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Robust Distance Metric Learning via Simultaneous ℓ1Norm Minimization and Maximization
"... Traditional distance metric learning with side information usually formulates the objectives using the covariance matrices of the data point pairs in the two constraint sets of mustlinks and cannotlinks. Because the covariance matrix computes the sum of the squared ℓ2norm distances, it is prone ..."
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Traditional distance metric learning with side information usually formulates the objectives using the covariance matrices of the data point pairs in the two constraint sets of mustlinks and cannotlinks. Because the covariance matrix computes the sum of the squared ℓ2norm distances, it is prone to both outlier samples and outlier features. To develop a robust distance metric learning method, we propose a new objective for distance metric learning using the ℓ1norm distances. The resulted objective is challenging to solve, because it simultaneously minimizes and maximizes (minmax) a number of nonsmooth ℓ1norm terms. As an important theoretical contribution of this paper, we systematically derive an efficient iterative algorithm to solve the general ℓ1norm minmax problem. We performed extensive empirical evaluations, where our new distance metric learning method outperforms related stateoftheart methods in a variety of experimental settings. 1.
Optimal Mean Robust Principal Component Analysis
"... Principal Component Analysis (PCA) is the most widely used unsupervised dimensionality reduction approach. In recent research, several robust PCA algorithms were presented to enhance the robustness of PCA model. However, the existing robust PCA methods incorrectly center the data using the `2norm ..."
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Principal Component Analysis (PCA) is the most widely used unsupervised dimensionality reduction approach. In recent research, several robust PCA algorithms were presented to enhance the robustness of PCA model. However, the existing robust PCA methods incorrectly center the data using the `2norm distance to calculate the mean, which actually is not the optimal mean due to the `1norm used in the objective functions. In this paper, we propose novel robust PCA objective functions with removing optimal mean automatically. Both theoretical analysis and empirical studies demonstrate our new methods can more effectively reduce data dimensionality than previous robust PCA methods. 1.
Robust Tensor Clustering with NonGreedy Maximization
 BERG, BELHUMEUR: TOMVSPETE CLASSIFIERS AND IDENTITYPRESERVING ALIGNMENT
"... Tensors are increasingly common in several areas such as data mining, computer graphics, and computer vision. Tensor clustering is a fundamental tool for data analysis and pattern discovery. However, there usually exist outlying data points in realworld datasets, which will reduce the performance o ..."
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Tensors are increasingly common in several areas such as data mining, computer graphics, and computer vision. Tensor clustering is a fundamental tool for data analysis and pattern discovery. However, there usually exist outlying data points in realworld datasets, which will reduce the performance of clustering. This motivates us to develop a tensor clustering algorithm that is robust to the outliers. In this paper, we propose an algorithm of Robust Tensor Clustering (RTC). The RTC firstly finds a lower rank approximation of the original tensor data using a L1 norm optimization function. Because the L1 norm doesn’t exaggerate the effect of outliers compared with L2 norm, the minimization of the L1 norm approximation function makes RTC robust to outliers. Then we compute the HOSVD decomposition of this approximate tensor to obtain the final clustering results. Different from the traditional algorithm solving the approximation function with a greedy strategy, we utilize a nongreedy strategy to obtain a better solution. Experiments demonstrate that RTC has better performance than the stateoftheart algorithms and is more robust to outliers.
Some options for L1subspace signal processing
 in Proc. IEEE ISWCS 2013
, 2013
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1Principal Component Analysis by Lpnorm
"... This paper proposes several principal component analysis methods based on Lpnorm optimization techniques. In doing so, the objective function is defined using the Lpnorm with an arbitrary p value, and the gradient of the objective function is computed on the basis of the fact that the number of t ..."
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This paper proposes several principal component analysis methods based on Lpnorm optimization techniques. In doing so, the objective function is defined using the Lpnorm with an arbitrary p value, and the gradient of the objective function is computed on the basis of the fact that the number of training samples is finite. In the first part, an easier problem of extracting only one feature is dealt with. In this case, principal components are searched for either by a gradient ascent method or by a Lagrangian multiplier method. When more than one feature is needed, features can be extracted one by one greedily, based on the proposed method. Secondly, a more difficult problem is tackled that simultaneously extracts more than one feature. The proposed methods are shown to find a local optimal solution. In addition, they are easy to implement without significantly increasing computational complexity. Lastly, the proposed methods are applied to several datasets with different values of p, and their performances are compared with those of conventional PCA methods.
Optimal Algorithms forsubspace Signal Processing
"... Abstract—We describe ways to define and calculatenorm signal subspaces that are less sensitive to outlying data thancalculated subspaces. We start with the computation of the maximumprojection principal component of a data matrix containing signal samples of dimension. We show that while the ge ..."
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Abstract—We describe ways to define and calculatenorm signal subspaces that are less sensitive to outlying data thancalculated subspaces. We start with the computation of the maximumprojection principal component of a data matrix containing signal samples of dimension. We show that while the general problem is formally NPhard in asymptotically large, , the case of engineering interest of fixed dimension and asymptotically large sample size is not. In particular, for the case where the sample size is less than the fixed dimension, we present in explicit form an optimal algorithm of computational cost. For the case, we present an optimal algorithm of complexity. We generalize to multiplemaxprojection components and present an explicit optimal subspace calculation algorithm of complexity where is the desired number of principal components (subspace rank). We conclude with illustrations ofsubspace signal processing in the fields of data dimensionality reduction, directionofarrival estimation, and image conditioning/restoration. Index Terms — norm, norm, dimensionality reduction, directionofarrival estimation, eigendecomposition, erroneous data, faulty measurements, machine learning, outlier resistance, subspace signal processing. I.
Shifted Subspaces Tracking on Sparse Outlier for Motion Segmentation
 PROCEEDINGS OF THE TWENTYTHIRD INTERNATIONAL JOINT CONFERENCE ON ARTIFICIAL INTELLIGENCE
"... In lowrank & sparse matrix decomposition, the entries of the sparse part are often assumed to be i.i.d. sampled from a random distribution. But the structure of sparse part, as the central interest of many problems, has been rarely studied. One motivating problem is tracking multiple sparse obj ..."
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In lowrank & sparse matrix decomposition, the entries of the sparse part are often assumed to be i.i.d. sampled from a random distribution. But the structure of sparse part, as the central interest of many problems, has been rarely studied. One motivating problem is tracking multiple sparse object flows (motions) in video. We introduce “shifted subspaces tracking (SST)” to segment the motions and recover their trajectories by exploring the lowrank property of background and the shifted subspace property of each motion. SST is composed of two steps, background modeling and flow tracking. In step 1, we propose “semisoft GoDec ” to separate all the motions from the lowrank background L as a sparse outlier S. Its softthresholding in updating S significantly speeds up GoDec and facilitates the parameter tuning. In step 2, we update X as S obtained in step 1 and develop “SST algorithm” further decomposing X as X = ∑ k i=1 L(i)◦τ(i)+ S+G, wherein L(i) is a lowrank matrix storing the i th flow after transformation τ(i). SST algorithm solves k subproblems in sequel by alternating minimization, each of which recovers one L(i) and its τ(i) by randomized method. Sparsity of L(i) and betweenframe affinity are leveraged to save computations. We justify the effectiveness of SST on surveillance video sequences.
Creative Commons License This work is licensed under a Creative Commons Attribution 3.0 License Principal Component Analysis and Optimization: A Tutorial
"... Abstract Principal component analysis (PCA) is one of the most widely used multivariate techniques in statistics. It is commonly used to reduce the dimensionality of data in order to examine its underlying structure and the covariance/correlation structure of a set of variables. While singular valu ..."
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Abstract Principal component analysis (PCA) is one of the most widely used multivariate techniques in statistics. It is commonly used to reduce the dimensionality of data in order to examine its underlying structure and the covariance/correlation structure of a set of variables. While singular value decomposition provides a simple means for identification of the principal components (PCs) for classical PCA, solutions achieved in this manner may not possess certain desirable properties including robustness, smoothness, and sparsity. In this paper, we present several optimization problems related to PCA by considering various geometric perspectives. New techniques for PCA can be developed by altering the optimization problems to which principal component loadings are the optimal solutions.
1Robust Bayesian Tensor Factorization for Incomplete Multiway Data
"... Abstract—We propose a generative model for robust tensor factorization in the presence of both missing data and outliers. The objective is to explicitly infer the underlying lowCPrank tensor capturing the global information and a sparse tensor capturing the local information (also considered as ou ..."
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Abstract—We propose a generative model for robust tensor factorization in the presence of both missing data and outliers. The objective is to explicitly infer the underlying lowCPrank tensor capturing the global information and a sparse tensor capturing the local information (also considered as outliers), thus providing the robust predictive distribution over missing entries. The lowCPrank tensor is modeled by multilinear interactions between multiple latent factors on which the column sparsity is enforced by a hierarchical prior, while the sparse tensor is modeled by a hierarchical view of Studentt distribution that associates an individual hyperparameter with each element independently. For model learning, we develop an efficient closedform variational inference under a fully Bayesian treatment, which can effectively prevent the overfitting problem and scales linearly with data size. In contrast to existing related works, our method can perform model selection automatically and implicitly without need of tuning parameters. More specifically, it can discover the groundtruth of CP rank and automatically adapt the sparsity inducing priors to various types of outliers. In addition, the tradeoff between the lowrank approximation and the sparse representation can be optimized in the sense of maximum model evidence. The extensive experiments and comparisons with many stateoftheart algorithms on both synthetic and realworld datasets demonstrate the superiorities of our method from several perspectives. Index Terms—Tensor factorization, tensor completion, robust factorization, rank determination, variational Bayesian inference, video background modeling F 1