  We introduce a novel method to perform linear optical random projections
without the need for holography. Our method consists of a computationally
trivial combination of multiple intensity measurements to mitigate the
information loss usually associated with the absolute-square non-linearity
imposed by optical intensity measurements. Both experimental and numerical
findings demonstrate that the resulting matrix consists of real-valued,
independent, and identically distributed (i.i.d.) Gaussian random entries. Our
optical setup is simple and robust, as it does not require interference between
two beams. We demonstrate the practical applicability of our method by
performing dimensionality reduction on high-dimensional data, a common task in
randomized numerical linear algebra with relevant applications in machine
learning.
