# Non-Parametric Inference Adaptive to Intrinsic Dimension

## TL;DR

This paper introduces a non-parametric method for high-dimensional conditional inference that adapts to the intrinsic low-dimensional structure of data, enabling accurate estimation even when the ambient dimension exceeds the sample size.

## Contribution

It develops an adaptive sub-sampled $k$-NN estimator that achieves optimal rates based on intrinsic dimension, with a data-driven method for tuning parameters.

## Key findings

- Estimation error scales as $n^{-1/(d+2)}$ with intrinsic dimension $d$.
- Estimator is asymptotically normal regardless of ambient dimension.
- Proposed method adapts to unknown intrinsic dimension, improving high-dimensional inference.

## Abstract

We consider non-parametric estimation and inference of conditional moment models in high dimensions. We show that even when the dimension $D$ of the conditioning variable is larger than the sample size $n$, estimation and inference is feasible as long as the distribution of the conditioning variable has small intrinsic dimension $d$, as measured by locally low doubling measures. Our estimation is based on a sub-sampled ensemble of the $k$-nearest neighbors ($k$-NN) $Z$-estimator. We show that if the intrinsic dimension of the covariate distribution is equal to $d$, then the finite sample estimation error of our estimator is of order $n^{-1/(d+2)}$ and our estimate is $n^{1/(d+2)}$-asymptotically normal, irrespective of $D$. The sub-sampling size required for achieving these results depends on the unknown intrinsic dimension $d$. We propose an adaptive data-driven approach for choosing this parameter and prove that it achieves the desired rates. We discuss extensions and applications to heterogeneous treatment effect estimation.