  There is a resurging interest in developing a neural-network-based solution
to the supervised machine learning problem. The convolutional neural network
(CNN) will be studied in this note. To begin with, we introduce a RECOS
transform as a basic building block of CNNs. The "RECOS" is an acronym for
"REctified-COrrelations on a Sphere". It consists of two main concepts: 1) data
clustering on a sphere and 2) rectification. Afterwards, we interpret a CNN as
a network that implements the guided multi-layer RECOS transform with three
highlights. First, we compare the traditional single-layer and modern
multi-layer signal analysis approaches, point out key ingredients that enable
the multi-layer approach, and provide a full explanation to the operating
principle of CNNs. Second, we discuss how guidance is provided by labels
through backpropagation (BP) in the training. Third, we show that a trained
network can be greatly simplified in the testing stage demanding only one-bit
representation for both filter weights and inputs.
