Structure of Indicator Function Classes With Finite Vapnik–Chervonenkis Dimensions

The Vapnik–Chervonenkis (VC) dimension is used to measure the complexity of a function class and plays an important role in a variety of fields, including artificial neural networks and machine learning. One major concern is the relationship between the VC dimension and inherent characteristics of the corresponding function class. According to Sauer's lemma, if the VC dimension of an indicator function class ${\cal F}$ is equal to $D$, the cardinality of the set ${\cal F}_{S_{1}^{N}}$ will not be larger than $\sum_{d=0}^{D}C_{N}^{d}$. Therefore, there naturally arises a question about the VC dimension of an indicator function class: what kinds of elements will be contained in the function class ${\cal F}$ if ${\cal F}$ has a finite VC dimension? In this brief, we answer the above question. First, we investigate the structure of the function class ${\cal F}$ when the cardinality of the set ${\cal F}_{S_{1}^{N}}$ reaches the maximum value $\sum_{d=0}^{D}C_{N}^{d}$. Based on the derived result, we then figure out what kinds of elements will be contained in ${\cal F}$ if ${\cal F}$ has a finite VC dimension.