  We consider the critical problem of distributed learning over data while
keeping it private from the computational servers. The state-of-the-art
approaches to this problem rely on quantizing the data into a finite field, so
that the cryptographic approaches for secure multiparty computing can then be
employed. These approaches, however, can result in substantial accuracy losses
due to fixed-point representation of the data and computation overflows. To
address these critical issues, we propose a novel algorithm to solve the
problem when data is in the analog domain, e.g., the field of real/complex
numbers. We characterize the privacy of the data from both
information-theoretic and cryptographic perspectives, while establishing a
connection between the two notions in the analog domain. More specifically, the
well-known connection between the distinguishing security (DS) and the mutual
information security (MIS) metrics is extended from the discrete domain to the
continues domain. This is then utilized to bound the amount of information
about the data leaked to the servers in our protocol, in terms of the DS
metric, using well-known results on the capacity of single-input
multiple-output (SIMO) channel with correlated noise. It is shown how the
proposed framework can be adopted to do computation tasks when data is
represented using floating-point numbers. We then show that this leads to a
fundamental trade-off between the privacy level of data and accuracy of the
result. As an application, we also show how to train a machine learning model
while keeping the data as well as the trained model private. Then numerical
results are shown for experiments on the MNIST dataset. Furthermore,
experimental advantages are shown comparing to fixed-point implementations over
finite fields.
