9 research outputs found
Lower Bounds for Tolerant Junta and Unateness Testing via Rejection Sampling of Graphs
We introduce a new model for testing graph properties which we call the rejection sampling model. We show that testing bipartiteness of n-nodes graphs using rejection sampling queries requires complexity Omega~(n^2). Via reductions from the rejection sampling model, we give three new lower bounds for tolerant testing of Boolean functions of the form f : {0,1}^n -> {0,1}:
- Tolerant k-junta testing with non-adaptive queries requires Omega~(k^2) queries.
- Tolerant unateness testing requires Omega~(n) queries.
- Tolerant unateness testing with non-adaptive queries requires Omega~(n^{3/2}) queries.
Given the O~(k^{3/2})-query non-adaptive junta tester of Blais [Eric Blais, 2008], we conclude that non-adaptive tolerant junta testing requires more queries than non-tolerant junta testing. In addition, given the O~(n^{3/4})-query unateness tester of Chen, Waingarten, and Xie [Xi Chen et al., 2017] and the O~(n)-query non-adaptive unateness tester of Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova, and Seshadhri [Roksana Baleshzar et al., 2017], we conclude that tolerant unateness testing requires more queries than non-tolerant unateness testing, in both adaptive and non-adaptive settings. These lower bounds provide the first separation between tolerant and non-tolerant testing for a natural property of Boolean functions
Mildly Exponential Lower Bounds on Tolerant Testers for Monotonicity, Unateness, and Juntas
We give the first super-polynomial (in fact, mildly exponential) lower bounds
for tolerant testing (equivalently, distance estimation) of monotonicity,
unateness, and juntas with a constant separation between the "yes" and "no"
cases. Specifically, we give
A -query lower bound for
non-adaptive, two-sided tolerant monotonicity testers and unateness testers
when the "gap" parameter is equal to
, for any ;
A -query lower bound for non-adaptive,
two-sided tolerant junta testers when the gap parameter is an absolute
constant.
In the constant-gap regime no non-trivial prior lower bound was known for
monotonicity, the best prior lower bound known for unateness was
queries, and the best prior lower bound known for
juntas was queries.Comment: 20 pages, 1 figur
On Tolerant Testing and Tolerant Junta Testing
Over the past few decades property testing has became an active field of study in theoretical computer science. The algorithmic task is to determine, given access to an unknown large object (e.g., function, graph, probability distribution), whether it has some fixed property, or it is far from any object having the property. The approximate nature of these algorithms allows in many cases to achieve a significant saving in running time, and obtain \emph{sublinear} running time. Nevertheless, in various settings and applications, accepting only inputs that exactly have a certain property is too restrictive, and it is more beneficial to distinguish between inputs that are close to having the property, and those that are far from it. The framework of \emph{tolerant} testing tackles this exact problem. In this thesis, we will focus on one of the most fundamental properties of Boolean functions: the property of being a \emph{-junta} (i.e., being dependent on at most variables).
The first chapter focuses on algorithms for tolerant junta testing. In particular, we show that there exists a \poly(k) query algorithm distinguishing functions close to -juntas and functions that are far from -juntas. We also show how to obtain a trade-off between the ``tolerance" of the algorithm and its query complexity.
The second chapter focuses on establishing a query lower bound for tolerant junta testing. In particular, we show that any non-adaptive tolerant junta tester, is required to make at least \Omega(k^2/\polylog k) queries.
The third chapter considers tolerant testing in a more general context, and asks whether tolerant testing is strictly harder than standard testing. In particular, we show that for any constant , there exists a property \calP_\ell such that \calP_\ell can be tested in queries, but any tolerant tester for \calP_\ell is required to make at least queries (where denote the times iterated log function).
The final chapter focuses on applications. We show how to leverage the techniques developed in previous chapters to obtain results on tolerant isomorphism testing, unateness testing, and erasure resilient testing
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New Methods in Sublinear Computation for High Dimensional Problems
We study two classes of problems within sublinear algorithms: data structures for approximate nearest neighbor search, and property testing of Boolean functions. We develop algorithmic and analytical tools for proving upper and lower bounds on the complexity of these problems, and obtain the following results:
* We give data structures for approximate nearest neighbor search achieving state-of-the-art approximations for various high-dimensional normed spaces. For example, our data structure for normed spaces over R answers queries in sublinear time while using nearly linear space and achieves approximation which is sub-polynomial in the dimension.
* We prove query complexity lower bounds for property testing of three fundamental properties: -juntas, monotonicity, and unateness. Our lower bounds for non-adaptive junta testing and adaptive unateness testing are nearly optimal, and the lower bound for adaptive monotonicity testing is the best that is currently known.
* We give an algorithm for testing unateness with nearly optimal query complexity. The algorithm is crucially adaptive and based on a novel analysis of binary search over long paths of the hypercube
Junta Distance Approximation with Sub-Exponential Queries
Leveraging tools of De, Mossel, and Neeman [FOCS, 2019], we show two
different results pertaining to the \emph{tolerant testing} of juntas. Given
black-box access to a Boolean function , we give a
query algorithm that distinguishes between
functions that are -close to -juntas and -far
from -juntas, where .
In the non-relaxed setting, we extend our ideas to give a
(adaptive) query algorithm that
distinguishes between functions that are -close to -juntas and
-far from -juntas. To the best of our knowledge, this
is the first subexponential-in- query algorithm for approximating the
distance of to being a -junta (previous results of Blais, Canonne, Eden,
Levi, and Ron [SODA, 2018] and De, Mossel, and Neeman [FOCS, 2019] required
exponentially many queries in ).
Our techniques are Fourier analytical and make use of the notion of
"normalized influences" that was introduced by Talagrand [AoP, 1994].Comment: To appear in CCC 202
Nearly Optimal Algorithms for Testing and Learning Quantum Junta Channels
We consider the problems of testing and learning quantum -junta channels,
which are -qubit to -qubit quantum channels acting non-trivially on at
most out of qubits and leaving the rest of qubits unchanged. We show
the following.
1. An -query algorithm to distinguish
whether the given channel is -junta channel or is far from any -junta
channels, and a lower bound on the number of
queries;
2. An -query algorithm to learn a -junta
channel, and a lower bound on the number of queries.
This answers an open problem raised by Chen et al. (2023). In order to settle
these problems, we develop a Fourier analysis framework over the space of
superoperators and prove several fundamental properties, which extends the
Fourier analysis over the space of operators introduced in Montanaro and
Osborne (2010)
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum