Open Questions from the Workshop on Algorithms for Data

1 " For all " guarantee for computationally bounded adversaries (Martin Strauss) There are two types of compressed sensing guarantees, illustrated using two players: 1. " For all " : Charlie constructs the sensing matrix φ, and then Mallory constructs the signal x = x(φ) as a function of φ. The Compressed Sensing question is to recover the approximate signal˜x from the measurement φx. The best guarantee possible is the following 2 // 1 guarantee: ||˜x − x|| 2 ≤ / √ k||x opt − x|| 1. 2. " For each " : Charlie construct a distribution D over sensing matrices φ. Then Mallory constructs a vector x = x(D) dependent on the distribution only. Finally, a sensing matrix φ is sampled from the distribution D. The goal is again to recover˜x, with good probability over the choice of φ. It turns out a stronger guarantee, termed 2 // 2 , is possible: ||˜x − x|| 2 ≤ (1 +)||x opt − x|| 2 In some sense the two " worlds " are incomparable: the first one works for all x but obtains weaker error guarantee, and the second one works for each x with some probability but gets better error guarantee. Question is: How can we get the best of both worlds (" for all " with 2 // 2 error) ? Once we require " for all " , it is provably impossible to obtain 2 // 2 guarantee. But what if Mallory has bounded computational resources to construct a " bad " x? A preliminary result considers the following setting. Mallory sees φ and writes down a sketch of φ (in bounded space). Then Mallory produces x from this sketch only. Then 2 // 2 is possible for such x's. Generally, we would like to allow Mallory to be probabilistic polynomial time, and have a φ so that Mallory still cannot find an input x = x(φ) that breaks the recovery algorithm.