  For their ability to capture non-linearities in the data and to scale to
large training sets, local Support Vector Machines (SVMs) have received a
special attention during the past decade. In this paper, we introduce a new
local SVM method, called L$^3$-SVMs, which clusters the input space, carries
out dimensionality reduction by projecting the data on landmarks, and jointly
learns a linear combination of local models. Simple and effective, our
algorithm is also theoretically well-founded. Using the framework of Uniform
Stability, we show that our SVM formulation comes with generalization
guarantees on the true risk. The experiments based on the simplest
configuration of our model (i.e. landmarks randomly selected, linear
projection, linear kernel) show that L$^3$-SVMs is very competitive w.r.t. the
state of the art and opens the door to new exciting lines of research.
