  Array processing performance depends on the number of microphones available.
Virtual microphone estimation (VME) has been proposed to increase the number of
microphone signals artificially. Neural network-based VME (NN-VME) trains an NN
with a VM-level loss to predict a signal at a microphone location that is
available during training but not at inference. However, this training
objective may not be optimal for a specific array processing back-end, such as
beamforming. An alternative approach is to use a training objective considering
the array-processing back-end, such as a loss on the beamformer output. This
approach may generate signals optimal for beamforming but not physically
grounded. To combine the advantages of both approaches, this paper proposes a
multi-task loss for NN-VME that combines both VM-level and beamformer-level
losses. We evaluate the proposed multi-task NN-VME on multi-talker
underdetermined conditions and show that it achieves a 33.1 % relative WER
improvement compared to using only real microphones and 10.8 % compared to
using a prior NN-VME approach.
