencies while preserving the details. Furthermore, we propose a highly efficient linear-time algorithm to reduce resource consumption. Thus, the designed modules can be plugged into existing deep neural networks conveniently. To this end, tree filtering modules are embedded to formulate a unified framework for semantic segmentation. We conduct extensive ablation studies to elaborate on the effectiveness and efficiency of the proposed method. Specifically, it attains better performance with much less overhead compared with the classic PSP block and Non-local operation under the same backbone. Our approach is proved to achieve consistent improvements on several benchmarks without bells-and-whistles. Code and models are available at https://github.com/StevenGrove/TreeFilter-Torch.
We study the problem of learning sequential decision-making policies in settings with multiple state-action representations. Such settings naturally arise in many domains, such as planning (e.g., multiple integer programming formulations) and various
combinatorial optimization problems (e.g., those with both integer programming and graph-based formulations). Inspired by the classical co-training framework for classification, we study the problem of co-training for policy learning. We present sufficient conditions under which learning from two views can improve upon learning from a single view alone. Motivated by these theoretical insights, we present a meta-algorithm for co-training for sequential decision making. Our framework is compatible with both reinforcement learning and imitation learning. We validate the effectiveness of our approach across a wide range of tasks, including discrete/continuous control and combinatorial optimization.
We apply numerical methods in combination with finite-difference-time-domain (FDTD) simulations to optimize transmission properties of plasmonic mirror color filters using a multi-objective figure of merit over a five-dimensional parameter space by u
tilizing novel multi-fidelity Gaussian processes approach. We compare these results with conventional derivative-free global search algorithms, such as (single-fidelity) Gaussian Processes optimization scheme, and Particle Swarm Optimization---a commonly used method in nanophotonics community, which is implemented in Lumerical commercial photonics software. We demonstrate the performance of various numerical optimization approaches on several pre-collected real-world datasets and show that by properly trading off expensive information sources with cheap simulations, one can more effectively optimize the transmission properties with a fixed budget.
