Subspace Clustering with Irrelevant Features via Robust Dantzig Selector
This paper considers the subspace clustering problem where the data contains irrelevant or corrupted features. We propose a method termed "robust Dantzig selector" which can successfully identify the clustering structure even with the presence of irrelevant features. The idea is simple yet powerful: we replace the inner product by its robust counterpart, which is insensitive to the irrelevant features given an upper bound of the number of irrelevant features. We establish theoretical guarantees for the algorithm to identify the correct subspace, and demonstrate the effectiveness of the algorithm via numerical simulations. To the best of our knowledge, this is the first method developed to tackle subspace clustering with irrelevant features.
The authors define a convex optimization problem for each data point and aim to recover the support of a data point in terms of other data points. The main contribution is to replace the standard inner product between two vectors with its robust counterpart, by selecting the smallest (hopefully) non-irrelevant features.Comments - Could the authors please define 'robustness' in the abstract or briefly describe it.- line 37 'motivates'- line 77, lin 82, can club WLOG together, or show a picture.- line 149, where does the robustness come from ? What is intuitively the benefit of neglecting the larger dot-products ?- what happens when all irrelevant features are really small ? they would satisfy line 207 as well enter the dot-products, what would fail then ?- The current set of numerical simulations although limited, show some insight. Could the authors also please the add the result of choosing the top 'D - k' for various values of 'k', rather than just D1 ? This would show the benefit of robustness in the first place.