Matrix Completion with Hypergraphs:Sharp Thresholds and Efficient Algorithms

This paper considers the problem of completing a rating matrix based on sub-sampled matrix entries as well as observed social graphs and hypergraphs. We show that there exists a \emph{sharp threshold} on the sample probability for the task of exactly completing the rating matrix -- the task is achievable when the sample probability is above the threshold, and is impossible otherwise -- demonstrating a phase transition phenomenon. The threshold can be expressed as a function of the ``quality'' of hypergraphs, enabling us to \emph{quantify} the amount of reduction in sample probability due to the exploitation of hypergraphs. This also highlights the usefulness of hypergraphs in the matrix completion problem. En route to discovering the sharp threshold, we develop a computationally efficient matrix completion algorithm that effectively exploits the observed graphs and hypergraphs. Theoretical analyses show that our algorithm succeeds with high probability as long as the sample probability exceeds the aforementioned threshold, and this theoretical result is further validated by synthetic experiments. Moreover, our experiments on a real social network dataset (with both graphs and hypergraphs) show that our algorithm outperforms other state-of-the-art matrix completion algorithms.

Community Detection in the Multi-View Stochastic Block Model

This paper considers the problem of community detection on multiple potentially correlated graphs from an information-theoretical perspective. We first put forth a random graph model, called the multi-view stochastic block model (MVSBM), designed to generate correlated graphs on the same set of nodes (with cardinality $n$). The $n$ nodes are partitioned into two disjoint communities of equal size. The presence or absence of edges in the graphs for each pair of nodes depends on whether the two nodes belong to the same community or not. The objective for the learner is to recover the hidden communities with observed graphs. Our technical contributions are two-fold: (i) We establish an information-theoretic upper bound (Theorem~1) showing that exact recovery of community is achievable when the model parameters of MVSBM exceed a certain threshold. (ii) Conversely, we derive an information-theoretic lower bound (Theorem~2) showing that when the model parameters of MVSBM fall below the aforementioned threshold, then for any estimator, the expected number of misclassified nodes will always be greater than one. Our results for the MVSBM recover several prior results for community detection in the standard SBM as well as in multiple independent SBMs as special cases.

MVPatch: More Vivid Patch for Adversarial Camouflaged Attacks on Object Detectors in the Physical World

Recent investigations demonstrate that adversarial patches can be utilized to manipulate the result of object detection models. However, the conspicuous patterns on these patches may draw more attention and raise suspicions among humans. Moreover, existing works have primarily focused on enhancing the efficacy of attacks in the physical domain, rather than seeking to optimize their stealth attributes and transferability potential. To address these issues, we introduce a dual-perception-based attack framework that generates an adversarial patch known as the More Vivid Patch (MVPatch). The framework consists of a model-perception degradation method and a human-perception improvement method. To derive the MVPatch, we formulate an iterative process that simultaneously constrains the efficacy of multiple object detectors and refines the visual correlation between the generated adversarial patch and a realistic image. Our method employs a model-perception-based approach that reduces the object confidence scores of several object detectors to boost the transferability of adversarial patches. Further, within the human-perception-based framework, we put forward a lightweight technique for visual similarity measurement that facilitates the development of inconspicuous and natural adversarial patches and eliminates the reliance on additional generative models. Additionally, we introduce the naturalness score and transferability score as metrics for an unbiased assessment of various adversarial patches' natural appearance and transferability capacity. Extensive experiments demonstrate that the proposed MVPatch algorithm achieves superior attack transferability compared to similar algorithms in both digital and physical domains while also exhibiting a more natural appearance. These findings emphasize the remarkable stealthiness and transferability of the proposed MVPatch attack algorithm.

Mutual Information Learned Regressor: an Information-theoretic Viewpoint of Training Regression Systems

As one of the central tasks in machine learning, regression finds lots of applications in different fields. An existing common practice for solving regression problems is the mean square error (MSE) minimization approach or its regularized variants which require prior knowledge about the models. Recently, Yi et al., proposed a mutual information based supervised learning framework where they introduced a label entropy regularization which does not require any prior knowledge. When applied to classification tasks and solved via a stochastic gradient descent (SGD) optimization algorithm, their approach achieved significant improvement over the commonly used cross entropy loss and its variants. However, they did not provide a theoretical convergence analysis of the SGD algorithm for the proposed formulation. Besides, applying the framework to regression tasks is nontrivial due to the potentially infinite support set of the label. In this paper, we investigate the regression under the mutual information based supervised learning framework. We first argue that the MSE minimization approach is equivalent to a conditional entropy learning problem, and then propose a mutual information learning formulation for solving regression problems by using a reparameterization technique. For the proposed formulation, we give the convergence analysis of the SGD algorithm for solving it in practice. Finally, we consider a multi-output regression data model where we derive the generalization performance lower bound in terms of the mutual information associated with the underlying data distribution. The result shows that the high dimensionality can be a bless instead of a curse, which is controlled by a threshold. We hope our work will serve as a good starting point for further research on the mutual information based regression.

Mutual Information Learned Classifiers: an Information-theoretic Viewpoint of Training Deep Learning Classification Systems

Deep learning systems have been reported to acheive state-of-the-art performances in many applications, and one of the keys for achieving this is the existence of well trained classifiers on benchmark datasets which can be used as backbone feature extractors in downstream tasks. As a main-stream loss function for training deep neural network (DNN) classifiers, the cross entropy loss can easily lead us to find models which demonstrate severe overfitting behavior when no other techniques are used for alleviating it such as data augmentation. In this paper, we prove that the existing cross entropy loss minimization for training DNN classifiers essentially learns the conditional entropy of the underlying data distribution of the dataset, i.e., the information or uncertainty remained in the labels after revealing the input. In this paper, we propose a mutual information learning framework where we train DNN classifiers via learning the mutual information between the label and input. Theoretically, we give the population error probability lower bound in terms of the mutual information. In addition, we derive the mutual information lower and upper bounds for a concrete binary classification data model in $\mbR^n$, and also the error probability lower bound in this scenario. Besides, we establish the sample complexity for accurately learning the mutual information from empirical data samples drawn from the underlying data distribution. Empirically, we conduct extensive experiments on several benchmark datasets to support our theory. Without whistles and bells, the proposed mutual information learned classifiers (MILCs) acheive far better generalization performances than the state-of-the-art classifiers with an improvement which can exceed more than 10\% in testing accuracy.

Exact Recovery in the General Hypergraph Stochastic Block Model

This paper investigates fundamental limits of exact recovery in the general d-uniform hypergraph stochastic block model (d-HSBM), wherein n nodes are partitioned into k disjoint communities with relative sizes (p1,..., pk). Each subset of nodes with cardinality d is generated independently as an order-d hyperedge with a certain probability that depends on the ground-truth communities that the d nodes belong to. The goal is to exactly recover the k hidden communities based on the observed hypergraph. We show that there exists a sharp threshold such that exact recovery is achievable above the threshold and impossible below the threshold (apart from a small regime of parameters that will be specified precisely). This threshold is represented in terms of a quantity which we term as the generalized Chernoff-Hellinger divergence between communities. Our result for this general model recovers prior results for the standard SBM and d-HSBM with two symmetric communities as special cases. En route to proving our achievability results, we develop a polynomial-time two-stage algorithm that meets the threshold. The first stage adopts a certain hypergraph spectral clustering method to obtain a coarse estimate of communities, and the second stage refines each node individually via local refinement steps to ensure exact recovery.

MC2G: An Efficient Algorithm for Matrix Completion with Social and Item Similarity Graphs

We consider a discrete-valued matrix completion problem for recommender systems in which both the social and item similarity graphs are available as side information. We develop and analyze MC2G (Matrix Completion with 2 Graphs), a quasilinear-time algorithm which is based on spectral clustering and local refinement steps. We show that the sample complexity of MC2G meets an information-theoretic limit that is derived using maximum likelihood estimation and is also order-optimal. We demonstrate that having both graphs as side information outperforms having just a single graph, thus the availability of two graphs results in a synergistic effect. Experiments on synthetic datasets corroborate our theoretical results. Finally, experiments on a sub-sampled version of the Netflix dataset show that MC2G significantly outperforms other state-of-the-art matrix completion algorithms that leverage graph side information.

Optimal Resolution of Change-Point Detection with Empirically Observed Statistics and Erasures

This paper revisits the offline change-point detection problem from a statistical learning perspective. Instead of assuming that the underlying pre- and post-change distributions are known, it is assumed that we have partial knowledge of these distributions based on empirically observed statistics in the form of training sequences. Our problem formulation finds a variety of real-life applications from detecting when climate change occurred to detecting when a virus mutated. Using the training sequences as well as the test sequence consisting of a single-change and allowing for the erasure or rejection option, we derive the optimal resolution between the estimated and true change-points under two different asymptotic regimes on the undetected error probability---namely, the large and moderate deviations regimes. In both regimes, strong converses are also proved. In the moderate deviations case, the optimal resolution is a simple function of a symmetrized version of the chi-square distance.