  Recently, many works have focused on the characterization of non-linear
dimensionality reduction methods obtained by quantizing linear embeddings,
e.g., to reach fast processing time, efficient data compression procedures,
novel geometry-preserving embeddings or to estimate the information/bits stored
in this reduced data representation. In this work, we prove that many linear
maps known to respect the restricted isometry property (RIP) can induce a
quantized random embedding with controllable multiplicative and additive
distortions with respect to the pairwise distances of the data points beings
considered. In other words, linear matrices having fast matrix-vector
multiplication algorithms (e.g., based on partial Fourier ensembles or on the
adjacency matrix of unbalanced expanders) can be readily used in the definition
of fast quantized embeddings with small distortions. This implication is made
possible by applying right after the linear map an additive and random "dither"
that stabilizes the impact of the uniform scalar quantization operator applied
afterwards. For different categories of RIP matrices, i.e., for different
linear embeddings of a metric space $(\mathcal K \subset \mathbb R^n, \ell_q)$
in $(\mathbb R^m, \ell_p)$ with $p,q \geq 1$, we derive upper bounds on the
additive distortion induced by quantization, showing that it decays either when
the embedding dimension $m$ increases or when the distance of a pair of
embedded vectors in $\mathcal K$ decreases. Finally, we develop a novel
"bi-dithered" quantization scheme, which allows for a reduced distortion that
decreases when the embedding dimension grows and independently of the
considered pair of vectors.
