Ming Gu

Amal Khabou, James W. Demmel, Laura Grigori et al.

We present the LU decomposition with panel rank revealing pivoting (LU_PRRP), an LU factorization algorithm based on strong rank revealing QR panel factorization. LU_PRRP is more stable than Gaussian elimination with partial pivoting (GEPP). Our extensive numerical experiments show that the new factorization scheme is as numerically stable as GEPP in practice, but it is more resistant to pathological cases and easily solves the Wilkinson matrix and the Foster matrix. We also present CALU_PRRP, a communication avoiding version of LU_PRRP that minimizes communication. CALU_PRRP is based on tournament pivoting, with the selection of the pivots at each step of the tournament being performed via strong rank revealing QR factorization. CALU_PRRP is more stable than CALU, the communication avoiding version of GEPP. CALU_PRRP is also more stable in practice and is resistant to pathological cases on which GEPP and CALU fail.

Ming Gu

A classical problem in matrix computations is the efficient and reliable approximation of a given matrix by a matrix of lower rank. The truncated singular value decomposition (SVD) is known to provide the best such approximation for any given fixed rank. However, the SVD is also known to be very costly to compute. Among the different approaches in the literature for computing low-rank approximations, randomized algorithms have attracted researchers' recent attention due to their surprising reliability and computational efficiency in different application areas. Typically, such algorithms are shown to compute with very high probability low-rank approximations that are within a constant factor from optimal, and are known to perform even better in many practical situations. In this paper, we present a novel error analysis that considers randomized algorithms within the subspace iteration framework and show with very high probability that highly accurate low-rank approximations as well as singular values can indeed be computed quickly for matrices with rapidly decaying singular values. Such matrices appear frequently in diverse application areas such as data analysis, fast structured matrix computations and fast direct methods for large sparse linear systems of equations and are the driving motivation for randomized methods. Furthermore, we show that the low-rank approximations computed by these randomized algorithms are actually rank-revealing approximations, and the special case of a rank-1 approximation can also be used to correctly estimate matrix 2-norms with very high probability. Our numerical experiments are in full support of our conclusions.

Jed A. Duersch, Meiyue Shao, Chao Yang et al.

Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is widely used to compute eigenvalues of large sparse symmetric matrices. The algorithm can suffer from numerical instability if it is not implemented with care. This is especially problematic when the number of eigenpairs to be computed is relatively large. In this paper we propose an improved basis selection strategy based on earlier work by Hetmaniuk and Lehoucq as well as a robust convergence criterion which is backward stable to enhance the robustness. We also suggest several algorithmic optimizations that improve performance of practical LOBPCG implementations. Numerical examples confirm that our approach consistently and significantly outperforms previous competing approaches in both stability and speed.

Qiaochu Yuan, Ming Gu, Bo Li

The low rank approximation of matrices is a crucial component in many data mining applications today. A competitive algorithm for this class of problems is the randomized block Lanczos algorithm - an amalgamation of the traditional block Lanczos algorithm with a randomized starting matrix. While empirically this algorithm performs quite well, there has been scant new theoretical results on its convergence behavior and approximation accuracy, and past results have been restricted to certain parameter settings. In this paper, we present a unified singular value convergence analysis for this algorithm, for all valid choices of the block size parameter. We present novel results on the rate of singular value convergence and show that under certain spectrum regimes, the convergence is superlinear. Additionally, we provide results from numerical experiments that validate our analysis.

Zhangxuan Gu, Changhua Meng, Ke Wang et al.

Recently, various multimodal networks for Visually-Rich Document Understanding(VRDU) have been proposed, showing the promotion of transformers by integrating visual and layout information with the text embeddings. However, most existing approaches utilize the position embeddings to incorporate the sequence information, neglecting the noisy improper reading order obtained by OCR tools. In this paper, we propose a robust layout-aware multimodal network named XYLayoutLM to capture and leverage rich layout information from proper reading orders produced by our Augmented XY Cut. Moreover, a Dilated Conditional Position Encoding module is proposed to deal with the input sequence of variable lengths, and it additionally extracts local layout information from both textual and visual modalities while generating position embeddings. Experiment results show that our XYLayoutLM achieves competitive results on document understanding tasks.

Jianwei Xiao, Ming Gu, Julien Langou

Factorizing large matrices by QR with column pivoting (QRCP) is substantially more expensive than QR without pivoting, owing to communication costs required for pivoting decisions. In contrast, randomized QRCP (RQRCP) algorithms have proven themselves empirically to be highly competitive with high-performance implementations of QR in processing time, on uniprocessor and shared memory machines, and as reliable as QRCP in pivot quality. We show that RQRCP algorithms can be as reliable as QRCP with failure probabilities exponentially decaying in oversampling size. We also analyze efficiency differences among different RQRCP algorithms. More importantly, we develop distributed memory implementations of RQRCP that are significantly better than QRCP implementations in ScaLAPACK. As a further development, we introduce the concept of and develop algorithms for computing spectrum-revealing QR factorizations for low-rank matrix approximations, and demonstrate their effectiveness against leading low-rank approximation methods in both theoretical and numerical reliability and efficiency.

Ming Gu, Gaoming Yang, Sheng Zhou et al.

Graph clustering is a fundamental task in graph analysis, and recent advances in utilizing graph neural networks (GNNs) have shown impressive results. Despite the success of existing GNN-based graph clustering methods, they often overlook the quality of graph structure, which is inherent in real-world graphs due to their sparse and multifarious nature, leading to subpar performance. Graph structure learning allows refining the input graph by adding missing links and removing spurious connections. However, previous endeavors in graph structure learning have predominantly centered around supervised settings, and cannot be directly applied to our specific clustering tasks due to the absence of ground-truth labels. To bridge the gap, we propose a novel method called \textbf{ho}mophily-enhanced structure \textbf{le}arning for graph clustering (HoLe). Our motivation stems from the observation that subtly enhancing the degree of homophily within the graph structure can significantly improve GNNs and clustering outcomes. To realize this objective, we develop two clustering-oriented structure learning modules, i.e., hierarchical correlation estimation and cluster-aware sparsification. The former module enables a more accurate estimation of pairwise node relationships by leveraging guidance from latent and clustering spaces, while the latter one generates a sparsified structure based on the similarity matrix and clustering assignments. Additionally, we devise a joint optimization approach alternating between training the homophily-enhanced structure learning and GNN-based clustering, thereby enforcing their reciprocal effects. Extensive experiments on seven benchmark datasets of various types and scales, across a range of clustering metrics, demonstrate the superiority of HoLe against state-of-the-art baselines.

Jianwei Xiao, Ming Gu

Kernel methods represent some of the most popular machine learning tools for data analysis. Since exact kernel methods can be prohibitively expensive for large problems, reliable low-rank matrix approximations and high-performance implementations have become indispensable for practical applications of kernel methods. In this work, we introduce spectrum-revealing Cholesky factorization, a reliable low-rank matrix factorization, for kernel matrix approximation. We also develop an efficient and effective randomized algorithm for computing this factorization. Our numerical experiments demonstrate that this algorithm is as effective as other Cholesky factorization based kernel methods on machine learning problems, but significantly faster.

Zhuoer Xu, Guanghui Zhu, Changhua Meng et al.

Based on the significant improvement of model robustness by AT (Adversarial Training), various variants have been proposed to further boost the performance. Well-recognized methods have focused on different components of AT (e.g., designing loss functions and leveraging additional unlabeled data). It is generally accepted that stronger perturbations yield more robust models. However, how to generate stronger perturbations efficiently is still missed. In this paper, we propose an efficient automated attacker called A2 to boost AT by generating the optimal perturbations on-the-fly during training. A2 is a parameterized automated attacker to search in the attacker space for the best attacker against the defense model and examples. Extensive experiments across different datasets demonstrate that A2 generates stronger perturbations with low extra cost and reliably improves the robustness of various AT methods against different attacks.

Yuehua Feng, Jianwei Xiao, Ming Gu

The Bunch-Kaufman algorithm and Aasen's algorithm are two of the most widely used methods for solving symmetric indefinite linear systems, yet they both are known to suffer from occasional numerical instability due to potentially exponential element growth or unbounded entries in the matrix factorization. In this work, we develop a randomized complete pivoting (RCP) algorithm for solving symmetric indefinite linear systems. RCP is comparable to the Bunch-Kaufman algorithm and Aasen's algorithm in computational efficiency, yet enjoys theoretical element growth and bounded entries in the factorization comparable to that of complete-pivoting, up to a theoretical failure probability that exponentially decays with an oversampling parameter. Our finite precision analysis shows that RCP is as numerically stable as Gaussian elimination with complete pivoting, and RCP has been observed to be numerically stable in our extensive numerical experiments.

Shengtao Li, Ge Gao, Yudong Liu et al.

Neural signed distance functions (SDFs) have shown powerful ability in fitting the shape geometry. However, inferring continuous signed distance fields from discrete unoriented point clouds still remains a challenge. The neural network typically fits the shape with a rough surface and omits fine-grained geometric details such as shape edges and corners. In this paper, we propose a novel non-linear implicit filter to smooth the implicit field while preserving high-frequency geometry details. Our novelty lies in that we can filter the surface (zero level set) by the neighbor input points with gradients of the signed distance field. By moving the input raw point clouds along the gradient, our proposed implicit filtering can be extended to non-zero level sets to keep the promise consistency between different level sets, which consequently results in a better regularization of the zero level set. We conduct comprehensive experiments in surface reconstruction from objects and complex scene point clouds, the numerical and visual comparisons demonstrate our improvements over the state-of-the-art methods under the widely used benchmarks.

Hairong Luo, Ge Gao, Han Huang et al.

Interoperability issue is a significant problem in Building Information Modeling (BIM). Object type, as a kind of critical semantic information needed in multiple BIM applications like scan-to-BIM and code compliance checking, also suffers when exchanging BIM data or creating models using software of other domains. It can be supplemented using deep learning. Current deep learning methods mainly learn from the shape information of BIM objects for classification, leaving relational information inherent in the BIM context unused. To address this issue, we introduce a two-branch geometric-relational deep learning framework. It boosts previous geometric classification methods with relational information. We also present a BIM object dataset IFCNet++, which contains both geometric and relational information about the objects. Experiments show that our framework can be flexibly adapted to different geometric methods. And relational features do act as a bonus to general geometric learning methods, obviously improving their classification performance, thus reducing the manual labor of checking models and improving the practical value of enriched BIM models.

Shengtao Li, Ge Gao, Yudong Liu et al.

Implicit neural networks have emerged as a crucial technology in 3D surface reconstruction. To reconstruct continuous surfaces from discrete point clouds, encoding the input points into regular grid features (plane or volume) has been commonly employed in existing approaches. However, these methods typically use the grid as an index for uniformly scattering point features. Compared with the irregular point features, the regular grid features may sacrifice some reconstruction details but improve efficiency. To take full advantage of these two types of features, we introduce a novel and high-efficiency attention mechanism between the grid and point features named Point-Grid Transformer (GridFormer). This mechanism treats the grid as a transfer point connecting the space and point cloud. Our method maximizes the spatial expressiveness of grid features and maintains computational efficiency. Furthermore, optimizing predictions over the entire space could potentially result in blurred boundaries. To address this issue, we further propose a boundary optimization strategy incorporating margin binary cross-entropy loss and boundary sampling. This approach enables us to achieve a more precise representation of the object structure. Our experiments validate that our method is effective and outperforms the state-of-the-art approaches under widely used benchmarks by producing more precise geometry reconstructions. The code is available at https://github.com/list17/GridFormer.

Han Liu, Xiaoyu Song, Ge Gao et al.

Semantic rule checking on RDFS/OWL data has been widely used in the construction industry. At present, semantic rule checking is mainly performed on static models. There are still challenges in integrating temporal models and semantic models for combined rule checking. In this paper, Semantic Petri-Net (SPN) is proposed as a novel temporal modeling and validating method, which implements the states and transitions of the Colored Petri-Net directly based on RDFS and SPARQL, and realizes two-way sharing of knowledge between domain semantic webs and temporal models in the runtime. Several cases are provided to demonstrate the possible applications in digital twins with concurrent state changes and dependencies.

Ming Gu, Zhuonan Zheng, Sheng Zhou et al.

Graph Neural Networks (GNNs) have achieved great success but are often considered to be challenged by varying levels of homophily in graphs. Recent \textit{empirical} studies have surprisingly shown that homophilic GNNs can perform well across datasets of different homophily levels with proper hyperparameter tuning, but the underlying theory and effective architectures remain unclear. To advance GNN universality across varying homophily, we theoretically revisit GNN message passing and uncover a novel \textit{smoothness-generalization dilemma}, where increasing hops inevitably enhances smoothness at the cost of generalization. This dilemma hinders learning in high-order homophilic neighborhoods and all heterophilic ones, where generalization is critical due to complex neighborhood class distributions that are sensitive to shifts induced by noise or sparsity. To address this, we introduce the Inceptive Graph Neural Network (IGNN) built on three simple yet effective design principles, which alleviate the dilemma by enabling distinct hop-wise generalization alongside improved overall generalization with adaptive smoothness. Benchmarking against 30 baselines demonstrates IGNN's superiority and reveals notable universality in certain homophilic GNN variants. Our code and datasets are available at \href{https://github.com/galogm/IGNN}{https://github.com/galogm/IGNN}.

Ming Gu, Ziwei Wang, Sicen Lai et al.

Ensuring web accessibility is crucial for advancing social welfare, justice, and equality in digital spaces, yet the vast majority of website user interfaces remain non-compliant, due in part to the resource-intensive and unscalable nature of current auditing practices. While WCAG-EM offers a structured methodology for site-wise conformance evaluation, it involves great human efforts and lacks practical support for execution at scale. In this work, we present an auditing framework, AAA, which operationalizes WCAG-EM through a human-AI partnership model. AAA is anchored by two key innovations: GRASP, a graph-based multimodal sampling method that ensures representative page coverage via learned embeddings of visual, textual, and relational cues; and MaC, a multimodal large language model-based copilot that supports auditors through cross-modal reasoning and intelligent assistance in high-effort tasks. Together, these components enable scalable, end-to-end web accessibility auditing, empowering human auditors with AI-enhanced assistance for real-world impact. We further contribute four novel datasets designed for benchmarking core stages of the audit pipeline. Extensive experiments demonstrate the effectiveness of our methods, providing insights that small-scale language models can serve as capable experts when fine-tuned.

Han Liu, Zhizhong Han, Yu-Shen Liu et al.

Low-rank metric learning aims to learn better discrimination of data subject to low-rank constraints. It keeps the intrinsic low-rank structure of datasets and reduces the time cost and memory usage in metric learning. However, it is still a challenge for current methods to handle datasets with both high dimensions and large numbers of samples. To address this issue, we present a novel fast low-rank metric learning (FLRML) method.FLRML casts the low-rank metric learning problem into an unconstrained optimization on the Stiefel manifold, which can be efficiently solved by searching along the descent curves of the manifold.FLRML significantly reduces the complexity and memory usage in optimization, which makes the method scalable to both high dimensions and large numbers of samples.Furthermore, we introduce a mini-batch version of FLRML to make the method scalable to larger datasets which are hard to be loaded and decomposed in limited memory. The outperforming experimental results show that our method is with high accuracy and much faster than the state-of-the-art methods under several benchmarks with large numbers of high-dimensional data. Code has been made available at https://github.com/highan911/FLRML

Han Huang, Yulun Wu, Junsheng Zhou et al.

Recently, neural implicit functions have demonstrated remarkable results in the field of multi-view reconstruction. However, most existing methods are tailored for dense views and exhibit unsatisfactory performance when dealing with sparse views. Several latest methods have been proposed for generalizing implicit reconstruction to address the sparse view reconstruction task, but they still suffer from high training costs and are merely valid under carefully selected perspectives. In this paper, we propose a novel sparse view reconstruction framework that leverages on-surface priors to achieve highly faithful surface reconstruction. Specifically, we design several constraints on global geometry alignment and local geometry refinement for jointly optimizing coarse shapes and fine details. To achieve this, we train a neural network to learn a global implicit field from the on-surface points obtained from SfM and then leverage it as a coarse geometric constraint. To exploit local geometric consistency, we project on-surface points onto seen and unseen views, treating the consistent loss of projected features as a fine geometric constraint. The experimental results with DTU and BlendedMVS datasets in two prevalent sparse settings demonstrate significant improvements over the state-of-the-art methods.

Meihan Liu, Zeyu Fang, Zhen Zhang et al.

Unsupervised Graph Domain Adaptation (UGDA) aims to transfer knowledge from a labelled source graph to an unlabelled target graph in order to address the distribution shifts between graph domains. Previous works have primarily focused on aligning data from the source and target graph in the representation space learned by graph neural networks (GNNs). However, the inherent generalization capability of GNNs has been largely overlooked. Motivated by our empirical analysis, we reevaluate the role of GNNs in graph domain adaptation and uncover the pivotal role of the propagation process in GNNs for adapting to different graph domains. We provide a comprehensive theoretical analysis of UGDA and derive a generalization bound for multi-layer GNNs. By formulating GNN Lipschitz for k-layer GNNs, we show that the target risk bound can be tighter by removing propagation layers in source graph and stacking multiple propagation layers in target graph. Based on the empirical and theoretical analysis mentioned above, we propose a simple yet effective approach called A2GNN for graph domain adaptation. Through extensive experiments on real-world datasets, we demonstrate the effectiveness of our proposed A2GNN framework.

Shengguo Li, Ming Gu, Beresford N. Parlett

In this paper we present an improved dqds algorithm for computing all the singular values of a bidiagonal matrix to high relative accuracy. There are two key contributions: a novel deflation strategy that improves the convergence for badly scaled matrices, and some modifications to certain shift strategies that accelerate the convergence for most bidiagonal matrices. These techniques together ensure linear worst case complexity of the improved algorithm (denoted by V5). Our extensive numerical experiments indicate that V5 is typically 1.2x--4x faster than DLASQ (the LAPACK-3.4.0 implementation of dqds) without any degradation in accuracy. On matrices for which DLASQ shows very slow convergence, V5 can be 3x--10x faster. At the end of this paper, a hybrid algorithm (HDLASQ) is developed by combining our improvements with the aggressive early deflation strategy (AggDef2 in [SIAM J. Matrix Anal. Appl., 33(2012), 22-51]). Numerical results show that HDLASQ is the fastest among these different versions.

Han Huang, Yulun Wu, Chao Deng et al.

Recently, Gaussian Splatting has sparked a new trend in the field of computer vision. Apart from novel view synthesis, it has also been extended to the area of multi-view reconstruction. The latest methods facilitate complete, detailed surface reconstruction while ensuring fast training speed. However, these methods still require dense input views, and their output quality significantly degrades with sparse views. We observed that the Gaussian primitives tend to overfit the few training views, leading to noisy floaters and incomplete reconstruction surfaces. In this paper, we present an innovative sparse-view reconstruction framework that leverages intra-view depth and multi-view feature consistency to achieve remarkably accurate surface reconstruction. Specifically, we utilize monocular depth ranking information to supervise the consistency of depth distribution within patches and employ a smoothness loss to enhance the continuity of the distribution. To achieve finer surface reconstruction, we optimize the absolute position of depth through multi-view projection features. Extensive experiments on DTU and BlendedMVS demonstrate that our method outperforms state-of-the-art methods with a speedup of 60x to 200x, achieving swift and fine-grained mesh reconstruction without the need for costly pre-training.

Yulun Wu, Han Huang, Wenyuan Zhang et al.

In recent years, reconstructing indoor scene geometry from multi-view images has achieved encouraging accomplishments. Current methods incorporate monocular priors into neural implicit surface models to achieve high-quality reconstructions. However, these methods require hundreds of images for scene reconstruction. When only a limited number of views are available as input, the performance of monocular priors deteriorates due to scale ambiguity, leading to the collapse of the reconstructed scene geometry. In this paper, we propose a new method, named Sparis, for indoor surface reconstruction from sparse views. Specifically, we investigate the impact of monocular priors on sparse scene reconstruction, introducing a novel prior based on inter-image matching information. Our prior offers more accurate depth information while ensuring cross-view matching consistency. Additionally, we employ an angular filter strategy and an epipolar matching weight function, aiming to reduce errors due to view matching inaccuracies, thereby refining the inter-image prior for improved reconstruction accuracy. The experiments conducted on widely used benchmarks demonstrate superior performance in sparse-view scene reconstruction.

Ming Gu, Yan Yang

Dialogue state tracking (DST) is evaluated by exact matching methods, which rely on large amounts of labeled data and ignore semantic consistency, leading to over-evaluation. Currently, leveraging large language models (LLM) in evaluating natural language processing tasks has achieved promising results. However, using LLM for DST evaluation is still under explored. In this paper, we propose a two-dimensional zero-shot evaluation method for DST using GPT-4, which divides the evaluation into two dimensions: accuracy and completeness. Furthermore, we also design two manual reasoning paths in prompting to further improve the accuracy of evaluation. Experimental results show that our method achieves better performance compared to the baselines, and is consistent with traditional exact matching based methods.

Ming Gu, Yan Yang

Data augmentation methods have been a promising direction to improve the performance of small models for low-resource dialogue state tracking. However, traditional methods rely on pre-defined user goals and neglect the importance of data complexity in this task. In this paper, we propose EDZ-DA, an Easy-to-Difficult Zero-shot Data Augmentation framework for low-resource dialogue state tracking that utilizes large language models to automatically catch the relationships of different domains and then generate the dialogue data. We also complicate the dialogues based on the domain relation to enhance the model's capability for co-reference slot tracking. Furthermore, we permute slot values to mitigate the influence of output orders and the problem of incomplete value generation. Experimental results illustrate the superiority of our proposed method compared to previous strong data augmentation baselines on MultiWOZ.

Zeyu Fang, Ming Gu, Sheng Zhou et al.

Unsupervised Anomaly Detection (UAD) plays a crucial role in identifying abnormal patterns within data without labeled examples, holding significant practical implications across various domains. Although the individual contributions of representation learning and clustering to anomaly detection are well-established, their interdependencies remain under-explored due to the absence of a unified theoretical framework. Consequently, their collective potential to enhance anomaly detection performance remains largely untapped. To bridge this gap, in this paper, we propose a novel probabilistic mixture model for anomaly detection to establish a theoretical connection among representation learning, clustering, and anomaly detection. By maximizing a novel anomaly-aware data likelihood, representation learning and clustering can effectively reduce the adverse impact of anomalous data and collaboratively benefit anomaly detection. Meanwhile, a theoretically substantiated anomaly score is naturally derived from this framework. Lastly, drawing inspiration from gravitational analysis in physics, we have devised an improved anomaly score that more effectively harnesses the combined power of representation learning and clustering. Extensive experiments, involving 17 baseline methods across 30 diverse datasets, validate the effectiveness and generalization capability of the proposed method, surpassing state-of-the-art methods.

Gang Hu, Ming Gu

Investment portfolios, central to finance, balance potential returns and risks. This paper introduces a hybrid approach combining Markowitz's portfolio theory with reinforcement learning, utilizing knowledge distillation for training agents. In particular, our proposed method, called KDD (Knowledge Distillation DDPG), consist of two training stages: supervised and reinforcement learning stages. The trained agents optimize portfolio assembly. A comparative analysis against standard financial models and AI frameworks, using metrics like returns, the Sharpe ratio, and nine evaluation indices, reveals our model's superiority. It notably achieves the highest yield and Sharpe ratio of 2.03, ensuring top profitability with the lowest risk in comparable return scenarios.

Ming Gu, Yan Yang, Chengcai Chen et al.

Recently, low-resource dialogue state tracking (DST) has received increasing attention. First obtaining state values then based on values to generate slot types has made great progress in this task. However, obtaining state values is still an under-studied problem. Existing extraction-based approaches cannot capture values that require the understanding of context and are not generalizable either. To address these issues, we propose a novel State VAlue Generation based framework (SVAG), decomposing DST into state value generation and domain slot generation. Specifically, we propose to generate state values and use self-training to further improve state value generation. Moreover, we design an estimator aiming at detecting incomplete generation and incorrect generation for pseudo-labeled data selection during self-training. Experimental results on the MultiWOZ 2.1 dataset show that our method which has only less than 1 billion parameters achieves state-of-the-art performance under the data ratio settings of 5%, 10%, and 25% when limited to models under 100 billion parameters. Compared to models with more than 100 billion parameters, SVAG still reaches competitive results.

David G. Anderson, Ming Gu

Low-rank matrix approximation is a fundamental tool in data analysis for processing large datasets, reducing noise, and finding important signals. In this work, we present a novel truncated LU factorization called Spectrum-Revealing LU (SRLU) for effective low-rank matrix approximation, and develop a fast algorithm to compute an SRLU factorization. We provide both matrix and singular value approximation error bounds for the SRLU approximation computed by our algorithm. Our analysis suggests that SRLU is competitive with the best low-rank matrix approximation methods, deterministic or randomized, in both computational complexity and approximation quality. Numeric experiments illustrate that SRLU preserves sparsity, highlights important data features and variables, can be efficiently updated, and calculates data approximations nearly as accurately as possible. To the best of our knowledge this is the first practical variant of the LU factorization for effective and efficient low-rank matrix approximation.

Jed A. Duersch, Ming Gu

The dominant contribution to communication complexity in factorizing a matrix using QR with column pivoting is due to column-norm updates that are required to process pivot decisions. We use randomized sampling to approximate this process which dramatically reduces communication in column selection. We also introduce a sample update formula to reduce the cost of sampling trailing matrices. Using our column selection mechanism we observe results that are comparable in quality to those obtained from the QRCP algorithm, but with performance near unpivoted QR. We also demonstrate strong parallel scalability on shared memory multiple core systems using an implementation in Fortran with OpenMP. This work immediately extends to produce low-rank truncated approximations of large matrices. We propose a truncated QR factorization with column pivoting that avoids trailing matrix updates which are used in current implementations of level-3 BLAS QR and QRCP. Provided the truncation rank is small, avoiding trailing matrix updates reduces approximation time by nearly half. By using these techniques and employing a variation on Stewart's QLP algorithm, we develop an approximate truncated SVD that runs nearly as fast as truncated QR.

Hui Kong, Fei He, Xiaoyu Song et al.

A barrier certificate is an inductive invariant function which can be used for the safety verification of a hybrid system. Safety verification based on barrier certificate has the benefit of avoiding explicit computation of the exact reachable set which is usually intractable for nonlinear hybrid systems. In this paper, we propose a new barrier certificate condition, called Exponential Condition, for the safety verification of semi-algebraic hybrid systems. The most important benefit of Exponential Condition is that it has a lower conservativeness than the existing convex condition and meanwhile it possesses the property of convexity. On the one hand, a less conservative barrier certificate forms a tighter over-approximation for the reachable set and hence is able to verify critical safety properties. On the other hand, the property of convexity guarantees its solvability by semidefinite programming method. Some examples are presented to illustrate the effectiveness and practicality of our method.

Ming Gu, Lek-Heng Lim, Cinna Julie Wu

In this article, we propose an algorithm, NESTA-LASSO, for the LASSO problem, i.e., an underdetermined linear least-squares problem with a 1-norm constraint on the solution. We prove under the assumption of the restricted isometry property (RIP) and a sparsity condition on the solution, that NESTA-LASSO is guaranteed to be almost always locally linearly convergent. As in the case of the algorithm NESTA proposed by Becker, Bobin, and Candes, we rely on Nesterov's accelerated proximal gradient method, which takes O(e^{-1/2}) iterations to come within e > 0 of the optimal value. We introduce a modification to Nesterov's method that regularly updates the prox-center in a provably optimal manner, and the aforementioned linear convergence is in part due to this modification. In the second part of this article, we attempt to solve the basis pursuit denoising BPDN problem (i.e., approximating the minimum 1-norm solution to an underdetermined least squares problem) by using NESTA-LASSO in conjunction with the Pareto root-finding method employed by van den Berg and Friedlander in their SPGL1 solver. The resulting algorithm is called PARNES. We provide numerical evidence to show that it is comparable to currently available solvers.