Property Testing Bounds for Linear and Quadratic Functions via Parity Decision Trees

Abstract

In this paper, we study linear and quadratic Boolean functions in the context of property testing. We do this by observing that the query complexity of testing properties of linear and quadratic functions can be characterized in terms of complexity in another model of computation called parity decision trees.

The observation allows us to characterize testable properties of linear functions in terms of the approximate l ₁ norm of the Fourier spectrum of an associated function. It also allows us to reprove the Ω(k) lower bound for testing k-linearity due to Blais et al [8]. More interestingly, it rekindles the hope of closing the gap of Ω(k) vs O(klogk) for testing k-linearity by analyzing the randomized parity decision tree complexity of a fairly simple function called E _k that evaluates to 1 if and only if the number of 1s in the input is exactly k. The approach of Blais et al. using communication complexity is unlikely to give anything better than Ω(k) as a lower bound.

In the case of quadratic functions, we prove an adaptive two-sided Ω(n ²) lower bound for testing affine isomorphism to the inner product function. We remark that this bound is tight and furnishes an example of a function for which the trivial algorithm for testing affine isomorphism is the best possible. As a corollary, we obtain an Ω(n ²) lower bound for testing the class of Bent functions.

We believe that our techniques might be of independent interest and may be useful in proving other testing bounds.