Low-rank Projection Research Articles

A Framework of Joint Low-Rank and Sparse Regression for Image Memorability Prediction

Image memorability is to measure the degree to which an image is remembered. Generally image memorability prediction involves two steps: feature representation and prediction. Most previous work just focused on addressing the first step by investigating the factors of making an image memorable. They not only lack the use of a learning mechanism in feature representation, but also often neglect the second step. In this paper, we first propose a joint low-rank and sparse regression (JLRSR) framework to address this problem. JLRSR aims to jointly learn: 1) a low-rank projection matrix that enables us to decompose the original data into a component part and an error part and 2) a sparse regression coefficient vector for image memorability prediction. The projection matrix and the regression coefficients are bound by a sparse constraint to make our approach invariant to training samples. Moreover, a graph regularizer is constructed to improve the generalization performance and prevent overfitting. We then extend JLRSR to a multi-view version called Mv-JLRSR by imposing the block-wise constraint to ensure the group effect and the view correlation constraint to eliminate the heterogeneity among views. Experiment results validate the effectiveness of our proposed approaches.

Latent graph-regularized inductive robust principal component analysis

Recovering low-rank subspaces for data sets becomes an attractive problem in recent years. We proposed a new low-rank subspace learning algorithm, termed latent graph-regularized inductive robust principal component analysis (LGIRPCA), in this paper. Different from the existing low-rank subspace learning methods, LGIRPCA considers the feature manifold structure of a given data set and designs a new Laplacian regularizer to characterize the structure information. We proved that the devised Laplacian regularizer could be transferred to be a weighted sparse constraint for the required low-rank projection matrix. Moreover, the relationships between LGIRPCA and some related algorithms were also discussed. An optimization algorithm based on augmented Lagrange multiplier method was presented to solve LGIRPCA problem. Finally, extensive experiments on several benchmark databases demonstrate the effectiveness of our method for image recover and image classification.

Improving children’s mismatched ASR using structured low-rank feature projection

The work presented in this paper explores the issues in automatic speech recognition (ASR) of children’s speech on acoustic models trained on adults’ speech. In such contexts, due to a large acoustic mismatch between training and test data, highly degraded recognition rates are noted. Even with the use of vocal tract length normalization (VTLN), the mismatched case recognition performance is still much below that for the matched case. Our earlier studies have shown that, for commonly used mel-filterbank-based cepstral features, the acoustic mismatch is exacerbated by insufficient smoothing of pitch harmonics for child speakers. To address this problem, a structured low-rank projection of the features vectors prior to learning the acoustic models as well as before decoding is proposed in this paper. To accomplish this, first a low-rank transform is learned on the training data (adults’ speech). Any dimensionality reduction technique which depends on the variance of the training data may be used for this purpose. In this work, principal component analysis and heteroscedastic linear discriminant analysis have been explored for the same. When the derived low-rank projection is applied in the mismatched testing case, it alleviates the pitch-dependent mismatch. The proposed approach provides a relative recognition performance improvement of 35% over the VTLN included baseline for the children’s mismatched ASR employing acoustic modeling based on hidden Markov models (HMM) with observation densities modeled using Gaussian mixture models (GMM). In addition to that, other acoustic modeling approaches based on subspace GMM (SGMM) and deep neural networks (DNN) have also been explored. Projecting the data to a lower-dimensional subspace is found to be effective in those frameworks as well. In the case of SGMM and DNN-based systems, the proposed approach is noted to result in relative recognition performance improvements of 33% and 21%, respectively, over their corresponding baselines.

Low-Rank Projection Learning via Graph Embedding

With robustness to various corruptions, it is the local geometrical relationship among data that plays an important role in the recognition/clustering task of subspace learning (SL). However, a lot of previous SL methods cannot take into consideration both of the local neighborhood and the robustness, which results in poor performance in image classification and feature extraction. In this paper, a robust SL method is proposed to solve the feature extraction problem, named as Low-Rank Projection Learning via Graph Embedding (LRP-GE). The proposed algorithm enjoys two merits. First, it preserves the local neighborhood information among data by introducing the graph embedding (GE). Second, it alleviates the impact of noise and corruption by learning a robust subspace based on the low-rank projection. We cast the problem as a convex optimization problem and provide an iterative solution that can be solved efficiently in polynomial time. Extensive experiments performed on four benchmark data sets demonstrate that the proposed method performs favorably against other well-established SL methods in image classification.

Low-rank magnetic resonance fingerprinting.

Magnetic resonance fingerprinting (MRF) is a relatively new approach that provides quantitative MRI measures using randomized acquisition. Extraction of physical quantitative tissue parameters is performed offline, without the need of patient presence, based on acquisition with varying parameters and a dictionary generated according to the Bloch equationsimulations. MRF uses hundreds of radio frequency (RF) excitation pulses for acquisition, and therefore, a high undersampling ratio in the sampling domain (k-space) is required for reasonable scanning time. This undersampling causes spatial artifacts that hamper the ability to accurately estimate the tissue's quantitative values. In this work, we introduce a new approach for quantitative MRI using MRF, called magnetic resonance fingerprinting with low rank (FLOR). We exploit the low-rank property of the concatenated temporal imaging contrasts, on top of the fact that the MRF signal is sparsely represented in the generated dictionary domain. We present an iterative recovery scheme that consists of a gradient step followed by a low-rank projection using the singular value decomposition. Experimental results consist of retrospective sampling that allows comparison to a well defined reference, and prospective sampling that shows the performance of FLOR for a real-data sampling scenario. Both experiments demonstrate improved parameter accuracy compared to other compressed-sensing and low-rank based methods for MRF at 5% and 9% sampling ratios for the retrospective and prospective experiments, respectively. We have shown through retrospective and prospective experiments that by exploiting the low-rank nature of the MRF signal, FLOR recovers the MRF temporal undersampled images and provides more accurate parameter maps compared to previous iterative approaches.

Anomaly Detection in Moving-Camera Video Sequences Using Principal Subspace Analysis

This paper presents a family of algorithms based on sparse decompositions that detect anomalies in video sequences obtained from slow moving cameras. These algorithms start by computing the union of subspaces that best represents all the frames from a reference (anomaly free) video as a low-rank projection plus a sparse residue. Then, they perform a low-rank representation of a target (possibly anomalous) video by taking advantage of both the union of subspaces and the sparse residue computed from the reference video. Such algorithms provide good detection results while at the same time obviating the need for previous video synchronization. However, this is obtained at the cost of a large computational complexity, which hinders their applicability. Another contribution of this paper approaches this problem by using intrinsic properties of the obtained data representation in order to restrict the search space to the most relevant subspaces, providing computational complexity gains of up to two orders of magnitude. The developed algorithms are shown to cope well with videos acquired in challenging scenarios, as verified by the analysis of 59 videos from the VDAO database that comprises videos with abandoned objects in a cluttered industrial scenario.

Approximate Low-Rank Projection Learning for Feature Extraction.

Feature extraction plays a significant role in pattern recognition. Recently, many representation-based feature extraction methods have been proposed and achieved successes in many applications. As an excellent unsupervised feature extraction method, latent low-rank representation (LatLRR) has shown its power in extracting salient features. However, LatLRR has the following three disadvantages: 1) the dimension of features obtained using LatLRR cannot be reduced, which is not preferred in feature extraction; 2) two low-rank matrices are separately learned so that the overall optimality may not be guaranteed; and 3) LatLRR is an unsupervised method, which by far has not been extended to the supervised scenario. To this end, in this paper, we first propose to use two different matrices to approximate the low-rank projection in LatLRR so that the dimension of obtained features can be reduced, which is more flexible than original LatLRR. Then, we treat the two low-rank matrices in LatLRR as a whole in the process of learning. In this way, they can be boosted mutually so that the obtained projection can extract more discriminative features. Finally, we extend LatLRR to the supervised scenario by integrating feature extraction with the ridge regression. Thus, the process of feature extraction is closely related to the classification so that the extracted features are discriminative. Extensive experiments are conducted on different databases for unsupervised and supervised feature extraction, and very encouraging results are achieved in comparison with many state-of-the-arts methods.

Sufficient Dimension Reduction via Direct Estimation of the Gradients of Logarithmic Conditional Densities.

Sufficient dimension reduction (SDR) is aimed at obtaining the low-rank projection matrix in the input space such that information about output data is maximally preserved. Among various approaches to SDR, a promising method is based on the eigendecomposition of the outer product of the gradient of the conditional density of output given input. In this letter, we propose a novel estimator of the gradient of the logarithmic conditional density that directly fits a linear-in-parameter model to the true gradient under the squared loss. Thanks to this simple least-squares formulation, its solution can be computed efficiently in a closed form. Then we develop a new SDR method based on the proposed gradient estimator. We theoretically prove that the proposed gradient estimator, as well as the SDR solution obtained from it, achieves the optimal parametric convergence rate. Finally, we experimentally demonstrate that our SDR method compares favorably with existing approaches in both accuracy and computational efficiency on a variety of artificial and benchmark data sets.

Robust Low Rank Subspace Segmentation via Joint $$\ell _{21} $$ ℓ 21 -Norm Minimization

In the real world, data samples are often contaminated. Using these contaminated data samples for subspace segmentation usually leads to segmentation results distorted. To remove error, many existing subspace segmentation algorithms directly use different regularization to model the error of the corresponding type in the objective. However, the priori of errors is difficult to obtain in practice, which leads to the degradation of segmentation performance. In this work, we propose to jointly learn the representation matrix and eliminate the effect of errors in the low rank projection spaces via joint the nuclear norm and $$\ell _{21} $$ -norm minimization on the representation matrix for subspace segmentation, termed as robust low rank subspace segmentation via joint $$\ell _{21} $$ -norm minimization (LR-L21). Numerical experiments indicate that the proposed method can effectively deal with different types of errors possibly existing in data, even without the priori of errors. A simple and efficient algorithm is presented based on the alternating direction method, which is convergent. Extensive experiments on three real datasets demonstrate the effectiveness of the proposed approach.

A Sparse and Low-Rank Near-Isometric Linear Embedding Method for Feature Extraction in Hyperspectral Imagery Classification

A sparse and low-rank near-isometric linear embedding (SLRNILE) method has been proposed to make dimensionality reduction and extract proper features for hyperspectral imagery (HSI) classification. The SLRNILE stands on the theory of the John-Lindenstrauss lemma, and tries to estimate a sparse and low-rank projection matrix that satisfies the restricted isometric property (RIP) condition on all secants of the HSI data. The RIP condition guarantees that the desired linear mapping near-isometrically preserves nearest neighbor points of all HSI pixels. Seeking the desired mapping is then modeled into minimizing a Lagrange multipliers formulation. The alternating direction method of multipliers framework is utilized to solve the above convex program, and column generation techniques are adopted to alleviate the computation memory burden during the optimization procedure. Five experiments on three widely used HSI data sets are designed to completely test the performance of SLRNILE, and experimental results are compared against those of six state-of-the-art feature extraction methods, including principal component analysis, Laplacian eigenmaps, locality preserving projections, neighborhood preserving embedding, sparse nonnegative matrix underapproximation, and random projections. The results show that SLRNILE performs best among all the seven methods, and its computational time is longest of all but still bearable for regular users. Therefore, the SLRNILE can be a good choice for feature extraction in HSI classification.

A Fast Adaptation Approach for Enhanced Automatic Recognition of Children’s Speech with Mismatched Acoustic Models

This study explores issues in automatic speech recognition (ASR) of children’s speech on acoustic models trained using adults’ speech. For acoustic modeling in ASR, the employed front-end features capture the characteristics of the vocal filter while smoothing out those of the source (excitation). Adults’ and children’s speech differ significantly due to large deviation in the acoustic correlates such as pitch, formants, speaking rate, etc. In the context of children’s speech recognition on mismatched acoustic models, the recognition rates remain highly degraded despite use of the vocal tract length normalization (VTLN) for addressing formant mismatch. For commonly used mel-filterbank-based cepstral features, earlier studies have shown that the acoustic mismatch is exacerbated by insufficient smoothing of pitch harmonics for child speakers. To address this problem, a structured low-rank projection of the test features as well as that of the mean and the covariance parameters of the acoustic models was explored in an earlier work. In this paper, a low-latency adaptation scheme is presented for children’s mismatched ASR. The presented fast adaptation approach exploits the earlier reported low-rank projection technique in order to reduce the computational cost. In the proposed approach, developmental data from the children’s domain is partitioned into separate groups on the basis of their estimated VTLN warp factors. A set of adapted acoustic models is then created by combining the low-rank projection with the model space adaptation technique for each of the warp factors. Given the children’s test utterance, first an appropriate pre-adapted model mean supervector is chosen based on its estimated warp factor. The chosen supervector is then optimally scaled. Consequently, only two parameters are required to be estimated, i.e., a warp factor and a model mean scaling factor. Even with such stringent constraints, the proposed adaptation technique results in a relative improvement of about $$44\%$$ over the VTLN included baseline.

Robust Multiview Data Analysis Through Collective Low-Rank Subspace.

Multiview data are of great abundance in real-world applications, since various viewpoints and multiple sensors desire to represent the data in a better way. Conventional multiview learning methods aimed to learn multiple view-specific transformations meanwhile assumed the view knowledge of training, and test data were available in advance. However, they would fail when we do not have any prior knowledge for the probe data's view information, since the correct view-specific projections cannot be utilized to extract effective feature representations. In this paper, we develop a collective low-rank subspace (CLRS) algorithm to deal with this problem in multiview data analysis. CLRS attempts to reduce the semantic gap across multiple views through seeking a view-free low-rank projection shared by multiple view-specific transformations. Moreover, we exploit low-rank reconstruction to build a bridge between the view-specific features and those view-free ones transformed with the CLRS. Furthermore, a supervised cross-view regularizer is developed to couple the within-class data across different views to make the learned collective subspace more discriminative. Our CLRS makes our algorithm more flexible when addressing the challenging issue without any prior knowledge of the probe data's view information. To that end, two different settings of experiments on several multiview benchmarks are designed to evaluate the proposed approach. Experimental results have verified the effective performance of our proposed method by comparing with the state-of-the-art algorithms.

A Preconditioned Low-Rank Projection Method with a Rank-Reduction Scheme for Stochastic Partial Differential Equations

In this study, we consider the numerical solution of large systems of linear equations obtained from the stochastic Galerkin formulation of stochastic partial differential equations (PDEs). We propose an iterative algorithm that exploits the Kronecker product structure of the linear systems. The proposed algorithm efficiently approximates the solutions in low-rank tensor format. Using standard Krylov subspace methods for the data in tensor format is computationally prohibitive due to the rapid growth of tensor ranks during the iterations. To keep tensor ranks low over the entire iteration process, we devise a rank-reduction scheme that can be combined with the iterative algorithm. The proposed rank-reduction scheme identifies an important subspace in the stochastic domain and compresses tensors of high rank on-the-fly during the iterations. The proposed reduction scheme is a coarse-grid method in that the important subspace can be identified inexpensively in a coarse spatial grid setting. The efficiency of the proposed method is illustrated by numerical experiments on benchmark problems.

Learning Fast Low-Rank Projection for Image Classification

Rooted in a basic hypothesis that a data matrix is strictly drawn from some independent subspaces, the low-rank representation (LRR) model and its variations have been successfully applied in various image classification tasks. However, this hypothesis is very strict to the LRR model as it cannot always be guaranteed in real images. Moreover, the hypothesis also prevents the sub-dictionaries of different subspaces from collaboratively representing an image. Fortunately, in supervised image classification, low-rank signal can be extracted from the independent label subspaces (ILS) instead of the independent image subspaces (IIS). Therefore, this paper proposes a projective low-rank representation (PLR) model by directly training a projective function to approximate the LRR derived from the labels. To the best of our knowledge, PLR is the first attempt to use the ILS hypothesis to relax the rigorous IIS hypothesis in the LRR models. We further prove a low-rank effect that the representations learned by PLR have high intraclass similarities and large interclass differences, which are beneficial to the classification tasks. The effectiveness of our proposed approach is validated by the experimental results on three databases.

Low rank magnetic resonance fingerprinting.

Magnetic Resonance Fingerprinting (MRF) is a relatively new approach that provides quantitative MRI using randomized acquisition. Extraction of physical quantitative tissue values is preformed off-line, based on acquisition with varying parameters and a dictionary generated according to the Bloch equations. MRF uses hundreds of radio frequency (RF) excitation pulses for acquisition, and therefore high under-sampling ratio in the sampling domain (k-space) is required. This under-sampling causes spatial artifacts that hamper the ability to accurately estimate the quantitative tissue values. In this work, we introduce a new approach for quantitative MRI using MRF, called Low Rank MRF. We exploit the low rank property of the temporal domain, on top of the well-known sparsity of the MRF signal in the generated dictionary domain. We present an iterative scheme that consists of a gradient step followed by a low rank projection using the singular value decomposition. Experiments on real MRI data demonstrate superior results compared to conventional implementation of compressed sensing for MRF at 15% sampling ratio.

Learning Better Word Embedding by Asymmetric Low-Rank Projection of Knowledge Graph

Word embedding, which refers to low-dimensional dense vector representations of natural words, has demon-strated its power in many natural language processing tasks. However, it may suffer from the inaccurate and incomplete information contained in the free text corpus as training data. To tackle this challenge, there have been quite a few studies that leverage knowledge graphs as an additional information source to improve the quality of word embedding. Although these studies have achieved certain success, they have neglected some important facts about knowledge graphs: 1) many relationships in knowledge graphs are many-to-one, one-to-many or even many-to-many, rather than simply one-to-one; 2) most head entities and tail entities in knowledge graphs come from very different semantic spaces. To address these issues, in this paper, we propose a new algorithm named ProjectNet. ProjectNet models the relationships between head and tail entities after transforming them with different low-rank projection matrices. The low-rank projection can allow non one-to-one relationships between entities, while different projection matrices for head and tail entities allow them to originate in different semantic spaces. The experimental results demonstrate that ProjectNet yields more accurate word embedding than previous studies, and thus leads to clear improvements in various natural language processing tasks.

Sparse error via reweighted Low Rank Representation for face recognition with various illumination and occlusion

In this paper, we present a novel algorithm which use sparse error via reweighted Low Rank Representation (SE_RLRR) for face recognition with various illumination and occlusion. The SE_RLRR is divided into two steps: The first step is to construct low rank projection for each type of training samples with RLRR. The second step is to calculate sparse error matrix on low rank projection for test sample and then classify the test sample with classification criterion which fuse smoothness with edge information of sparse error matrix using weighted sum rules. Experiment results on face database of AR and Extended Yale B confirm that our method is robust to various illumination and occlusion, and has better recognition rate than many other methods.

Image classification based on low-rank matrix recovery and Naive Bayes collaborative representation

Most image classification methods require an expensive learning/training phase to gain high performances. But they frequently encounter problems such as overfitting of parameters and scarcity of training data. In this paper, we present a novel learning-free image classification algorithm under the framework of Naive-Bayes Nearest-Neighbor (NBNN) and collaborative representation, where non-negative sparse coding, low-rank matrix recovery and collaborative representation are jointly employed to obtain more robust and discriminative representation. First, instead of using general sparse coding, non-negative sparse coding combined with max pooling is introduced to further reduce information loss. Second, we use the low-rank matrix recovery technique to decompose the training data of the same class into a discriminative low-rank matrix, in which more structurally correlated information is preserved. As for testing images, a low-rank projection matrix is also learned to remove possible image corruptions. Finally, the classification process is implemented by simply comparing the responses over the different bases. Experimental results on several image datasets demonstrate the effectiveness of our method.

Sparse representation for face recognition by discriminative low-rank matrix recovery

This paper proposes a discriminative low-rank representation (DLRR) method for face recognition in which both the training and test samples are corrupted owing to variations in occlusion and disguise. The proposed method extends the sparse representation-based classification algorithm by incorporating the low-rank structure of data representation. The DLRR algorithm recovers a clean dictionary with enhanced discrimination ability from the corrupted training samples for sparse representation. Simultaneously, it learns a low-rank projection matrix to correct corrupted test samples by projecting them onto their corresponding underlying subspaces. The dictionary elements from different classes are encouraged to be as independent as possible by regularizing the structural incoherence of the original training samples. This leads to a compact representation of a corrected test sample by a linear combination of more dictionary elements from the corrected class. The experimental results on benchmark databases show the effectiveness and robustness of our face recognition technique.

Large Margin Low Rank Tensor Analysis

We present a supervised model for tensor dimensionality reduction, which is called large margin low rank tensor analysis (LMLRTA). In contrast to traditional vector representation-based dimensionality reduction methods, LMLRTA can take any order of tensors as input. And unlike previous tensor dimensionality reduction methods, which can learn only the low-dimensional embeddings with a priori specified dimensionality, LMLRTA can automatically and jointly learn the dimensionality and the low-dimensional representations from data. Moreover, LMLRTA delivers low rank projection matrices, while it encourages data of the same class to be close and of different classes to be separated by a large margin of distance in the low-dimensional tensor space. LMLRTA can be optimized using an iterative fixed-point continuation algorithm, which is guaranteed to converge to a local optimal solution of the optimization problem. We evaluate LMLRTA on an object recognition application, where the data are represented as 2D tensors, and a face recognition application, where the data are represented as 3D tensors. Experimental results show the superiority of LMLRTA over state-of-the-art approaches.