Testing k-Monotonicity

@article{Canonne2016TestingK,
  title={Testing k-Monotonicity},
  author={Cl{\'e}ment L. Canonne and Elena Grigorescu and Siyao Guo and Akash Kumar and Karl Wimmer},
  journal={Electron. Colloquium Comput. Complex.},
  year={2016},
  volume={TR16}
}
A Boolean $k$-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 monotone functions, which are the $1$-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… 

Tables from this paper

Downsampling for Testing and Learning in Product Distributions

We study the domain reduction problem of eliminating dependence on $n$ from the complexity of property testing and learning algorithms on domain $[n]^d$, and the related problem of establishing

K-Monotonicity is Not Testable on the Hypercube

This work disproves the conjecture that k-monotonicity can be tested with poly(n) queries, and shows that even 2-monotone requires an exponential in √ n number of queries.

Improved Bounds for Testing Forbidden Order Patterns

An adaptivity hierarchy for $\pi=(1,3,2)$ is shown by proving upper and lower bounds for (one- and two-sided) testing of $\pi$-freeness with $r$ rounds of adaptivity and demonstrating a surprising behavior for non-adaptive tests with one-sided error.

Flipping out with Many Flips: Hardness of Testing k-Monotonicity

This work studies k-monotone functions in the more relaxed parametrized property testing model, introduced by Parnas et al. (JCSS, 72(6), 2006), and resolves a problem left open in previous work.

Almost Optimal Distribution-Free Sample-Based Testing of k-Modality

This work studies one-sided error property testing of k-modality in the distribution-free sample-based model and proves an upper bound of1 O (√ kn log k ) on the sample complexity, and an almost matching lower bound of Ω ( √ kn ) .

Properly learning monotone functions via local reconstruction

—We give a 2 ˜ O ( √ n/ε ) -time algorithm for properly learning monotone Boolean functions under the uni- form distribution over { 0 , 1 } n . Our algorithm is robust to adversarial label noise and

Testing Hereditary Properties of Ordered Graphs and Matrices

The proof bridges the gap between techniques related to the regularity lemma, used in the long chain of papers investigating graph testing, and string testing techniques and develops a Ramsey-type lemma for multipartite graphs with undesirable edges.

Earthmover Resilience and Testing in Ordered Structures

A wide class of properties of ordered structures - the earthmover resilient (ER) properties - are identified and it is shown that the "good behavior" of such properties allows us to obtain general testability results that are similar to (and more general than) those of unordered graphs.

Proofs of Proximity for Distribution Testing

The main results include showing that MA distribution testers can be quadratically stronger than standard distribution testers, but no stronger than that; in contrast, IP distribution testers are shown to be exponentially stronger than normal distributions, but when restricted to public coins they can be at best quadratic stronger.

Testing distributional assumptions of learning algorithms

A model by which to systematically study the design of tester-learner pairs is proposed, such that if the distribution on examples in the data passes the tester T then one can safely trust the output of the agnostic learner A on the data.

References

SHOWING 1-10 OF 57 REFERENCES

Sensitivity Conjecture and Log-Rank Conjecture for Functions with Small Alternating Numbers

These results are extended to functions which alternate values for a relatively small number of times on any monotone path from 0-n to 1-n, and contribute to the recent line of research on functions with small alternating numbers.

Monotonicity testing over general poset domains

It is shown that in its most general setting, testing that Boolean functions are close to monotone is equivalent, with respect to the number of required queries, to several other testing problems in logic and graph theory.

Transitive-Closure Spanners

The common task implicitly tackled in these diverse applications as the problem of constructing sparse TC-spanners is abstracted asThe study of approximability of the size of the sparsest of a given directed graph is initiated.

Learning monotone decision trees in polynomial time

  • R. O'DonnellR. Servedio
  • Computer Science, Mathematics
    21st Annual IEEE Conference on Computational Complexity (CCC'06)
  • 2006
This is the first algorithm that can learn arbitrary monotone Boolean functions to high accuracy, using random examples only, in time polynomial in a reasonable measure of the complexity of f.

Monotonicity testing and shortest-path routing on the cube

It is shown that for any δ > 0, the n-dimensional hypercube is not n-realizable with shortest paths, while previously it was only known that hypercubes are not 1- realizable with longest paths.

New Algorithms and Lower Bounds for Monotonicity Testing

A new lower bound is proved on the query complexity of any non-adaptive two-sided error algorithm for testing whether an unknown Boolean function f is monotone versus constant-far from monotones and an algorithm is presented that makes O(n<sup>7/8</sup>) poly(1/ε) queries.

Learning circuits with few negations

This paper studies the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and the class of all functions.

On learning monotone DNF under product distributions

  • R. Servedio
  • Mathematics, Computer Science
    Inf. Comput.
  • 2001
We show that the class of monotone 2O(√ n)-term DNF formulae can be PAC learned in polynomial time under the uniform distribution from random examples only. This is an exponential improvement over

Estimating the distance to a monotone function

The running time of the algorithm is O(εf−1 log log εf− 1 log n), which is optimal within a factor of loglog ε f−1 and represents a substantial improvement over previous work.

Correlation Bounds Against Monotone NC^1

  • Benjamin Rossman
  • Mathematics, Computer Science
    Computational Complexity Conference
  • 2015
The main theorem, proved using the pathset complexity framework introduced in [56], shows that the average-case k-CYCLE problem (on Erdos-Renyi random graphs with an appropriate edge density) is [EQUATION] hard for mNC1.
...