This paper presents a bias-variance tradeoff of graph Laplacian regularizer, which is widely used in graph signal processing and semi-supervised learning tasks. The scaling law of the optimal regularization parameter is specified in terms of the spectral graph properties and a novel signal-to-noise ratio parameter, which suggests selecting a mediocre regularization parameter is often suboptimal. The analysis is applied to three applications, including random, band-limited, and multiple-sampled graph signals. Experiments on synthetic and real-world graphs demonstrate near-optimal performance of the established analysis.
Graph-based semi-supervised learning is one of the most popular methods in machine learning. Some of its theoretical properties such as bounds for the generalization error and the convergence of the graph Laplacian regularizer have been studied in computer science and statistics literatures. However, a fundamental statistical property, the consistency of the estimator from this method has not been proved. In this article, we study the consistency problem under a non-parametric framework. We prove the consistency of graph-based learning in the case that the estimated scores are enforced to be e…
IMT Atlantique（IMT大西洋学院）；University of Southern California（南加州大学）
For the past few years, Deep Neural Network (DNN) robustness has become a question of paramount importance. As a matter of fact, in sensitive settings misclassification can lead to dramatic consequences. Such misclassifications are likely to occur when facing adversarial attacks, hardware failures or limitations, and imperfect signal acquisition. To address this question, authors have proposed different approaches aiming at increasing the robustness of DNNs, such as adding regularizers or training using noisy examples. In this paper we propose a new regularizer built upon the Laplacian of simi…
Institute for Infocomm Research, A*STAR（新加坡科技研究局资讯通信研究院）
Generative Adversarial Networks are powerful generative models that are able to model the manifold of natural images. We leverage this property to perform manifold regularization by approximating a variant of the Laplacian norm using a Monte Carlo approximation that is easily computed with the GAN. When incorporated into the semi-supervised feature-matching GAN we achieve state-of-the-art results for GAN-based semi-supervised learning on CIFAR-10 and SVHN benchmarks, with a method that is significantly easier to implement than competing methods. We also find that manifold regularization improv…
Given a weighted graph with $N$ vertices, consider a real-valued regression problem in a semi-supervised setting, where one observes $n$ labeled vertices, and the task is to label the remaining ones. We present a theoretical study of $\ell_p$-based Laplacian regularization under a $d$-dimensional geometric random graph model. We provide a variational characterization of the performance of this regularized learner as $N$ grows to infinity while $n$ stays constant, the associated optimality conditions lead to a partial differential equation that must be satisfied by the associated function estim…
Microsoft Research（微软研究院）；University of Washington（华盛顿大学）；Stanford University（斯坦福大学）
We propose a new framework for differentially private optimization of convex functions which are Lipschitz in an arbitrary norm $\|\cdot\|$. Our algorithms are based on a regularized exponential mechanism which samples from the density $\propto \exp(-k(F+μr))$ where $F$ is the empirical loss and $r$ is a regularizer which is strongly convex with respect to $\|\cdot\|$, generalizing a recent work of [Gopi, Lee, Liu '22] to non-Euclidean settings. We show that this mechanism satisfies Gaussian differential privacy and solves both DP-ERM (empirical risk minimization) and DP-SCO (stochastic convex…
Indian Institute of Technology, Madras（印度理工学院马德拉斯分校）
Regularization methods, specifically those which directly alter weights like $L_1$ and $L_2$, are an integral part of many learning algorithms. Both the regularizers mentioned above are formulated by assuming certain priors in the parameter space and these assumptions, in some cases, induce sparsity in the parameter space. Regularizers help in transferring beliefs one has on the dataset or the parameter space by introducing adequate terms in the loss function. Any kind of formulation represents a specific set of beliefs: $L_1$ regularization conveys that the parameter space should be sparse wh…
Department of Science Education（科学教育系）；Center for Theoretical Physics and Artificial Intelligence Institute（理论物理与人工智能研究所）；School of Computational Sciences（计算科学学院）
Multiview canonical correlation analysis (MCCA) seeks latent low-dimensional representations encountered with multiview data of shared entities (a.k.a. common sources). However, existing MCCA approaches do not exploit the geometry of the common sources, which may be available \emph{a priori}, or can be constructed using certain domain knowledge. This prior information about the common sources can be encoded by a graph, and be invoked as a regularizer to enrich the maximum variance MCCA framework. In this context, the present paper's novel graph-regularized (G) MCCA approach minimizes the dista…
Oden Institute of Computational Engineering and Sciences, University of Texas（德克萨斯大学奥登计算工程与科学研究所）；University of Minnesota（明尼苏达大学）
We show that uncertainty sampling is sufficient to achieve exploration versus exploitation in graph-based active learning, as long as the measure of uncertainty properly aligns with the underlying model and the model properly reflects uncertainty in unexplored regions. In particular, we use a recently developed algorithm, Poisson ReWeighted Laplace Learning (PWLL) for the classifier and we introduce an acquisition function designed to measure uncertainty in this graph-based classifier that identifies unexplored regions of the data. We introduce a diagonal perturbation in PWLL which produces ex…
Wangxuan Institute of Computer Technology, Peking University（北京大学王选计算机研究所）
Graph Convolutional Neural Networks (GCNNs) are generalizations of CNNs to graph-structured data, in which convolution is guided by the graph topology. In many cases where graphs are unavailable, existing methods manually construct graphs or learn task-driven adaptive graphs. In this paper, we propose Graph Learning Neural Networks (GLNNs), which exploit the optimization of graphs (the adjacency matrix in particular) from both data and tasks. Leveraging on spectral graph theory, we propose the objective of graph learning from a sparsity constraint, properties of a valid adjacency matrix as wel…
Ecole Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
In sparse signal representation, the choice of a dictionary often involves a tradeoff between two desirable properties -- the ability to adapt to specific signal data and a fast implementation of the dictionary. To sparsely represent signals residing on weighted graphs, an additional design challenge is to incorporate the intrinsic geometric structure of the irregular data domain into the atoms of the dictionary. In this work, we propose a parametric dictionary learning algorithm to design data-adapted, structured dictionaries that sparsely represent graph signals. In particular, we model grap…
STARK denoises spatial transcriptomics images via adaptive regularization
Department of Mathematics, University of British Columbia（数学系，不列颠哥伦比亚大学）；Department of Computer Science, University of British Columbia（计算机科学系，不列颠哥伦比亚大学）；Department of Mathematics and Mathematical Statistics, Umeå University（数学与统计学系，乌梅大学）；Premium Research Institute for Human Metaverse Medicine (WPI-PRIMe), The University of Osaka（人类元宇宙医学研究学院（WPI-PRIMe），大阪大学）
