Minimax lower bounds for linear independence testing

Linear independence testing is a fundamental information-theoretic and statistical problem that can be posed as follows: given n points {(X
_i
, Y
_i
)}
_i=1
ⁿ
from a p+q dimensional multivariate distribution where X
_i
∈ ℝ
^p
and Y
_i
∈ ℝ
^q
, determine whether a
^T
X and b
^T
Y are uncorrelated for every a ∈ ℝ
^p
; b ∈ ℝ
^q
or not. We give minimax lower bound for this problem (when p + q, n → ∞, (p + q)/n ≥ κ <; ∞, without sparsity assumptions). In summary, our results imply that n must be at least as large as √pq/∥Σ
_XY
∥
_F
²
for any procedure (test) to have non-trivial power, where Σ
_XY
is the cross-covariance matrix of X, Y. We also provide evidence that the lower bound is tight, by connections to two-sample testing and regression when q = 1.