arXiv subjects

Lei Song

Publications and source records attributed to Lei Song.

At least 73 records · Page 4Linked to original sources

arXiv2023

A higher-dimensional Chevalley restriction theorem for orthogonal groups

We prove a higher-dimensional Chevalley restriction theorem for orthogonal groups, which was conjectured by Chen and Ngô for reductive groups. In characteristic $p>2$, we also prove a weaker statement. In characteristic $0$, the theorem implies that the categorical quotient of a commuting scheme by the diagonal adjoint action of the group is integral and normal. As applications, we deduce some trace identities and a certain multiplicative property of the Pfaffian over an arbitrary commutative algebra.

math.RT↗

arXiv2023

H-TSP: Hierarchically Solving the Large-Scale Travelling Salesman Problem

We propose an end-to-end learning framework based on hierarchical reinforcement learning, called H-TSP, for addressing the large-scale Travelling Salesman Problem (TSP). The proposed H-TSP constructs a solution of a TSP instance starting from the scratch relying on two components: the upper-level policy chooses a small subset of nodes (up to 200 in our experiment) from all nodes that are to be traversed, while the lower-level policy takes the chosen nodes as input and outputs a tour connecting them to the existing partial route (initially only containing the depot). After jointly training the upper-level and lower-level policies, our approach can directly generate solutions for the given TSP instances without relying on any time-consuming search procedures. To demonstrate effectiveness of the proposed approach, we have conducted extensive experiments on randomly generated TSP instances with different numbers of nodes. We show that H-TSP can achieve comparable results (gap 3.42% vs. 7.32%) as SOTA search-based approaches, and more importantly, we reduce the time consumption up to two orders of magnitude (3.32s vs. 395.85s). To the best of our knowledge, H-TSP is the first end-to-end deep reinforcement learning approach that can scale to TSP instances of up to 10000 nodes. Although there are still gaps to SOTA results with respect to solution quality, we believe that H-TSP will be useful for practical applications, particularly those that are time-sensitive e.g., on-call routing and ride hailing service.

cs.AI↗

arXiv2023

Pointerformer: Deep Reinforced Multi-Pointer Transformer for the Traveling Salesman Problem

Traveling Salesman Problem (TSP), as a classic routing optimization problem originally arising in the domain of transportation and logistics, has become a critical task in broader domains, such as manufacturing and biology. Recently, Deep Reinforcement Learning (DRL) has been increasingly employed to solve TSP due to its high inference efficiency. Nevertheless, most of existing end-to-end DRL algorithms only perform well on small TSP instances and can hardly generalize to large scale because of the drastically soaring memory consumption and computation time along with the enlarging problem scale. In this paper, we propose a novel end-to-end DRL approach, referred to as Pointerformer, based on multi-pointer Transformer. Particularly, Pointerformer adopts both reversible residual network in the encoder and multi-pointer network in the decoder to effectively contain memory consumption of the encoder-decoder architecture. To further improve the performance of TSP solutions, Pointerformer employs both a feature augmentation method to explore the symmetries of TSP at both training and inference stages as well as an enhanced context embedding approach to include more comprehensive context information in the query. Extensive experiments on a randomly generated benchmark and a public benchmark have shown that, while achieving comparative results on most small-scale TSP instances as SOTA DRL approaches do, Pointerformer can also well generalize to large-scale TSPs.

cs.AI↗

arXiv2023

On the image of Hitchin morphism for algebraic surfaces: The case ${\rm GL}_n$

The Hitchin morphism is a map from the moduli space of Higgs bundles $\mathscr{M}_X$ to the Hitchin base $\mathscr{B}_X$, where $X$ is a smooth projective variety. When $X$ has dimension at least two, this morphism is not surjective in general. Recently, Chen-Ngô introduced a closed subscheme $\mathscr{A}_X$ of $\mathscr{B}_X$, which is called the space of spectral data. They proved that the Hitchin morphism factors through $\mathscr{A}_X$ and conjectured that $\mathscr{A}_X$ is the image of the Hitchin morphism. We prove that when $X$ is a smooth projective surface, this conjecture is true for vector bundles. Moreover, we show that $\mathscr{A}_X$, for any dimension, is invariant under proper birational morphisms, and apply the result to study $\mathscr{A}_X$ for ruled surfaces.

math.AG↗

arXiv2022

Multi-Agent Reinforcement Learning with Shared Resources for Inventory Management

In this paper, we consider the inventory management (IM) problem where we need to make replenishment decisions for a large number of stock keeping units (SKUs) to balance their supply and demand. In our setting, the constraint on the shared resources (such as the inventory capacity) couples the otherwise independent control for each SKU. We formulate the problem with this structure as Shared-Resource Stochastic Game (SRSG)and propose an efficient algorithm called Context-aware Decentralized PPO (CD-PPO). Through extensive experiments, we demonstrate that CD-PPO can accelerate the learning procedure compared with standard MARL algorithms.

cs.AI↗

arXiv2022

TD3 with Reverse KL Regularizer for Offline Reinforcement Learning from Mixed Datasets

We consider an offline reinforcement learning (RL) setting where the agent need to learn from a dataset collected by rolling out multiple behavior policies. There are two challenges for this setting: 1) The optimal trade-off between optimizing the RL signal and the behavior cloning (BC) signal changes on different states due to the variation of the action coverage induced by different behavior policies. Previous methods fail to handle this by only controlling the global trade-off. 2) For a given state, the action distribution generated by different behavior policies may have multiple modes. The BC regularizers in many previous methods are mean-seeking, resulting in policies that select out-of-distribution (OOD) actions in the middle of the modes. In this paper, we address both challenges by using adaptively weighted reverse Kullback-Leibler (KL) divergence as the BC regularizer based on the TD3 algorithm. Our method not only trades off the RL and BC signals with per-state weights (i.e., strong BC regularization on the states with narrow action coverage, and vice versa) but also avoids selecting OOD actions thanks to the mode-seeking property of reverse KL. Empirically, our algorithm can outperform existing offline RL algorithms in the MuJoCo locomotion tasks with the standard D4RL datasets as well as the mixed datasets that combine the standard datasets.

stat.ML↗

arXiv2022

Monte Carlo Tree Search based Variable Selection for High Dimensional Bayesian Optimization

Bayesian optimization (BO) is a class of popular methods for expensive black-box optimization, and has been widely applied to many scenarios. However, BO suffers from the curse of dimensionality, and scaling it to high-dimensional problems is still a challenge. In this paper, we propose a variable selection method MCTS-VS based on Monte Carlo tree search (MCTS), to iteratively select and optimize a subset of variables. That is, MCTS-VS constructs a low-dimensional subspace via MCTS and optimizes in the subspace with any BO algorithm. We give a theoretical analysis of the general variable selection method to reveal how it can work. Experiments on high-dimensional synthetic functions and real-world problems (i.e., NAS-bench problems and MuJoCo locomotion tasks) show that MCTS-VS equipped with a proper BO optimizer can achieve state-of-the-art performance.

cs.LG↗

arXiv2022

Curves contracted by the Gauss map

Given a singular projective variety in some projective space, we characterize the smooth curves contracted by the Gauss map in terms of normal bundles. As a consequence, we show that if the variety is normal, then a contracted line always has local obstruction for the embedded deformation and each component of the Hilbert scheme where the line lies is non-reduced everywhere.

math.AG↗

arXiv2022

Symmetric Transformer-based Network for Unsupervised Image Registration

Medical image registration is a fundamental and critical task in medical image analysis. With the rapid development of deep learning, convolutional neural networks (CNN) have dominated the medical image registration field. Due to the disadvantage of the local receptive field of CNN, some recent registration methods have focused on using transformers for non-local registration. However, the standard Transformer has a vast number of parameters and high computational complexity, which causes Transformer can only be applied at the bottom of the registration models. As a result, only coarse information is available at the lowest resolution, limiting the contribution of Transformer in their models. To address these challenges, we propose a convolution-based efficient multi-head self-attention (CEMSA) block, which reduces the parameters of the traditional Transformer and captures local spatial context information for reducing semantic ambiguity in the attention mechanism. Based on the proposed CEMSA, we present a novel Symmetric Transformer-based model (SymTrans). SymTrans employs the Transformer blocks in the encoder and the decoder respectively to model the long-range spatial cross-image relevance. We apply SymTrans to the displacement field and diffeomorphic registration. Experimental results show that our proposed method achieves state-of-the-art performance in image registration. Our code is publicly available at \url{https://github.com/MingR-Ma/SymTrans}.

cs.CV↗

arXiv2021

On direct images of twisted pluricanonical sheaves on normal varieties

We study the depth properties of certain direct image sheaves on normal varieties. Let $f: Y\rightarrow X$ be a proper morphism of relative dimension $d$ from a smooth variety onto a normal variety such that the preimage $E$ of the singular locus of $X$ is a divisor. We show that for any integer $m>0$, the higher direct image $R^df_*ω^{\otimes m}_Y(aE)$ modulo the torsion subsheaf is $S_2$, provided that $a$ is sufficiently large. In case $f$ is birational, we give criteria on $a$ for the direct image $f_*ω_Y(aE)$ to coincide with $ω_X$. We also introduce an index measuring the singularities of normal varieties.

math.AG↗

arXiv2020

Tunneling Rate Constants for H2CO+H on Amorphous Solid Water Surfaces

Formaldehyde (H2CO) is one of the most abundant molecules observed in the icy mantle covering interstellar grains. Studying its evolution can contribute to our understanding of the formation of complex organic molecules in various interstellar environments. In this work, we investigated the hydrogenation reactions of H2CO yielding CH3O, CH2OH, and the hydrogen abstraction resulting in H2+HCO on an amorphous solid water (ASW) surface using a quantum mechanics/molecular mechanics (QM/MM) model. The binding energies of H2CO on the ASW surface vary broadly, from 1000 to 9370 K. No correlation was found between binding energies and activation energies of hydrogenation reactions. Combining instanton theory with QM/MM modeling, we calculated rate constants for the Langmuir Hinshelwood and the Eley Rideal mechanisms for the three product channels of H+H2CO surface reactions down to 59 K. We found that the channel producing CH2OH can be ignored, owing to its high activation barrier leading to significantly lower rates than the other two channels. The ASW surface influences the reactivity in favor of formation of CH3O (branching ratio 80%) and hinders the H2CO dissociation into H2+HCO. In addition, kinetic isotope effects are strong in all reaction channels and vary strongly between the channels. Finally, we provide fits of the rate constants to be used in astrochemical models.

astro-ph.GA↗

arXiv2019

Regularity of structure sheaves of varieties with isolated singularities

Let $X\subseteq \mathbb{P}^N$ be a non-degenerate normal projective variety of codimension $e$ and degree $d$ with isolated $\mathbb{Q}$-Gorenstein singularities. We prove that the Castelnuovo-Mumford regularity $\text{reg}(\mathcal{O}_{X})\le d-e$, as predicted by the Eisenbud-Goto regularity conjecture. Such a bound fails for general projective varieties. The main techniques are Noma's classification of non-degenerated projective varieties and Nadel vanishing for multiplier ideals. We also classify the extremal and the next to extremal cases.

math.AG↗

arXiv2019

On the projective normality of cyclic coverings over a rational surface

Let $S$ be a rational surface with $\dim|-K_S|\ge 1$ and let $π: X\rightarrow S$ be a ramified cyclic covering from a nonruled smooth surface $X$. We show that for any integer $k\ge 3$ and ample divisor $A$ on $S$, the adjoint divisor $K_X+kπ^*A$ is very ample and normally generated. Similar result holds for minimal (possibly singular) coverings.

math.AG↗

arXiv2018

Kodaira vanishing for singular varieties revisited

We correct the proof and slightly strengthen a Kodaira-type vanishing theorem for singular varieties originally due to Jaffe and the first author. Specifically, we show that if $L$ is a nef and big line bundle on a projective variety of characteristic zero, the $i^{\text{th}}$ cohomology of $L^{-1}$ vanishes for $i$ in a range determined by the depth and dimension of the singular locus.

math.AG↗

arXiv2017

Singularities of secant varieties

We study the singularities of the secant variety $Σ(X,L)$ associated to a smooth variety $X$ embedded by a sufficiently positive adjoint bundle $L$. We show that $Σ(X,L)$ is always Du Bois singular. Examples of secant varieties with worse singularities when $L$ has weak positivity are provided. We also give a necessary and sufficient condition for $Σ(X, L)$ to have rational singularities.

math.AG↗

arXiv2016

Formation of the prebiotic molecule NH$_2$CHO on astronomical amorphous solid water surfaces: accurate tunneling rate calculations

Investigating how formamide forms in the interstellar medium is a hot topic in astrochemistry, which can contribute to our understanding of the origin of life on Earth. We have constructed a QM/MM model to simulate the hydrogenation of isocyanic acid on amorphous solid water surfaces to form formamide. The binding energy of HNCO on the ASW surface varies significantly between different binding sites, we found values between $\sim$0 and 100 kJ mol$^{-1}$. The barrier for the hydrogenation reaction is almost independent of the binding energy, though. We calculated tunneling rate constants of H + HNCO $\rightarrow$ NH$_2$CO at temperatures down to 103 K combining QM/MM with instanton theory. Tunneling dominates the reaction at such low temperatures. The tunneling reaction is hardly accelerated by the amorphous solid water surface compared to the gas phase for this system, even though the activation energy of the surface reaction is lower than the one of the gas-phase reaction. Both the height and width of the barrier affect the tunneling rate in practice. Strong kinetic isotope effects were observed by comparing to rate constants of D + HNCO $\rightarrow$ NHDCO. At 103 K we found a KIE of 231 on the surface and 146 in the gas phase. Furthermore, we investigated the gas-phase reaction NH$_2$ + H$_2$CO $\rightarrow$ NH$_2$CHO + H and found it unlikely to occur at cryogenic temperatures. The data of our tunneling rate constants are expected to significantly influence astrochemical models.

physics.chem-ph↗

arXiv2016

The Commutativity Problem of the MapReduce Framework: A Transducer-based Approach

MapReduce is a popular programming model for data parallel computation. In MapReduce, the reducer produces an output from a list of inputs. Due to the scheduling policy of the platform, the inputs may arrive at the reducers in different order. The commutativity problem of reducers asks if the output of a reducer is independent of the order of its inputs. Although the problem is undecidable in general, the MapReduce programs in practice are usually used for data analytics and thus require very simple control flow. By exploiting the simplicity, we propose a programming language for reducers where the commutativity problem is decidable. The main idea of the reducer language is to separate the control and data flow of programs and disallow arithmetic operations in the control flow. The decision procedure for the commutativity problem is obtained through a reduction to the equivalence problem of streaming numerical transducers (SNTs), a novel automata model over infinite alphabets introduced in this paper. The design of SNTs is inspired by streaming transducers (Alur and Cerny, POPL 2011). Nevertheless, the two models are intrinsically different since the outputs of SNTs are integers while those of streaming transducers are data words. The decidability of the equivalence of SNTs is achieved with an involved combinatorial analysis of the evolvement of the values of the integer variables during the runs of SNTs.

cs.FL↗

arXiv2016

On the projective normality of double coverings over a rational surface

We study the projective normality of a minimal surface $X$ which is a ramified double covering over a rational surface $S$ with $\dim|-K_S|\ge 1$. In particular Horikawa surfaces, the minimal surfaces of general type with $K^2_X=2p_g(X)-4$, are of this type, up to resolution of singularities. Let $π$ be the covering map from $X$ to $S$. We show that the $\mathbb{Z}_2$-invariant adjoint divisors $K_X+rπ^*A$ are normally generated, where the integer $r\ge 3$ and $A$ is an ample divisor on $S$.

math.AG↗