  This paper investigates reduced complexity neural network (NN) based
architectures for equalization over the two-dimension magnetic recording (TDMR)
digital communication channel for data storage. We use realistic waveforms
measured from a hard disk drive (HDD) with TDMR technology. We show that the
multilayer perceptron (MLP) non-linear equalizer achieves a $10.91\%$ reduction
in bit error rate (BER) over the linear equalizer with cross-entropy-based
optimization. However, the MLP equalizer's complexity is $6.6$ times the linear
equalizer's complexity. Thus, we propose reduced complexity MLP (RC-MLP)
equalizers. Each RC-MLP variant consists of finite-impulse response filters, a
non-linear activation, and a hidden delay line. A proposed RC-MLP variant
entails only $1.59$ times the linear equalizer's complexity while achieving a
$8.23\%$ reduction in BER over the linear equalizer.
