  We address the problem of partial index tracking, replicating a benchmark
index using a small number of assets. Accurate tracking with a sparse portfolio
is extensively studied as a classic finance problem. However in practice, a
tracking portfolio must also be diverse in order to minimise risk -- a
requirement which has only been dealt with by ad-hoc methods before. We
introduce the first index tracking method that explicitly optimises both
diversity and sparsity in a single joint framework. Diversity is realised by a
regulariser based on pairwise similarity of assets, and we demonstrate that
learning similarity from data can outperform some existing heuristics. Finally,
we show that the way we model diversity leads to an easy solution for sparsity,
allowing both constraints to be optimised easily and efficiently. we run
out-of-sample backtesting for a long interval of 15 years (2003 -- 2018), and
the results demonstrate the superiority of the proposed algorithm.
