  Training model to generate data has increasingly attracted research attention
and become important in modern world applications. We propose in this paper a
new geometry-based optimization approach to address this problem. Orthogonal to
current state-of-the-art density-based approaches, most notably VAE and GAN, we
present a fresh new idea that borrows the principle of minimal enclosing ball
to train a generator G\left(\bz\right) in such a way that both training and
generated data, after being mapped to the feature space, are enclosed in the
same sphere. We develop theory to guarantee that the mapping is bijective so
that its inverse from feature space to data space results in expressive
nonlinear contours to describe the data manifold, hence ensuring data generated
are also lying on the data manifold learned from training data. Our model
enjoys a nice geometric interpretation, hence termed Geometric Enclosing
Networks (GEN), and possesses some key advantages over its rivals, namely
simple and easy-to-control optimization formulation, avoidance of mode
collapsing and efficiently learn data manifold representation in a completely
unsupervised manner. We conducted extensive experiments on synthesis and
real-world datasets to illustrate the behaviors, strength and weakness of our
proposed GEN, in particular its ability to handle multi-modal data and quality
of generated data.
