Density estimation in high-dimensional settings is an important and challenging statistical problem.Traditional methods based on kernel smoothing are inefficient in high dimensions due to the difficulties in specifying appropriate location-adaptive kernels. In this work, we introduce pre-training, a key idea behind many cutting-edge AI technologies, to the context of non-parametric density estimation. By establishing a pre-trained neural network that can recommend an appropriate location-adaptive kernel for each sample point, efficient density estimation with adaptive kernels is achieved in high dimensions. A wide range of numerical experiments show that this strategy is highly effective for improving density-estimation accuracy, when the target distribution is close to the distribution family for pre-training. When the target distribution is substantially different from the pre-training distribution family, the benefit from the proposed pre-training strategy may be diluted, but can be reactivated by an additional fine-tuning procedure.

We propose a new conformal prediction method for time-series data with a guaranteed asymptotic conditional coverage rate, Sequential Conformalized Density Regions (SCDR), which is flexible enough to produce both prediction intervals and disconnected prediction sets, signifying the emergence of bifurcations. Our approach uses existing estimated conditional highest density predictive regions to form initial predictive regions. We then use a quantile random forest conformal adjustment to provide guaranteed coverage while adaptively changing to take the non-exchangeable nature of time-series data into account. We show that the proposed method achieves the guaranteed coverage rate asymptotically under certain regularity conditions. In particular, the method is doubly robust -- it works if the predictive density model is correctly specified and/or if the scores follow a nonlinear autoregressive model with the correct order specified. Simulations reveal that the proposed method outperforms existing methods in terms of empirical coverage rates and set sizes. We illustrate the method using two real datasets, the Old Faithful geyser dataset and the Australian electricity usage dataset. Prediction sets formed using SCDR for the geyser eruption durations include both single intervals and unions of two intervals, whereas existing methods produce wider, less informative, single-interval prediction sets.

Given iidobservations from an unknown absolute continuous distribution defined on some domain Ω, we propose a nonparametric method to learn a piecewise constant function to approximate the underlying probability density function. Our density estimate is a piecewise constant function defined on a binary partition of Ω. The key ingredient of the algorithm is to use discrepancy, a concept originates from Quasi Monte Carlo analysis, to control the partition process. The resulting algorithm is simple, efficient, and has a provable convergence rate. We empirically demonstrate its efficiency as a density estimation method. We also show how it can be utilized to find good initializations for k-means.

We study a class of non-parametric density estimators under Bayesian settings. The estimators are obtained by adaptively partitioning the sample space. Under a suitable prior, we analyze the concentration rate of the posterior distribution, and demonstrate that the rate does not directly depend on the dimension of the problem in several special cases. Another advantage of this class of Bayesian density estimators is that it can adapt to the unknown smoothness of the true density function, thus achieving the optimal convergence rate without artificial conditions on the density.

We introduce a novel framework for uncertainty quantification in clustering. By combining the martingale posterior paradigm with density-based clustering, uncertainty in the estimated density is naturally propagated to the clustering structure. The approach scales effectively to high-dimensional and irregularly shaped data by leveraging modern neural density estimators and GPU-friendly parallel computation. We establish frequen-tist consistency guarantees and validate the methodology on synthetic and real data.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

In unpaired image translation setting, we are given two collections of samples without pairing information and we need to learn a proper mapping from one domain to another.