Data-driven discovery of partial differential equations (PDEs) has attracted increasing attention in recent years. Although significant progress has been made, certain unresolved issues remain. For example, for PDEs with high-order derivatives, the performance of existing methods is unsatisfactory, especially when the data are sparse and noisy. It is also difficult to discover heterogeneous parametric PDEs where heterogeneous parameters are embedded in the partial differential operators. In this work, a new framework combining deep-learning and integral form is proposed to handle the above-mentioned problems simultaneously, and improve the accuracy and stability of PDE discovery. In the framework, a deep neural network is firstly trained with observation data to generate meta-data and calculate derivatives. Then, a unified integral form is defined, and the genetic algorithm is employed to discover the best structure. Finally, the value of parameters is calculated, and whether the parameters are constants or variables is identified. Numerical experiments proved that our proposed algorithm is more robust to noise and more accurate compared with existing methods due to the utilization of integral form. Our proposed algorithm is also able to discover PDEs with high-order derivatives or heterogeneous parameters accurately with sparse and noisy data.
Baidu Research（百度研究院）；Arizona State University（亚利桑那州立大学）
Sparse subspace clustering methods with sparsity induced by $\ell^{0}$-norm, such as $\ell^{0}$-Sparse Subspace Clustering ($\ell^{0}$-SSC)~\citep{YangFJYH16-L0SSC-ijcv}, are demonstrated to be more effective than its $\ell^{1}$ counterpart such as Sparse Subspace Clustering (SSC)~\citep{ElhamifarV13}. However, the theoretical analysis of $\ell^{0}$-SSC is restricted to clean data that lie exactly in subspaces. Real data often suffer from noise and they may lie close to subspaces. In this paper, we show that an optimal solution to the optimization problem of noisy $\ell^{0}$-SSC achieves subsp…
Central China Normal University（华中师范大学）；The University of Hong Kong（香港大学）
In this paper, we study the sparse nonnegative tensor factorization and completion problem from partial and noisy observations for third-order tensors. Because of sparsity and nonnegativity, the underlying tensor is decomposed into the tensor-tensor product of one sparse nonnegative tensor and one nonnegative tensor. We propose to minimize the sum of the maximum likelihood estimation for the observations with nonnegativity constraints and the tensor $\ell_0$ norm for the sparse factor. We show that the error bounds of the estimator of the proposed model can be established under general noise o…
Peking University（北京大学）；Southern University of Science and Technology（南方科技大学）；Peng Cheng Laboratory（鹏城实验室）
In recent years, data-driven methods have been developed to learn dynamical systems and partial differential equations (PDE). The goal of such work is discovering unknown physics and the corresponding equations. However, prior to achieving this goal, major challenges remain to be resolved, including learning PDE under noisy data and limited discrete data. To overcome these challenges, in this work, a deep-learning based data-driven method, called DL-PDE, is developed to discover the governing PDEs of underlying physical processes. The DL-PDE method combines deep learning via neural networks an…
Noisy Max and Sparse Vector are selection algorithms for differential privacy and serve as building blocks for more complex algorithms. In this paper we show that both algorithms can release additional information for free (i.e., at no additional privacy cost). Noisy Max is used to return the approximate maximizer among a set of queries. We show that it can also release for free the noisy gap between the approximate maximizer and runner-up. This free information can improve the accuracy of certain subsequent counting queries by up to 50%. Sparse Vector is used to return a set of queries that a…
Peking University（北京大学）；Southern University of Science and Technology（南方科技大学）；Peng Cheng Laboratory（鹏城实验室）
Data-driven methods have recently made great progress in the discovery of partial differential equations (PDEs) from spatial-temporal data. However, several challenges remain to be solved, including sparse noisy data, incomplete candidate library, and spatially- or temporally-varying coefficients. In this work, a new framework, which combines neural network, genetic algorithm and adaptive methods, is put forward to address all of these challenges simultaneously. In the framework, a trained neural network is utilized to calculate derivatives and generate a large amount of meta-data, which solve…
Subspace clustering is the problem of clustering data points into a union of low-dimensional linear/affine subspaces. It is the mathematical abstraction of many important problems in computer vision, image processing and machine learning. A line of recent work (4, 19, 24, 20) provided strong theoretical guarantee for sparse subspace clustering (4), the state-of-the-art algorithm for subspace clustering, on both noiseless and noisy data sets. It was shown that under mild conditions, with high probability no two points from different subspaces are clustered together. Such guarantee, however, is …
Sparse regression on a library of candidate features has developed as the prime method to discover the partial differential equation underlying a spatio-temporal data-set. These features consist of higher order derivatives, limiting model discovery to densely sampled data-sets with low noise. Neural network-based approaches circumvent this limit by constructing a surrogate model of the data, but have to date ignored advances in sparse regression algorithms. In this paper we present a modular framework that dynamically determines the sparsity pattern of a deep-learning based surrogate using any…
We study the problem of learning an unknown mixture of $k$ rankings over $n$ elements, given access to noisy samples drawn from the unknown mixture. We consider a range of different noise models, including natural variants of the "heat kernel" noise framework and the Mallows model. For each of these noise models we give an algorithm which, under mild assumptions, learns the unknown mixture to high accuracy and runs in $n^{O(\log k)}$ time. The best previous algorithms for closely related problems have running times which are exponential in $k$.
This paper studies early-stopped mirror descent applied to noisy sparse phase retrieval, which is the problem of recovering a $k$-sparse signal $\mathbf{x}^\star\in\mathbb{R}^n$ from a set of quadratic Gaussian measurements corrupted by sub-exponential noise. We consider the (non-convex) unregularized empirical risk minimization problem and show that early-stopped mirror descent, when equipped with the hyperbolic entropy mirror map and proper initialization, achieves a nearly minimax-optimal rate of convergence, provided the sample size is at least of order $k^2$ (modulo logarithmic term) and …
Scientific machine learning has been successfully applied to inverse problems and PDE discovery in computational physics. One caveat concerning current methods is the need for large amounts of ("clean") data, in order to characterize the full system response and discover underlying physical models. Bayesian methods may be particularly promising for overcoming these challenges, as they are naturally less sensitive to the negative effects of sparse and noisy data. In this paper, we propose to use Bayesian neural networks (BNN) in order to: 1) Recover the full system states from measurement data …
Harbin Institute of Technology（哈尔滨工业大学）；Peng Cheng Laboratory（鹏城实验室）；Tsinghua University（清华大学）
Learning with noisy labels is an important and challenging task for training accurate deep neural networks. Some commonly-used loss functions, such as Cross Entropy (CE), suffer from severe overfitting to noisy labels. Robust loss functions that satisfy the symmetric condition were tailored to remedy this problem, which however encounter the underfitting effect. In this paper, we theoretically prove that \textbf{any loss can be made robust to noisy labels} by restricting the network output to the set of permutations over a fixed vector. When the fixed vector is one-hot, we only need to constra…
We formulate sparse support recovery as a salient set identification problem and use information-theoretic analyses to characterize the recovery performance and sample complexity. We consider a very general model where we are not restricted to linear models or specific distributions. We state non-asymptotic bounds on recovery probability and a tight mutual information formula for sample complexity. We evaluate our bounds for applications such as sparse linear regression and explicitly characterize effects of correlation or noisy features on recovery performance. We show improvements upon previ…
Exact matrix completion and low rank matrix estimation problems has been studied in different underlying conditions. In this work we study exact low-rank completion under non-degenerate noise model. Non-degenerate random noise model has been previously studied by many researchers under given condition that the noise is sparse and existing in some of the columns. In this paper, we assume that each row can receive random noise instead of columns and propose an interactive algorithm that is robust to this noise. We show that we use a parametrization technique to give a condition when the underlyi…
Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterion
College of Engineering, Peking University（北京大学工学院）；Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）；Southern University of Science and Technology（南方科技大学）；Peng Cheng Laboratory（鹏城实验室）
In this paper we present a linear programming solution for sign pattern recovery of a sparse signal from noisy random projections of the signal. We consider two types of noise models, input noise, where noise enters before the random projection; and output noise, where noise enters after the random projection. Sign pattern recovery involves the estimation of sign pattern of a sparse signal. Our idea is to pretend that no noise exists and solve the noiseless $\ell_1$ problem, namely, $\min \|β\|_1 ~ s.t. ~ y=G β$ and quantizing the resulting solution. We show that the quantized solution perfect…
University of Cambridge（剑桥大学）；Google Research（谷歌研究院）
Capturing the microscopic interactions that determine molecular reactivity poses a challenge across the physical sciences. Even a basic understanding of the underlying reaction mechanisms can substantially accelerate materials and compound design, including the development of new catalysts or drugs. Given the difficulties routinely faced by both experimental and theoretical investigations that aim to improve our mechanistic understanding of a reaction, recent advances have focused on data-driven routes to derive structure-property relationships directly from high-throughput screens. However, e…
Yahoo! Research（雅虎研究院）；University of Minnesota – Twin Cities（明尼苏达大学双城分校）
Inductive Matrix Completion (IMC) is an important class of matrix completion problems that allows direct inclusion of available features to enhance estimation capabilities. These models have found applications in personalized recommendation systems, multilabel learning, dictionary learning, etc. This paper examines a general class of noisy matrix completion tasks where the underlying matrix is following an IMC model i.e., it is formed by a mixing matrix (a priori unknown) sandwiched between two known feature matrices. The mixing matrix here is assumed to be well approximated by the product of …
Chalmers University of Technology（查尔姆斯理工大学）；AstraZeneca AB（阿斯利康公司）
We study the problem of performing automated experiment design for drug screening through Bayesian inference and optimisation. In particular, we compare and contrast the behaviour of linear-Gaussian models and Gaussian processes, when used in conjunction with upper confidence bound algorithms, Thompson sampling, or bounded horizon tree search. We show that non-myopic sophisticated exploration techniques using sparse tree search have a distinct advantage over methods such as Thompson sampling or upper confidence bounds in this setting. We demonstrate the significant superiority of the approach …
Singapore University of Technology and Design（新加坡科技设计大学）；University of Edinburgh（爱丁堡大学）
We propose a novel topic discovery algorithm for unlabeled images based on the bag-of-words (BoW) framework. We first extract a dictionary of visual words and subsequently for each image compute a visual word occurrence histogram. We view these histograms as rows of a large matrix from which we extract sparse principal components (PCs). Each PC identifies a sparse combination of visual words which co-occur frequently in some images but seldom appear in others. Each sparse PC corresponds to a topic, and images whose interference with the PC is high belong to that topic, revealing the common par…
Sparse subspace clustering (SSC) using greedy-based neighbor selection, such as matching pursuit (MP) and orthogonal matching pursuit (OMP), has been known as a popular computationally-efficient alternative to the conventional L1-minimization based methods. Under deterministic bounded noise corruption, in this paper we derive coherence-based sufficient conditions guaranteeing correct neighbor identification using MP/OMP. Our analyses exploit the maximum/minimum inner product between two noisy data points subject to a known upper bound on the noise level. The obtained sufficient condition clear…
The Pennsylvania State University（宾夕法尼亚州立大学）；Michigan State University（密歇根州立大学）
Graph Neural Networks (GNNs) have shown their great ability in modeling graph structured data. However, real-world graphs usually contain structure noises and have limited labeled nodes. The performance of GNNs would drop significantly when trained on such graphs, which hinders the adoption of GNNs on many applications. Thus, it is important to develop noise-resistant GNNs with limited labeled nodes. However, the work on this is rather limited. Therefore, we study a novel problem of developing robust GNNs on noisy graphs with limited labeled nodes. Our analysis shows that both the noisy edges …
Uppsala University（乌普萨拉大学）；Qatar Computing Research Institute（卡塔尔计算研究所）
This paper provides an analysis of character-level machine translation models used in pivot-based translation when applied to sparse and noisy datasets, such as crowdsourced movie subtitles. In our experiments, we find that such character-level models cut the number of untranslated words by over 40% and are especially competitive (improvements of 2-3 BLEU points) in the case of limited training data. We explore the impact of character alignment, phrase table filtering, bitext size and the choice of pivot language on translation quality. We further compare cascaded translation models to the use…
Symbolic Graph Networks for Robust PDE Discovery from Noisy Sparse Data
School of Software Engineering, Chengdu University of Information Technology（成都信息科技学院软件工程学院）；School of Statistics, Chengdu University of Information Technology（成都信息科技学院统计学院）；College of Applied Mathematics, Chengdu University of Information Technology（成都信息科技学院应用数学学院）；Key Laboratory of Mathematical Meteorology, Chengdu University of Information Technology（成都信息科技学院数学气象重点实验室）
We propose a hierarchical learning strategy aimed at generating sparse representations and associated models for large noisy datasets. The hierarchy follows from approximation spaces identified at successively finer scales. For promoting model generalization at each scale, we also introduce a novel, projection based penalty operator across multiple dimension, using permutation operators for incorporating proximity and ordering information. The paper presents a detailed analysis of approximation properties in the reconstruction Reproducing Kernel Hilbert Spaces (RKHS) with emphasis on optimalit…
We study the performance of sparse regression methods and propose new techniques to distill the governing equations of dynamical systems from data. We first look at the generic methodology of learning interpretable equation forms from data, proposed by Brunton et al., followed by performance of LASSO for this purpose. We then propose a new algorithm that uses the dual of LASSO optimization for higher accuracy and stability. In the second part, we propose a novel algorithm that learns the candidate function library in a completely data-driven manner to distill the governing equations of the dyn…
Concepts are key building blocks of higher level human understanding. Explainable AI (XAI) methods have shown tremendous progress in recent years, however, local attribution methods do not allow to identify coherent model behavior across samples and therefore miss this essential component. In this work, we study concept-based explanations and put forward a new definition of concepts as low-dimensional subspaces of hidden feature layers. We novelly apply sparse subspace clustering to discover these concept subspaces. Moving forward, we derive insights from concept subspaces in terms of localize…
University of Southern California（南加州大学）；University of California, Irvine（加州大学欧文分校）
We propose an efficient ADMM method with guarantees for high-dimensional problems. We provide explicit bounds for the sparse optimization problem and the noisy matrix decomposition problem. For sparse optimization, we establish that the modified ADMM method has an optimal convergence rate of $\mathcal{O}(s\log d/T)$, where $s$ is the sparsity level, $d$ is the data dimension and $T$ is the number of steps. This matches with the minimax lower bounds for sparse estimation. For matrix decomposition into sparse and low rank components, we provide the first guarantees for any online method, and pro…
Machine Learning Hamiltonian Dynamical Systems with Sparse and Noisy Data
The Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom（牛津大学数学研究所）；Center for Theoretical Physics, St. Stephen's College, University of Delhi, Delhi 110007, India（德里大学斯特金学院理论物理中心）
Sparse PCA with False Discovery Rate Controlled Variable Selection
Technische Universität Darmstadt（达姆施塔特工业大学）；Conservatoire National des Arts et Métiers（法国国立工艺学院）；The Hong Kong University of Science and Technology（香港科技大学）
Non-negative blind source separation (BSS) has raised interest in various fields of research, as testified by the wide literature on the topic of non-negative matrix factorization (NMF). In this context, it is fundamental that the sources to be estimated present some diversity in order to be efficiently retrieved. Sparsity is known to enhance such contrast between the sources while producing very robust approaches, especially to noise. In this paper we introduce a new algorithm in order to tackle the blind separation of non-negative sparse sources from noisy measurements. We first show that sp…
Center for Research and Interdisciplinarity (CRI)（跨学科研究中心（CRI））；Université de Paris, INSERM U1284（巴黎大学，法国国家健康与医学研究院U1284研究所）；F-75006 Paris, France（法国巴黎75006）
Discovering the partial differential equations underlying spatio-temporal datasets from very limited and highly noisy observations is of paramount interest in many scientific fields. However, it remains an open question to know when model discovery algorithms based on sparse regression can actually recover the underlying physical processes. In this work, we show the design matrices used to infer the equations by sparse regression can violate the irrepresentability condition (IRC) of the Lasso, even when derived from analytical PDE solutions (i.e. without additional noise). Sparse regression te…
L3S Research Center（L3S研究中心）；Leibniz Universität Hannover（莱布尼茨汉诺威大学）
Embedded markup of Web pages has seen widespread adoption throughout the past years driven by standards such as RDFa and Microdata and initiatives such as schema.org, where recent studies show an adoption by 39% of all Web pages already in 2016. While this constitutes an important information source for tasks such as Web search, Web page classification or knowledge graph augmentation, individual markup nodes are usually sparsely described and often lack essential information. For instance, from 26 million nodes describing events within the Common Crawl in 2016, 59% of nodes provide less than s…
We consider the problem of using a factor model we call {\em spike-and-slab sparse coding} (S3C) to learn features for a classification task. The S3C model resembles both the spike-and-slab RBM and sparse coding. Since exact inference in this model is intractable, we derive a structured variational inference procedure and employ a variational EM training algorithm. Prior work on approximate inference for this model has not prioritized the ability to exploit parallel architectures and scale to enormous problem sizes. We present an inference procedure appropriate for use with GPUs which allows u…
Microsoft Research（微软研究院）；University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）
In the problem of learning mixtures of linear regressions, the goal is to learn a collection of signal vectors from a sequence of (possibly noisy) linear measurements, where each measurement is evaluated on an unknown signal drawn uniformly from this collection. This setting is quite expressive and has been studied both in terms of practical applications and for the sake of establishing theoretical guarantees. In this paper, we consider the case where the signal vectors are sparse; this generalizes the popular compressed sensing paradigm. We improve upon the state-of-the-art results as follows…
University of Waterloo（滑铁卢大学）；ATS Automation（ATS自动化公司）
Causal discovery between collections of time-series data can help diagnose causes of symptoms and hopefully prevent faults before they occur. However, reliable causal discovery can be very challenging, especially when the data acquisition rate varies (i.e., non-uniform data sampling), or in the presence of missing data points (e.g., sparse data sampling). To address these issues, we proposed a new system comprised of two parts, the first part fills missing data with a Gaussian Process Regression, and the second part leverages an Echo State Network, which is a type of reservoir computer (i.e., …
Subspace clustering is the problem of partitioning unlabeled data points into a number of clusters so that data points within one cluster lie approximately on a low-dimensional linear subspace. In many practical scenarios, the dimensionality of data points to be clustered are compressed due to constraints of measurement, computation or privacy. In this paper, we study the theoretical properties of a popular subspace clustering algorithm named sparse subspace clustering (SSC) and establish formal success conditions of SSC on dimensionality-reduced data. Our analysis applies to the most general …
NeRF-enabled Analysis-Through-Synthesis for ISAR Imaging of Small Everyday Objects with Sparse and Noisy UWB Radar Data
Manning College of Information and Computer Sciences, University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校信息与计算机科学学院）；Kitware INC（Kitware公司）；Arizona State University（亚利桑那州立大学）；Halıcıoğlu Data Science Institute, University of California San Diego（加利福尼亚大学圣地亚哥分校哈利西奥卢数据科学研究所）
We study a practical algorithm for sparse principal component analysis (PCA) of incomplete and noisy data. Our algorithm is based on the semidefinite program (SDP) relaxation of the non-convex $l_1$-regularized PCA problem. We provide theoretical and experimental evidence that SDP enables us to exactly recover the true support of the sparse leading eigenvector of the unknown true matrix, despite only observing an incomplete (missing uniformly at random) and noisy version of it. We derive sufficient conditions for exact recovery, which involve matrix incoherence, the spectral gap between the la…
We propose a novel deep neural network, coined DeepFPC-$\ell_2$, for solving the 1-bit compressed sensing problem. The network is designed by unfolding the iterations of the fixed-point continuation (FPC) algorithm with one-sided $\ell_2$-norm (FPC-$\ell_2$). The DeepFPC-$\ell_2$ method shows higher signal reconstruction accuracy and convergence speed than the traditional FPC-$\ell_2$ algorithm. Furthermore, we compare its robustness to noise with the previously proposed DeepFPC network---which stemmed from unfolding the FPC-$\ell_1$ algorithm---for different signal to noise ratio (SNR) and si…
