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Sparse approximation

About: Sparse approximation is a research topic. Over the lifetime, 18037 publications have been published within this topic receiving 497739 citations. The topic is also known as: Sparse approximation.


Papers
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Journal ArticleDOI
TL;DR: This paper explores the effectiveness of sparse representations obtained by learning a set of overcomplete basis (dictionary) in the context of action recognition in videos and presents the idea of a new local spatio-temporal feature that is distinctive, scale invariant, and fast to compute.
Abstract: This paper explores the effectiveness of sparse representations obtained by learning a set of overcomplete basis (dictionary) in the context of action recognition in videos. Although this work concentrates on recognizing human movements-physical actions as well as facial expressions-the proposed approach is fairly general and can be used to address other classification problems. In order to model human actions, three overcomplete dictionary learning frameworks are investigated. An overcomplete dictionary is constructed using a set of spatio-temporal descriptors (extracted from the video sequences) in such a way that each descriptor is represented by some linear combination of a small number of dictionary elements. This leads to a more compact and richer representation of the video sequences compared to the existing methods that involve clustering and vector quantization. For each framework, a novel classification algorithm is proposed. Additionally, this work also presents the idea of a new local spatio-temporal feature that is distinctive, scale invariant, and fast to compute. The proposed approach repeatedly achieves state-of-the-art results on several public data sets containing various physical actions and facial expressions.

376 citations

Journal Article
TL;DR: The procedure has a complexity linear, or close to linear, in the number of atoms, and allows the use of accelerated gradient techniques to solve the tree-structured sparse approximation problem at the same computational cost as traditional ones using the l1-norm.
Abstract: Sparse coding consists in representing signals as sparse linear combinations of atoms selected from a dictionary. We consider an extension of this framework where the atoms are further assumed to be embedded in a tree. This is achieved using a recently introduced tree-structured sparse regularization norm, which has proven useful in several applications. This norm leads to regularized problems that are difficult to optimize, and in this paper, we propose efficient algorithms for solving them. More precisely, we show that the proximal operator associated with this norm is computable exactly via a dual approach that can be viewed as the composition of elementary proximal operators. Our procedure has a complexity linear, or close to linear, in the number of atoms, and allows the use of accelerated gradient techniques to solve the tree-structured sparse approximation problem at the same computational cost as traditional ones using the l1-norm. Our method is efficient and scales gracefully to millions of variables, which we illustrate in two types of applications: first, we consider fixed hierarchical dictionaries of wavelets to denoise natural images. Then, we apply our optimization tools in the context of dictionary learning, where learned dictionary elements naturally self-organize in a prespecified arborescent structure, leading to better performance in reconstruction of natural image patches. When applied to text documents, our method learns hierarchies of topics, thus providing a competitive alternative to probabilistic topic models.

369 citations

Journal ArticleDOI
TL;DR: This work applies the Laplacian sparse coding to feature quantization in Bag-of-Words image representation, and it outperforms sparse coding and achieves good performance in solving the image classification problem and is successfully used to solve the semi-auto image tagging problem.
Abstract: Sparse coding exhibits good performance in many computer vision applications. However, due to the overcomplete codebook and the independent coding process, the locality and the similarity among the instances to be encoded are lost. To preserve such locality and similarity information, we propose a Laplacian sparse coding (LSc) framework. By incorporating the similarity preserving term into the objective of sparse coding, our proposed Laplacian sparse coding can alleviate the instability of sparse codes. Furthermore, we propose a Hypergraph Laplacian sparse coding (HLSc), which extends our Laplacian sparse coding to the case where the similarity among the instances defined by a hypergraph. Specifically, this HLSc captures the similarity among the instances within the same hyperedge simultaneously, and also makes the sparse codes of them be similar to each other. Both Laplacian sparse coding and Hypergraph Laplacian sparse coding enhance the robustness of sparse coding. We apply the Laplacian sparse coding to feature quantization in Bag-of-Words image representation, and it outperforms sparse coding and achieves good performance in solving the image classification problem. The Hypergraph Laplacian sparse coding is also successfully used to solve the semi-auto image tagging problem. The good performance of these applications demonstrates the effectiveness of our proposed formulations in locality and similarity preservation.

366 citations

Journal ArticleDOI
Fred G. Gustavson1
TL;DR: An O(M) algorithm is produced to solve A x = b where M is the number of multiplications needed to factor A into L U and the concept of an unordered merge plays a key role in obtaining this algorithm.
Abstract: Let A and B be two sparse matrices whose orders are p by q and q by r. Their product C -A B requires N nontrlvial multiplications where 0 <_ N <_ pqr. The operation count of our algorithm is usually proportional to N; however, its worse case is O(p, r, NA, N) where N A is the number of elements in A This algorithm can be used to assemble the sparse matrix arising from a finite element problem from the basic elements, using ~-1 [order (g)]2 operations where m is the total number of basic elements and order(g) is the order of the ~th element matrix. The concept of an unordered merge plays a key role m obtaining our fast multiplication algorithm It forces us to accept an unordered sparse row-wise format as output for the product C The permuted transposition algorithm computes ( R A ) T i n O(p, q, NA) operations where R is a permutation matrix It also orders an unordered sparse row-wise representation. We can combine these algorithms to produce an O(M) algorithm to solve A x = b where M is the number of multiplications needed to factor A into L U

366 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023193
2022454
2021641
2020924
20191,208
20181,371