  We study the complexity of Non-Gaussian Component Analysis (NGCA) in the
Statistical Query (SQ) model. Prior work developed a general methodology to
prove SQ lower bounds for this task that have been applicable to a wide range
of contexts. In particular, it was known that for any univariate distribution
$A$ satisfying certain conditions, distinguishing between a standard
multivariate Gaussian and a distribution that behaves like $A$ in a random
hidden direction and like a standard Gaussian in the orthogonal complement, is
SQ-hard. The required conditions were that (1) $A$ matches many low-order
moments with the standard univariate Gaussian, and (2) the chi-squared norm of
$A$ with respect to the standard Gaussian is finite. While the moment-matching
condition is necessary for hardness, the chi-squared condition was only
required for technical reasons. In this work, we establish that the latter
condition is indeed not necessary. In particular, we prove near-optimal SQ
lower bounds for NGCA under the moment-matching condition only. Our result
naturally generalizes to the setting of a hidden subspace. Leveraging our
general SQ lower bound, we obtain near-optimal SQ lower bounds for a range of
concrete estimation tasks where existing techniques provide sub-optimal or even
vacuous guarantees.
