7 research outputs found
Testing Junta Truncation
We consider the basic statistical problem of detecting truncation of the
uniform distribution on the Boolean hypercube by juntas. More concretely, we
give upper and lower bounds on the problem of distinguishing between i.i.d.
sample access to either (a) the uniform distribution over , or (b)
the uniform distribution over conditioned on the satisfying
assignments of a -junta .
We show that (up to constant factors) samples suffice for this task and also show
that a dependence on sample complexity is unavoidable. Our
results suggest that testing junta truncation requires learning the set of
relevant variables of the junta
Testing and Learning Quantum Juntas Nearly Optimally
We consider the problem of testing and learning quantum -juntas: -qubit
unitary matrices which act non-trivially on just of the qubits and as
the identity on the rest. As our main algorithmic results, we give (a) a
-query quantum algorithm that can distinguish quantum
-juntas from unitary matrices that are "far" from every quantum -junta;
and (b) a -query algorithm to learn quantum -juntas. We complement
our upper bounds for testing quantum -juntas and learning quantum -juntas
with near-matching lower bounds of and
, respectively. Our techniques are Fourier-analytic and
make use of a notion of influence of qubits on unitaries
Quantitative Correlation Inequalities via Semigroup Interpolation
Most correlation inequalities for high-dimensional functions in the
literature, such as the Fortuin-Kasteleyn-Ginibre (FKG) inequality and the
celebrated Gaussian Correlation Inequality of Royen, are qualitative statements
which establish that any two functions of a certain type have non-negative
correlation. In this work we give a general approach that can be used to
bootstrap many qualitative correlation inequalities for functions over product
spaces into quantitative statements. The approach combines a new extremal
result about power series, proved using complex analysis, with harmonic
analysis of functions over product spaces. We instantiate this general approach
in several different concrete settings to obtain a range of new and
near-optimal quantitative correlation inequalities, including:
A quantitative version of Royen's celebrated Gaussian Correlation
Inequality. Royen (2014) confirmed a conjecture, open for 40 years, stating
that any two symmetric, convex sets must be non-negatively correlated under any
centered Gaussian distribution. We give a lower bound on the correlation in
terms of the vector of degree-2 Hermite coefficients of the two convex sets,
analogous to the correlation bound for monotone Boolean functions over
obtained by Talagrand (1996).
A quantitative version of the well-known FKG inequality for
monotone functions over any finite product probability space, generalizing the
quantitative correlation bound for monotone Boolean functions over
obtained by Talagrand (1996). The only prior generalization of which we are
aware is due to Keller (2008, 2009, 2012), which extended Talagrand's result to
product distributions over . We also give two different quantitative
versions of the FKG inequality for monotone functions over the continuous
domain , answering a question of Keller (2009).Comment: 37 pages, conference version to appear in ITCS 202
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
Testing Convex Truncation
We study the basic statistical problem of testing whether normally
distributed -dimensional data has been truncated, i.e. altered by only
retaining points that lie in some unknown truncation set . As our main algorithmic results,
(1) We give a computationally efficient -sample algorithm that can
distinguish the standard normal distribution from
conditioned on an unknown and arbitrary convex set .
(2) We give a different computationally efficient -sample algorithm
that can distinguish from conditioned on an unknown and
arbitrary mixture of symmetric convex sets.
These results stand in sharp contrast with known results for learning or
testing convex bodies with respect to the normal distribution or learning
convex-truncated normal distributions, where state-of-the-art algorithms
require essentially samples. An easy argument shows that no
finite number of samples suffices to distinguish from an unknown and
arbitrary mixture of general (not necessarily symmetric) convex sets, so no
common generalization of results (1) and (2) above is possible.
We also prove that any algorithm (computationally efficient or otherwise)
that can distinguish from conditioned on an unknown
symmetric convex set must use samples. This shows that the sample
complexity of each of our algorithms is optimal up to a constant factor.Comment: Preliminary version in SODA 2023; the current version includes a
strengthened lower bound. 26 page
Gaussian Approximation of Convex Sets by Intersections of Halfspaces
We study the approximability of general convex sets in by
intersections of halfspaces, where the approximation quality is measured with
respect to the standard Gaussian distribution and the complexity of
an approximation is the number of halfspaces used. While a large body of
research has considered the approximation of convex sets by intersections of
halfspaces under distance metrics such as the Lebesgue measure and Hausdorff
distance, prior to our work there has not been a systematic study of convex
approximation under the Gaussian distribution.
We establish a range of upper and lower bounds, both for general convex sets
and for specific natural convex sets that are of particular interest. Our
results demonstrate that the landscape of approximation is intriguingly
different under the Gaussian distribution versus previously studied distance
measures. For example, we show that halfspaces are both
necessary and sufficient to approximate the origin-centered ball of
Gaussian volume 1/2 to any constant accuracy, and that for , the
origin-centered ball of Gaussian volume 1/2 can be approximated to any
constant accuracy as an intersection of many
halfspaces. These bounds are quite different from known approximation results
under more commonly studied distance measures.
Our results are proved using techniques from many different areas. These
include classical results on convex polyhedral approximation, Cram\'er-type
bounds on large deviations from probability theory, and -- perhaps surprisingly
-- a range of topics from computational complexity, including computational
learning theory, unconditional pseudorandomness, and the study of influences
and noise sensitivity in the analysis of Boolean functions.Comment: 64 pages, 3 figure