A rep-tile is a polygon that can be dissected into smaller copies (of the same size) of the original polygon. A polyomino is a polygon that is formed by joining one or more unit squares edge to edge. These two notions were first introduced and investigated by Solomon W. Golomb in the 1950s and popularized by Martin Gardner in the 1960s. Since then, dozens of studies have been made in communities of recreational mathematics and puzzles. In this study, we first focus on the specific rep-tiles that have been investigated in these communities. Since the notion of rep-tiles is so simple that can be formulated mathematically in a natural way, we can apply a representative puzzle solver, a MIP solver, and SAT-based solvers for solving the rep-tile problem in common. In comparing their performance, we can conclude that the puzzle solver is the weakest while the SAT-based solvers are the strongest in the context of simple puzzle solving. We then turn to analyses of the specific rep-tiles. Using some properties of the rep-tile patterns found by a solver, we can complete analyses of specific rep-tiles up to certain sizes. That is, up to certain sizes, we can determine the existence of solutions, clarify the number of the solutions, or we can enumerate all the solutions for each size. In the last case, we find new series of solutions for the rep-tiles which have never been found in the communities.
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Some researchers speculate that intelligent reinforcement learning (RL) agents would be incentivized to seek resources and power in pursuit of their objectives. Other researchers point out that RL agents need not have human-like power-seeking instincts. To clarify this discussion, we develop the first formal theory of the statistical tendencies of optimal policies. In the context of Markov decision processes, we prove that certain environmental symmetries are sufficient for optimal policies to tend to seek power over the environment. These symmetries exist in many environments in which the agent can be shut down or destroyed. We prove that in these environments, most reward functions make it optimal to seek power by keeping a range of options available and, when maximizing average reward, by navigating towards larger sets of potential terminal states.
Repeatedly solving the parameterized optimal mass transport (pOMT) problem is a frequent task in applications such as image registration and adaptive grid generation. It is thus critical to develop a highly efficient reduced solver that is equally accurate as the full order model. In this paper, we propose such a machine learning-like method for pOMT by adapting a new reduced basis (RB) technique specifically designed for nonlinear equations, the reduced residual reduced over-collocation (R2-ROC) approach, to the parameterized Monge-Amp$\grave{\rm e}$re equation. It builds on top of a narrow-stencil finite different method (FDM), a so-called truth solver, which we propose in this paper for the Monge-Amp$\grave{\rm e}$re equation with a transport boundary. Together with the R2-ROC approach, it allows us to handle the strong and unique nonlinearity pertaining to the Monge-Amp$\grave{\rm e}$re equation achieving online efficiency without resorting to any direct approximation of the nonlinearity. Several challenging numerical tests demonstrate the accuracy and high efficiency of our method for solving the Monge-Amp$\grave{\rm e}$re equation with various parametric boundary conditions.
The K-way vertex cut problem} consists in, given a graph G, finding a subset of vertices of a given size, whose removal partitions G into the maximum number of connected components. This problem has many applications in several areas. It has been proven to be NP-complete on general graphs, as well as on split and planar graphs. In this paper, we enrich its complexity study with two new results. First, we prove that it remains NP-complete even when restricted on the class of bipartite graphs. This is unlike what it is expected, given that the K-way vertex cut problem is a generalization of the Maximum Independent set problem which is polynomially solvable on bipartite graphs. We also provide its equivalence to the wellknown problem, namely the Critical Node Problem (CNP), On split graphs. Therefore, any solving algorithm for the CNP on split graphs is a solving algorithm for the K-way vertex cut problem and vice versa.
In this paper, we formulate and study substructuring type algorithm for the Cahn-Hilliard (CH) equation, which was originally proposed to describe the phase separation phenomenon for binary melted alloy below the critical temperature and since then it has appeared in many fields ranging from tumour growth simulation, image processing, thin liquid films, population dynamics etc. Being a non-linear equation, it is important to develop robust numerical techniques to solve the CH equation. Here we present the formulation of Dirichlet-Neumann (DN) and Neumann-Neumann (NN) methods applied to CH equation and study their convergence behaviour. We consider the domain-decomposition based DN and NN methods in one and two space dimension for two subdomains and extend the study for multi-subdomain setting for NN method. We verify our findings with numerical results.
Due to the redundant nature of DNA synthesis and sequencing technologies, a basic model for a DNA storage system is a multi-draw "shuffling-sampling" channel. In this model, a random number of noisy copies of each sequence is observed at the channel output. Recent works have characterized the capacity of such a DNA storage channel under different noise and sequencing models, relying on sophisticated typicality-based approaches for the achievability. Here, we consider a multi-draw DNA storage channel in the setting of noise corruption by a binary erasure channel. We show that, in this setting, the capacity is achieved by linear coding schemes. This leads to a considerably simpler derivation of the capacity expression of a multi-draw DNA storage channel than existing results in the literature.
Leveraging biased click data for optimizing learning to rank systems has been a popular approach in information retrieval. Because click data is often noisy and biased, a variety of methods have been proposed to construct unbiased learning to rank (ULTR) algorithms for the learning of unbiased ranking models. Among them, automatic unbiased learning to rank (AutoULTR) algorithms that jointly learn user bias models (i.e., propensity models) with unbiased rankers have received a lot of attention due to their superior performance and low deployment cost in practice. Despite their differences in theories and algorithm design, existing studies on ULTR usually use uni-variate ranking functions to score each document or result independently. On the other hand, recent advances in context-aware learning-to-rank models have shown that multivariate scoring functions, which read multiple documents together and predict their ranking scores jointly, are more powerful than uni-variate ranking functions in ranking tasks with human-annotated relevance labels. Whether such superior performance would hold in ULTR with noisy data, however, is mostly unknown. In this paper, we investigate existing multivariate scoring functions and AutoULTR algorithms in theory and prove that permutation invariance is a crucial factor that determines whether a context-aware learning-to-rank model could be applied to existing AutoULTR framework. Our experiments with synthetic clicks on two large-scale benchmark datasets show that AutoULTR models with permutation-invariant multivariate scoring functions significantly outperform those with uni-variate scoring functions and permutation-variant multivariate scoring functions.
Graph convolution is the core of most Graph Neural Networks (GNNs) and usually approximated by message passing between direct (one-hop) neighbors. In this work, we remove the restriction of using only the direct neighbors by introducing a powerful, yet spatially localized graph convolution: Graph diffusion convolution (GDC). GDC leverages generalized graph diffusion, examples of which are the heat kernel and personalized PageRank. It alleviates the problem of noisy and often arbitrarily defined edges in real graphs. We show that GDC is closely related to spectral-based models and thus combines the strengths of both spatial (message passing) and spectral methods. We demonstrate that replacing message passing with graph diffusion convolution consistently leads to significant performance improvements across a wide range of models on both supervised and unsupervised tasks and a variety of datasets. Furthermore, GDC is not limited to GNNs but can trivially be combined with any graph-based model or algorithm (e.g. spectral clustering) without requiring any changes to the latter or affecting its computational complexity. Our implementation is available online.
Driven by the visions of Internet of Things and 5G communications, the edge computing systems integrate computing, storage and network resources at the edge of the network to provide computing infrastructure, enabling developers to quickly develop and deploy edge applications. Nowadays the edge computing systems have received widespread attention in both industry and academia. To explore new research opportunities and assist users in selecting suitable edge computing systems for specific applications, this survey paper provides a comprehensive overview of the existing edge computing systems and introduces representative projects. A comparison of open source tools is presented according to their applicability. Finally, we highlight energy efficiency and deep learning optimization of edge computing systems. Open issues for analyzing and designing an edge computing system are also studied in this survey.
Bidirectional Encoder Representations from Transformers (BERT) reach state-of-the-art results in a variety of Natural Language Processing tasks. However, understanding of their internal functioning is still insufficient and unsatisfactory. In order to better understand BERT and other Transformer-based models, we present a layer-wise analysis of BERT's hidden states. Unlike previous research, which mainly focuses on explaining Transformer models by their attention weights, we argue that hidden states contain equally valuable information. Specifically, our analysis focuses on models fine-tuned on the task of Question Answering (QA) as an example of a complex downstream task. We inspect how QA models transform token vectors in order to find the correct answer. To this end, we apply a set of general and QA-specific probing tasks that reveal the information stored in each representation layer. Our qualitative analysis of hidden state visualizations provides additional insights into BERT's reasoning process. Our results show that the transformations within BERT go through phases that are related to traditional pipeline tasks. The system can therefore implicitly incorporate task-specific information into its token representations. Furthermore, our analysis reveals that fine-tuning has little impact on the models' semantic abilities and that prediction errors can be recognized in the vector representations of even early layers.
The study of algorithms to automatically answer visual questions currently is motivated by visual question answering (VQA) datasets constructed in artificial VQA settings. We propose VizWiz, the first goal-oriented VQA dataset arising from a natural VQA setting. VizWiz consists of over 31,000 visual questions originating from blind people who each took a picture using a mobile phone and recorded a spoken question about it, together with 10 crowdsourced answers per visual question. VizWiz differs from the many existing VQA datasets because (1) images are captured by blind photographers and so are often poor quality, (2) questions are spoken and so are more conversational, and (3) often visual questions cannot be answered. Evaluation of modern algorithms for answering visual questions and deciding if a visual question is answerable reveals that VizWiz is a challenging dataset. We introduce this dataset to encourage a larger community to develop more generalized algorithms that can assist blind people.
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