Regularization is one of the most fundamental topics in optimization, statistics and machine learning. To get sparsity in estimating a parameter $u\in\mathbb{R}^d$, an $\ell_q$ penalty term, $\Vert u\Vert_q$, is usually added to the objective function. What is the probabilistic distribution corresponding to such $\ell_q$ penalty? What is the correct stochastic process corresponding to $\Vert u\Vert_q$ when we model functions $u\in L^q$? This is important for statistically modeling large dimensional objects, e.g. images, with penalty to preserve certainty properties, e.g. edges in the image. In this work, we generalize the $q$-exponential distribution (with density proportional to) $\exp{(- \frac{1}{2}|u|^q)}$ to a stochastic process named $Q$-exponential (Q-EP) process that corresponds to the $L_q$ regularization of functions. The key step is to specify consistent multivariate $q$-exponential distributions by choosing from a large family of elliptic contour distributions. The work is closely related to Besov process which is usually defined by the expanded series. Q-EP can be regarded as a definition of Besov process with explicit probabilistic formulation and direct control on the correlation length. From the Bayesian perspective, Q-EP provides a flexible prior on functions with sharper penalty ($q<2$) than the commonly used Gaussian process (GP). We compare GP, Besov and Q-EP in modeling functional data, reconstructing images, and solving inverse problems and demonstrate the advantage of our proposed methodology.
相關內容
Processing 是(shi)一門開源編(bian)程語言(yan)和與之配套(tao)的(de)集成開發環境（IDE）的(de)名稱。Processing 在電子藝(yi)術和視覺設計社區被用(yong)來教授(shou)編(bian)程基礎，并(bing)運用(yong)于大量的(de)新媒體和互(hu)動(dong)藝(yi)術作品(pin)中。
Transformers have achieved promising results on a variety of tasks. However, the quadratic complexity in self-attention computation has limited the applications, especially in low-resource settings and mobile or edge devices. Existing works have proposed to exploit hand-crafted attention patterns to reduce computation complexity. However, such hand-crafted patterns are data-agnostic and may not be optimal. Hence, it is likely that relevant keys or values are being reduced, while less important ones are still preserved. Based on this key insight, we propose a novel deformable audio Transformer for audio recognition, named DATAR, where a deformable attention equipping with a pyramid transformer backbone is constructed and learnable. Such an architecture has been proven effective in prediction tasks,~\textit{e.g.}, event classification. Moreover, we identify that the deformable attention map computation may over-simplify the input feature, which can be further enhanced. Hence, we introduce a learnable input adaptor to alleviate this issue, and DATAR achieves state-of-the-art performance.
False discovery rate (FDR) is commonly used for correction for multiple testing in neuroimaging studies. However, when using two-tailed tests, making directional inferences about the results can lead to vastly inflated error rate, even approaching 100\% in some cases. This happens because FDR only provides weak control over the error rate, meaning that the proportion of error is guaranteed only globally over all tests, not within subsets, such as among those in only one or another direction. Here we consider and evaluate different strategies for FDR control with two-tailed tests, using both synthetic and real imaging data. Approaches that separate the tests by direction of the hypothesis test, or by the direction of the resulting test statistic, more properly control the directional error rate and preserve FDR benefits, albeit with a doubled risk of errors under complete absence of signal. Strategies that combine tests in both directions, or that use simple two-tailed p-values, can lead to invalid directional conclusions, even if these tests remain globally valid. To enable valid thresholding for directional inference, we suggest that imaging software should allow the possibility that the user sets asymmetrical thresholds for the two sides of the statistical map. While FDR continues to be a valid, powerful procedure for multiple testing correction, care is needed when making directional inferences for two-tailed tests, or more broadly, when making any localized inference.
Missing data is a widespread problem in many domains, creating challenges in data analysis and decision making. Traditional techniques for dealing with missing data, such as excluding incomplete records or imputing simple estimates (e.g., mean), are computationally efficient but may introduce bias and disrupt variable relationships, leading to inaccurate analyses. Model-based imputation techniques offer a more robust solution that preserves the variability and relationships in the data, but they demand significantly more computation time, limiting their applicability to small datasets. This work enables efficient, high-quality, and scalable data imputation within a database system using the widely used MICE method. We adapt this method to exploit computation sharing and a ring abstraction for faster model training. To impute both continuous and categorical values, we develop techniques for in-database learning of stochastic linear regression and Gaussian discriminant analysis models. Our MICE implementations in PostgreSQL and DuckDB outperform alternative MICE implementations and model-based imputation techniques by up to two orders of magnitude in terms of computation time, while maintaining high imputation quality.
Triangle counting in networks under LDP (Local Differential Privacy) is a fundamental task for analyzing connection patterns or calculating a clustering coefficient while strongly protecting sensitive friendships from a central server. In particular, a recent study proposes an algorithm for this task that uses two rounds of interaction between users and the server to significantly reduce estimation error. However, this algorithm suffers from a prohibitively high communication cost due to a large noisy graph each user needs to download. In this work, we propose triangle counting algorithms under LDP with a small estimation error and communication cost. We first propose two-rounds algorithms consisting of edge sampling and carefully selecting edges each user downloads so that the estimation error is small. Then we propose a double clipping technique, which clips the number of edges and then the number of noisy triangles, to significantly reduce the sensitivity of each user's query. Through comprehensive evaluation, we show that our algorithms dramatically reduce the communication cost of the existing algorithm, e.g., from 6 hours to 8 seconds or less at a 20 Mbps download rate, while keeping a small estimation error.
This study examines a resource-sharing problem involving multiple parties that agree to use a set of capacities together. We start with modeling the whole problem as a mathematical program, where all parties are required to exchange information to obtain the optimal objective function value. This information bears private data from each party in terms of coefficients used in the mathematical program. Moreover, the parties also consider the individual optimal solutions as private. In this setting, the concern for the parties is the privacy of their data and their optimal allocations. We propose a two-step approach to meet the privacy requirements of the parties. In the first step, we obtain a reformulated model that is amenable to a decomposition scheme. Although this scheme eliminates almost all data exchanges, it does not provide a formal privacy guarantee. In the second step, we provide this guarantee with a locally differentially private algorithm, which does not need a trusted aggregator, at the expense of deviating slightly from the optimality. We provide bounds on this deviation and discuss the consequences of these theoretical results. We also propose a novel modification to increase the efficiency of the algorithm in terms of reducing the theoretical optimality gap. The study ends with a numerical experiment on a planning problem that demonstrates an application of the proposed approach. As we work with a general linear optimization model, our analysis and discussion can be used in different application areas including production planning, logistics, and revenue management.
The essence of multivariate sequential learning is all about how to extract dependencies in data. These data sets, such as hourly medical records in intensive care units and multi-frequency phonetic time series, often time exhibit not only strong serial dependencies in the individual components (the "marginal" memory) but also non-negligible memories in the cross-sectional dependencies (the "joint" memory). Because of the multivariate complexity in the evolution of the joint distribution that underlies the data generating process, we take a data-driven approach and construct a novel recurrent network architecture, termed Memory-Gated Recurrent Networks (mGRN), with gates explicitly regulating two distinct types of memories: the marginal memory and the joint memory. Through a combination of comprehensive simulation studies and empirical experiments on a range of public datasets, we show that our proposed mGRN architecture consistently outperforms state-of-the-art architectures targeting multivariate time series.
Data augmentation has been widely used to improve generalizability of machine learning models. However, comparatively little work studies data augmentation for graphs. This is largely due to the complex, non-Euclidean structure of graphs, which limits possible manipulation operations. Augmentation operations commonly used in vision and language have no analogs for graphs. Our work studies graph data augmentation for graph neural networks (GNNs) in the context of improving semi-supervised node-classification. We discuss practical and theoretical motivations, considerations and strategies for graph data augmentation. Our work shows that neural edge predictors can effectively encode class-homophilic structure to promote intra-class edges and demote inter-class edges in given graph structure, and our main contribution introduces the GAug graph data augmentation framework, which leverages these insights to improve performance in GNN-based node classification via edge prediction. Extensive experiments on multiple benchmarks show that augmentation via GAug improves performance across GNN architectures and datasets.
Graph Convolutional Networks (GCNs) have received increasing attention in recent machine learning. How to effectively leverage the rich structural information in complex graphs, such as knowledge graphs with heterogeneous types of entities and relations, is a primary open challenge in the field. Most GCN methods are either restricted to graphs with a homogeneous type of edges (e.g., citation links only), or focusing on representation learning for nodes only instead of jointly optimizing the embeddings of both nodes and edges for target-driven objectives. This paper addresses these limitations by proposing a novel framework, namely the GEneralized Multi-relational Graph Convolutional Networks (GEM-GCN), which combines the power of GCNs in graph-based belief propagation and the strengths of advanced knowledge-base embedding methods, and goes beyond. Our theoretical analysis shows that GEM-GCN offers an elegant unification of several well-known GCN methods as specific cases, with a new perspective of graph convolution. Experimental results on benchmark datasets show the advantageous performance of GEM-GCN over strong baseline methods in the tasks of knowledge graph alignment and entity classification.
Representation learning on a knowledge graph (KG) is to embed entities and relations of a KG into low-dimensional continuous vector spaces. Early KG embedding methods only pay attention to structured information encoded in triples, which would cause limited performance due to the structure sparseness of KGs. Some recent attempts consider paths information to expand the structure of KGs but lack explainability in the process of obtaining the path representations. In this paper, we propose a novel Rule and Path-based Joint Embedding (RPJE) scheme, which takes full advantage of the explainability and accuracy of logic rules, the generalization of KG embedding as well as the supplementary semantic structure of paths. Specifically, logic rules of different lengths (the number of relations in rule body) in the form of Horn clauses are first mined from the KG and elaborately encoded for representation learning. Then, the rules of length 2 are applied to compose paths accurately while the rules of length 1 are explicitly employed to create semantic associations among relations and constrain relation embeddings. Besides, the confidence level of each rule is also considered in optimization to guarantee the availability of applying the rule to representation learning. Extensive experimental results illustrate that RPJE outperforms other state-of-the-art baselines on KG completion task, which also demonstrate the superiority of utilizing logic rules as well as paths for improving the accuracy and explainability of representation learning.
Multi-relation Question Answering is a challenging task, due to the requirement of elaborated analysis on questions and reasoning over multiple fact triples in knowledge base. In this paper, we present a novel model called Interpretable Reasoning Network that employs an interpretable, hop-by-hop reasoning process for question answering. The model dynamically decides which part of an input question should be analyzed at each hop; predicts a relation that corresponds to the current parsed results; utilizes the predicted relation to update the question representation and the state of the reasoning process; and then drives the next-hop reasoning. Experiments show that our model yields state-of-the-art results on two datasets. More interestingly, the model can offer traceable and observable intermediate predictions for reasoning analysis and failure diagnosis.
小貼士
登錄享
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191