The Intrinsic Dimensionality of Data

We consider the problem of determining the intrinsic dimensionality of data which is important for optimizing the organization and processing of large data sets in classical machines, quantum decision theory, and observations of natural phenomena. We prove a theorem that determines the minimum dimensions associated with the data and this result is consistent with the result that base- e is optimal for number representation. The dimension value be viewed as coding the structure in the most efficient representation of the data and has relevance for natural and engineered systems. Since the optimal intrinsic dimensionality is shown to be noninteger, this paper provides a rationale for fractals in natural data.