#### DMCA

## Sparse Embedding: A Framework For Sparsity Promoting Dimensionality Reduction

Citations: | 6 - 4 self |

### Citations

1856 | Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories
- Lazebnik, Schmid, et al.
- 2006
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Citation Context ... many important applications. For example, in [10–12] it has been shown that by taking into account non-linearity, one can do better in reconstruction and classification. In addition, spatial pyramid =-=[13]-=-, a popular descriptor for object and scene classification, and region of covariance [14], a popular descriptor for object detection and tracking, both have non-linear distance measures thus making th... |

921 | K-svd: An algorithm for designing overcomplete dictionaries for sparse representation
- Aharon, Elad, et al.
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Citation Context ...ace of output signals after dimensionality reduction.A Framework For Sparsity Promoting Dimensionality Reduction 3 2 Sparse Embedding Framework The classical approach to learn sparse representations =-=[15]-=- is by minimizing the reconstruction error over a finite set of signals subject to some sparsity constraint. Let Y = [y1, . . . , yN] ∈ R n×N denotes the matrix of N input signals, where yi ∈ R n . A ... |

905 | Robust Face Recognition via Sparse Representation
- Wright, Yang, et al.
- 2009
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Citation Context ... 59.1 62.0 - 66.2 Lazebnik [13] - - 56.4 - - 64.6 Griffin [19] 44.2 54.5 59.0 63.3 65.8 67.6 Irani [20] - - 65.0 - - 70.4 Yang [21] - - 67.0 - - 73.2 Wang [22] 51.15 59.77 65.43 67.74 70.16 73.44 SRC =-=[6]-=- 48.8 60.1 64.9 67.7 69.2 70.7 KSVD [15] 49.8 59.8 65.2 68.7 71.0 73.2 D-KSVD [23] 49.6 59.5 65.1 68.6 71.1 73.0 LC-KSVD [24] 54.0 63.1 67.7 70.5 72.3 73.6 LASERC 55.2 65.6 69.5 73.1 75.8 77.3 # train... |

619 | Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition
- Pati, Rezaiifar, et al.
- 1993
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Citation Context ...e similar performances for applications like object classification. Sparse coding that solves for X can be done by any off-the-shelves pursuit algorithms. We use the orthogonal matching pursuit (OMP) =-=[17]-=- due to its high efficiency and effectiveness. Note that sparse coding is the most expensive step in many dictionary learning algorithms. Kernel KSVD [11] is an extreme example where sparse coding has... |

593 |
Image denoising via sparse and redundant representations over learned dictionaries
- Elad, Aharon
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Citation Context ..., it is sufficient to seek an optimal solution for the optimization in Eq. (3) through A and B. By substituting Eq. (4) into Eq. (3), we have: CY(P, D, X) = ∥A T K(I − BX)∥ 2 F + λ∥Y(I − AA T K)∥ 2 F =-=(5)-=- where K = Y T Y and λ is a regularization parameter. The equality constraint becomes PP T = A T KA = I (6) The solution can be derived as {A ∗ , B ∗ , X ∗ } = argmin A,B,X ( T 2 ∥A K(I − BX)∥F + λ∥Y(... |

481 | Linear spatial pyramid matching using sparse coding for image classification
- Yang, Yu, et al.
- 2009
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Citation Context ...tom row) of LASERC. # train samples 5 10 15 20 25 30 Malik [18] 46.6 55.8 59.1 62.0 - 66.2 Lazebnik [13] - - 56.4 - - 64.6 Griffin [19] 44.2 54.5 59.0 63.3 65.8 67.6 Irani [20] - - 65.0 - - 70.4 Yang =-=[21]-=- - - 67.0 - - 73.2 Wang [22] 51.15 59.77 65.43 67.74 70.16 73.44 SRC [6] 48.8 60.1 64.9 67.7 69.2 70.7 KSVD [15] 49.8 59.8 65.2 68.7 71.0 73.2 D-KSVD [23] 49.6 59.5 65.1 68.6 71.1 73.0 LC-KSVD [24] 54... |

434 |
Caltech-256 object category dataset
- Griffin, Holub, et al.
- 2007
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Citation Context ...onding to the highest accuracy (top row) and the lowest accuracy (bottom row) of LASERC. # train samples 5 10 15 20 25 30 Malik [18] 46.6 55.8 59.1 62.0 - 66.2 Lazebnik [13] - - 56.4 - - 64.6 Griffin =-=[19]-=- 44.2 54.5 59.0 63.3 65.8 67.6 Irani [20] - - 65.0 - - 70.4 Yang [21] - - 67.0 - - 73.2 Wang [22] 51.15 59.77 65.43 67.74 70.16 73.44 SRC [6] 48.8 60.1 64.9 67.7 69.2 70.7 KSVD [15] 49.8 59.8 65.2 68.... |

429 | Locality-constrained linear coding for image classification
- Wang, Yang, et al.
- 2010
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Citation Context ...samples 5 10 15 20 25 30 Malik [18] 46.6 55.8 59.1 62.0 - 66.2 Lazebnik [13] - - 56.4 - - 64.6 Griffin [19] 44.2 54.5 59.0 63.3 65.8 67.6 Irani [20] - - 65.0 - - 70.4 Yang [21] - - 67.0 - - 73.2 Wang =-=[22]-=- 51.15 59.77 65.43 67.74 70.16 73.44 SRC [6] 48.8 60.1 64.9 67.7 69.2 70.7 KSVD [15] 49.8 59.8 65.2 68.7 71.0 73.2 D-KSVD [23] 49.6 59.5 65.1 68.6 71.1 73.0 LC-KSVD [24] 54.0 63.1 67.7 70.5 72.3 73.6 ... |

340 |
Optimal global rates of convergence for nonparametric regression
- Stone
- 1982
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Citation Context ...onal manifold embedded in a high dimensional space. Dealing with the high-dimension is not practical for both learning and inference tasks. As an example of the effect of dimension on learning, Stone =-=[1]-=- showed that, under certain regularity assumption including that samples are identically independent distributed, the optimal rate of convergence for non-parametric regression decreases exponentially ... |

331 | SVM-KNN: Discriminative nearest neighbor classification for visual category recognition
- Zhang, Berg, et al.
- 2007
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Citation Context ... (35%) (f) Seahorse (42%) Fig. 6. Sample images from the classes corresponding to the highest accuracy (top row) and the lowest accuracy (bottom row) of LASERC. # train samples 5 10 15 20 25 30 Malik =-=[18]-=- 46.6 55.8 59.1 62.0 - 66.2 Lazebnik [13] - - 56.4 - - 64.6 Griffin [19] 44.2 54.5 59.0 63.3 65.8 67.6 Irani [20] - - 65.0 - - 70.4 Yang [21] - - 67.0 - - 73.2 Wang [22] 51.15 59.77 65.43 67.74 70.16 ... |

270 | Region covariance: A fast descriptor for detection and classification
- Tuzel, Porikli, et al.
- 2006
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Citation Context ...to account non-linearity, one can do better in reconstruction and classification. In addition, spatial pyramid [13], a popular descriptor for object and scene classification, and region of covariance =-=[14]-=-, a popular descriptor for object detection and tracking, both have non-linear distance measures thus making the current sparse representation inappropriate. In this paper, we propose a novel framewor... |

260 | In defense of nearest-neighbor based image classification
- Boiman, Shechtman, et al.
- 2008
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Citation Context ...and the lowest accuracy (bottom row) of LASERC. # train samples 5 10 15 20 25 30 Malik [18] 46.6 55.8 59.1 62.0 - 66.2 Lazebnik [13] - - 56.4 - - 64.6 Griffin [19] 44.2 54.5 59.0 63.3 65.8 67.6 Irani =-=[20]-=- - - 65.0 - - 70.4 Yang [21] - - 67.0 - - 73.2 Wang [22] 51.15 59.77 65.43 67.74 70.16 73.44 SRC [6] 48.8 60.1 64.9 67.7 69.2 70.7 KSVD [15] 49.8 59.8 65.2 68.7 71.0 73.2 D-KSVD [23] 49.6 59.5 65.1 68... |

253 | On feature combination for multiclass object classification
- Gehler, Nowozin
- 2009
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Citation Context ...cted from Internet. The database contains a diverse and challenging set of images from buildings, musical instruments, animals and natural scenes, etc. We used the combination of 39 descriptors as in =-=[28]-=-. We follow the suggested protocol in [13, 29], namely, we train on m images, where m ∈ {5, 10, 15, 20, 25, 30}, and test on the rest. The corresponding parameters settings of SE are: T0 = {3, 4, 5, 7... |

251 |
Sparse and Redundant Representations. From Theory to Applications
- Elad
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Citation Context ...e such that non-informative or irrelevant information in the data are discarded. Recently, there has been an explosion of activities in modeling a signal using appropriate sparse representations (see =-=[4]-=- and references therein). This approach is motivated by the observation that most signals encountered in practical applications are compressible. In other words, their sorted coefficient magnitudes in... |

155 |
Nonlinear Dimensionality Reduction
- Lee, Verleysen
- 2007
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Citation Context ... the inference task harder. This is known as the concentration phenomenon [2]. To address these issues, various linear and non-linear dimensionality reduction (DR) techniques have been developed (see =-=[3]-=- and references therein). In general, these techniques map data to a lower-dimensional space such that non-informative or irrelevant information in the data are discarded. Recently, there has been an ... |

122 | Kernel codebooks for scene categorization
- Gemert, Geusebroek, et al.
- 2008
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Citation Context ... 59.8 65.2 68.7 71.0 73.2 D-KSVD [23] 49.6 59.5 65.1 68.6 71.1 73.0 LC-KSVD [24] 54.0 63.1 67.7 70.5 72.3 73.6 LASERC 55.2 65.6 69.5 73.1 75.8 77.3 # train samples 15 30 Griffin [19] 28.3 34.1 Gemert =-=[25]-=- - 27.2 Yang [26] 34.4 41.2 LASERC 35.2 43.6 Time Train (s) Test (ms) SVM 2.1 8.1 Ker. KSVD 2598 3578 SRC N/A 520 D-KSVD 450 12.8 LASERC 7.2 9.4 Table 1. Comparison of recognition results on Caltech-1... |

102 | Discriminative K-SVD for dictionary learning in face recognition
- Zhang, Li
- 2010
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Citation Context ...3 65.8 67.6 Irani [20] - - 65.0 - - 70.4 Yang [21] - - 67.0 - - 73.2 Wang [22] 51.15 59.77 65.43 67.74 70.16 73.44 SRC [6] 48.8 60.1 64.9 67.7 69.2 70.7 KSVD [15] 49.8 59.8 65.2 68.7 71.0 73.2 D-KSVD =-=[23]-=- 49.6 59.5 65.1 68.6 71.1 73.0 LC-KSVD [24] 54.0 63.1 67.7 70.5 72.3 73.6 LASERC 55.2 65.6 69.5 73.1 75.8 77.3 # train samples 15 30 Griffin [19] 28.3 34.1 Gemert [25] - 27.2 Yang [26] 34.4 41.2 LASER... |

101 | Fast Image Search for Learned Metrics
- Jain, Kulis, et al.
- 2008
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Citation Context ...a diverse and challenging set of images from buildings, musical instruments, animals and natural scenes, etc. We used the combination of 39 descriptors as in [28]. We follow the suggested protocol in =-=[13, 29]-=-, namely, we train on m images, where m ∈ {5, 10, 15, 20, 25, 30}, and test on the rest. The corresponding parameters settings of SE are: T0 = {3, 4, 5, 7, 8, 9}, d = {5, 10, 15, 20, 25, 30}, and λ = ... |

94 |
G.: Classification and clustering via dictionary learning with structured incoherence and shared features. CVPR
- Ramirez, Sprechmann, et al.
- 2010
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Citation Context ...d λ is a regularization parameter. The equality constraint becomes PP T = A T KA = I (6) The solution can be derived as {A ∗ , B ∗ , X ∗ } = argmin A,B,X ( T 2 ∥A K(I − BX)∥F + λ∥Y(I − AA T K)∥ 2 ) F =-=(7)-=- subject to: A T KA = I, and ∥xi∥0 ≤ T0 The advantage of this formulation will become clear later. Basically, this formulation allows a joint update of P and D via A. As we shall see later in section ... |

86 | Task-driven dictionary learning
- Mairal, Bach, et al.
- 2012
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Citation Context ...The cost function in (1) promotes a dictionary D that can best represent Y by linearly combining only a few of its columns. This type of optimization can be done efficiently using the current methods =-=[12, 15]-=-. Different from the classical approaches, we develop an algorithm that embeds input signals into a low-dimensional space, and simultaneously learns an optimized dictionary. Let M denote the mapping t... |

85 | Learning A Discriminative Dictionary for Sparse Coding via Label Consistent K-SVD
- Jiang, Lin, et al.
- 2011
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Citation Context ...ng [21] - - 67.0 - - 73.2 Wang [22] 51.15 59.77 65.43 67.74 70.16 73.44 SRC [6] 48.8 60.1 64.9 67.7 69.2 70.7 KSVD [15] 49.8 59.8 65.2 68.7 71.0 73.2 D-KSVD [23] 49.6 59.5 65.1 68.6 71.1 73.0 LC-KSVD =-=[24]-=- 54.0 63.1 67.7 70.5 72.3 73.6 LASERC 55.2 65.6 69.5 73.1 75.8 77.3 # train samples 15 30 Griffin [19] 28.3 34.1 Gemert [25] - 27.2 Yang [26] 34.4 41.2 LASERC 35.2 43.6 Time Train (s) Test (ms) SVM 2.... |

42 |
Multi-frame compression: Theory and design
- Engan, Aase, et al.
- 2000
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Citation Context ...B T K T QKBX ) (15) A possible way of solving for B is by taking the derivative of the objective function with respect to B and setting it to zero, which yields: −2(K T QK)X T + 2(K T QK)B(XX T ) = 0 =-=(16)-=- B = X T (XX T ) † (17) This is similar to the method of optimal direction (MOD) [16] updating step except that B is not the dictionary but its representation coefficients over the training data Y. It... |

22 |
Ramakrishnan R., Shaft U..: ‘When Is “Nearest Neighbor” Meaningful?‘, submitted for publication
- Beyer, Goldstein
- 1998
(Show Context)
Citation Context ...on of the data. As the dimension increases, the Euclidean distances between feature vectors become closer to each other making the inference task harder. This is known as the concentration phenomenon =-=[2]-=-. To address these issues, various linear and non-linear dimensionality reduction (DR) techniques have been developed (see [3] and references therein). In general, these techniques map data to a lower... |

9 |
Discriminative affine sparse codes for image classification
- Kulkarni, Li
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Citation Context ...1.0 73.2 D-KSVD [23] 49.6 59.5 65.1 68.6 71.1 73.0 LC-KSVD [24] 54.0 63.1 67.7 70.5 72.3 73.6 LASERC 55.2 65.6 69.5 73.1 75.8 77.3 # train samples 15 30 Griffin [19] 28.3 34.1 Gemert [25] - 27.2 Yang =-=[26]-=- 34.4 41.2 LASERC 35.2 43.6 Time Train (s) Test (ms) SVM 2.1 8.1 Ker. KSVD 2598 3578 SRC N/A 520 D-KSVD 450 12.8 LASERC 7.2 9.4 Table 1. Comparison of recognition results on Caltech-101 dataset (left)... |

8 | Kernel dictionary learning
- Nguyen, Patel, et al.
(Show Context)
Citation Context ... We use the orthogonal matching pursuit (OMP) [17] due to its high efficiency and effectiveness. Note that sparse coding is the most expensive step in many dictionary learning algorithms. Kernel KSVD =-=[11]-=- is an extreme example where sparse coding has to be done in the high-dimensional feature space. Our algorithm performs sparse coding in the reduced space leading to a huge computational advantage, ye... |

8 |
Learning Generative Visual Models from Few Training Examples: An Incremental Bayesian Approach Tested on
- Fei-FeiL, Perona
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Citation Context ...256 dataset (upper right), and the computing time (lower right). Caltech-101 and Caltech-256 Object Recognition We perform the second set of object recognition experiments on the Caltech-101 database =-=[27]-=-. This database comprises of 101 object classes, and 1 background class collected from Internet. The database contains a diverse and challenging set of images from buildings, musical instruments, anim... |

7 | On the dimensionality reduction for sparse representation based face recognition
- Zhang, Yang, et al.
- 2010
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Citation Context ...iques are not designed to respect and promote underlying sparse structures of data. Therefore, they cannot help the process of learning the dictionary D. Note that the recently developed DR technique =-=[8, 9]-=- based on the sparse linear model is also not suitable for the purpose of sparse learning since it assumes that the dictionary is given. Ideally, we want an algorithm that can discard non-informative ... |

4 | Using the kernel trick in compressive sensing: Accurate signal recovery from fewer measurements
- Qi, Hughes
- 2011
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Citation Context ...] subject to: G T G = I (9) where H = S 1 2 V T ((I − BX)(I − BX) T − λI)VS 1 2 ∈ R N×N . Proof. The cost function can be expanded as follows: CY(P, D, X) = ∥A T K(I − BX)∥ 2 F + λ∥Y(I − AA T K)∥ 2 F =-=(10)-=- = tr [ (I − BX)(I − BX) T K T Q T K + λ(K − 2K T Q T K + K T Q T KQK) ] (11) where Q = AAT ∈ RN×N . The constraint AT KA = I leads to the new constraint AAT KAAT = AAT or QKQT = Q. Using this equalit... |

3 | Dimensionality reduction using the sparse linear model
- IA, Zickler
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Citation Context ...int of (7). The following proposition shows that A can be solved efficiently after some algebraic manipulation: Proposition 2 The optimal solution of (7) when B and X are fixed is: A ∗ 1 − = VS 2 G ∗ =-=(8)-=- where V and S come from the eigendecomposition of K = VSV T , and G ∗ ∈ R N×d is the optimal solution of the following minimum eigenvalues problem: {G ∗ } = argmin G tr [ G T HG ] subject to: G T G =... |