Given a training set drawn from an unknown $d$-variate probability distribution, QuantTree constructs a histogram by recursively splitting $\mathbb{R}^d$. The splits are defined by a stochastic process so that each bin contains a certain proportion of the training set. These histograms can be used to define test statistics (e.g., the Pearson statistic) to tell whether a batch of data is drawn from $\phi_0$ or not. The most crucial property of QuantTree is that the distribution of any statistic based on QuantTree histograms is independent of $\phi_0$, thus enabling nonparametric statistical testing.

**Feature Matching** is a regularizing objective for a generator in [generative adversarial networks](https://paperswithcode.com/methods/category/generative-adversarial-networks) that prevents it from overtraining on the current discriminator. Instead of directly maximizing the output of the discriminator, the new objective requires the generator to generate data that matches the statistics of the real data, where we use the discriminator only to specify the statistics that we think are worth matching. Specifically, we train the generator to match the expected value of the features on an intermediate layer of the discriminator. This is a natural choice of statistics for the generator to match, since by training the discriminator we ask it to find those features that are most discriminative of real data versus data generated by the current model.

Letting $\mathbf{f}\left(\mathbf{x}\right)$ denote activations on an intermediate layer of the discriminator, our new objective for the generator is defined as: $ ||\mathbb{E}\_{x\sim p\_{data} } \mathbf{f}\left(\mathbf{x}\right) − \mathbb{E}\_{\mathbf{z}∼p\_{\mathbf{z}}\left(\mathbf{z}\right)}\mathbf{f}\left(G\left(\mathbf{z}\right)\right)||^{2}\_{2} $. The discriminator, and hence
$\mathbf{f}\left(\mathbf{x}\right)$, are trained as with vanilla GANs. As with regular [GAN](https://paperswithcode.com/method/gan) training, the objective has a fixed point where G exactly matches the distribution of training data.

GAN Feature Matching

Improved Techniques for Training GANs

QuantTree

QuantTree: Histograms for Change Detection in Multivariate Data Streams

**GPT-4** is a transformer based model pre-trained to predict the next token in a document.

Source	QuantTree: Histograms for Change Detection in Multivariate Data Streams
Year	2000
Data Source	CC BY-SA - https://paperswithcode.com