Canyi Lu
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View article: Tensor Q-Rank: New Data Dependent Definition of Tensor Rank
Tensor Q-Rank: New Data Dependent Definition of Tensor Rank Open
Recently, the \textit{Tensor Nuclear Norm~(TNN)} regularization based on t-SVD has been widely used in various low tubal-rank tensor recovery tasks. However, these models usually require smooth change of data along the third dimension to e…
View article: Exact Recovery of Tensor Robust Principal Component Analysis under Linear Transforms
Exact Recovery of Tensor Robust Principal Component Analysis under Linear Transforms Open
This work studies the Tensor Robust Principal Component Analysis (TRPCA) problem, which aims to exactly recover the low-rank and sparse components from their sum. Our model is motivated by the recently proposed linear transforms based tens…
View article: Exact Low Tubal Rank Tensor Recovery from Gaussian Measurements
Exact Low Tubal Rank Tensor Recovery from Gaussian Measurements Open
The recent proposed Tensor Nuclear Norm (TNN) [Lu et al., 2016; 2018a] is an interesting convex penalty induced by the tensor SVD [Kilmer and Martin, 2011]. It plays a similar role as the matrix nuclear norm which is the convex surrogate o…
View article: Tensor-Tensor Product Toolbox
Tensor-Tensor Product Toolbox Open
The tensor-tensor product (t-product) [M. E. Kilmer and C. D. Martin, 2011] is a natural generalization of matrix multiplication. Based on t-product, many operations on matrix can be extended to tensor cases, including tensor SVD, tensor s…
View article: Exact Low Tubal Rank Tensor Recovery from Gaussian Measurements
Exact Low Tubal Rank Tensor Recovery from Gaussian Measurements Open
The recent proposed Tensor Nuclear Norm (TNN) [Lu et al., 2016; 2018a] is an interesting convex penalty induced by the tensor SVD [Kilmer and Martin, 2011]. It plays a similar role as the matrix nuclear norm which is the convex surrogate o…
View article: Subspace Clustering by Block Diagonal Representation
Subspace Clustering by Block Diagonal Representation Open
This paper studies the subspace clustering problem. Given some data points approximately drawn from a union of subspaces, the goal is to group these data points into their underlying subspaces. Many subspace clustering methods have been pr…
View article: Nonconvex Sparse Spectral Clustering by Alternating Direction Method of Multipliers and Its Convergence Analysis
Nonconvex Sparse Spectral Clustering by Alternating Direction Method of Multipliers and Its Convergence Analysis Open
Spectral Clustering (SC) is a widely used data clustering method which first learns a low-dimensional embedding U of data by computing the eigenvectors of the normalized Laplacian matrix, and then performs k-means on UT to get the final cl…
View article: Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm
Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm Open
In this paper, we consider the Tensor Robust Principal Component Analysis (TRPCA) problem, which aims to exactly recover the low-rank and sparse components from their sum. Our model is based on the recently proposed tensor-tensor product (…
View article: Nonconvex Sparse Spectral Clustering by Alternating Direction Method of Multipliers and Its Convergence Analysis
Nonconvex Sparse Spectral Clustering by Alternating Direction Method of Multipliers and Its Convergence Analysis Open
Spectral Clustering (SC) is a widely used data clustering method which first learns a low-dimensional embedding $U$ of data by computing the eigenvectors of the normalized Laplacian matrix, and then performs k-means on $U^\top$ to get the …
View article: Tensor Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Tensors via Convex Optimization
Tensor Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Tensors via Convex Optimization Open
This paper studies the Tensor Robust Principal Component (TRPCA) problem which extends the known Robust PCA (Candes et al. 2011) to the tensor case. Our model is based on a new tensor Singular Value Decomposition (t-SVD) (Kilmer and Martin…
View article: Connections Between Nuclear-Norm and Frobenius-Norm-Based Representations
Connections Between Nuclear-Norm and Frobenius-Norm-Based Representations Open
A lot of works have shown that frobenius-norm-based representation (FNR) is competitive to sparse representation and nuclear-norm-based representation (NNR) in numerous tasks such as subspace clustering. Despite the success of FNR in exper…
View article: A Unified Alternating Direction Method of Multipliers by Majorization Minimization
A Unified Alternating Direction Method of Multipliers by Majorization Minimization Open
Accompanied with the rising popularity of compressed sensing, the Alternating Direction Method of Multipliers (ADMM) has become the most widely used solver for linearly constrained convex problems with separable objectives. In this work, w…
View article: Accelerated Stochastic Mirror Descent Algorithms For Composite Non-strongly Convex Optimization
Accelerated Stochastic Mirror Descent Algorithms For Composite Non-strongly Convex Optimization Open
We consider the problem of minimizing the sum of the average function consisting of a large number of smooth convex component functions and a general convex function that can be non-differentiable. Although many methods have been proposed …
View article: Accelerated Randomized Mirror Descent Algorithms For Composite Non-strongly Convex Optimization
Accelerated Randomized Mirror Descent Algorithms For Composite Non-strongly Convex Optimization Open
We consider the problem of minimizing the sum of an average function of a large number of smooth convex components and a general, possibly non-differentiable, convex function. Although many methods have been proposed to solve this problem …
View article: Convex Sparse Spectral Clustering: Single-View to Multi-View
Convex Sparse Spectral Clustering: Single-View to Multi-View Open
Spectral clustering (SC) is one of the most widely used methods for data clustering. It first finds a low-dimensional embedding U of data by computing the eigenvectors of the normalized Laplacian matrix, and then performs k-means on UT to …
View article: Fast Proximal Linearized Alternating Direction Method of Multiplier with Parallel Splitting
Fast Proximal Linearized Alternating Direction Method of Multiplier with Parallel Splitting Open
The Augmented Lagragian Method (ALM) and Alternating Direction Method of Multiplier (ADMM) have been powerful optimization methods for general convex programming subject to linear constraint. We consider the convex problem whose objective …
View article: Fast Proximal Linearized Alternating Direction Method of Multiplier with Parallel Splitting
Fast Proximal Linearized Alternating Direction Method of Multiplier with Parallel Splitting Open
The Augmented Lagragian Method (ALM) and Alternating Direction Method of Multiplier (ADMM) have been powerful optimization methods for general convex programming subject to linear constraint. We consider the convex problem whose objective …
View article: Optimized Projections for Compressed Sensing via Direct Mutual Coherence Minimization
Optimized Projections for Compressed Sensing via Direct Mutual Coherence Minimization Open
Compressed Sensing (CS) is a novel technique for simultaneous signal sampling and compression based on the existence of a sparse representation of signal and a projected dictionary $PD$, where $P\in\mathbb{R}^{m\times d}$ is the projection…
View article: Adaptive Nonparametric Image Parsing
Adaptive Nonparametric Image Parsing Open
In this paper, we present an adaptive nonparametric solution to the image parsing task, namely annotating each image pixel with its corresponding category label. For a given test image, first, a locality-aware retrieval set is extracted fr…
View article: Projection onto the capped simplex
Projection onto the capped simplex Open
We provide a simple and efficient algorithm for computing the Euclidean projection of a point onto the capped simplex---a simplex with an additional uniform bound on each coordinate---together with an elementary proof. Both the MATLAB and …
View article: Generalized Singular Value Thresholding
Generalized Singular Value Thresholding Open
This work studies the Generalized Singular Value Thresholding (GSVT) operator associated with a nonconvex function g defined on the singular values of X. We prove that GSVT can be obtained by performing the proximal operator of g on the si…
View article: Optimized Projection for Sparse Representation Based Classification
Optimized Projection for Sparse Representation Based Classification Open
Dimensionality reduction (DR) methods have been commonly used as a principled way to understand the high-dimensional data such as facial images. In this paper, we propose a new supervised DR method called Optimized Projection for Sparse Re…
View article: Correlation Adaptive Subspace Segmentation by Trace Lasso
Correlation Adaptive Subspace Segmentation by Trace Lasso Open
This paper studies the subspace segmentation problem. Given a set of data points drawn from a union of subspaces, the goal is to partition them into their underlying subspaces they were drawn from. The spectral clustering method is used as…
View article: Correntropy Induced L2 Graph for Robust Subspace Clustering
Correntropy Induced L2 Graph for Robust Subspace Clustering Open
In this paper, we study the robust subspace clustering problem, which aims to cluster the given possibly noisy data points into their underlying subspaces. A large pool of previous subspace clustering methods focus on the graph constructio…