
Submitted by Assigned_Reviewer_1
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions)
 Glad to see simple simulations illustrating that the SSC and Lasso SSC don't solve this kind of problem well  thanks.
 Surprised and a bit confused the lassoSSC does worse than the original SSC in Figure 3. This appears to be for a fixed (that is, nonoptimized) lambda=2 on the soft constraint for Lasso, that doesn't seem like a great comparison, I would expect a larger lambda to do better, but would have expected any nonzero lambda to do better than the original SSC. A comment on this would have been useful.
 It is unfortunate the method breaks around 20% irrelevant features in the simulations. An interesting case would be if there are many many (like 90% or 99%) irrelevant features.
 Generally wellwritten, but paper needs a careful comb: many minor typos.
Q2: Please summarize your review in 12 sentences
Paper extends subspace clustering to handle irrelevant features using a robust inner product (drops k largest terms). They recast the resulting nonconvex optimization problem as a linear program using the robust Dantzig selector.
Paper more theoretical than experimental, which is ok, but limits its awesomeness.
Submitted by Assigned_Reviewer_2
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions)
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) nonirrelevant 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 dotproducts ?
 what happens when all irrelevant features are really small ? they would satisfy line 207 as well enter the dotproducts, 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.
Q2: Please summarize your review in 12 sentences
This paper proposes to perform subspace clustering by formulating a convex optimization problem for each data point and using a 'robust' definition of inner product. Unfortunately, I am not very familiar with this area and am not able to judge the quality of this work very well.
Submitted by Assigned_Reviewer_3
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions)
The paper casts the subspace clustering problem as a Dantzig Selector estimator (along the line of CandesTao's work) and claims for its robustness w.r.t. irrelevant features. The paper is essentially theoretical, which provides guarantees for the algorithm to find the correct subspace. As I am not expert in the field of robust estimator for sparse recovery, it is difficult for me to evaluate the quality of this work.
Q2: Please summarize your review in 12 sentences
The paper casts the subspace clustering problem as a Dantzig Selector estimator (along the line of CandesTao's work) and claims for its robustness w.r.t. irrelevant features.
Submitted by Assigned_Reviewer_4
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions)
The paper proposed to use robust Dantzig selector to perform subspace clustering with irrelevant features. The idea presented in the paper is straightforward and intuitive. The authors also provided analysis to show when the Robust Dantzig selector can detect the true subspace clustering. My concern on this work is the usefulness of the approach in real application. In the method, each sample is used as the target of the robust Dantzig selector. If the data is large, too many models need to be fit, which makes the methods impractical.
Q2: Please summarize your review in 12 sentences
The paper proposed to use robust Dantzig selector to perform subspace clustering with irrelevant features. The idea presented in the paper is straightforward and intuitive, but when data size is large the proposed method might not be able to handle it.
Submitted by Assigned_Reviewer_5
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions)
For my understanding, there is no silver bullet for feature selection clustering algorithms: their performances are ultimately affected by the nature of datasets and tasks. Therefore it is very important for clustering works to show concrete examples where the proposed model works well, and where it does not work (limitations). In that sense, the paper does not validate the proposed model in real world datasets, and no such discussions are provided. This greatly reduces the reliability concerning the usefulness of the proposed model.
A number of irrelevant features D1 = 20 within D=200 is too "mild" to use for some applications such as sensor networks. In such a small setting, we may simply conduct "leaveonefeatureout" studies to identify irreverent features. It may make the experiments more convincing if you can work on more large D cases, or 99% of features are indeed irrelevant.
Q2: Please summarize your review in 12 sentences
This paper proposes a variant of the sparse subspace clustering, to deal with the existence of the irrelevant feature attributes. The objective introduces a sparse error term into the SSC objective.
I think the proposed model is simple and reasonable. However, experimental validation is too weak to recommend for NIPS publication.
Q1:Author
rebuttal: Please respond to any concerns raised in the reviews. There are
no constraints on how you want to argue your case, except for the fact
that your text should be limited to a maximum of 5000 characters. Note
however, that reviewers and area chairs are busy and may not read long
vague rebuttals. It is in your own interest to be concise and to the
point.
We thank all reviewers for their detailed and
constructive comments.
Reviewer 1 and 3 commented that it would be
better if the algorithm can handle many (e.g., 90% or 99%) irrelevant
features. We certainly agree with this. However, we want to point out that
the subspace clustering problem with corrupted feature is an open problem,
even when the fraction of irrelevant features are mild. Indeed, as shown
in the paper, SSC and LASSOSSC break down even with few irrelevant
features. In this sense, our work opens a door to the area of subspace
clustering with corrupted data, and hopefully may inspire methods that can
eventually handle the much more challenging case of many irrelevant
features, maybe under additional assumptions on the irrelevant features
(recall that in this paper we made no assumptions on the irrelevant
features).
We now respond to other comments:
Reviewer
1
Thank you for your suggestion on LASSOSSC. We will finetune
the parameter \lambda of LASSOSSC and compare the results in the final
version of the paper.
Reviewer 3
We admit that
simulations on real world datasets would improve the paper. However, we
want to point out that our work focuses on providing methods with
theoretical guarantees (deterministic model and fully random model) to
solve the subspace clustering problem with corrupted features. It is based
upon the sparse subspace clustering, an algorithm that has been widely
applied in many real problems.
As for your suggestion on
"leaveonefeatureout " strategy, we think it is not easy to apply this
to the subspace clustering problem with corrupted features. Even with
D1=20, and D=200 (which is the setting in our simulation), if we follow
"leaveonefeatureout " strategy, we need to enumerate C_{200}^{20}
(i..e., approximately 10^46) combinations, and solve a Lassolike
algorithm for each. This is way beyond the existing computation power of
the world. Moreover, scalability of such a strategy is clearly a
problem.
Reviewer 4
Thank you for your comments. We now
provide an intuitive explanation why we use this robust inner product. The
essence of SSC, LASSOSSC and our formulation is to write a sample as a
sparse linear combination of other samples (plus some small noise) .
Intuitively, the irrelevant features with large magnitude may affect the
correct subspace clustering, so we introduce this truncation process of
robust innerproduct. When all irrelevant features are really small, they
do enter the dotproducts as you said. But since they are small, they
would not affect the correct clustering, as they will be treated as "small
noise" by the Lassolike algorithm of LassoSSC. By exploiting this
intuition, we bound the error terms \delta_1, \delta_2 in Lemma2, which
only depends on the number of irrelevant features but not the magnitude,
which lead to our theoretical guarantees.
Thank you for your
suggestion on choosing top 'Dk' for various value of 'k' to illustrate
robustness, we will add it in the final version of this paper.
Reviewer 6
Thank you for your comments. As for your concern
about large computations in real application, we believe this should not
be an issue. Notice that in terms of computation cost our method is
similar to the original sparse subspace clustering (SSC) algorithm, which
also needs to solve a linear programming for each sample. Yet, SSC has
been widely used in computer vision and other real applications (where
datasets are typically pretty large) and has seen many successes.
Moreover, we can easily accelerate computations by parallel computing,
since for each sample we solve an independent linear optimization problem.

