Spectral clustering methods are widely used for detecting clusters in networks for community detection, while a small change on the graph Laplacian matrix could bring a dramatic improvement. In this paper, we propose a dual regularized graph Laplacian matrix and then employ it to three classical spectral clustering approaches under the degree-corrected stochastic block model. If the number of communities is known as $K$, we consider more than $K$ leading eigenvectors and weight them by their corresponding eigenvalues in the spectral clustering procedure to improve the performance. Three improved spectral clustering methods are dual regularized spectral clustering (DRSC) method, dual regularized spectral clustering on Ratios-of-eigenvectors (DRSCORE) method, and dual regularized symmetrized Laplacian inverse matrix (DRSLIM) method. Theoretical analysis of DRSC and DRSLIM show that under mild conditions DRSC and DRSLIM yield stable consistent community detection, moreover, DRSCORE returns perfect clustering under the ideal case. We compare the performances of DRSC, DRSCORE and DRSLIM with several spectral methods by substantial simulated networks and eight real-world networks.

Based on the classical Degree Corrected Stochastic Blockmodel (DCSBM) model for network community detection problem, we propose two novel approaches: principal component clustering (PCC) and normalized principal component clustering (NPCC). Without any parameters to be estimated, the PCC method is simple to be implemented. Under mild conditions, we show that PCC yields consistent community detection. NPCC is designed based on the combination of the PCC and the RSC method (Qin & Rohe 2013). Population analysis for NPCC shows that NPCC returns perfect clustering for the ideal case under DCSBM. PCC and NPCC is illustrated through synthetic and real-world datasets. Numerical results show that NPCC provides a significant improvement compare with PCC and RSC. Moreover, NPCC inherits nice properties of PCC and RSC such that NPCC is insensitive to the number of eigenvectors to be clustered and the choosing of the tuning parameter. When dealing with two weak signal networks Simmons and Caltech, by considering one more eigenvectors for clustering, we provide two refinements PCC+ and NPCC+ of PCC and NPCC, respectively. Both two refinements algorithms provide improvement performances compared with their original algorithms. Especially, NPCC+ provides satisfactory performances on Simmons and Caltech, with error rates of 121/1137 and 96/590, respectively.

The estimation of covariance matrices of multiple classes with limited training data is a difficult problem. The sample covariance matrix (SCM) is known to perform poorly when the number of variables is large compared to the available number of samples. In order to reduce the mean squared error (MSE) of the SCM, regularized (shrinkage) SCM estimators are often used. In this work, we consider regularized SCM (RSCM) estimators for multiclass problems that couple together two different target matrices for regularization: the pooled (average) SCM of the classes and the scaled identity matrix. Regularization toward the pooled SCM is beneficial when the population covariances are similar, whereas regularization toward the identity matrix guarantees that the estimators are positive definite. We derive the MSE optimal tuning parameters for the estimators as well as propose a method for their estimation under the assumption that the class populations follow (unspecified) elliptical distributions with finite fourth-order moments. The MSE performance of the proposed coupled RSCMs are evaluated with simulations and in a regularized discriminant analysis (RDA) classification set-up on real data. The results based on three different real data sets indicate comparable performance to cross-validation but with a significant speed-up in computation time.

We propose a greedy strategy to spectrally train a deep network for multi-class classification. Each layer is defined as a composition of linear weights with the feature map of a Gaussian kernel acting as the activation function. At each layer, the linear weights are learned by maximizing the dependence between the layer output and the labels using the Hilbert Schmidt Independence Criterion (HSIC). By constraining the solution space on the Stiefel Manifold, we demonstrate how our network construct (Kernel Dependence Network or KNet) can be solved spectrally while leveraging the eigenvalues to automatically find the width and the depth of the network. We theoretically guarantee the existence of a solution for the global optimum while providing insight into our network's ability to generalize. This workshop paper is only part one of the full paper. For the full paper, see https://arxiv.org/abs/2006.08539.

The framework of differential privacy (DP) upper bounds the information disclosure risk involved in using sensitive datasets for statistical analysis. A DP mechanism typically operates by adding carefully calibrated noise to the data release procedure. Generalized linear models (GLMs) are among the most widely used arms in data analyst's repertoire. In this work, with logistic and Poisson regression as running examples, we propose a generic noise-aware Bayesian framework to quantify the parameter uncertainty for a GLM at hand, given noisy sufficient statistics. We perform a tight privacy analysis and experimentally demonstrate that the posteriors obtained from our model, while adhering to strong privacy guarantees, are similar to the non-private posteriors.

We present a probabilistic programmed deep kernel learning approach to personalized, predictive modeling of neurodegenerative diseases. Our analysis considers a spectrum of neural and symbolic machine learning approaches, which we assess for predictive performance and important medical AI properties such as interpretability, uncertainty reasoning, data-efficiency, and leveraging domain knowledge. Our Bayesian approach combines the flexibility of Gaussian processes with the structural power of neural networks to model biomarker progressions, without needing clinical labels for training. We run evaluations on the problem of Alzheimer's disease prediction, yielding results surpassing deep learning and with the practical advantages of Bayesian non-parametrics and probabilistic programming.

For sake of reliability, it is necessary for models in real-world applications to be both powerful and globally interpretable. Simple classifiers, e.g., Logistic Regression (LR), are globally interpretable, but not powerful enough to model complex nonlinear interactions among features in tabular data. Meanwhile, Deep Neural Networks (DNNs) have shown great effectiveness for modeling tabular data, but is not globally interpretable. In this work, we find local piece-wise interpretations in DNN of a specific feature are usually inconsistent in different samples, which is caused by feature interactions in the hidden layers. Accordingly, we can design an automatic feature crossing method to find feature interactions in DNN, and use them as cross features in LR. We give definition of the interpretation inconsistency in DNN, based on which a novel feature crossing method called DNN2LR is proposed. Extensive experiments have been conducted on four public datasets and two real-world datasets. The final model, i.e., a LR model empowered with cross features, generated by DNN2LR can outperform the complex DNN model, as well as several state-of-the-art feature crossing methods. The experimental results strongly verify the effectiveness and efficiency of DNN2LR, especially on real-world datasets with large numbers of feature fields.

Domain Adaptation (DA) enables transferring a learning machine from a labeled source domain to an unlabeled target one. While remarkable advances have been made, most of the existing DA methods focus on improving the target accuracy at inference. How to estimate the predictive uncertainty of DA models is vital for decision-making in safety-critical scenarios but remains the boundary to explore. In this paper, we delve into the open problem of Calibration in DA, which is extremely challenging due to the coexistence of domain shift and the lack of target labels. We first reveal the dilemma that DA models learn higher accuracy at the expense of well-calibrated probabilities. Driven by this finding, we propose Transferable Calibration (TransCal) to achieve more accurate calibration with lower bias and variance in a unified hyperparameter-free optimization framework. As a general post-hoc calibration method, TransCal can be easily applied to recalibrate existing DA methods. Its efficacy has been justified both theoretically and empirically.

Scientific observations may consist of a large number of variables (features). Identifying a subset of meaningful features is often ignored in unsupervised learning, despite its potential for unraveling clear patterns hidden in the ambient space. In this paper, we present a method for unsupervised feature selection, and we demonstrate its use for the task of clustering. We propose a differentiable loss function that combines the Laplacian score, which favors low-frequency features, with a gating mechanism for feature selection. We improve the Laplacian score, by replacing it with a gated variant computed on a subset of features. This subset is obtained using a continuous approximation of Bernoulli variables whose parameters are trained to gate the full feature space. We mathematically motivate the proposed approach and demonstrate that in the high noise regime, it is crucial to compute the Laplacian on the gated inputs, rather than on the full feature set. Experimental demonstration of the efficacy of the proposed approach and its advantage over current baselines is provided using several real-world examples.

Random Reshuffling (RR) is an algorithm for minimizing finite-sum functions that utilizes iterative gradient descent steps in conjunction with data reshuffling. Often contrasted with its sibling Stochastic Gradient Descent (SGD), RR is usually faster in practice and enjoys significant popularity in convex and non-convex optimization. The convergence rate of RR has attracted substantial attention recently and, for strongly convex and smooth functions, it was shown to converge faster than SGD if 1) the stepsize is small, 2) the gradients are bounded, and 3) the number of epochs is large. We remove these 3 assumptions, improve the dependence on the condition number from $\kappa^2$ to $\kappa$ (resp. from $\kappa$ to $\sqrt{\kappa}$) and, in addition, show that RR has a different type of variance. We argue through theory and experiments that the new variance type gives an additional justification of the superior performance of RR. To go beyond strong convexity, we present several results for non-strongly convex and non-convex objectives. We show that in all cases, our theory improves upon existing literature. Finally, we prove fast convergence of the Shuffle-Once (SO) algorithm, which shuffles the data only once, at the beginning of the optimization process. Our theory for strongly-convex objectives tightly matches the known lower bounds for both RR and SO and substantiates the common practical heuristic of shuffling once or only a few times. As a byproduct of our analysis, we also get new results for the Incremental Gradient algorithm (IG), which does not shuffle the data at all.