We apply the concept of distance correlation for testing independence of long-range dependent time series. For this, we establish a non-central limit theorem for stochastic processes with values in an $L_2$-Hilbert space. This limit theorem is of a general theoretical interest that goes beyond the context of this article. For the purpose of this article, it provides the basis for deriving the asymptotic distribution of the distance covariance of subordinated Gaussian processes. Depending on the dependence in the data, the standardization and the limit of distance correlation vary. In any case, the limit is not feasible, such that test decisions are based on a subsampling procedure. We prove the validity of the subsampling procedure and assess the finite sample performance of a hypothesis test based on the distance covariance. In particular, we compare its finite sample performance to that of a test based on Pearson's sample correlation coefficient. For this purpose, we additionally establish convergence results for this dependence measure. Different dependencies between the vectors are considered. It turns out that only linear correlation is better detected by Pearson's sample correlation coefficient, while all other dependencies are better detected by distance correlation. An analysis with regard to cross-dependencies between the mean monthly discharges of three different rivers provides an application of the theoretical results established in this article.
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Structural features are important features in a geometrical graph. Although there are some correlation analysis of features based on covariance, there is no relevant research on structural feature correlation analysis with graph neural networks. In this paper, we introuduce graph feature to feature (Fea2Fea) prediction pipelines in a low dimensional space to explore some preliminary results on structural feature correlation, which is based on graph neural network. The results show that there exists high correlation between some of the structural features. An irredundant feature combination with initial node features, which is filtered by graph neural network has improved its classification accuracy in some graph-based tasks. We compare differences between concatenation methods on connecting embeddings between features and show that the simplest is the best. We generalize on the synthetic geometric graphs and certify the results on prediction difficulty between structural features.
A novel procedure to perform fuzzy clustering of multivariate time series generated from different dependence models is proposed. Different amounts of dissimilarity between the generating models or changes on the dynamic behaviours over time are some arguments justifying a fuzzy approach, where each series is associated to all the clusters with specific membership levels. Our procedure considers quantile-based cross-spectral features and consists of three stages: (i) each element is characterized by a vector of proper estimates of the quantile cross-spectral densities, (ii) principal component analysis is carried out to capture the main differences reducing the effects of the noise, and (iii) the squared Euclidean distance between the first retained principal components is used to perform clustering through the standard fuzzy C-means and fuzzy C-medoids algorithms. The performance of the proposed approach is evaluated in a broad simulation study where several types of generating processes are considered, including linear, nonlinear and dynamic conditional correlation models. Assessment is done in two different ways: by directly measuring the quality of the resulting fuzzy partition and by taking into account the ability of the technique to determine the overlapping nature of series located equidistant from well-defined clusters. The procedure is compared with the few alternatives suggested in the literature, substantially outperforming all of them whatever the underlying process and the evaluation scheme. Two specific applications involving air quality and financial databases illustrate the usefulness of our approach.
We provide sufficient conditions for the asymptotic normality of the generalized correlation coefficient $\sum a_{ij}b_{ij}$ under the permutation null distribution when $a_{ij}$'s are symmetric and $b_{ij}$'s are symmetric.
Much real-world data come with explicitly defined domain orders; e.g., lexicographic order for strings, numeric for integers, and chronological for time. Our goal is to discover implicit domain orders that we do not already know; for instance, that the order of months in the Chinese Lunar calendar is Corner < Apricot < Peach. To do so, we enhance data profiling methods by discovering implicit domain orders in data through order dependencies. We enumerate tractable special cases and proceed towards the most general case, which we prove is NP-complete. We show that the general case nevertheless can be effectively handled by a SAT solver. We also devise an interestingness measure to rank the discovered implicit domain orders, which we validate with a user study. Based on an extensive suite of experiments with real-world data, we establish the efficacy of our algorithms, and the utility of the domain orders discovered by demonstrating significant added value in three applications (data profiling, query optimization, and data mining).
Recent advances in Transformer models allow for unprecedented sequence lengths, due to linear space and time complexity. In the meantime, relative positional encoding (RPE) was proposed as beneficial for classical Transformers and consists in exploiting lags instead of absolute positions for inference. Still, RPE is not available for the recent linear-variants of the Transformer, because it requires the explicit computation of the attention matrix, which is precisely what is avoided by such methods. In this paper, we bridge this gap and present Stochastic Positional Encoding as a way to generate PE that can be used as a replacement to the classical additive (sinusoidal) PE and provably behaves like RPE. The main theoretical contribution is to make a connection between positional encoding and cross-covariance structures of correlated Gaussian processes. We illustrate the performance of our approach on the Long-Range Arena benchmark and on music generation.
In many real-world applications, we want to exploit multiple source datasets of similar tasks to learn a model for a different but related target dataset -- e.g., recognizing characters of a new font using a set of different fonts. While most recent research has considered ad-hoc combination rules to address this problem, we extend previous work on domain discrepancy minimization to develop a finite-sample generalization bound, and accordingly propose a theoretically justified optimization procedure. The algorithm we develop, Domain AggRegation Network (DARN), is able to effectively adjust the weight of each source domain during training to ensure relevant domains are given more importance for adaptation. We evaluate the proposed method on real-world sentiment analysis and digit recognition datasets and show that DARN can significantly outperform the state-of-the-art alternatives.
Generative Adversarial Networks (GAN) have shown great promise in tasks like synthetic image generation, image inpainting, style transfer, and anomaly detection. However, generating discrete data is a challenge. This work presents an adversarial training based correlated discrete data (CDD) generation model. It also details an approach for conditional CDD generation. The results of our approach are presented over two datasets; job-seeking candidates skill set (private dataset) and MNIST (public dataset). From quantitative and qualitative analysis of these results, we show that our model performs better as it leverages inherent correlation in the data, than an existing model that overlooks correlation.
In this paper, we propose to tackle the problem of reducing discrepancies between multiple domains referred to as multi-source domain adaptation and consider it under the target shift assumption: in all domains we aim to solve a classification problem with the same output classes, but with labels' proportions differing across them. We design a method based on optimal transport, a theory that is gaining momentum to tackle adaptation problems in machine learning due to its efficiency in aligning probability distributions. Our method performs multi-source adaptation and target shift correction simultaneously by learning the class probabilities of the unlabeled target sample and the coupling allowing to align two (or more) probability distributions. Experiments on both synthetic and real-world data related to satellite image segmentation task show the superiority of the proposed method over the state-of-the-art.
Methods that align distributions by minimizing an adversarial distance between them have recently achieved impressive results. However, these approaches are difficult to optimize with gradient descent and they often do not converge well without careful hyperparameter tuning and proper initialization. We investigate whether turning the adversarial min-max problem into an optimization problem by replacing the maximization part with its dual improves the quality of the resulting alignment and explore its connections to Maximum Mean Discrepancy. Our empirical results suggest that using the dual formulation for the restricted family of linear discriminators results in a more stable convergence to a desirable solution when compared with the performance of a primal min-max GAN-like objective and an MMD objective under the same restrictions. We test our hypothesis on the problem of aligning two synthetic point clouds on a plane and on a real-image domain adaptation problem on digits. In both cases, the dual formulation yields an iterative procedure that gives more stable and monotonic improvement over time.
Attention mechanism has been used as an ancillary means to help RNN or CNN. However, the Transformer (Vaswani et al., 2017) recently recorded the state-of-the-art performance in machine translation with a dramatic reduction in training time by solely using attention. Motivated by the Transformer, Directional Self Attention Network (Shen et al., 2017), a fully attention-based sentence encoder, was proposed. It showed good performance with various data by using forward and backward directional information in a sentence. But in their study, not considered at all was the distance between words, an important feature when learning the local dependency to help understand the context of input text. We propose Distance-based Self-Attention Network, which considers the word distance by using a simple distance mask in order to model the local dependency without losing the ability of modeling global dependency which attention has inherent. Our model shows good performance with NLI data, and it records the new state-of-the-art result with SNLI data. Additionally, we show that our model has a strength in long sentences or documents.
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