  We consider the problem of clustering in the presence of noise. That is, when
on top of cluster structure, the data also contains a subset of
\emph{unstructured} points. Our goal is to detect the clusters despite the
presence of many unstructured points. Any algorithm that achieves this goal is
noise-robust. We consider a regularisation method which converts any
center-based clustering objective into a noise-robust one. We focus on the
$k$-means objective and we prove that the regularised version of $k$-means is
NP-Hard even for $k=1$. We consider two algorithms based on the convex (sdp and
lp) relaxation of the regularised objective and prove robustness guarantees for
both.
  The sdp and lp relaxation of the standard (non-regularised) $k$-means
objective has been previously studied by [ABC+15]. Under the stochastic ball
model of the data they show that the sdp-based algorithm recovers the
underlying structure as long as the balls are separated by $\delta > 2\sqrt{2}
+ \epsilon$. We improve upon this result in two ways. First, we show recovery
even for $\delta > 2 + \epsilon$. Second, our regularised algorithm recovers
the balls even in the presence of noise so long as the number of noisy points
is not too large. We complement our theoretical analysis with simulations and
analyse the effect of various parameters like regularization constant,
noise-level etc. on the performance of our algorithm. In the presence of noise,
our algorithm performs better than $k$-means++ on MNIST.
