Keep it Tighter -- A Story on Analytical Mean Embeddings
This work addresses a specific bottleneck in kernel-based divergence estimation for financial data, offering incremental improvements in theoretical guarantees and practical applications.
The paper tackles the problem of estimating maximum mean discrepancy (MMD) when the mean embedding of one distribution is known analytically, proving tighter concentration bounds for the estimator and extending results to unbounded kernels with minimax-optimal lower bounds. It demonstrates efficiency in real-world financial applications, such as index replication and calibration on loss-given-default ratios and S&P 500 data.
Kernel techniques are among the most popular and flexible approaches in data science allowing to represent probability measures without loss of information under mild conditions. The resulting mapping called mean embedding gives rise to a divergence measure referred to as maximum mean discrepancy (MMD) with existing quadratic-time estimators (w.r.t. the sample size) and known convergence properties for bounded kernels. In this paper we focus on the problem of MMD estimation when the mean embedding of one of the underlying distributions is available analytically. Particularly, we consider distributions on the real line (motivated by financial applications) and prove tighter concentration for the proposed estimator under this semi-explicit setting; we also extend the result to the case of unbounded (exponential) kernel with minimax-optimal lower bounds. We demonstrate the efficiency of our approach beyond synthetic example in three real-world examples relying on one-dimensional random variables: index replication and calibration on loss-given-default ratios and on S&P 500 data.