We study a graph-theoretic problem in the Penrose P2-graphs which are the dual graphs of Penrose tilings by kites and darts. Using substitutions, local isomorphism and other properties of Penrose tilings, we construct a family of arbitrarily large induced subtrees of Penrose graphs with the largest possible number of leaves for a given number $n$ of vertices. These subtrees are called fully leafed induced subtrees. We denote their number of leaves $L_{P2}(n)$ for any non-negative integer $n$, and the sequence $\left(L_{P2}(n)\right)_{n\in\mathbb{N}}$ is called the leaf function of Penrose P2-graphs. We present exact and recursive formulae for $L_{P2}(n)$, as well as an infinite sequence of fully leafed induced subtrees, which are caterpillar graphs. In particular, our proof relies on the construction of a finite graded poset of 3-internal-regular subtrees.
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The Sinkhorn algorithm is the state-of-the-art to approximate solutions of entropic optimal transport (OT) distances between discrete probability distributions. We show that meticulously training a neural network to learn initializations to the algorithm via the entropic OT dual problem can significantly speed up convergence, while maintaining desirable properties of the Sinkhorn algorithm, such as differentiability and parallelizability. We train our predictive network in an adversarial fashion using a second, generating network and a self-supervised bootstrapping loss. The predictive network is universal in the sense that it is able to generalize to any pair of distributions of fixed dimension and cost at inference, and we prove that we can make the generating network universal in the sense that it is capable of producing any pair of distributions during training. Furthermore, we show that our network can even be used as a standalone OT solver to approximate regularized transport distances to a few percent error, which makes it the first meta neural OT solver.
A process algebra is proposed, whose semantics maps a term to a nondeterministic finite automaton (NFA, for short). We prove a representability theorem: for each NFA $N$, there exists a process algebraic term $p$ such that its semantics is an NFA isomorphic to $N$. Moreover, we provide a concise axiomatization of language equivalence: two NFAs $N_1$ and $N_2$ recognize the same language if and only if the associated terms $p_1$ and $p_2$, respectively, can be equated by means of a set of axioms, comprising 7 axioms plus 3 conditional axioms, only.
This study explores four methods of generating paraphrases in Malayalam, utilizing resources available for English paraphrasing and pre-trained Neural Machine Translation (NMT) models. We evaluate the resulting paraphrases using both automated metrics, such as BLEU, METEOR, and cosine similarity, as well as human annotation. Our findings suggest that automated evaluation measures may not be fully appropriate for Malayalam, as they do not consistently align with human judgment. This discrepancy underscores the need for more nuanced paraphrase evaluation approaches especially for highly agglutinative languages.
We study the probabilistic modeling performed by Autoregressive Large Language Models through the angle of time directionality. We empirically find a time asymmetry exhibited by such models in their ability to model natural language: a difference in the average log-perplexity when trying to predict the next token versus when trying to predict the previous one. This difference is at the same time subtle and very consistent across various modalities (language, model size, training time, ...). Theoretically, this is surprising: from an information-theoretic point of view, there should be no such difference. We provide a theoretical framework to explain how such an asymmetry can appear from sparsity and computational complexity considerations, and outline a number of perspectives opened by our results.
An effective and efficient architecture performance evaluation scheme is essential for the success of Neural Architecture Search (NAS). To save computational cost, most of existing NAS algorithms often train and evaluate intermediate neural architectures on a small proxy dataset with limited training epochs. But it is difficult to expect an accurate performance estimation of an architecture in such a coarse evaluation way. This paper advocates a new neural architecture evaluation scheme, which aims to determine which architecture would perform better instead of accurately predict the absolute architecture performance. Therefore, we propose a \textbf{relativistic} architecture performance predictor in NAS (ReNAS). We encode neural architectures into feature tensors, and further refining the representations with the predictor. The proposed relativistic performance predictor can be deployed in discrete searching methods to search for the desired architectures without additional evaluation. Experimental results on NAS-Bench-101 dataset suggests that, sampling 424 ($0.1\%$ of the entire search space) neural architectures and their corresponding validation performance is already enough for learning an accurate architecture performance predictor. The accuracies of our searched neural architectures on NAS-Bench-101 and NAS-Bench-201 datasets are higher than that of the state-of-the-art methods and show the priority of the proposed method.
Graph Neural Networks (GNNs) have been shown to be effective models for different predictive tasks on graph-structured data. Recent work on their expressive power has focused on isomorphism tasks and countable feature spaces. We extend this theoretical framework to include continuous features - which occur regularly in real-world input domains and within the hidden layers of GNNs - and we demonstrate the requirement for multiple aggregation functions in this context. Accordingly, we propose Principal Neighbourhood Aggregation (PNA), a novel architecture combining multiple aggregators with degree-scalers (which generalize the sum aggregator). Finally, we compare the capacity of different models to capture and exploit the graph structure via a novel benchmark containing multiple tasks taken from classical graph theory, alongside existing benchmarks from real-world domains, all of which demonstrate the strength of our model. With this work, we hope to steer some of the GNN research towards new aggregation methods which we believe are essential in the search for powerful and robust models.
Knowledge graph completion aims to predict missing relations between entities in a knowledge graph. While many different methods have been proposed, there is a lack of a unifying framework that would lead to state-of-the-art results. Here we develop PathCon, a knowledge graph completion method that harnesses four novel insights to outperform existing methods. PathCon predicts relations between a pair of entities by: (1) Considering the Relational Context of each entity by capturing the relation types adjacent to the entity and modeled through a novel edge-based message passing scheme; (2) Considering the Relational Paths capturing all paths between the two entities; And, (3) adaptively integrating the Relational Context and Relational Path through a learnable attention mechanism. Importantly, (4) in contrast to conventional node-based representations, PathCon represents context and path only using the relation types, which makes it applicable in an inductive setting. Experimental results on knowledge graph benchmarks as well as our newly proposed dataset show that PathCon outperforms state-of-the-art knowledge graph completion methods by a large margin. Finally, PathCon is able to provide interpretable explanations by identifying relations that provide the context and paths that are important for a given predicted relation.
How can we estimate the importance of nodes in a knowledge graph (KG)? A KG is a multi-relational graph that has proven valuable for many tasks including question answering and semantic search. In this paper, we present GENI, a method for tackling the problem of estimating node importance in KGs, which enables several downstream applications such as item recommendation and resource allocation. While a number of approaches have been developed to address this problem for general graphs, they do not fully utilize information available in KGs, or lack flexibility needed to model complex relationship between entities and their importance. To address these limitations, we explore supervised machine learning algorithms. In particular, building upon recent advancement of graph neural networks (GNNs), we develop GENI, a GNN-based method designed to deal with distinctive challenges involved with predicting node importance in KGs. Our method performs an aggregation of importance scores instead of aggregating node embeddings via predicate-aware attention mechanism and flexible centrality adjustment. In our evaluation of GENI and existing methods on predicting node importance in real-world KGs with different characteristics, GENI achieves 5-17% higher NDCG@100 than the state of the art.
We introduce an effective model to overcome the problem of mode collapse when training Generative Adversarial Networks (GAN). Firstly, we propose a new generator objective that finds it better to tackle mode collapse. And, we apply an independent Autoencoders (AE) to constrain the generator and consider its reconstructed samples as "real" samples to slow down the convergence of discriminator that enables to reduce the gradient vanishing problem and stabilize the model. Secondly, from mappings between latent and data spaces provided by AE, we further regularize AE by the relative distance between the latent and data samples to explicitly prevent the generator falling into mode collapse setting. This idea comes when we find a new way to visualize the mode collapse on MNIST dataset. To the best of our knowledge, our method is the first to propose and apply successfully the relative distance of latent and data samples for stabilizing GAN. Thirdly, our proposed model, namely Generative Adversarial Autoencoder Networks (GAAN), is stable and has suffered from neither gradient vanishing nor mode collapse issues, as empirically demonstrated on synthetic, MNIST, MNIST-1K, CelebA and CIFAR-10 datasets. Experimental results show that our method can approximate well multi-modal distribution and achieve better results than state-of-the-art methods on these benchmark datasets. Our model implementation is published here: //github.com/tntrung/gaan
Image segmentation is an important component of many image understanding systems. It aims to group pixels in a spatially and perceptually coherent manner. Typically, these algorithms have a collection of parameters that control the degree of over-segmentation produced. It still remains a challenge to properly select such parameters for human-like perceptual grouping. In this work, we exploit the diversity of segments produced by different choices of parameters. We scan the segmentation parameter space and generate a collection of image segmentation hypotheses (from highly over-segmented to under-segmented). These are fed into a cost minimization framework that produces the final segmentation by selecting segments that: (1) better describe the natural contours of the image, and (2) are more stable and persistent among all the segmentation hypotheses. We compare our algorithm's performance with state-of-the-art algorithms, showing that we can achieve improved results. We also show that our framework is robust to the choice of segmentation kernel that produces the initial set of hypotheses.
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