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## Graph Connectivity in Noisy Sparse Subspace Clustering (2016)

### Citations

1699 | On spectral clustering: Analysis and an algorithm
- Ng, Jordan, et al.
(Show Context)
Citation Context ...e construct a counter-example, which shows that our success condition cannot be significantly improved under the adversarial noise model. 3.1 The noiseless case We first review the procedure of vanilla noiseless Sparse Subspace Clustering (SSC, [4, 19]). The first step is to solve the following `1 optimization problem for each data point xi in the input matrix X: min ci∈RN ‖ci‖1, s.t. xi = Xci, cii = 0. (3.1) Afterwards, a similarity graph C ∈ RN×N is constructed as Cij = |[c∗i ]j |+ |[c∗j ]i|, where {c∗i }Ni=1 are optimal solutions to Eq. (3.1). Finally, spectral clustering algorithms (e.g., [16]) are applied on the similarity graph C to cluster the N data points into L clusters as desired. Much work has shown that the similarity graph C satisfies SEP under various data and noise regimes [4, 19, 24, 20]. However, as we remarked earlier, SEP alone does not guarantee perfect clustering because the obtained similarity graph C could be poorly connected [15]. In fact, little is known prov540 Graph Connectivity in Noisy Sparse Subspace Clustering Algorithm 1 Clustering consistent noiseless SSC 1: Input: the noiseless data matrix X. 2: Initialization: Normalize each column of X so that it ha... |

888 | Matrix Perturbation Theory - STEWART, SUN - 1990 |

467 | Simultaneous analysis of lasso and dantzig selector. The Annals of Statistics
- Bickel, Ritov, et al.
- 2009
(Show Context)
Citation Context ...r the adversarial noise model. Remark 3 Some components of Algorithm 2 can be revised to make the method more robust in practical applications. For example, instead of randomly picking d points and computing their range, one could apply robust PCA on all points in the connected component, which is more robust to potential outliers. In addition, the single linkage clustering step could be replaced by k-means clustering, which is more robust to false connections in practice. Remark 4 There has been extensive study of using restricted eigenvalue assumptions in the analysis of Lasso-type problems [1, 13, 2, 18]. However, in our problem the assumption is used in a very different manner. In particular, we used the restricted eigenvalue assumption to prove one key lemma (Lemma C.2) that lower bounds the support size of the optimal solution to a Lasso problem. Such results might be of independent interest as a nice contribution to the analysis of Lasso in general. 3.3 Discussion on Assumption 3.1 Assumption 3.1 requires a spectral gap for every subset of data points in each subspace. This seems a very strong assumption that restricts the maximum tolerable noise magnitude to be very small. In this sectio... |

205 | Generalized principal component analysis (GPCA
- Vidal, Ma, et al.
(Show Context)
Citation Context ...additional postprocessing step the algorithm achieves consistent clustering under mild “general-position” conditions. This simple observation completes previous theoretical analysis of SSC by bridging the gap between SEP and clustering consistency. The general position condition is formally defined in Definition 3.1, which concerns the distribution of data points within a single subspace. Intuitively, it requires that no subspace contains data points that are in “degenerate” positions. Similar assumptions were made for the analysis of some algebraic subspace clustering algorithms such as GPCA [23]. The generally positioned data assumption is very mild and is almost always satisfied in practice. For example, it is satisfied almost surely if data points are i.i.d. generated from any continuous underlying distribution. Definition 3.1 (General position). Fix ` ∈ {1, · · · , L}. We say X(`) is in general position if for all k ≤ d`, any subset of k data points (columns) in X(`) are linearly independent. We say X is in general position if X(`) is in general position for all ` = 1, · · · , L. With the self-expressiveness property and the additional assumption that the data matrix X is in gener... |

145 | Robust subspace segmentation by low-rank representation
- Liu, Lin, et al.
- 2010
(Show Context)
Citation Context ...4], articulated moving parts of a human body [27], illumination of different convex objects under Lambertian model [9] and so on. Subspace clustering is also more generically used in agnostic learning of the best linear mixture structures in the data. For instance, it is used for images/video compression [10], hybrid system identification, disease identification [14] as well as modeling social network communities [3], studying privacy in movie recommendations [28] and inferring router network topology [5]. There is rich literature on algorithmic and theoretical analysis of subspace clustering [4, 12, 8, 17]. Among the many algorithms, sparse subspace clustering (SSC) [4] is arguably the most well-studied due to its elegant formulation, strong empirical performance and provable guarantees to work under relatively weak conditions. The algorithm involves constructing a sparse linear representation of each data point using the remaining dataset as a dictionary. This approach embeds the relationship of the data points into a sparse graph and the intuition is that the data points are likely to choose only those points on the same subspace to linearly represent itself. Then clustering can be obtained b... |

139 | A general framework for motion segmentation: Independent, articulated, rigid, non-rigid, degenerate and nondegenerate
- Yan, Pollefeys
- 2006
(Show Context)
Citation Context ...DUCTION The problem of subspace clustering originates from numerous applications in computer vision and image processing, where there are either physical laws or empirical evidence that ensure a given set of data points to form a union of linear or affine subspaces. Such data Appearing in Proceedings of the 19th International Conference on Artificial Intelligence and Statistics (AISTATS) 2016, Cadiz, Spain. JMLR: W&CP volume 51. Copyright 2016 by the authors. points could be feature trajectories of rigid moving objects captured by an affine camera [4], articulated moving parts of a human body [27], illumination of different convex objects under Lambertian model [9] and so on. Subspace clustering is also more generically used in agnostic learning of the best linear mixture structures in the data. For instance, it is used for images/video compression [10], hybrid system identification, disease identification [14] as well as modeling social network communities [3], studying privacy in movie recommendations [28] and inferring router network topology [5]. There is rich literature on algorithmic and theoretical analysis of subspace clustering [4, 12, 8, 17]. Among the many algorithms, sparse... |

115 | Clustering appearances of objects under varying illumination conditions
- Ho, Yang, et al.
- 2003
(Show Context)
Citation Context ...plications in computer vision and image processing, where there are either physical laws or empirical evidence that ensure a given set of data points to form a union of linear or affine subspaces. Such data Appearing in Proceedings of the 19th International Conference on Artificial Intelligence and Statistics (AISTATS) 2016, Cadiz, Spain. JMLR: W&CP volume 51. Copyright 2016 by the authors. points could be feature trajectories of rigid moving objects captured by an affine camera [4], articulated moving parts of a human body [27], illumination of different convex objects under Lambertian model [9] and so on. Subspace clustering is also more generically used in agnostic learning of the best linear mixture structures in the data. For instance, it is used for images/video compression [10], hybrid system identification, disease identification [14] as well as modeling social network communities [3], studying privacy in movie recommendations [28] and inferring router network topology [5]. There is rich literature on algorithmic and theoretical analysis of subspace clustering [4, 12, 8, 17]. Among the many algorithms, sparse subspace clustering (SSC) [4] is arguably the most well-studied due ... |

95 | Sparse subspace clustering: Algorithm, theory, and applications
- Elhamifar, Vidal
- 2013
(Show Context)
Citation Context ...subspaces of dimension greater then 3. 1 INTRODUCTION The problem of subspace clustering originates from numerous applications in computer vision and image processing, where there are either physical laws or empirical evidence that ensure a given set of data points to form a union of linear or affine subspaces. Such data Appearing in Proceedings of the 19th International Conference on Artificial Intelligence and Statistics (AISTATS) 2016, Cadiz, Spain. JMLR: W&CP volume 51. Copyright 2016 by the authors. points could be feature trajectories of rigid moving objects captured by an affine camera [4], articulated moving parts of a human body [27], illumination of different convex objects under Lambertian model [9] and so on. Subspace clustering is also more generically used in agnostic learning of the best linear mixture structures in the data. For instance, it is used for images/video compression [10], hybrid system identification, disease identification [14] as well as modeling social network communities [3], studying privacy in movie recommendations [28] and inferring router network topology [5]. There is rich literature on algorithmic and theoretical analysis of subspace clustering [4... |

66 |
Oracle inequalities and optimal inference under group sparsity. Annals of Statistics
- Lounici, Pontil, et al.
- 2012
(Show Context)
Citation Context ...r the adversarial noise model. Remark 3 Some components of Algorithm 2 can be revised to make the method more robust in practical applications. For example, instead of randomly picking d points and computing their range, one could apply robust PCA on all points in the connected component, which is more robust to potential outliers. In addition, the single linkage clustering step could be replaced by k-means clustering, which is more robust to false connections in practice. Remark 4 There has been extensive study of using restricted eigenvalue assumptions in the analysis of Lasso-type problems [1, 13, 2, 18]. However, in our problem the assumption is used in a very different manner. In particular, we used the restricted eigenvalue assumption to prove one key lemma (Lemma C.2) that lower bounds the support size of the optimal solution to a Lasso problem. Such results might be of independent interest as a nice contribution to the analysis of Lasso in general. 3.3 Discussion on Assumption 3.1 Assumption 3.1 requires a spectral gap for every subset of data points in each subspace. This seems a very strong assumption that restricts the maximum tolerable noise magnitude to be very small. In this sectio... |

63 | A geometric analysis of subspace clustering with outliers
- Soltanolkotabi, Candes
(Show Context)
Citation Context ...o work under relatively weak conditions. The algorithm involves constructing a sparse linear representation of each data point using the remaining dataset as a dictionary. This approach embeds the relationship of the data points into a sparse graph and the intuition is that the data points are likely to choose only those points on the same subspace to linearly represent itself. Then clustering can be obtained by finding connected components of the graph, or more robustly, using spectral clustering [4]. Assuming data lie exactly or approximately on a union of linear subspaces, 1 it is shown in [4, 19, 24, 20] that under certain separation conditions, this embedded graph will have no edges between any two points in different subspaces. This criterion of success is referred to as the “Self-Expressiveness Property (SEP)” [4, 24] and “Subspace Detection Property (SDP)” [19]. The drawback is that there is no guarantee that the vertices within one cluster form a connected component. Therefore, the solution may potentially over segment the data points. This subtle point was originally raised and partially addressed in [15], reaching an answer that when data are noiseless and intrinsic subspace dimension ... |

52 | Restricted eigenvalue properties for correlated gaussian designs
- Raskutti, Wainwright, et al.
- 2010
(Show Context)
Citation Context ...r the adversarial noise model. Remark 3 Some components of Algorithm 2 can be revised to make the method more robust in practical applications. For example, instead of randomly picking d points and computing their range, one could apply robust PCA on all points in the connected component, which is more robust to potential outliers. In addition, the single linkage clustering step could be replaced by k-means clustering, which is more robust to false connections in practice. Remark 4 There has been extensive study of using restricted eigenvalue assumptions in the analysis of Lasso-type problems [1, 13, 2, 18]. However, in our problem the assumption is used in a very different manner. In particular, we used the restricted eigenvalue assumption to prove one key lemma (Lemma C.2) that lower bounds the support size of the optimal solution to a Lasso problem. Such results might be of independent interest as a nice contribution to the analysis of Lasso in general. 3.3 Discussion on Assumption 3.1 Assumption 3.1 requires a spectral gap for every subset of data points in each subspace. This seems a very strong assumption that restricts the maximum tolerable noise magnitude to be very small. In this sectio... |

45 | Clustering partially observed graphs via convex optimization
- Jalali, Chen, et al.
- 2011
(Show Context)
Citation Context ...atistics (AISTATS) 2016, Cadiz, Spain. JMLR: W&CP volume 51. Copyright 2016 by the authors. points could be feature trajectories of rigid moving objects captured by an affine camera [4], articulated moving parts of a human body [27], illumination of different convex objects under Lambertian model [9] and so on. Subspace clustering is also more generically used in agnostic learning of the best linear mixture structures in the data. For instance, it is used for images/video compression [10], hybrid system identification, disease identification [14] as well as modeling social network communities [3], studying privacy in movie recommendations [28] and inferring router network topology [5]. There is rich literature on algorithmic and theoretical analysis of subspace clustering [4, 12, 8, 17]. Among the many algorithms, sparse subspace clustering (SSC) [4] is arguably the most well-studied due to its elegant formulation, strong empirical performance and provable guarantees to work under relatively weak conditions. The algorithm involves constructing a sparse linear representation of each data point using the remaining dataset as a dictionary. This approach embeds the relationship of the dat... |

41 | Y.: Multiscale hybrid linear models for lossy image representation
- Hong, Wright, et al.
(Show Context)
Citation Context ...aces. Such data Appearing in Proceedings of the 19th International Conference on Artificial Intelligence and Statistics (AISTATS) 2016, Cadiz, Spain. JMLR: W&CP volume 51. Copyright 2016 by the authors. points could be feature trajectories of rigid moving objects captured by an affine camera [4], articulated moving parts of a human body [27], illumination of different convex objects under Lambertian model [9] and so on. Subspace clustering is also more generically used in agnostic learning of the best linear mixture structures in the data. For instance, it is used for images/video compression [10], hybrid system identification, disease identification [14] as well as modeling social network communities [3], studying privacy in movie recommendations [28] and inferring router network topology [5]. There is rich literature on algorithmic and theoretical analysis of subspace clustering [4, 12, 8, 17]. Among the many algorithms, sparse subspace clustering (SSC) [4] is arguably the most well-studied due to its elegant formulation, strong empirical performance and provable guarantees to work under relatively weak conditions. The algorithm involves constructing a sparse linear representation of... |

30 | Degrees of Freedom in Lasso Problems - Tibshirani, Taylor - 2012 |

22 | Robust subspace clustering.
- Soltanolkotabi, Elhamifar, et al.
- 2014
(Show Context)
Citation Context ...o work under relatively weak conditions. The algorithm involves constructing a sparse linear representation of each data point using the remaining dataset as a dictionary. This approach embeds the relationship of the data points into a sparse graph and the intuition is that the data points are likely to choose only those points on the same subspace to linearly represent itself. Then clustering can be obtained by finding connected components of the graph, or more robustly, using spectral clustering [4]. Assuming data lie exactly or approximately on a union of linear subspaces, 1 it is shown in [4, 19, 24, 20] that under certain separation conditions, this embedded graph will have no edges between any two points in different subspaces. This criterion of success is referred to as the “Self-Expressiveness Property (SEP)” [4, 24] and “Subspace Detection Property (SDP)” [19]. The drawback is that there is no guarantee that the vertices within one cluster form a connected component. Therefore, the solution may potentially over segment the data points. This subtle point was originally raised and partially addressed in [15], reaching an answer that when data are noiseless and intrinsic subspace dimension ... |

21 | Turning big data into tiny data: Constant-size coresets for kmeans, pca and projective clustering.
- Feldman, Schmidt, et al.
- 2013
(Show Context)
Citation Context ...]). Their results however rely critically on the semi-random model assumption. For instance, [7] uses the connectivity of a random k-nearest neighbor graph on a sphere to facilitate an argument for clustering consistency. In addition, these approaches do not easily generalize to SSC even under the semi-random model since the solution of SSC is considerably harder to characterize. In contrast, our results are much simpler and work generically without any probabilistic assumptions. Lastly, there is a long line of research on “projective clustering” in the theoretical computer science literature [11, 6]. Unlike subspace clustering that posits an approximate union-of-subspace model, projective clustering makes no assumption on the data points and is completely agnostic. The algorithms [11, 6] are typTable 3: Summary of existing theoretical guarantees. (*) denotes results from this paper. Algorithm SEP Exact clustering LRR [12] A-2-a A-2-a SSC [4] B-2-a - SSC [19] C-{1,2}-a - Noisy SSC [24] C-{1,2}-{a,b,c} - Robust SSC [20] C-1-{a,b} - LRSSC [26] C-{1,2}-a A-{1,2}-a Thresh. SC [8] C-1-a - Robust TSC [7] C-1-{a,b} C-1-{a,b} Greedy SC [17] C-1-a C-1-a SSC (*) C-{1,2}-{a,b,c} C-{1,2}-{a,b,c} ical... |

14 | Noisy sparse subspace clustering.
- Wang, Xu
- 2013
(Show Context)
Citation Context ..., such over-segmentation will not occur as long as all points within the same subspace are 1affine subspaces are handled by augmenting 1 to every data point. 538 Graph Connectivity in Noisy Sparse Subspace Clustering in general position; but when d ≥ 4, a counter example was provided, showing that this weak “general position” condition is no longer sufficient. In this paper, we revisit the graph connectivity problem for noisy sparse subspace clustering. Inspired by the post-merging step presented in [4] for noiseless data, we propose in this paper a variant of noisy sparse subspace clustering [25] that provably produces perfect clustering with high probability, under certain “general position” or “restricted eigenvalue” assumptions. We also provide a counter-example to show that our derived success conditions are almost tight under the adversarial noise perturbation model. This is the first time a subspace clustering algorithm is proven to give correct clustering under no statistical assumptions on data corrupted by noise. To the best of our knowledge, this is also the first guarantee for Lasso that lower bounds the number of discoveries, which might be of independent interest for othe... |

11 | Graph connectivity in sparse subspace clustering.
- Nasihatkon, Hartley
- 2011
(Show Context)
Citation Context .../affine subspaces. It is the mathematical abstraction of many important problems in computer vision, image processing and machine learning. A line of recent work [4, 19, 24, 20] provided strong theoretical guarantee for sparse subspace clustering [4], the state-of-the-art algorithm for subspace clustering, on both noiseless and noisy data sets. It was shown that under mild conditions, with high probability no two points from different subspaces are clustered together. Such guarantee, however, is not sufficient for the clustering to be correct, due to the notorious “graph connectivity problem” [15]. In this paper, we investigate the graph connectivity problem for noisy sparse subspace clustering and show that a simple postprocessing procedure is capable of delivering consistent clustering under certain “general position” or “restricted eigenvalue” assumptions. We also show that our condition is almost tight with adversarial noise perturbation by constructing a counter-example. These results provide the first exact clustering guarantee of noisy SSC for subspaces of dimension greater then 3. 1 INTRODUCTION The problem of subspace clustering originates from numerous applications in compute... |

10 | Robust subspace clustering via thresholding. arXiv:1307.4891,
- Heckel, Bolcskei
- 2013
(Show Context)
Citation Context ...r the original noiseless SSC, the problem becomes trickier since the solution is more constrained. In [15] it was shown that when subspace dimension is no larger than 3, SSC outputs block-wise connected similarity graph under very mild conditions; however, the graph connectivity is easily broken when subspace dimension exceeds 3. Though a simple post-processing step was remarked in [4, Footnote 6 in Section 5] to alleviate the graph connectivity issue on noiseless data, it is unclear how to extend their method when data are corrupted by noise. Among other subspace clustering methods, [17] and [7] are the only two papers that provide provable exact clustering guarantees for problems beyond independent subspaces (for which LRR provably gives dense graphs [26]). Their results however rely critically on the semi-random model assumption. For instance, [7] uses the connectivity of a random k-nearest neighbor graph on a sphere to facilitate an argument for clustering consistency. In addition, these approaches do not easily generalize to SSC even under the semi-random model since the solution of SSC is considerably harder to characterize. In contrast, our results are much simpler and work gen... |

8 | Fused sparsity and robust estimation for linear models with unknown variance.
- Chen, Dalalyan
- 2012
(Show Context)
Citation Context |

8 | Highrank matrix completion.
- Eriksson, Balzano, et al.
- 2012
(Show Context)
Citation Context .... points could be feature trajectories of rigid moving objects captured by an affine camera [4], articulated moving parts of a human body [27], illumination of different convex objects under Lambertian model [9] and so on. Subspace clustering is also more generically used in agnostic learning of the best linear mixture structures in the data. For instance, it is used for images/video compression [10], hybrid system identification, disease identification [14] as well as modeling social network communities [3], studying privacy in movie recommendations [28] and inferring router network topology [5]. There is rich literature on algorithmic and theoretical analysis of subspace clustering [4, 12, 8, 17]. Among the many algorithms, sparse subspace clustering (SSC) [4] is arguably the most well-studied due to its elegant formulation, strong empirical performance and provable guarantees to work under relatively weak conditions. The algorithm involves constructing a sparse linear representation of each data point using the remaining dataset as a dictionary. This approach embeds the relationship of the data points into a sparse graph and the intuition is that the data points are likely to choos... |

7 | Subspace clustering of high-dimensional data: a predictive approach.
- McWilliams, Montana
- 2014
(Show Context)
Citation Context ...ational Conference on Artificial Intelligence and Statistics (AISTATS) 2016, Cadiz, Spain. JMLR: W&CP volume 51. Copyright 2016 by the authors. points could be feature trajectories of rigid moving objects captured by an affine camera [4], articulated moving parts of a human body [27], illumination of different convex objects under Lambertian model [9] and so on. Subspace clustering is also more generically used in agnostic learning of the best linear mixture structures in the data. For instance, it is used for images/video compression [10], hybrid system identification, disease identification [14] as well as modeling social network communities [3], studying privacy in movie recommendations [28] and inferring router network topology [5]. There is rich literature on algorithmic and theoretical analysis of subspace clustering [4, 12, 8, 17]. Among the many algorithms, sparse subspace clustering (SSC) [4] is arguably the most well-studied due to its elegant formulation, strong empirical performance and provable guarantees to work under relatively weak conditions. The algorithm involves constructing a sparse linear representation of each data point using the remaining dataset as a dictionar... |

7 | Guess who rated this movie: Identifying users through subspace clustering.
- Zhang, Fawaz, et al.
- 2012
(Show Context)
Citation Context ...CP volume 51. Copyright 2016 by the authors. points could be feature trajectories of rigid moving objects captured by an affine camera [4], articulated moving parts of a human body [27], illumination of different convex objects under Lambertian model [9] and so on. Subspace clustering is also more generically used in agnostic learning of the best linear mixture structures in the data. For instance, it is used for images/video compression [10], hybrid system identification, disease identification [14] as well as modeling social network communities [3], studying privacy in movie recommendations [28] and inferring router network topology [5]. There is rich literature on algorithmic and theoretical analysis of subspace clustering [4, 12, 8, 17]. Among the many algorithms, sparse subspace clustering (SSC) [4] is arguably the most well-studied due to its elegant formulation, strong empirical performance and provable guarantees to work under relatively weak conditions. The algorithm involves constructing a sparse linear representation of each data point using the remaining dataset as a dictionary. This approach embeds the relationship of the data points into a sparse graph and the intuition i... |

6 | Subspace clustering via thresholding and spectral clustering.
- Heckel, Bolcskei
- 2013
(Show Context)
Citation Context ...4], articulated moving parts of a human body [27], illumination of different convex objects under Lambertian model [9] and so on. Subspace clustering is also more generically used in agnostic learning of the best linear mixture structures in the data. For instance, it is used for images/video compression [10], hybrid system identification, disease identification [14] as well as modeling social network communities [3], studying privacy in movie recommendations [28] and inferring router network topology [5]. There is rich literature on algorithmic and theoretical analysis of subspace clustering [4, 12, 8, 17]. Among the many algorithms, sparse subspace clustering (SSC) [4] is arguably the most well-studied due to its elegant formulation, strong empirical performance and provable guarantees to work under relatively weak conditions. The algorithm involves constructing a sparse linear representation of each data point using the remaining dataset as a dictionary. This approach embeds the relationship of the data points into a sparse graph and the intuition is that the data points are likely to choose only those points on the same subspace to linearly represent itself. Then clustering can be obtained b... |

5 | Provable subspace clustering: When LRR meets SSC.
- Wang, Xu, et al.
- 2013
(Show Context)
Citation Context ... row is weaker than its previous row. Except for the independent subspace assumption, which on its own is sufficient, results for more general models typically require additional conditions on the subspaces and data points in each subspaces. For instance, the “semi-random model” assumes data points to be drawn i.i.d. uniformly at random from the unit sphere in each subspace and the more generic “deterministic model” places assumptions on the radius of the smallest inscribing sphere of the symmetric polytope spanned by data points [19] or the smallest non-zero singular value of the data matrix [26]. Related theoretical guarantees of subspace clustering algorithms in the literature are summarized in Table 3 where the assumptions about subspaces are denoted with capital letters “A, B, C”; different noise settings are referred to using lowercase letters “a,b,c” in Table 2. Results that are applicable to SSC are highlighted. As we can see from the second column of Table 3, 539 Yining Wang, Yu-Xiang Wang and Aarti Singh SEP guarantees have been quite exhaustively studied and now we understand very well the conditions under which it holds. Specifically, most of the results are now near optima... |

3 | Greedy subspace clustering.
- Park, Caramanis, et al.
- 2014
(Show Context)
Citation Context ...4], articulated moving parts of a human body [27], illumination of different convex objects under Lambertian model [9] and so on. Subspace clustering is also more generically used in agnostic learning of the best linear mixture structures in the data. For instance, it is used for images/video compression [10], hybrid system identification, disease identification [14] as well as modeling social network communities [3], studying privacy in movie recommendations [28] and inferring router network topology [5]. There is rich literature on algorithmic and theoretical analysis of subspace clustering [4, 12, 8, 17]. Among the many algorithms, sparse subspace clustering (SSC) [4] is arguably the most well-studied due to its elegant formulation, strong empirical performance and provable guarantees to work under relatively weak conditions. The algorithm involves constructing a sparse linear representation of each data point using the remaining dataset as a dictionary. This approach embeds the relationship of the data points into a sparse graph and the intuition is that the data points are likely to choose only those points on the same subspace to linearly represent itself. Then clustering can be obtained b... |

3 |
Noisy sparse subspace clustering. arXiv:1309.1233,
- Wang, Xu
- 2013
(Show Context)
Citation Context ...o work under relatively weak conditions. The algorithm involves constructing a sparse linear representation of each data point using the remaining dataset as a dictionary. This approach embeds the relationship of the data points into a sparse graph and the intuition is that the data points are likely to choose only those points on the same subspace to linearly represent itself. Then clustering can be obtained by finding connected components of the graph, or more robustly, using spectral clustering [4]. Assuming data lie exactly or approximately on a union of linear subspaces, 1 it is shown in [4, 19, 24, 20] that under certain separation conditions, this embedded graph will have no edges between any two points in different subspaces. This criterion of success is referred to as the “Self-Expressiveness Property (SEP)” [4, 24] and “Subspace Detection Property (SDP)” [19]. The drawback is that there is no guarantee that the vertices within one cluster form a connected component. Therefore, the solution may potentially over segment the data points. This subtle point was originally raised and partially addressed in [15], reaching an answer that when data are noiseless and intrinsic subspace dimension ... |

1 | Approximation and streaming algorithms for projective clustering via random projections. arXiv:1407.2063,
- Kerber, Raghvendra
- 2014
(Show Context)
Citation Context ...]). Their results however rely critically on the semi-random model assumption. For instance, [7] uses the connectivity of a random k-nearest neighbor graph on a sphere to facilitate an argument for clustering consistency. In addition, these approaches do not easily generalize to SSC even under the semi-random model since the solution of SSC is considerably harder to characterize. In contrast, our results are much simpler and work generically without any probabilistic assumptions. Lastly, there is a long line of research on “projective clustering” in the theoretical computer science literature [11, 6]. Unlike subspace clustering that posits an approximate union-of-subspace model, projective clustering makes no assumption on the data points and is completely agnostic. The algorithms [11, 6] are typTable 3: Summary of existing theoretical guarantees. (*) denotes results from this paper. Algorithm SEP Exact clustering LRR [12] A-2-a A-2-a SSC [4] B-2-a - SSC [19] C-{1,2}-a - Noisy SSC [24] C-{1,2}-{a,b,c} - Robust SSC [20] C-1-{a,b} - LRSSC [26] C-{1,2}-a A-{1,2}-a Thresh. SC [8] C-1-a - Robust TSC [7] C-1-{a,b} C-1-{a,b} Greedy SC [17] C-1-a C-1-a SSC (*) C-{1,2}-{a,b,c} C-{1,2}-{a,b,c} ical... |