  Intelligent reflecting surface (IRS) is a promising technology for the 6th
generation of wireless systems, realizing the smart radio environment concept.
In this paper, we present a novel tensor-based receiver for IRS-assisted
multiple-input multiple-output communications capable of jointly estimating the
channels and the transmitted data streams in a semi-blind fashion. Assuming a
fully passive IRS architecture and introducing a simple space-time coding
scheme at the transmitter, the received signal model can be advantageously
built using the PARATUCK tensor model, which can be seen as a hybrid of
parallel factor analysis and Tucker models. Exploiting the algebraic structure
of the PARATUCK tensor model, a semi-blind receiver is derived. The proposed
receiver is based on a trilinear alternating least squares method that
iteratively estimates the two involved - IRS- base station and user
terminal-IRS-communication channels and the transmitted symbol matrix. We
discuss identifiability conditions that ensure the joint semi-blind recovery of
the involved channel and symbol matrices, and propose a joint design of the
coding and IRS reflection matrices to optimize the receiver performance. For
the proposed semi-blind receiver, the derivation of the expected Cram\'er-Rao
lower bound is also provided. A numerical performance evaluation of the
proposed receiver design corroborates its superior performance in terms of the
normalized mean squared error of the estimated channels and the achieved symbol
error rate.
