The Sparse Fourier Transform. Haitham Hassanieh

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holds iff Image; and

      • “Large noise” event Enoise(i): holds Image.

      By Claims 4.1 and 4.2, Image and Image for any iS

      Claim 4.4 For any iS, Image.

      Proof For each ji, Image by Lemma 2.1.

      Then

Image

      The result follows by Markov’s inequality.

      We will show for iS that if none of Ecoll(i), Eoff(i), and Enoise(i) hold then SPARSEFFTINNER recovers Image with 1 − O(α) probability.

      Lemma 4.5 Let a ∈ [n] uniformly at random, B divide n, and the other parameters be arbitrary in

Image

      Then for any i ∈ [n] with Image and none of Ecoll(i), Eoff(i), or Enoise(i) holding,

Image

      Proof The proof can be found in Appendix A.5.

      Properties of LOCATESIGNAL

      In our intuition, we made a claim that if Image uniformly at random, and i > 16w, then Image is “roughly uniformly distributed about the circle” and hence not concentrated in any small region. This is clear if β is chosen as a random real number; it is less clear in our setting where β is a random integer in this range. We now prove a lemma that formalizes this claim.

      Lemma 4.6 Let T ⊂ [m] consist of t consecutive integers, and suppose βT uniformly at random. Then for any i ∈ [n] and set S ⊂ [n] of l consecutive integers,

Image

      Proof Note that any interval of length l can cover at most Image elements of any arithmetic sequence of common difference i. Then Image is such a sequence, and there are at most Image intervals an + S overlapping this sequence. Hence, at most Image of the β ∈ [m] have βi mod nS. Hence, Image.

      Lemma 4.7 Let iS. Suppose none of Ecoll(i), Eoff(i), and Enoise(i) hold, and let j = hσ,b(i). Consider any run of LOCATEINNER with Image. Let f > 0 be a parameter such that

Image

      for C larger than some fixed constant. Then Image with probability at least Image,

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