Multidimensional scaling (MDS) is the act of embedding proximity information about a set of $n$ objects in $d$-dimensional Euclidean space. As originally conceived by the psychometric community, MDS was concerned with embedding a fixed set of proximities associated with a fixed set of objects. Modern concerns, e.g., that arise in developing asymptotic theories for statistical inference on random graphs, more typically involve studying the limiting behavior of a sequence of proximities associated with an increasing set of objects. Standard results from the theory of point-to-set maps imply that, if $n$ is fixed and a sequence of proximities converges, then the limit of the embedded structures is the embedded structure of the limiting proximities. But what if $n$ increases? It then becomes necessary to reformulate MDS so that the entire sequence of embedding problems can be viewed as a sequence of optimization problems in a fixed space. We present such a reformulation and derive some consequences.
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Strategy logic (SL) is a powerful temporal logic that enables strategic reasoning in multi-agent systems. SL supports explicit (first-order) quantification over strategies and provides a logical framework to express many important properties such as Nash equilibria, dominant strategies, etc. While in SL the same strategy can be used in multiple strategy profiles, each such profile is evaluated w.r.t. a path-property, i.e., a property that considers the single path resulting from a particular strategic interaction. In this paper, we present Hyper Strategy Logic (HyperSL), a strategy logic where the outcome of multiple strategy profiles can be compared w.r.t. a hyperproperty, i.e., a property that relates multiple paths. We show that HyperSL can capture important properties that cannot be expressed in SL, including non-interference, quantitative Nash equilibria, optimal adversarial planning, and reasoning under imperfect information. On the algorithmic side, we identify an expressive fragment of HyperSL with decidable model checking and present a model-checking algorithm. We contribute a prototype implementation of our algorithm and report on encouraging experimental results.
The inadequate mixing of conventional Markov Chain Monte Carlo (MCMC) methods for multi-modal distributions presents a significant challenge in practical applications such as Bayesian inference and molecular dynamics. Addressing this, we propose Diffusive Gibbs Sampling (DiGS), an innovative family of sampling methods designed for effective sampling from distributions characterized by distant and disconnected modes. DiGS integrates recent developments in diffusion models, leveraging Gaussian convolution to create an auxiliary noisy distribution that bridges isolated modes in the original space and applying Gibbs sampling to alternately draw samples from both spaces. Our approach exhibits a better mixing property for sampling multi-modal distributions than state-of-the-art methods such as parallel tempering. We demonstrate that our sampler attains substantially improved results across various tasks, including mixtures of Gaussians, Bayesian neural networks and molecular dynamics.
A $\textit{resolving set}$ $R$ in a graph $G$ is a set of vertices such that every vertex of $G$ is uniquely identified by its distances to the vertices of $R$. Introduced in the 1970s, this concept has been since then extensively studied from both combinatorial and algorithmic point of view. We propose a generalization of the concept of resolving sets to temporal graphs, i.e., graphs with edge sets that change over discrete time-steps. In this setting, the $\textit{temporal distance}$ from $u$ to $v$ is the earliest possible time-step at which a journey with strictly increasing time-steps on edges leaving $u$ reaches $v$, i.e., the first time-step at which $v$ could receive a message broadcast from $u$. A $\textit{temporal resolving set}$ of a temporal graph $\mathcal{G}$ is a subset $R$ of its vertices such that every vertex of $\mathcal{G}$ is uniquely identified by its temporal distances from vertices of $R$. We study the problem of finding a minimum-size temporal resolving set, and show that it is NP-complete even on very restricted graph classes and with strong constraints on the time-steps: temporal complete graphs where every edge appears in either time-step 1 or 2, temporal trees where every edge appears in at most two consecutive time-steps, and even temporal subdivided stars where every edge appears in at most two (not necessarily consecutive) time-steps. On the other hand, we give polynomial-time algorithms for temporal paths and temporal stars where every edge appears in exactly one time-step, and give a combinatorial analysis and algorithms for several temporal graph classes where the edges appear in periodic time-steps.
We define and investigate the Fr\'{e}chet edit distance problem. Given two polygonal curves $\pi$ and $\sigma$ and a threshhold value $\delta>0$, we seek the minimum number of edits to $\sigma$ such that the Fr\'{e}chet distance between the edited $\sigma$ and $\pi$ is at most $\delta$. For the edit operations we consider three cases, namely, deletion of vertices, insertion of vertices, or both. For this basic problem we consider a number of variants. Specifically, we provide polynomial time algorithms for both discrete and continuous Fr\'{e}chet edit distance variants, as well as hardness results for weak Fr\'{e}chet edit distance variants.
We present an algorithm to solve the dispersive depth-averaged Serre-Green-Naghdi (SGN) equations using patch-based adaptive mesh refinement. These equations require adding additional higher derivative terms to the nonlinear shallow water equations. This has been implemented as a new component of the open source GeoClaw software that is widely used for modeling tsunamis, storm surge, and related hazards, improving its accuracy on shorter wavelength phenomena. We use a formulation that requires solving an elliptic system of equations at each time step, making the method implicit. The adaptive algorithm allows different time steps on different refinement levels, and solves the implicit equations level by level. Computational examples are presented to illustrate the stability and accuracy on a radially symmetric test case and two realistic tsunami modeling problems, including a hypothetical asteroid impact creating a short wavelength tsunami for which dispersive terms are necessary.
Message passing mechanism contributes to the success of GNNs in various applications, but also brings the oversquashing problem. Recent works combat oversquashing by improving the graph spectrums with rewiring techniques, disrupting the structural bias in graphs, and having limited improvement on oversquashing in terms of oversquashing measure. Motivated by unitary RNN, we propose Graph Unitary Message Passing (GUMP) to alleviate oversquashing in GNNs by applying unitary adjacency matrix for message passing. To design GUMP, a transformation is first proposed to make general graphs have unitary adjacency matrix and keep its structural bias. Then, unitary adjacency matrix is obtained with a unitary projection algorithm, which is implemented by utilizing the intrinsic structure of unitary adjacency matrix and allows GUMP to be permutation-equivariant. Experimental results show the effectiveness of GUMP in improving the performance on various graph learning tasks.
The information bottleneck (IB) method is a technique for extracting information that is relevant for predicting the target random variable from the source random variable, which is typically implemented by optimizing the IB Lagrangian that balances the compression and prediction terms. However, the IB Lagrangian is hard to optimize, and multiple trials for tuning values of Lagrangian multiplier are required. Moreover, we show that the prediction performance strictly decreases as the compression gets stronger during optimizing the IB Lagrangian. In this paper, we implement the IB method from the perspective of supervised disentangling. Specifically, we introduce Disentangled Information Bottleneck (DisenIB) that is consistent on compressing source maximally without target prediction performance loss (maximum compression). Theoretical and experimental results demonstrate that our method is consistent on maximum compression, and performs well in terms of generalization, robustness to adversarial attack, out-of-distribution detection, and supervised disentangling.
Knowledge graph (KG) embedding encodes the entities and relations from a KG into low-dimensional vector spaces to support various applications such as KG completion, question answering, and recommender systems. In real world, knowledge graphs (KGs) are dynamic and evolve over time with addition or deletion of triples. However, most existing models focus on embedding static KGs while neglecting dynamics. To adapt to the changes in a KG, these models need to be re-trained on the whole KG with a high time cost. In this paper, to tackle the aforementioned problem, we propose a new context-aware Dynamic Knowledge Graph Embedding (DKGE) method which supports the embedding learning in an online fashion. DKGE introduces two different representations (i.e., knowledge embedding and contextual element embedding) for each entity and each relation, in the joint modeling of entities and relations as well as their contexts, by employing two attentive graph convolutional networks, a gate strategy, and translation operations. This effectively helps limit the impacts of a KG update in certain regions, not in the entire graph, so that DKGE can rapidly acquire the updated KG embedding by a proposed online learning algorithm. Furthermore, DKGE can also learn KG embedding from scratch. Experiments on the tasks of link prediction and question answering in a dynamic environment demonstrate the effectiveness and efficiency of DKGE.
It is always well believed that modeling relationships between objects would be helpful for representing and eventually describing an image. Nevertheless, there has not been evidence in support of the idea on image description generation. In this paper, we introduce a new design to explore the connections between objects for image captioning under the umbrella of attention-based encoder-decoder framework. Specifically, we present Graph Convolutional Networks plus Long Short-Term Memory (dubbed as GCN-LSTM) architecture that novelly integrates both semantic and spatial object relationships into image encoder. Technically, we build graphs over the detected objects in an image based on their spatial and semantic connections. The representations of each region proposed on objects are then refined by leveraging graph structure through GCN. With the learnt region-level features, our GCN-LSTM capitalizes on LSTM-based captioning framework with attention mechanism for sentence generation. Extensive experiments are conducted on COCO image captioning dataset, and superior results are reported when comparing to state-of-the-art approaches. More remarkably, GCN-LSTM increases CIDEr-D performance from 120.1% to 128.7% on COCO testing set.
We investigate a lattice-structured LSTM model for Chinese NER, which encodes a sequence of input characters as well as all potential words that match a lexicon. Compared with character-based methods, our model explicitly leverages word and word sequence information. Compared with word-based methods, lattice LSTM does not suffer from segmentation errors. Gated recurrent cells allow our model to choose the most relevant characters and words from a sentence for better NER results. Experiments on various datasets show that lattice LSTM outperforms both word-based and character-based LSTM baselines, achieving the best results.
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