A Boolean {\em $k$-monotone} function defined over a finite poset domain ${\cal D}$
alternates between the values $0$ and $1$ at most $k$ times on any ascending chain in ${\cal D}$. Therefore, $k$-monotone functions are natural generalizations of the classical {\em monotone} functions, which are the {\em $1$-monotone} functions.
Motivated by the recent interest in $k$-monotone functions in the context of circuit complexity and learning theory, and by the central role that monotonicity testing plays in the context of property testing, we initiate a systematic study of $k$-monotone functions, in the property testing model. In this model, the goal is to distinguish functions that are $k$-monotone (or are close to being $k$-monotone) from functions that are far from being $k$-monotone.
Our results include the following:
\begin{enumerate}
\item We demonstrate a separation between testing $k$-monotonicity and testing monotonicity, on the hypercube domain $\{0,1\}^d$, for $k\geq 3$;
\item We demonstrate a separation between testing and learning on $\{0,1\}^d$, for $k=\omega(\log d)$: testing $k$-monotonicity can be performed with $2^{O(\sqrt d \cdot \log d\cdot \log{1/\eps})}$ queries, while learning $k$-monotone functions requires $2^{\Omega(k\cdot \sqrt d\cdot{1/\eps})}$ queries (Blais et al. (RANDOM 2015)).
\item We present a tolerant test for functions $f\colon[n]^d\to \{0,1\}$ with complexity independent of $n$, which makes progress on a problem left open by Berman et al. (STOC 2014).
\end{enumerate}
Our techniques exploit the testing-by-learning paradigm, use novel applications of Fourier analysis on the grid $[n]^d$, and draw connections to distribution testing techniques.
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A Statistically and Numerically Efficient Independence Test Based on Random Projections and Distance Covariance
Testing for independence plays a fundamental role in many statistical techniques. Among the nonparametric approaches, the distance-based methods (such as the distance correlation-based hypotheses testing for independence) have many advantages, compared with many other alternatives. A known limitation of the distance-based method is that its computational complexity can be high. In general, when the sample size is n , the order of computational complexity of a distance-based method, which typically requires computing of all pairwise distances, can be O ( n 2 ). Recent advances have discovered that in the univariate cases, a fast method with O ( n log n ) computational complexity and O ( n ) memory requirement exists. In this paper, we introduce a test of independence method based on random projection and distance correlation, which achieves nearly the same power as the state-of-the-art distance-based approach, works in the multivariate cases, and enjoys the O ( nK log n ) computational complexity and O ( max{ n , K }) memory requirement, where K is the number of random projections. Note that saving is achieved when K < n / log n . We name our method a Randomly Projected Distance Covariance (RPDC). The statistical theoretical analysis takes advantage of some techniques on the random projection which are rooted in contemporary machine learning. Numerical experiments demonstrate the efficiency of the proposed method, relative to numerous competitors.
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- Award ID(s):
- 2015363
- PAR ID:
- 10461721
- Date Published:
- Journal Name:
- Frontiers in Applied Mathematics and Statistics
- Volume:
- 7
- ISSN:
- 2297-4687
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3389/fams.2021.779841
