In 1975 the first author proved that every finite tight two-person game form $g$ is Nash-solvable, that is, for every payoffs $u$ and $w$ of two players the obtained game $(g;u,w)$, in normal form, has a Nash equilibrium (NE) in pure strategies. This result was extended in several directions; here we strengthen it further. We construct two special NE realized by a lexicographically safe (lexsafe) strategy of one player and a best response of the other. We obtain a polynomial algorithm computing these lexsafe NE. This is trivial when game form $g$ is given explicitly. Yet, in applications $g$ is frequently realized by an oracle $\cO$ such that size of $g$ is exponential in size $|\cO|$ of $\cO$. We assume that game form $g = g(\cO)$ generated by $\cO$ is tight and that an arbitrary {\em win-lose game} $(g;u,w)$ (in which payoffs $u$ and $w$ are zero-sum and take only values $\pm 1$) can be solved, in time polynomial in $|\cO|$. These assumptions allow us to construct an algorithm computing two (one for each player) lexsafe NE in time polynomial in $|\cO|$. We consider four types of oracles known in the literature and show that all four satisfy the above assumptions.
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A matrix-free parallel two-level deflation preconditioner for the two-dimensional Helmholtz problems
We propose a matrix-free parallel two-level-deflation preconditioner combined with the Complex Shifted Laplacian preconditioner(CSLP) for the two-dimensional Helmholtz problems. The Helmholtz equation is widely studied in seismic exploration, antennas, and medical imaging. It is one of the hardest problems to solve both in terms of accuracy and convergence, due to scalability issues of the numerical solvers. Motivated by the observation that for large wavenumbers, the eigenvalues of the CSLP-preconditioned system shift towards zero, deflation with multigrid vectors, and further high-order vectors were incorporated to obtain wave-number-independent convergence. For large-scale applications, high-performance parallel scalable methods are also indispensable. In our method, we consider the preconditioned Krylov subspace methods for solving the linear system obtained from finite-difference discretization. The CSLP preconditioner is approximated by one parallel geometric multigrid V-cycle. For the two-level deflation, the matrix-free Galerkin coarsening as well as high-order re-discretization approaches on the coarse grid are studied. The results of matrix-vector multiplications in Krylov subspace methods and the interpolation/restriction operators are implemented based on the finite-difference grids without constructing any coefficient matrix. These adjustments lead to direct improvements in terms of memory consumption. Numerical experiments of model problems show that wavenumber independence has been obtained for medium wavenumbers. The matrix-free parallel framework shows satisfactory weak and strong parallel scalability.
Characterizing shapes of high-dimensional objects via Ricci curvatures plays a critical role in many research areas in mathematics and physics. However, even though several discretizations of Ricci curvatures for discrete combinatorial objects such as networks have been proposed and studied by mathematicians, the computational complexity aspects of these discretizations have escaped the attention of theoretical computer scientists to a large extent. In this paper, we study one such discretization, namely the Ollivier-Ricci curvature, from the perspective of efficient computation by fine-grained reductions and local query-based algorithms. Our main contributions are the following. (a) We relate our curvature computation problem to minimum weight perfect matching problem on complete bipartite graphs via fine-grained reduction. (b) We formalize the computational aspects of the curvature computation problems in suitable frameworks so that they can be studied by researchers in local algorithms. (c) We provide the first known lower and upper bounds on queries for query-based algorithms for the curvature computation problems in our local algorithms framework. En route, we also illustrate a localized version of our fine-grained reduction. We believe that our results bring forth an intriguing set of research questions, motivated both in theory and practice, regarding designing efficient algorithms for curvatures of objects.
The direct deep learning simulation for multi-scale problems remains a challenging issue. In this work, a novel higher-order multi-scale deep Ritz method (HOMS-DRM) is developed for thermal transfer equation of authentic composite materials with highly oscillatory and discontinuous coefficients. In this novel HOMS-DRM, higher-order multi-scale analysis and modeling are first employed to overcome limitations of prohibitive computation and Frequency Principle when direct deep learning simulation. Then, improved deep Ritz method are designed to high-accuracy and mesh-free simulation for macroscopic homogenized equation without multi-scale property and microscopic lower-order and higher-order cell problems with highly discontinuous coefficients. Moreover, the theoretical convergence of the proposed HOMS-DRM is rigorously demonstrated under appropriate assumptions. Finally, extensive numerical experiments are presented to show the computational accuracy of the proposed HOMS-DRM. This study offers a robust and high-accuracy multi-scale deep learning framework that enables the effective simulation and analysis of multi-scale problems of authentic composite materials.
L. Klebanov proved the following theorem. Let $\xi_1, \dots, \xi_n$ be independent random variables. Consider linear forms $L_1=a_1\xi_1+\cdots+a_n\xi_n,$ $L_2=b_1\xi_1+\cdots+b_n\xi_n,$ $L_3=c_1\xi_1+\cdots+c_n\xi_n,$ $L_4=d_1\xi_1+\cdots+d_n\xi_n,$ where the coefficients $a_j, b_j, c_j, d_j$ are real numbers. If the random vectors $(L_1,L_2)$ and $(L_3,L_4)$ are identically distributed, then all $\xi_i$ for which $a_id_j-b_ic_j\neq 0$ for all $j=\overline{1,n}$ are Gaussian random variables. The present article is devoted to an analog of the Klebanov theorem in the case when random variables take values in a locally compact Abelian group and the coefficients of the linear forms are integers.
In this work, we first develop a general mesoscopic multiple-relaxation-time lattice Boltzmann (MRT-LB) model for the two-dimensional diffusion equation with the constant diffusion coefficient and source term, where the D2Q5 (five discrete velocities in two-dimensional space) lattice structure is considered. Then we exactly derive the equivalent macroscopic finite-difference scheme of the MRT-LB model. Additionally, we also propose a proper MRT-LB model for the diffusion equation with a linear source term, and obtain an equivalent macroscopic six-level finite-difference scheme. After that, we conduct the accuracy and stability analysis of the finite-difference scheme and the mesoscopic MRT-LB model. It is found that at the diffusive scaling, both of them can achieve a fourth-order accuracy in space based on the Taylor expansion. The stability analysis also shows that they are both unconditionally stable. Finally, some numerical experiments are conducted, and the numerical results are also consistent with our theoretical analysis.
We address the challenge of sound propagation simulations in $3$D virtual rooms with moving sources, which have applications in virtual/augmented reality, game audio, and spatial computing. Solutions to the wave equation can describe wave phenomena such as diffraction and interference. However, simulating them using conventional numerical discretization methods with hundreds of source and receiver positions is intractable, making stimulating a sound field with moving sources impractical. To overcome this limitation, we propose using deep operator networks to approximate linear wave-equation operators. This enables the rapid prediction of sound propagation in realistic 3D acoustic scenes with moving sources, achieving millisecond-scale computations. By learning a compact surrogate model, we avoid the offline calculation and storage of impulse responses for all relevant source/listener pairs. Our experiments, including various complex scene geometries, show good agreement with reference solutions, with root mean squared errors ranging from 0.02 Pa to 0.10 Pa. Notably, our method signifies a paradigm shift as no prior machine learning approach has achieved precise predictions of complete wave fields within realistic domains. We anticipate that our findings will drive further exploration of deep neural operator methods, advancing research in immersive user experiences within virtual environments.
Comparative analysis of mathematical formulations for the two-dimensional guillotine cutting problem
About ten years ago, a paper proposed the first integer linear programming formulation for the constrained two-dimensional guillotine cutting problem (with unlimited cutting stages). Since, six other formulations followed, five of them in the last two years. This spike of interest gave no opportunity for a comprehensive comparison between the formulations. We review each formulation and compare their empirical results over instance datasets of the literature. We adapt most formulations to allow for piece rotation. The possibility of adaptation was already predicted but not realized by the prior work. The results show the dominance of pseudo-polynomial formulations until the point instances become intractable by them, while more compact formulations keep achieving good primal solutions. Our study also reveals a small but consistent advantage of the Gurobi solver over the CPLEX solver in our context; that the choice of solver hardly benefits one formulation over another; and a mistake in the generation of the T instances, which should have the same optima with or without guillotine cuts. Our study also proposes hybridising the most recent formulation with a prior formulation for a restricted version of the problem. The hybridisations show a reduction of about 20% of the branch-and-bound time thanks to the symmetries broken by the hybridisation.
In the past few years, the emergence of pre-training models has brought uni-modal fields such as computer vision (CV) and natural language processing (NLP) to a new era. Substantial works have shown they are beneficial for downstream uni-modal tasks and avoid training a new model from scratch. So can such pre-trained models be applied to multi-modal tasks? Researchers have explored this problem and made significant progress. This paper surveys recent advances and new frontiers in vision-language pre-training (VLP), including image-text and video-text pre-training. To give readers a better overall grasp of VLP, we first review its recent advances from five aspects: feature extraction, model architecture, pre-training objectives, pre-training datasets, and downstream tasks. Then, we summarize the specific VLP models in detail. Finally, we discuss the new frontiers in VLP. To the best of our knowledge, this is the first survey on VLP. We hope that this survey can shed light on future research in the VLP field.
Recently pre-trained language representation models such as BERT have shown great success when fine-tuned on downstream tasks including information retrieval (IR). However, pre-training objectives tailored for ad-hoc retrieval have not been well explored. In this paper, we propose Pre-training with Representative wOrds Prediction (PROP) for ad-hoc retrieval. PROP is inspired by the classical statistical language model for IR, specifically the query likelihood model, which assumes that the query is generated as the piece of text representative of the "ideal" document. Based on this idea, we construct the representative words prediction (ROP) task for pre-training. Given an input document, we sample a pair of word sets according to the document language model, where the set with higher likelihood is deemed as more representative of the document. We then pre-train the Transformer model to predict the pairwise preference between the two word sets, jointly with the Masked Language Model (MLM) objective. By further fine-tuning on a variety of representative downstream ad-hoc retrieval tasks, PROP achieves significant improvements over baselines without pre-training or with other pre-training methods. We also show that PROP can achieve exciting performance under both the zero- and low-resource IR settings. The code and pre-trained models are available at //github.com/Albert-Ma/PROP.
Recent work pre-training Transformers with self-supervised objectives on large text corpora has shown great success when fine-tuned on downstream NLP tasks including text summarization. However, pre-training objectives tailored for abstractive text summarization have not been explored. Furthermore there is a lack of systematic evaluation across diverse domains. In this work, we propose pre-training large Transformer-based encoder-decoder models on massive text corpora with a new self-supervised objective. In PEGASUS, important sentences are removed/masked from an input document and are generated together as one output sequence from the remaining sentences, similar to an extractive summary. We evaluated our best PEGASUS model on 12 downstream summarization tasks spanning news, science, stories, instructions, emails, patents, and legislative bills. Experiments demonstrate it achieves state-of-the-art performance on all 12 downstream datasets measured by ROUGE scores. Our model also shows surprising performance on low-resource summarization, surpassing previous state-of-the-art results on 6 datasets with only 1000 examples. Finally we validated our results using human evaluation and show that our model summaries achieve human performance on multiple datasets.
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