Although Prop. 2 follows from Prop. 1, it follows the idea An upper bound on the Euclidean OT[...] The We will insist more on the importance of sampling tree metrics randomly, both for low-dimensional in 6.1 Definite-negativity is mentioned and highlighted[...] explain why is it important Is this to ensure that the kernel is positive-definite? This is why kernel methods kick in from .6 (or Gaussian processes as per Reviewer #2's suggestion). Indeed, averaging of negative definite functions is trivially negative definite. We used the farthest-point clustering due to its fast computation, i.e.

Comparing bridging annotations across coreference resources is difficult, largely due to a lack of standardization across definitions and annotation schemas and narrow coverage of disparate text domains across resources. To alleviate domain coverage issues and consolidate schemas, we compare guidelines and use interpretable predictive models to examine the bridging instances annotated in the GUM, GENTLE and ARRAU corpora. Examining these cases, we find that there is a large difference in types of phenomena annotated as bridging. Beyond theoretical results, we release a harmonized, subcategorized version of the test sets of GUM, GENTLE and the ARRAU Wall Street Journal data to promote meaningful and reliable evaluation of bridging resolution across domains.

In this paper, we propose a new class of positive definite kernels based on the spectral truncation, which has been discussed in the fields of noncommutative geometry and $C^*$-algebra. We focus on kernels whose inputs and outputs are functions and generalize existing kernels, such as polynomial, product, and separable kernels, by introducing a truncation parameter $n$ that describes the noncommutativity of the products appearing in the kernels. When $n$ goes to infinity, the proposed kernels tend to the existing commutative kernels. If $n$ is finite, they exhibit different behavior, and the noncommutativity induces interactions along the data function domain. We show that the truncation parameter $n$ is a governing factor leading to performance enhancement: by setting an appropriate $n$, we can balance the representation power and the complexity of the representation space. The flexibility of the proposed class of kernels allows us to go beyond previous commutative kernels.

This study examines the global behavior of dynamics in learning in games between two players, X and Y. We consider the simplest situation for memory asymmetry between two players: X memorizes the other Y's previous action and uses reactive strategies, while Y has no memory. Although this memory complicates the learning dynamics, we discover two novel quantities that characterize the global behavior of such complex dynamics. One is an extended Kullback-Leibler divergence from the Nash equilibrium, a well-known conserved quantity from previous studies. The other is a family of Lyapunov functions of X's reactive strategy. These two quantities capture the global behavior in which X's strategy becomes more exploitative, and the exploited Y's strategy converges to the Nash equilibrium. Indeed, we theoretically prove that Y's strategy globally converges to the Nash equilibrium in the simplest game equipped with an equilibrium in the interior of strategy spaces. Furthermore, our experiments also suggest that this global convergence is universal for more advanced zero-sum games than the simplest game. This study provides a novel characterization of the global behavior of learning in games through a couple of indicators.

Theoretical linguists have suggested that some languages (e.g., Chinese and Japanese) are "cooler" than other languages based on the observation that the intended meaning of phrases in these languages depends more on their contexts. As a result, many expressions in these languages are shortened, and their meaning is inferred from the context. In this paper, we focus on the omission of the plurality and definiteness markers in Chinese noun phrases (NPs) to investigate the predictability of their intended meaning given the contexts. To this end, we built a corpus of Chinese NPs, each of which is accompanied by its corresponding context, and by labels indicating its singularity/plurality and definiteness/indefiniteness. We carried out corpus assessments and analyses. The results suggest that Chinese speakers indeed drop plurality and definiteness markers very frequently. Building on the corpus, we train a bank of computational models using both classic machine learning models and state-of-the-art pre-trained language models to predict the plurality and definiteness of each NP. We report on the performance of these models and analyse their behaviours.

We present an analysis tool based on joint matrix factorization for comparing latent representations of multilingual and monolingual models. An alternative to probing, this tool allows us to analyze multiple sets of representations in a joint manner. Using this tool, we study to what extent and how morphosyntactic features are reflected in the representations learned by multilingual pre-trained models. We conduct a large-scale empirical study of over 33 languages and 17 morphosyntactic categories. Our findings demonstrate variations in the encoding of morphosyntactic information across upper and lower layers, with category-specific differences influenced by language properties. Hierarchical clustering of the factorization outputs yields a tree structure that is related to phylogenetic trees manually crafted by linguists. Moreover, we find the factorization outputs exhibit strong associations with performance observed across different cross-lingual tasks. We release our code to facilitate future research.

We introduce a multi-fidelity estimator of covariance matrices that employs the log-Euclidean geometry of the symmetric positive-definite manifold. The estimator fuses samples from a hierarchy of data sources of differing fidelities and costs for variance reduction while guaranteeing definiteness, in contrast with previous approaches. The new estimator makes covariance estimation tractable in applications where simulation or data collection is expensive; to that end, we develop an optimal sample allocation scheme that minimizes the mean-squared error of the estimator given a fixed budget. Guaranteed definiteness is crucial to metric learning, data assimilation, and other downstream tasks. Evaluations of our approach using data from physical applications (heat conduction, fluid dynamics) demonstrate more accurate metric learning and speedups of more than one order of magnitude compared to benchmarks.

Symmetric positive definite matrices (symmetric matrices with positive eigenvalues) are ubiquitous, not least because they arise in the solution of many minimization problems. However, a matrix that is supposed to be positive definite may fail to be so for a variety of reasons. Missing or inconsistent data in forming a covariance matrix or a correlation matrix can cause a loss of definiteness, and rounding errors can cause a tiny positive eigenvalue to go negative. The best way to check definiteness is to compute a Cholesky factorization, which is often needed anyway. The MATLAB function chol returns an error message if the factorization fails, and a second output argument can be requested, which is set to the number of the stage on which the factorization failed, or to zero if the factorization succeeded.

Graph kernels are an instance of the class of $\mathcal{R}$-Convolution kernels, which measure the similarity of objects by comparing their substructures. Despite their empirical success, most graph kernels use a naive aggregation of the final set of substructures, usually a sum or average, thereby potentially discarding valuable information about the distribution of individual components. Furthermore, only a limited instance of these approaches can be extended to continuously attributed graphs. We propose a novel method that relies on the Wasserstein distance between the node feature vector distributions of two graphs, which allows to find subtler differences in data sets by considering graphs as high-dimensional objects, rather than simple means. We further propose a Weisfeiler-Lehman inspired embedding scheme for graphs with continuous node attributes and weighted edges, enhance it with the computed Wasserstein distance, and thus improve the state-of-the-art prediction performance on several graph classification tasks.