  We present a generalization of the well-known problem of learning k-juntas in
R^n, and a novel tensor algorithm for unraveling the structure of
high-dimensional distributions. Our algorithm can be viewed as a higher-order
extension of Principal Component Analysis (PCA).
  Our motivating problem is learning a labeling function in R^n, which is
determined by an unknown k-dimensional subspace. This problem of learning a
k-subspace junta is a common generalization of learning a k-junta (a function
of k coordinates in R^n) and learning intersections of k halfspaces. In this
context, we introduce an irrelevant noisy attributes model where the
distribution over the "relevant" k-dimensional subspace is independent of the
distribution over the (n-k)-dimensional "irrelevant" subspace orthogonal to it.
  We give a spectral tensor algorithm which identifies the relevant subspace,
and thereby learns k-subspace juntas under some additional assumptions. We do
this by exploiting the structure of local optima of higher moment tensors over
the unit sphere; PCA finds the global optima of the second moment tensor
(covariance matrix). Our main result is that when the distribution in the
irrelevant (n-k)-dimensional subspace is any Gaussian, the complexity of our
algorithm is T(k,\epsilon) + \poly(n), where T is the complexity of learning
the concept in k dimensions, and the polynomial is a function of the
k-dimensional concept class being learned. This substantially generalizes
existing results on learning low-dimensional concepts.
