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DeepTensor is a computationally efficient framework for low-rank decomposition of matrices and tensors using deep generative networks. We decompose a tensor as the product of low-rank tensor factors where each low-rank tensor is generated by a deep network (DN) that is trained in a self-supervised manner to minimize the mean-square approximation error. Our key observation is that the implicit regularization inherent in DNs enables them to capture nonlinear signal structures that are out of the reach of classical linear methods like the singular value decomposition (SVD) and principal components analysis (PCA). We demonstrate that the performance of DeepTensor is robust to a wide range of distributions and a computationally efficient drop-in replacement for the SVD, PCA, nonnegative matrix factorization (NMF), and similar decompositions by exploring a range of real-world applications, including hyperspectral image denoising, 3D MRI tomography, and image classification. 

We take a random matrix theory approach to random sketching and show an asymptotic first-order equivalence of the regularized sketched pseudoinverse of a positive semidefinite matrix to a certain evaluation of the resolvent of the same matrix. We focus on real-valued regularization and extend previous results on an asymptotic equivalence of random matrices to the real setting, providing a precise characterization of the equivalence even under negative regularization, including a precise characterization of the smallest nonzero eigenvalue of the sketched matrix. We then further characterize the second-order equivalence of the sketched pseudoinverse. We also apply our results to the analysis of the sketch-and-project method and to sketched ridge regression. Last, we prove that these results generalize to asymptotically free sketching matrices, obtaining the resulting equivalence for orthogonal sketching matrices and comparing our results to several common sketches used in practice. 

The first step toward investigating the effectiveness of a treatment via a randomized trial is to split the population into control and treatment groups then compare the average response of the treatment group receiving the treatment to the control group receiving the placebo. To ensure that the difference between the two groups is caused only by the treatment, it is crucial that the control and the treatment groups have similar statistics. Indeed, the validity and reliability of a trial are determined by the similarity of two groups’ statistics. Covariate balancing methods increase the similarity between the distributions of the two groups’ covariates. However, often in practice, there are not enough samples to accurately estimate the groups’ covariate distributions. In this article, we empirically show that covariate balancing with the standardized means difference (SMD) covariate balancing measure, as well as Pocock and Simon’s sequential treatment assignment method, are susceptible to worst case treatment assignments. Worst case treatment assignments are those admitted by the covariate balance measure, but result in highest possible ATE estimation errors. We developed an adversarial attack to find adversarial treatment assignment for any given trial. Then, we provide an index to measure how close the given trial is to the worst case. To this end, we provide an optimization-based algorithm, namely adversarial treatment assignment in treatment effect trials (ATASTREET), to find the adversarial treatment assignments. 

Abstract We develop new theoretical results on matrix perturbation to shed light on the impact of architecture on the performance of a deep network. In particular, we explain analytically what deep learning practitioners have long observed empirically: the parameters of some deep architectures (e.g., residual networks, ResNets, and Dense networks, DenseNets) are easier to optimize than others (e.g., convolutional networks, ConvNets). Building on our earlier work connecting deep networks with continuous piecewise-affine splines, we develop an exact local linear representation of a deep network layer for a family of modern deep networks that includes ConvNets at one end of a spectrum and ResNets, DenseNets, and other networks with skip connections at the other. For regression and classification tasks that optimize the squared-error loss, we show that the optimization loss surface of a modern deep network is piecewise quadratic in the parameters, with local shape governed by the singular values of a matrix that is a function of the local linear representation. We develop new perturbation results for how the singular values of matrices of this sort behave as we add a fraction of the identity and multiply by certain diagonal matrices. A direct application of our perturbation results explains analytically why a network with skip connections (such as a ResNet or DenseNet) is easier to optimize than a ConvNet: thanks to its more stable singular values and smaller condition number, the local loss surface of such a network is less erratic, less eccentric, and features local minima that are more accommodating to gradient-based optimization. Our results also shed new light on the impact of different nonlinear activation functions on a deep network’s singular values, regardless of its architecture. 

Is overparameterization a privacy liability? In this work, we study the effect that the number of parameters has on a classifier's vulnerability to membership inference attacks. We first demonstrate how the number of parameters of a model can induce a privacy--utility trade-off: increasing the number of parameters generally improves generalization performance at the expense of lower privacy. However, remarkably, we then show that if coupled with proper regularization, increasing the number of parameters of a model can actually simultaneously increase both its privacy and performance, thereby eliminating the privacy--utility trade-off. Theoretically, we demonstrate this curious phenomenon for logistic regression with ridge regularization in a bi-level feature ensemble setting. Pursuant to our theoretical exploration, we develop a novel leave-one-out analysis tool to precisely characterize the vulnerability of a linear classifier to the optimal membership inference attack. We empirically exhibit this "blessing of dimensionality" for neural networks on a variety of tasks using early stopping as the regularizer.