lp-Recovery of the Most Significant Subspace among Multiple Subspaces with Outliers
University of Minnesota（明尼苏达大学）
Gilad Lerman, Teng Zhang
备注：
This is a revised version of the part of 1002.1994 that deals with single subspace recovery. V3: Improved estimates (in particular for Lemma 3.1 and for estimates relying on it), asymptotic dependence of probabilities and constants on D and d and further clarifications; for simplicity it assumes uniform distributions on spheres. V4: minor revision for the published version
发表信息：
Constructive Approximation, December 2014, Volume 40, Issue 3, pp 329-385
We assume data sampled from a mixture of d-dimensional linear subspaces with spherically symmetric distributions within each subspace and an additional outlier component with spherically symmetric distribution within the ambient space (for simplicity we may assume that all distributions are uniform on their corresponding unit spheres). We also assume mixture weights for the different components. We say that one of the underlying subspaces of the model is most significant if its mixture weight is higher than the sum of the mixture weights of all other subspaces. We study the recovery of the most significant subspace by minimizing the lp-averaged distances of data points from d-dimensional subspaces, where p>0. Unlike other lp minimization problems, this minimization is non-convex for all p>0 and thus requires different methods for its analysis. We show that if 0<p<=1, then for any fraction of outliers the most significant subspace can be recovered by lp minimization with overwhelming probability (which depends on the generating distribution and its parameters). We show that when adding small noise around the underlying subspaces the most significant subspace can be nearly recovered by lp minimization for any 0<p<=1 with an error proportional to the noise level. On the other hand, if p>1 and there is more than one underlying subspace, then with overwhelming probability the most significant subspace cannot be recovered or nearly recovered. This last result does not require spherically symmetric outliers.
Many computer vision tasks involve processing large amounts of data contaminated by outliers, which need to be detected and rejected. While outlier detection methods based on robust statistics have existed for decades, only recently have methods based on sparse and low-rank representation been developed along with guarantees of correct outlier detection when the inliers lie in one or more low-dimensional subspaces. This paper proposes a new outlier detection method that combines tools from sparse representation with random walks on a graph. By exploiting the property that data points can be ex…
Rare data in a large-scale database are called outliers that reveal significant information in the real world. The subspace-based outlier detection is regarded as a feasible approach in very high dimensional space. However, the outliers found in subspaces are only part of the true outliers in high dimensional space, indeed. The outliers hidden in normal-clustered points are sometimes neglected in the projected dimensional subspace. In this paper, we propose a robust subspace method for detecting such inner outliers in a given dataset, which uses two dimensional-projections: detecting outliers …
We introduce and develop a novel approach to outlier detection based on adaptation of random subspace learning. Our proposed method handles both high-dimension low-sample size and traditional low-dimensional high-sample size datasets. Essentially, we avoid the computational bottleneck of techniques like minimum covariance determinant (MCD) by computing the needed determinants and associated measures in much lower dimensional subspaces. Both theoretical and computational development of our approach reveal that it is computationally more efficient than the regularized methods in high-dimensional…
Georgia State University（佐治亚州立大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
Due to diverse nature of data acquisition and modern applications, many contemporary problems involve high dimensional datum $\x \in \R^\d$ whose entries often lie in a union of subspaces and the goal is to find out which entries of $\x$ match with a particular subspace $\sU$, classically called \emph {matched subspace detection}. Consequently, entries that match with one subspace are considered as inliers w.r.t the subspace while all other entries are considered as outliers. Proportion of outliers relative to each subspace varies based on the degree of coordinates from subspaces. This problem…
We study the problem of robust subspace recovery (RSR) in the presence of adversarial outliers. That is, we seek a subspace that contains a large portion of a dataset when some fraction of the data points are arbitrarily corrupted. We first examine a theoretical estimator that is intractable to calculate and use it to derive information-theoretic bounds of exact recovery. We then propose two tractable estimators: a variant of RANSAC and a simple relaxation of the theoretical estimator. The two estimators are fast to compute and achieve state-of-the-art theoretical performance in a noiseless RS…
The subspace approximation problem with outliers, for given $n$ points in $d$ dimensions $x_{1},\ldots, x_{n} \in R^{d}$, an integer $1 \leq k \leq d$, and an outlier parameter $0 \leq α\leq 1$, is to find a $k$-dimensional linear subspace of $R^{d}$ that minimizes the sum of squared distances to its nearest $(1-α)n$ points. More generally, the $\ell_{p}$ subspace approximation problem with outliers minimizes the sum of $p$-th powers of distances instead of the sum of squared distances. Even the case of robust PCA is non-trivial, and previous work requires additional assumptions on the input. …
We study the robustness properties of $\ell_1$ norm minimization for the classical linear regression problem with a given design matrix and contamination restricted to the dependent variable. We perform a fine error analysis of the $\ell_1$ estimator for measurements errors consisting of outliers coupled with noise. We introduce a new estimation technique resulting from a regularization of $\ell_1$ minimization by inf-convolution with the $\ell_2$ norm. Concerning robustness to large outliers, the proposed estimator keeps the breakdown point of the $\ell_1$ estimator, and reduces to least squa…
We propose a neural network for unsupervised anomaly detection with a novel robust subspace recovery layer (RSR layer). This layer seeks to extract the underlying subspace from a latent representation of the given data and removes outliers that lie away from this subspace. It is used within an autoencoder. The encoder maps the data into a latent space, from which the RSR layer extracts the subspace. The decoder then smoothly maps back the underlying subspace to a "manifold" close to the original inliers. Inliers and outliers are distinguished according to the distances between the original and…
This paper investigates differentially private analysis of distance-based outliers. The problem of outlier detection is to find a small number of instances that are apparently distant from the remaining instances. On the other hand, the objective of differential privacy is to conceal presence (or absence) of any particular instance. Outlier detection and privacy protection are thus intrinsically conflicting tasks. In this paper, instead of reporting outliers detected, we present two types of differentially private queries that help to understand behavior of outliers. One is the query to count …
University of Edinburgh（爱丁堡大学）；Amazon Development Centre Scotland（亚马逊苏格兰开发中心）；Google AI（谷歌人工智能）；Alan Turing Institute（艾伦·图灵研究所）
Data cleaning often comprises outlier detection and data repair. Systematic errors result from nearly deterministic transformations that occur repeatedly in the data, e.g. specific image pixels being set to default values or watermarks. Consequently, models with enough capacity easily overfit to these errors, making detection and repair difficult. Seeing as a systematic outlier is a combination of patterns of a clean instance and systematic error patterns, our main insight is that inliers can be modelled by a smaller representation (subspace) in a model than outliers. By exploiting this, we pr…
Iowa State University（爱荷华州立大学）；Université de La Rochelle（拉罗谢尔大学）；University of Warwick（华威大学）
PCA is one of the most widely used dimension reduction techniques. A related easier problem is "subspace learning" or "subspace estimation". Given relatively clean data, both are easily solved via singular value decomposition (SVD). The problem of subspace learning or PCA in the presence of outliers is called robust subspace learning or robust PCA (RPCA). For long data sequences, if one tries to use a single lower dimensional subspace to represent the data, the required subspace dimension may end up being quite large. For such data, a better model is to assume that it lies in a low-dimensional…
This paper considers the problem of robust subspace recovery: given a set of $N$ points in $\mathbb{R}^D$, if many lie in a $d$-dimensional subspace, then can we recover the underlying subspace? We show that Tyler's M-estimator can be used to recover the underlying subspace, if the percentage of the inliers is larger than $d/D$ and the data points lie in general position. Empirically, Tyler's M-estimator compares favorably with other convex subspace recovery algorithms in both simulations and experiments on real data sets.
Given an overcomplete dictionary $A$ and a signal $b = Ac^*$ for some sparse vector $c^*$ whose nonzero entries correspond to linearly independent columns of $A$, classical sparse signal recovery theory considers the problem of whether $c^*$ can be recovered as the unique sparsest solution to $b = A c$. It is now well-understood that such recovery is possible by practical algorithms when the dictionary $A$ is incoherent or restricted isometric. In this paper, we consider the more general case where $b$ lies in a subspace $\mathcal{S}_0$ spanned by a subset of linearly dependent columns of $A$,…
This paper will serve as an introduction to the body of work on robust subspace recovery. Robust subspace recovery involves finding an underlying low-dimensional subspace in a dataset that is possibly corrupted with outliers. While this problem is easy to state, it has been difficult to develop optimal algorithms due to its underlying nonconvexity. This work emphasizes advantages and disadvantages of proposed approaches and unsolved problems in the area.
We study a data model in which the data matrix D can be expressed as D = L + S + C, where L is a low rank matrix, S an element-wise sparse matrix and C a matrix whose non-zero columns are outlying data points. To date, robust PCA algorithms have solely considered models with either S or C, but not both. As such, existing algorithms cannot account for simultaneous element-wise and column-wise corruptions. In this paper, a new robust PCA algorithm that is robust to simultaneous types of corruption is proposed. Our approach hinges on the sparse approximation of a sparsely corrupted column so that…
This paper considers the problem of clustering a collection of unlabeled data points assumed to lie near a union of lower-dimensional planes. As is common in computer vision or unsupervised learning applications, we do not know in advance how many subspaces there are nor do we have any information about their dimensions. We develop a novel geometric analysis of an algorithm named sparse subspace clustering (SSC) [In IEEE Conference on Computer Vision and Pattern Recognition, 2009. CVPR 2009 (2009) 2790-2797. IEEE], which significantly broadens the range of problems where it is provably effecti…
Recent work has demonstrated the effectiveness of gradient descent for directly recovering the factors of low-rank matrices from random linear measurements in a globally convergent manner when initialized properly. However, the performance of existing algorithms is highly sensitive in the presence of outliers that may take arbitrary values. In this paper, we propose a truncated gradient descent algorithm to improve the robustness against outliers, where the truncation is performed to rule out the contributions of samples that deviate significantly from the {\em sample median} of measurement re…
We assume data independently sampled from a mixture distribution on the unit ball of the D-dimensional Euclidean space with K+1 components: the first component is a uniform distribution on that ball representing outliers and the other K components are uniform distributions along K d-dimensional linear subspaces restricted to that ball. We study both the simultaneous recovery of all K underlying subspaces and the recovery of the best l0 subspace (i.e., with largest number of points) by minimizing the lp-averaged distances of data points from d-dimensional subspaces of the D-dimensional space. U…
In this work, we study the robust subspace tracking (RST) problem and obtain one of the first two provable guarantees for it. The goal of RST is to track sequentially arriving data vectors that lie in a slowly changing low-dimensional subspace, while being robust to corruption by additive sparse outliers. It can also be interpreted as a dynamic (time-varying) extension of robust PCA (RPCA), with the minor difference that RST also requires a short tracking delay. We develop a recursive projected compressive sensing algorithm that we call Nearly Optimal RST via ReProCS (ReProCS-NORST) because it…
The K-subspaces (KSS) method is a generalization of the K-means method for subspace clustering. In this work, we present local convergence analysis and a recovery guarantee for KSS, assuming data are generated by the semi-random union of subspaces model, where $N$ points are randomly sampled from $K \ge 2$ overlapping subspaces. We show that if the initial assignment of the KSS method lies within a neighborhood of a true clustering, it converges at a superlinear rate and finds the correct clustering within $Θ(\log\log N)$ iterations with high probability. Moreover, we propose a thresholding in…
This paper explores and analyzes two randomized designs for robust Principal Component Analysis (PCA) employing low-dimensional data sketching. In one design, a data sketch is constructed using random column sampling followed by low dimensional embedding, while in the other, sketching is based on random column and row sampling. Both designs are shown to bring about substantial savings in complexity and memory requirements for robust subspace learning over conventional approaches that use the full scale data. A characterization of the sample and computational complexity of both designs is deriv…
This work presents a fast and non-convex algorithm for robust subspace recovery. The data sets considered include inliers drawn around a low-dimensional subspace of a higher dimensional ambient space, and a possibly large portion of outliers that do not lie nearby this subspace. The proposed algorithm, which we refer to as Fast Median Subspace (FMS), is designed to robustly determine the underlying subspace of such data sets, while having lower computational complexity than existing methods. We prove convergence of the FMS iterates to a stationary point. Further, under a special model of data,…
Many applications in data analysis rely on the decomposition of a data matrix into a low-rank and a sparse component. Existing methods that tackle this task use the nuclear norm and L1-cost functions as convex relaxations of the rank constraint and the sparsity measure, respectively, or employ thresholding techniques. We propose a method that allows for reconstructing and tracking a subspace of upper-bounded dimension from incomplete and corrupted observations. It does not require any a priori information about the number of outliers. The core of our algorithm is an intrinsic Conjugate Gradien…
For the degree corrected stochastic block model in the presence of arbitrary or even adversarial outliers, we develop a convex-optimization-based clustering algorithm that includes a penalization term depending on the positive deviation of a node from the expected number of edges to other inliers. We prove that under mild conditions, this method achieves exact recovery of the underlying clusters. Our synthetic experiments show that our algorithm performs well on heterogeneous networks, and in particular those with Pareto degree distributions, for which outliers have a broad range of possible d…
University of Iowa（爱荷华大学）；University of Texas, Austin（德克萨斯大学奥斯汀分校）
This paper considers the problem of recovering signals from compressed measurements contaminated with sparse outliers, which has arisen in many applications. In this paper, we propose a generative model neural network approach for reconstructing the ground truth signals under sparse outliers. We propose an iterative alternating direction method of multipliers (ADMM) algorithm for solving the outlier detection problem via $\ell_1$ norm minimization, and a gradient descent algorithm for solving the outlier detection problem via squared $\ell_1$ norm minimization. We establish the recovery guaran…
We assume i.i.d. data sampled from a mixture distribution with K components along fixed d-dimensional linear subspaces and an additional outlier component. For p>0, we study the simultaneous recovery of the K fixed subspaces by minimizing the l_p-averaged distances of the sampled data points from any K subspaces. Under some conditions, we show that if $0<p\leq1$, then all underlying subspaces can be precisely recovered by l_p minimization with overwhelming probability. On the other hand, if K>1 and p>1, then the underlying subspaces cannot be recovered or even nearly recovered by l_p minimizat…
UT-Battelle, LLC（UT-巴特勒有限责任公司）；US Department of Energy（美国能源部）
Robust Subspace Recovery (RoSuRe) algorithm was recently introduced as a principled and numerically efficient algorithm that unfolds underlying Unions of Subspaces (UoS) structure, present in the data. The union of Subspaces (UoS) is capable of identifying more complex trends in data sets than simple linear models. We build on and extend RoSuRe to prospect the structure of different data modalities individually. We propose a novel multi-modal data fusion approach based on group sparsity which we refer to as Robust Group Subspace Recovery (RoGSuRe). Relying on a bi-sparsity pursuit paradigm and…
We consider the problem of minimizing a difference-of-convex (DC) function, which can be written as the sum of a smooth convex function with Lipschitz gradient, a proper closed convex function and a continuous possibly nonsmooth concave function. We refine the convergence analysis in [38] for the proximal DC algorithm with extrapolation (pDCA$_e$) and show that the whole sequence generated by the algorithm is convergent when the objective is level-bounded, {\em without} imposing differentiability assumptions in the concave part. Our analysis is based on a new potential function and we assume s…
In applications ranging from communications to genetics, signals can be modeled as lying in a union of subspaces. Under this model, signal coefficients that lie in certain subspaces are active or inactive together. The potential subspaces are known in advance, but the particular set of subspaces that are active (i.e., in the signal support) must be learned from measurements. We show that exploiting knowledge of subspaces can further reduce the number of measurements required for exact signal recovery, and derive universal bounds for the number of measurements needed. The bound is universal in …
High-dimensional data often lie in low-dimensional subspaces corresponding to different classes they belong to. Finding sparse representations of data points in a dictionary built using the collection of data helps to uncover low-dimensional subspaces and address problems such as clustering, classification, subset selection and more. In this paper, we address the problem of recovering sparse representations for noisy data points in a dictionary whose columns correspond to corrupted data lying close to a union of subspaces. We consider a constrained $\ell_1$-minimization and study conditions un…
Outlying Aspect Mining (OAM) aims to find the subspaces (a.k.a. aspects) in which a given query is an outlier with respect to a given dataset. Existing OAM algorithms use traditional distance/density-based outlier scores to rank subspaces. Because these distance/density-based scores depend on the dimensionality of subspaces, they cannot be compared directly between subspaces of different dimensionality. $Z$-score normalisation has been used to make them comparable. It requires to compute outlier scores of all instances in each subspace. This adds significant computational overhead on top of al…
Dynamic robust PCA refers to the dynamic (time-varying) extension of robust PCA (RPCA). It assumes that the true (uncorrupted) data lies in a low-dimensional subspace that can change with time, albeit slowly. The goal is to track this changing subspace over time in the presence of sparse outliers. We develop and study a novel algorithm, that we call simple-ReProCS, based on the recently introduced Recursive Projected Compressive Sensing (ReProCS) framework. Our work provides the first guarantee for dynamic RPCA that holds under weakened versions of standard RPCA assumptions, slow subspace chan…
Homomorphic sensing is a recent algebraic-geometric framework that studies the unique recovery of points in a linear subspace from their images under a given collection of linear maps. It has been successful in interpreting such a recovery in the case of permutations composed by coordinate projections, an important instance in applications known as unlabeled sensing, which models data that are out of order and have missing values. In this paper, we provide tighter and simpler conditions that guarantee the unique recovery for the single-subspace case, extend the result to the case of a subspace…
Massachusetts Institute of Technology（麻省理工学院）；University of Central Florida（中佛罗里达大学）；University of Minnesota（明尼苏达大学）
We present a mathematical analysis of a non-convex energy landscape for robust subspace recovery. We prove that an underlying subspace is the only stationary point and local minimizer in a specified neighborhood under a deterministic condition on a dataset. If the deterministic condition is satisfied, we further show that a geodesic gradient descent method over the Grassmannian manifold can exactly recover the underlying subspace when the method is properly initialized. Proper initialization by principal component analysis is guaranteed with a simple deterministic condition. Under slightly str…
Dalhousie University（达尔豪斯大学）；University of Waterloo（滑铁卢大学）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
Learning non-linear systems from noisy, limited, and/or dependent data is an important task across various scientific fields including statistics, engineering, computer science, mathematics, and many more. In general, this learning task is ill-posed; however, additional information about the data's structure or on the behavior of the unknown function can make the task well-posed. In this work, we study the problem of learning nonlinear functions from corrupted and dependent data. The learning problem is recast as a sparse robust linear regression problem where we incorporate both the unknown c…
National University of Defense Technology（国防科技大学）；Shanxi University（山西大学）
Multi-view subspace clustering has been applied to applications such as image processing and video surveillance, and has attracted increasing attention. Most existing methods learn view-specific self-representation matrices, and construct a combined affinity matrix from multiple views. The affinity construction process is time-consuming, and the combined affinity matrix is not guaranteed to reflect the whole true subspace structure. To overcome these issues, the Latent Complete Row Space Recovery (LCRSR) method is proposed. Concretely, LCRSR is based on the assumption that the multi-view obser…
We consider the problem of clustering noisy high-dimensional data points into a union of low-dimensional subspaces and a set of outliers. The number of subspaces, their dimensions, and their orientations are unknown. A probabilistic performance analysis of the thresholding-based subspace clustering (TSC) algorithm introduced recently in [1] shows that TSC succeeds in the noisy case, even when the subspaces intersect. Our results reveal an explicit tradeoff between the allowed noise level and the affinity of the subspaces. We furthermore find that the simple outlier detection scheme introduced …
Given an overcomplete dictionary $A$ and a signal $b$ that is a linear combination of a few linearly independent columns of $A$, classical sparse recovery theory deals with the problem of recovering the unique sparse representation $x$ such that $b = A x$. It is known that under certain conditions on $A$, $x$ can be recovered by the Basis Pursuit (BP) and the Orthogonal Matching Pursuit (OMP) algorithms. In this work, we consider the more general case where $b$ lies in a low-dimensional subspace spanned by some columns of $A$, which are possibly linearly dependent. In this case, the sparsest s…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Domain generalization asks for models trained over a set of training environments to perform well in unseen test environments. Recently, a series of algorithms such as Invariant Risk Minimization (IRM) has been proposed for domain generalization. However, Rosenfeld et al. (2021) shows that in a simple linear data model, even if non-convexity issues are ignored, IRM and its extensions cannot generalize to unseen environments with less than $d_s+1$ training environments, where $d_s$ is the dimension of the spurious-feature subspace. In this paper, we propose to achieve domain generalization with…
We introduce a two step algorithm with theoretical guarantees to recover a jointly sparse and low-rank matrix from undersampled measurements of its columns. The algorithm first estimates the row subspace of the matrix using a set of common measurements of the columns. In the second step, the subspace aware recovery of the matrix is solved using a simple least square algorithm. The results are verified in the context of recovering CINE data from undersampled measurements; we obtain good recovery when the sampling conditions are satisfied.
National University of Defense Technology（国防科技大学）；University of Sydney（悉尼大学）；Google（谷歌）
Non-negative matrix factorization (NMF) minimizes the Euclidean distance between the data matrix and its low rank approximation, and it fails when applied to corrupted data because the loss function is sensitive to outliers. In this paper, we propose a Truncated CauchyNMF loss that handle outliers by truncating large errors, and develop a Truncated CauchyNMF to robustly learn the subspace on noisy datasets contaminated by outliers. We theoretically analyze the robustness of Truncated CauchyNMF comparing with the competing models and theoretically prove that Truncated CauchyNMF has a generaliza…
Recovering a planted vector $v$ in an $n$-dimensional random subspace of $\mathbb{R}^N$ is a generic task related to many problems in machine learning and statistics, such as dictionary learning, subspace recovery, principal component analysis, and non-Gaussian component analysis. In this work, we study computationally efficient estimation and detection of a planted vector $v$ whose $\ell_4$ norm differs from that of a Gaussian vector with the same $\ell_2$ norm. For instance, in the special case where $v$ is an $N ρ$-sparse vector with Bernoulli-Gaussian or Bernoulli-Rademacher entries, our r…
Recovering linear subspaces from data is a fundamental and important task in statistics and machine learning. Motivated by heterogeneity in Federated Learning settings, we study a basic formulation of this problem: the principal component analysis (PCA), with a focus on dealing with irregular noise. Our data come from $n$ users with user $i$ contributing data samples from a $d$-dimensional distribution with mean $μ_i$. Our goal is to recover the linear subspace shared by $μ_1,\ldots,μ_n$ using the data points from all users, where every data point from user $i$ is formed by adding an independe…
We study the basic problem of robust subspace recovery. That is, we assume a data set that some of its points are sampled around a fixed subspace and the rest of them are spread in the whole ambient space, and we aim to recover the fixed underlying subspace. We first estimate "robust inverse sample covariance" by solving a convex minimization procedure; we then recover the subspace by the bottom eigenvectors of this matrix (their number correspond to the number of eigenvalues close to 0). We guarantee exact subspace recovery under some conditions on the underlying data. Furthermore, we propose…
Group-Aware Matrix Estimation and Latent Subspace Recovery
Department of Statistics, Columbia University（哥伦比亚大学统计系）；Irving Institute for Cellular Dynamics, Zuckerman Institute Columbia University（Zuckerman研究所细胞动力学院，哥伦比亚大学）；School of Data and Information Sciences, University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校数据与信息科学学院）
New York University（纽约大学）；University of Denver（丹佛大学）；Chinese University of Hong Kong（香港中文大学）；ShanghaiTech University（上海科技大学）；Columbia University（哥伦比亚大学）；Johns Hopkins University（约翰斯·霍普金斯大学）
The problem of finding the sparsest vector (direction) in a low dimensional subspace can be considered as a homogeneous variant of the sparse recovery problem, which finds applications in robust subspace recovery, dictionary learning, sparse blind deconvolution, and many other problems in signal processing and machine learning. However, in contrast to the classical sparse recovery problem, the most natural formulation for finding the sparsest vector in a subspace is usually nonconvex. In this paper, we overview recent advances on global nonconvex optimization theory for solving this problem, r…
Diffusion models achieve state-of-the-art performance in various generation tasks. However, their theoretical foundations fall far behind. This paper studies score approximation, estimation, and distribution recovery of diffusion models, when data are supported on an unknown low-dimensional linear subspace. Our result provides sample complexity bounds for distribution estimation using diffusion models. We show that with a properly chosen neural network architecture, the score function can be both accurately approximated and efficiently estimated. Furthermore, the generated distribution based o…
Asymptotic Signal Subspace Recovery in Softmax Attention Models
Faculty of Computer Science and Engineering（计算机科学与工程学院）；Ho Chi Minh City University of Technology（胡志明市技术大学）；Vietnam National University Ho Chi Minh City（越南胡志明市国家大学）；VNU-HCM
