On Numerosity of Deep Convolutional Neural Networks

Subitizing, or the sense of small natural numbers, is a cognitive construct so primary and critical to the survival and well-being of humans and primates that is considered and proven to be innate; it responds to visual stimuli prior to the development of any symbolic skills, language or arithmetic. Given highly acclaimed successes of deep convolutional neural networks (DCNN) in tasks of visual intelligence, one would expect that DCNNs can learn subitizing. But somewhat surprisingly, our carefully crafted extensive experiments, which are similar to those of cognitive psychology, demonstrate that DCNNs cannot, even with strong supervision, see through superficial variations in visual representations and distill the abstract notion of natural number, a task that children perform with high accuracy and confidence. The DCNN black box learners driven by very large training sets are apparently still confused by geometric variations and fail to grasp the topological essence in subitizing. In sharp contrast to the failures of the black box learning, by incorporating a mechanism of mathematical morphology into convolutional kernels, we are able to construct a recurrent convolutional neural network that can perform subitizing deterministically. Our findings in this study of cognitive computing, without and with prior of human knowledge, are discussed; they are, we believe, significant and thought-provoking in the interests of AI research, because visual-based numerosity is a benchmark of minimum sort for human cognition.