  Recovering a low-rank tensor from incomplete information is a recurring
problem in signal processing and machine learning. The most popular convex
relaxation of this problem minimizes the sum of the nuclear norms of the
unfoldings of the tensor. We show that this approach can be substantially
suboptimal: reliably recovering a $K$-way tensor of length $n$ and Tucker rank
$r$ from Gaussian measurements requires $\Omega(r n^{K-1})$ observations. In
contrast, a certain (intractable) nonconvex formulation needs only $O(r^K +
nrK)$ observations. We introduce a very simple, new convex relaxation, which
partially bridges this gap. Our new formulation succeeds with $O(r^{\lfloor K/2
\rfloor}n^{\lceil K/2 \rceil})$ observations. While these results pertain to
Gaussian measurements, simulations strongly suggest that the new norm also
outperforms the sum of nuclear norms for tensor completion from a random subset
of entries.
  Our lower bound for the sum-of-nuclear-norms model follows from a new result
on recovering signals with multiple sparse structures (e.g. sparse, low rank),
which perhaps surprisingly demonstrates the significant suboptimality of the
commonly used recovery approach via minimizing the sum of individual sparsity
inducing norms (e.g. $l_1$, nuclear norm). Our new formulation for low-rank
tensor recovery however opens the possibility in reducing the sample complexity
by exploiting several structures jointly.
