Title: Geared Rotationally Identical and Invariant Convolutional Neural Network Systems,
Abstract: Theorems and techniques to form different types of transformationally
invariant processing and to produce the same output quantitatively based on
either transformationally invariant operators or symmetric operations have
recently been introduced by the authors. In this study, we further propose to
compose a geared rotationally identical CNN system (GRI-CNN) with a small step
angle by connecting networks of participated processes at the first flatten
layer. Using an ordinary CNN structure as a base, requirements for constructing
a GRI-CNN include the use of either symmetric input vector or kernels with an
angle increment that can form a complete cycle as a "gearwheel". Four basic
GRI-CNN structures were studied. Each of them can produce quantitatively
identical output results when a rotation angle of the input vector is evenly
divisible by the step angle of the gear. Our study showed when an input vector
rotated with an angle does not match to a step angle, the GRI-CNN can also
produce a highly consistent result. With a design of using an ultra-fine
gear-tooth step angle (e.g., 1 degree or 0.1 degree), all four GRI-CNN systems
can be constructed virtually isotropically.