# End-to-End Semantic Video Transformer for Zero-Shot Action Recognition

Keval Doshi111Performed the work while at University of South Florida.
<EMAIL_ADDRESS>Yasin Yilmaz<EMAIL_ADDRESS>
#

Keval Doshi111Performed the work while at University of South Florida.
<EMAIL_ADDRESS>Yasin Yilmaz<EMAIL_ADDRESS>
###### Abstract

While video action recognition has been an active area of research for several
years, zero-shot action recognition has only recently started gaining
traction. In this work, we propose a novel end-to-end trained transformer
model which is capable of capturing long range spatiotemporal dependencies
efficiently, contrary to existing approaches which use 3D-CNNs. Moreover, to
address a common ambiguity in the existing works about classes that can be
considered as previously unseen, we propose a new experimentation setup that
satisfies the zero-shot learning premise for action recognition by avoiding
overlap between the training and testing classes. The proposed approach
significantly outperforms the state of the arts in zero-shot action
recognition in terms of the the top-1 accuracy on UCF-101, HMDB-51 and
ActivityNet datasets. The code and proposed experimentation setup are
available in GitHub: https://github.com/Secure-and-Intelligent-Systems-
Lab/SemanticVideoTransformer

[1]organization=Amazon, addressline=Prime Video, postcode=98109, city=Seattle,
state=Washington, country=USA [2]organization=University of South Florida,
addressline=Department of Electrical Engineering, postcode=33620, city=Tampa,
state=Florida, country=USA

Keywords: zero-shot learning , action recognition , video transformer ,
semantic embedding

##  1 Introduction

Several visual recognition tasks, such as image classification and video
action recognition, have made tremendous progress in recent years, thanks to
the availability of extensively annotated datasets and enhanced deep learning
architectures. However, collecting and annotating video samples for every
possible interaction between objects is impractical, therefore recognizing
previously unseen actions remains a challenging task. On the other hand,
humans are exceptionally good at recognizing new categories without seeing any
visual samples. For example, if a person is familiar with _ice skating_ and
understands the concept of _dancing_ , (s)he will have no trouble recognizing
the action of _ice dancing_. In the recent literature, this problem has drawn
considerable attention and is known as Zero-Shot Learning (ZSL) for video
action recognition. While several approaches have shown promising results in
the image domain, zero-shot video action recognition remains largely
unexplored.

In the existing research, ZSL is characterized as a classification problem in
which a model is trained on a collection of known classes and then uses
semantic attributes to identify unknown classes. Most existing approaches
employ a 3D Convolutional Neural Network (3D-CNN) to extract visual features
from videos. Because the utilized 3D-CNNs are pretrained on a variety of
large-scale datasets, there is no obvious demarcation between what defines
seen and unseen classes. Several recent techniques, for example, are
pretrained on the Kinetics-400/600/700 [1, 2, 3] datasets and evaluated on the
UCF-101 [4], HMDB-51 [5], Olympics [6] and ActivityNet [7] datasets. However,
as shown in Fig. 1, several classes that are considered as “unseen” are
already present in the Kinetics dataset, which clearly violates the zero-shot
paradigm. On the other hand, even a human is incapable of recognizing
_archery_ as an activity if he has never seen a _bow_ and _arrow_ before and
is oblivious to the concept of _shooting_. Thus, it can be argued that it is
redundant to include activities in the test set that are significantly
dissimilar to the activities present in the training set. Furthermore, the
existing evaluation setup consists of randomly splitting the test set classes
into half and evaluating the proposed approach on that set. However, this
causes an unfair comparison since modifying the split can significantly alter
the result. To this end, we propose a new ZSL experimentation framework for
action recognition, that addresses these issues and thus would provide
guidance for future algorithm design.

Fig. 1: In the recent state-of-the-art approaches CLASTER [8], E2E-664 [9], OD
[10], there is a significant overlap between the training classes and the
testing classes. The vertical axis shows the percentage of test classes in the
benchmark datasets (UCF-101, HMDB, Olympics, ActivityNet) that overlap with
the training classes in the Kinetics dataset, which is significantly larger
than the benchmark datasets used for testing. Since the test classes are
supposed to be previously unseen, such high-percentage overlap (30%-70%)
violates the ZSL premise.

Since the majority of existing works employ 2D or 3D convolutions as the
principal operator for learning spatiotemporal features from videos, they
suffer from several challenges [11]. First, inductive biases such as local
connectivity, translation invariance, and a locally restricted receptive field
severely limit the learning capability of convolutional models on large
datasets. Moreover, convolutional kernels are incapable of capturing
spatiotemporal relations that span multiple time instances [11]. Finally, even
with advances in hardware acceleration, training and evaluating deep CNNs on
large video datasets remain computationally expensive.

Motivated by these observations, we propose leveraging self-attention
architectures, in particular a spatiotemporal transformer model, for
extracting semantic embeddings from videos. The self-attention approach, in
contrast to convolutional kernels, can capture long-range dependencies and is
permutation invariant while being computationally efficient during training
and inference. Our end-to-end trained transformer based approach is able to
learn semantically more separable features as compared to 3D-CNNs. Our
contributions can be summarized as follows:

  * 1.

_A novel end-to-end transformer to learn visual-semantic representations._ To
the best of our knowledge, this is the first work to propose a _spatiotemporal
transformer_ for _zero-shot video action recognition_ although there are
existing transformer methods for zero-shot image tasks.

  * 2.

_A thorough analysis of the shortcomings in the existing ZSL framework for
action recognition._ We propose a new evaluation framework that satisfies the
ZSL criterion for action recognition.

  * 3.

_An extensive evaluation of the proposed transformer method on several
benchmark datasets._ The proposed method outperforms several state-of-the-art
approaches by a wide margin under different experimental setups.

##  2 Related Works

Video action recognition has been extensively studied over the past several
years [12, 13, 14, 15, 16, 17]. In contrast, ZSL for video action recognition
has only recently started gaining attention. Broadly, ZSL can be classified
into the inductive setting [18, 19, 20, 21, 22], where the test data is
completely unknown during training, and the transductive setting [23, 24, 25,
26, 27, 28, 29], where the test data without class labels is available. In
this work, we only focus on the inductive setting.

Existing approaches have been predominantly dependent on word embeddings to
tackle the problem of ZSL. Specifically, these approaches use a pretrained
model to extract visual features from training videos and map them to a
semantic space and hypothesize that a good robust generalization on the
semantic space can lead to improved performance on unseen classes [19, 9, 8,
18, 30, 31, 32, 21, 26, 29, 23]. For extracting the visual features, most
recent approaches propose using a 3D-CNN, which takes 16 frames sampled from a
video as input. In [9], Brattoli et al. propose training a C3D [33] and a
R(2+1)D model [34] in an end-to-end fashion for ZSL. On the other hand, Gowda
et al. [8] propose a reinforcement learning based clustering approach, which
uses a two-stream I3D [16] model for learning visual features. However, we
argue that for realistic applications, sampling 16 frames from a video might
not always be sufficient, especially when it comes to large-scale datasets. In
[16, 35], it is shown that increasing the number of input frames only
marginally increases the performance of convolutional models. On the contrary,
recent transformer based approaches [11, 36] have shown considerable
performance gains when number of input frames are increased from 8 to 96,
especially on tasks that require longer temporal reasoning. While earlier
approaches use hand-crafted semantic features [37], recent works have
primarily use Word2Vec [38] for generating the semantic embeddings from the
class labels. However, such approaches are prone to the domain shift problem,
which occurs when a model trained on the seen semantic labels is unable to
generalize well to the unseen class labels [18].

Recently, Brattoli et al. [9] propose an extension on the work of Roitberg et
al. [39] and formulate a novel evaluation protocol for satisfying the ZSL
paradigm, which involves removing certain classes from the training set which
overlap with the test set by using semantic embedding matching. However, in
this paper, we show that such an approach fails to remove a significant
portion of the overlapping classes, thus still violates the ZSL premise.
Alternatively, Gowda et al. [40] propose a deterministic “TruZe” split for the
UCF-101 [4] and HMDB-51 [5] datasets, by manually removing all classes which
overlap with the Kinetics-400 dataset. While it is a promising approach, we
show that it quickly becomes obsolete since several recent approaches use
Kinetics-600/700 for training, which includes a majority of the classes from
the “TruZe” test set. Furthermore, neither of the proposed approaches remove
classes which are significantly different from any of the classes in the
training set. Hence, we propose a novel test set (Section 3.3) which addresses
all of the above mentioned issues.

##  3 Proposed Approach

In this section, we present the proposed approach for zero-shot video action
recognition. We begin by carefully defining the problem formulation. Next, we
introduce a novel end-to-end trained semantic video transformer (SVT), which
leverages divided space-time attention to learn improved semantically
differentiable visual features (Fig. 2). Then, we present a new data split for
more suitable ZSL experimentation. Finally, we propose to enhance the
available semantic embeddings using class descriptions instead of labels.

###  3.1 Problem Definition

Traditionally, zero-shot action recognition has been defined as a
classification problem, where given a training set of videos $X^{s}$ and
labels $S$ from seen classes
$\\{(x^{s}_{1},s_{1}),\dots,(x^{s}_{N},s_{N})\\}$, we aim to accurately
classify a set of videos $X^{u}=\\{x^{u}_{1},\dots,x^{u}_{M}\\}$ from
previously unseen classes $U=\\{u_{1},\dots,u_{M}\\}$, where $N$ and $M$ are
the number of training and testing videos respectively. To satisfy the ZSL
premise, there should be no overlap between the seen and unseen classes
$(S\cap U=\varnothing)$.

A broad generalization capability is needed to succeed in the ZSL action
recognition problem. The challenge compared to the regular action recognition
task lies in the fact that no direct mapping from the input videos to the
output unseen class labels can be learned during training. Typically, semantic
embeddings are used to bridge the input videos and the output unseen class
labels, which consist of words. The idea behind this mainstream ZSL approach
is to learn a semantic embedding model $f(x)$ for the input videos and choose
the class that is semantically most similar.

Fig. 2: The proposed spatiotemporal transformer architecture, SVT, for video
semantic embedding. Fig. 3: Visualization of the overlapping classes between
UCF-101 and Kinetics-600/700. The Kinetics classes in green are considered as
overlapping by the metric given by Eq. (3) and removed in [9]. The Kinetics
classes in red are given by Word2Vec as the semantically nearest classes to
the corresponding UCF-101 class, but not removed in [9] according to the
criterion in Eq. (3). For several cases, the actual closest Kinetics classes,
shown in blue, are missed by Word2Vec. They are almost identical to the
corresponding UCF-101 classes, and thus violate the ZSL idea.

###  3.2 Semantic Video Transformer (SVT)

We consider the inductive ZSL approach for training our model, i.e., during
training the model only has access to the videos and corresponding class
labels from the seen classes. Most existing approaches extract visual features
using pretrained 3D-CNNs. Recently, it was shown in [9] that end-to-end
training a 3D-CNN performs significantly better than directly using a
pretrained 3D-CNN model. However, the inability of 3D-CNNs to capture
spatiotemporal information over a long time span make them unsuitable for
large scale datasets such as Kinetics and ActivityNet. Hence, we propose a
novel end-to-end trained transformer model, called _Semantic Video Transformer
(SVT)_. The overall structure of SVT is shown in Fig. 2, which is based on the
recently proposed TimeSformer [11] architecture. Specifically, we leverage the
self-attention blocks from the Vision transformer (ViT) [41] model and space-
time attention from the TimeSformer model. However, contrary to the existing
video transformers, we train our model to learn visual-semantic
representations.

Input: Even with a computationally efficient architecture, end-to-end training
on the entire video is impractical due to GPU limitations. Following the
existing video classification approaches, we first sample a clip $y$ of $F$
frames and size $H\times W$ from the input video $x$. The standard NLP
transformer architecture [42] requires a 1D sequence of token embeddings as an
input. Hence, as shown in Fig. 2, we first breakdown the entire video clip $y$
into a sequence of 2D patches, $e_{t}^{p}\in\mathbb{R}^{3\times P^{2}}$, where
$p=1,\dots,N$ represents the spatial locations (i.e., patch index),
$t=1,\dots,F$ denotes the temporal index, $3$ is the number of color channels,
and $P\times P$ is the patch size. Next, we flatten each patch $e^{p}_{t}$
into $v_{t}^{p}\in\mathbb{R}^{3P^{2}}$ and linearly map it into a score vector
using a trainable linear projection $E\in\mathbb{R}^{q\times 3P^{2}}$:

$z^{p}_{t}(0)=Ev_{t}^{p}+\mu^{p}_{t},$ (1)

where $\mu^{p}_{t}\in\mathbb{R}^{q}$ is a latent vector learned to encode the
spatiotemporal position of each $(p,t)$ pair. Following the NLP transformer
BERT [43] we add a latent vector $z^{0}_{0}(0)\in\mathbb{R}^{q}$ for an
additional fictional patch which will be learned to represent the score vector
for the entire video by interacting with all patches through time and space
self-attention. $\\{z^{0}_{0}(0),z^{p}_{t}(0)\\}_{p,t}$ is the input to the
transformer model.

Overall Structure: The transformer model consists of $L$ sequential encoding
blocks, each of which includes $A$ parallel self-attention heads. The first
encoding block processes the input $\\{z^{0}_{0}(0),z^{p}_{t}(0)\\}$ in
parallel in its self-attention heads, as explained next, and outputs
$\\{z^{0}_{0}(1),z^{p}_{t}(1)\\}$ to the second encoding block for each patch
$(p,t)$. Similarly, each block $l$ gets $\\{z^{0}_{0}(l-1),z^{p}_{t}(l-1)\\}$
and outputs $\\{z^{0}_{0}(l),z^{p}_{t}(l)\\}$. Finally, the output of the last
block $z^{0}_{0}(L)$ is used to obtain the semantic embedding for the video.
The role of $z^{0}_{0}(L)$ is to learn an effective summary score for the
entire video, as an alternative to the straightforward approach of simply
averaging $z^{p}_{t}(L)$ over all patches and frames. $z^{0}_{0}(L)$ is passed
through an MLP with three hidden layers and ReLU activation function to obtain
the semantic embedding $f(x)$ for video $x$.

Training and Inference: The whole transformer model is trained end-to-end by
minimizing the loss function

$C=\|f(x^{s}_{i})-\phi(s_{i})\|^{2},$ (2)

where $\phi(s_{i})$ is the semantic embedding of the class label/description
$s_{i}$ for the training video $x_{i}^{s}$ from a Sent2Vec model [44]. After
the model is trained, for ZSL inference, the semantic embedding of the test
video $f(x^{u}_{i})$ is found using the transformer model, and the class
closest to the video in the semantic space is chosen,

$\arg\min_{j}D_{\cos}(f(x^{u}_{i}),\phi(u_{j})),$

where $D_{\cos}$ is the cosine distance and $\phi(u_{j})$ is the Sent2Vec
semantic embedding of the class label/description $u_{j}$.

Class Descriptions: Due to the unavailability of class descriptions or
attributes, existing methods in the literature alternatively use the class
labels and Word2Vec to extract the semantic embeddings $\phi(u_{j})$. However,
we argue that such an approach would severely limit the performance of a
model. Specifically, several class labels in all the datasets are not
sufficiently distinctive and could refer to various different activities. For
example, _diving_ in UCF-101 could either refer to _scuba diving_ , _cliff
diving_ , _sky diving_ or _springboard diving_. Furthermore, using Word2Vec on
multi-word labels is not an efficient technique since averaging the semantic
embedding over several words might lose the context. This can be also seen in
Fig. 3, where the semantic embedding of several multi-word class labels is
mapped incorrectly. To circumvent these issues, we first manually annotate all
the training and testing datasets with one line class descriptions. We then
use a Sent2Vec [44] model to better capture the semantic information with
respect to the context of the class description sentence. The annotated class
descriptions are available in GitHub 222https://github.com/Secure-and-
Intelligent-Systems-Lab/SemanticVideoTransformer. In Section 4.4, we evaluate
the contribution of class descriptions by comparing the performance of the
proposed SVT method with class labels and descriptions in an ablation study.

###  3.3 Fair Zero-Shot Action Recognition

While several recent works have shown promising results on the benchmark
datasets, there are still several shortcomings. In contrast to earlier works,
recent approaches [9, 8, 40, 45] use large-scale external datasets such as the
Kinetics 400/600/700 dataset to pretrain the visual feature extractors.
However, as shown in Fig. 1, there is a significant overlap between the seen
and unseen classes, $S\cap U\neq\varnothing$, which clearly violates the ZSL
paradigm. Recently, Brattoli et al. [9] proposed a novel training protocol
which involves removing classes from the training set if

$\min_{s_{i}\in S,u_{j}\in U}D_{\cos}(\phi(s_{i}),\phi(u_{j}))<\tau,$ (3)

where $\tau$ is set as $0.05$. However, in Fig. 3, we show that even after
applying such a constraint, there are several overlapping classes which are
not removed. This can be partly attributed to semantic mismatch due to domain
shift, where two very similar classes are called by slightly different names.
For example, _blowing out candles_ and _blowing candles_ refer to the same
class and yet the cosine distance between them in the semantic space is much
greater than $\tau$. Moreover, removing overlapping classes from the training
set is a cumbersome approach since it requires retraining computationally
expensive models for every individual test set. On the contrary, it would be
far easier to remove overlapping classes from the test set as the class
information is already available in the training data.

Furthermore, it is also worth considering whether a given test class is
completely irrelevant with respect to all the seen classes. Intuitively, even
humans cannot comprehend a new activity if it involves interactions and
objects that are never seen before. Hence, we also propose removing classes
which are significantly different from the activities present in the training
dataset. While keeping such classes does not necessarily violate the ZSL
premise, it introduces a source of randomness since no reasonable algorithm
can be expected to recognize such activities.

In the current literature, most approaches randomly split a single dataset and
evaluate performance on it over several trials. Such an evaluation setup is
not practical in a real-world scenario since most of the videos would have
disjoint sources. Hence, to ensure a realistic ZSL setting, we extend the
setup discussed in [9], which suggests using independent datasets for training
and testing. However, [9] does not fully account for overlapping classes
between the train and test sets, as seen in Fig. 3. We propose a new ZSL
experimentation setup for action recognition, called _Fair ZSL_. For each
dataset, completely removing classes that either overlap or completely
irrelevant with respect to the train set (Kinetics 600/700) leaves us with a
very small number of test classes per test set (as shown for UCF-101 in Fig.
3). Hence, we propose pooling the valid test classes from all benchmark
datasets to form a novel test set. In the proposed Fair ZSL setup, there are
30 unique classes333The detailed list of all the classes in the proposed split
is available at https://github.com/Secure-and-Intelligent-Systems-
Lab/SemanticVideoTransformer. from the UCF-101, HMDB-51, and ActivityNet
datasets, as shown in Table 1. Each class was carefully handpicked such that
it does not violate the ZSL premise. Additionally, the proposed test set is
also more robust since it evaluates how well a single model accounts for
domain shifts in addition to the ZSL performance.

Benchmark Dataset | Selected Classes
---|---
UCF-101 | 8
HMDB-51 | 3
ActivityNet-101 | 19
Table 1: Number of classes from each benchmark dataset in the proposed test
set for Fair ZSL.

###  3.4 Implementation Details

The proposed SVT model is built upon the PySlowFast [46] package. For training
the model, the shorter side of the input video is first resized to 256 pixels
and then randomly cropped to form a $224\times 224$ ($H\times W$) video
snippet. The patch size is chosen as $16\times 16$, resulting in $N=196$
patches in a frame. The size of the score vectors ($z^{p}_{t}(l)$) at each
encoding block for each patch is $q=768$, and the number of self-attention
heads is $A=12$. The size of the learned semantic embedding $f(x)$ and the
Sent2Vec embedding for class descriptions is $600$. We train two versions of
the proposed transformer, SVT-8 and SVT-96 for $F=8$ and $F=96$ frames in the
input video snippet, respectively. In all our experiments, we use the SVT-96
model unless explicitly stated otherwise. We train the models on 4 NVIDIA A40
GPUs with a batch size of 16 for SVT-8 and 4 for SVT-96. The loss function is
minimized via synchronized SGD with a learning rate of 0.002. To extract
semantic embeddings, we use the Sent2Vec algorithm proposed in [44].

###  3.5 Computational Efficiency

Thanks to the scalability of the proposed SVT model, we are able to vary the
length of the input video snippet (i.e., number of frames $F$), which also
leads to an increase in the number of input tokens. In Table 3, we see a
significant increase in the performance when the number of input frames are
increased from 8 to 96. Increasing the number of video frames is intuitive
since it allows a model to better capture the spatiotemporal activities that
span several frames. However, due to the current GPU limitations, we are
unable to further increase the input length. On the other hand, even after
increasing our model complexity to accommodate 96 input frames, our model is
still more computationally efficient as compared to the I3D model with 8 input
frames, which requires 10.8 TFLOPS for inference, in contrast to the proposed
SVT-8 model, which only requires 0.79 TFLOPS, and SVT-96, which requires 7.57
TFLOPS.

##  4 Experiments

###  4.1 Datasets

Most of the recent works evaluate their performance on three publicly
available benchmark datasets, namely the UCF-101, HMDB-51 and Olympics
dataset. The UCF-101 dataset consists of 13,320 videos from 101 classes,
primarily focusing on five types of actions. The HMDB-51 dataset consists of
6767 videos from 51 classes based on daily human actions. The Olympics dataset
consists of 16 categories, related to an Olympic sport. Recently, Brattoli et
al. [9] evaluated the performance of their approach on the ActivityNet dataset
by extracting labelled frames from every video. As compared to the other
benchmark datasets, ActivityNet is considerably more comprehensive, consisting
of 27,801 videos from 200 classes related to daily activities. The
Kinetics-700 dataset is the largest dataset available for training video
action recognition models, with more than 500K videos in 700 categories
sourced from YouTube. Since several classes from Kinetics-700 were not
available or had files corrupted, we use the Kinetics-600 dataset in our
experimental setup. Due to its small size and significant overlap with
Kinetics, we do not consider the Olympics dataset in our evaluations.

###  4.2 Experimental Setup

To analyze the performance of our proposed SVT model and provide a fair
comparison with benchmark algorithms, we first follow the training and
evaluation protocols used in the existing papers. The existing training
protocols can be broadly classified into two categories, open-ended and
restrictive. Most existing works are based on the open-ended formulation,
whereas a few recent approaches [9, 39, 40] use the restrictive one.

Open-Ended: In this setup, a model is first trained on a large-scale visual
dataset such as the Kinetics dataset without removing any classes, and then
evaluated on a smaller application specific dataset such as UCF or HMDB. A few
approaches also further fine-tune their models on the smaller dataset by
randomly splitting it into train and test sets; however doing so does not
necessarily improve performance. Moreover, as also suggested in [9], a model
ideally should have separate video sources for training and testing to
evaluate its generalization capability.

Restrictive: In the restrictive training approach proposed in [9], we remove
all classes from Kinetics-600 whose distance to any class in UCF $\cup$ HMDB
is smaller than $\tau$ when testing on UCF or HMDB, which results in a subset
of Kinetics with 564 classes. For testing on ActivityNet, an even more
restrictive approach is proposed which involves removing all classes whose
distance to any class in ActivityNet $\cup$ UCF $\cup$ HMDB is smaller than
$\tau$. This setting leads to an even smaller subset of Kinetics with 505
classes.

For the open-ended formulation, we train our model on the entire Kinetics-600
dataset and choose to forgo fine-tuning on UCF or HMDB since transformer based
models require significant amount of data to learn meaningful representations.
To make our evaluation comparable to the existing approaches, we first
randomly split the test dataset in half and evaluate our proposed approach on
it over 10 trials. For the restrictive approach, in addition to the random
split, we also evaluate our model on the entire UCF and HMDB datasets (Table
3) since it allows for a more robust evaluation due to lack of randomness.

Fair ZSL: Since neither of the existing training protocols provide the true
essence of ZSL, we propose a new setup where a model is trained on the entire
Kinetics-600/700 dataset and evaluated on the proposed test set discussed in
Section 3.3. For evaluation, we compare our approach in Table 4 with the
recently proposed E2E model [9] and our implementation of the CLASTER model
[8]. Recently, [47] proposed an elaborate rehearsal approach which, in
addition to a 3D-CNN, explicitly uses an object classifier trained on ImageNet
to learn objects detected in videos. Since their approach and problem setup is
significantly different from the existing approaches, we do not compare our
approach with them.

###  4.3 Results

In the model name “SVT-X(Y) + Z”, X refers to the number of frames used in the
analysis of each video, Y denotes the number of the classes from Kinetics-600
used in training, and Z represents the data type used for class semantic
embedding (CD for class descriptions and CL for class labels). The impacts of
frame number and class descriptions are analyzed in Section 4.4.

We first compare the proposed method with the state-of-the art approaches
under the dominant random split setup in Table 2. However, [48] have presented
their results using a different split (e.g., 78-23 for UCF-101), so we do not
include their results in Table 2. The test datasets are randomly split into
half using the seed 10, as in [9]. It is seen that the proposed spatiotemporal
transformer based ZSL approach consistently outperforms all other state-of-
the-art approaches under all settings. On the UCF-101 dataset, we notice a
significant improvement of 21.9% and 5.2% over the next best results, for the
OE and R protocols, respectively. On the HMDB-51 dataset, the improvements of
the proposed method are 3.4% and 1.8% for the OE and R protocols,
respectively. None of the existing approaches present their results on the
ActivityNet dataset for the OE protocol, hence we could not compare our result
on it. However, for the R protocol, we outperform the E2E approach by a wide
margin of 9.2%.

Method | Protocol | UCF | HMDB | ActivityNet
---|---|---|---|---
DataAug [23] | OE | 18.3 | 19.7 | -
InfDem [45] | OE | 17.8 | 21.3 | -
Bidirectional [28] | OE | 21.4 | 18.9 | -
TARN [19] | OE | 19 | 19.5 | -
Action2Vec [18] | OE | 22.1 | 23.5 | -
OD [10] | OE | 26.9 | 30.2 | -
CLASTER [8] | OE | 46.4 | 36.8 | -
DASZL [49] | OE | 48.9 | - | -
GGM [10] | OE | 20.3 | 20.7 | -
SVT-96(600) + CD (Ours) | OE | 68.3 | 40.2 | 44.8
E2E (605classes) | R | 44.1 | 29.8 | 26.6
E2E (664classes) | R | 48 | 32.7 | -
PS-ZSAR (662 classes) [50] | R | 49.2 | 33.8 | -
SVT-96(505) + CD (Ours) | R | 45.6 | 31.3 | 35.8
SVT-96(564) + CD (Ours) | R | 53.2 | 34.5 | -

Table 2: Comparison with the state-of-the-art methods on standard benchmark
datasets using the open-ended (OE) and Restrictive (R) protocols. All the
methods are evaluated by randomly splitting the dataset in half and averaging
the results over 10 trials.

A disjoint training and testing split allows us to evaluate our model on the
entire UCF-101, HMDB-51 and ActivityNet datasets without any random split. As
shown in Table 3, the performances drop compared to the random split case in
Table 2. However, we still considerably outperform the E2E based approach
under the R protocol while using the class descriptions and class labels.
Since the rest of the existing state-of-the-art approaches use some part of
UCF-101 or HMDB-51 for fine tuning their models, we could not compare our
results with them in this setup.

Finally, we also consider the Fair ZSL protocol proposed in Section 4.2. In
Table 4, we compare our model with the E2E [9] method and our implementation
of the CLASTER [8] method. We again notice a significant improvement of 5.9%
over E2E, which shows the robustness and generalization capability of the
proposed method. It should be noted that we use the E2E model trained on 664
classes for comparison since the model trained on all 700 classes is not
available. However, since we train on Kinetics-600, which has even less number
of classes, we believe it is a fair comparison.

Method | Embedding Type | Protocol | UCF | HMDB | ActivityNet
---|---|---|---|---|---
E2E (605classes) | CL | R | 35.3 | 24.8 | 20.0
E2E (664classes) | CL | R | 37.6 | 26.9 | -
SVT-96(505) | CD | R | 36.1 | 25.7 | 26.3
SVT-96(564) | CD | R | 40.2 | 30.5 | -
SVT-8(564) | CD | R | 37.6 | 27.9 | -
SVT-96(505) | CL | R | 33.9 | 25.2 | 24.7
SVT-96(564) | CL | R | 38.3 | 27.6 | -

Table 3: Comparison with the E2E [9] approach under the Restrictive (R) protocol using the entire datasets for testing without any random split. E2E and SVT are trained on Kinetics-700 and Kinetics-600, respectively. Method | Protocol | FZSL Split
---|---|---
CLASTER | FZSL | 24.3
E2E (664 classes) | FZSL | 30.8
SVT-96(600) + CD | FZSL | 36.7
Table 4: Comparison with E2E [9] and our implementation of CLASTER [8] using
the proposed Fair ZSL protocol. The models are tested on the test set proposed
in Section 3.3 (Table 1).

###  4.4 Ablation Study

In this section, we analyze the contributions of different components of the
proposed approach by performing empirical studies.

Fig. 4: t-SNE visualization of the I3D (left) and SVT (right) features
extracted from the UCF-101 dataset. Each point represents a video and various
classes are represented with different colors. We see that the features
learned by the proposed SVT model (right) are semantically more separable than
the I3D features (left). Best viewed in color.

Impact of the Spatiotemporal Transformer: Here, we study how well the proposed
SVT model is able to learn the spatiotemporal visual features. For comparison,
we consider the a state-of-the-art 3D-CNN model called I3D [16], which has
been a popular choice for video action recognition, and has been used by
several existing approaches [10, 8, 51] for zero-shot action recognition. For
SVT, we extract the output of the last block $z_{0}^{0}(L)$, which serves as a
visual embedding for the entire video. To extract I3D features, we follow the
approach employed by [10, 8, 51] and average the output from the _Mixed 5c_
layer across the temporal dimension followed by pooling by four in the spatial
dimension and finally flattening it to a vector of size 4096. We use both RGB
and flow features and concatenate them to form a vector of size 8192. In Fig.
4, we present the t-SNE visualization for the I3D and SVT features on UCF-101,
where each point represents a video in the UCF dataset. It is clearly seen
that, as compared to the I3D features, SVT learns more semantically separable
features with more well-defined clusters. We also quantitatively compare them
in Table 5 by using various statistical metrics. The average silhouette score
measures how tightly grouped all the points in the cluster are. The adjusted
rand index computes a similarity measure between the clusterings and the
ground truth. The homogeneity score checks if a cluster contains samples
belonging to a single class. Finally, we also apply a simple $k$-NN
classification algorithm on the extracted features to compute the accuracy for
traditional video classification. In all metrics, the SVT features provide a
better separation.

Method | I3D | SVT |
---|---|---|---
Average Silhouette | 0.119 | 0.196 |
Adjusted Rand Index | 0.80 | 0.88 |
Homogeneity Score | 0.92 | 0.96 |
Classification | 0.93 | 0.96 |
Table 5: Comparison between the SVT and I3D features in terms of clustering
and classification performance using different metrics. The classification
accuracy is for the $k$-NN classifier.

Impact of Class Descriptions: In Table 6, we compare the performance of our
approach when it uses class labels (CL) and class descriptions (CD) for
semantic embedding. Specifically, we train the SVT model on the semantic
embeddings extracted using the Sent2Vec model on class labels/descriptions,
and test on the entire UCF, HMDB and ActivityNet datasets using the
restrictive (R) setting. We see that there is a noticeable improvement in the
performance when class descriptions are used, which ascertains our conjecture
that descriptive semantic embeddings are crucial in improving the learned
class representations. Moreover, our performance using class label embeddings
is still better than existing state-of-the-art, which shows the efficacy of
our proposed SVT model. While generating class descriptions require some human
involvement, we argue that the cost is still significantly less as compared to
video level annotations required for supervised learning. Furthermore,
manually defined attributes are already being used for zero-shot image
classification [52] since class labels are not always discriminative enough to
distinguish the context.

Method | Embedding Type | UCF | HMDB | ActivityNet
---|---|---|---|---
SVT-96(505) | CL | 33.9 | 25.2 | 24.7
SVT-96(564) | CL | 38.3 | 27.6 | -
SVT-96(505) | CD | 36.1 | 25.7 | 26.3
SVT-96(564) | CD | 40.2 | 30.5 | -
Table 6: Comparison between the performance of SVT while using class labels
(CL) and class descriptions (CD) under the restrictive protocol.

Number of Input Frames: Finally, we analyze the impact of the number of input
frames on the SVT performance. As shown in Table 3, there is a 2.6% gain in
both UCF and HMDB datasets when 96 frames are used instead of 8.

##  5 Conclusion

In this work, we introduce a spatiotemporal transformer architecture for zero-
shot video action recognition, called SVT. Moreover, we highlight several
areas where the existing approaches either violate the zero-shot learning
(ZSL) premise or are unable to perform well due to the limited capabilities of
3D-CNN based visual extractors. We propose a new evaluation protocol, Fair
ZSL, that strictly adheres to the ZSL premise. Through several experiments, we
show that the proposed approach consistently outperforms the existing
approaches under various experimental setups, including the existing ones in
the literature and the proposed Fair ZSL setup.

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Strong coupling at the $\tau$-mass scale from an improved vector isovector
spectral function

Diogo Boito,1,2$\star$ Maarten Golterman,3,4 Kim Maltman,5,6 Santiago Peris,4

Marcus V. Rodrigues,1 and Wilder Schaaf.3,7

1 Instituto de Física de São Carlos, Universidade de São Paulo, CP 369,
13560-970,

São Carlos, SP, Brazil

2 University of Vienna, Faculty of Physics, Boltzmanngasse 5, A-1090 Wien,
Austria

3 Department of Physics and Astronomy, San Francisco State University,

San Francisco, CA 94132, USA

4 Department of Physics and IFAE-BIST, Universitat Autònoma de Barcelona

E-08193 Bellaterra, Barcelona, Spain

5 Department of Mathematics and Statistics, York University, Toronto, ON
Canada M3J 1P3

6 CSSM, University of Adelaide, Adelaide, SA 5005 Australia

7 Department of Physics, University of Washington, Seattle, WA 98195

*<EMAIL_ADDRESS>
16th International Workshop on Tau Lepton Physics (TAU2021),
September 27 – October 1, 2021

## Abstract

We perform a precise extraction of the QCD coupling at the $\tau$-mass scale,
$\alpha_{s}(m_{\tau})$, from a new vector isovector spectral function which
combines ALEPH and OPAL distributions for the dominant channels,
$\tau\to\pi\pi^{0}\nu_{\tau}$, $\tau\to 3\pi\pi^{0}\nu_{\tau}$ and $\tau\to\pi
3\pi^{0}\nu_{\tau}$, with estimates of sub-leading contributions obtained from
electroproduction cross-sections using CVC, as well as BaBar results for
$\tau\to K^{-}K^{0}\nu_{\tau}$. The fully inclusive spectral function thus
obtained is entirely based on experimental data, without Monte Carlo input.
From this new data set, we obtain $\alpha_{s}(m_{\tau})=0.3077\pm 0.0075$,
which corresponds to $\alpha_{s}(m_{Z})=0.1171\pm 0.0010$. This analysis can
be improved on the experimental side with new measurements of the dominant
$\pi\pi^{0}$, $\pi 3\pi^{0}$, and $3\pi\pi^{0}$ $\tau$ decay modes.

###### Contents

  1. 1 Introduction
  2. 2 Data combination and the new spectral function
  3. 3 Results for the strong coupling
  4. 4 Conclusion

##  1 Introduction

The extraction of the strong coupling, $\alpha_{s}$, from weighted integrals
of inclusive hadronic tau decay data ranks among the most precise
determinations of this coupling based on experiment. Since 2008, the
$\alpha_{s}^{4}$ term in the perturbative QCD description of these inclusive
quantities is exactly known [1, 2], which has renewed the interest in this
process. Here, we report on the work of Ref. [3] where an updated, more
precise, extraction of $\alpha_{s}$ has been performed, using a new vector
isovector spectral function.

In order to perform reliable, self-consistent, QCD analyzes of tau decay data,
the hadronic $\tau$ decay width is not sufficient — it is crucial to have
information about the spectral functions. Any realistic analysis includes
several integrated spectral function moments, which allows for the extraction
of $\alpha_{s}$ and non-perturbative contributions in a combined fit.
Obtaining the necessary experimental data is intrinsically difficult since, in
principle, the theoretical description in QCD requires fully inclusive
spectral functions, with measurements of all subleading hadronic decay
channels. The most complete spectral functions have been produced and made
publicly available by the ALEPH [4, 5] and OPAL [6] collaborations.111The
spectral functions measured by CLEO [7] were never made public, to the best of
our knowledge. Later, a number of updates of the ALEPH spectral functions have
been produced, reflecting new experimental information on the different
branching ratios and improving the separation in the vector (V) and axial-
vector (A) components, among other changes. The most recent version was
published in 2013 [8], correcting the correlation matrices that were
underestimated in an earlier iteration [9]. The OPAL data, in turn, were
updated to reflect more recent branching ratio measurements in Ref. [10].

Extractions of $\alpha_{s}$ from the ALEPH data have smaller experimental
uncertainties [11], but extractions from the OPAL data are consistent within
errors [10]. For the lack of a combined analysis, a weighted average between
values obtained from the two data sets, assumed to be uncorrelated, has been
quoted as the final result for $\alpha_{s}$ in Ref. [11]. This procedure,
however, is not ideal since it does not combine the experimental information
used to obtain the integrated moments in an optimal way. As is well
established in the dispersive studies of the hadronic vacuum polarization
contribution to the muon $g-2$, for example, one should first combine the
experimental results following a rigorous statistical procedure and then
employ this new combined data set to perform discretized integrals [12, 13].
This procedure also quantitatively tests the compatibility of the data sets
and reveals potential local discrepancies, should they exist. This is one of
the questions addressed in the work of Ref.[3].

Another reason for revisiting the $\alpha_{s}$ extraction from $\tau$ decay
data is related to the subleading hadronic channels, to which we often refer
as “residual modes". The ALEPH and OPAL spectral functions include detailed
measurements of the decay spectrum of dominant channels. These are the
$\pi\pi^{0}$, $\pi 2\pi^{0}$, $3\pi$, $\pi 3\pi^{0}$, and $3\pi\pi^{0}$ tau
decay channels. However, these are not sufficient to obtain fully inclusive
spectral functions and other subleading channels, such as
$\tau\to\omega(\to{\rm non}\,3\pi)\pi^{-}\nu_{\tau}$, $\tau\to
6\pi\nu_{\tau}$, and $\tau\to K^{-}K^{0}\nu_{\tau}$, were included by ALEPH
and OPAL using Monte Carlo simulations [6, 5]. These residual modes, although
subleading, do play an important role in QCD studies: removing them makes the
spectral function not inclusive enough which can lead to non-physical results.
Recently, however, many new experimental results have appeared which allows
for a significant improvement in the description of the residual modes in the
vector channel. First, BaBar has measured the spectrum of $\tau\to
K^{-}K^{0}\nu_{\tau}$ [14]. Second, thanks to a wealth of new experimental
measurements of many exclusive-channel cross sections in $e^{+}e^{-}\to{\rm
hadrons}$ [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29] by the
CMD-3, BaBar, and SND collaborations, conserved vector current (CVC) relations
allow essentially all remaining vector isovector mode contributions to be
obtained using exclusive mode electroproduction cross-section data (with
negligible isospin-breaking corrections, as we discuss below). In Ref.[3],
this recent experimental information was used to build a new vector isovector
spectral function solely based on data, without the use of Monte Carlo
simulations for any of the residual modes.

This new vector isovector spectral function has two main features. First, it
relies on a combination of ALEPH and OPAL measurements of the dominant vector
tau decay channels, namely $\tau\to\pi^{-}\pi^{0}\nu_{\tau}$, $\tau\to
2\pi^{-}\pi^{+}\pi^{0}\nu_{\tau}$ and $\tau\to\pi^{-}3\pi^{0}\nu_{\tau}$,
which had never been done in this context. Second, as explained above, all
subleading modes are obtained from experiment. This new spectral function has
smaller errors and is more inclusive than the ALEPH and OPAL results, since it
contains more residual modes, covering 99.95% of the total vector branching
ratio (BF). With this new data set, we have performed an updated analysis of
$\alpha_{s}$ at the $\tau$-mass scale, following the framework of Refs. [30,
10, 11]. Below we discuss in more detail the work of Ref. [3] and summarize
its main results.

##  2 Data combination and the new spectral function

The first ingredient in the new vector isovector spectral function is the
combined ALEPH and OPAL data for the dominant $\tau$ decay channels:
$\tau\to\pi^{-}\pi^{0}\nu_{\tau}$, $\tau\to 2\pi^{-}\pi^{+}\pi^{0}\nu_{\tau}$,
and $\tau\to\pi^{-}3\pi^{0}\nu_{\tau}$. Ideally, one would like to combine
unit normalized spectral distributions for each channel individually, multiply
them by up-to-date values of the respective BFs, and subsequently add them up
to build the dominant channel contributions. In practice, however, this
procedure is not possible because of the presence of 100% correlations in
parts of the covariance matrices of the $3\pi\pi^{0}$ and $\pi 3\pi^{0}$
spectral distributions, which prevents the corresponding covariance matrices
from being inverted (a crucial step in the data combination, as we will
describe below). Without additional information on how these covariances are
obtained and what they represent in terms of statistical and systematic
errors, the solution to this problem is to first add all dominant channels,
already normalized by the respective BFs, for each experiment separately and
later combine the sums of the three dominant channels in a single data set.
Since in the sum of the dominant channels the well-behaved $\pi\pi^{0}$ mode
gives the largest contribution, this procedure leads to a well-defined
covariance matrix and we therefore employ this route in our data combination.

For the data combination algorithm, we follow closely Refs. [31, 12]. In
summary, we first define $N_{\rm cl}$ clusters which roughly play the role of
bins for the combined data. Each cluster contains a number of data points. The
representative value of $s$ for each cluster, where $s$ is the invariant mass
of the final-state hadrons, is given by the weighted average of the $s$ values
of the data points contained in that cluster (the weights in this average are
the inverses of the data-point uncertainty squared). The description of the
combined data set is done by a piece-wise linear function given by the linear
interpolation between the values of the spectral function in each cluster,
that we denote $\rho^{(m)}$. Extrapolation is used for the few data points
whose energy is smaller (larger) than the smallest (largest) cluster energy.

The $N_{\rm cl}$ values of the spectral function at each cluster,
$\rho^{(m)}$, are then obtained by minimizing the following $\chi^{2}$
function

$\chi^{2}=\sum_{i,j=1}^{N_{\rm
data}}[\,d_{i}-R(s_{i};\rho)\,]\,(C^{-1})_{ij}\,[d_{j}-R(s_{j};\rho)],$ (1)

where $N_{\rm data}$ is the total number of experimental data points, $d_{i}$
is $i$-th experimental value of the spectral function, and $R(s_{i};\rho)$,
which plays the role of “theory" in the minimization, is the value of the
combined spectral function at the energy $s_{i}$, obtained as the linear
interpolation between $\rho^{(m)}$ at each cluster center (and using
extrapolation at the extremes). Finally, $C$ is the total covariance matrix of
the $N_{\rm data}$ experimental data points.

With the procedure outlined above one obtains the result displayed in the left
panel of Fig. 1. In this figure, we show the total ALEPH, OPAL, and combined
data for $\tau\to\pi^{-}\pi^{0}\nu_{\tau}$, $\tau\to
2\pi^{-}\pi^{+}\pi^{0}\nu_{\tau}$, and $\tau\to\pi^{-}3\pi^{0}\nu_{\tau}$.
These dominant channels are responsible for $98.0\%$ of the total inclusive V
channel BF. The $\chi^{2}/dof$ of the combination, 1.144, is very reasonable
and we have also checked that for each cluster, locally, the $\chi^{2}$ is, in
general, very good. There are a handful of clusters with not-so-good
$\chi^{2}$ but they are so few that local error inflation makes no difference
in our final results.

Figure 1: (Left panel) ALEPH (orange), OPAL (green), and combined data (blue)
for the sum of the dominant vector isovector channels:
$\tau\to\pi^{-}\pi^{0}\nu_{\tau}$, $\tau\to 2\pi^{-}\pi^{+}\pi^{0}\nu_{\tau}$,
and $\tau\to\pi^{-}3\pi^{0}\nu_{\tau}$. (Right panel) Residual mode
contributions, channel by channel.

The next step is the inclusion of the residual modes. We include, in our new
spectral function, the following decay modes: (i)
$\pi^{-}\omega(\rightarrow{\rm\mbox{non-}3}\pi)$, $K^{-}K^{0}$,
$\eta\pi^{-}\pi^{0}$, $K\bar{K}\pi$, $3\pi^{-}2\pi^{+}\pi^{0}$, and
$2\pi^{-}\pi^{+}3\pi^{0}$, which were also included by both the ALEPH and OPAL
collaborations, (ii) $(3\pi)^{-}\omega(\rightarrow{\rm\mbox{non-}3}\pi)$ and
$K\bar{K}\pi\pi$, included only by ALEPH, and (iii) the small additional
contributions from $\pi^{-}5\pi^{0}$ and $\eta\omega\pi+\eta 4\pi$, not
included in either of the ALEPH or OPAL analyses. With the exception of the
contributions from $\tau\to K^{-}K^{0}\nu_{\tau}$, which were measured by
BaBar [14], all the other modes are obtained from the respective $e^{+}e^{-}$
cross sections with CVC [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29], applying the necessary corrections for vacuum polarization effects.
The individual channels are normalized to the respective BFs described in the
2019 HFLAV compilation [32] (correlations among these values are taken into
account). Since the residual modes contribute at energies outside the region
of narrow resonances, isospin-breaking corrections, which are expected to be
of the order of a percent, are very small given the experimental errors and
the smallness of the residual mode contributions to the inclusive spectral
function. A much more detailed description of each of the individual residual
modes can be found in Sec. III.C of[3]. The contribution of the residual modes
is shown, channel by channel, in the right-hand panel of Fig. 1.

Figure 2: Total vector isovector spectral function (green) obtained from the
addition of the dominant modes (blue) and the residual modes (yellow).

The final result for the vector isovector spectral function is then obtained
by the addition of the dominant and residual channels, and is shown in Fig. 2
(green data points). This new vector isovector spectral function, entirely
based on experimental data, covers 99.95% of the total inclusive vector
isovector BF and has smaller uncertainties than its predecessors [8, 6].

##  3 Results for the strong coupling

Here we describe the main results of the QCD finite-energy-sum-rule (FESR)
analysis of Ref. [3]. We follow the analysis strategy developed in [30] and
previously employed in the analysis of ALEPH and OPAL data [10, 11]. For
further details of the analysis we refer to [3] and to the previous
publications [30, 10, 11].

As is well established [33], the QCD analyses employ FESRs where, on the
experimental side, one has integrated spectral function moments, obtained from
a discretized integral over the data set, and on the theory side one has an
integral of the relevant current-current correlation function, $\Pi(s)$, along
a closed contour in the complex plane, such that the value of the coupling is
kept in the perturbative domain:

$\frac{1}{s_{0}}\int_{0}^{s_{0}}ds\,w(s)\,\rho(s)=-\frac{1}{2\pi
i\,s_{0}}\oint_{|z|=s_{0}}dz\,w(z)\,\Pi(z),$ (2)

where $\rho(s)$ is the total vector isovector spectral function of Fig. 2. In
the FESRs any analytical weight function $w(s)$ can be employed and this
freedom is used in order to emphasize or suppress the different theoretical
contributions. The QCD contributions on the theoretical side can be split into
a perturbative component, cast in terms of the perturbative Adler function
(known to $\alpha_{s}^{4}$ [1, 2]), and non-perturbative contributions: the
operator product expansion (OPE) condensates and duality violations (DVs),
which go beyond the usual OPE. The results that we report here are based on
Fixed Order Perturbation Theory (FOPT) (see [34], for example) which can be
more directly compared with determinations of $\alpha_{s}$ from other sources.
We do not quote values for Contour Improved Perturbation Theory [35] (CIPT)
since there are solid indications of a potential inconsistency between this
prescription and the standard OPE corrections [36, 37].

We follow the DV strategy, where one choses weight functions that strongly
suppress higher-order OPE condensates, which avoids completely any problem
related to the truncation of OPE series. (Assumptions about the truncation of
the OPE can be dangerous in this context, and may lead to uncontrolled
systematic effects [38, 39, 40].) The price to pay is an unavoidable
enhancement of DV contributions which must be included in the theoretical
description. We use the parametrization of Refs. [41, 42] which is based on
generally accepted assumptions about the QCD resonance spectrum. We then fit
simultaneously several FESRs, in each case integrating the data from threshold
up to $s_{0}$, with $s_{0}$ varied between $s_{\rm min}$ and its highest
possible value (that of the right-most point in Fig. 2, which is very close to
$m_{\tau}^{2}$). The parameters of the fits are $\alpha_{s}$, the four DV
parameters and, depending on the moment used, the relevant OPE condensates.

Figure 3: $\alpha_{s}(m_{\tau})$ as a function of $s_{\rm min}$ from the fits
described in the text. The horizontal red line and the yellow band give the
final result with uncertainties, Eq.(4). Points in red are included in the
final weighted average (see text).

In Fig. 3 we show results for $\alpha_{s}(m_{\tau})$ from FESRs using $w(x)=1$
obtained from fits with different $s_{\rm min}$, as a function of $s_{\rm
min}$. As one can see, as long as $s_{\rm min}$ is not too low, the results
obtained are very consistent and stable. We have performed other consistency
checks, such as using other weight functions $w(x)$, as well as using
combinations of moments, always with very consistent results. To obtain a
final value of $\alpha_{s}(m_{\tau})$ we perform a weighted average of every
second result within the yellow band of Fig. 3. In this average, the strong
correlation between the different values of $\alpha_{s}(m_{\tau})$ is properly
taken into account. Our final result is then

$\displaystyle\alpha_{s}(m_{\tau})$ $\displaystyle=$ $\displaystyle 0.3077\pm
0.0065_{\rm stat}\pm 0.0038_{\rm pert}$ (3) $\displaystyle=$ $\displaystyle
0.3077\pm 0.0075\qquad\qquad(\overline{{\rm MS}},n_{f}=3\ ,\ \mbox{FOPT})\ $
(4)

where we quote a statistical (stat) and a perturbative (pert) error.222For
details about the error associated with perturbation theory see Sec. IV.C of
[3]. Evolving this result to the $Z$-mass scale with five-loop running [43]
and four-loop matching [44, 45] we find

$\alpha_{s}(m_{Z})=0.1171\pm 0.0010\qquad(\overline{{\rm MS}},n_{f}=5).$ (5)

This result is compared, in Fig. 4, with our previous determinations using the
same strategy with other data sets [10, 11, 46].

A comparison of our results with other analyses found in the literature is not
completely straightforward, since they are based on different data sets and
different analysis strategies. The results from Refs. [8, 47] employ variants
of the so-called “truncated OPE strategy" (tOPE), aimed at suppressing the DV
contributions, but in which several OPE condensates have to be neglected in
the fits. This is a strong assumption and there is evidence that it leads to
an uncontrolled systematics [38, 39, 40]. With this caveat in mind, the
results of [8, 47] are significantly higher than ours (and than the PDG world
average [48]) and read $\alpha_{s}(m_{Z})=0.1199\pm 0.0015$ [8] and
$\alpha_{s}(m_{Z})=0.1197\pm 0.0015$. [47]. A more recent analysis [49], which
employs a variant of the tOPE strategy that has not yet been subjected to the
same level of scrutiny as those of Refs. [10, 11, 8, 47], gives a value of
$\alpha_{s}$ from FOPT essentially identical to the one we quote in Eq. (5).

Figure 4: Comparison of $\alpha_{s}(m_{Z})$ within FOPT from the analyses of
OPAL hadronic tau decay data [10], ALEPH hadronic tau decay data [11], from
$e^{+}e^{-}\to{\rm hadrons}$ below charm threshold [46], and from the new
vector isovector spectral function [3] (this work). The yellow band gives the
Particle Data Group world average [48].

##  4 Conclusion

We have recently produced a new vector isovector spectral function based on a
combination of the ALEPH and OPAL spectra for the dominant $\pi\pi^{0}$, $\pi
3\pi^{0}$, and $3\pi\pi^{0}$ $\tau$ decay channels, supplemented by the
addition of contributions from all numerically significant residual decay
modes. The residual modes contributions are obtained entirely from
experimental data, using the recent BaBAR spectrum for $\tau\to
K^{-}K^{0}\nu_{\tau}$ [14] and a wealth of recent CVC-related cross-section
results from $e^{+}e^{-}\to{\rm hadrons}$ obtained by CMD-3, SND and BaBar
[15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. This new
spectral function is more inclusive than its predecessors, since it contains
more residual modes, is solely based on experimental data (with no Monte Carlo
simulated channels), and has smaller errors. Isospin-breaking corrections for
the CVC related channels are expected to be very small given the size of
experimental errors and the smallness of the contributions from the residual
modes. It is important to observe that an analogously improved spectral
function cannot be built for the axial channel, since the $e^{+}e^{-}\to{\rm
hadrons}$ cross-sections are purely vector at this energy scale.

We analyze FESRs based on this new data set in order to extract the value of
the strong coupling at the tau mass scale, $\alpha_{s}(m_{\tau})$. Our result,
$\alpha_{s}(m_{\tau})=0.3077\pm 0.0075$, is very competitive and corresponds
to $\alpha_{s}(m_{Z})=0.1171\pm 0.0010$. The results of the present analysis
could be improved with new measurements for the spectral distributions of the
dominant vector $\tau$ decay channels.

## Acknowledgements

We would like to thank the organizers of the Tau2021 workshop for keeping
alive this conference series during difficult times. We thank Alex Keshavarzi
for valuable discussions.

### Funding information

DB was supported by the São Paulo Research Foundation (FAPESP) Grant No.
2015/20689-9, by CNPq Grant No. 309847/2018-4, and by Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code
001. MG and WS are supported by the U.S. Department of Energy, Office of
Science, Office of High Energy Physics, under Award No. DE-SC0013682. MVR is
supported by FAPESP grant No. 2019/16957-9. KM is supported by a grant from
the Natural Sciences and Engineering Research Council of Canada. SP is
supported by CICYTFEDER-FPA2017-86989-P and by Grant No. 2017 SGR 1069. IFAE
is partially funded by the CERCA program of the Generalitat de Catalunya.

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