Uniform Distribution
The continuous uniform distribution on \([a, b]\) is defined by:
\[p(x) = \frac{1}{b - a}, \quad a \leq x \leq b\]
CDF
\[F(x) = \frac{x - a}{b - a}, \quad a \leq x \leq b\]
Entropy
For the multivariate case with independent marginals on \([a_d, b_d]\):
\[H(x) = \log \left[ \prod_{d=1}^{D}(b_d - a_d) \right]\]
For the standard uniform on \([0, 1]^D\), this simplifies to \(H(x) = 0\).
Role in RBIG
The uniform distribution is the intermediate target in the uniformization step of RBIG. Applying the marginal CDF \(F_d\) to each dimension maps the data to \([0, 1]\), producing a uniform marginal distribution before the subsequent Gaussianization step.