  We propose two practical non-convex approaches for learning near-isometric,
linear embeddings of finite sets of data points. Given a set of training points
$\mathcal{X}$, we consider the secant set $S(\mathcal{X})$ that consists of all
pairwise difference vectors of $\mathcal{X}$, normalized to lie on the unit
sphere. The problem can be formulated as finding a symmetric and positive
semi-definite matrix $\boldsymbol{\Psi}$ that preserves the norms of all the
vectors in $S(\mathcal{X})$ up to a distortion parameter $\delta$. Motivated by
non-negative matrix factorization, we reformulate our problem into a Frobenius
norm minimization problem, which is solved by the Alternating Direction Method
of Multipliers (ADMM) and develop an algorithm, FroMax. Another method solves
for a projection matrix $\boldsymbol{\Psi}$ by minimizing the restricted
isometry property (RIP) directly over the set of symmetric, postive
semi-definite matrices. Applying ADMM and a Moreau decomposition on a proximal
mapping, we develop another algorithm, NILE-Pro, for dimensionality reduction.
FroMax is shown to converge faster for smaller $\delta$ while NILE-Pro
converges faster for larger $\delta$. Both non-convex approaches are then
empirically demonstrated to be more computationally efficient than prior convex
approaches for a number of applications in machine learning and signal
processing.
