  In mixed multi-view data, multiple sets of diverse features are measured on
the same set of samples. By integrating all available data sources, we seek to
discover common group structure among the samples that may be hidden in
individualistic cluster analyses of a single data-view. While several
techniques for such integrative clustering have been explored, we propose and
develop a convex formalization that will inherit the strong statistical,
mathematical and empirical properties of increasingly popular convex clustering
methods. Specifically, our Integrative Generalized Convex Clustering
Optimization (iGecco) method employs different convex distances, losses, or
divergences for each of the different data views with a joint convex fusion
penalty that leads to common groups. Additionally, integrating mixed multi-view
data is often challenging when each data source is high-dimensional. To perform
feature selection in such scenarios, we develop an adaptive shifted group-lasso
penalty that selects features by shrinking them towards their loss-specific
centers. Our so-called iGecco+ approach selects features from each data-view
that are best for determining the groups, often leading to improved integrative
clustering. To fit our model, we develop a new type of generalized multi-block
ADMM algorithm using sub-problem approximations that more efficiently fits our
model for big data sets. Through a series of numerical experiments and real
data examples on text mining and genomics, we show that iGecco+ achieves
superior empirical performance for high-dimensional mixed multi-view data.
