  In this paper, we propose an implicit gradient descent algorithm for the
classic $k$-means problem. The implicit gradient step or backward Euler is
solved via stochastic fixed-point iteration, in which we randomly sample a
mini-batch gradient in every iteration. It is the average of the fixed-point
trajectory that is carried over to the next gradient step. We draw connections
between the proposed stochastic backward Euler and the recent entropy
stochastic gradient descent (Entropy-SGD) for improving the training of deep
neural networks. Numerical experiments on various synthetic and real datasets
show that the proposed algorithm provides better clustering results compared to
$k$-means algorithms in the sense that it decreased the objective function (the
cluster) and is much more robust to initialization.
