This paper studies determinantal point processes (DPP) with non-symmetric kernels. Most of the machine learning literature on DPP assumes symmetric kernels, and the prior work that studies non-symmetric kernels have assumed a quite restricted class of non-symmetric kernels. The novelty of this paper is in proposing the learning algorithm for a fairly general class of non-symmetric kernels. The proposed approach assumes a particular representation of non-symmetric kernels. This representation follows from two known results in a rather straightforward manner, as I also summarize in "1.

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.