Title: ON COMPLEX-DOMAIN CNN REPRESENTATIONS FOR CLASSIFYING REAL/COMPLEX-VALUED DATA. Abstract: This paper is about complex-valued CNNs (CV-CNNs) for computer vision that use  representations that are complex-valued instead of real-valued. We divide input data into three categories: inherently real-valued, inherently complex-valued, and  complex-valued obtained by transforming real-valued. We study the question whether complex-valued representation of CV-CNNs  offers any advantages over the commonly used  real-valued CNNs (RV-CNNs). For concreteness, we focus on the classification task.  The existing literature offers contradictory answers to our question. We find that this is mainly because (a) they seldom employ a common performance measure (e.g., CV-CNN compared against RV-CNN with similar network structure vs similar number of parameters) (b) diversity of evaluation datasets used are limited (e.g., datasets in which  magnitude information is more, less or as important as phase information) (c) less effort has been devoted to reduce the randomness in training between CV-CNN and RV-CNN.  Towards this, we propose  performance measures based on similar network structure, number of parameters and number of MAC operations. Also, we consider diverse datasets with varying magnitude/phase information, and deal with the randomness in training. As a result, we expect that any observed performance differences will be independent of the above disparities, and arise from the use of real vs complex representations.  Theoretically, we show that, unlike RV-CNNs,  CV-CNNs can preserve magnitude and phase through intermediate stages of processing. Our main experimental findings are the following. (1) As network depth decreases ---   the performance of CV-CNNs  improves  with respect to similar  network structure;   the performances of CV-CNN and RV-CNN having a similar number of parameters become more comparable; and   the performance of RV-CNNs improves with respect to similar number of MAC operations; (2) The above performance differences  diminish as the network depth increases. (3) With respect to data diversity,  performance depends on whether the dataset has dominant magnitude or phase, i.e., whether reconstruction error is lower using only magnitude or only phase.  If a complex-valued data has dominant magnitude,  instead of providing real and imaginary parts as input, providing the  magnitude part produces significant performance gain, whereas if the data has dominant phase, providing both real and imaginary parts is important. This holds true for different network depths.