Abstract. Compressive sampling offers a new paradigm for acquiring signals that are compressible with respect to an orthonormal basis. The major algorithmic challenge in compressive sampling is to approximate a compressible signal from noisy samples. This paper describes a new iterative recovery algorithm called CoSaMP that delivers the same guarantees as the best optimization-based approaches. Moreover, this algorithm offers rigorous bounds on computational cost and storage. It is likely to be extremely efficient for practical problems because it requires only matrix–vector multiplies with the sampling matrix. For compressible signals, the running time is just O(N log 2 N), where N is the length of the signal. 1.

We study the problem of estimating the best k term Fourier representation for a given frequency-sparse signal (i.e., vector) A of length N ≫ k. More explicitly, we investigate how to deterministically identify k of the largest magnitude frequencies of Â, and estimate their coefficients, in polynomial(k, log N) time. Randomized sublinear time algorithms which have a small (controllable) probability of failure for each processed signal exist for solving this problem [24, 25]. In this paper we develop the first known deterministic sublinear time sparse Fourier Transform algorithm which is guaranteed to produce accurate results. As an added bonus, a simple relaxation of our deterministic Fourier result leads to a new Monte Carlo Fourier algorithm with similar runtime/sampling bounds to the current best randomized Fourier method [25]. Finally, the Fourier algorithm we develop here implies a simpler optimized version of the deterministic compressed sensing method previously developed in [30]. 1

Identification of defective members of large populations has been widely studied in the statistics community under the name of group testing. It involves grouping subsets of items into different pools and detecting defective members based on the set of test results obtained for each pool. In a classical noiseless group testing setup, it is assumed that the sampling procedure is fully known to the reconstruction algorithm, in the sense that the existence of a defective member in a pool results in the test outcome of that pool to be positive. However, this may not be always a valid assumption in some cases of interest. In particular, we consider the case where the defective items in a pool can become independently inactive with a certain probability. Hence, one may obtain a negative test result in a pool despite containing some defective items. As a result, any sampling and reconstruction method should be able to cope with two different types of uncertainty, i.e., the unknown set of defective items and the partially unknown, probabilistic testing procedure. In this work, motivated by the application of detecting infected people in viral epidemics, we design non-adaptive sampling procedures that allow successful identification of the defective items through a set of probabilistic tests. Our design requires only a small number of tests to single out the defective items.

A significant Fourier transform (SFT) algorithm, given a threshold τ and oracle access to a function f, outputs (the frequencies and approximate values of) all the τ-significant Fourier coefficients of f, i.e., the Fourier coefficients whose magnitude exceeds τ‖f ‖ 2 2. In this paper we present the first deterministic SFT algorithm for functions f over ZN which is: (1) Local, i.e., its running time is polynomial in log N, 1/τ and L1 ( ̂ f) (the L1 norm of f’s Fourier transform). (2) Robust to random noise. This strictly extends the class of compressible/Fourier sparse functions over ZN efficiently handled by prior deterministic algorithms. As a corollary we obtain deterministic and robust algorithms for sparse Fourier approximation, compressed sensing and sketching. As a central tool, we prove that there are: 1. Explicit sets A of size poly((ln N) d, 1/ε) with ε-discrepancy in all rank d Bohr sets in ZN. This extends the Razborov-Szemeredi-Wigderson result on ε-discrepancy in arithmetic progressions to Bohr sets, which are their higher rank analogue. 2. Explicit sets AP of size poly(ln N, 1/ε) that ε-approximate the uniform distribution over a given arithmetic progression P in ZN, in the sense that |Ex∈A χ(x) − Ex∈P χ(x) | < ε for all linear tests χ in ZN. This extends results on small biased sets, which are sets approximating the uniform distribution over the entire domain, to sets approximating uniform distributions over (arbitrary size) arithmetic progressions. These results may be of independent interest. 1

Abstract—We present a deterministic number theoretic construction for matrices with the Restricted Isometry Property (RIP). Furthermore, we show that the number theoretic properties of our RIP matrices allow their products with Discrete Fourier Transform (DFT) matrices to be well approximated via a few highly sparse matrix multiplications. Hence, our RIP matrices may be approximately multiplied by the DFT of any input vector in sublinear-time by reading only a small fraction of its entries. As a consequence, we obtain small deterministic sample sets which are guaranteed to allow the recovery of near-optimal sparse Fourier representations for all periodic functions having an integrable second derivative over a single period. Explicit bounds are provided for the sizes of our RIP matrices, the sizes of their associated sublinear Fourier sampling sets, and the errors incurred by quickly approximating their products with DFT matrices. The Fourier sampling requirements obtained herein improve on previous deterministic Fourier sampling results in [1], [2]. Index Terms—Fourier transforms, Discrete Fourier transforms,

Abstract. In the Hidden Number Problem (HNP), the goal is to find a hidden number s, whengivenp, g and access to an oracle that on query a returns the k most significant bits of s · g a mod p. We present an algorithm solving HNP, when given an advice depending only on p and g; the running time and advice length are polynomial in log p. This algorithm improves over prior HNP algorithms in achieving: (1) optimal number of bits k ≥ 1(comparedwithk ≥ Ω(log log p)); (2) robustness to random noise; and (3) handling a wide family of predicates on top of the most significant bit. As a central tool we present an algorithm that, given oracle access to a function f over ZN, outputs all the significant Fourier coefficients of f (i.e., those occupying, say, at least 1 % of the energy). This algorithm improves over prior works in being: – Local. Its running time is polynomial in log N and L1 ( ̂ f)(forL1 ( ̂ f) the sum of f’s Fourier coefficients, in absolute value). – Universal. For any N, t, thesame oracle queries are asked for all functions f over ZN s.t. L1 ( ̂ f) ≤ t. – Robust. The algorithm succeeds with high probability even if the oracle to f is corrupted by random noise. 1

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This paper empirically evaluates a recently proposed Deterministic Sparse Fourier Transform algorithm (hereafter called DSFT) for the first time. Our experiments indicate that DSFT is capable of guaranteed general frequency-sparse signal recovery using subNyquist sampling for realistic bandwidth values. Furthermore, we show that both variants of DSFT have fast reconstruction runtimes. In fact, the sublinear-time DSFT variant is shown to be faster than a traditional Fast Fourier Transform (FFT) for highly-sparse wideband signals.

Computing the Fourier transform is a basic building block used in numerous applications. For data intensive applications, even the O(N logN) running time of the Fast Fourier Transform (FFT) algorithm may be too slow, and sub-linear running time is necessary. Clearly, outputting the entire Fourier transform in sub-linear time is infeasible, nevertheless, in many applications it suffices to find only the τ-significant Fourier transform coefficients, that is, the Fourier coefficients whose magnitude is at least τ-fraction (say, 1%) of the energy (i.e., the sum of squared Fourier coefficients). We call algorithms achieving the latter SFT algorithms. In this work we present a deterministic algorithm that finds the τ-significant Fourier coefficients of functions f over any finite abelian group G in time polynomial in log|G|, 1/τ and L1 ( ̂f) (for L1 ( ̂f) denoting the sum of absolute values of the Fourier coefficients of f). Our algorithm is robust to random noise. Our algorithm is the first deterministic and efficient (i.e., polynomial in log|G|) SFT algorithm to handle functions over any finite abelian groups, as well as the first such algorithm to handle functions over ZN that are neither compressible nor Fourier-sparse. Our analysis is the first to show robustness to noise in the context of deterministic SFT algorithms. Using our SFT algorithm we obtain (1) deterministic (universal and explicit) algorithms for sparse Fourier approximation, compressed sensing and sketching; (2) an algorithm solving the Hidden Number Problem with advice, with cryptographic bit security implications; and (3) an efficient decoding algorithm in the random noise model for polynomial rate variants of Homomorphism codes and any other concentrated & recoverable codes.

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