# FG-Net: Facial Action Unit Detection with Generalizable Pyramidal Features

Yufeng Yin1, Di Chang1, Guoxian Song2, Shen Sang2, Tiancheng Zhi2,
Jing Liu2, Linjie Luo2, Mohammad Soleymani1
1 University of Southern California, 2 ByteDance
{yufengy, dichang<EMAIL_ADDRESS>
{guoxiansong, shen.sang, tiancheng.zhi, jing.liu<EMAIL_ADDRESS>Work done during internship at ByteDance.

###### Abstract

Automatic detection of facial Action Units (AUs) allows for objective facial
expression analysis. Due to the high cost of AU labeling and the limited size
of existing benchmarks, previous AU detection methods tend to overfit the
dataset, resulting in a significant performance loss when evaluated across
corpora. To address this problem, we propose FG-Net for generalizable facial
action unit detection. Specifically, FG-Net extracts feature maps from a
StyleGAN2 model pre-trained on a large and diverse face image dataset. Then,
these features are used to detect AUs with a Pyramid CNN Interpreter, making
the training efficient and capturing essential local features. The proposed
FG-Net achieves a strong generalization ability for heatmap-based AU detection
thanks to the generalizable and semantic-rich features extracted from the pre-
trained generative model. Extensive experiments are conducted to evaluate
within- and cross-corpus AU detection with the widely-used DISFA and BP4D
datasets. Compared with the state-of-the-art, the proposed method achieves
superior cross-domain performance while maintaining competitive within-domain
performance. In addition, FG-Net is data-efficient and achieves competitive
performance even when trained on 1000 samples. Our code will be released at
https://github.com/ihp-lab/FG-Net

##  1 Introduction

Automatic detection of facial action units is a fundamental block for
objective facial expression analysis [6]. Manual annotations for facial action
units are cumbersome and costly, as they require trained coders to label each
frame individually. Common AU datasets, _i.e_., DISFA [22] and BP4D [33], only
contain a limited number of subjects (27 and 41 subjects respectively). Recent
methods for AU detection [36, 25, 21] focus on deep representation learning,
requiring a large number of samples. Existing AU detection methods are often
evaluated with within-domain cross-validation, with training and testing data
from the same dataset, and the generalization to other datasets (model trained
and tested on different datasets) has not been widely investigated. As within-
domain performance can be due to overfitting, cross-corpus performance can
suffer a considerable loss (see comparisons in Figure 1).

Figure 1: Performance (F1 score $\uparrow$) gap between the within- and cross-
domain AU detection for DRML [36], J$\hat{\text{A}}$A-Net [25], ME-GraphAU
[21], and the proposed FG-Net. The within-domain performance is averaged
between DISFA and BP4D, while the cross-domain performance is averaged between
BP4D to DISFA and DISFA to BP4D. The proposed FG-Net has the highest cross-
domain performance, thus, superior generalization ability.

In the field of semantic segmentation, recent studies [34, 2] leverage a well-
trained generative model to synthesize image-annotation pairs from only a few
labeled examples (around 30 training samples). They show that the intermediate
features of generative models exhibit semantic-rich representations that are
well-suited for pixel-wise segmentation tasks in a few-shot manner. Li _et
al_. [16] showcase extreme out-of-domain generalization ability from such
approaches. However, to the best of our knowledge, no existing work adapts
such architectures to AU detection, potentially due to the following
limitations: (i) the high dimensionality of the pixel-wise features results in
inefficient training, and (ii) inference with pixel-level features lacks the
information from the nearby regions, which is crucial to AU detection [36,
25].

In this paper, inspired by the success of GAN features in semantic
segmentation, we propose FG-Net, a facial action unit detection method that
can better generalize across domains. The general idea of FG-Net is to extract
the generalizable and semantic-rich deep representations from a well-trained
generative model (see Figure 2). Specifically, FG-Net first encodes and
decodes the input image with a StyleGAN2 encoder (pSp) [24], and a StyleGAN2
generator [14], trained on the FFHQ dataset [13]. Then, FG-Net extracts
feature maps from the generator during decoding. To take advantage of the
informative pixel-wise representations from the generator, FG-Net detects the
AUs through heatmap regression. We propose a Pyramid CNN Interpreter which
incorporates the multi-resolution feature maps in a hierarchical manner. The
proposed module makes the training efficient and captures essential
information from nearby regions. Thanks to the powerful features from the
generative model pre-trained on a large and diverse facial image dataset, the
proposed FG-Net obtains a strong generalization ability and data efficiency
for AU detection.

To demonstrate the effectiveness of our proposed method, we conduct extensive
experiments with the widely-used DISFA [22] and BP4D [33] for AU detection.
The results show that the proposed FG-Net method has a strong generalization
ability and achieves state-of-the-art cross-domain performance (see Figure 1).
In addition, FG-Net achieves comparable or superior within-domain performance
to the existing methods. Finally, we showcase that FG-Net is a data-efficient
approach. With only 100 training samples, it can achieve decent performance.

Our major contributions are as follows. (i) We propose FG-Net, a data-
efficient method for generalizable facial action unit detection. To the best
of our knowledge, we are the first to utilize StyleGAN model features for AU
detection. (ii) Extensive experiments on the widely-used DISFA and BP4D
datasets show that FG-Net has a strong generalization ability for heatmap-
based AU detection achieving superior cross-domain performance and maintaining
competitive within-domain performance compared to the state-of-the-art. (iii)
FG-Net is data-efficient. The performance of FG-Net trained on 1k samples is
close to the whole set ($\sim$100k).

##  2 Related Work

Facial Action Unit Detection. A facial action unit is an indicator of
activation of an individual or a group of muscles, _e.g_., cheek raiser (AU6).
AUs are formalized by Paul Ekman in Facial Action Coding System (FACS) [6].
Previous studies explore attention mechanism [25, 28, 12] or self-supervised
learning [4] to get discriminative representations for AU detection.

Shao _et al_. [25] propose J$\hat{\text{A}}$A-Net for joint AU detection and
face alignment. J$\hat{\text{A}}$A-Net uses adaptive attention learning to
refine the attention map for each AU. Tang _et al_. [28] propose a joint
strategy called PIAP-DF for AU representation learning. PIAP-DF involves
pixel-level attention for each AU and individual feature elimination and
utilizes the unlabeled data to mitigate the negative impacts of wrong labels.
Jacob _et al_. [12] combine transformer-based architectures with region of
interest (ROI) attention module, per-AU embeddings, and correlation module to
capture relationships between different AUs. Chang _et al_. [4] propose a
knowledge-driven self-supervised representation learning framework. AU
labeling rules are leveraged to design facial partition manners and determine
correlations between facial regions.

Recent work on AU detection use graph neural networks [35, 27, 21]. Zhang _et
al_. [35] utilize a heatmap regression-based approach for AU detection. The
ground-truth heatmaps are defined based on the ROI for each AU. Besides, the
authors utilize graph convolution for feature refinement. Song _et al_. [27]
propose a hybrid message-passing neural network with performance-driven
structures (HMP-PS). A performance-driven Monte Carlo Markov Chain sampling
method is proposed for generating the graph structures. Besides, hybrid
message passing is proposed to combine different types of messages. Luo _et
al_. [21] propose an AU relationship modeling approach that learns a unique
graph to explicitly describe the relationship between each pair of AUs of the
target facial display. Previous studies on AU detection achieve promising
within-domain performance. However, the generalization ability, _i.e_., cross-
domain performance, for AU detection has not been widely investigated.

Figure 2: Overview of our proposed pipeline. FG-Net first encodes the input
image into a latent code using a StyleGAN2 encoder (e.g. pSp [24] here). In
the decoding stage [14], we extract the intermediate multi-resolution feature
maps and pass them through our Pyramid CNN Interpreter to detect AUs coded in
the form of heatmaps. Mean Squared Error (MSE) loss is used for optimization
between the ground truth and predicted heatmaps.

Ertugrul _et al_. [7, 8] demonstrate that the deep-learning-based AU detectors
achieve poor cross-domain performance due to the variations in the cameras,
environments, and subjects. Tu _et al_. [29] propose Identity-Aware Facial
Action Unit Detection (IdenNet). IdenNet is jointly trained by AU detection
and face clustering datasets that contain numerous subjects to improve the
model’s generalization ability. Yin _et al_. [32] propose to use domain
adaptation and self-supervised patch localization to improve the cross-corpora
performance for AU detection. However, this method requires data from the
target domain for domain adaptation. Hernandez _et al_. [10] conduct an in-
depth analysis of performance differences across subjects, genders, skin
types, and databases. To address this gap, they propose deep face
normalization (DeepFN) that transfers the facial expressions of different
people onto a common facial template.

In this paper, without using any data from the target domain, we improve the
cross-corpus AU detection with the semantic-rich features from a generative
model trained on a large-scale and diverse dataset.

Face Understanding with Generative Models. Generative models provide an
estimate of the distribution of training samples [3]. Prior work utilizing
generative models for face understanding has mainly focused on semantic
segmentation [34, 2, 16] and landmark detection [34, 31].

Zhang _et al_. [34] introduce DatasetGAN, an automatic procedure to generate
massive datasets of high-quality semantically segmented images requiring
minimal human effort. The authors show how the GAN latent code can be decoded
to produce a semantic segmentation of the image and allow the decoder to be
trained with only a few labeled examples. Baranchuk _et al_. [2] demonstrate
that feature maps from diffusion models [5] can capture the semantic
information and appear to be excellent pixel-wise representations. Li _et al_.
[16] propose semanticGAN, a generative adversarial network that captures the
joint image-label distribution. The proposed semanticGAN showcases an extreme
out-of-domain generalization ability, such as transferring from real faces to
paintings, sculptures, and even cartoons and animal faces. Xu _et al_. [31]
consider the pre-trained StyleGAN generator as a learned loss function and
train a hierarchical encoder to get visual representations, namely GH-Feat,
for input images. GH-Feat has strong transferability to both generative and
discriminative tasks.

Previous studies show that the hidden states from the generative models are
powerful representations for face understanding. However, to the best of our
knowledge, no existing work adapts such architectures to AU detection. Zhang
_et al_. [34] and Baranchuk _et al_. [2] extract pixel-wise features and treat
each pixel as a training sample, leading to extreme inefficiency due to the
per-sample computational overhead. More importantly, inference with singe-
pixel features lacks the inductive bias (local features), crucial to AU
detection shown in the previous studies [36, 25]. In addition, semanticGAN
[16] has to encode the input image to the latent space in an optimization-
based manner for inference, which is extremely time-consuming. Thus
semanticGAN can be only tested with a few samples. The limitation of this
approach does not allow for training or testing with larger datasets. In this
paper, we propose a Pyramid CNN Interpreter to detect the heatmaps,
representing activated AUs, for AU detection, which is more efficient and can
capture both local and global information. GH-Feat [31] is the closest method
to ours that extracts latent code representations from generative models. The
major differences between GH-Feat and our method are: (i) GH-Feat extracts the
1D latent code features while FG-Net further decodes the latent codes to
images and gets the 2D feature maps. (ii) GH-Feat is trained in a multi-stage
manner while our method is end-to-end. GH-Feat utilizes the StyleGAN generator
as a learned loss function and trains a hierarchical encoder and then uses
this encoder to extract visual representations for downstream tasks. The whole
pipeline requires more than 700 GPU hours due to the complicated training
process while FG-Net only needs 10 GPU hours for training.

Figure 3: Visualizations of the ROI centers for DISFA (left) and BP4D (right).
AU indices are labeled above or below. Figure 4: Visualization of the
generated ground-truth heatmaps on DISFA (first row) and BP4D (second row). We
generate one heatmap for every AU which has two Gaussian windows with the
maximum values at the two ROI centers (see Figure 3). The peak value is either
$1$ (red, AU is active) or $-1$ (blue, AU is inactive).

##  3 Methods

###  3.1 Problem Formulation

Facial Action Unit Detection. Given a video set $S$, for each frame $x\in S$,
the goal is to detect the occurrence for each AU $a_{i}$ $(i=1,2,...,n)$ using
function $\text{F}(\cdot)$.

$a_{1},a_{2},...,a_{n}=\text{F}(x),$ (1)

where $n$ is the number of AUs. $a_{i}=1$ if the AU is active otherwise
$a_{i}=0$.

###  3.2 Overview

Figure 2 illustrates an overview of the proposed FG-Net. FG-Net first encodes
and decodes the input image with the pSp encoder [24] and the StyleGAN2
generator [14] pre-trained on the FFHQ dataset [13].

During the decoding, FG-Net extracts feature maps from the generator.
Leveraging the features extracted from a generative model trained on a large-
scale and diverse dataset, FG-Net offers a higher generalizability for AU
detection.

To take advantage of the pixel-wise representations from the generator, FG-Net
is designed to detect the AUs using a heatmap regression. To keep the training
efficient and capture both local and global information, a Pyramid-CNN
Interpreter is proposed to incorporate the multi-resolution feature maps in a
hierarchical manner and detect the heatmaps representing facial action units.

###  3.3 Model

Prerequisites. Our proposed method is built on top of the StyleGAN2 generator
[14]. The StyleGAN2 generator decodes a latent code $z\in\mathcal{Z}$ sampled
from $\mathcal{N}(0,I)$ to an image. The latent code $z$ is first mapped to a
style code $w\in\mathcal{W}$ by a mapping function. Both $z$ and $w$ have
$512$ dimensions. There are $k$ synthesis blocks (in practice $k=9$) and each
block has two convolution layers and one upsampling layer. Each convolution
layer is followed by an adaptive instance normalization (AdaIN) layer [11]
which is controlled by the style code $w$. However, for image inversion which
encodes the images into the latent space, $\mathcal{W}$-space has limited
expressiveness and thus can not fully reconstruct the input [30]. Therefore,
prior works [1, 24] extend $\mathcal{W}$-space to $\mathcal{W}^{+}$-space
where a different style code $w$ is fed to each AdaIN layer.
$\mathcal{W}^{+}$-space alleviates the reconstruction distortion. The
dimension of $w^{+}\in\mathcal{W}^{+}$ is $18\times 512$.

Feature Extraction from StyleGAN2. To extract features from the StyleGAN2
generator, we first encode the input image to the latent space and then decode
the latent code.

Prior work [16] encodes the input image in an optimization-based manner.
Optimization-based methods iteratively optimize a reconstruction objective
which is extremely time-consuming. Instead, we utilize the pSp encoder E [24]
to encode the input image $x\in\mathcal{X}$ and get the latent code
$w^{+}\in\mathcal{W}^{+}$ via $w^{+}=\text{E}(x)$.

Despite the efficient encoding of the pSp encoder, the generator features may
not capture the key facial features for AU detection. To address the problem,
we fine-tune the encoder and the generator during training. Then, we decode
the latent code with the StyleGAN2 generator G [14] to obtain image
$x^{\prime}=\text{G}(w^{+})$. During decoding, we extract the intermediate
activations from the generator. To keep the training efficient, unlike the
previous work [34] which extracts the outputs from all the AdaIN layers [11],
we only extract the hidden states after the second AdaIN layer in each block.
We denote the feature maps we get from the $k$ blocks as
$\\{f_{1},f_{2},...,f_{k}\\}=\text{G}^{\prime}(w^{+})=\text{G}^{\prime}(\text{E}(x))$.

Heatmap Detection. The proposed method detects the AU occurrences in a heatmap
regression-based approach. We generate the ground-truth heatmaps following the
previous work [35]. We first define the Region of Interest (ROI) for each AU.
We select two points on the face based on the most representative landmarks
(see Figure 3, detailed positions are provided in the supplementary).

Then, we generate the ground-truth heatmaps with the definition of ROI. Figure
4 gives the visualization of the ground-truth heatmaps on DISFA and BP4D.
Formally, given a face image $x\in\mathbb{R}^{w\times h\times 3}$, we generate
$n$ ground-truth heatmaps $m_{1},m_{2},...,m_{n}\in\mathbb{R}^{w\times h}$
with the AU labels, where $n$ is the number of AUs. Specifically, for heatmap
$m_{i}$, we add two Gaussian windows $g_{i}^{1}$ and $g_{i}^{2}$ with the
maximum value at the two ROI centers $c_{i}^{1}$ and $c_{i}^{2}$ following
[35].

$g_{i}^{j}(p)=\lambda_{i}\exp(-\frac{||p-c_{i}^{j}||_{2}^{2}}{2\sigma^{2}}),\quad
j=1,2,$ (2) $m_{i}(p)=g_{i}^{1}(p)+g_{i}^{2}(p).$ (3)

where $p$ is the pixel location ($p\in[1,w]\times[1,h]$). $\lambda_{i}$ is the
indicator denoting whether the $i$-th AU is active. $\lambda_{i}=1$ if
$a_{i}=1$ otherwise $\lambda_{i}=-1$. $\sigma$ is the standard derivation. We
clip the heatmaps into the range of $[-1,1]$ to make sure the peak value is
either $1$ or $-1$.

After feature extraction from the generative models, prior work [34, 2]
upsamples the features to the input resolution and concatenates them according
to the channel dimension. Then, each pixel is treated as a training sample and
a multi-layer perceptron (MLP) is trained to detect the semantic class. Simply
upsampling and concatenating all the feature maps results in redundant and
high dimensional features (in practice the number of channels is $6080$), thus
leading to inefficient training and inference. More importantly, using single-
pixel features for inference lacks the spatial context from nearby regions,
which is crucial to AU detection, as demonstrated in the previous studies [36,
25].

To address these problems, we propose a multi-scale Pyramid-CNN Interpreter H
for heatmap-based AU detection which incorporates the multi-resolution feature
maps in a hierarchical manner (see Figure 2). Specifically, the Pyramid-CNN
Interpreter H contains $k$ pyramid levels, where $k$ is the number of feature
maps extracted from the generator. In each pyramid level, the hidden states
from the last pyramid level $c_{i-1}$ are first summed with the feature map
from the generator $f_{i}$ and then passed through an interpreter block
$\text{C}_{i}$. Each interpreter block consists of one Interpolate, one
Convolution, one ReLU, and one BatchNorm layer. $m=c_{k}$ is the ultimate AU
heatmap. specifically,

$c_{0}=0,c_{i}=\text{C}_{i}(c_{i-1}+f_{i}),i=1,2,...,k,$ (4)
$m=c_{k}=\text{H}(f_{1},f_{2},...,f_{k}).$ (5)

###  3.4 Training and Inference

Training. The learning objective is the Mean Squared Error (MSE) loss between
the ground-truth heatmap $m$ and the detected heatmap $\hat{m}$:
$\mathcal{L}=||m-\hat{m}||_{2}^{2}$.

Inference. For each detected heatmap $\hat{m}_{i}$, we sum up the whole
heatmap. If the sum is greater than $0$, the corresponding AU is active
otherwise the AU is inactive.

##  4 Experiments and Discussions

###  4.1 Datasets

We select two publicly available datasets, _i.e_., DISFA [22] and BP4D [33].
These two datasets are widely used for AU detection and are captured from
different subjects with different backgrounds and lighting conditions.

DISFA [22] includes videos from 27 subjects, with around 130,000 frames. Each
frame has labels for eight AU intensities (1, 2, 4, 6, 9, 12, 25, and 26).
Following the settings of previous studies [36, 25, 21], we map the AU
intensity greater than 1 to the positive class.

BP4D [33] consists of videos from 41 subjects with around 146,000 frames. Each
frame has labels for 12 AU occurrences (1, 2, 4, 6, 7, 10, 12, 14, 15, 17, 23,
and 24).

Table 1: Within-domain evaluation in terms of F1 score ($\uparrow$). Except for GH-Feat and ME-GraphAU + FFHQ pre-train, all the baseline numbers are from the original papers. Our method achieves competitive performance compared to the state-of-the-art.  Methods | DISFA | BP4D
---|---|---
DRML [36] | 26.7 | 48.3
IdenNet [29] | 52.6 | 59.3
SRERL [17] | 55.9 | 62.9
UGN-B [26] | 60.0 | 63.3
HMP-PS [27] | 61.0 | 63.4
FAT [12] | 61.5 | 64.2
Zhang _et al_. [35] | 62.0 | 63.5
J$\hat{\text{A}}$A-Net [25] | 63.5 | 62.4
PIAP [28] | 63.8 | 64.1
Chang _et al_. [4] | 64.5 | 64.5
ME-GraphAU [21] | 63.1 | 65.5
ME-GraphAU + FFHQ pre-train | 59.5 | 61.1
GH-Feat [31] | 36.9 | 56.7
Ours | 65.4 | 64.3
Table 2: Cross-domain evaluation between DISFA and BP4D in terms of F1 scores
($\uparrow$). Our model achieves superior performance compared to the
baselines. ∗ The numbers are reported from the original paper.

Direction | DISFA $\rightarrow$ BP4D | BP4D $\rightarrow$ DISFA
---|---|---
AU | 1 | 2 | 4 | 6 | 12 | Avg. | 1 | 2 | 4 | 6 | 12 | Avg.
DRML [36] | 19.4 | 16.9 | 22.4 | 58.0 | 64.5 | 36.3 | 10.4 | 7.0 | 16.9 | 14.4 | 22.0 | 14.1
J$\hat{\text{A}}$A-Net [25] | 10.9 | 6.7 | 42.4 | 52.9 | 68.3 | 36.2 | 12.5 | 13.2 | 27.6 | 19.2 | 46.7 | 23.8
ME-GraphAU [21] | 36.5 | 30.3 | 35.8 | 48.8 | 62.2 | 42.7 | 43.3 | 22.5 | 41.7 | 23.0 | 34.9 | 33.1
ME-GraphAU + FFHQ pre-train | 20.1 | 32.9 | 38.0 | 64.0 | 73.0 | 45.6 | 51.2 | 14.4 | 54.4 | 17.7 | 30.6 | 33.7
GH-Feat [31] | 29.4 | 30.0 | 37.1 | 64.0 | 73.5 | 46.8 | 18.9 | 15.2 | 27.5 | 52.7 | 50.1 | 32.9
Patch-MCD∗ [32] | - | - | - | - | - | - | 34.3 | 16.6 | 52.1 | 33.5 | 50.4 | 37.4
IdenNet∗ [29] | - | - | - | - | - | - | 20.1 | 25.5 | 37.3 | 49.6 | 66.1 | 39.7
Ours | 51.4 | 46.0 | 36.0 | 49.6 | 61.8 | 49.0 | 61.3 | 70.5 | 36.3 | 42.2 | 61.5 | 54.4

We use dlib [15] to detect the 68 facial landmarks for all the frames and
FFHQ-alignment to align them. The detected landmarks are also used for
generating the ground-truth heatmaps (see Figure 4).

###  4.2 Implementation and Training Details

All methods are implemented in PyTorch [23]. Code and model weights are
available, for the sake of reproducibility.111https://github.com/ihp-lab/FG-
Net We use a machine with two Intel(R) Xeon(R) Gold 5218 (2.30GHz) CPUs with
eight NVIDIA Quadro RTX8000 GPUs for all the experiments. Each image is
resized into $128\times 128$. We train the proposed model with the AdamW
optimizer [20] for 15 epochs with a batch size of $8$ on a single GPU. The
learning rate is $5e-5$. The weight decay is $5e-4$. The gradient clipping is
set to $0.1$. $\sigma$ for the heatmaps (Equation 2) is $20.0$. The dropout
rate is $0.1$.

###  4.3 Experimental Results

The models are evaluated for within-domain and cross-domain performance in
addition to data efficiency. Cross-domain evaluation enables us to measure the
generalization ability of our AU detection method. For all the experiments, F1
score ($\uparrow$) is the evaluation metric.

Within-domain Evaluation. We perform within-domain evaluation on widely used
DISFA and BP4D datasets. We follow the same evaluation protocols as the
previous studies [25, 27, 21]. Both datasets are evaluated with subject-
independent 3-fold cross-validation. We use two folds for training and one
fold for validation and iterate three times. We compare FG-Net to the state-
of-the-art AU detection methods, including DRML [36], IdenNet [29], SRERL
[17], UGN-B [26], HMP-PS [27], FAT [12], Zhang _et al_. [35],
J$\hat{\text{A}}$A-Net [25], PIAP [28], Chang _et al_. [4], and ME-GraphAU
[21]. These baseline numbers are reported from the original papers.

Previous methods do not use the FFHQ dataset for training. Thus, to make the
comparison fair, two baselines are implemented and compared, _e.g_., ME-
GraphAU + FFHQ pre-train and GH-Feat [31]. Specifically, we first pre-train
the ME-GraphAU’s backbones (ResNet and Swin Transformer) with the FFHQ dataset
and its facial expression labels. Then we train the ME-GraphAU with the pre-
trained backbones. GH-Feat extracts features from generative models and it is
pre-trained on the FFHQ in a self-supervised manner. For both baselines, we
implement with the officially released source codes.

Table 1 reports the within-domain results regarding the average performance of
AUs. We provide detailed results for every individual AU in the supplementary
material. On DISFA, FG-Net outperforms all the baseline methods and achieves
an average F1 score of 65.4. The major improvement comes from AU1 and AU2. On
BP4D, FG-Net achieves competitive performance. These results demonstrate that
the pixel-wise features extracted from StyleGAN2 are beneficial for heatmap-
based AU detection.

Cross-domain Evaluation. We perform two directions of cross-domain evaluation,
_i.e_., DISFA to BP4D and BP4D to DISFA. For each direction, we use two folds
and one fold of data from the source domain as the training and validation set
and use the target data as the testing set. We compare the proposed method
with DRML [36], J$\hat{\text{A}}$A-Net [25], ME-GraphAU [21], ME-GraphAU +
FFHQ pre-train, and GH-Feat [31] since they are open-source, and we can use
the officially released source codes and model weights to conduct the
experiments. In addition, we compare with Patch-MCD [32] and IdenNet [29]. The
numbers are reported from the original paper. Ertugrul _et al_. [7, 8] and
Hernandez _et al_. [10] do not report F1 scores for the cross-domain
performance of the aforementioned directions.

We report the cross-domain results in Table 2. As expected, compared to the
within-domain performance, all the baseline methods suffer a considerable
performance loss when evaluated across corpora. In particular, when evaluated
from BP4D to DISFA, the baseline methods’ performance (average F1) drops by
more than 30%, which demonstrates the challenging nature of cross-domain AU
detection and the importance of developing generalizable AU detection.

Compared with DRML and J$\hat{\text{A}}$A-Net, ME-GraphAU achieves higher
cross-domain performance. We suspect it is because it utilizes the pre-trained
models (ResNet [9] and Swin Transformer [18]) as the backbones. In addition,
when we continue pre-training ME-GraphAU with the FFHQ dataset, we observe a
further performance boost in both directions of cross-domain evaluations.
Similarly, GH-Feat, which is trained on the FFHQ dataset, also obtains
superior performance than DRML and J$\hat{\text{A}}$A-Net. The experimental
results show the effectiveness of pre-training on the FFHQ dataset since it is
a large and diverse facial image dataset. Moreover, Patch-MCD utilizes
unsupervised domain adaptation with unlabeled target data while IdenNet is
jointly trained by AU detection and face cluster datasets (CelebA [19]). Thus,
with additional face data, these two methods have better cross-domain
performance than the aforementioned baselines.

For both directions of cross-domain evaluation, our proposed method achieves
superior performance compared to the baselines. Specifically, when evaluated
from BP4D to DISFA, FG-Net can outperform the baselines by 15% in terms of the
average F1 score. The major improvement comes from AU1 and AU2, which is
consistent with the findings in within-domain evaluation. Overall, the results
showcase that features extracted from the StyleGAN2 generator are
generalizable and thus improve the performance for cross-domain AU detection,
showing its potential to solve AU detection in a real-life scenario.

We present two qualitative examples of cross-domain prediction in Figure 5.
The models are trained on the BP4D dataset and tested on the DISFA dataset.
ME-GraphAU fails in those two cases while the proposed FG-Net method
accurately predicts the action units.

Figure 5: Case analysis on ME-GraphAU [21] and FG-Net. Models are trained on
BP4D and tested on DISFA. Orange means active AU while blue means inactive AU.
FG-Net is more accurate than ME-GraphAU.

(a) Evaluation on DISFA.

(b) Evaluation on BP4D.

Figure 6: Data efficiency evaluation with different numbers of samples. Our
method is data-efficient and its performance trained on 1k samples is close to
the whole set.

Data Efficiency Evaluation. To further evaluate the generalization capacity of
the proposed approach, an investigation of its learning capability with
limited samples is conducted through within-domain evaluation. In this
evaluation, a subset of the training data is randomly selected, while the
testing data remains unchanged to facilitate assessment. The model is trained
using four different sample sizes: 100, 1k, 5k, and 10k. A comparative
analysis is performed between our method and two other approaches, namely ME-
GraphAU [21] and ME-GraphAU + FFHQ pre-train. It is noteworthy that ME-GraphAU
+ FFHQ pre-train and our method employ the same pre-training dataset.

The efficiency evaluation results, depicted in Figure 6, demonstrate the
impact of data scarcity on performance for both datasets. Notably, ME-GraphAU
[21] exhibits remarkably low F1 scores when trained with 100 and 1k samples on
the DISFA dataset, as well as with 100 samples on the BP4D dataset. This
outcome can be attributed to the limited and sparse nature of the training
set, causing ME-GraphAU to predict inactive AUs predominantly. By contrast,
the performance of ME-GraphAU improves when pre-trained on the FFHQ dataset,
underscoring the effectiveness of utilizing this extensive and diverse facial
dataset for pre-training. However, even with 100 samples from the DISFA
dataset, the performance of ME-GraphAU remains at 0.

In comparison, FG-Net outperforms ME-GraphAU + FFHQ pre-train when trained
with partial training data for both datasets. Notably, FG-Net trained on 1k
samples achieves performance levels approaching those of the full training
set. Furthermore, even with a mere 100 training samples, FG-Net manages to
achieve commendable performance. These results serve as evidence of the robust
generalization ability exhibited by our proposed method when confronted with
limited data.

Table 3: Ablation study for FG-Net. F1 score ($\uparrow$) is the metric. D and
B stand for DISFA and BP4D. D $\rightarrow$ B means the model is trained on
DISFA and tested on BP4D and similar to B $\rightarrow$ D. (i) Our method gets
better performance than GH-Feat [31]. (ii) With every component, our method
achieves the highest within-domain performance while removing late features
gets the best cross-domain performance.

| D | B | Avg. | D $\rightarrow$ B | B $\rightarrow$ D | Avg.
---|---|---|---|---|---|---
Upscale & concat | 64.2 | 62.7 | 63.4 | 42.5 | 35.9 | 39.2
Latent code | 68.4 | 58.8 | 63.6 | 46.4 | 47.3 | 46.9
\- Early | 68.1 | 61.7 | 64.9 | 37.9 | 47.2 | 42.6
\- Middle | 67.4 | 63.1 | 65.3 | 48.5 | 38.0 | 43.3
\- Late | 67.4 | 62.8 | 65.1 | 51.2 | 56.6 | 53.9
FG-Net | 68.9 | 63.6 | 66.3 | 49.0 | 54.4 | 51.7

Ablation Study. We conduct three ablation experiments: (i) We compare to the
existing upscaling and concatenating features proposed in [34, 2] (upscale &
concat). (ii) We directly compare to using latent code to predict the
activations of AUs (latent code). (iii) We explore the best blocks for
extracting feature maps. Specifically, we divide the features extracted from
the nine synthesis blocks into three groups, where each group has three
feature maps, and denote them as the early, middle, and late groups. Each
time, we remove one group. We perform both within- and cross-domain
evaluations for the ablation study. Note that for within-domain evaluation, we
use two folds for training and one fold for validation.

Table 3 shows the within- and cross-domain performance on DISFA and BP4D. (i)
We observe that FG-Net outperforms Upscale & concat for both within- and
cross-domain settings. We believe inference with singe-pixel features lacks
the inductive bias, considering local features, necessary for AU detection.
(ii) FG-Net outperforms latent code for predicting AU activations for both
within- and cross-domain experiments. We think using the heatmap regression
allows the model to localize where the AUs occur and improves the model’s
capacity. In addition, compared with the 2D feature maps, the latent codes
lose the semantic-rich representations. (iii) For the contributions of
different feature maps, we observe that removing any one of the feature groups
lowers the within-domain performance. Surprisingly, removing late features
achieves the highest cross-domain performance. We suspect it is because the
late features contain more high-frequency and domain-specific information
which reduces the model’s generalization ability.

Figure 7: Visualization of the detected heatmaps for ablation study. With all
the components, FG-Net detects the most similar heatmaps to the ground-truth
(GT) for within-domain evaluation. Removing late features results in the best
cross-domain evaluation.

We visualize the ground-truth and detected heatmaps for ablation study in
Figure 7. For the within-domain evaluation, models are trained and tested with
BP4D; For the cross-domain evaluation, models are trained with BP4D and tested
with DISFA. For latent code, we directly use it to predict the AU activations,
thus, we do not have the detected heatmaps for latent code. For within-domain
evaluation, FG-Net detects all AUs correctly, whereas the other methods output
the wrong prediction for AU2 (outer brow raiser), showing that FG-Net achieves
the best within-domain performance with every component. For cross-domain
evaluation, both using all features and removing late features detect all AUs
correctly. However, removing late features results in a more accurate heatmap
for AU12 than using all features.

###  4.4 Limitations

In the within-domain evaluation, FG-Net achieves inferior results on AU9 (nose
wrinkler), AU15 (lip corner depressor), and AU26 (jaw drop). Failure cases are
shown in Figure 8. We suspect it is because the FFHQ dataset lacks faces with
such active AUs, and thus the StyleGAN2 features can not capture the
corresponding information well. In addition, these failure AUs are not common
in DISFA and BP4D thus they do not appear in the cross-domain evaluations and
we can not evaluate the generalization for them.

FG-Net addresses the AU detection problem using a heatmap regression. Though
our method can be extended to AU intensity estimation, there are only three
common AUs for intensity estimation between BP4D and DISFA (6, 12, and 17)
with no AU on the eyebrows. Thus, we can not properly evaluate the
generalization ability of FG-Net for AU intensity estimation.

Figure 8: Visualization of the failure cases. FG-Net achieves inferior
performance on AU9, AU15, and AU26.

##  5 Conclusion

In this paper, we propose FG-Net, a data-efficient method for generalizable
facial action unit detection. FG-Net extracts the generalizable and semantic-
rich features from the generative model. A Pyramid CNN-Interpreter is proposed
to detect AUs coded as heatmaps which makes the training efficient and
captures essential information from the nearby regions. The experimental
results demonstrate the challenging nature of cross-domain AU detection and
the importance of developing generalizable AU detection. We show that the
proposed FG-Net method has a strong generalization ability when evaluated
across corpora or trained with limited data, demonstrating its strong
potential to solve action unit detection in a real-life scenario.

Social Implications. Our work falls within the broad domain of facial
expression analysis. Despite potential benefits, any surveillance technology
can be misused, and sensitive private information may be revealed by malicious
actors. Mitigation strategies for such misuses include restrictive licensing
and government regulations.

##  6 Acknowledgement

The work of Soleymani, Yin and Chang was sponsored by the Army Research Office
and was accomplished under Cooperative Agreement Number W911NF-20-2-0053. The
views and conclusions contained in this document are those of the authors and
should not be interpreted as representing the official policies, either
expressed or implied, of the Army Research Office or the U.S. Government. The
U.S. Government is authorized to reproduce and distribute reprints for
Government purposes notwithstanding any copyright notation herein.

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# Theoretical predictions for inclusive $B\to X_{u}\tau\bar{\nu}$ decay

Zoltan Ligeti  Ernest Orlando Lawrence Berkeley National Laboratory,
University of California, Berkeley, CA 94720  Michael Luke  Department of
Physics, University of Toronto, 60 St. George Street, Toronto, Ontario, Canada
M5S 1A7  Frank J. Tackmann  Deutsches Elektronen-Synchrotron DESY, Notkestr.
85, 22607 Hamburg, Germany

###### Abstract

With the expected large increase in data sets, previously not measured decays
will be studied at Belle II. We derive standard model predictions for the
$B\to X_{u}\tau\bar{\nu}$ decay rate and distributions. The region in the
lepton energy spectrum where higher-dimension operators in the local OPE need
to be resummed into the $b$-quark light-cone distribution function is a
significantly greater fraction of the phase space than for massless leptons.
The finite $\tau$ mass has the novel effect of shifting and squeezing how the
distribution function enters the lepton energy spectrum. We also derive new
predictions for the $\tau$ polarization.

††preprint:  DESY 21-215

##  I Introduction

The more than $3\sigma$ deviation of the measured $B\to D^{(*)}\tau\bar{\nu}$
rates [1, 2, 3, 4, 5, 6, 7, 8, 9] from the standard model (SM) predictions
motivates the study of all possible semileptonic decays with $\tau$ leptons in
the final state, both experimentally and theoretically. Comparisons of
measured spectra and rates to different hadronic final states can give
information on the structure of contributing four-fermion operators.
Comparisons of $b\to c\ell\bar{\nu}$ and $b\to u\ell\bar{\nu}$ decays give
constraints on the flavor structure of beyond standard model scenarios at
play.

In this paper we study the inclusive decay $B\to X_{u}\tau\bar{\nu}$, which
has been much less explored theoretically. Precise predictions for this decay
are naturally interesting as a signal channel to measure in the future. In the
near term, reliably modelling this decay as a background is interesting both
to SM measurements and analyses aimed at more precisely measuring $R(D^{(*)})$
and clarifying the current tension with the SM. The Belle Collaboration set
the first bound on a $b\to u\tau\bar{\nu}$ mediated decay, ${\cal
B}(B\to\pi\tau\bar{\nu})<2.5\times 10^{-4}$ [10], at a level several times
higher than SM predictions, and recent theoretical studies [11, 12, 13] also
focused on exclusive decays.

Inclusive semileptonic decays of hadrons containing a heavy quark allow for a
systematic expansion of nonperturbative effects in powers of $\Lambda_{\rm
QCD}/m_{Q}$ [14]. The inclusive decay rates computed in the
$m_{Q}\gg\Lambda_{\rm QCD}$ limit coincide with the free-quark decay rate,
while corrections of order $\Lambda_{\rm QCD}/m_{Q}$ vanish [14, 15]. The
leading nonperturbative corrections are of order $\Lambda_{\rm
QCD}^{2}/m_{Q}^{2}$ and depend on only two hadronic quantities, $\lambda_{1}$
and $\lambda_{2}$, which describe certain forward matrix elements of local
dimension-five operators. These corrections have been computed for a number of
processes [16, 17, 18, 19, 20, 21, 22]. For $B\to X_{u}\tau\bar{\nu}$ decay,
expressions for the total rate and leptonic $q^{2}$ spectra are
straightforward to derive by taking the $m_{q}\to 0$ limit of the $B\to
X_{c}\tau\bar{\nu}$ results [22, 23], but this limit is singular for the
lepton energy spectrum and has not been given in the literature. Similarly,
the perturbative ${\cal O}(\alpha_{s})$ corrections to the total $B\to
X_{u}\tau\bar{\nu}$ semileptonic decay rate [24], the dilepton $q^{2}$
spectrum [25], and the doubly differential ${\rm d}\Gamma/{\rm d}q^{2}{\rm
d}y$ spectrum [26, 27] are known analytically. However, no closed form
expressions have thus far been derived for the ${\cal O}(\alpha_{s})$
corrections to the $\tau$ lepton energy spectrum. We present the results of
the local OPE to ${\cal O}(\Lambda_{\rm QCD}^{2}/m_{b}^{2},\,\alpha_{s})$ in
Sec. II.

Phase space regions in inclusive $B\to X_{u}e\bar{\nu}$ decay, when kinematic
cuts restrict the invariant mass of the hadronic final state to be small
(i.e., $m_{X}<m_{D}$), are relevant for the determination of $|V_{ub}|$. Decay
rates in such regions are subject to large corrections, both perturbative and
nonperturbative. In the region near maximal lepton energy the OPE breaks down
and a resummation of the series of leading nonperturbative corrections is
required [28, 29]. The lepton energy spectrum in a region of width $\Delta
E_{\ell}\sim\Lambda_{\rm QCD}$ near the endpoint is determined by a
nonperturbative $b$-quark distribution function in the $B$ meson. Similarly,
the local OPE for $B\to X_{u}\tau\bar{\nu}$ breaks down near the endpoint of
the $\tau$ energy spectrum; however, since in $B\to X_{u}\tau\bar{\nu}$ decay,
$m_{\tau}<E_{\tau}<(m_{B}^{2}+m_{\tau}^{2})/(2m_{B})$ amounts to $1.78\,{\rm
GeV}<E_{\tau}<2.94\,{\rm GeV}$, the distribution function is important over a
much greater fraction of the available phase space than in $B\to
X_{u}e\bar{\nu}$, where $0<E_{e}<m_{B}/2$. We consider the effects of the
$b$-quark distribution function in Sec. III and explore its effect on the
spectrum. Since the distribution of the measurable $\tau$ decay products
(e.g., the charged lepton energy) are sensitive to the $\tau$ polarization, we
also present results for decays to each polarization state.

To appreciate the mass suppressions in the decay rates, simply using the
${\cal O}(\Lambda_{\rm QCD}^{2}/m_{b}^{2})$ [20, 21, 22] and ${\cal
O}(\alpha_{s})$ contributions [24, 22] in the $1S$ scheme [30, 31, 32], one
finds [31]

$\frac{\Gamma(B\to X_{u}\ell\bar{\nu})}{\Gamma(B\to
X_{u}\tau\bar{\nu})}\,=2.97\,,\quad\frac{\Gamma(B\to
X_{c}\ell\bar{\nu})}{\Gamma(B\to X_{c}\tau\bar{\nu})}\,=4.50\,,$ (1)

Thus, the suppression of the rate due to finite $m_{\tau}$ is less strong in
$b\to u$ than in $b\to c$ decays. Correspondingly, the suppression due to
finite $m_{c}$ is clearly greater in $B\to\tau$ than in $B\to e$ semileptonic
decays,

$\displaystyle\frac{\Gamma(B\to X_{u}\tau\bar{\nu})}{\Gamma(B\to
X_{c}\tau\bar{\nu})}\,\frac{|V_{cb}|^{2}}{|V_{ub}|^{2}}$ $\displaystyle=$
$\displaystyle 3.13\,,$ $\displaystyle\frac{\Gamma(B\to
X_{u}\ell\bar{\nu})}{\Gamma(B\to
X_{c}\ell\bar{\nu})}\,\frac{|V_{cb}|^{2}}{|V_{ub}|^{2}}$ $\displaystyle=$
$\displaystyle 1.83\,.$ (2)

##  II local ope results

###  II.1 Nonperturbative Corrections

The inclusive $B\to X_{q}\,\ell\bar{\nu}$ decay ($q=u,c$ ; $\ell=e,\mu,\tau$)
has been considered to order $1/m_{b}^{2}$ in the heavy quark expansion [20,
21, 22], including effects of the finite lepton mass. For $m_{q}=0$ the lepton
energy spectrum becomes singular, and the limit must be taken with care. We
find for $B\to X_{u}\tau\bar{\nu}$ decay,111The results in this section apply,
with obvious changes of hadron masses and matrix elements, to inclusive
$B_{c}\to X_{c}\tau\bar{\nu}$ decay, just like exclusive $B_{c}$ decays can be
calculated using HQET methods [33]. Treating charm as a heavy quark, the
$B_{c}$ has a size parametrically smaller than $\Lambda_{\rm QCD}$, and the
$b$ quark distribution function in $B_{c}$ is in principle calculable in
NRQCD. This decay might be observable in the tera-$Z$ phase of a future
$e^{+}e^{-}$ collider.

$\displaystyle\frac{1}{\Gamma_{u}}\frac{{\rm d}\Gamma}{{\rm d}y}$
$\displaystyle=$ $\displaystyle
2\sqrt{y^{2}-4\rho_{\tau}}\,\bigg{[}3y-2y^{2}-4\rho_{\tau}+3\rho_{\tau}y+\frac{\lambda_{2}}{m_{b}^{2}}\,6y$
(3)
$\displaystyle{}+\frac{\lambda_{1}+3\lambda_{2}}{3m_{b}^{2}}\,(5y^{2}-14\rho_{\tau})\bigg{]}\,\theta(1+\rho_{\tau}-y)$
$\displaystyle-$
$\displaystyle\bigg{[}\frac{\lambda_{1}}{3m_{b}^{2}}\,(1+\rho_{\tau})+\frac{\lambda_{2}}{m_{b}^{2}}\,(11-5\rho_{\tau})\bigg{]}$
$\displaystyle{}\times(1-\rho_{\tau})^{3}\,\delta(1+\rho_{\tau}-y)$
$\displaystyle-$
$\displaystyle\frac{\lambda_{1}}{3m_{b}^{2}}\,(1-\rho_{\tau})^{5}\,\delta^{\prime}(1+\rho_{\tau}-y)\,,$

where we use the dimensionless variables

$y=\frac{2E_{\tau}}{m_{b}}\,,\qquad\hat{q}^{2}=\frac{q^{2}}{m_{b}^{2}}\,,\qquad\rho_{\tau}=\frac{m_{\tau}^{2}}{m_{b}^{2}}\,,$
(4)

and

$\Gamma_{u}=\frac{|V_{ub}|^{2}\,G_{F}^{2}\,m_{b}^{5}}{192\,\pi^{3}}\,.$ (5)

This agrees with the more complicated expression given in Ref. [21]. Here
$\lambda_{1}$ and $\lambda_{2}$ are matrix elements in the heavy quark
effective theory (HQET), defined by

$\displaystyle\frac{1}{2m_{B}}\langle
B|\,\bar{b}_{v}\,(iD)^{2}\,b_{v}|B\rangle$ $\displaystyle=$ $\displaystyle
2\,\lambda_{1}\,,$ $\displaystyle\frac{1}{2m_{B}}\langle
B|\,\frac{g_{s}}{2}\,\bar{b}_{v}\,\sigma_{\mu\nu}G^{\mu\nu}\,b_{v}|B\rangle$
$\displaystyle=$ $\displaystyle 6\,\lambda_{2}\,,$ (6)

and $b_{v}$ is the heavy $b$-quark field of HQET [34] with velocity $v$.

The $\tau$ can have spin up ($s=+$) or spin down ($s=-$) relative to the
direction of its three-momentum, and it is convenient to decompose the
corresponding decay rates as

$\Gamma(B\to
X_{u}\,\tau(s=\pm)\,\bar{\nu})=\frac{1}{2}\,\Gamma\pm\tilde{\Gamma}\,.$ (7)

The rate, summed over the tau polarizations, is given by $\Gamma$, while the
average tau polarization is $A_{\rm pol}=2\tilde{\Gamma}/\Gamma$. The $\tau$
polarization gives complementary sensitivity to BSM physics [35]. We obtain
for its lepton energy dependence,

$\displaystyle\frac{1}{\Gamma_{u}}\frac{{\rm d}\tilde{\Gamma}}{{\rm d}y}$
$\displaystyle=$ $\displaystyle-(y^{2}-4\rho_{\tau})\bigg{[}3-2y+\rho_{\tau}$
(8)
$\displaystyle{}\quad+\frac{6\lambda_{2}}{m_{b}^{2}}+\frac{\lambda_{1}+3\lambda_{2}}{3m_{b}^{2}}5y\bigg{]}\,\theta(1+\rho_{\tau}-y)$
$\displaystyle+$
$\displaystyle\bigg{[}\frac{\lambda_{1}}{6m_{b}^{2}}\,(1-3\rho_{\tau})+\frac{\lambda_{2}}{2m_{b}^{2}}\,(11-5\rho_{\tau})\bigg{]}$
$\displaystyle{}\quad\times(1-\rho_{\tau})^{3}\,\delta(1+\rho_{\tau}-y)$
$\displaystyle+$
$\displaystyle\frac{\lambda_{1}}{6m_{b}^{2}}\,(1-\rho_{\tau})^{5}\,\delta^{\prime}(1+\rho_{\tau}-y)\,.$

Note that for $\rho_{\tau}=0$, $-2\,{\rm d}\tilde{\Gamma}={\rm d}\Gamma$,
since the massless lepton is purely left-handed. Angular momentum conservation
in $B\to X_{u}\tau\bar{\nu}$ implies that the $\tau$ polarization is fully
left-handed at maximal $E_{\tau}$. This holds at the parton level to all
orders in $\alpha_{s}$, and our results indeed satisfy it at order
$\alpha_{s}^{0}$ and order $\alpha_{s}^{1}$; i.e., $\Gamma/2=-\tilde{\Gamma}$
at $y=1+\rho_{\tau}$. However, the power-suppressed terms that start at order
$\Lambda_{\rm QCD}^{2}/m_{b}^{2}$ incorporate nonperturbative corrections
between the $E_{\tau}$ endpoint at the parton level and at the hadron level.
As a result, the physical rate at maximal $E_{\tau}$ vanishes (it is nonzero
at the parton level). In a small region very close to the endpoint the most
singular terms of the form $\lambda_{1}\,\delta^{\prime}(1+\rho_{\tau}-y)$ are
the most important, and these also obey the $\Gamma/2=-\tilde{\Gamma}$
relation.

For ${\rm d}\Gamma/{\rm d}\hat{q}^{2}$, the $m_{q}\to 0$ limit the of $B\to
X_{c}\tau\bar{\nu}$ expression is smooth, which gives the known result [23],

$\displaystyle\frac{1}{\Gamma_{u}}\frac{{\rm d}\Gamma}{{\rm d}\hat{q}^{2}}$
$\displaystyle=$
$\displaystyle\frac{(\hat{q}^{2}-\rho_{\tau})^{2}}{\hat{q}^{6}}\bigg{\\{}\bigg{(}1+\frac{\lambda_{1}}{2m_{b}^{2}}\bigg{)}2(1-\hat{q}^{2})^{2}$
$\displaystyle{}\times\Big{[}\hat{q}^{2}(1+2\hat{q}^{2})+\rho_{\tau}(2+\hat{q}^{2})\Big{]}$
$\displaystyle+$
$\displaystyle\frac{3\lambda_{2}}{m_{b}^{2}}\Big{[}\hat{q}^{2}(1-15\hat{q}^{4}+10\hat{q}^{6})+\rho_{\tau}(2-3\hat{q}^{2}+5\hat{q}^{6})\Big{]}\\!\bigg{\\}}.$

Integrating over phase space, the $B\to X_{u}\tau\bar{\nu}$ rate is

$\displaystyle\frac{\Gamma}{\Gamma_{u}}$ $\displaystyle=$
$\displaystyle\bigg{(}1+\frac{\lambda_{1}}{2m_{b}^{2}}\bigg{)}\big{(}1-8\rho_{\tau}+8\rho_{\tau}^{3}-\rho_{\tau}^{4}-12\rho_{\tau}^{2}\ln\rho_{\tau}\big{)}$
$\displaystyle-\frac{3\lambda_{2}}{2m_{b}^{2}}\big{(}3-8\rho_{\tau}+24\rho_{\tau}^{2}-24\rho_{\tau}^{3}+5\rho_{\tau}^{4}+12\rho_{\tau}^{2}\ln\rho_{\tau}\big{)}\,,$

and the polarization is given by

$\displaystyle\frac{\tilde{\Gamma}}{\Gamma_{u}}$ $\displaystyle=$
$\displaystyle-\frac{(1-\hat{m}_{\tau})^{3}}{2}\bigg{[}\frac{(1-\hat{m}_{\tau})^{2}}{3}\,\big{(}3+15\hat{m}_{\tau}+5\hat{m}_{\tau}^{2}+\hat{m}_{\tau}^{3}\big{)}$
$\displaystyle{}+\frac{\lambda_{1}}{6m_{b}^{2}}\,(1+\hat{m}_{\tau})^{3}\,(3+\hat{m}_{\tau}^{2})$
$\displaystyle-\frac{\lambda_{2}}{2m_{b}^{2}}\big{(}9+27\hat{m}_{\tau}+70\hat{m}_{\tau}^{2}+10\hat{m}_{\tau}^{3}-15\hat{m}_{\tau}^{4}-5\hat{m}_{\tau}^{5}\big{)}\bigg{]},$

where $\hat{m}_{\tau}=\sqrt{\rho_{\tau}}$.

###  II.2 Perturbative Corrections

Analytic results for the doubly differential ${\rm d}\Gamma/{\rm d}q^{2}{\rm
d}y$ spectra (including the $\tau$ polarization dependence) were given in
Refs. [26, 27] 222We corrected some typos in the $m_{c}\to 0$ limit in these
references., but only numerical results were presented for the $\tau$ energy
spectrum. Integrating the doubly differential spectra over $q^{2}$ gives the
charged lepton energy spectra for both unpolarized and polarized $\tau$
leptons. In the unpolarized case, writing

$\frac{1}{\Gamma_{0}}\frac{{\rm d}\Gamma_{\tau}}{{\rm
d}y}=\Big{[}F_{0}(y)-\frac{\alpha_{s}C_{F}}{2\pi}F_{1}(y)\Big{]}\,\theta(1+\rho_{\tau}-y)\,,$
(12)

where $C_{F}=4/3$, we find

$F_{0}(y)=2\sqrt{y^{2}-4\rho_{\tau}}\,\big{[}(3-2y)y+\rho_{\tau}(3y-4)\big{]}\,,$
(13)

and

$\displaystyle F_{1}(y)={}$ $\displaystyle
F_{0}(y)\left[\text{Li}_{2}(\tau_{+})+\text{Li}_{2}(\tau_{-})+4Y_{p}^{2}\right]+\left(6y^{2}-4y^{3}+6\rho_{\tau}y^{2}-12\rho_{\tau}^{2}\right)\left[\text{Li}_{2}(\tau_{+})-\text{Li}_{2}(\tau_{-})\right]$
(14)
$\displaystyle-2Y_{p}\left(\frac{5\rho_{\tau}^{3}}{3}+\rho_{\tau}(7y^{2}-6y+7)-6y^{3}+10y^{2}+\rho_{\tau}^{2}(4y-23)+6y-\frac{41}{3}\right)$
$\displaystyle+\sqrt{y^{2}-4\rho_{\tau}}\left(-\frac{34y^{2}}{3}+\rho_{\tau}\left(15y-\frac{74}{3}\right)+24y-6\right)\ln(1-y+\rho_{\tau})$
$\displaystyle+\sqrt{y^{2}-4\rho_{\tau}}\left(21\rho_{\tau}^{2}+\frac{1}{6}\left[\left(86-16\pi^{2}\right)y^{2}+\left(24\pi^{2}-153\right)y+82\right]+\frac{\rho_{\tau}}{6}\left(24\pi^{2}y-167y-32\pi^{2}+64\right)\right),$

where $Y_{p}=\frac{1}{2}\ln[(1-\tau_{+})/(1-\tau_{-})]$ is the rapidity of all
decay products (combined) against which the $\tau$ recoils, and

$\tau_{\pm}=\frac{1}{2}\left(y\pm\sqrt{y^{2}-4\rho_{\tau}}\right).$ (15)

Similarly, defining the polarization dependence of the lepton energy spectrum
as

$\frac{{\rm d}\Gamma^{\pm}_{\tau}}{{\rm d}y}=\frac{1}{2}\frac{{\rm
d}\Gamma_{\tau}}{{\rm d}y}\pm\frac{{\rm d}\tilde{\Gamma}_{\tau}}{{\rm d}y}\,,$
(16)

we write the polarization dependence of the rate to produce a $\tau$ lepton as

$\frac{1}{\Gamma_{0}}\frac{{\rm d}\tilde{\Gamma}_{\tau}}{{\rm
d}y}=\Big{[}\tilde{F}_{0}(y)-\frac{\alpha_{s}C_{F}}{2\pi}\tilde{F}_{1}(y)\Big{]}\,\theta(1+\rho_{\tau}-y)\,.$
(17)

At tree level,

$\tilde{F}_{0}(y)=\left(y^{2}-4\rho_{\tau}\right)(2y-3-\rho_{\tau})\,,$ (18)

while at one loop,

$\displaystyle\tilde{F}_{1}(y)={}$
$\displaystyle\tilde{F}_{0}(y)\left[\text{Li}_{2}(\tau_{+})+\text{Li}_{2}(\tau_{-})+4Y_{p}^{2}\right]+\frac{12\rho_{\tau}^{2}-\rho_{\tau}\left(y^{3}+6y^{2}-6y\right)+2y^{4}-3y^{3}}{\sqrt{y^{2}-4\rho_{\tau}}}\left[\text{Li}_{2}(\tau_{+})-\text{Li}_{2}(\tau_{-})\right]$
$\displaystyle-\frac{Y_{p}}{3\sqrt{y^{2}-4\rho_{\tau}}}\bigg{[}\rho_{\tau}\left(y^{3}-210y^{2}+405y-260\right)+\left(18y^{4}-12y^{3}-36y^{2}+41y\right)+\rho_{\tau}^{3}(y-24)+\rho_{\tau}^{2}(93y-12)\bigg{]}$
$\displaystyle+\left[\frac{34\rho_{\tau}^{2}}{3}+\rho_{\tau}\left(-\frac{23y^{2}}{6}-\frac{53y}{3}+30\right)+\frac{17y^{3}}{3}-9y^{2}\right]\ln(1-y+\rho_{\tau})+\frac{\rho_{\tau}^{3}}{3}+\rho_{\tau}^{2}\left(-\frac{11y}{6}+\frac{8\pi^{2}}{3}-21\right)$
$\displaystyle+\frac{\rho_{\tau}}{12}\left(-8\pi^{2}y^{2}+149y^{2}-64\pi^{2}y-48y+96\pi^{2}+32\right)+\frac{1}{12}\left(16\pi^{2}y^{3}-86y^{3}-24\pi^{2}y^{2}+153y^{2}-82y\right).$
(19)

##  III The lepton energy endpoint region

Near the endpoint of the lepton energy spectrum $y\sim 1+\rho_{\tau}$, a class
of higher-order terms in the local OPE in Eq. (3) is no longer suppressed, and
instead the differential rate is given by a nonlocal OPE in terms of the
light-cone momentum distribution function of the $b$ quark [36, 28, 37, 38,
29, 39].

This endpoint region has been extensively studied in the context of massless
leptons. It is straightforward to extend this to nonzero $\tau$ mass. At the
parton level the lepton energy endpoint is determined by the $\theta$ function

$\theta\big{(}(p_{b}-p_{\tau})^{2}\big{)}=\theta\left(m_{b}^{2}+m_{\tau}^{2}-2\,p_{\tau}\cdot
p_{b}\right).$ (20)

Writing

$p_{\tau}^{\mu}=\frac{m_{b}}{2}\big{(}\tau_{-}\,n^{\mu}+\tau_{+}\,\bar{n}^{\mu}\big{)}\,,$
(21)

where $\tau_{\pm}$ are given in Eq. (15), defines the light-like vectors
$n^{\mu}$ and $\bar{n}^{\mu}=2v^{\mu}-n^{\mu}$. Taking
$p_{b}^{\mu}=m_{b}v^{\mu}+k^{\mu}$, expanding in powers of $k^{\mu}/m_{b}$ and
applying the HQET onshell condition $k\cdot v=0$, the $\theta$ function
becomes

$\theta\left(1+\rho_{\tau}-y+\frac{k\cdot
n}{m_{b}}\,\sqrt{y^{2}-4\rho_{\tau}}+{\cal O}(k^{2})\right).$ (22)

Over most of the spectrum, the ${\cal O}(k\cdot n)$ term may be neglected at
leading order in $1/m_{b}$ and we recover the OPE result in Eq. (3). However,
when $E_{\tau}$ is near the partonic endpoint, i.e., $1+\rho_{\tau}-y={\cal
O}(\Lambda_{\rm QCD}/m_{b})$, $p_{b}-p_{\tau}$ approaches a light-like vector
in the $n$ direction. In this region the ${\cal O}(k\cdot n)$ term is the same
order as the leading term, and so must be included in the leading-order
expression. Defining

$\Delta\equiv\frac{1+\rho_{\tau}-y}{1-\rho_{\tau}}\,,$ (23)

taking $\Delta\sim{\cal O}(\Lambda_{\rm QCD}/m_{b})$, and expanding (22) in
powers of $\Delta$ then gives

$\theta\bigg{(}\Delta+\frac{k\cdot n}{m_{b}}\bigg{)}+{\cal O}(\Lambda_{\rm
QCD}/m_{b})\,.$ (24)

Comparing with the $\rho_{\tau}\to 0$ limit, the nonzero $\tau$ mass shifts
the endpoint of the lepton spectrum and squeezes it by a factor of
$1-\rho_{\tau}$. This is also reflected by the fact that the lepton energy
endpoint changes between the parton- and hadron-level kinematics, at leading
order, by $(1-\rho_{\tau})\,\bar{\Lambda}/2$, where
$m_{B}=m_{b}+\bar{\Lambda}+{\cal O}(\Lambda_{\rm QCD}^{2}/m_{b})$.

At the hadron level, matrix elements of the $\theta$ function may be expressed
as an integral over the light-cone momentum distribution function of the $b$
quark in the $B$ meson,

$f(\omega,\mu)=\frac{1}{2m_{B}}\,\langle B|\bar{b}_{v}\delta(\omega+iD\cdot
n)b_{v}|B\rangle\,.$ (25)

Following [40], it is convenient to define the nonperturbative function $F(k)$
via the convolution

$f(\omega,\mu)=\int{\rm d}k\,C_{0}(\omega-k,\mu)\,F(k)\,,$ (26)

where, at one loop [41],

$C_{0}(\omega,\mu)=\delta(\omega)-\frac{\alpha_{s}C_{F}}{4\pi}\Bigg{(}\frac{\pi^{2}}{6}\delta(\omega)+\frac{4}{\mu}\bigg{[}\frac{\mu}{\omega}\bigg{]}_{+}+\frac{8}{\mu}\bigg{[}\frac{\ln\frac{\omega}{\mu}}{\omega/\mu}\bigg{]}_{+}\Bigg{)}.$
(27)

The convolution (26) factors out the perturbative corrections to the parton-
level matrix element of $f(\omega)$. With this definition, $F(k)$ is a
nonperturbative function with support from $k=-\bar{\Lambda}$ to $k=\infty$,
whose moments are related to the matrix elements of local operators. The
$\tau$ energy spectrum may then be written in the endpoint region as the
convolution

$\frac{1}{\Gamma_{\tau}}\frac{{\rm d}\Gamma_{\tau}}{{\rm d}y}=\int{\rm
d}\omega\,G_{\tau}\bigg{(}\Delta-\frac{\omega}{m_{b}}\bigg{)}F(\omega)+{\cal
O}(\Delta,\Lambda_{\rm QCD}/m_{b})\,,$ (28)

where $G_{\tau}(x)$ is obtained by expanding the parton level perturbative
results (14) in the limit $\Delta\to 0$,

$\displaystyle G_{\tau}(x)=\theta(x)\,$
$\displaystyle\bigg{\\{}1-\frac{\alpha_{s}C_{F}}{2\pi}\bigg{[}\ln^{2}x$ (29)
$\displaystyle+\bigg{(}\frac{31}{6}-2\ln(1-\rho_{\tau})\bigg{)}\ln
x+C(\rho_{\tau})\bigg{]}\bigg{\\}}\,,$

and $C(\rho_{\tau})=\pi^{2}+5/4+\rho_{\tau}(\pi^{2}-6)+{\cal
O}(\rho_{\tau}^{2})$. Note that in Eq. (29) the $m_{\tau}$ dependent terms at
${\cal O}(\alpha_{s})$ are small corrections: $C(\rho_{\tau})/C(0)$ is within
5% of unity, and the $2\ln(1-\rho_{\tau})$ term is less than a 6% correction
relative to the “31/6” term. $G_{\tau}(x)$ therefore has very weak
$\rho_{\tau}$ dependence: none at tree level, and only about
$5\%\times\alpha_{s}C_{F}/(2\pi)$ at one loop. The large difference between
the shapes arises almost entirely from the kinematic rescaling in Eq. (23).
SCET techniques may be used to sum logarithms of $\Delta$ in this expression
(as in Refs. [41] and [40]), but this is beyond the scope of this paper or the
accuracy we desire.

Figure 1: The $B\to X_{u}\ell\bar{\nu}$ lepton energy spectrum for (a)
$\ell=e,\,\mu$ and (b) $\ell=\tau$ in the parton model (blue, dashed), and
incorporating the leading order $b$-quark distribution function (red, solid).
Figure 2: The $\tau$ energy dependence of its polarization in $B\to
X_{u}\tau\bar{\nu}$ in the parton model (blue, dashed), and incorporating the
leading order $b$-quark distribution function (red, solid).

The expression (28) is only valid in the region $\Delta\sim\Lambda_{\rm
QCD}/m_{b}$; in order to have an expression which smoothly interpolates with
the local OPE away from the endpoint, it is convenient instead to incorporate
distribution function effects by redefining the $b$-quark mass $m_{b}\to
m_{b}^{\prime}=m_{b}+k\cdot n$ [36, 28]. Writing
$p_{b}^{\mu}=m_{b}^{\prime}v^{\mu}+k^{\prime\mu}$, where
$k^{\prime\mu}=k^{\mu}-k\cdot n\,v^{\mu}$, the residual momentum
$k^{\prime\mu}$ satisfies $k^{\prime}\cdot n=0$, and so the effects of nonzero
$k\cdot n$ are automatically incorporated into the leading-order spectrum with
this mass definition. The $\tau$ energy spectrum in the endpoint region may
then be written as the convolution

$\frac{{\rm d}\Gamma_{\tau}}{{\rm d}E_{\tau}}=2\int{\rm
d}\omega\,\frac{1}{m_{b}}\frac{{\rm d}\Gamma_{\tau}}{{\rm
d}y}(y^{\prime},\rho_{\tau}^{\prime})\,F(\omega)\,,$ (30)

where we have defined the scaled variables

$y^{\prime}\equiv\frac{2E_{\tau}}{m_{b}-\omega}\,,\qquad\rho_{\tau}^{\prime}\equiv\frac{m_{\tau}^{2}}{(m_{b}-\omega)^{2}}\,,$
(31)

and ${\rm d}\Gamma/{\rm d}y$ is the parton level spectrum in Eq. (12). An
analogous formula holds for the polarized spectrum Eq. (17). For simplicity,
we have written the prefactor in Eq. (30) as $1/m_{b}$, not
$1/(m_{b}-\omega)$, since the difference is higher order everywhere in the
spectrum. In this form, Eq. (30) includes subleading terms suppressed by
powers of $\Delta$ in the endpoint region, but which are leading order when
$\Delta$ is not small, so are required to reproduce the local OPE away from
the endpoint.

$F(k)$ has been extracted from the measured $B\to X_{s}\gamma$ spectra by the
SIMBA collaboration [42]. At leading order in $\Lambda_{\rm QCD}/m_{b}$, it
can be used to make predictions for $B\to X_{u}\ell\bar{\nu}$ decays. Figure 1
shows the $B\to X_{u}\ell\bar{\nu}$ lepton spectra for $\ell=e$ and
$\ell=\tau$ in the parton model and including the effects of the $b$-quark
distribution function. It is clear from this plot that the distribution
function is indeed important in a greater fraction of the $\tau$ energy
spectrum than in the massless lepton channels; the fraction of the lepton
energy spectrum where the distribution function is important is enhanced by
$(1-\rho_{\tau})/(1-\sqrt{}\rho_{\tau})^{2}\sim 2.2$. Figure 2 shows the
$E_{\tau}$ spectra separately for left- and right-handed $\tau$ leptons in
$B\to X_{u}\tau\bar{\nu}$. The average $\tau$ polarization, including order
$\alpha_{s}$ and $\Lambda_{\rm QCD}^{2}/m_{b}^{2}$ corrections, is
$2\tilde{\Gamma}/\Gamma=-0.77$.

##  IV Conclusions

We presented theoretical predictions for inclusive $B\to X_{u}\tau\bar{\nu}$
decay. We derived previously unknown results at order $\Lambda_{\rm
QCD}^{2}/m_{b}^{2}$ and analytic expressions for the order $\alpha_{s}$
corrections for the $\tau$ energy spectrum and polarization. We also
incorporated the effects of the $b$-quark light-cone distribution function to
the case of nonzero lepton mass. Due to the suppressed kinematic range, the
$b$-quark distribution function is more important in determining the lepton
energy spectrum in $B\to X_{u}\tau\bar{\nu}$ than in $B\to X_{u}e\bar{\nu}$
decay.

It will probably take many ab-1 of data at Belle II to have sensitivity to
$B\to X_{u}\tau\bar{\nu}$. While it is clearly a challenging decay to measure,
the rate according to Eqs. (1) and (I) is only about 3 times smaller than
$B\to X_{u}e\bar{\nu}$, and about $|V_{cb}|^{2}/(3\,|V_{ub}|^{2})$ times
smaller than $B\to X_{c}\tau\bar{\nu}$. One may, for example, try to utilize
the fact that electrons or muons from the $\tau$ decay with maximal allowed
energies correspond to the most energetic $\tau$ leptons. We hope that Belle
II will be able to make measurements of this decay.

###### Acknowledgements.

This paper is dedicated to the memory of Sheldon Stone, with whom we had
countless inspiring and entertaining discussions, e.g., related to the papers
[43, 44, 45]; he’d surely be appalled knowing how long this one took to get
out. We thank Mark Wise for helpful conversations about these decays in the
previous millennium, and Florian Bernlochner and Aneesh Manohar in this one.
ZL thanks the Aspen Center for Physics (supported by the NSF Grant
PHY-1607611) for hospitality while some of this work was carried out. This
work was supported in part by the Office of High Energy Physics of the U.S.
Department of Energy under contract DE-AC02-05CH11231 and by the Natural
Sciences and Engineering Research Council of Canada.

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# Moments of the Distribution of Pearson’s Correlation over Permutations of
Data

Marc Jaffrey111University of Washington<EMAIL_ADDRESS>222RootFault LLC
Michael Dushkoff333Rochester Institute of Technology<EMAIL_ADDRESS>††footnotemark:

## Introduction

Let $D_{n}=\\{(x_{i},y_{i})|\ i=1,..,n\\}\subset\mathbb{R}^{2}$, be a data
sample of size $n$. Let $\Pi=Perm(n)$ denote the set of all permutation on the
indices $(1,..,n)$. Given $\pi=(\pi_{1},..,\pi_{n})\in\Pi$, let
$\pi(D_{n})=\\{(x_{i},y_{\pi_{i}})|i=1,..,n\\}$ where the permutation $\pi$
acts on the $y$-coordinate alone, and $\Pi(D_{n})=\\{\pi(D_{n})|\pi\in\Pi\\}$.
Next, take Pearson’s correlation $\rho$ and let
$\rho_{\pi}=\rho(\pi(D_{n}))=\rho(x,y_{\pi})$ and
$\rho_{\Pi}=\\{\rho(\pi(D_{n}))|\pi\in\Pi\\}$.

Presented is an inductive formula for computing the exact moments of the
distribution of Pearson’s correlation over $\Pi(D_{n})$, denoted
$\langle\rho_{\Pi}^{k}\rangle$, $k\in\mathbb{N}$. These exact formulas for the
moments open the door to the possibility of more precise and computationally
efficient methods of evaluating the p-value for a hypothesis test of Pearson’s
correlation. We have carefully reviewed the literature and are currently
unaware of these findings being shown elsewhere.

## General Inductive Formula for the $k^{th}$ moment

Given $m$ non-zero positive integers $n_{1}+...+n_{m}=k$, define

$\displaystyle X_{(n_{1},..,n_{m})}^{m,k}=$ $\displaystyle\ n\langle
x^{n_{m}}\rangle
X_{(n_{1},...,n_{m-1})}^{m-1,k-n_{m}}-\sum_{j}X_{(n_{1}+n_{m}\delta_{ij},...,n_{m-1}+n_{m}\delta_{(m-1)j})}^{m-1,k}$
where $\delta_{ij}$ is $\displaystyle\text{ the standard delta function.
Similarly define is}\ Y_{(n_{1},..,n_{m})}^{m,k},$ $\displaystyle\ \
\text{where}\ X_{(k)}^{1,k}\ $ $\displaystyle=\ n\langle x^{k}\rangle\
\text{and}\ \ Y_{(k)}^{1,k}=n\langle y^{k}\rangle.$

Then

$\displaystyle\langle\rho_{\Pi}^{k}\rangle=$
$\displaystyle\frac{1}{n!}\sum_{\Pi}\Bigg{[}\frac{\sum_{i}(x_{i}-\mu_{x})(y_{\pi_{i}}-\mu_{y})}{\sqrt{\sum_{k}(x_{k}-\mu_{x})^{2}\sum_{j}(y_{\pi_{j}}-\mu_{y})^{2}}}\Bigg{]}^{k}$
$\displaystyle=$
$\displaystyle\frac{1}{n^{k}n!\sigma_{x}^{k}\sigma_{y}^{k}}\sum_{\Pi}\sum_{i_{i},..,i_{k}}(x_{i_{1}}-\mu_{x})..(x_{i_{k}}-\mu_{x})(y_{\pi_{i_{i}}}-\mu_{y})...(y_{\pi_{i_{k}}}-\mu_{y})$
$\displaystyle=$ $\displaystyle\
\frac{1}{n^{k}n!\sigma_{x}^{k}\sigma_{y}^{k}}\sum_{m=1}^{k}\ \
\sum_{n_{1}+...+n_{m}=k}\binom{k}{n_{1},..,n_{m}}^{*}X_{(n_{1},..,n_{m})}^{m,k}(n-m)!Y_{(n_{1},..,n_{m})}^{m,k}\cdot
h_{n,m}$

where

$h_{n,m}=\left\\{\begin{array}[]{rcl}0&\mbox{for}&n-m<0\\\
0&\mbox{for}&n-m\geq 1\end{array}\right.$

and

$\binom{k}{n_{1},\dots,n_{m}}^{*}=\frac{1}{d_{1}!}\cdots\frac{1}{d_{r}!}\binom{k}{n_{1},\dots,n_{m}}$

The multinomial coefficient has to be adjusted for degeneracy in the exponents
$(n_{1},...,n_{m})$. Partition the exponents $(n_{1},..,n_{m})$ into
subgroups, $g_{1},..,g_{r}$, via the equivalence relation $n_{i}\equiv
n_{j}\iff n_{i}=n_{j}$, and let $d_{i}=|g_{i}|$. The multiplicity that arises
from this degeneracy is equal to $d_{1}!\cdots d_{r}!$, which is divided out
of the multinomial coefficient as above.

The term $h_{n,m}$ is included for completeness. If there are fewer data
points than the order of the moment being computed then higher order terms
begin to drop out as they are no longer relevant.

As the formula presented can be validated numerically and the derivation does
not add anything to the application of these results, the derivation will be
presented in future works.

Note: All the formulas in this paper assume that the standard deviation is
computed as

$\sigma_{z}=\frac{1}{n}\sum(z_{i}-\mu_{z})^{2},$

a result of how sample correlations are computed,

$\rho(x,y)=\frac{\sum_{i}(x_{i}-\mu_{x})(y_{i}-\mu_{y})}{\sqrt{\sum_{k}(x_{k}-\mu_{x})^{2}\sum_{j}(y_{j}-\mu_{y})^{2}}}.$

## Exact Formulas for $k=1,..,5$

From the induction formula every moment can be analytically determined. Below
are the exact formulas for the first five moments in terms of the easily
computed moments of the data. For ease of notion let $\sigma_{x}=\sqrt{\langle
x^{2}\rangle}$, $\sigma_{y}=\sqrt{\langle y^{2}\rangle}$, $\chi_{k}=\langle
x^{k}\rangle$, $\nu_{k}=\langle y^{k}\rangle$, and $\mu_{k,j}=\langle
x^{k}\rangle\langle y^{j}\rangle$. The first five moments are as follows:

$\displaystyle\langle\rho_{\Pi}^{1}\rangle=$ $\displaystyle\ 0$
$\displaystyle\langle\rho_{\Pi}^{2}\rangle=$ $\displaystyle\ \frac{1}{(n-1)}$
$\displaystyle\langle\rho_{\Pi}^{3}\rangle=$ $\displaystyle\
\frac{\mu_{3,3}}{\sigma_{x}^{3}\sigma_{y}^{3}}\bigg{[}\frac{1}{n^{2}}h_{n,1}+\frac{3}{n^{2}(n-1)}h_{n,2}+\frac{4}{n^{2}(n-1)(n-2)}h_{n,3}\bigg{]}$
$\displaystyle\langle\rho_{\Pi}^{4}\rangle=$ $\displaystyle\
\frac{1}{\sigma_{x}^{4}\sigma_{y}^{4}}\bigg{[}\frac{\mu_{4,4}}{n^{3}}h_{n,1}+\frac{4\chi_{4}\nu_{4}+3[n^{2}\sigma_{x}^{4}-n\chi_{4}][n^{2}\sigma_{y}^{4}-n\nu_{4}]}{n^{3}(n-1)}h_{n,2}$
$\displaystyle+\frac{6[2n\chi_{4}-n^{2}\sigma_{x}^{4}][2n\nu_{4}-n^{2}\sigma_{y}^{4}]}{n^{5}(n-1)(n-2)}h_{n,3}+\frac{9[2n\chi_{4}-n^{2}\sigma_{x}^{4}][2n\nu_{4}-n^{2}\sigma_{y}^{4}]}{n^{5}(n-1)(n-2)(n-3)}h_{n,4}\bigg{]}$
$\displaystyle\langle\rho_{\Pi}^{5}\rangle=$ $\displaystyle\
\frac{1}{\sigma_{x}^{5}\sigma_{y}^{5}}\bigg{[}\frac{\mu_{5,5}}{n^{4}}h_{n,1}+5\frac{\mu_{5,5}}{n^{4}(n-1)}h_{n,2}+10\frac{[n^{2}\chi_{3}\chi_{2}-n\chi_{5}][n^{2}\nu_{3}\nu_{2}-n\nu_{5}]}{n^{6}(n-1)}h_{n,2}$
$\displaystyle+10\frac{[2n\chi_{5}-n^{2}\chi_{3}\chi_{2}][2n\nu_{5}-n^{2}\nu_{3}\nu_{2}]}{n^{6}(n-1)(n-2)}h_{n,3}$
$\displaystyle+60\frac{[n\chi_{5}-n^{2}\chi_{3}\chi_{2}][n\nu_{5}-n^{2}\nu_{3}\nu_{2}]}{n^{6}(n-1)(n-2)}h_{n,3}$
$\displaystyle+10\frac{[6n\chi_{5}-5n^{2}\chi_{3}\chi_{2}][6n\nu_{5}-5n^{2}\nu_{3}\nu_{2}]}{n^{6}(n-1)(n-2)(n-3)}h_{n,4}$
$\displaystyle+\frac{16[6n\chi_{5}-5n^{2}\chi_{3}\chi_{2}][6n\nu_{5}-5n^{2}\nu_{3}\nu_{2}]}{n^{6}(n-1)(n-2)(n-3)(n-4)}h_{n,5}\bigg{]}$

## Validation

While the exact formulas speak for themselves, we demonstrate for completeness
sake the validity of these moments numerically for randomly generated datasets
of sizes $n=\\{3,\dots,8\\}$ over $100$ trials at each size. For all derived
moments above, we compare them with the actual moments as computed directly
from $\Pi(D_{n})$, which we are able to compute in the case of small datasets.
Beyond $n=8$, the computational cost to compute the moments directly from
$\Pi(D_{n})$ becomes onerous and does not add anything more to the discussion.

The mean squared error was computed as:

$\frac{1}{N_{trials}}\sum\left(\langle\rho^{k}\rangle_{\Pi(D_{n})}-\langle\rho^{k}\rangle_{exact}\right)^{2}$

Validation was done in MATLAB using double-precision floating point
operations. Table 1 clearly shows that all of the errors are well within
machine error.

| MSE of $k^{th}$ Moment
---|---
Sample Size | 2 | 3 | 4 | 5
$n=3$ | $1.20\mathrm{e}{-32}$ | $2.57\mathrm{e}{-32}$ | $3.99\mathrm{e}{-32}$ | $4.82\mathrm{e}{-32}$
$n=4$ | $6.81\mathrm{e}{-33}$ | $2.46\mathrm{e}{-33}$ | $1.11\mathrm{e}{-32}$ | $3.18\mathrm{e}{-33}$
$n=5$ | $6.06\mathrm{e}{-33}$ | $1.23\mathrm{e}{-33}$ | $5.75\mathrm{e}{-32}$ | $1.55\mathrm{e}{-33}$
$n=6$ | $2.42\mathrm{e}{-32}$ | $8.99\mathrm{e}{-34}$ | $6.00\mathrm{e}{-33}$ | $4.37\mathrm{e}{-34}$
$n=7$ | $1.31\mathrm{e}{-31}$ | $3.50\mathrm{e}{-33}$ | $1.78\mathrm{e}{-32}$ | $1.29\mathrm{e}{-33}$
$n=8$ | $7.38\mathrm{e}{-31}$ | $1.05\mathrm{e}{-32}$ | $5.37\mathrm{e}{-32}$ | $3.17\mathrm{e}{-33}$
Table 1: Validation Error

## Application: Moment-Derived Testing for p-value

One direct application of these exact moments presented herein is towards the
development of a more accurate and computationally efficient estimation method
of p-value of Pearson’s correlation, as compared with existing approximation
methods and permutation testing algorithms.

The theoretical problem of approximating the cumulative distribution function
given a fixed set of moments has been explored fairly thoroughly[2][3]. Since
the distribution of all permutations of the Pearson’s Correlation is on a
fixed interval $[-1,1]$, the uniqueness of this distribution from the moments
is related to the Hausdorff Moment Problem, a topic for further
investigation[1].

Using such methods of estimation, one can obtain a valid approximation of the
CDF with sufficiently less effort, especially since the distribution itself
has no known tractable closed-form solution, while the moments are relatively
inexpensive to obtain, depending on the moments of the data alone. Given a
sufficient number of moments for Pearson’s correlation over the permutations
of the data, one can estimate the distribution of Pearson’s correlation over
the set of permutations to any desired degree of accuracy. Such an analysis
could potentially be more efficient, certainly more efficient than computing
all the permutations of the correlation coefficient. We posit that such a
method exists for determining a direct p-value estimate and is a topic to be
present in a separate paper.

## Conclusion

On a final note, for free we obtain exact formulas for the moments for the
distribution of Spearman’s correlation over permutations of data. Converting
$x$ and $y$ to their rank or partial rank ordering and inputting these into
the formulas for the moments of Pearson’s correlation gives the exact moments
of Spearman’s correlation. It should be noted that as Spearman’s correlation
is independent of the exact form of the data, the moments being a function of
$n$ alone, the moments of Spearman’s correlation $\Pi(D_{n})$ can be tabulated
for computing its p-value with a moment approximation to any desired level of
accuracy with minimal numerical cost.

## References

  * [1] Naum Iljič Aheizer and N Kemmer.  The classical moment problem and some related questions in analysis.  Oliver & Boyd Edinburgh, 1965.
  * [2] V John, I Angelov, AA Öncül, and D Thévenin.  Techniques for the reconstruction of a distribution from a finite number of its moments.  Chemical Engineering Science, 62(11):2890–2904, 2007.
  * [3] Robert M Mnatsakanov.  Hausdorff moment problem: reconstruction of distributions.  Statistics & Probability Letters, 78(12):1612–1618, 2008.