First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. The set in question in this paper is the set of function with bounded partial derivatives up to order K. The authors' technique mimics the work of Thaler et al (ICALP 2012) only the authors decompose the queries not into regular polynomials (Chebyshev polynomials in the case of Thaler et al), but rather to trigonometric polynomial in this case. The bulk of the work is indeed to show that the abovementioned set of queries can be well-approximated by trigonometric polynomials. Having established that, adding Laplace noise to each monomial suffices to guarantee differential privacy.

We consider accurately answering smooth queries while preserving differential privacy. A query is said to be $K$-smooth if it is specified by a function defined on $[-1,1]^d$ whose partial derivatives up to order $K$ are all bounded. We develop an $\epsilon$-differentially private mechanism for the class of $K$-smooth queries. The major advantage of the algorithm is that it outputs a synthetic database. In real applications, a synthetic database output is appealing. Our mechanism achieves an accuracy of $O (n^{-\frac{K}{2d+K}}/\epsilon )$, and runs in polynomial time. We also generalize the mechanism to preserve $(\epsilon, \delta)$-differential privacy with slightly improved accuracy. Extensive experiments on benchmark datasets demonstrate that the mechanisms have good accuracy and are efficient.