Andrea Castellani, Sebastian Schmitt, Barbara Hammer

11institutetext: Bielefeld University,
11email<EMAIL_ADDRESS>
22institutetext: Honda Research Institute Europe,
22email<EMAIL_ADDRESS>
# Estimating the Electrical Power Output of Industrial Devices with End-to-End
Time-Series Classification in the Presence of Label Noise

Andrea Castellani (✉) 11 0000-0003-0476-5978 Sebastian Schmitt 22
0000-0001-7130-5483 Barbara Hammer 11 0000-0002-0935-5591

###### Abstract

In complex industrial settings, it is common practice to monitor the operation
of machines in order to detect undesired states, adjust maintenance schedules,
optimize system performance or collect usage statistics of individual
machines. In this work, we focus on estimating the power output of a Combined
Heat and Power (CHP) machine of a medium-sized company facility by analyzing
the total facility power consumption. We formulate the problem as a time-
series classification problem, where the class label represents the CHP power
output. As the facility is fully instrumented and sensor measurements from the
CHP are available, we generate the training labels in an automated fashion
from the CHP sensor readings. However, sensor failures result in mislabeled
training data samples which are hard to detect and remove from the dataset.
Therefore, we propose a novel multi-task deep learning approach that jointly
trains a classifier and an autoencoder with a shared embedding representation.
The proposed approach targets to gradually correct the mislabelled data
samples during training in a self-supervised fashion, without any prior
assumption on the amount of label noise. We benchmark our approach on several
time-series classification datasets and find it to be comparable and sometimes
better than state-of-the-art methods. On the real-world use-case of predicting
the CHP power output, we thoroughly evaluate the architectural design choices
and show that the final architecture considerably increases the robustness of
the learning process and consistently beats other recent state-of-the-art
algorithms in the presence of unstructured as well as structured label noise.

###### Keywords:

Time-series Deep Learning Label noise Self-supervision Non-Intrusive Load
Monitoring Time-Series Classification

##  1 Introduction

It is common to monitor multiple machines in complex industrial settings for
many diverse reasons, such as to detect undesired operational states, adjust
maintenance schedules or optimize system performance. In situations where the
installation of many sensors for individual devices is not feasible due to
cost or technical reasons, Non-Intrusive Load Monitoring (NILM)[18] is able to
identify the utilization of individual machines based on the analysis of
cumulative electrical load profiles. The problem of the generation of labelled
training data sets is a cornerstone of data-driven approaches to NILM. In this
context, industry relies mostly on manually annotated data [11] and less often
the training labels can be automatically generated from the sensors [13],
which is often unreliable because of sensor failure and human
misinterpretation. Data cleaning techniques are often hard to implement [42]
which unavoidably leads to the presence of wrongly annotated instances in
automatically generated datasets i.e. label noise [12]. Many machine learning
methods, and in particular deep neural networks, are able to overfit training
data with noisy labels [48], thus it is challenging to apply data-driven
approaches successfully in complex industrial settings.

We consider a medium-sized company facility and target the problem of
estimating the electrical power output of a Combined Heat and Power (CHP)
machine by only analyzing the facility electrical power consumption. The
electrical power output of the CHP is sufficient to supply a substantial share
of the total electricity demand of the facility. Therefore, knowing the CHP’s
electrical power output is very helpful for distributing the electrical energy
in the facility, for example when scheduling the charging of electrical
vehicles (EVs) or reducing total peak-load [28]. We propose a data-driven deep
learning-based approach to this problem, which is modelled as time-series
classification challenge in the presence of label noise, where the class label
of each time series represents the estimated CHP power output level. As the
facility is fully instrumented, and sensor measurements from the CHP are
available, we generate the training labels in an automated fashion from the
CHP sensor readings. However, these sensors fail from time to time which
resulting wrong labels. To tackle this problem, we propose a novel multi-task
deep learning approach named Self-Re-Labeling with Embedding Analysis (SREA),
which targets the detection and re-labeling of wrongly labeled instances in a
self-supervised training fashion.

In the following, after a formal introduction to SREA, we empirically validate
it with several benchmarks data sets for time-series classification and
compare against state-of-the-art (SotA) algorithms. In order to evaluate the
performance, we create a training data set from clean sensor readings and we
corrupt it by introducing three types of artificial noise in a controlled
fashion. After that, we apply the proposed method to a real-world use-case
including real sensor failures and show that those are properly detected and
corrected by our algorithm. Finally, we perform an extensive ablation study to
investigate the sensitivity of the SREA to its hyper-parameters.

##  2 Related work

Deep learning based techniques constitute a promising approach to solve the
NILM problem [30, 34]. Since the requirement of large annotated data sets is
very challenging, the problem is often addressed as a semi-supervised learning
task [20, 4, 46], where only a part of the data is correctly labeled, and the
other is left without any label. Here we take a different stance to the
problem of energy disaggregation: we assume that labels for all data are
given, but not all labels are correct, as is often the case in complex sensor
networks [12].

Learning noisy labels is an important topic in machine learning research [38,
16]. Several approaches are based on the fact that deep neural networks tend
to first learn clean data statistics during early stages of training [36, 48,
2]. Methods can be based on a loss function which is robust to label noise
[50, 40], or they can introduce an explicit [5] or implicit [35]
regularization term. Another adaptation of the loss function is achieved by
using different bootstrapping methods based on the predicted class labels [16,
33, 41]. Some other common approaches are based on labeling samples with
smaller loss as clean ones [1, 15, 23, 27], and fitting a two-component
mixture model on the loss distribution, or using cross-validation techniques
[7] in order to separate clean and mislabeled samples. Several existing
methods [39, 29, 15] require knowledge about the percentage of noisy samples,
or estimate this quantity based on a given. The suitability of these
approaches is unclear for real-world applications. Since our proposed method,
SREA, targets the correction of mislabeled training samples based on their
embedding representation, we do not rely on specific assumptions on the amount
of label noise in advance. Our modeling approach is based on the observation
that self-supervision has proven to provide an effective representation for
downstream tasks without requiring labels [17, 19], this leading to an
improvement of performance in the main supervised task [22].

Most applications of models which deal with label noise come from the domain
of image data, where noise can be induced by e.g. crowd-working annotations
[25, 16]. Some approaches consider label noise in other domains such as human
activity detection [3], sound event classification [10], or malware detection
[14]. An attempt to analyze the effect of noisy data on applications for real-
world data sets is made in [44], but the authors do not compare to the SotA.
Up to the authors knowledge, we are the first to report a detailed evaluation
of time-series classification in the presence of label noise, which is a
crucial data characteristics in the domain of NILM.

##  3 CHP electrical power output estimation

Figure 1: Sensory data from the CHP and total electrical power. Upper: normal
operation. Lower: an example of $P_{CHP}$ sensor malfunction is highlighted in
yellow.

The CHP is a complex industrial machinery which burns natural gas in order to
produce heat and electrical power. It is controlled by an algorithm where only
some aspects are known, so its behavior is mostly unclear and not well
predictable. It is known that important control signals are the ambient
outside temperature $T_{amb}$, the internal water temperature $T_{water}$, the
generated electrical power output $P_{CHP}$, and the total electricity demand
of the facility $P_{tot}$. Fig. 1 shows examples of recorded data from the CHP
as well as $P_{tot}$ of the facility. $T_{amb}$ has a known strong influence
as the CHP is off in the summer period and more or less continuously on in
winter and cold periods. In the transition seasons (spring and fall), the CHP
sometimes turns on (night of 25th Sept.), sometimes just heats up its internal
water (nights of 20st, 21st, and 23rd Sept.), or exhibits a fast switching
behavior (e.g. 20st, and 27th Oct.). Even though the CHP usually turns on for
a couple of hours (e.g. 25th Sept.) at rare instances it just turns on for a
very short time (e.g. 28th Oct.).

Due to the complicated operational pattern, it is already hard to make a
detector for the CHP operational state even with full access to the
measurement data. Additionally, the sensors measuring the CHP output power are
prone to failure which can be observed in the yellow highlighted area in the
bottom side of Fig. 1. During that 10-hour period, the CHP did produce
electrical power, even though the sensor reading does not indicate this. The
total electrical power drawn from the grid, $P_{tot}$, provides a much more
stable measurement signal. The signature of the CHP is clearly visible in the
total power signal and we propose to estimate $P_{CHP}$ from this signal, but
many variables also affects the total load, e.g. PV system, changing
workloads, etc.

We focus on estimating the power output but where the estimate power value
should only be accurate within a certain range. Thus, the problem is
formulated as time-series classification instead of regression, where each
class represents a certain range of output values. The class labels are
calculated directly from the $P_{CHP}$ sensor measurement as the mean power
output of a fixed-length sliding window. Due to frequent sensor
malfunctioning, as displayed in Fig. 1, the resulting classification problem
is subject to label noise.

##  4 Self-Re-Labeling with Embedding Analysis (SREA)

####  4.0.1 Architecture and loss function

Figure 2: SREA processing architecture (a) and the dynamics of the parameters
$\alpha$ and $w$ during the training epochs (b).

In this work, column vectors are denoted in bold (e.g. $\bm{x}$). As described
in Sec. 3, we treat the challenge to model time series data as a
classification problem to predict averaged characteristics of the process
using windowing techniques. Hence, we deal with a supervised k-class
classification problem setting with a dataset of $n$ training examples
$\mathcal{D}=\\{(\bm{x}_{i},\bm{y}_{i}),i=1,...,n\\}$ with
$\bm{y}_{i}\in\\{0,1\\}^{k}$ being the one-hot encoding label for sample
$\bm{x}_{i}$. Thereby, label noise is present, i.e. we expect that
$\bm{y}_{i}$ is wrong for a substantial (and possibly skewed) amount of
instances in the training set.

The overall processing architecture of the proposed approach is shown in Fig.
2(a). The autoencoder ($f_{ae}$), represented by the encoder ($e$) and the
decoder, provides a strong surrogate supervisory signal for feature learning
[22] that is not affected by the label noise. Two additional components, a
classifier network ($f_{c}$) and a constrained clustering module ($f_{cc}$)
are introduced, which independently propose class labels as output. Each of
the three processing pipelines share the same embedding representation,
created by the encoder, and each output module is associated with one separate
contribution to the total loss function.

For the autoencoder, a typical reconstruction loss is utilized:

$\mathcal{L}_{ae}=\frac{1}{n}\sum_{i=1}^{n}\left(\bm{\hat{x}}_{i}-\bm{x}_{i}\right)^{2},$
(1)

where $\bm{\hat{x}}_{i}$ is the output of the autoencoder given the input
$\bm{x}_{i}$.

Cross entropy is used as loss function for the classification network output,

$\mathcal{L}_{c}=-\frac{1}{n}\sum_{i=1}^{n}\bm{y}_{i}^{T}\cdot\log(\bm{p}^{c}_{i}),$
(2)

where $\bm{p}^{c}_{i}$ are the k-class softmax probabilities produced by the
model for the training sample $i$, i.e.
$\bm{p}^{c}_{i}=\text{softmax}(f_{c}(\bm{x}_{i}))$.

For the constraint clustering loss, we first initialize the cluster center
$\bm{C}\in\mathbb{R}^{d\times k}$ in the d-dimensional embedding space, with
the $k$-means clustering of the training samples. Then, inspired by the good
results achieved in [47], we constrain the embedding space to have small
intra-class and large inter-class distances, by iteratively adapting $\bm{C}$.
The resulting clustering loss is given by:

$\mathcal{L}_{cc}=\frac{1}{n}\sum_{i=1}^{n}\Bigg{[}{\underbrace{\left\|\bm{e}(\bm{x}_{i})-\bm{C}_{\bm{y}_{i}}\right\|^{2}_{2}}_{\text{intra-
class}}}+{\underbrace{\log\sum_{j=1}^{k}\exp\left(-\left\|\bm{e}(\bm{x}_{i})-\bm{C}_{j}\right\|_{2}\right)}_{\text{inter-
class}}}\Bigg{]}+\ell_{reg}.$ (3)

The entropy regularization $\ell_{reg}=-\sum_{i}^{k}\min_{i\neq
j}\log\left\|\bm{C}_{i}-\bm{C}_{j}\right\|_{2}$ is aimed to create well
separated embedding for different classes [37].

The final total loss function is given by the sum of those contributions,

$\mathcal{L}=\mathcal{L}_{ae}+\alpha\left(\mathcal{L}_{c}+\mathcal{L}_{cc}+\mathcal{L}_{\rho}\right),$
(4)

where we introduced the dynamic parameter $0\leq\alpha\leq 1$ which changes
during the training (explained below). We also add the regularization loss
$\mathcal{L}_{\rho}$ to prevent the assignment of all labels to a single
class,
$\mathcal{L}_{\rho}=\sum_{j=1}^{k}\bm{h}_{j}\cdot\log\frac{\bm{h}_{j}}{\bm{p}^{\rho}_{j}}$,
where $\bm{h}_{j}$ denotes the prior probability distribution for class j,
which assume to be uniformly distributed to $1/k$. The term
$\bm{p}^{\rho}_{j}$ is the mean softmax probability of the model for class j
across all samples in dataset which we approximate using mini-batches as done
in previous works [1].

####  4.0.2 Re-Labeling strategy

We do not compute the total loss function in Eq. (4) with the given (noisy)
label $\bm{y}_{i}$. But, we estimate the true label for each data sample
$\bm{y^{*}}_{i}$ by taking the weighted average of the given training label
$\bm{y}_{i}$, and the pseudo-labels proposed by the classifier
$\bm{y}^{c}_{i}$ and the constraint clustering $\bm{y}^{cc}_{i}$.

In order to increase robustness of the labels proposed by the classifier
($\bm{y}^{c}_{i}$), we take for each data sample the exponentially averaged
probabilities during the last five training epochs, with the weighting factor
$\tau_{t}\sim e^{\frac{t-5}{2}}$:

$\bm{y}^{c}_{i}=\sum_{\mathrm{last\>5\>epochs\>}t}\tau_{t}\;[\bm{p}_{i}^{c}]_{t}.$
(5)

The label from constraint clustering ($\bm{y}^{cc}_{i}$) is determined by the
distances of the samples to the cluster centers in the embedding space:

$\bm{y}^{cc}_{i}=\text{softmin}_{j}\big{(}\|\bm{e}(\bm{x}_{i})-\bm{C}_{j}\|_{2}\big{)}$
(6)

Then, the corrected label $\bm{y^{*}}_{i}$ (in one hot-encoding) is produced
by selecting the class corresponding to the maximum entry:

$\bm{y}^{*}_{i}=\text{argmax}\big{[}(1-w)\,\bm{y}_{i}+w\,(\bm{y}^{c}_{i}+\bm{y}^{cc}_{i})\big{]}$
(7)

where the dynamic weighting factor $0\leq w\leq 1$ is function of the training
epoch $t$ and to be discussed below when explaining the training dynamics.

####  4.0.3 Training Dynamics

A key aspect of our proposed approach is to dynamically change the loss
function, as well as the label correction mechanism, during the training. This
is achieved by changing the parameters $\alpha$ (loss function) and $w$ (label
correction mechanism) as depicted in Fig. 2(b). The training dynamics is
completely defined by the three hyper-parameters $\lambda_{init}$,
$\Delta_{start}$ and $\Delta_{end}$.

Initially, we start with $\alpha=0$ and $w=0$, and only train the autoencoder
from epoch $t=0$ to epoch $t=\lambda_{init}$. At the training epoch
$t=\lambda_{init}$, $\alpha$ is ramped up linearly until it reaches $\alpha=1$
in training epoch $t=\lambda_{start}=\lambda_{init}+\Delta_{start}$. The
purpose of this first warm-up period is an unsupervised initialization of the
embedding space with slowly turning on the supervision of the given labels.
The dominant structure of the clean labels is learned, as neural networks tend
to learn the true labels, rather than overfit to the noisy ones, at early
training stages [48, 2]. Then, we also increase $w$ linearly from zero to one,
between epochs $t=\lambda_{start}$ to
$t=\lambda_{start}+\Delta_{end}=\lambda_{end}$, thereby turning on the label
correction mechanism (re-labeling). After training epoch $t=\lambda_{end}$
until the rest of the training, we keep $\alpha=1$ and $w=1$ which means we
are fully self-supervised where the given training labels do not enter
directly anymore (fine-tuning). We summarize the SREA and display the pseudo-
code in Algorithm 1.

Require: Data $\\{(\bm{x}_{i},\bm{y}_{i})\\}_{n}$, autoencoder $f_{ae}$,
classifier $f_{c}$, constraint clustering $f_{cc}$, hyper-parameters:
$\lambda_{init}$, $\Delta_{start}$, $\Delta_{end}$.

Init hyper-parameter ramp-up functions $w_{t}$ and $\alpha_{t}$

// see Fig. 2(b)

1 for  _training epoch $t=0$ to $t_{end}$ _ do

2 Fetch mini-batch data $\\{(\bm{x}_{i},\bm{y}_{i})\\}_{b}$ at current epoch
$t$

3 for  _$i=1$ to $b$_ do

4 if  _$t==\lambda_{init}$_ then

$f_{cc}\leftarrow$ $k$-means$(\bm{e}(\bm{x}_{i}))$

// Initialize constraint clustering

5

6 end if

$\bm{\hat{x}}_{i}=f_{ae}(\bm{x}_{i})$

// Auto-encoder forward pass

$\bm{y}^{c}_{i}\leftarrow$ Eq.(5)

// Classifier forward pass

$\bm{y}^{cc}_{i}\leftarrow$ Eq.(6)

// Constraint clustering output

Adjust $w\leftarrow w_{t}$, $\alpha\leftarrow\alpha_{t}$

// Label correction and loss parameters

$\bm{y^{*}}_{i}\leftarrow$ Eq.(7)

// Re-labeling

$\mathcal{L}\leftarrow$ Eq.(4)

// Evaluate loss function

7 Update $f_{ae},f_{c},f_{cc}$ by SGD on $\mathcal{L}$

8 end for

9

10 end for

Algorithm 1 SREA: Self-Re-Labeling with Embedding Analysis.

##  5 Experimental setup

####  5.0.1 Label Noise

True labels are corrupted by a label transition matrix T [38], where $T_{ij}$
is the probability of the label $i$ being flipped into label $j$. For all the
experiments, we corrupt the labels with symmetric (unstructured) and
asymmetric (structured) noise with noise ratio $\epsilon\in[0,1]$. For
symmetric noise, a true label is randomly assigned to other labels with equal
probability, i.e. $T_{ii}=1-\epsilon$ and $T_{ij}=\frac{\epsilon}{k-1}$
($i\neq j$), with $k$ the number of classes. For asymmetric noise, a true
label is mislabelled by shifting it by one, i.e. $T_{ii}=1-\epsilon$ and
$T_{(j+1\mathrm{mod}k)j}=\epsilon$ ($i\neq j$). For the estimation of the CHP
power, we also analyze another kind of structured noise which we call flip
noise, where a true label is only flipped to zero, i.e. $T_{ii}=1-\epsilon$
and $T_{i0}=\epsilon$. This mimics sensor failures, where a broken sensor
produces a constant output regardless of the real value. Note that learning
with structured noise is much harder than with unstructured noise [12].

####  5.0.2 Network architecture

Since SREA is model agnostic, we use CNNs in the experiments, as these are
currently the SotA deep learning network topology for time-series
classification [43, 9]. The encoder and decoder have a symmetric structure
with 4 convolutional blocks. Each block is composed by a 1D-conv layer
followed by batch normalization [21], a ReLU activation and a dropout layer
with probability 0.2. The dimension of the shared embedding space is 32. For
the classifier we use a fully connected network with 128 hidden units and
#classes outputs. We use the Adam optimizer [26] with an initial learning rate
of 0.01 for 100 epochs. Such high value for the initial learning rate helps to
avoid overfitting of noisy data in the early stages of training [48, 2]. We
halve the learning rate every $20\%$ of training (20 epochs). In the
experiments, we assume to not have access to any clean data, thus it is not
possible to use a validation set, and the models are trained without early
stopping. The SREA hyper-parameters $\lambda_{init}=0$, $\Delta_{start}=25$
and $\Delta_{end}=30$ are used if not specified otherwise. Further
implementation details are reported in the supplementary material [6],
including references to the availability of code and data sets.

####  5.0.3 Comparative methods

In order to make a fair comparison, we use the same neural network topology
throughout all the experiments. A baseline method, which does not take in
accout any label noise correction criteria, is a CNN classifier [43] trained
with cross-entropy loss function of Eq. (2), which we refer to as CE. We
compare to MixUp [49], which is a data augmentation technique that exhibits
strong robustness to label noise. In MixUp-BMM [1] a two-component beta-
mixture model is fitted to the loss distribution and training with
bootstrapping loss is implemented. SIGUA [15] implements stochastic gradient
ascent on likely mislabeled data, thereby trying to reduce the effect of noisy
labels. Finally, in Co-teaching [39] two networks are simultaneously trained
which inform each other about which training examples to keep. The algorithms
SIGUA and Co-teaching assume that the noise level $\epsilon$ is known. In our
experiments, we use the true value of $\epsilon$ for those approaches, in
order to create an upper-bound of their performance. All the hyper-parameters
of the investigated algorithm are set to their default and recommended values.

####  5.0.4 Implementation details

For the problem of estimating CHP power output, the raw data consists of 78
days of measurement with a sampling rate of 1 sample/minute. As preprocessing,
we do a re-sampling to 6 samples/hour. The CHP should have a minimal on-time
of one hour, in order to avoid too rapid switching which would damage the
machine. However, during normal operation, the CHP is controlled in a way that
on- and off-time periods are around 4 to 8 hours. Due to these time-scales, we
are interested in the power output on a scale of 6 hours, which means we use a
sliding window with a size of 6 hours (36 samples) and a stride of 10 minutes
(1 sample). Therefore, the preprocessing of the three input variables
$P_{tot}$, $T_{water}$, and $T_{amb}$ lead to $\mathbb{R}^{(36\times
3)}$-dimensional data samples. For generating the labels, we use 5 power
output levels, linearly spaced from 0 to $P_{CHP,max}$, and correspondingly to
a five-dimensional one-hot encoded label vector $\bm{y}_{i}\in\mathbb{R}^{5}$.
For every dataset investigated, we normalize the dataset to have zero mean and
unit standard deviation, and we randomly divide the total available data in
train-set and test-set with a ratio 80:20.

####  5.0.5 Evaluation measures

To evaluate the performance, we report the averaged $\mathcal{F}_{1}$-score on
the test-set, where the well-known $F_{1}$-scores are calculated for each
class separately and then averaged via arithmetic mean,
$\mathcal{F}_{1}=\frac{1}{k}\sum_{j=1}^{k}F_{1,j}$. This formulation of the
$F_{1}$-score results in a larger penalization when the model do not perform
well in the minority class, in cases with class imbalance. Other metrics, such
as the accuracy, show qualitatively similar results and are therefore not
reported in this manuscript. In order to get performance statistics of the
methods, all the experiments have been repeated 10 times with different random
initialization. The non-parametric statistical Mann-Whitney U Test [32] is
used to compare the SREA against the SotA algorithms.

##  6 Results and discussion

####  6.0.1 Benchmarks Datasets

We evaluate the proposed SREA on publicly available time-series classification
datasets from UCR repository [8]. We randomly choose 10 datasets with
different size, length, number of classes and dimensions in order to try to
avoid bias in the data. A summary of the datasets is given in Table 1.

Table 1: UCR Single-variate and Multi-variate dataset description. Dataset | Size | Length | #classes | #dim
---|---|---|---|---
ArrowHead† | 211 | 251 | 3 | 1
CBF | 930 | 128 | 3 | 1
Epilepsy | 275 | 206 | 4 | 3
FaceFour† | 112 | 350 | 4 | 1
MelbourneP. | 3633 | 24 | 10 | 1
NATOPS | 360 | 51 | 6 | 24
OSULeaf† | 442 | 427 | 6 | 1
Plane | 210 | 144 | 7 | 1
Symbols† | 1020 | 398 | 6 | 1
Trace† | 200 | 275 | 4 | 1
$\dagger$ results reported in the supplementary material [6].

We show a representative selection of all comparisons of the results in Table
2. Without any label noise, CE is expected to provide very good results. We
observe that our SREA gives similar or better $\mathcal{F}_{1}$ scores than
the CE method on 9 out of 10 datasets without label noise. Considering all
algorithms and datasets, with symmetric noise we achieve statistically
significantly better scores in 62, similar scores in 105, and worse scores in
33 experiments out of a total of 200 experiments111Number of experiments: 10
datasets $\times$ 4 noise levels $\times$ 5 algorithms = 200. Each experiment
consists of 10 independent runs. . For the more challenging case of asymmetric
noise, the SREA-results are 86 times significantly better, 97 times equal, and
17 times worse than SotA algorithms.

Table 2: $\mathcal{F}_{1}$ test scores on UCR datasets. The best results per noise level are underlined. In parenthesis the results of a Mann–Whitney U test with $\alpha=0.05$ of SREA against the other approaches: SREA $\mathcal{F}_{1}$ is significantly higher ($+$), lower ($-$) or not significant ($\approx$). Dataset | Noise | % | CE | MixUp | M-BMM | SIGUA | Co-teach | SREA
---|---|---|---|---|---|---|---|---
CBF | - | 0 |  1.000 $(+)$ |  0.970 $(+)$ |  0.886 $(+)$ |  1.000 $(+)$ |  0.997 $(+)$ | 1.000
Symm | 15 |  0.943 $(+)$ |  0.923 $(+)$ |  0.941 $(+)$ |  0.976 $(+)$ |  0.923 $(+)$ | 1.000
30 |  0.780 $(+)$ |  0.799 $(+)$ |  0.932 $(+)$ |  0.923 $(+)$ |  0.833 $(+)$ | 0.998
Asymm | 10 |  0.973 $(+)$ |  0.956 $(+)$ |  0.920 $(+)$ |  0.989 $(+)$ |  0.963 $(+)$ | 1.000
20 |  0.905 $(+)$ |  0.897 $(+)$ |  0.949 $(+)$ |  0.980 $(+)$ |  0.900 $(+)$ | 1.000
Epilepsy | - | 0 |  0.974 $(\approx)$ |  0.955 $(+)$ |  0.926 $(+)$ |  0.978 $(\approx)$ |  0.971 $(+)$ | 0.973
Symm | 15 |  0.890 $(\approx)$ |  0.913 $(\approx)$ |  0.899 $(\approx)$ |  0.884 $(\approx)$ |  0.861 $(\approx)$ | 0.861
30 |  0.784 $(-)$ |  0.823 $(-)$ |  0.805 $(-)$ |  0.741 $(\approx)$ |  0.744 $(\approx)$ | 0.708
Asymm | 10 |  0.919 $(\approx)$ |  0.930 $(\approx)$ |  0.847 $(\approx)$ |  0.905 $(\approx)$ |  0.919 $(\approx)$ | 0.888
20 |  0.861 $(\approx)$ |  0.894 $(-)$ |  0.891 $(-)$ |  0.826 $(\approx)$ |  0.863 $(\approx)$ | 0.825
Melbourne | - | 0 |  0.923 $(\approx)$ |  0.879 $(+)$ |  0.773 $(+)$ |  0.918 $(\approx)$ |  0.913 $(\approx)$ | 0.911
Symm | 15 |  0.869 $(+)$ |  0.870 $(+)$ |  0.856 $(+)$ |  0.883 $(\approx)$ |  0.886 $(\approx)$ | 0.883
30 |  0.826 $(+)$ |  0.858 $(\approx)$ |  0.870 $(\approx)$ |  0.855 $(\approx)$ |  0.876 $(-)$ | 0.862
Asymm | 10 |  0.898 $(+)$ |  0.877 $(+)$ |  0.860 $(+)$ |  0.899 $(+)$ |  0.897 $(+)$ | 0.911
20 |  0.865 $(+)$ |  0.861 $(+)$ |  0.851 $(+)$ |  0.858 $(+)$ |  0.893 $(\approx)$ | 0.903
NATOPS | - | 0 |  0.858 $(\approx)$ |  0.801 $(\approx)$ |  0.711 $(+)$ |  0.848 $(\approx)$ |  0.835 $(\approx)$ | 0.866
Symm | 15 |  0.779 $(\approx)$ |  0.718 $(+)$ |  0.702 $(+)$ |  0.754 $(\approx)$ |  0.761 $(\approx)$ | 0.796
30 |  0.587 $(\approx)$ |  0.580 $(+)$ |  0.602 $(+)$ |  0.593 $(+)$ |  0.673 $(+)$ | 0.670
Asymm | 10 |  0.798 $(+)$ |  0.822 $(\approx)$ |  0.756 $(+)$ |  0.764 $(+)$ |  0.790 $(+)$ | 0.829
20 |  0.703 $(\approx)$ |  0.763 $(\approx)$ |  0.762 $(\approx)$ |  0.698 $(\approx)$ |  0.733 $(\approx)$ | 0.762
Plane | - | 0 |  0.995 $(\approx)$ |  0.962 $(+)$ |  0.577 $(+)$ |  0.981 $(+)$ |  0.990 $(\approx)$ | 0.998
Symm | 15 |  0.930 $(+)$ |  0.953 $(+)$ |  0.873 $(+)$ |  0.971 $(\approx)$ |  0.981 $(\approx)$ | 0.983
30 |  0.887 $(+)$ |  0.902 $(+)$ |  0.943 $(\approx)$ |  0.862 $(+)$ |  0.941 $(\approx)$ | 0.944
Asymm | 10 |  0.981 $(\approx)$ |  0.986 $(\approx)$ |  0.648 $(+)$ |  0.986 $(\approx)$ |  0.990 $(\approx)$ | 0.976
20 |  0.952 $(\approx)$ |  0.923 $(\approx)$ |  0.751 $(+)$ |  0.976 $(\approx)$ |  0.990 $(\approx)$ | 0.966

####  6.0.2 CHP power estimation

Table 3:  $\mathcal{F}_{1}$ test scores of the CHP power estimation. Same notation as Table 2. Noise | % | CE | MixUp | M-BMM | SIGUA | Co-teach | SREA
---|---|---|---|---|---|---|---
- | 0 |  0.980 $(\approx)$ |  0.957 $(+)$ |  0.882 $(+)$ |  0.979 $(\approx)$ |  0.974 $(\approx)$ | 0.979
Symmetric | 15 |  0.931 $(+)$ |  0.934 $(+)$ |  0.903 $(+)$ |  0.954 $(\approx)$ |  0.950 $(\approx)$ | 0.960
30 |  0.856 $(+)$ |  0.910 $(+)$ |  0.897 $(+)$ |  0.912 $(+)$ |  0.920 $(+)$ | 0.938
45 |  0.763 $(+)$ |  0.883 $(+)$ |  0.895 $(+)$ |  0.867 $(+)$ |  0.886 $(+)$ | 0.918
60 |  0.661 $(+)$ |  0.761 $(\approx)$ |  0.692 $(+)$ |  0.817 $(\approx)$ |  0.839 $(\approx)$ | 0.800
Asymmetric | 10 |  0.954 $(\approx)$ |  0.945 $(+)$ |  0.893 $(+)$ |  0.959 $(\approx)$ |  0.964 $(\approx)$ | 0.961
20 |  0.924 $(+)$ |  0.925 $(+)$ |  0.899 $(+)$ |  0.935 $(\approx)$ |  0.938 $(\approx)$ | 0.946
30 |  0.895 $(\approx)$ |  0.909 $(\approx)$ |  0.873 $(+)$ |  0.916 $(\approx)$ |  0.923 $(\approx)$ | 0.919
40 |  0.807 $(-)$ |  0.848 $(-)$ |  0.784 $(-)$ |  0.836 $(-)$ |  0.876 $(-)$ | 0.287
Flip | 10 |  0.970 $(\approx)$ |  0.950 $(+)$ |  0.860 $(+)$ |  0.965 $(+)$ |  0.973 $(\approx)$ | 0.971
20 |  0.942 $(+)$ |  0.942 $(+)$ |  0.860 $(+)$ |  0.945 $(+)$ |  0.962 $(\approx)$ | 0.963
30 |  0.908 $(+)$ |  0.919 $(+)$ |  0.880 $(+)$ |  0.923 $(+)$ |  0.792 $(+)$ | 0.956
40 |  0.868 $(+)$ |  0.791 $(+)$ |  0.696 $(+)$ |  0.779 $(+)$ |  0.623 $(+)$ | 0.945

Figure 3: Confusion matrix of the corrected labels (left) and embedding space
of the train-set (right) of the CHP power estimation, corrupted with 30% flip
noise.

Table 3 shows the results of the estimation of the CHP output power level.
Without labels noise, the proposed approach has comparable performance to CE,
with an average $\mathcal{F}_{1}$-score of 0.979. This implies that we
successfully solve the CHP power estimating problem by analyzing the total
load with an error rate less than 2%. When the training labels are corrupted
with low level of symmetric label noise, $\epsilon\leq 0.4$, SREA consistently
outperforms the other algorithms. With higher level of symmetric noise we
achieve a comparable performance to the other algorithms. Under the presence
of asymmetric label noise, SREA shows a performance comparable to the other
SotA algorithms. Only for a highly unrealistic asymmetric noise level of 40%,
the performance is significantly worse than the SotA. This indicates that,
during the warm-up and relabelling phase, the network is not able to learn the
true labels but also learns the wrong labels induced by the structured noise.
During the fine-tuning phase, the feedback of the wrongly labeled instances is
amplified and the wrong labels a reinforced. For flip noise, which reflects
sensors failures, SREA retains a high performance and outperforms all other
SotA algorithms up to noise levels of 40%. For low noise levels up to 20%,
SREA has similar performance to Co-teaching, but without the need to know the
amount of noise.

In Fig. 3 (left) we show the label confusion matrix (with in-class percentage
in parentheses) of SREA for the resulting corrected labels for the case of 30%
of flip label noise. The corrected labels have 99% and 98% accuracy for the
fully off- (0) and on-state (4), respectively. The intermediate power values’
accuracies are 90% for state 1, 86% for state 2 and 92% for state 3. As an
example, a visualization of the 32 dimensional embedding using the UMAP [31]
dimension reduction technique is shown in Fig. 3 for the cases of 30% flip
noise. The clusters representing the classes are very well separated, and we
can see that the majority of the noisy label samples have been corrected and
assigned to their correct class. Similar plots for other noise types as well
as critical distance plots can be found in the supplementary material [6].

Finally, we run SREA on a different real-world test-set which includes a
sensor failure, and the corresponding noisy label, as shown in Fig. 1. The
method was able to correctly re-label the period of the sensor failure.

###  6.1 Ablation studies

####  6.1.1 Hyper-parameter sensitivity

We investigate the effect of the hyper-parameters of SREA on both, benchmarks
and CHP datasets. The observed trends were similar in all datasets, and
therefore we only report the result for the CHP dataset. The effect of the
three hyper-parameters related to the training dynamics ($\lambda_{init}$,
$\Delta_{start}$, and $\Delta_{end}$) are reported in Fig. 4 for the cases of
unstructured symmetric label noise (results for the other noise types can be
found in the supplementary material [6]). For the variation of
$\Delta_{start}$ and $\Delta_{end}$ there is a clear pattern for every noise
level as the performance increases with the values of the hyper-parameters,
and best performance is achieved by $\Delta_{start}=25$ and $\Delta_{end}=30$.
This shows that both the warm-up and re-labeling periods should be rather long
and last about 55% of the training time, before fully self-supervised
training. The effect of $\lambda_{init}$ is not as clear, but it seems that
either a random initialization of the cluster centers ($\lambda_{init}$ close
to zero) or an extended period of unsupervised training of the autoencoder
($\lambda_{init}$ between 20 and 40) is beneficial.

Figure 4:  SREA sensitivity to hyper-parameters $\Delta_{start}$ (left),
$\Delta_{end}$ (middle), $\lambda_{init}$ (right) for the CHP data and
symmetric noise.

####  6.1.2 Loss function components

We study what effect each of the major loss function components of Eq. 4 has
on the performance of SREA and report the results in Table 4. It can be
observed, that not including the constrained clustering, i.e. using only
$\mathcal{L}_{c}$ or $\mathcal{L}_{c}+\mathcal{L}_{ae}$, gives rather poor
performance for all noise types and levels. This is explicitly observed as the
performance decreases again during training in the self-supervision phase
without $\mathcal{L}_{cc}$ (not shown). This seems understandable as the label
correction method repeatedly bootstraps itself by using only the labels
provided by the classifier, without any anchor to preserve the information
from the training labels. This emphasizes the necessity of constraining the
data in the embedding space during self-supervision.

Table 4: Ablation studies on loss function components of SREA. Noise | % | $\bm{\mathcal{L}_{c}}$ | $\bm{\mathcal{L}_{c}+\mathcal{L}_{ae}}$ | $\bm{\mathcal{L}_{c}+\mathcal{L}_{cc}}$ | $\bm{\mathcal{L}_{c}+\mathcal{L}_{ae}+\mathcal{L}_{cc}}$
---|---|---|---|---|---
- | 0 |  0.472$\scriptstyle\pm$0.060 |  0.504$\scriptstyle\pm$0.012 |  0.974$\scriptstyle\pm$0.003 | 0.980$\scriptstyle\pm$0.003
Symmetric | 15 |  0.388$\scriptstyle\pm$0.027 |  0.429$\scriptstyle\pm$0.023 |  0.943$\scriptstyle\pm$0.004 | 0.957$\scriptstyle\pm$0.007
30 |  0.355$\scriptstyle\pm$0.038 |  0.366$\scriptstyle\pm$0.041 |  0.919$\scriptstyle\pm$0.006 | 0.930$\scriptstyle\pm$0.008
45 |  0.290$\scriptstyle\pm$0.014 |  0.318$\scriptstyle\pm$0.010 |  0.892$\scriptstyle\pm$0.009 | 0.902$\scriptstyle\pm$0.008
Asymmetric | 10 |  0.400$\scriptstyle\pm$0.026 |  0.407$\scriptstyle\pm$0.012 |  0.949$\scriptstyle\pm$0.003 | 0.957$\scriptstyle\pm$0.003
20 |  0.348$\scriptstyle\pm$0.003 |  0.358$\scriptstyle\pm$0.003 |  0.930$\scriptstyle\pm$0.009 | 0.941$\scriptstyle\pm$0.009
30 |  0.342$\scriptstyle\pm$0.002 |  0.349$\scriptstyle\pm$0.004 |  0.901$\scriptstyle\pm$0.007 | 0.922$\scriptstyle\pm$0.010
Flip | 10 |  0.460$\scriptstyle\pm$0.030 |  0.468$\scriptstyle\pm$0.030 |  0.961$\scriptstyle\pm$0.009 | 0.973$\scriptstyle\pm$0.006
20 |  0.415$\scriptstyle\pm$0.028 |  0.424$\scriptstyle\pm$0.037 |  0.951$\scriptstyle\pm$0.008 | 0.962$\scriptstyle\pm$0.005
30 |  0.405$\scriptstyle\pm$0.030 |  0.411$\scriptstyle\pm$0.040 |  0.943$\scriptstyle\pm$0.007 | 0.957$\scriptstyle\pm$0.003

####  6.1.3 Input variables

We investigate the influence of the selection of the input variables on the
estimation of the CHP power level. The results for all possible combination of
the input signals are reported in Fig 5. Unsurprisingly, using only the
ambient temperature gives by far the worst results, while utilizing all
available inputs results in the highest scores. Without the inclusion of the
$P_{tot}$, we still achieve a $\mathcal{F}_{1}$-score above 0.96 with only
using the $T_{water}$ as input signal.

Figure 5: SREA sensitivity to the input variables for CHP data and asymmetric
noise.

##  7 Conclusion and future work

In this work, we presented the problem of estimating the electrical power
output of a Combined Heat and Power (CHP) machine by analyzing the total
electrical power consumption of a medium size company facility. We presented
an approach to estimate the CHP power output by analyzing the total load, the
ambient temperature and the water temperature of the CHP, all of which are
known to be control variables of the CHP. The training dataset for the deep-
learning based approach was automatically derived from sensor measurements of
the CHP power output, and sensor failures create noisy samples in the
generated class labels. The proposed Self-Re-Labeling with Embedding Analysis
(SREA)incorporates an autoencoder, a classifier and a constraint clustering
which all share and operate on a common low-dimensional embedding
representation. During the network training, the loss function and the label
correction mechanism are adjusted in a way that a robust relabeling of noisy
training labels is possible. We compare SREA to five SotA label noise
correction approaches on ten time-series classification benchmarks and observe
mostly comparable or better performance for various noise levels and types. We
also observe superior performance on the CHP use-case for a wide range of
noise levels and all studied noise types. We thoroughly analyzed the
dependence of the proposed methods on the (hyper-)parameters and architecture
choices.

The proposed approach is straight-forward to realize without any
(hyper-)parameter tuning, as there are clear insights on how to set the
parameters and the method is not sensitive to details. It also has the strong
benefit that the amount of label noise need not be known or guessed. We used
CNNs as building blocks of the proposed algorithm, but since SREA is model
agnostic, it is possible to utilize other structures, such as recurrent neural
networks [24] or transformers [45], which would also utilize the time-
structure of the problem. The application of such dynamic models is left for
future work.

Estimating the CHP output as shown in this work will be used in the future in
energy optimization scenarios to arrive at more reliable and robust EV
charging schedules. But, due to the robustness of the proposed method and the
ability to exchange the neural networks with arbitrary other machine learning
modules, we see a high potential for this architecture to be used for label
noise correction in other application domains. We also see a high potential
for an application in anomaly detection scenarios where sensor failures need
to be detected. A thorough evaluation in these application areas is left for
future work.

## References

  * [1] Arazo, E., Ortego, D., Albert, P., O’Connor, N., McGuinness, K.: Unsupervised label noise modeling and loss correction. In: International Conference on Machine Learning. pp. 312–321 (2019)
  * [2] Arpit, D., Jastrzebski, S., Ballas, N., Krueger, D., Bengio, E., Kanwal, M.S., Maharaj, T., Fischer, A., Courville, A., Bengio, Y., et al.: A closer look at memorization in deep networks. In: International Conference on Machine Learning. pp. 233–242 (2017)
  * [3] Atkinson, G., Metsis, V.: Identifying label noise in time-series datasets. In: Adjunct Proceedings of the 2020 ACM International Joint Conference on Pervasive and Ubiquitous Computing and Proceedings of the 2020 ACM International Symposium on Wearable Computers. pp. 238–243 (2020)
  * [4] Barsim, K.S., Yang, B.: Toward a semi-supervised non-intrusive load monitoring system for event-based energy disaggregation. In: 2015 IEEE global conference on signal and information processing (GlobalSIP). pp. 58–62 (2015)
  * [5] Berthelot, D., Carlini, N., Goodfellow, I., Papernot, N., Oliver, A., Raffel, C.: Mixmatch: A holistic approach to semi-supervised learning. arXiv:1905.02249 (2019)
  * [6] Castellani, A., Schmitt, S., Hammer, B.: Supplementary material for: Estimating the electrical power output of industrial devices with end-to-end time-series classification in the presence of label noise. arXiv:2105.00349 (2021)
  * [7] Chen, P., Liao, B.B., Chen, G., Zhang, S.: Understanding and utilizing deep neural networks trained with noisy labels. In: International Conference on Machine Learning. pp. 1062–1070 (2019)
  * [8] Dau, H.A., Keogh, E., Kamgar, K., Yeh, C.C.M., Zhu, Y., Gharghabi, S., Ratanamahatana, C.A., Yanping, Hu, B., Begum, N., Bagnall, A., Mueen, A., Batista, G., Hexagon-ML: The ucr time series classification archive (2018)
  * [9] Fawaz, H.I., Forestier, G., Weber, J., Idoumghar, L., Muller, P.A.: Deep learning for time series classification: a review. Data Mining and Knowledge Discovery 33(4), 917–963 (2019)
  * [10] Fonseca, E., Plakal, M., Ellis, D.P., Font, F., Favory, X., Serra, X.: Learning sound event classifiers from web audio with noisy labels. In: ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). pp. 21–25 (2019)
  * [11] Fredriksson, T., Mattos, D.I., Bosch, J., Olsson, H.H.: Data labeling: An empirical investigation into industrial challenges and mitigation strategies. In: PROFES (2020)
  * [12] Frénay, B., Verleysen, M.: Classification in the presence of label noise: a survey. IEEE transactions on neural networks and learning systems 25(5), 845–869 (2013)
  * [13] Gan, O.P.: Automatic labeling for personalized iot wearable monitoring. IECON 2018 - 44th Annual Conference of the IEEE Industrial Electronics Society pp. 2861–2866 (2018)
  * [14] Gavrilut, D., Ciortuz, L.: Dealing with class noise in large training datasets for malware detection. 2011 13th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing pp. 401–407 (2011)
  * [15] Han, B., Niu, G., Yu, X., Yao, Q., Xu, M., Tsang, I., Sugiyama, M.: Sigua: Forgetting may make learning with noisy labels more robust. In: International Conference on Machine Learning. pp. 4006–4016 (2020)
  * [16] Han, B., Yao, Q., Liu, T., Niu, G., Tsang, I.W., Kwok, J.T., Sugiyama, M.: A survey of label-noise representation learning: Past, present and future. arXiv:2011.04406 (2020)
  * [17] Hendrycks, D., Mazeika, M., Kadavath, S., Song, D.: Using self-supervised learning can improve model robustness and uncertainty. arXiv:1906.12340 (2019)
  * [18] Holmegaard, E., Kjærgaard, M.B.: Nilm in an industrial setting: A load characterization and algorithm evaluation. 2016 IEEE SMARTCOMP pp. 1–8 (2016)
  * [19] Huang, L., Zhang, C., Zhang, H.: Self-adaptive training: Bridging the supervised and self-supervised learning. arXiv:2101.08732 (2021)
  * [20] Humala, B., Nambi, A.S.U., Prasad, V.R.: Universalnilm: A semi-supervised energy disaggregation framework using general appliance models. In: Proceedings of the Ninth International Conference on Future Energy Systems. pp. 223–229 (2018)
  * [21] Ioffe, S., Szegedy, C.: Batch normalization: Accelerating deep network training by reducing internal covariate shift. In: International conference on machine learning. pp. 448–456 (2015)
  * [22] Jawed, S., Grabocka, J., Schmidt-Thieme, L.: Self-supervised learning for semi-supervised time series classification. In: Pacific-Asia Conference on Knowledge Discovery and Data Mining. pp. 499–511 (2020)
  * [23] Jiang, L., Zhou, Z., Leung, T., Li, L.J., Fei-Fei, L.: Mentornet: Learning data-driven curriculum for very deep neural networks on corrupted labels. In: International Conference on Machine Learning. pp. 2304–2313 (2018)
  * [24] Karim, F., Majumdar, S., Darabi, H., Chen, S.: Lstm fully convolutional networks for time series classification. IEEE Access 6, 1662–1669 (2018)
  * [25] Karimi, D., Dou, H., Warfield, S., Gholipour, A.: Deep learning with noisy labels: exploring techniques and remedies in medical image analysis. Medical image analysis 65, 101759 (2020)
  * [26] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv:1412.6980 (2014)
  * [27] Li, J., Socher, R., Hoi, S.C.: Dividemix: Learning with noisy labels as semi-supervised learning. arXiv:2002.07394 (2020)
  * [28] Limmer, S.: Evaluation of optimization-based ev charging scheduling with load limit in a realistic scenario. Energies 12(24) (2019)
  * [29] Mandal, D., Bharadwaj, S., Biswas, S.: A novel self-supervised re-labeling approach for training with noisy labels. In: Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision. pp. 1381–1390 (2020)
  * [30] Massidda, L., Marrocu, M., Manca, S.: Non-intrusive load disaggregation by convolutional neural network and multilabel classification. Applied Sciences 10(4), 1454 (2020)
  * [31] McInnes, L., Healy, J.: Umap: Uniform manifold approximation and projection for dimension reduction. arXiv:1802.03426 (2018)
  * [32] McKnight, P.E., Najab, J.: Mann-whitney u test. The Corsini encyclopedia of psychology p. 1 (2010)
  * [33] Nguyen, D.T., Mummadi, C.K., Ngo, T.P.N., Nguyen, T.H.P., Beggel, L., Brox, T.: Self: Learning to filter noisy labels with self-ensembling. In: International Conference on Learning Representations (2019)
  * [34] Paresh, S., Thokala, N., Majumdar, A., Chandra, M.: Multi-label auto-encoder based electrical load disaggregation. 2020 International Joint Conference on Neural Networks (IJCNN) pp. 1–6 (2020)
  * [35] Reed, S.E., Lee, H., Anguelov, D., Szegedy, C., Erhan, D., Rabinovich, A.: Training deep neural networks on noisy labels with bootstrapping. In: ICLR (2015)
  * [36] Rolnick, D., Veit, A., Belongie, S., Shavit, N.: Deep learning is robust to massive label noise. arXiv:1705.10694 (2017)
  * [37] Sablayrolles, A., Douze, M., Schmid, C., Jégou, H.: Spreading vectors for similarity search. In: ICLR 2019-7th International Conference on Learning Representations. pp. 1–13 (2019)
  * [38] Song, H., Kim, M., Park, D., Lee, J.G.: Learning from noisy labels with deep neural networks: A survey. arXiv:2007.08199 (2020)
  * [39] Sugiyama, M.: Co-teaching: Robust training of deep neural networks with extremely noisy labels. In: NeurIPS (2018)
  * [40] Van Rooyen, B., Menon, A.K., Williamson, R.C.: Learning with symmetric label noise: The importance of being unhinged. arXiv:1505.07634 (2015)
  * [41] Wang, J., Ma, Y., Gao, S.: Self-semi-supervised learning to learn from noisylabeled data. arXiv:2011.01429 (2020)
  * [42] Wang, X., Wang, C.: Time series data cleaning: A survey. IEEE Access 8, 1866–1881 (2020)
  * [43] Wang, Z., Yan, W., Oates, T.: Time series classification from scratch with deep neural networks: A strong baseline. In: 2017 International joint conference on neural networks (IJCNN). pp. 1578–1585 (2017)
  * [44] Wang, Z., yi Luo, X., Liang, J.: A label noise robust stacked auto-encoder algorithm for inaccurate supervised classification problems. Mathematical Problems in Engineering 2019, 1–19 (2019)
  * [45] Wu, N., Green, B., Ben, X., O’Banion, S.: Deep transformer models for time series forecasting: The influenza prevalence case. arXiv:2001.08317 (2020)
  * [46] Yang, Y., Zhong, J., Li, W., Gulliver, T.A., Li, S.: Semisupervised multilabel deep learning based nonintrusive load monitoring in smart grids. IEEE Transactions on Industrial Informatics 16(11), 6892–6902 (2019)
  * [47] Zeghidour, N., Grangier, D.: Wavesplit: End-to-end speech separation by speaker clustering. arXiv:2002.08933 (2020)
  * [48] Zhang, C., Bengio, S., Hardt, M., Recht, B., Vinyals, O.: Understanding deep learning requires rethinking generalization. arXiv:1611.03530 (2016)
  * [49] Zhang, H., Cisse, M., Dauphin, Y.N., Lopez-Paz, D.: mixup: Beyond empirical risk minimization. arXiv:1710.09412 (2017)
  * [50] Zhang, Z., Sabuncu, M.R.: Generalized cross entropy loss for training deep neural networks with noisy labels. arXiv:1805.07836 (2018)


Hesse Pencils and 3-Torsion Structures

Hesse Pencils and 3-Torsion Structures††This paper is a contribution to the
Special Issue on Modular Forms and String Theory in honor of Noriko Yui. The
full collection is available at http://www.emis.de/journals/SIGMA/modular-
forms.html

Ane S.I. ANEMA, Jaap TOP and Anne TUIJP

A.S.I. Anema, J. Top and A. Tuijp

Bernoulli Institute for Mathematics, Computer Science and Artificial
Intelligence,
University of Groningen, P.O. Box 407, 9700 AK Groningen, The Netherlands
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Received May 08, 2018, in final form September 18, 2018; Published online
September 21, 2018

This paper intends to focus on the universal property of this Hesse pencil and
of its twists. The main goal is to do this as explicit and elementary as
possible, and moreover to do it in such a way that it works in every
characteristic different from three.

Hesse pencil; Galois representation; torsion points; elliptic curves

14D10; 14G99

##  1 Introduction

In a paper with Noriko Yui [14], explicit equations for all elliptic modular
surfaces corresponding to genus zero torsion-free subgroups of
$\mbox{PSL}_{2}({\mathbb{Z}})$ were presented. Arguably the most famous and
classical one of these surfaces is the Hesse pencil, usually described as the
family of plane cubics

$\displaystyle x^{3}+y^{3}+z^{3}+6txyz=0$

(see, e.g., [14, Table 2] and references given there). The current paper
intends to focus on the universal property of this Hesse pencil and of its
twists (see Theorem 1.1). The main goal is to do this as explicit and
elementary as possible, and moreover to do it in such a way that it works in
every characteristic different from three. We are not aware of any earlier
publication of these results in the special case of characteristic two, so
although admittedly not very difficult, those appear to be new. The first
author of this paper worked on the present results as a (small) part of his
Ph.D. Thesis [1] and the third author did the same as part of her Bachelor’s
Thesis [15]. Both were supervised by the second author.

Let $k$ be a perfect field of characteristic different from three. Denote the
absolute Galois group of $k$ by $G_{k}$. Given an elliptic curve $E$ defined
over $k$, one obtains a Galois representation on the 3-torsion group
$E{\left[{3}\right]}$ of $E$. This paper describes the family of all elliptic
curves that have equivalent Galois representations on $E{\left[{3}\right]}$.
Recall that elliptic curves $E$ and $E^{\prime}$ over $k$ yield equivalent
Galois representations on their 3-torsion if and only if $E{\left[{3}\right]}$
and $E^{\prime}{\left[{3}\right]}$ are isomorphic as $G_{k}$-modules. To be
more specific, we demand that the equivalence is symplectic: a symplectic
homomorphism $\phi\colon E[3]\to E^{\prime}[3]$ is defined as in [10], as
follows. If

$\displaystyle
e_{3}{\left({S,T}\right)}=e^{\prime}_{3}{\left({\phi{\left({S}\right)},\phi{\left({T}\right)}}\right)}$

for all $S,T\in E{\left[{3}\right]}$ where $e_{3}$ and $e^{\prime}_{3}$ are
the Weil-pairings on the 3-torsion of $E$ and $E^{\prime}$ respectively, then
$\phi$ is called a  _symplectic_ homomorphism, otherwise $\phi$ is called an
_anti-symplectic_ homomorphism.

Next, recall the definition of the Hessian of a polynomial. Let $F\in
k{\left[{X,Y,Z}\right]}$ be a homogeneous polynomial of degree $n$. The
_Hessian_ $\operatorname{Hess}{\left({F}\right)}$ of $F$ is the determinant of
the Hessian matrix of $F$, that is

$\displaystyle\operatorname{Hess}{\left({F}\right)}=\det\begin{pmatrix}\dfrac{\partial^{2}F}{\partial
X^{2}}&\dfrac{\partial^{2}F}{\partial X\partial
Y}&\dfrac{\partial^{2}F}{\partial X\partial Z}\vspace{1mm}\\\
\dfrac{\partial^{2}F}{\partial X\partial Y}&\dfrac{\partial^{2}F}{\partial
Y^{2}}&\dfrac{\partial^{2}F}{\partial Y\partial Z}\vspace{1mm}\\\
\dfrac{\partial^{2}F}{\partial X\partial Z}&\dfrac{\partial^{2}F}{\partial
Y\partial Z}&\dfrac{\partial^{2}F}{\partial Z^{2}}\end{pmatrix},$

which is either a homogeneous polynomial of degree $3n-6$ or zero.

Given a curve $C=Z{\left({F}\right)}$ with $F\in k{\left[{X,Y,Z}\right]}$
homogeneous of degree three, the _Hesse pencil_ of $C$ is defined as

$\displaystyle\mathcal{C}=Z{\left({tF+\operatorname{Hess}{\left({F}\right)}}\right)}$

over $k{\left({t}\right)}$. Recall that the discrete valuations on
$k{\left({t}\right)}$ correspond to the points in
${{\mathbb{P}}^{1}{\left({k}\right)}}$, where we usually write
${\left({t_{0}:1}\right)}$ as $t_{0}$ and ${\left({1:0}\right)}$ as $\infty$.
We denote the reduced curve of $\mathcal{C}$ at
$t_{0}\in{{\mathbb{P}}^{1}{\left({k}\right)}}$ by $C_{t_{0}}$. Notice that
$C_{\infty}=C$ and for $t_{0}\neq\infty$

$\displaystyle
C_{t_{0}}=Z{\left({t_{0}F+\operatorname{Hess}{\left({F}\right)}}\right)}.$

In the special case that $C=E$ is an elliptic curve given by a Weierstrass
equation, we have (see Section 2) that the point $O$ at infinity is a point on
$E_{t_{0}}$ for every $t_{0}\in{{\mathbb{P}}^{1}{\left({k}\right)}}$. If
$E_{t_{0}}$ is a smooth curve, then this makes it an elliptic curve with unit
element $O$.

In the case of characteristic two, the standard definition of the Hessian does
not lead to a satisfactory theory. In Sections 9.2 and 9.3 a modified Hessian
is introduced for this case; in fact this modification was already used by
Dickson [5] in 1915. The goal of this paper is to provide an elementary proof
of the following theorem:

######  Theorem 1.1.

If $E$ and $E^{\prime}$ are elliptic curves over $k$, with $E$ given by some
Weierstrass equation over $k$, then there exists a symplectic isomorphism
$E{\left[{3}\right]}\rightarrow E^{\prime}{\left[{3}\right]}$ if and only if
$E^{\prime}$ appears in the Hesse pencil of $E$, i.e.,
$E_{t_{0}}\cong_{k}E^{\prime}$ for some
$t_{0}\in{{\mathbb{P}}^{1}{\left({k}\right)}}$.

We note that both Fisher’s paper [6] and Kuwata’s paper [10] discuss, apart
from the result above (although not in characteristic two) also the case of
anti-symplectic isomorphisms between the $3$-torsion groups of elliptic
curves.

In Sections 2, 3 and 4 we show that the 3-torsion groups of an elliptic curve
in Weierstrass form and its Hesse pencil are identical not only as sets, but
also have the same group structure and Weil-pairings. Using the Weierstrass
form of the Hesse pencil computed in Section 5 and the relation between a
linear change of coordinates and its restriction to the 3-torsion group
described in Section 6, we prove in Section 7 essentially by a counting
argument that an isomorphism of the 3-torsion groups respecting the Weil-
pairings is the restriction of a linear change of coordinates. The proof of
the theorem is completed in Section 8. After this, we adapt the argument in
order to conclude the same result in characteristic $2$ (where a slightly
adapted notion of Hesse pencil is required). We compare our results with
existing literature in Section 10.

##  2 The flex points

Let $C=Z{\left({F}\right)}$ be a plane curve with $F\in
k{\left[{X,Y,Z}\right]}$ homogeneous of degree $n$ and irreducible. A point
$P$ on $C$ is called a _flex point_ if there exists a line $L$ such that the
intersection number of $C$ and $L$ at $P$ is at least three. Notice that in
our definition $P$ is allowed to be a singular point on $C$.

The _Hessian curve_ of $C$ is defined as
$\operatorname{Hess}{\left({C}\right)}=Z{\left({\operatorname{Hess}{\left({F}\right)}}\right)}$.

######  Proposition 2.1.

If $P$ is a point on $C$ and
$\operatorname{char}{\left({k}\right)}\nmid(n-1)$, then $P$ is a flex point if
and only if $P\in C\cap\operatorname{Hess}{\left({C}\right)}$.

###### Proof.

See [8, Exercise 5.23]. ∎

From now on we will only work with curves of degree three, so the proposition
above is only usable for fields $k$ of characteristic different from two. This
is the reason for why we exclude characteristic two for now; see Section 9 for
the excluded case.

######  Corollary 2.2.

If $P$ is a flex point on $C$, then it is also a point on the Hesse pencil
$\mathcal{C}$ and it is again a flex point of each curve of the Hesse pencil.

This is a well-known and old result in the case of $F=X^{3}+Y^{3}+Z^{3}$, see
for example [11, Section VII.1].

###### Proof.

A computation using Magma [4] shows that

$\displaystyle\operatorname{Hess}{\left({tF+\operatorname{Hess}{\left({F}\right)}}\right)}=\alpha
F+\beta\operatorname{Hess}{\left({F}\right)}$

with $\alpha,\beta\in k{\left[{t}\right]}$.

Assume that $P$ is a flex point, then $P\in
C\cap\operatorname{Hess}{\left({C}\right)}$ by Proposition 2.1, that is
$F{\left({P}\right)}=0$ and
$\operatorname{Hess}{\left({F}\right)}{\left({P}\right)}=0$. So
${\left({tF+\operatorname{Hess}{\left({F}\right)}}\right)}{\left({P}\right)}=0$,
which implies that $P\in\mathcal{C}$. The computation above also implies that

$\displaystyle\operatorname{Hess}{\left({tF+\operatorname{Hess}{\left({F}\right)}}\right)}{\left({P}\right)}=0,$

that is $P\in\operatorname{Hess}{\left({\mathcal{C}}\right)}$. Therefore
$P\in\mathcal{C}\cap\operatorname{Hess}{\left({\mathcal{C}}\right)}$. Hence
$P$ is a flex point on $\mathcal{C}$ by Proposition 2.1. ∎

######  Corollary 2.3.

Let $P\in C_{t_{0}}\cap C_{t_{1}}$. If $t_{0}\neq t_{1}$, then $P$ is a flex
point on $C$.

###### Proof.

Suppose that $t_{0}={\left({t_{00}:t_{01}}\right)}$ and
$t_{1}={\left({t_{10}:t_{11}}\right)}$, then

$\displaystyle\begin{pmatrix}t_{00}&t_{01}\\\
t_{10}&t_{11}\end{pmatrix}\begin{pmatrix}F{\left({P}\right)}\\\
\operatorname{Hess}{\left({F}\right)}{\left({P}\right)}\end{pmatrix}=\begin{pmatrix}0\\\
0\end{pmatrix},$

with the matrix being invertible since $t_{0}\neq t_{1}$. Thus
$F{\left({P}\right)}=0$ and
$\operatorname{Hess}{\left({F}\right)}{\left({P}\right)}=0$, that is $P\in
C\cap\operatorname{Hess}{\left({C}\right)}$. Hence Proposition 2.1 implies
that $P$ is a flex point on $C$. ∎

##  3 The 3-torsion group

Let $E=Z{\left({F}\right)}$ be an elliptic curve with unit element $O$ and
$F\in k{\left[{X,Y,Z}\right]}$ homogeneous of degree 3. Recall the following
well known fact.

######  Proposition 3.1.

Let $S$ and $T$ be points on $E$. If $S$ is a flex point, then $T$ is a flex
point if and only if $S-T\in E{\left[{3}\right]}$.

###### Proof.

Let $L_{S}$ and $L_{T}$ be the tangent lines to $E$ at $S$ and $T$
respectively. Assume that $T$ is also a flex point. Consider the function
$\frac{L_{S}}{L_{T}}$ on $E$ which has divisor
$3{\left({S}\right)}-3{\left({T}\right)}$. From [13, Corollary III.3.5] it
follows that $3S-3T=O$. Hence $S-T\in E{\left[{3}\right]}$.

Assume that $T$ is not a flex point. Now the divisor of the function
$\frac{L_{S}}{L_{T}}$ is
$3{\left({S}\right)}-2{\left({T}\right)}-{\left({T^{\prime}}\right)}$ with
$T^{\prime}\neq T$. From this it follows that $3S-2T-T^{\prime}=O$, thus
$3S-3T=T^{\prime}-T\neq O$. Hence $S-T\notin E{\left[{3}\right]}$. ∎

This result tells us that if $O$ is a flex point on $E$, then the concepts of
flex point and 3-torsion point coincide. In the previous section we learned
that a flex point on $E$ is also a flex point on
$\mathcal{E}=\mathcal{Z}(tF+\operatorname{Hess}{\left({F}\right)})$. Hence if
we combine these statements, then we obtain
$E{\left[{3}\right]}\subset\mathcal{E}{\left[{3}\right]}$. Since the
characteristic of $k$ is different from three, these sets are equal in size,
thus the same. Moreover suppose that $E_{t_{0}}$ for some
$t_{0}\in{{\mathbb{P}}^{1}{\left({k}\right)}}$ is non-singular. Provide
$\mathcal{E}$ and $E_{t_{0}}$ with a group structure by taking $O$ as the unit
element. Since the flex points of $\mathcal{E}$ (considered as a plane cubic
over $k(t)$) and of $E_{t_{0}}$ (a cubic curve over $k$) are the same and a
line that intersects an elliptic curve at two flex points will also intersect
the curve at a third flex point, the group structures on
$\mathcal{E}{\left[{3}\right]}$ and $E_{t_{0}}{\left[{3}\right]}$ are equal as
well.

Recall that if the unit element $O$ is a flex point on $E$, then we can find a
projective linear transformation in
${\operatorname{PGL}_{3}}{\left({k}\right)}$ such that $E$ is given by a
Weierstrass equation in the new coordinates. Moreover since the characteristic
of $k$ is different from two and three, we may even assume that $E\colon
y^{2}=x^{3}+ax+b$ for some $a,b\in k$.

##  4 The Weil-pairing

In the previous section we saw that
$\mathcal{E}{\left[{3}\right]}=E_{t_{0}}{\left[{3}\right]}$ for all
$t_{0}\in{{\mathbb{P}}^{1}{\left({{\overline{k}}}\right)}}$ such that
$E_{t_{0}}$ is non-singular. Denote the Weil-pairing on the 3-torsion of
$\mathcal{E}$ by $e_{3}$ and on the 3-torsion of $E_{t_{0}}$ by
$e_{3}^{t_{0}}$. An introduction to Weil-pairings can be found in [13, Section
III.8] and [16, Sections 3.3 and 11.2].

######  Proposition 4.1.

Let $E$ be an elliptic curve given by a Weierstrass equation and let
$\mathcal{E}$ be its Hesse pencil. The Weil-pairings $e_{3}$ and
$e_{3}^{t_{0}}$ on $E{\left[{3}\right]}$ are equal.

###### Proof.

Let $S,T\in E{\left[{3}\right]}$ generate $E{\left[{3}\right]}$. The Weil-
pairing is determined by its value on ${\left({S,T}\right)}$. Follow [13,
Exercise 3.16] to construct the Weil-pairings. Recall that $O$ is a flex point
on $E$.

Let $L_{O}$, $L_{S}$, $L_{T}$ and $L_{-T}$ be the tangent lines to
$\mathcal{E}$ at $O$, $S$, $T$ and $-T$ respectively. Define
$D_{S}={\left({S}\right)}-{\left({O}\right)}$ and
$D_{T}=2{\left({T}\right)}-2{\left({-T}\right)}$. Notice that $D_{S}$ and
$D_{T}$ have disjoint support. Since $2T-2{\left({-T}\right)}=T$ in
$\mathcal{E}$, it follows that
$D_{T}\sim{\left({T}\right)}-{\left({O}\right)}$. Consider the functions
$f_{S}=\frac{L_{S}}{L_{O}}$ and
$f_{T}=\big{(}\frac{L_{T}}{L_{-T}}\big{)}^{2}$, then
$\operatorname{div}{\left({f_{S}}\right)}=3D_{S}$ and
$\operatorname{div}{\left({f_{T}}\right)}=3D_{T}$. The Weil-pairing on
$\mathcal{E}$ is defined as

$\displaystyle
e_{3}{\left({S,T}\right)}=\frac{f_{S}{\left({D_{T}}\right)}}{f_{T}{\left({D_{S}}\right)}}={\left({\frac{f_{S}{\left({T}\right)}}{f_{S}{\left({-T}\right)}}}\right)}^{2}\frac{f_{T}{\left({O}\right)}}{f_{T}{\left({S}\right)}}={\left({\frac{L_{S}{\left({T}\right)}L_{O}{\left({-T}\right)}L_{T}{\left({O}\right)}L_{-T}{\left({S}\right)}}{L_{O}{\left({T}\right)}L_{S}{\left({-T}\right)}L_{-T}{\left({O}\right)}L_{T}{\left({S}\right)}}}\right)}^{2}.$

Let $s\in k{\left({S,T}\right)}{\left({t}\right)}$ be a local coordinate at
$t_{0}$. Choose the equations of the tangent lines such that they are also
defined over $k{\left({S,T}\right)}{\left[\left[{s}\right]\right]}$ and are
non-zero modulo $s$. Notice that $L_{O}$, $L_{S}$, $L_{T}$ and $L_{-T}$ modulo
$s$ are tangent lines to $E_{t_{0}}$ at $O$, $S$, $T$ and $-T$ respectively.
Follow the construction above to obtain the Weil-pairing
$e_{3}^{t_{0}}{\left({S,T}\right)}$ on $E_{t_{0}}$.

Now $L_{O}{\left({T}\right)}$ is a unit in
$k{\left({S,T}\right)}{\left[\left[{s}\right]\right]}$, because $T$ is not
contained in the tangent line $L_{O}$ modulo $s$ to $E_{t_{0}}$ at $O$.
Similarly the other terms in the expression of $e_{3}{\left({S,T}\right)}$ are
units as well. Thus by construction $e_{3}{\left({S,T}\right)}\\!\\!\mod
s=e_{3}^{t_{0}}{\left({S,T}\right)}$. Recall that $e_{3}{\left({S,T}\right)}$
is a root of unity. Hence $e_{3}=e_{3}^{t_{0}}$. ∎

##  5 The Weierstrass form

######  Proposition 5.1.

Let $E$ be an elliptic curve given by the Weierstrass equation
$y^{2}z=x^{3}+axz^{2}+bz^{3}$ with $a,b\in k$. Then the Hesse pencil
$\mathcal{E}$ can be given by

$\displaystyle
ty^{2}z+3xy^{2}=tx^{3}-3ax^{2}z+{\left({at-9b}\right)}xz^{2}+\big{(}bt+a^{2}\big{)}z^{3}$

over $k{\left({t}\right)}$. The linear change of coordinates

$\displaystyle\begin{pmatrix}x\\\ y\\\ z\end{pmatrix}=A\begin{pmatrix}\xi\\\
\eta\\\ \zeta\end{pmatrix}\qquad\text{with}\qquad
A=\begin{pmatrix}t&0&3at^{2}-27bt-9a^{2}\\\ 0&1&0\\\
-3&0&t^{3}+9at-27b\end{pmatrix}$

transforms this into the Weierstrass form
$\mathcal{E}^{W}\colon\eta^{2}\zeta=\xi^{3}+a_{t}\xi\zeta^{2}+b_{t}\zeta^{3}$,
with

$\displaystyle
a_{t}=at^{4}-18bt^{3}-18a^{2}t^{2}+54abt-\big{(}27a^{3}+243b^{2}\big{)},$
$\displaystyle
b_{t}=bt^{6}+4a^{2}t^{5}-45abt^{4}+270b^{2}t^{3}+135a^{2}bt^{2}+\big{(}108a^{4}+486ab^{2}\big{)}t-\big{(}243a^{3}b+1458b^{3}\big{)}.$

Moreover
$\Delta{\left({\mathcal{E}^{W}}\right)}=\Delta{\left({E}\right)}{\left({\det
A}\right)}^{3}$ and $\det A=t^{4}+18at^{2}-108bt-27a^{2}$.

Observe that the $t$ in the proposition is equal to $8t$ in the previous
sections.

###### Proof.

The proof boils down to computing the map $A$, which can be found in three
steps. First map the tangent line to $\mathcal{E}$ at $O$ to the line at
infinity. Next scale the $z$-coordinate so that the coefficient in front of
$x^{3}$ and $y^{2}z$ are equal up to minus sign. Finally shift the $y$ – resp.
the $x$-coordinate so that the $xyz$, $yz^{2}$ resp. the $x^{2}z$ terms
vanish. ∎

This proposition shows that

$\displaystyle
j{\left({\mathcal{E}}\right)}=1728\frac{4}{4a^{3}+27b^{2}}{\left({\frac{at^{4}-18bt^{3}-18a^{2}t^{2}+54abt-{\left({27a^{3}+243b^{2}}\right)}}{t^{4}+18at^{2}-108bt-27a^{2}}}\right)}^{3}.$

##  6 Linear change of coordinates I

######  Proposition 6.1.

Let $P_{i}\in{{\mathbb{P}}^{2}{\left({k}\right)}}$ for $i=1,\dots,4$ be points
such that no three of them are collinear. If
$Q_{i}\in{{\mathbb{P}}^{2}{\left({k}\right)}}$ for $i=1,\dots,4$ is another
such set of points, then there exists a unique
$A\in{\operatorname{PGL}_{3}}{\left({k}\right)}$ such that
$A{\left({P_{i}}\right)}=Q_{i}$ for all $i=1,\dots,4$.

This is a well-known result which is easily proved using some elementary
linear algebra. Observe that an analogous result holds for two sets of $n+2$
points in ${{\mathbb{P}}^{n}{\left({k}\right)}}$ such that no $n+1$ of them
lie on a hyperplane.

######  Proposition 6.2.

Let $E$ be an elliptic curve given by a Weierstrass equation defined over $k$.
If $E{\left[{3}\right]}={\left\langle{S,T}\right\rangle}$, then any line in
${{\mathbb{P}}^{2}{\left({{\overline{k}}}\right)}}$ contains at most two of
the following points: $O$, $S$, $T$, $S+T$.

###### Proof.

Suppose that $L$ is a line in
${{\mathbb{P}}^{2}{\left({{\overline{k}}}\right)}}$ containing three of the
points $O$, $S$, $T$ and $S+T$. Denote these by $P_{1}$, $P_{2}$ and $P_{3}$.
Since $E$ is given by a Weierstrass equation, $O$ is a flex point, thus
$P_{1}+P_{2}+P_{3}=O$. However this is impossible for the points mentioned
above. Hence such a line $L$ does not exist. ∎

Suppose that we are given two elliptic curves $E$ and $E^{\prime}$ as in the
proposition above with $E{\left[{3}\right]}={\left\langle{S,T}\right\rangle}$
and
$E^{\prime}{\left[{3}\right]}={\left\langle{S^{\prime},T^{\prime}}\right\rangle}$,
then Propositions 6.1 and 6.2 imply that there exists an
$A\in{\operatorname{PGL}_{3}}{\left({{\overline{k}}}\right)}$ such that
$O\mapsto O^{\prime}$, $S\mapsto S^{\prime}$, $T\mapsto T^{\prime}$ and
$S+T\mapsto S^{\prime}+T^{\prime}$ and that this $A$ is unique.

##  7 Linear change of coordinates II

######  Proposition 7.1.

Let $E$ and $E^{\prime}$ be elliptic curves given by a Weierstrass equation
defined over $k$. If $\phi\colon E{\left[{3}\right]}\rightarrow
E^{\prime}{\left[{3}\right]}$ is an isomorphism which respects the Weil-
pairings, then there exists some
$t_{0}\in{{\mathbb{P}}^{1}{\left({{\overline{k}}}\right)}}$ such that the
fiber $E_{t_{0}}$ of the Hesse pencil of $E$ admits a linear change of
coordinates $\Phi\colon E_{t_{0}}\rightarrow E^{\prime}$ with
${\left.{\Phi}\right|_{E{\left[{3}\right]}}}=\phi$.

The essence of the proof of this proposition is the following: We determine
the $t_{i}\in{{\mathbb{P}}^{1}{\left({{\overline{k}}}\right)}}$ for which the
$j$-invariant of $E_{t_{i}}$ is equal to the $j$-invariant of $E^{\prime}$.
For each of these $t_{i}$’s we obtain a number of linear changes of
coordinates $E_{t_{i}}\rightarrow E$. A counting argument shows that $\phi$ is
the restriction of one of those maps. The following observation is used in the
counting argument:

######  Lemma 7.2.

Let $E$ and $E^{\prime}$ be elliptic curves. Then $24$ out of the $48$
isomorphisms $E{\left[{3}\right]}\rightarrow E^{\prime}{\left[{3}\right]}$
respect the Weil-pairings.

###### Proof.

Let $S,T\in E{\left[{3}\right]}$ be such that
$E{\left[{3}\right]}={\left\langle{S,T}\right\rangle}$ and
$e_{3}{\left({S,T}\right)}=\zeta_{3}$ with $\zeta_{3}$ a fixed primitive third
root of unity. Choose $S^{\prime},T^{\prime}\in E^{\prime}{\left[{3}\right]}$
likewise. Since $E{\left[{3}\right]}$ and $E^{\prime}{\left[{3}\right]}$ are
two-dimensional vector spaces over ${\mathbb{F}}_{3}$, there exists a
bijection

$\displaystyle{\operatorname{GL}_{2}}{\left({{\mathbb{F}}_{3}}\right)}\longrightarrow\operatorname{Iso}{\left({E{\left[{3}\right]},E^{\prime}{\left[{3}\right]}}\right)},$
$\displaystyle A=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\longmapsto\phi_{A}{\left({\alpha S+\beta
T}\right)}={\left({\alpha a+\beta b}\right)}S^{\prime}+{\left({\alpha c+\beta
d}\right)}T^{\prime}.$

Notice that

$\displaystyle
e_{3}^{\prime}{\left({\phi_{A}{\left({S}\right)},\phi_{A}{\left({T}\right)}}\right)}=e_{3}^{\prime}{\left({aS^{\prime}+bT^{\prime},cS^{\prime}+dT^{\prime}}\right)}=e_{3}^{\prime}{\left({S^{\prime},T^{\prime}}\right)}^{ad-
bc}=\zeta_{3}^{\det{A}}.$

So $\phi_{A}$ respects the Weil-pairings if and only if $\det{A}=1$, that is
$A\in{\operatorname{SL}_{2}}{\left({{\mathbb{F}}_{3}}\right)}$. Now
${\left|{{\operatorname{GL}_{2}}{\left({{\mathbb{F}}_{3}}\right)}}\right|}=48$
and
${\left[{{\operatorname{GL}_{2}}{\left({{\mathbb{F}}_{3}}\right)}\colon{\operatorname{SL}_{2}}{\left({{\mathbb{F}}_{3}}\right)}}\right]}=2$.
Hence there are 48 isomorphisms $E{\left[{3}\right]}\rightarrow
E^{\prime}{\left[{3}\right]}$ of which 24 respect the Weil-pairings. ∎

Next we prove the proposition.

###### Proof of Proposition 7.1.

Let $j_{0}$ and $j_{0}^{\prime}$ be the $j$-invariants of $E$ and $E^{\prime}$
respectively. Denote the specialization of $\mathcal{E}^{W}$ at
$t_{0}\in{{\mathbb{P}}^{1}{\left({{\overline{k}}}\right)}}$ by
$E_{t_{0}}^{W}$. If $E_{t_{0}}^{W}$ is non-singular, then let $A_{t_{0}}\colon
E_{t_{0}}\rightarrow E_{t_{0}}^{W}$ be the isomorphism induced by the linear
change of coordinates $A$ from Proposition 5.1 at $t_{0}$.

Assume that $j_{0}^{\prime}\neq j_{0},0,1728$ and take $a_{t}$ as defined in
Proposition 5.1. Consider the polynomial

$\displaystyle
G=-1728{\left({4a_{t}}\right)}^{3}-j_{0}^{\prime}\Delta\big{(}\mathcal{E}^{W}\big{)}={\left({j_{0}-j_{0}^{\prime}}\right)}\Delta{\left({E}\right)}\;t^{12}+2^{13}3^{6}a^{2}bt^{11}+\cdots.$

in $k{\left[{t}\right]}$, whose roots give $E_{t_{0}}^{W}$’s with
$j$-invariant equal to $j_{0}^{\prime}$. The polynomial $G$ has degree 12 and
its discriminant is

$\displaystyle-3^{147}{j_{0}^{\prime}}^{8}{\left({j_{0}^{\prime}-1728}\right)}^{6}\Delta{\left({E}\right)}^{44},$

which is non-zero, so $G$ has distinct roots $t_{1},\ldots,t_{12}$ in
${\overline{k}}$. Since the $j$-invariant of $E_{t_{i}}^{W}$ is equal to
$j_{0}^{\prime}$, there exists an isomorphism $\Psi_{i}\colon
E_{t_{i}}^{W}\rightarrow E^{\prime}$. An isomorphism respects the Weil-
pairings, see [13, Proposition III.8.2] or [16, Theorem 3.9]. From Sections 3
and 4 it follows that $E_{t_{i}}{\left[{3}\right]}=E{\left[{3}\right]}$ as
groups with identical Weil-pairings. Therefore for every $i=1,\ldots,12$ and
$\sigma\in\operatorname{Aut}{\left({E^{\prime}}\right)}\cong{\mathbb{Z}}/2{\mathbb{Z}}$

$\displaystyle\phi_{i,\sigma}={\left.{{\left({\sigma\circ\Psi_{i}\circ
A_{t_{i}}}\right)}}\right|_{E_{t_{i}}{\left[{3}\right]}}}\colon\
E{\left[{3}\right]}\rightarrow E^{\prime}{\left[{3}\right]}$

is an isomorphism respecting the Weil-pairings. Notice that
$\sigma\circ\Psi_{i}\circ A_{t_{i}}$ is an element of
${\operatorname{PGL}_{3}}{\left({{\overline{k}}}\right)}$, because
$E_{t_{i}}^{W}$ and $E^{\prime}$ are in Weierstrass form and $A$ is a linear
change of coordinates. All 24 isomorphisms $\phi_{i,\sigma}$ are distinct as
the following argument shows. Suppose that $\phi_{i,\sigma}=\phi_{j,\tau}$,
then $\sigma\circ\Psi_{i}\circ A_{t_{i}}=\tau\circ\Psi_{j}\circ A_{t_{j}}$
according to Section 6. Let $P\in E^{\prime}\setminus
E^{\prime}{\left[{3}\right]}$, then $Q={\left({\sigma\circ\Psi_{i}\circ
A_{t_{i}}}\right)}^{-1}{\left({P}\right)}$ is a point in $E_{t_{i}}\cap
E_{t_{j}}$, so Corollary 2.3 implies that $t_{i}=t_{j}$, that is $i=j$. Since
$\Psi_{i}$ and $A_{t_{i}}$ are isomorphisms, $\sigma=\tau$. Thus
$\phi_{i,\sigma}=\phi_{j,\tau}$ if and only if $i=j$ and $\sigma=\tau$. Since
the $\phi_{i,\sigma}$’s respect the Weil-pairings, Lemma 7.2 implies that
these are all the possible isomorphisms $E{\left[{3}\right]}\rightarrow
E^{\prime}{\left[{3}\right]}$ that respect the Weil-pairings. Hence
$\phi=\phi_{i,\sigma}$ for some $i=1,\ldots,12$ and
$\sigma\in\operatorname{Aut}{\left({E^{\prime}}\right)}$, which proves the
proposition in this case.

Suppose that $j_{0}^{\prime}=j_{0}$ and $j_{0}^{\prime}\neq 0,1728$, then the
$G$ above has degree 11 and the discriminant of $G$ is

$\displaystyle-2^{130}3^{195}a^{20}b^{10}\Delta{\left({E}\right)}^{30},$

which is again non-zero, so $G$ has distinct roots $t_{1},\ldots,t_{11}$ in
${\overline{k}}$. In this case the $j$-invariant of $E_{\infty}$ is also equal
to $j_{0}^{\prime}$, so let $t_{12}=\infty$. The argument presented before now
finishes the proof in this case.

Assume that $j_{0}^{\prime}=0$. This case is the same as before with the
exception of the polynomial $G$, which in this case should be replaced by
$a_{t}$. The four distinct $t_{i}$’s and the six elements in
$\operatorname{Aut}{\left({E^{\prime}}\right)}$ again give 24 isomorphisms
$\phi_{i,\sigma}$.

Finally, if $j_{0}^{\prime}=1728$, then replace $G$ by $b_{t}$ (defined in
Proposition 5.1) and proceed as before. ∎

##  8 Proof of the theorem

In the proof of Theorem 1.1 we need a result from Galois cohomology, namely:

######  Lemma 8.1.

If $k$ is a perfect field, then
${\operatorname{PGL}_{3}}{\left({{\overline{k}}}\right)}^{G_{k}}={\operatorname{PGL}_{3}}{\left({k}\right)}$.

###### Proof.

Consider the short exact sequence of $G_{k}$-groups

$\displaystyle
1\longrightarrow{\overline{k}}^{\ast}\longrightarrow{\operatorname{GL}_{3}}{\left({{\overline{k}}}\right)}\longrightarrow{\operatorname{PGL}_{3}}{\left({{\overline{k}}}\right)}\longrightarrow
1,$

which induces the exact sequence in the first row of the diagram

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\overline{k}}^{\ast}}^{G_{k}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{GL}_{3}}{\left({{\overline{k}}}\right)}^{G_{k}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{PGL}_{3}}{\left({{\overline{k}}}\right)}^{G_{k}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
H}^{1}\big{(}G_{k},{\overline{k}}^{\ast}\big{)}}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{GL}_{3}}{\left({k}\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{PGL}_{3}}{\left({k}\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1.}$

The second row is the definition of
${\operatorname{PGL}_{3}}{\left({k}\right)}$ and the vertical maps are the
inclusion maps. Hilbert’s Theorem 90 gives that ${\rm
H}^{1}\big{(}G_{k},{\overline{k}}^{\ast}\big{)}={\left\\{{1}\right\\}}$. Since
${{\overline{k}}^{\ast}}^{G_{k}}=k^{\ast}$ and
${\operatorname{GL}_{3}}{\left({{\overline{k}}}\right)}^{G_{k}}={\operatorname{GL}_{3}}{\left({k}\right)}$,
also
${\operatorname{PGL}_{3}}{\left({{\overline{k}}}\right)}^{G_{k}}={\operatorname{PGL}_{3}}{\left({k}\right)}$.
∎

###### Proof of Theorem 1.1.

Assume that $\Phi\colon E_{t_{0}}\rightarrow E^{\prime}$ for some
$t_{0}\in{{\mathbb{P}}^{1}{\left({k}\right)}}$ is an isomorphism defined over
$k$. This map respects the Weil-pairings according to [13, Proposition
III.8.2]. So ${\left.{\Phi}\right|_{E_{t_{0}}{\left[{3}\right]}}}\colon
E_{t_{0}}{\left[{3}\right]}\rightarrow E^{\prime}{\left[{3}\right]}$ is a
symplectic isomorphism. In Sections 3 and 4 it was shown that
$E{\left[{3}\right]}=E_{t_{0}}{\left[{3}\right]}$ as groups and have identical
Weil-pairings. Thus ${\left.{\Phi}\right|_{E_{t_{0}}{\left[{3}\right]}}}$ can
be considered as a symplectic isomorphism $E{\left[{3}\right]}\rightarrow
E^{\prime}{\left[{3}\right]}$. Hence
${\left.{\Phi}\right|_{E_{t_{0}}{\left[{3}\right]}}}$ is the desired map.

Suppose that there exists a symplectic isomorphism $\phi\colon
E{\left[{3}\right]}\rightarrow E^{\prime}{\left[{3}\right]}$, then Proposition
7.1 implies that there exists a
$\Phi\in{\operatorname{PGL}_{3}}{\left({{\overline{k}}}\right)}$ and a
$t_{0}\in{{\mathbb{P}}^{1}{\left({{\overline{k}}}\right)}}$ such that
$\Phi\colon E_{t_{0}}\rightarrow E^{\prime}$ and
$\phi={\left.{\Phi}\right|_{E{\left[{3}\right]}}}$. Since
$\sigma\circ\phi=\phi\circ\sigma$ for all $\sigma\in G_{k}$,

$\displaystyle\sigma{\left({\Phi}\right)}{\left({\sigma{\left({S}\right)}}\right)}=\sigma\circ\Phi{\left({S}\right)}=\sigma\circ\phi{\left({S}\right)}=\phi\circ\sigma{\left({S}\right)}=\Phi{\left({\sigma{\left({S}\right)}}\right)}$

for all $S\in E{\left[{3}\right]}$, so Propositions 6.1 and 6.2 imply that
$\sigma{\left({\Phi}\right)}=\Phi$. Therefore Lemma 8.1 implies that
$\Phi\in{\operatorname{PGL}_{3}}{\left({k}\right)}$. Hence
$t_{0}\in{{\mathbb{P}}^{1}{\left({k}\right)}}$ and
$E^{\prime}\cong_{k}E_{t_{0}}$. ∎

##  9 Characteristic two

So far we assumed $k$ to be a perfect field of characteristic different from
two and three. There is a natural idea how to adapt the proof of Theorem 1.1
to characteristic two: replace the explicitly given Hesse pencil by what it
actually describes, namely the pencil of cubics with the nine points of order
$3$ on the initial elliptic curve as base points. This was done by one of us
in her bachelor’s project [15], and we briefly describe the results here.

###  9.1 Elliptic curves in characteristic two

Any elliptic curve $E$ over a field of characteristic $2$ can be given as
follows, see [13, p. 409]:

$\displaystyle j(E)\neq 0\colon\
y^{2}+xy=x^{3}+a_{2}x^{2}+a_{6},\qquad\Delta=a_{6},\qquad j(E)=1/a_{6},$
$\displaystyle j(E)=0\colon\
y^{2}+a_{3}y=x^{3}+a_{4}x+a_{6},\qquad\Delta=a_{3}^{4},\qquad j(E)=0.$

If $k$ is a field of characteristic 2, then the Hessian of any homogeneous
polynomial $F\in k[X,Y,Z]$ of degree 3 equals zero, as is easily verified. We
will show that given an elliptic curve $E$ over $k$, say by a special equation
as above, the curves in the pencil with base points $E[3]$ all have $E[3]$ as
flex points, compare Corollary 2.2 for the classical situation. In [9], Glynn
defines a Hessian for any curve $C=\mathcal{Z}(F)$ with $F\in k[X,Y,Z]$
homogeneous of degree 3 (characteristic two). In fact our construction
coincides with his, although we put more emphasis on how it is obtained from
considering $3$-division polynomials. Note that the subject of flex points on
cubic curves in characteristic two is in fact very classical: compare with,
e.g., Dickson’s paper [5] published in 1915.

###  9.2 The case $\boldsymbol{j(E)\neq 0}$

We may and will assume that $E$ is given by

$\displaystyle y^{2}+xy=x^{3}+a_{2}x^{2}+a_{6},\qquad\Delta=a_{6},\qquad
j=1/a_{6}.$

Define Hess($E$) as the plane curve defined by

$\displaystyle y^{2}+xy^{2}+x^{2}y+xy+a_{2}x^{3}+a_{2}x^{2}+a_{6}x=0.$

The Hesse pencil $\mathcal{E}$ in this case is given by

$\displaystyle
t\big{(}y^{2}+xy+x^{3}+a_{2}x^{2}+a_{6}\big{)}+y^{2}+xy^{2}+x^{2}y+xy+a_{2}x^{3}+a_{2}x^{2}+a_{6}x=0.$

In the next paragraphs, we will show that the Hessian and Hesse pencil have
the desired properties. Firstly, the analog of Proposition 2.1 holds:

######  Proposition 9.1.

If $P$ is a point on an elliptic curve $E$ with equation
$y^{2}+xy=x^{3}+a_{2}x^{2}+a_{6}$, then $P$ is a flex point of $E$ if and only
if $P\in E\cap\operatorname{Hess}(E)$.

###### Proof.

The point $O$ is a flex point on both $E$ and $\operatorname{Hess}(E)$ hence
the result holds for $O$. Next take any other flex point of $E$, i.e., a point
$P\neq O$ with $3P=O$. Put $P=(x,y)$. Note that $x\neq 0$, because any point
$(0,y)\in E$ has order two. A small calculations (compare $x$-coordinates of
$-P$ and $2P$) shows $P$ is a flex point precisely when

$\displaystyle(E1)\quad
0=x^{2}+\left(\frac{y}{x}\right)^{2}+\frac{y}{x}+a_{2},$
$\displaystyle(E2)\quad
y+x=\left(x+\frac{y}{x}+1\right)\left(x^{2}+\left(\frac{y}{x}\right)^{2}+x+\frac{y}{x}+a_{2}\right)+x^{2}.$

Using the equation defining $E$, one rewrites $(E1)$ as

$\displaystyle 0=y^{2}+xy^{2}+x^{2}y+xy+a_{2}x^{3}+a_{2}x^{2}+a_{6}x.$ (9.1)

The proposition now follows from a straightforward calculation. ∎

Note that in fact $P=(x,y)$ is a flex point on $E$ if and only if $P\in E$
satisfies equation (9.1).

Now we show the analog of Corollary 2.2.

######  Proposition 9.2.

If $P$ is a flex point on an elliptic curve $E$ given by
$y^{2}+xy=x^{3}+a_{2}x^{2}+a_{6}$, then $P$ is also a flex point on the Hesse
pencil $\mathcal{E}$.

###### Proof.

It follows directly from the construction of $\mathcal{E}$ that $P$ is indeed
a point on it. To prove that $P$ is also a flex point on $\mathcal{E}$, one
shows that the tangent line to $\mathcal{E}$ at $P$ intersects $\mathcal{E}$
at $P$ with multiplicity 3. This is a straightforward calculation for which we
refer to [15]. Clearly the point $O$ is also a point on the Hesse pencil and
it is also a flex point, as can be shown in the same way. ∎

###  9.3 The case $\boldsymbol{j(E)=0}$

In the remaining case $j(E)=0$ we may and will assume that $E$ is given as

$\displaystyle y^{2}+a_{3}y=x^{3}+a_{4}x+a_{6},\qquad\Delta=a_{3}^{4},\qquad
j=0.$

Now define Hess($E$) by

$\displaystyle
xy^{2}+a_{3}xy+a_{4}x^{2}+\big{(}a_{3}^{2}+a_{6}\big{)}x+a_{4}^{2}=0,$

so the Hesse pencil $\mathcal{E}$ becomes

$\displaystyle
t\big{(}y^{2}+a_{3}y+x^{3}+a_{4}x+a_{6}\big{)}+xy^{2}+a_{3}xy+a_{4}x^{2}+\big{(}a_{3}^{2}+a_{6}\big{)}x+a_{4}^{2}=0.$

In this case as well, the analogs of Proposition 2.1 and Corollary 2.2 hold:

######  Proposition 9.3.

If $P$ is a point on the elliptic curve $E$ given by
$y^{2}+a_{3}y=x^{3}+a_{4}x+a_{6}$, then $P$ is a flex point if and only if
$P\in E\cap\operatorname{Hess}(E)$.

###### Proof.

For $O$ the exact same argument holds as when $j(E)\neq 0$. A calculation
shows that $P=(x,y)\in E$ is a flex point precisely when

$\displaystyle a_{3}^{2}x=x^{4}+a_{4}^{2}.$

Using the equation of the elliptic curve this is rewritten as

$\displaystyle
0=a_{3}^{2}x+x\big{(}y^{2}+a_{3}y+a_{4}x+a_{6}\big{)}+a_{4}^{2}=xy^{2}+\big{(}a_{3}^{2}+a_{6}\big{)}x+a_{4}x^{2}+a_{3}xy+a_{4}^{2}.$
∎

######  Proposition 9.4.

If $P$ is a flex point on the elliptic curve $E$ given by
$y^{2}+a_{3}y=x^{3}+a_{4}x+a_{6}$, then it is also a flex point on the Hesse
pencil $\mathcal{E}$.

###### Proof.

The reasoning is the same as for the case $j(E)\neq 0$. The straightforward
calculation is presented in detail in [15]. ∎

Using the properties shown above of our Hesse pencil in characteristic two, we
can now almost completely follow the reasoning of the earlier sections since
most arguments do not involve the characteristic of $k$. Only for the analog
of Proposition 7.1 the proof needs to be adjusted in characteristic two,
because here actual calculations are done with the Hesse pencil. We state it
in the present situation.

######  Proposition 9.5.

Let $E$ and $E^{\prime}$ be elliptic curves given by a Weierstrass equation
defined over $k$. If $\phi\colon E[3]\rightarrow E^{\prime}[3]$ is an
isomorphism which respects the Weil-pairings, then there exists a linear
change of coordinates $\Phi\colon E_{t_{0}}\rightarrow E^{\prime}$ for some
$t_{0}\in\mathbb{P}^{1}(\bar{k})$ such that $\Phi|_{E[3]}=\phi$.

The remainder of this section consists of proving Proposition 9.5.

###  9.4 The case $\boldsymbol{j(E)\neq 0}$

We first determine the Weierstrass form of the Hesse pencil in the present
case.

Rewrite its equation (as introduced in Section 9.2) as

$\displaystyle(t+1)y^{2}z+(t+1)xyz+xy^{2}+x^{2}y+(t+a_{2})x^{3}+a_{2}(t+1)x^{2}z+a_{6}xz^{2}+ta_{6}z^{3}=0.$

By a suitable change of coordinates this can be brought in Weierstrass form
$\eta^{2}\zeta+b_{1}\xi\eta\zeta=\xi^{3}+b_{2}\xi^{2}\zeta+b_{6}\zeta^{3}$.
This is quite analogous to what was sketched in the proof of Proposition 5.1;
see [15] for a detailed description and [3] for a more general setup. The
stated equation is found with

$\displaystyle b_{1}=(t+1)^{2},\qquad b_{2}=a_{2}(t+1)^{4}+a_{6}t(t+1),\qquad
b_{6}={a_{6}}(t^{4}+t^{3}+t^{2}+t+a_{6})^{3}.$

Denote this family of curves by $\mathcal{E}^{W}$ and let an individual curve
in the pencil be denoted by $E_{t}^{W}$, then

$\displaystyle
j\big{(}E^{W}_{t}\big{)}=b_{1}^{6}/b_{6}=\frac{(t+1)^{12}}{a_{6}\big{(}t^{4}+t^{3}+t^{2}+t+a_{6}\big{)}^{3}}.$

If $t=1$, the transformations needed to obtain the above Weierstrass form do
not work. In this case one transforms the fiber of given pencil, so the curve
$E_{1}$ with equation

$\displaystyle xy^{2}+x^{2}y+(1+a_{2})x^{3}+a_{6}xz^{2}+a_{6}z^{3}=0,$

into the other Weierstrass form in characteristic 2:

$\displaystyle\eta^{2}\zeta+b_{3}\eta\zeta^{2}=\xi^{3}+b_{4}\xi\zeta^{2}+b_{6}\zeta^{3}.$

Explicitly, this results in the equation

$\displaystyle\eta^{2}\zeta+a_{6}\eta\zeta^{2}+\xi^{3}+a_{6}^{2}\xi\zeta^{2}+a_{6}^{2}(1+a_{2})\zeta^{3}=0.$

In this way one obtains for every $t$ a projective, linear transformation
$E_{t}\rightarrow E_{t}^{W}$. Let us denote this transformation by $A_{t}$.

###### Proof of Proposition 9.5 for the case $\boldsymbol{j(E)\neq 0}$.

Given another elliptic curve $E^{\prime}$ with $j$-invariant $j_{0}^{\prime}$,
we want to determine $t$ for which our Hesse pencil has the same
$j$-invariant. First, let us assume that $j_{0}^{\prime}$ is nonzero and not
equal to $j_{0}$. Then

$\displaystyle j_{0}^{\prime}=j\big{(}E^{W}_{t}\big{)}\ \Leftrightarrow\
(t+1)^{12}=j_{0}^{\prime}a_{6}\big{(}t^{4}+t^{3}+t^{2}+t+a_{6}\big{)}^{3}.$

Define the polynomial

$\displaystyle
G=(t+1)^{12}+j_{0}^{\prime}a_{6}\big{(}t^{4}+t^{3}+t^{2}+t+a_{6}\big{)}^{3}.$

The zeros of this polynomial are precisely all $t_{0}$ such that
$j\big{(}E_{t_{0}}^{W}\big{)}=j_{0}^{\prime}$. The discriminant of $G$ equals
$a_{6}^{44}j_{0}^{\prime 14}$, which is nonzero, because $j_{0}^{\prime}$ and
$a_{6}$ are nonzero. We conclude that precisely $12$ values $t_{0}\in\bar{k}$
exist which give the desired $j$-invariant.

For every $t_{0}$, there is an isomorphism $A_{t_{0}}$ between $E_{t_{0}}$ and
$E_{t_{0}}^{W}$, induced by the change of coordinates seen above. For every
$t_{0}$ which is moreover a zero of $G$, there is an isomorphism
$\Psi_{t_{0}}$ between $E_{t_{0}}^{W}$ and $E^{\prime}$, because these curves
have equal $j$-invariants. Lastly, there exist 2 automorphisms $\sigma$ of
$E^{\prime}$ [13, p. 410]. Taking the composition of these three isomorphisms
and restricting to the 3-torsion group $E_{t_{0}}[3]$, which equals $E[3]$, we
obtain $12\times 2=24$ isomorphisms $\phi_{t_{0},\sigma}$; they are described
as

$\displaystyle\phi_{t_{0},\sigma}=\sigma\circ\Psi_{t_{0}}\circ
A_{t_{0}}|_{E_{t_{0}}[3]}\colon\ E[3]\rightarrow E^{\prime}[3].$

These 24 isomorphisms are pairwise distinct and respect the Weil-pairing (see
Section 7, observe that this argument is independent of the characteristic of
$k$).

Now consider the case $j_{0}^{\prime}=j_{0}\neq 0$. Then
$j_{0}^{\prime}a_{6}=1$ since $j_{0}=1/a_{6}$. Our polynomial $G$ therefore
has degree 11 and discriminant $a_{6}^{30}\neq 0$. So this gives us 11
pairwise distinct $t\in\bar{k}$ such that $j\big{(}E_{t}^{W}\big{)}=j_{0}$.
Another curve with this $j$-invariant is $E_{\infty}=E$. So again we find 12
distinct $t$’s and in the same way as above, we find 24 isomorphisms
respecting the Weil-pairing.

If $j_{0}^{\prime}=0$, the only $t$-value with $j(E_{t})=0$ is $t=1$. Because
$E^{\prime}$ has $j$-invariant zero and $k$ has characteristic 2, its
automorphism group has 24 elements [13, p. 410]. So again we find 24
isomorphisms respecting the Weil-pairing.

We now complete the proof of Proposition 9.5 for the case $j(E)\neq 0$ by the
exact same argument as presented in the proof of Proposition 7.1. ∎

###  9.5 The case $\boldsymbol{j(E)=0}$

For $j(E)=0$ the calculations are slightly more involved. Bringing the Hesse
pencil

$\displaystyle
t\big{(}y^{2}z+a_{3}yz^{2}+x^{3}+a_{4}xz^{2}+a_{6}z^{3}\big{)}=xy^{2}+a_{3}xyz+a_{4}x^{2}z+\big{(}a_{3}^{2}+a_{6}\big{)}xz^{2}+a_{4}^{2}z^{3}$

in Weierstrass form, one obtains $\mathcal{E}^{W}$ of the form

$\displaystyle\eta^{2}\zeta+\xi\eta\zeta+\xi^{3}+b_{2}\xi^{2}\zeta+b_{6}\zeta^{3}=0$

with $b_{2}$, $b_{6}$ explicit rational expressions in the $a_{j}$. The
$j$-invariant of $E_{t}^{W}$ for $t\neq 0$ is

$\displaystyle
j\big{(}E^{W}_{t}\big{)}=\frac{{a_{3}}^{8}}{\big{(}t^{4}+a_{3}^{2}t+a_{4}^{2}\big{)}^{3}}.$

If $t=0$, so if $E_{t}=E_{0}$ is the Hessian curve, the transformations needed
here are not valid. Therefore we treat this case separately. The Hessian here
is given by

$\displaystyle
xy^{2}+a_{3}xyz+a_{4}x^{2}z+\big{(}a_{3}^{2}+a_{6}\big{)}xz^{2}+a_{4}^{2}z^{3}=0.$

This results in the Weierstrass equation

$\displaystyle\eta^{2}\zeta+a_{3}^{2}\xi\eta\zeta+\xi^{3}+\big{(}{a_{3}^{4}+a_{6}}{a_{3}^{2}}\big{)}\xi^{2}\zeta+{a_{3}^{4}a_{4}^{6}}\zeta^{3}=0$

with $j$-invariant $\frac{a_{3}^{8}}{a_{4}^{6}}$. We conclude that for every
$t$ including $t=0$, the $j$-invariant of $E_{t}^{W}$ is given by

$\displaystyle
j\big{(}E_{t}^{W}\big{)}=\frac{{a_{3}}^{8}}{\big{(}t^{4}+a_{3}^{2}t+a_{4}^{2}\big{)}^{3}}.$

Moreover, again we have a projective linear automorphism $A_{t}\colon E_{t}\to
E_{t}^{W}$.

###### Proof of Proposition 9.5 if $\boldsymbol{j(E)=0}$.

Again, we want to show that for every $j_{0}^{\prime}$, there exist 24
different isomorphisms. First, assume that $j_{0}^{\prime}\neq 0$. Define

$\displaystyle
G:=a_{3}^{8}+j_{0}^{\prime}\big{(}t^{4}+a_{3}^{2}t+a_{4}^{2}\big{)}^{3},$

which has degree 12 and discriminant $a_{3}^{176}j_{0}^{\prime
14}=\Delta(E)^{44}j_{0}^{\prime 14}$. Therefore $G$ has $12$ pairwise distinct
zeros, which are all solutions $t_{0}$ such that
$j\big{(}E^{W}_{t_{0}}\big{)}=j_{0}^{\prime}$. Again, together with the 2
automorphisms $\sigma$, we find 24 isomorphisms.

Now assume that $j_{0}^{\prime}=0$. In this case, the curve in the Hesse
pencil we are looking for, is $E$ itself: this curve has $j$-invariant zero.
And again the automorphism group has order 24 so also in this case, we find 24
isomorphisms again, and the proof of Proposition 9.5 in this case is completed
using the same reasoning as before (compare the proof of Proposition 7.1). ∎

Using Proposition 9.5 one concludes that Theorem 1.1 holds in characteristic
two as well, using the reasoning as presented in Section 8.

##  10 Comparison with the literature

Theorem 1.1 is part of a more general problem: Given an elliptic curve $E$
over a field $k$ and an integer $n$, describe the universal family of elliptic
curves $\mathcal{E}$ such that for each member $\mathcal{E}_{t_{0}}$ the
Galois representations on $E{\left[{n}\right]}$ and
$\mathcal{E}_{t_{0}}{\left[{n}\right]}$ are isomorphic and the isomorphism is
symplectic. For various $n$ explicit families are known in the literature.

In [12] Rubin and Silverberg construct for any elliptic curve over
${\mathbb{Q}}$ such an explicit family for $n=3$ and $n=5$. Their proofs are
motivated by the theory of modular curves. Our Theorem 1.1 corresponds roughly
to [12, Theorem 4.1] and [12, Remark 4.2].

Using invariant theory and a generalization of the classical Hesse pencil,
Fisher in [6] describes such families for elliptic curves defined over a
perfect field of characteristic not dividing $6n$ with $n=2,3,4,5$. Theorem
1.1 is a special case of [6, Theorem 13.2]. It is unclear whether Fisher’s
proof of [6, Theorem 13.2] can be adapted to the case of characteristic two.
In [7] Fisher moreover treats the cases $n=7$ and $n=11$.

The Hesse pencil is used by Kuwata in [10]. For any elliptic curve $E$ over a
number field he constructs two families of elliptic curves such that for each
member the Galois representation on its 3-torsion is equivalent to the one on
$E{\left[{3}\right]}$. In the first family the isomorphism of the 3-torsion
groups is symplectic, whereas in the second family the isomorphism is anti-
symplectic. The proofs use classical projective geometry and the
classification of rational elliptic surfaces. Theorem 1.1 is essentially [10,
Theorem 4.2] (although our proof is more detailed and totally elementary, and
moreover we extend the result to characteristic two). Notice that the
Weierstrass form of the Hesse pencil in [10, Remark 4.4] is the same as the
one in Proposition 5.1 with $t$ replaced by $t^{-1}$ and the $x$ and $y$
coordinates scaled by some power of $t$.

An overview of results on the classical Hesse pencil is given by Artebani and
Dolgachev in [2].

### Acknowledgements

It is a pleasure to thank the referees of an earlier draft of this text. Their
comments and suggestions were greatly appreciated. We also thank Matthias
Schütt for helpful remarks.

## References

  * [1] Anema A.S.I., The arithmetic of maximal curves, the Hesse pencil and the Mestre curve, Ph.D. Thesis, Rijksuniversiteit Groningen, 2016, available at http://hdl.handle.net/11370/0ef530b1-709b-4285-b68d-016a67e6e928.
  * [2] Artebani M., Dolgachev I., The Hesse pencil of plane cubic curves, Enseign. Math. 55 (2009), 235–273, math.AG/0611590.
  * [3] Artin M., Rodriguez-Villegas F., Tate J., On the Jacobians of plane cubics, Adv. Math. 198 (2005), 366–382.
  * [4] Bosma W., Cannon J., Playoust C., The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), 235–265.
  * [5] Dickson L.E., Invariantive theory of plane cubic curves modulo 2, Amer. J. Math. 37 (1915), 107–116.
  * [6] Fisher T., The Hessian of a genus one curve, Proc. Lond. Math. Soc. 104 (2012), 613–648, math.NT/0610403.
  * [7] Fisher T., On families of 7- and 11-congruent elliptic curves, LMS J. Comput. Math. 17 (2014), 536–564.
  * [8] Fulton W., Algebraic curves. An introduction to algebraic geometry, 2008, available at http://www.math.lsa.umich.edu/~wfulton/.
  * [9] Glynn D.G., On cubic curves in projective planes of characteristic two, Australas. J. Combin. 17 (1998), 1–20.
  * [10] Kuwata M., Constructing families of elliptic curves with prescribed mod 3 representation via Hessian and Cayleyan curves, arXiv:1112.6317.
  * [11] Pascal E., Repertorium der höheren Mathematik: II. Teil: Die Geometrie, B.G. Teubner, Leipzig, 1902 (German translation of Repertorio di matematiche superiori (definizioni, formole, teoremi, cenni bibliografici), II. Geometria, Ulrico Hoepli, Milano, 1900).
  * [12] Rubin K., Silverberg A., Families of elliptic curves with constant mod $p$ representations, in Elliptic Curves, Modular Forms, & Fermat’s Last Theorem (Hong Kong, 1993), Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, 148–161.
  * [13] Silverman J.H., The arithmetic of elliptic curves, Graduate Texts in Mathematics, Vol. 106, Springer-Verlag, New York, 1986.
  * [14] Top J., Yui N., Explicit equations of some elliptic modular surfaces, Rocky Mountain J. Math. 37 (2007), 663–687, math.AG/0307230.
  * [15] Tuijp A., Hesse pencil in characteristic two, Bachelor’s Thesis, Rijksuniversiteit Groningen, 2015, available at http://fse.studenttheses.ub.rug.nl/id/eprint/13074.
  * [16] Washington L.C., Elliptic curves. Number theory and cryptography, 2nd ed., Discrete Mathematics and its Applications, Chapman & Hall/CRC, Boca Raton, FL, 2008.