Statistical inference using pairwise comparison data is an effective approach to analyzing large-scale sparse networks. In this paper, we propose a general framework to model the mutual interactions in a network, which enjoys ample flexibility in terms of model parametrization. Under this setup, we show that the maximum likelihood estimator for the latent score vector of the subjects is uniformly consistent under a near-minimal condition on network sparsity. This condition is sharp in terms of the leading order asymptotics describing the sparsity. Our analysis utilizes a novel chaining technique and illustrates an important connection between graph topology and model consistency. Our results guarantee that the maximum likelihood estimator is justified for estimation in large-scale pairwise comparison networks where data are asymptotically deficient. Simulation studies are provided in support of our theoretical findings.
We consider learning a sparse pairwise Markov Random Field (MRF) with continuous-valued variables from i.i.d samples. We adapt the algorithm of Vuffray et al. (2019) to this setting and provide finite-sample analysis revealing sample complexity scaling logarithmically with the number of variables, as in the discrete and Gaussian settings. Our approach is applicable to a large class of pairwise MRFs with continuous variables and also has desirable asymptotic properties, including consistency and normality under mild conditions. Further, we establish that the population version of the optimizati…
In a bivariate setting, we consider the problem of detecting a sparse contamination or mixture component, where the effect manifests itself as a positive dependence between the variables, which are otherwise independent in the main component. We first look at this problem in the context of a normal mixture model. In essence, the situation reduces to a univariate setting where the effect is a decrease in variance. In particular, a higher criticism test based on the pairwise differences is shown to achieve the detection boundary defined by the (oracle) likelihood ratio test. We then turn to a Ga…
We consider high-dimensional distribution estimation through autoregressive networks. By combining the concepts of sparsity, mixtures and parameter sharing we obtain a simple model which is fast to train and which achieves state-of-the-art or better results on several standard benchmark datasets. Specifically, we use an L1-penalty to regularize the conditional distributions and introduce a procedure for automatic parameter sharing between mixture components. Moreover, we propose a simple distributed representation which permits exact likelihood evaluations since the latent variables are interl…
We characterize the effectiveness of a classical algorithm for recovering the Markov graph of a general discrete pairwise graphical model from i.i.d. samples. The algorithm is (appropriately regularized) maximum conditional log-likelihood, which involves solving a convex program for each node; for Ising models this is $\ell_1$-constrained logistic regression, while for more general alphabets an $\ell_{2,1}$ group-norm constraint needs to be used. We show that this algorithm can recover any arbitrary discrete pairwise graphical model, and also characterize its sample complexity as a function of…
The pseudo-likelihood method is one of the most popular algorithms for learning sparse binary pairwise Markov networks. In this paper, we formulate the $L_1$ regularized pseudo-likelihood problem as a sparse multiple logistic regression problem. In this way, many insights and optimization procedures for sparse logistic regression can be applied to the learning of discrete Markov networks. Specifically, we use the coordinate descent algorithm for generalized linear models with convex penalties, combined with strong screening rules, to solve the pseudo-likelihood problem with $L_1$ regularizatio…
Weizmann Institute of Science（魏茨曼科学研究所）；Princeton University（普林斯顿大学）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Texas A&M University（得克萨斯农工大学）
Several classification methods assume that the underlying distributions follow tree-structured graphical models. Indeed, trees capture statistical dependencies between pairs of variables, which may be crucial to attain low classification errors. The resulting classifier is linear in the log-transformed univariate and bivariate densities that correspond to the tree edges. In practice, however, observed data may not be well approximated by trees. Yet, motivated by the importance of pairwise dependencies for accurate classification, here we propose to approximate the optimal decision boundary by …
Tandon School of Engineering（坦顿工程学院）；New York University（纽约大学）
Deep neural networks are often highly overparameterized, prohibiting their use in compute-limited systems. However, a line of recent works has shown that the size of deep networks can be considerably reduced by identifying a subset of neuron indicators (or mask) that correspond to significant weights prior to training. We demonstrate that an simple iterative mask discovery method can achieve state-of-the-art compression of very deep networks. Our algorithm represents a hybrid approach between single shot network pruning methods (such as SNIP) with Lottery-Ticket type approaches. We validate ou…
John A. Paulson School of Engineering and Applied Sciences（约翰·A·保尔森工程与应用科学学院）；Harvard University（哈佛大学）；Department of Stem Cell and Regenerative Biology（干细胞与再生生物学系）；Department of Molecular and Cellular Biology（分子与细胞生物学系）
Extracting an understanding of the underlying system from high dimensional data is a growing problem in science. Discovering informative and meaningful features is crucial for clustering, classification, and low dimensional data embedding. Here we propose to construct features based on their ability to discriminate between clusters of the data points. We define a class of problems in which linear separability of clusters is hidden in a low dimensional space. We propose an unsupervised method to identify the subset of features that define a low dimensional subspace in which clustering can be co…
Attentional Graph Meta-Learning for Indoor Localization Using Extremely Sparse Fingerprints
School of Science & Engineering, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）理工学院）；Huawei Technologies Co., LTD.（华为技术有限公司）；School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）数据科学学院）
Crowdsourcing platforms are now extensively used for conducting subjective pairwise comparison studies. In this setting, a pairwise comparison dataset is typically gathered via random sampling, either \emph{with} or \emph{without} replacement. In this paper, we use tools from random graph theory to analyze these two random sampling methods for the HodgeRank estimator. Using the Fiedler value of the graph as a measurement for estimator stability (informativeness), we provide a new estimate of the Fiedler value for these two random graph models. In the asymptotic limit as the number of vertices …
Does higher interpretability imply better utility? A Pairwise Analysis on Sparse Autoencoders
School of Computing and Data Science, The University of Hong Kong（计算与数据科学学院，香港大学）；School of Data Science, The Chinese University of Hong Kong, Shenzhen（数据科学学院，香港中文大学（深圳））
CITEC cluster of excellence（CITEC卓越集群）；Bielefeld University（比勒费尔德大学）
We are interested in the decomposition of motion data into a sparse linear combination of base functions which enable efficient data processing. We combine two prominent frameworks: dynamic time warping (DTW), which offers particularly successful pairwise motion data comparison, and sparse coding (SC), which enables an automatic decomposition of vectorial data into a sparse linear combination of base vectors. We enhance SC as follows: an efficient kernelization which extends its application domain to general similarity data such as offered by DTW, and its restriction to non-negative linear rep…
University of Helsinki（赫尔辛基大学）；Aalto University（阿尔托大学）；University of Oslo（奥斯陆大学）；Tokyo Institute of Technology（东京工业大学）；Wellcome Sanger Institute（威康桑格研究所）
Learning the undirected graph structure of a Markov network from data is a problem that has received a lot of attention during the last few decades. As a result of the general applicability of the model class, a myriad of methods have been developed in parallel in several research fields. Recently, as the size of the considered systems has increased, the focus of new methods has been shifted towards the high-dimensional domain. In particular, introduction of the pseudo-likelihood function has pushed the limits of score-based methods which were originally based on the likelihood function. At th…
As a typical dimensionality reduction technique, random projection can be simply implemented with linear projection, while maintaining the pairwise distances of high-dimensional data with high probability. Considering this technique is mainly exploited for the task of classification, this paper is developed to study the construction of random matrix from the viewpoint of feature selection, rather than of traditional distance preservation. This yields a somewhat surprising theoretical result, that is, the sparse random matrix with exactly one nonzero element per column, can present better featu…
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China（上海交通大学数学科学学院）；Department of Mathematics, University of Massachusetts Dartmouth, North Dartmouth, MA 02747（大学马萨诸塞州达特茅斯分校数学系）；Department of Chemical and Biomolecular Engineering, University of Pennsylvania, Philadelphia, PA 19104, United States of America（宾夕法尼亚大学化学与生物分子工程系）
We present a novel approach to the formulation and the resolution of sparse Linear Discriminant Analysis (LDA). Our proposal, is based on penalized Optimal Scoring. It has an exact equivalence with penalized LDA, contrary to the multi-class approaches based on the regression of class indicator that have been proposed so far. Sparsity is obtained thanks to a group-Lasso penalty that selects the same features in all discriminant directions. Our experiments demonstrate that this approach generates extremely parsimonious models without compromising prediction performances. Besides prediction, the …
Hebrew University（希伯来大学）；Institute of Computer Science（计算机科学研究所）；Center for Neural Computations（神经计算中心）
Learning Bayesian networks is often cast as an optimization problem, where the computational task is to find a structure that maximizes a statistically motivated score. By and large, existing learning tools address this optimization problem using standard heuristic search techniques. Since the search space is extremely large, such search procedures can spend most of the time examining candidates that are extremely unreasonable. This problem becomes critical when we deal with data sets that are large either in the number of instances, or the number of attributes. In this paper, we introduce an …
