  Matrices satisfying the Restricted Isometry Property (RIP) play an important
role in the areas of compressed sensing and statistical learning. RIP matrices
with optimal parameters are mainly obtained via probabilistic arguments, as
explicit constructions seem hard. It is therefore interesting to ask whether a
fixed matrix can be incorporated into a construction of restricted isometries.
In this paper, we construct a new broad ensemble of random matrices with
dependent entries that satisfy the restricted isometry property. Our
construction starts with a fixed (deterministic) matrix $X$ satisfying some
simple stable rank condition, and we show that the matrix $XR$, where $R$ is a
random matrix drawn from various popular probabilistic models (including,
subgaussian, sparse, low-randomness, satisfying convex concentration property),
satisfies the RIP with high probability. These theorems have various
applications in signal recovery, random matrix theory, dimensionality
reduction, etc. Additionally, motivated by an application for understanding the
effectiveness of word vector embeddings popular in natural language processing
and machine learning applications, we investigate the RIP of the matrix
$XR^{(l)}$ where $R^{(l)}$ is formed by taking all possible (disregarding
order) $l$-way entrywise products of the columns of a random matrix $R$.
