Doubly-bounded data, where data values are limited to a fixed range,
play an important role in machine learning, where activation functions
typically clamp the data to a fixed range, either [-1, 1], or [0, 1].
We assume without any loss of generality that data is limited to .
The first operation in a neural network layer is linear transformation.
Therefore, linear transformation of doubly-bounded data is central to
the study of neural networks.
Linear transformation of doubly-bounded data