Consistent Kernel Mean Estimation for Functions of Random Variables
Authors: Carl-Johann Simon-Gabriel, Adam Scibior, Ilya O. Tolstikhin, Bernhard Schölkopf
NeurIPS 2016 | Conference PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | We ran experiments on synthetic data to show how accurately ˆµ1, ˆµ2 and ˆµ3 approximate µf(X,Y ) with growing sample size N. We considered three basic arithmetic operations: multiplication X Y , division X/Y , and exponentiation XY , with X N(3; 0.5) and Y N(4; 0.5). As the true embedding µf(X,Y ) is unknown, we approximated it by a U-statistic estimator based on a large sample (125 points). For ˆµ3, we used the simplest possible reduced set method: we randomly sampled subsets of size n = 0.01 N of the xi, and optimized the weights wi and ui to best approximate ˆµX and ˆµY . The results are summarised in Figure 1 and corroborate our expectations: (i) all estimators converge, (ii) ˆµ2 converges fastest and has the lowest variance, and (iii) ˆµ3 is worse than ˆµ2, but much better than the diagonal estimator ˆµ1. |
| Researcher Affiliation | Academia | Department of Empirical Inference, Max Planck Institute for Intelligent Systems Spemanstraße 38, 72076 Tübingen, Germany joint first authors; also with: Engineering Department, Cambridge University |
| Pseudocode | No | The paper does not contain any structured pseudocode or algorithm blocks. |
| Open Source Code | No | The paper does not contain an explicit statement or link indicating the release of source code for the described methodology. |
| Open Datasets | No | We ran experiments on synthetic data to show how accurately ˆµ1, ˆµ2 and ˆµ3 approximate µf(X,Y ) with growing sample size N. |
| Dataset Splits | No | The paper mentions approximating the true embedding with a large sample, but does not specify train/validation/test splits for their own experiments. It states: 'As the true embedding µf(X,Y ) is unknown, we approximated it by a U-statistic estimator based on a large sample (125 points).' |
| Hardware Specification | No | The paper does not provide specific details about the hardware used for running the experiments. |
| Software Dependencies | No | The paper does not list specific software dependencies with version numbers. |
| Experiment Setup | Yes | For ˆµ3, we used the simplest possible reduced set method: we randomly sampled subsets of size n = 0.01 N of the xi, and optimized the weights wi and ui to best approximate ˆµX and ˆµY . |