  We consider two problems that arise in machine learning applications: the
problem of recovering a planted sparse vector in a random linear subspace and
the problem of decomposing a random low-rank overcomplete 3-tensor. For both
problems, the best known guarantees are based on the sum-of-squares method. We
develop new algorithms inspired by analyses of the sum-of-squares method. Our
algorithms achieve the same or similar guarantees as sum-of-squares for these
problems but the running time is significantly faster.
  For the planted sparse vector problem, we give an algorithm with running time
nearly linear in the input size that approximately recovers a planted sparse
vector with up to constant relative sparsity in a random subspace of $\mathbb
R^n$ of dimension up to $\tilde \Omega(\sqrt n)$. These recovery guarantees
match the best known ones of Barak, Kelner, and Steurer (STOC 2014) up to
logarithmic factors.
  For tensor decomposition, we give an algorithm with running time close to
linear in the input size (with exponent $\approx 1.086$) that approximately
recovers a component of a random 3-tensor over $\mathbb R^n$ of rank up to
$\tilde \Omega(n^{4/3})$. The best previous algorithm for this problem due to
Ge and Ma (RANDOM 2015) works up to rank $\tilde \Omega(n^{3/2})$ but requires
quasipolynomial time.
