Density Estimation via Discrepancy
This work addresses density estimation for pattern recognition tasks, but appears incremental as it builds on existing discrepancy methods without claiming major breakthroughs.
The paper tackles non-parametric density estimation from i.i.d. samples on a hyper-rectangle by learning a piecewise constant function via adaptive discrepancy control, proving it preserves estimation power and applying it to tasks like mode seeking and density landscape exploration.
Given i.i.d samples from some unknown continuous density on hyper-rectangle $[0, 1]^d$, we attempt to learn a piecewise constant function that approximates this underlying density non-parametrically. Our density estimate is defined on a binary split of $[0, 1]^d$ and built up sequentially according to discrepancy criteria; the key ingredient is to control the discrepancy adaptively in each sub-rectangle to achieve overall bound. We prove that the estimate, even though simple as it appears, preserves most of the estimation power. By exploiting its structure, it can be directly applied to some important pattern recognition tasks such as mode seeking and density landscape exploration. We demonstrate its applicability through simulations and examples.