Causal Groupoid Symmetries and Big Data

Abstract

The big problem of Big Data is the lack of a machine learning process that scales and finds meaningful features. Humans fill in for the insufficient automation, but the complexity of the tasks outpaces the human mind's capacity to comprehend the data. Heuristic partition methods may help but still need humans to adjust the parameters. The same problems exist in many other disciplines and technologies that depend on Big Data or Machine Learning. Proposed here is a fractal groupoid-theoretical method that recursively partitions the problem and requires no heuristics or human intervention. It takes two steps. First, make explicit the fundamental causal nature of information in the physical world by encoding it as a causal set. Second, construct a functor F: C ⇒ C′ on the category of causal sets that morphs causal set C into smaller causal set C′ by partitioning C into a set of invariant groupoid-theoretical blocks. Repeating the construction, there arises a sequence of progressively smaller causal sets C, C′, C″, ··· The sequence defines a fractal hierarchy of features, with the features being invariant and hence endowed with a physical meaning, and the hierarchy being scale-free and hence ensuring proper scaling at all granularities. Fractals exist in nature nearly everywhere and at all physical scales, and invariants have long been known to be meaningful to us. The theory is also of interest for NP-hard combinatorial problems that can be expressed as a causal set, such as the Traveling Salesman problem. The recursive groupoid partition promoted by functor F works against their combinatorial complexity and appears to allow a low-order polynomial solution. A true test of this property requires special hardware, not yet available. However, as a proof of concept, a suite of sequential, non-heuristic algorithms were developed and used to solve a real-world 120-city problem of TSP on a personal computer. The results are reported.

Reproducing Kernel Hilbert Space Associated with a Unitary Representation of a Groupoid

“…The problem, often encountered in data processing modules (implemented in such fields as electronics or control engineering) is the selection of the optimal kernel regarding the distance between feature vectors in the multidimensional space. According to [20], such a measure (on the groupoids) can be used to solve the generalized version of the traveling salesman problem. The overall distance to minimize is given as: L = N i=1 d(c i , c i+1 ), where c i and c i+1 are two subsequent nodes from the graph in the optimized cycle.…”

Reproducing Kernel Hilbert Space Associated with a Unitary Representation of a Groupoid

“…The problem, often encountered in data processing modules (implemented in such fields as electronics or control engineering) is the selection of the optimal kernel regarding the distance between feature vectors in the multidimensional space. According to [19], such a measure (on the groupoids) can be used to solve the generalized version of the Traveling Salesman Problem. The overall distance to minimize is given as: L = N i=1 d(c i , c i+1 ), where c i and c i+1 are two subsequent nodes from the graph in the optimized cycle.…”