Problem 70: Open Problems in $L_p$-Testing

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Suggested by Grigory Yaroslavtsev
Source Baltimore 2016
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Extending the usual setting of property testing to functions $f\colon\{1,\dots,n\}^n\to[0,1]$, Berman et al. initiate in [BermanRY-12] study of property testing with regard to $L_p$ distances between functions. Namely, these distances are defined as $\operatorname{dist}_p(f,g) = \frac{\lVert f-g\rVert_p}{\lVert \mathbf{1}\rVert_p}$ (for $p > 0$), so that for instance $\operatorname{dist}_1(f,g) = \frac{\lVert f-g\rVert_1}{n^d}$ (and if the functions are Boolean, we get back the Hamming distance).

- Problem 1: [BermanRY-12] describe an $L_1$-tester for convexity whose query complexity, $O(\frac{1}{\varepsilon^{d/2}}+\frac{1}{\varepsilon})$, grows exponentially with the dimension $d$. Is this exponential dependence necessary, or is there a tester with query complexity $O(\frac{1}{\varepsilon^{o(d)}})$?

- Problem 2: Obtain a tolerant $L_1$ tester for monotonicity for $d\geq 3$. (There exist testers, albeit maybe non-optimal, in the case $d=1$ or $d=2$, from [BermanRY-12]; nothing non-trivial is known for higher dimensions).

Note: slides describing the setting and open problems can be found on Grigory's webpage.