Ashley Montanaro and Ronald de Wolf have written a comprehensive survey article on Quantum Property Testing (ArXiv). It would be nice to have more such surveys in our field.
Here’s a report on the new papers we saw in September. The list of accepted papers in SODA 2014 also came out in September. While we have covered one of the accepted papers in our “News for August 2013″ post we look at two more accepted paper in this post.
Non-Interactive Proofs of Proximity by Tom Gur and Ron D. Rothblum (ECCC). In a proof-systems a verifier tries to ascertain the validity of a statement by using the help of a proof. PCP, PCPP are classical examples of proof-systems. The kind of access a verifier has to the statement and the proof determines the power of the proof system. This paper considers the proof system when the verifier has full access to the proof (which is sub-linear) and oracle access to the statement. And the goal of the verifier is to accept a valid statement and reject a statement that is far from true while optimizing the queries to the statement. This paper tries to understand the power of this proof system (in terms of query complexity) in comparison to other proof systems.
Testing Surface Area by Pravesh Kothari, Amir Nayyeri, Ryan O’Donnell and Chenggang Wu (accepted in SODA 2014) [preprint]. Estimating the surface area of a set F ⊆ Rn given point-query access is a very fundamental problem in geometry. Usually one assumes some structure on the surface for estimating the area, like convexity. This paper considers general surface areas and the goal is to distinguish the case when the surface area is less than A from the case when the surface is far from having a surface area less than κA (κ a parameter ≥1). In the 1, 2 or 3 dimension constant query algorithm is obtained for certain κ.
A connection between surface area and noise sensitivity by Joe Neeman (ArXiv). This paper also considers the problem of estimating the surface area of a set. It improves some of the results from the above mentioned paper – “Testing Surface Area” by Pravesh Kothari, Amir Nayyeri, Ryan O’Donnell and Chenggang Wu.
Testing equivalence between distributions using conditional samples by Clement Canonne, Dana Ron and Rocco A. Servedio (accepted in SODA 2014) [preprint]. Testing whether two distributions are equivalent is an important and well studied problem. Usually a distribution is accessed by drawing random samples according to the distribution. But other variants of how to access the distributions has also been studied. One such is the conditional query: here one can specify a set S and draws a random sample according to the distribution conditioned on the fact that outcome comes from S. This paper shows that conditional queries can help in improving the query complexity by an exponential amount (or more) for testing equivalence of distributions (both when one distribution is known and when none of them are known). Some restricted versions of conditional queries (where the size of S is fixed) have also been studied.