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Deep Neural Networks (DNNs) trained for classification tasks are vulnerable to adversarial attacks. But not all the classes are equally vulnerable. Adversarial training does not make all classes or groups equally robust as well. For example, in classification tasks with long-tailed distributions, classes are asymmetrically affected during adversarial training, with lower robust accuracy for less frequent classes. In this regard, we propose a provable robustness method by leveraging the continuous piecewise-affine (CPA) nature of DNNs. Our method can impose linearity constraints on the decision boundary, as well as the DNN CPA partition, without requiring any adversarial training. Using such constraints, we show that the margin between the decision boundary and minority classes can be increased in a provable manner. We also present qualitative and quantitative validation of our method for class-specific robustness. Our code is available at https: //github.com/Josuelmet/CROP 

Current Deep Network (DN) visualization and inter-pretability methods rely heavily on data space visualizations such as scoring which dimensions of the data are responsible for their associated prediction or generating new data features or samples that best match a given DN unit or representation. In this paper, we go one step further by developing the first provably exact method for computing the geometry of a DN's mapping - including its decision boundary - over a specified region of the data space. By lever-aging the theory of Continuous Piece- Wise Linear (CPWL) spline DNs, SplineCam exactly computes a DN's geometry without resorting to approximations such as sampling or architecture simplification. SplineCam applies to any DN architecture based on CPWL activation nonlinearities, including (leaky) ReLU, absolute value, maxout, and max-pooling and can also be applied to regression DNs such as implicit neural representations. Beyond decision boundary visualization and characterization, SplineCam enables one to compare architectures, measure generalizability, and sample from the decision boundary on or off the data manifold. Project website: bit.ly/splinecam. 

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. 

We present Polarity Sampling, a theoretically justified plug-and-play method for controlling the generation quality and diversity of any pre-trained deep generative network (DGN). Leveraging the fact that DGNs are, or can be approximated by, continuous piecewise affine splines, we derive the analytical DGN output space distribution as a function of the product of the DGN's Jacobian singular values raised to a power rho. We dub rho the polarity parameter and prove that rho focuses the DGN sampling on the modes (rho< 0) or anti-modes (rho> 0) of the DGN output space probability distribution. We demonstrate that nonzero polarity values achieve a better precision-recall (quality-diversity) Pareto frontier than standard methods, such as truncation, for a number of state-of-the-art DGNs. We also present quantitative and qualitative results on the improvement of overall generation quality (eg, in terms of the Frechet Inception Distance) for a number of state-of-the-art DGNs, including StyleGAN3, BigGAN-deep, NVAE, for different conditional and unconditional image generation tasks. In particular, Polarity Sampling redefines the state-of-the-art for StyleGAN2 on the FFHQ Dataset to FID 2.57, StyleGAN2 on the LSUN Car Dataset to FID 2.27 and StyleGAN3 on the AFHQv2 Dataset to FID 3.95. Colab Demo: bit. ly/polarity-samp 

Computing or approximating the convex hull of a dataset plays a role in a wide range of applications, including economics, statistics, and physics, to name just a few. However, convex hull computation and approximation is exponentially complex, in terms of both memory and computation, as the ambient space dimension increases. In this paper, we propose DeepHull, a new convex hull approximation algorithm based on convex deep networks (DNs) with continuous piecewise-affine nonlinearities and nonnegative weights. The idea is that binary classification between true data samples and adversarially generated samples with such a DN naturally induces a polytope decision boundary that approximates the true data convex hull. A range of exploratory experiments demonstrates that DeepHull efficiently produces a meaningful convex hull approximation, even in a high-dimensional ambient space. 

Recurrent Neural Networks (RNNs) are important tools for processing sequential data such as time-series or video. Interpretability is defined as the ability to be understood by a person and is different from explainability, which is the ability to be explained in a mathematical formulation. A key interpretability issue with RNNs is that it is not clear how each hidden state per time step contributes to the decision-making process in a quantitative manner. We propose NeuroView-RNN as a family of new RNN architectures that explains how all the time steps are used for the decision-making process. Each member of the family is derived from a standard RNN architecture by concatenation of the hidden steps into a global linear classifier. The global linear classifier has all the hidden states as the input, so the weights of the classifier have a linear mapping to the hidden states. Hence, from the weights, NeuroView-RNN can quantify how important each time step is to a particular decision. As a bonus, NeuroView-RNN also offers higher accuracy in many cases compared to the RNNs and their variants. We showcase the benefits of NeuroView-RNN by evaluating on a multitude of diverse time-series datasets. 

Deep Generative Networks (DGNs) are extensively employed in Generative Adversarial Networks (GANs), Variational Autoencoders (VAEs), and their variants to approximate the data manifold and distribution. However, training samples are often distributed non-uniformly on the manifold, due to the cost or convenience of collection. For example, the CelebA dataset contains a large fraction of smiling faces. These inconsistencies will be reproduced when sampling from the trained DGN, which is not always preferred, e.g., for fairness or data augmentation. In response, we develop MaGNET, a novel and theoretically motivated latent space sampler for any pre-trained DGN that produces samples uniformly distributed on the learned manifold. We perform a range of experiments on several datasets and DGNs, e.g., for the state-of-the-art StyleGAN2 trained on the FFHQ dataset, uniform sampling via MaGNET increases distribution precision by 4.1% and recall by 3.0% and decreases gender bias by 41.2%, without requiring labels or retraining. Since uniform sample distribution does not imply uniform semantic distribution, we also explore how semantic attributes of generated samples vary under MaGNET sampling. Colab and codes at bit.ly/magnet-sampling 

The study of deep neural networks (DNNs) in the infinite-width limit, via the so-called neural tangent kernel (NTK) approach, has provided new insights into the dynamics of learning, generalization, and the impact of initialization. One key DNN architecture remains to be kernelized, namely, the recurrent neural network (RNN). In this paper we introduce and study the Recurrent Neural Tangent Kernel (RNTK), which provides new insights into the behavior of overparametrized RNNs. A key property of the RNTK should greatly benefit practitioners is its ability to compare inputs of different length. To this end, we characterize how the RNTK weights different time steps to form its output under different initialization parameters and nonlinearity choices. A synthetic and 56 real-world data experiments demonstrate that the RNTK offers significant performance gains over other kernels, including standard NTKs, across a wide array of data sets. 

Deep Generative Networks (DGNs) with probabilistic modeling of their output and latent space are currently trained via Variational Autoencoders (VAEs). In the absence of a known analytical form for the posterior and likelihood expectation, VAEs resort to approximations, including (Amortized) Variational Inference (AVI) and Monte-Carlo (MC) sampling. We exploit the Continuous Piecewise Affine (CPA) property of modern DGNs to derive their posterior and marginal distributions as well as the latter's first moments. These findings enable us to derive an analytical Expectation-Maximization (EM) algorithm that enables gradient-free DGN learning. We demonstrate empirically that EM training of DGNs produces greater likelihood than VAE training. Our findings will guide the design of new VAE AVI that better approximate the true posterior and open avenues to apply standard statistical tools for model comparison, anomaly detection, and missing data imputation. 

Abstract The continuously growing amount of seismic data collected worldwide is outpacing our abilities for analysis, since to date, such datasets have been analyzed in a human-expert-intensive, supervised fashion. Moreover, analyses that are conducted can be strongly biased by the standard models employed by seismologists. In response to both of these challenges, we develop a new unsupervised machine learning framework for detecting and clustering seismic signals in continuous seismic records. Our approach combines a deep scattering network and a Gaussian mixture model to cluster seismic signal segments and detect novel structures. To illustrate the power of the framework, we analyze seismic data acquired during the June 2017 Nuugaatsiaq, Greenland landslide. We demonstrate the blind detection and recovery of the repeating precursory seismicity that was recorded before the main landslide rupture, which suggests that our approach could lead to more informative forecasting of the seismic activity in seismogenic areas.