  This work proposes the continuous conditional generative adversarial network
(CcGAN), the first generative model for image generation conditional on
continuous, scalar conditions (termed regression labels). Existing conditional
GANs (cGANs) are mainly designed for categorical conditions (eg, class labels);
conditioning on regression labels is mathematically distinct and raises two
fundamental problems:(P1) Since there may be very few (even zero) real images
for some regression labels, minimizing existing empirical versions of cGAN
losses (aka empirical cGAN losses) often fails in practice;(P2) Since
regression labels are scalar and infinitely many, conventional label input
methods are not applicable. The proposed CcGAN solves the above problems,
respectively, by (S1) reformulating existing empirical cGAN losses to be
appropriate for the continuous scenario; and (S2) proposing a naive label input
(NLI) method and an improved label input (ILI) method to incorporate regression
labels into the generator and the discriminator. The reformulation in (S1)
leads to two novel empirical discriminator losses, termed the hard vicinal
discriminator loss (HVDL) and the soft vicinal discriminator loss (SVDL)
respectively, and a novel empirical generator loss. The error bounds of a
discriminator trained with HVDL and SVDL are derived under mild assumptions in
this work. Two new benchmark datasets (RC-49 and Cell-200) and a novel
evaluation metric (Sliding Fr\'echet Inception Distance) are also proposed for
this continuous scenario. Our experiments on the Circular 2-D Gaussians, RC-49,
UTKFace, Cell-200, and Steering Angle datasets show that CcGAN is able to
generate diverse, high-quality samples from the image distribution conditional
on a given regression label. Moreover, in these experiments, CcGAN
substantially outperforms cGAN both visually and quantitatively.
