Stanley J. Osher

David J. Arnold, Dayne Fernandez, Ruizhe Jia et al.

National parks often serve as hotspots for environmental crime such as illegal deforestation and animal poaching. Previous attempts to model environmental crime were either discrete and network-based or required very restrictive assumptions on the geometry of the protected region and made heavy use of radial symmetry. We formulate a level set method to track criminals inside a protected region which uses real elevation data to determine speed of travel, does not require any assumptions of symmetry, and can be applied to regions of arbitrary shape. In doing so, we design a Hamilton-Jacobi equation to describe movement of criminals while also incorporating the effects of patrollers who attempt to deter the crime. We discuss the numerical schemes that we use to solve this Hamilton-Jacobi equation. Finally, we apply our method to Yosemite National Park and Kangaroo Island, Australia and design practical patrol strategies with the goal of minimizing the area that is affected by criminal activity.

Tan Nguyen, Richard G. Baraniuk, Robert M. Kirby et al.

Transformers have achieved remarkable success in sequence modeling and beyond but suffer from quadratic computational and memory complexities with respect to the length of the input sequence. Leveraging techniques include sparse and linear attention and hashing tricks; efficient transformers have been proposed to reduce the quadratic complexity of transformers but significantly degrade the accuracy. In response, we first interpret the linear attention and residual connections in computing the attention map as gradient descent steps. We then introduce momentum into these components and propose the \emph{momentum transformer}, which utilizes momentum to improve the accuracy of linear transformers while maintaining linear memory and computational complexities. Furthermore, we develop an adaptive strategy to compute the momentum value for our model based on the optimal momentum for quadratic optimization. This adaptive momentum eliminates the need to search for the optimal momentum value and further enhances the performance of the momentum transformer. A range of experiments on both autoregressive and non-autoregressive tasks, including image generation and machine translation, demonstrate that the momentum transformer outperforms popular linear transformers in training efficiency and accuracy.

Liu Yang, Siting Liu, Tingwei Meng et al.

This paper introduces a new neural-network-based approach, namely In-Context Operator Networks (ICON), to simultaneously learn operators from the prompted data and apply it to new questions during the inference stage, without any weight update. Existing methods are limited to using a neural network to approximate a specific equation solution or a specific operator, requiring retraining when switching to a new problem with different equations. By training a single neural network as an operator learner, we can not only get rid of retraining (even fine-tuning) the neural network for new problems, but also leverage the commonalities shared across operators so that only a few demos in the prompt are needed when learning a new operator. Our numerical results show the neural network's capability as a few-shot operator learner for a diversified type of differential equation problems, including forward and inverse problems of ordinary differential equations (ODEs), partial differential equations (PDEs), and mean-field control (MFC) problems, and also show that it can generalize its learning capability to operators beyond the training distribution.

Michael A. Puthawala, Cory D. Hauck, Stanley J. Osher

We investigate overdetermined linear inverse problems for which the forward operator may not be given accurately. We introduce a new tool called the structure, based on the Wasserstein distance, and propose the use of this to diagnose and remedy forward operator error. Computing the structure turns out to use an easy calculation for a Euclidean homogeneous degree one distance, the Earth Mover's Distance, based on recently developed algorithms. The structure is proven to distinguish between noise and signals in the residual and gives a plan to help recover the true direct operator in some interesting cases. We expect to use this technique not only to diagnose the error, but also to correct it, which we do in some simple cases presented below.

Justin Baker, Hedi Xia, Yiwei Wang et al.

Learning neural ODEs often requires solving very stiff ODE systems, primarily using explicit adaptive step size ODE solvers. These solvers are computationally expensive, requiring the use of tiny step sizes for numerical stability and accuracy guarantees. This paper considers learning neural ODEs using implicit ODE solvers of different orders leveraging proximal operators. The proximal implicit solver consists of inner-outer iterations: the inner iterations approximate each implicit update step using a fast optimization algorithm, and the outer iterations solve the ODE system over time. The proximal implicit ODE solver guarantees superiority over explicit solvers in numerical stability and computational efficiency. We validate the advantages of proximal implicit solvers over existing popular neural ODE solvers on various challenging benchmark tasks, including learning continuous-depth graph neural networks and continuous normalizing flows.

Tan Nguyen, Minh Pham, Tam Nguyen et al.

Multi-head attention empowers the recent success of transformers, the state-of-the-art models that have achieved remarkable success in sequence modeling and beyond. These attention mechanisms compute the pairwise dot products between the queries and keys, which results from the use of unnormalized Gaussian kernels with the assumption that the queries follow a mixture of Gaussian distribution. There is no guarantee that this assumption is valid in practice. In response, we first interpret attention in transformers as a nonparametric kernel regression. We then propose the FourierFormer, a new class of transformers in which the dot-product kernels are replaced by the novel generalized Fourier integral kernels. Different from the dot-product kernels, where we need to choose a good covariance matrix to capture the dependency of the features of data, the generalized Fourier integral kernels can automatically capture such dependency and remove the need to tune the covariance matrix. We theoretically prove that our proposed Fourier integral kernels can efficiently approximate any key and query distributions. Compared to the conventional transformers with dot-product attention, FourierFormers attain better accuracy and reduce the redundancy between attention heads. We empirically corroborate the advantages of FourierFormers over the baseline transformers in a variety of practical applications including language modeling and image classification.

Alex Tong Lin, Adrian S. Wong, Robert Martin et al.

We provide a method to identify system parameters of dynamical systems, called ID-ODE -- Inference by Differentiation and Observing Delay Embeddings. In this setting, we are given a dataset of trajectories from a dynamical system with system parameter labels. Our goal is to identify system parameters of new trajectories. The given trajectories may or may not encompass the full state of the system, and we may only observe a one-dimensional time series. In the latter case, we reconstruct the full state by using delay embeddings, and under sufficient conditions, Taken's Embedding Theorem assures us the reconstruction is diffeomorphic to the original. This allows our method to work on time series. Our method works by first learning the velocity operator (as given or reconstructed) with a neural network having both state and system parameters as variable inputs. Then on new trajectories we backpropagate prediction errors to the system parameter inputs giving us a gradient. We then use gradient descent to infer the correct system parameter. We demonstrate the efficacy of our approach on many numerical examples: the Lorenz system, Lorenz96, Lotka-Volterra Predator-Prey, and the Compound Double Pendulum. We also apply our algorithm on a real-world dataset: propulsion of the Hall-effect Thruster (HET).

Liu Yang, Siting Liu, Stanley J. Osher

In the growing domain of scientific machine learning, in-context operator learning has shown notable potential in building foundation models, as in this framework the model is trained to learn operators and solve differential equations using prompted data, during the inference stage without weight updates. However, the current model's overdependence on function data overlooks the invaluable human insight into the operator. To address this, we present a transformation of in-context operator learning into a multi-modal paradigm. In particular, we take inspiration from the recent success of large language models, and propose using "captions" to integrate human knowledge about the operator, expressed through natural language descriptions and equations. Also, we introduce a novel approach to train a language-model-like architecture, or directly fine-tune existing language models, for in-context operator learning. We beat the baseline on single-modal learning tasks, and also demonstrated the effectiveness of multi-modal learning in enhancing performance and reducing function data requirements. The proposed method not only significantly enhanced the development of the in-context operator learning paradigm, but also created a new path for the application of language models.

Hedi Xia, Vai Suliafu, Hangjie Ji et al.

We propose heavy ball neural ordinary differential equations (HBNODEs), leveraging the continuous limit of the classical momentum accelerated gradient descent, to improve neural ODEs (NODEs) training and inference. HBNODEs have two properties that imply practical advantages over NODEs: (i) The adjoint state of an HBNODE also satisfies an HBNODE, accelerating both forward and backward ODE solvers, thus significantly reducing the number of function evaluations (NFEs) and improving the utility of the trained models. (ii) The spectrum of HBNODEs is well structured, enabling effective learning of long-term dependencies from complex sequential data. We verify the advantages of HBNODEs over NODEs on benchmark tasks, including image classification, learning complex dynamics, and sequential modeling. Our method requires remarkably fewer forward and backward NFEs, is more accurate, and learns long-term dependencies more effectively than the other ODE-based neural network models. Code is available at \url{https://github.com/hedixia/HeavyBallNODE}.

Tan M. Nguyen, Richard G. Baraniuk, Andrea L. Bertozzi et al.

Designing deep neural networks is an art that often involves an expensive search over candidate architectures. To overcome this for recurrent neural nets (RNNs), we establish a connection between the hidden state dynamics in an RNN and gradient descent (GD). We then integrate momentum into this framework and propose a new family of RNNs, called {\em MomentumRNNs}. We theoretically prove and numerically demonstrate that MomentumRNNs alleviate the vanishing gradient issue in training RNNs. We study the momentum long-short term memory (MomentumLSTM) and verify its advantages in convergence speed and accuracy over its LSTM counterpart across a variety of benchmarks. We also demonstrate that MomentumRNN is applicable to many types of recurrent cells, including those in the state-of-the-art orthogonal RNNs. Finally, we show that other advanced momentum-based optimization methods, such as Adam and Nesterov accelerated gradients with a restart, can be easily incorporated into the MomentumRNN framework for designing new recurrent cells with even better performance. The code is available at https://github.com/minhtannguyen/MomentumRNN.

Thu Dinh, Bao Wang, Andrea L. Bertozzi et al.

Deep neural nets (DNNs) compression is crucial for adaptation to mobile devices. Though many successful algorithms exist to compress naturally trained DNNs, developing efficient and stable compression algorithms for robustly trained DNNs remains widely open. In this paper, we focus on a co-design of efficient DNN compression algorithms and sparse neural architectures for robust and accurate deep learning. Such a co-design enables us to advance the goal of accommodating both sparsity and robustness. With this objective in mind, we leverage the relaxed augmented Lagrangian based algorithms to prune the weights of adversarially trained DNNs, at both structured and unstructured levels. Using a Feynman-Kac formalism principled robust and sparse DNNs, we can at least double the channel sparsity of the adversarially trained ResNet20 for CIFAR10 classification, meanwhile, improve the natural accuracy by $8.69$\% and the robust accuracy under the benchmark $20$ iterations of IFGSM attack by $5.42$\%. The code is available at \url{https://github.com/BaoWangMath/rvsm-rgsm-admm}.

Bao Wang, Stanley J. Osher

Improving the accuracy and robustness of deep neural nets (DNNs) and adapting them to small training data are primary tasks in deep learning research. In this paper, we replace the output activation function of DNNs, typically the data-agnostic softmax function, with a graph Laplacian-based high dimensional interpolating function which, in the continuum limit, converges to the solution of a Laplace-Beltrami equation on a high dimensional manifold. Furthermore, we propose end-to-end training and testing algorithms for this new architecture. The proposed DNN with graph interpolating activation integrates the advantages of both deep learning and manifold learning. Compared to the conventional DNNs with the softmax function as output activation, the new framework demonstrates the following major advantages: First, it is better applicable to data-efficient learning in which we train high capacity DNNs without using a large number of training data. Second, it remarkably improves both natural accuracy on the clean images and robust accuracy on the adversarial images crafted by both white-box and black-box adversarial attacks. Third, it is a natural choice for semi-supervised learning. For reproducibility, the code is available at \url{https://github.com/BaoWangMath/DNN-DataDependentActivation}.

Bao Wang, Quanquan Gu, March Boedihardjo et al.

Machine learning (ML) models trained by differentially private stochastic gradient descent (DP-SGD) have much lower utility than the non-private ones. To mitigate this degradation, we propose a DP Laplacian smoothing SGD (DP-LSSGD) to train ML models with differential privacy (DP) guarantees. At the core of DP-LSSGD is the Laplacian smoothing, which smooths out the Gaussian noise used in the Gaussian mechanism. Under the same amount of noise used in the Gaussian mechanism, DP-LSSGD attains the same DP guarantee, but in practice, DP-LSSGD makes training both convex and nonconvex ML models more stable and enables the trained models to generalize better. The proposed algorithm is simple to implement and the extra computational complexity and memory overhead compared with DP-SGD are negligible. DP-LSSGD is applicable to train a large variety of ML models, including DNNs. The code is available at \url{https://github.com/BaoWangMath/DP-LSSGD}.

Bao Wang, Binjie Yuan, Zuoqiang Shi et al.

Empirical adversarial risk minimization (EARM) is a widely used mathematical framework to robustly train deep neural nets (DNNs) that are resistant to adversarial attacks. However, both natural and robust accuracies, in classifying clean and adversarial images, respectively, of the trained robust models are far from satisfactory. In this work, we unify the theory of optimal control of transport equations with the practice of training and testing of ResNets. Based on this unified viewpoint, we propose a simple yet effective ResNets ensemble algorithm to boost the accuracy of the robustly trained model on both clean and adversarial images. The proposed algorithm consists of two components: First, we modify the base ResNets by injecting a variance specified Gaussian noise to the output of each residual mapping. Second, we average over the production of multiple jointly trained modified ResNets to get the final prediction. These two steps give an approximation to the Feynman-Kac formula for representing the solution of a transport equation with viscosity, or a convection-diffusion equation. For the CIFAR10 benchmark, this simple algorithm leads to a robust model with a natural accuracy of {\bf 85.62}\% on clean images and a robust accuracy of ${\bf 57.94 \%}$ under the 20 iterations of the IFGSM attack, which outperforms the current state-of-the-art in defending against IFGSM attack on the CIFAR10. Both natural and robust accuracies of the proposed ResNets ensemble can be improved dynamically as the building block ResNet advances. The code is available at: \url{https://github.com/BaoWangMath/EnResNet}.

Liu Yang, Stanley J. Osher

Can we build a single large model for a wide range of PDE-related scientific learning tasks? Can this model generalize to new PDEs, even of new forms, without any fine-tuning? In-context operator learning and the corresponding model In-Context Operator Networks (ICON) represent an initial exploration of these questions. The capability of ICON regarding the first question has been demonstrated previously. In this paper, we present a detailed methodology for solving PDE problems with ICON, and show how a single ICON model can make forward and reverse predictions for different equations with different strides, provided with appropriately designed data prompts. We show the positive evidence to the second question, i.e., ICON can generalize well to some PDEs with new forms without any fine-tuning. This is exemplified through a study on 1D scalar nonlinear conservation laws, a family of PDEs with temporal evolution. We also show how to broaden the range of problems that an ICON model can address, by transforming functions and equations to ICON's capability scope. We believe that the progress in this paper is a significant step towards the goal of training a foundation model for PDE-related tasks under the in-context operator learning framework.

Benjamin J. Zhang, Siting Liu, Wuchen Li et al.

We focus on the fundamental mathematical structure of score-based generative models (SGMs). We first formulate SGMs in terms of the Wasserstein proximal operator (WPO) and demonstrate that, via mean-field games (MFGs), the WPO formulation reveals mathematical structure that describes the inductive bias of diffusion and score-based models. In particular, MFGs yield optimality conditions in the form of a pair of coupled partial differential equations: a forward-controlled Fokker-Planck (FP) equation, and a backward Hamilton-Jacobi-Bellman (HJB) equation. Via a Cole-Hopf transformation and taking advantage of the fact that the cross-entropy can be related to a linear functional of the density, we show that the HJB equation is an uncontrolled FP equation. Second, with the mathematical structure at hand, we present an interpretable kernel-based model for the score function which dramatically improves the performance of SGMs in terms of training samples and training time. In addition, the WPO-informed kernel model is explicitly constructed to avoid the recently studied memorization effects of score-based generative models. The mathematical form of the new kernel-based models in combination with the use of the terminal condition of the MFG reveals new explanations for the manifold learning and generalization properties of SGMs, and provides a resolution to their memorization effects. Finally, our mathematically informed, interpretable kernel-based model suggests new scalable bespoke neural network architectures for high-dimensional applications.

Elisa Negrini, Yuxuan Liu, Liu Yang et al.

Neural networks are one tool for approximating non-linear differential equations used in scientific computing tasks such as surrogate modeling, real-time predictions, and optimal control. PDE foundation models utilize neural networks to train approximations to multiple differential equations simultaneously and are thus a general purpose solver that can be adapted to downstream tasks. Current PDE foundation models focus on either learning general solution operators and/or the governing system of equations, and thus only handle numerical or symbolic modalities. However, real-world applications may require more flexible data modalities, e.g. text analysis or descriptive outputs. To address this gap, we propose a novel multimodal deep learning approach that leverages a transformer-based architecture to approximate solution operators for a wide variety of ODEs and PDEs. Our method integrates numerical inputs, such as equation parameters and initial conditions, with text descriptions of physical processes or system dynamics. This enables our model to handle settings where symbolic representations may be incomplete or unavailable. In addition to providing accurate numerical predictions, our approach generates interpretable scientific text descriptions, offering deeper insights into the underlying dynamics and solution properties. The numerical experiments show that our model provides accurate solutions for in-distribution data (with average relative error less than 3.3%) and out-of-distribution data (average relative error less than 7.8%) together with precise text descriptions (with correct descriptions generated 100% of times). In certain tests, the model is also shown to be capable of extrapolating solutions in time.

Benjamin J. Zhang, Siting Liu, Stanley J. Osher et al.

In-context operator networks (ICON) are a class of operator learning methods based on the novel architectures of foundation models. Trained on a diverse set of datasets of initial and boundary conditions paired with corresponding solutions to ordinary and partial differential equations (ODEs and PDEs), ICON learns to map example condition-solution pairs of a given differential equation to an approximation of its solution operator. Here, we present a probabilistic framework that reveals ICON as implicitly performing Bayesian inference, where it computes the mean of the posterior predictive distribution over solution operators conditioned on the provided context, i.e., example condition-solution pairs. The formalism of random differential equations provides the probabilistic framework for describing the tasks ICON accomplishes while also providing a basis for understanding other multi-operator learning methods. This probabilistic perspective provides a basis for extending ICON to \emph{generative} settings, where one can sample from the posterior predictive distribution of solution operators. The generative formulation of ICON (GenICON) captures the underlying uncertainty in the solution operator, which enables principled uncertainty quantification in the solution predictions in operator learning.

Tan M. Nguyen, Tam Nguyen, Nhat Ho et al.

Self-attention is key to the remarkable success of transformers in sequence modeling tasks including many applications in natural language processing and computer vision. Like neural network layers, these attention mechanisms are often developed by heuristics and experience. To provide a principled framework for constructing attention layers in transformers, we show that the self-attention corresponds to the support vector expansion derived from a support vector regression problem, whose primal formulation has the form of a neural network layer. Using our framework, we derive popular attention layers used in practice and propose two new attentions: 1) the Batch Normalized Attention (Attention-BN) derived from the batch normalization layer and 2) the Attention with Scaled Head (Attention-SH) derived from using less training data to fit the SVR model. We empirically demonstrate the advantages of the Attention-BN and Attention-SH in reducing head redundancy, increasing the model's accuracy, and improving the model's efficiency in a variety of practical applications including image and time-series classification.

Raunak Manekar, Elisa Negrini, Minh Pham et al.

Phase retrieval (PR) is fundamentally important in scientific imaging and is crucial for nanoscale techniques like coherent diffractive imaging (CDI). Low radiation dose imaging is essential for applications involving radiation-sensitive samples. However, most PR methods struggle in low-dose scenarios due to high shot noise. Recent advancements in optical data acquisition setups, such as in-situ CDI, have shown promise for low-dose imaging, but they rely on a time series of measurements, making them unsuitable for single-image applications. Similarly, data-driven phase retrieval techniques are not easily adaptable to data-scarce situations. Zero-shot deep learning methods based on pre-trained and implicit generative priors have been effective in various imaging tasks but have shown limited success in PR. In this work, we propose low-dose deep image prior (LoDIP), which combines in-situ CDI with the power of implicit generative priors to address single-image low-dose phase retrieval. Quantitative evaluations demonstrate LoDIP's superior performance in this task and its applicability to real experimental scenarios.

Tam Nguyen, Tan M. Nguyen, Dung D. Le et al.

Multi-head attention is a driving force behind state-of-the-art transformers, which achieve remarkable performance across a variety of natural language processing (NLP) and computer vision tasks. It has been observed that for many applications, those attention heads learn redundant embedding, and most of them can be removed without degrading the performance of the model. Inspired by this observation, we propose Transformer with a Mixture of Gaussian Keys (Transformer-MGK), a novel transformer architecture that replaces redundant heads in transformers with a mixture of keys at each head. These mixtures of keys follow a Gaussian mixture model and allow each attention head to focus on different parts of the input sequence efficiently. Compared to its conventional transformer counterpart, Transformer-MGK accelerates training and inference, has fewer parameters, and requires fewer FLOPs to compute while achieving comparable or better accuracy across tasks. Transformer-MGK can also be easily extended to use with linear attention. We empirically demonstrate the advantage of Transformer-MGK in a range of practical applications, including language modeling and tasks that involve very long sequences. On the Wikitext-103 and Long Range Arena benchmark, Transformer-MGKs with 4 heads attain comparable or better performance to the baseline transformers with 8 heads.

Tan M. Nguyen, Vai Suliafu, Stanley J. Osher et al.

We propose FMMformers, a class of efficient and flexible transformers inspired by the celebrated fast multipole method (FMM) for accelerating interacting particle simulation. FMM decomposes particle-particle interaction into near-field and far-field components and then performs direct and coarse-grained computation, respectively. Similarly, FMMformers decompose the attention into near-field and far-field attention, modeling the near-field attention by a banded matrix and the far-field attention by a low-rank matrix. Computing the attention matrix for FMMformers requires linear complexity in computational time and memory footprint with respect to the sequence length. In contrast, standard transformers suffer from quadratic complexity. We analyze and validate the advantage of FMMformers over the standard transformer on the Long Range Arena and language modeling benchmarks. FMMformers can even outperform the standard transformer in terms of accuracy by a significant margin. For instance, FMMformers achieve an average classification accuracy of $60.74\%$ over the five Long Range Arena tasks, which is significantly better than the standard transformer's average accuracy of $58.70\%$.

Bao Wang, Tan M. Nguyen, Andrea L. Bertozzi et al.

Stochastic gradient descent (SGD) with constant momentum and its variants such as Adam are the optimization algorithms of choice for training deep neural networks (DNNs). Since DNN training is incredibly computationally expensive, there is great interest in speeding up the convergence. Nesterov accelerated gradient (NAG) improves the convergence rate of gradient descent (GD) for convex optimization using a specially designed momentum; however, it accumulates error when an inexact gradient is used (such as in SGD), slowing convergence at best and diverging at worst. In this paper, we propose Scheduled Restart SGD (SRSGD), a new NAG-style scheme for training DNNs. SRSGD replaces the constant momentum in SGD by the increasing momentum in NAG but stabilizes the iterations by resetting the momentum to zero according to a schedule. Using a variety of models and benchmarks for image classification, we demonstrate that, in training DNNs, SRSGD significantly improves convergence and generalization; for instance in training ResNet200 for ImageNet classification, SRSGD achieves an error rate of 20.93% vs. the benchmark of 22.13%. These improvements become more significant as the network grows deeper. Furthermore, on both CIFAR and ImageNet, SRSGD reaches similar or even better error rates with significantly fewer training epochs compared to the SGD baseline.

Alex Tong Lin, Samy Wu Fung, Wuchen Li et al.

We present APAC-Net, an alternating population and agent control neural network for solving stochastic mean field games (MFGs). Our algorithm is geared toward high-dimensional instances of MFGs that are beyond reach with existing solution methods. We achieve this in two steps. First, we take advantage of the underlying variational primal-dual structure that MFGs exhibit and phrase it as a convex-concave saddle point problem. Second, we parameterize the value and density functions by two neural networks, respectively. By phrasing the problem in this manner, solving the MFG can be interpreted as a special case of training a generative adversarial network (GAN). We show the potential of our method on up to 100-dimensional MFG problems.

Lisa Maria Kreusser, Stanley J. Osher, Bao Wang

Loss functions with a large number of saddle points are one of the major obstacles for training modern machine learning models efficiently. First-order methods such as gradient descent are usually the methods of choice for training machine learning models. However, these methods converge to saddle points for certain choices of initial guesses. In this paper, we propose a modification of the recently proposed Laplacian smoothing gradient descent [Osher et al., arXiv:1806.06317], called modified Laplacian smoothing gradient descent (mLSGD), and demonstrate its potential to avoid saddle points without sacrificing the convergence rate. Our analysis is based on the attraction region, formed by all starting points for which the considered numerical scheme converges to a saddle point. We investigate the attraction region's dimension both analytically and numerically. For a canonical class of quadratic functions, we show that the dimension of the attraction region for mLSGD is floor((n-1)/2), and hence it is significantly smaller than that of the gradient descent whose dimension is n-1.

Zehao Dou, Stanley J. Osher, Bao Wang

In this paper, we analyze efficacy of the fast gradient sign method (FGSM) and the Carlini-Wagner's L2 (CW-L2) attack. We prove that, within a certain regime, the untargeted FGSM can fool any convolutional neural nets (CNNs) with ReLU activation; the targeted FGSM can mislead any CNNs with ReLU activation to classify any given image into any prescribed class. For a special two-layer neural network: a linear layer followed by the softmax output activation, we show that the CW-L2 attack increases the ratio of the classification probability between the target and ground truth classes. Moreover, we provide numerical results to verify all our theoretical results.

Bao Wang, Alex T. Lin, Wei Zhu et al.

We improve the robustness of Deep Neural Net (DNN) to adversarial attacks by using an interpolating function as the output activation. This data-dependent activation remarkably improves both the generalization and robustness of DNN. In the CIFAR10 benchmark, we raise the robust accuracy of the adversarially trained ResNet20 from $\sim 46\%$ to $\sim 69\%$ under the state-of-the-art Iterative Fast Gradient Sign Method (IFGSM) based adversarial attack. When we combine this data-dependent activation with total variation minimization on adversarial images and training data augmentation, we achieve an improvement in robust accuracy by 38.9$\%$ for ResNet56 under the strongest IFGSM attack. Furthermore, We provide an intuitive explanation of our defense by analyzing the geometry of the feature space.

Zuoqiang Shi, Bao Wang, Stanley J. Osher

We analyze the convergence of the weighted nonlocal Laplacian (WNLL) on high dimensional randomly distributed data. The analysis reveals the importance of the scaling weight $μ\sim P|/|S|$ with $|P|$ and $|S|$ be the number of entire and labeled data, respectively. The result gives a theoretical foundation of WNLL for high dimensional data interpolation.

Bao Wang, Xiyang Luo, Zhen Li et al.

We replace the output layer of deep neural nets, typically the softmax function, by a novel interpolating function. And we propose end-to-end training and testing algorithms for this new architecture. Compared to classical neural nets with softmax function as output activation, the surrogate with interpolating function as output activation combines advantages of both deep and manifold learning. The new framework demonstrates the following major advantages: First, it is better applicable to the case with insufficient training data. Second, it significantly improves the generalization accuracy on a wide variety of networks. The algorithm is implemented in PyTorch, and code will be made publicly available.

Bao Wang, Penghang Yin, Andrea L. Bertozzi et al.

Real-time crime forecasting is important. However, accurate prediction of when and where the next crime will happen is difficult. No known physical model provides a reasonable approximation to such a complex system. Historical crime data are sparse in both space and time and the signal of interests is weak. In this work, we first present a proper representation of crime data. We then adapt the spatial temporal residual network on the well represented data to predict the distribution of crime in Los Angeles at the scale of hours in neighborhood-sized parcels. These experiments as well as comparisons with several existing approaches to prediction demonstrate the superiority of the proposed model in terms of accuracy. Finally, we present a ternarization technique to address the resource consumption issue for its deployment in real world. This work is an extension of our short conference proceeding paper [Wang et al, Arxiv 1707.03340].

Braxton Osting, Christoph Brune, Stanley J. Osher

Given a graph where vertices represent alternatives and arcs represent pairwise comparison data, the statistical ranking problem is to find a potential function, defined on the vertices, such that the gradient of the potential function agrees with the pairwise comparisons. Our goal in this paper is to develop a method for collecting data for which the least squares estimator for the ranking problem has maximal Fisher information. Our approach, based on experimental design, is to view data collection as a bi-level optimization problem where the inner problem is the ranking problem and the outer problem is to identify data which maximizes the informativeness of the ranking. Under certain assumptions, the data collection problem decouples, reducing to a problem of finding multigraphs with large algebraic connectivity. This reduction of the data collection problem to graph-theoretic questions is one of the primary contributions of this work. As an application, we study the Yahoo! Movie user rating dataset and demonstrate that the addition of a small number of well-chosen pairwise comparisons can significantly increase the Fisher informativeness of the ranking. As another application, we study the 2011-12 NCAA football schedule and propose schedules with the same number of games which are significantly more informative. Using spectral clustering methods to identify highly-connected communities within the division, we argue that the NCAA could improve its notoriously poor rankings by simply scheduling more out-of-conference games.