Structured Local Minima in Sparse Blind Deconvolution
Blind deconvolution is a ubiquitous problem of recovering two unknown signals from their convolution. Unfortunately, this is an ill-posed problem in general. This paper focuses on the short and sparse blind deconvolution problem, where the one unknown signal is short and the other one is sparsely and randomly supported. This variant captures the structure of the unknown signals in several important applications. We assume the short signal to have unit 2 norm and cast the blind deconvolution problem as a nonconvex optimization problem over the sphere. We demonstrate that (i) in a certain region of the sphere, every local optimum is close to some shift truncation of the ground truth, and (ii) for a generic short signal of length k, when the sparsity of activation signal θ k −2/3 and number of measurements m poly (k), a simple initialization method together with a descent algorithm which escapes strict saddle points recovers a near shift truncation of the ground truth kernel.
This paper considers sparse blind deconvolution. This is an inverse problem where we observe the (cyclic) convolution of two unknown signals: one sparse, and one short. The aim is to recover both signals, up to fundamental indeterminacies. This is done through non-convex optimization with a soft sparsity promoting cost function. The main contributions are an analysis of regimes where the non-convexity is benign. Importantly, there is no noise in the proposed model.  This is an interesting problem. Both the problem and the proposed approach I would deem fairly original. Exposition is satisfactory, and results are interesting.  There are two main points I would like the authors to clarify. A priori, I expect the authors will be able to resolve these two doubts I have, which is why my initial rating is high. But if I cannot be convinced, I will have to lower my grade.  First, does the proposed theory cover more than a subclass of this problem which is trivial? Specifically, see the rate at L69. What does this mean exactly? From the model detailed after L69, about m\theta entries of x_0 are nonzero. If k is such that there is a good probability that just one nonzero entry is separated from any other nonzero entry by at least k zeros, then the kernel a_0 can be read off directly from the observation just by spotting the presence of two exact zeros around the signal, which is then only scaled by the corresponding g_i. Thus, to make sense of the sparsity rate (and to know if it is "relatively dense" as stated in the paper), it should be shown that under the proposed setting, this situation does not occur with large probability. Otherwise, the problem appears to be trivial. This issue could also be resolved by adding noise to the observation y, in which case the exact zeros would not appear, and it wouldn't be all that easy to locate the separated occurrences. More generally, it would be nice to treat the presence of noise, but this is not a requirement.   Second, line 65, the word "benign" (and many other places where this sentiment is echoed): it is unclear to me that this effect is benign. Is it not the case that the shift truncation could eliminate most if not all of the signal? If that is the case, what is it really that we can hope to recover? In other words, does the measure proposed on line 253 really capture closeness to a meaningful version of a_0? Or are some of these shift truncated versions of a_0 somehow significantly shortened to the point that we didn't recover much?     Other comments:  Figure titles, legends, axis labels etc.: a lot of them are illegible. Please fix this.  Eq. (3): at this point, the modulo operator has not been defined yet. On that note: notation would be a lot nicer with 0 indexing, as this would ease all the modulo business.  Line 44: , -> .  L62,63: this is nice intuition.  Page 2, footnote 2: is it on purpose that y_1 is not moved? If so, please specify so readers are not left wondering if this is a typo.  L78: what is entry-wise \ell^p norm? Also, this could conceivably interfere with the definition of _2 for operator norm. I understand that this is why you use the Frobenius norm for vectors. An alternative is to write \|.\|_{op}. In any case: there is some confusion in the paper itself about these norms. See for example eq. (9) where the 2 norm is presumably a Frobenius norm, since the input is a vector.  Eq. (6): the matrix seems wrongly typeset  L108: This justification (footnote 8) is a bit weak: one needs to argue first that YY'/m concentrates around its mean, then to argue that the same goes for the inverse matrix square root of it, and this this all happens nicely considering YY' appears in a product with other random variables.  Footnote 5: you get m-1 such contributions -- then, how can I understand this as small?  L143 (and likely elsewhere): optima is the plural of optimum (and minima/minimum, maxima/maximum)  L148: small \kappa, but eq. (19) seems to say small kappa (condition number) is bad; what am I missing?  L204: semidefinite -> should be definite. L205: convex -> should be locally geodesically convex L209-211: I do not understand this statement. Please clarify. L220, end: that -> such that L249: what does big mean here? very sparse or not very sparse? L249: move footnote mark outside parenthesis (and after period) L253: recover+y   ####### Post rebuttal #######  I thank the reviewers for their thorough response to my two main concerns. Based on their convincing explanations, I maintain my assessment of 8. I strongly encourage the authors to include the points raised in their rebuttal in their paper in some form.