# Learning Beyond Similarities: Incorporating Dissimilarities between Positive
Pairs in Self-Supervised Time Series Learning

###### Abstract

By identifying similarities between successive inputs, Self-supervised
Learning (SSL) methods for time series analysis have demonstrated their
effectiveness in encoding the inherent static characteristics of temporal
data. However, an exclusive emphasis on similarities might result in
representations that overlook the dynamic attributes critical for modeling
cardiovascular diseases within a confined subject cohort. Introducing
Distilled Encoding Beyond Similarities (DEBS), this paper pioneers an SSL
approach that transcends mere similarities by integrating dissimilarities
among positive pairs. The framework is applied to electrocardiagram (ECG)
signals, leading to a notable enhancement of +10% in the detection accuracy of
Atrial Fibrillation (AFib) across diverse subjects. DEBS underscores the
potential of attaining a more refined representation by encoding the dynamic
characteristics of time series data, tapping into dissimilarities during the
optimization process. Broadly, the strategy delineated in this study holds the
promise of unearthing novel avenues for advancing SSL methodologies tailored
to temporal data.

Index Terms—  Self-Supervised Learning, Time Series, Electrocardiogram (ECG),
Signal Processing, Atrial Fibrillation (Afib)

##  1 Introduction

Self-supervised Learning (SSL) is a promising approach to machine learning
where the model learns to predict certain aspects of a dataset without
explicit supervision or labeling of the data. In the context of time series,
SSL has several benefits: (i) the model learns generic representations that
are useful across many tasks, (ii) in some cases, such as physiological
signals, the labeling is costly, or the specific task is not known apriori,
and (iii) the model learns representations that are more robust across
different perturbations incorporated during the recording of the data.

This paper presents a novel Self-supervised Learning (SSL) method for time
series analysis, namely the Distilled Encoding Beyond Similarities
(DEBS)111Throughout this paper, the term distilled is used in its idiomatic
sense rather than in the deep learning sense., with a specific focus on the
analysis of physiological signals. The underlying concept of this approach is
based on the categorization of signal characteristics into two types: (i)
inherent static features that account for individual characteristics such as
gender and age, and (ii) dynamic features that can reveal transitional states
or events experienced by the subjects during the recording, such as heart
arrhythmias in ECG.

Fig. 1: Similarity path. $X_{1}$ and $X_{2}$ belong to the same subject

Fig. 2: Dissimilarity path. $X_{t-i}$, $X_{t}$, $X_{t+j}$ belong to the same
subject

As such, DEBS not only captures what is common but also the changes between
two positive pairs. This is achieved by enhancing the dissimilarities between
the positive pairs. We extend the ongoing research trajectory that considers
the naturally obtained multiple views as an organic source of variance in
order to avoid data augmentation. In addition, we hypothesize that (i) looking
solely for similarities can lead the representations to neglect altogether the
variance and, therefore, not to encode meaningful dynamic features contained
in the data, and (ii) incorporating a focus on the dissimilarities between
positive pairs can result in the representation of the dynamic features and,
consequently, an improvement in the performance of downstream tasks such as
AFib detection.

In summary, the contributions of this paper are: (i) We introduce DEBS, the
first SSL method that enforces the representations of positive pairs to be
dissimilar as a part of the objective function. (ii) We show that by
incorporating dissimilarities during the optimization process, it increases
the AFib detection accuracy over 10%. (iii) We open a new approach in which
dissimilarities between positive pairs should be considered for learning the
dynamic features of the signals when handling temporal data, such as ECG.

##  2 Related Work

Different SSL methods have recently been designed in the domain of time series
analysis, exploiting the characteristics of this kind of data. Mixing-up [1]
utilizes the temporal characteristics of the data for a more tailored data
augmentation by creating a time series, product of mixing two time series from
the same subject. From an alternative point-of-view, the Time-Frequency
Consistency (TF-C) [2] produces two variations of time-domain and frequency-
domain pairs associated with the input signal. The SSL method is developed to
identify similarities inherent in these paired representations.

All of the aforementioned studies require the creation of at least one version
of the same input in order to train the model to learn an invariant
representation with respect to the artificial variance introduced through the
use of data augmentation. We consider it a bottleneck in the SSL domain, due
to (i) data augmentation methods specifically designed for physiological
signals are still an ongoing area of development. Recent studies [3, 4]
indicate that the optimal approach to data augmentation in this context has
yet to be determined. (ii) Despite achieving a consensus on effective data
augmentation procedures, as seen in the field of Computer Vision, the choice
of specific data augmentation techniques remains crucial for the success of
the SSL method being employed [5, 6].

In a more promising manner, Patient Contrastive Learning (PCLR) [7] obviates
the need of data augmentation by leveraging the inherent dynamic nature of
physiological signals. It avoids using data augmentation by considering the
two time series belonging to the same subject as positive pairs.

##  3  Distilled Encoding Beyond Similarities (DEBS)

Utilizing this organic multiple views and, thereby obviating the need for data
augmentation raises an important implicit question. The object of interest
evolves across time while exhibiting changes, i.e., the dynamic
characteristics of temporal data. A SSL which only considers similarities
between these multiple views, will neglect these dynamic characteristics,
resulting in a loss of information within the representations and a consequent
low performance in identifying events in downstream tasks. Therefore, learning
beyond similarities is essential for capturing the dynamic characteristics of
the temporal data. DEBS represents the first SSL technique to incorporate
dissimilarities between positive pairs as part of the objective in addition to
similarities during the training process, with the purpose of driving the
representations to reflect what has changed within the signal and therefore,
capturing the dynamic characteristics within the representation.

###  3.1 Description of DEBS

Non-Contrastive Method:  The omission of the negative pairs enable the
proposed SSL methods to surpass the conventional emphasis on similarity. By
not considering dissimilarity among negative pairs to avoid mode collapse,
dissimilarity can be incorporated among positive pairs. As Bootstrap Your Own
Latent (BYOL) [8] framework, DEBS incorporates both a teacher network and a
student network. While the student network is optimized using Stochastic
Gradient Descent (SGD) with respect to the loss function, the teacher network
serves as an exponential moving average (EMA) of the student network,
effectively operating as a delayed version. This EMA updating rule is
described below.

$\xi\leftarrow\tau\cdot\xi+(1-\tau)\cdot\theta,$ (1)

where $\tau$, $\xi$, and $\theta$ are the updating hyperparameter, the teacher
weights, and the student weights, respectively.

In contrast to the BYOL method, DEBS integrates two projectors within both the
student and teacher networks. Consequently, two predictors are also
incorporated into the student network, deviating from using a single
predictor. As a result, the encoder generates representations that traverse
two distinct paths, namely the similarity path and the dissimilarity path, as
termed in this work. The rationale behind this design is to enable the first
path to capture static features inherent in the representation while the
second path to capture the dynamic features.

Similarity path:  The objective of the student network’s predictor is to
produce a representation that closely aligns with the one generated by the
same path in the teacher network. It is illustrated in Figure 1. The degree of
similarity serves as one of the cost functions in the proposed method, termed
the “Similarity Loss ($\mathcal{L}_{sim}$)”, and it is described as the
following:

$\mathcal{L}_{sim}(\mathbf{z}_{2}^{t},\mathbf{q}_{s}(\mathbf{z}_{1}^{s}))=1-{C}_{Sim}(\mathbf{z}_{2}^{t},\mathbf{q}_{s}(\mathbf{z}_{1}^{s}),$
(2)

where $\mathbf{z}_{2}^{t}$ and $\mathbf{q}_{s}(\mathbf{z}_{1}^{s})$ are the
representations for $X_{2}$ and $X_{1}$, computed by the teacher and the
student network, respectively. $\mathcal{C}_{Sim}$ in Eq.(3) is defined as,

$\mathcal{C}_{Sim}(\mathbf{x}_{1},\mathbf{x}_{2})=\frac{x_{1}\cdot
x_{2}}{\max\left(\left\|x_{1}\right\|_{2}\cdot\left\|x_{2}\right\|_{2},1e-8\right)}\text{.
}$ (3)

Dissimilarity path: In contrast to the similarity path, the dissimilarity path
aims to predict representations that exhibit differences between two inputs.
To achieve this, the “Dissimilarity Loss ($\mathcal{L}_{dis}$) (Eq.(4))” is
introduced. It is a cost function specifically designed to guide the
optimization process and encourage the model to generate dissimilar
representations.

$\mathcal{L}_{dis}(\mathbf{z}_{t+j}^{t},\mathbf{q}_{s}(\mathbf{z}_{t-i}^{s}))=1+{C}_{Sim}(\mathbf{z}_{t+j}^{t},\mathbf{q}_{s}(\mathbf{z}_{t-i}^{s})),$
(4)

where $\mathbf{z}_{t+j}^{t}$ and $\mathbf{q}_{s}(\mathbf{z}_{t-i}^{s})$ are
the representation vector and the representation prediction for $X_{t+j}$ and
$X_{t-i}$, computed by the teacher and the student networks, respectively.

In addition to $\mathcal{L}_{dis}$, we introduce the “Gradual Loss
($\mathcal{L}_{gra}$)” as a part of the training objective. We consider that
it is not only essential for the representations of two time points drawn from
the same subject, $X_{t-i}$ and $X_{t+j}$, to be dissimilar, but also for the
representation of $X_{t}$ to lie between them. In other words, if a subject’s
state evolves from $X_{t-i}$ to $X_{t+j}$, the representation of $X_{t}$,
i.e., $z_{t}$, should approximate an intermediate point between these two
extremes. This ensures that the temporal evolution is properly captured within
the representations. It is described as:

$\begin{split}&\mathcal{L}_{gra}(\mathbf{q}_{s}(\mathbf{z}_{t}^{s}),\mathcal{PAR}(\mathbf{z}_{t-i}^{t},\mathbf{z}_{t+j}^{t}))=\\\
&1-{C}_{Sim}(\mathbf{q}_{s}(\mathbf{z}_{t}^{s}),\mathcal{PAR}(\mathbf{z}_{t-i}^{t},\mathbf{z}_{t+j}^{t})),\end{split}$
(5)

where Pondered Average Representation (PAR) is the approximation of
$\mathbf{z}_{t}$, drawn from $\mathbf{z}_{t-i}$ and $\mathbf{z}_{t+j}$. We do
not force $\mathbf{z}_{t}$ to be equally spaced from both $\mathbf{z}_{t-i}$
and $\mathbf{z}_{t+j}$, therefore, we calculate PAR as,

$\mathcal{PAR}(\mathbf{z}_{t-i}^{t},\mathbf{z}_{t+j}^{t})=\frac{\mathbf{z}_{t-i}^{t}\cdot
j+\mathbf{z}_{t+j}^{t}\cdot i}{i+j}.$ (6)

The overall loss function that is minimized in the dissimilarity path
($\mathcal{L}_{dispath}$) is described as:

$\begin{split}&\mathcal{L}_{dispath}(\mathbf{q}_{s}(\mathbf{z}_{t}^{s}),\mathbf{q}_{s}(\mathbf{z}_{t-i}^{s}),\mathbf{z}_{t-i}^{t},\mathbf{z}_{t+j}^{t})=\\\
&\mathcal{\alpha}\cdot\mathcal{L}_{dis}(\mathbf{z}_{t+j}^{t},\mathbf{q}_{s}(\mathbf{z}_{t-i}^{s}))\\\
&+\mathcal{L}_{gra}(\mathbf{q}_{s}(\mathbf{z}_{t}^{s}),\mathcal{PAR}(\mathbf{z}_{t-i}^{t},\mathbf{z}_{t+j}^{t})),\end{split}$
(7)

where $\mathcal{\alpha}$ is the dissimilarity coefficient. This dissimilarity
path is illustrated in Figure 2.

###  3.2 Intuitions behind DEBS

Intuitions behind a two-step procedure: Our belief is that understanding the
changes in the input requires first understanding what remains constant. This
principle drives DEBS to undergo two distinct learning phases. In the initial
phase, the method focuses on reducing the variance by ensuring similarity
among representations, i.e., encoding the static characteristics of the
signals. Once these are adequately captured, DEBS drives the model to encode
the remaining variance as a source information about the dynamic nature of the
data. By doing so, the method does not neglect but understands the remaining
variance, ultimately capturing the dynamic characteristics of the temporal
data.

Intuitions behind the dissimilarity coefficient ($\mathcal{\alpha}$): Even
$\mathcal{L}_{dis}$ is minimized when the representations are entirely
dissimilar, it is not what is intended since static features should remain
constant throughout the signal. Also the dynamic characteristics need to
maintain some level of relational information. Hence, DEBS deliberately
introduces $\mathcal{\alpha}$ as a regularization factor for lowering the
weight of this objective.

Intuitions behind the window size: An essential consideration in implementing
the method is determining the appropriate spacing window size between
$\mathbf{X}_{t-i}$ and $\mathbf{X}_{t+j}$, i.e., how much these inputs may be
separated in time. This spacing window size must be large enough to
accommodate signal changes, yet narrow enough to contain only a single one. If
successive changes occur within this window, it can lead to conflicting
directions in which these changes are reflected in the representations,
thereby $\mathcal{PAR}(\mathbf{z}_{t-i}^{t},\mathbf{z}_{t+j}^{t})$ may not be
aligned with $\mathbf{q}_{s}(\mathbf{z}_{t}^{s})$.

###  3.3 Implementation details

Architecture: We use an adaptation of the Vision Transformer (ViT) [9] model
for performing physiological signals. The input data is a time series of 1000
samples, which is split into patches of size 20. The model counts with 6
regular transformer blocks with 4 heads each. The model dimension is set to
128, for a total of 1,192,616 trainable parameters.

DEBS implementation: The projectors and predictors are implemented as a two-
layer Multilayer Perceptron (MLP), with a dimensionality of 256 and 64,
respectively. Batch normalization and rectified linear unit (ReLU) operations
are incorporated between the two layers of each structure. The EMA updating
factor ($\tau$), the window size and and the $\mathcal{L}_{diss}$ coefficient
are set to 0.995, 2 minutes and 0.1 respectively.

DEBS optimization: The model is trained with the Sleep Heart Health Study
(SHHS) dataset [10, 11]. The training procedure consists of 25,000 iterations.
After 15,000 iterations, dissimilarities are integrated into the objective
function, while similarities are no longer taken into account. Before starting
the second step, we update the teacher weights as a copy of the current
student weights. We use a batch size of 256, and Adam [12] with a learning
rate of $3e-4$ and a weight decay of $1.5e-6$ as the optimizer.

##  4 Experimental evaluation

Comparison against SOTA methods: In this experiment, we evaluate DEBS against
three different baselines: (i) PCLR [7], (ii) Mixing-Up [1] and (iii) TF-C
[2]. For this evaluation, a Support Vector Classificatier (SVC) [13] is fitted
on top of the representations, based on samples obtained from the MIT-BIH
Arrhythmia Database (MIT-ARR) database [14], and evaluated in two different
databases ( MIT-BIH Atrial Fibrillation Database (MIT-AFIB) [15] and Computing
in Cardiology Challenge 2017 (CINC2017) [16]). All used datasets are publicly
available in Physionet [17].

We have optimized the same model used in this work, under the same
configuration (optimizer, data, batch size and number of iterations), except
for the TF-C method, where their proposed model has been used. This is due to
the fact that it requires the use of two encoders instead of one. Note that
this model contains approximately 32 million parameters, which is 30x more
than our proposed model. To ensure that the model converges, the latter has
been optimized over 75K iterations, instead of the 25K iterations proposed in
this work. We have saved the model ater 25K, 50K and 75K iterations.

Table 1 shows that the proposed method clearly outperform all the baselines in
the different databases.

Table 1: Comparison agains SOTA SSL Methods

Dataset | SSL Method | Accuracy (%) | Sensitivity (%) | Specificity (%)
---|---|---|---|---
MIT AFIB | PCLR [7] | 72.7 | 65.6 | 78.9
Mixing-Up [1] | 65.0 | 60.5 | 67.2
TF-C (25K) [2] | 72.2 | 65.0 | 78.5
TF-C (50K) [2] | 69.8 | 62.3 | 76.2
TF_C (75K ) [2] | 71.3 | 65.9 | 76.4
DEBS | 77.5 | 75.6 | 79.5
CINC2017 (Training) | PCLR[7] | 63.5 | 21.2 | 93.0
Mixing-Up[1] | 68.0 | 20.5 | 90.9
TF-C (25K)[2] | 62.4 | 20.5 | 92.8
TF-C (50K)[2] | 62.0 | 20.3 | 92.7
TF-C (75K)[2] | 62.4 | 20.6 | 92.9
DEBS | 78.2 | 34.0 | 95.3
CINC2017 (Validation) | PCLR [7] | 70.6 | 43.3 | 89.2
Mixing-Up [1] | 66.0 | 36.0 | 82.8
TF-C (25K) [2] | 69.2 | 42.0 | 42.0
TF-C (50K) [2] | 65.5 | 38.0 | 87.0
TF-C (75K) [2] | 67.7 | 40.7 | 89.9
DEBS | 81.8 | 59.0 | 92.7

The effect of incorporating dissimilarities: To comprehend the impact of
integrating the dissimilarity path, an analysis is conducted on the
performance of the MIT-ARR $\rightarrow$ MIT-AFIB framework during the
optimization process, with evaluations performed every 500 iterations. The
results, illustrated in Figure 3, highlight that exclusive emphasis on
similarities leads to a degradation in model performance. Conversely, the
incorporation of dissimilarities contributes to a consistent enhancement in
model performance, which leads to a difference of + 10% at the end of the
training procedure.

(a) AFib classification accuracy over iterations

(b) Effect of incorporating dissimilarities

Fig. 3: Differences between incorporating and not incorporating
dissimilarities

Study on learned representations The fundamental concept underpinning this
approach revolves around the integration of dissimilarities to facilitate the
encoding of both static and dynamic features within the model. To ascertain
the validity of this premise, a Principal Component Analysis (PCA) [18] is
executed on the generated representations from MIT-AFIB database. As depicted
in Figure 4a, temporal segments originating from the same individual exhibit
analogous positions in Principal Components 1 and 2, irrespective of the
individual’s status. Conversely, Principal Component 5 yields distinct values
between AFib and normal rhythm (NR) classifications, regardless of the
individual, as shown in Figure 4b.

(a) Subject-based PCA

(b) Event-based PCA

Fig. 4: PCA Analysis on MIT-AFIB representations

Discussion of the results: This study has demonstrated that by incorporating
dissimilarities during the training process, we can enable both static and
dynamic characteristics to be captured within the representation, as they are
projected separately into distinct components. It leads to a significant
difference in the proposed downstream task at the end of the training
procedure. These results shows substantial support for hypothesis: (i) looking
solely for similarities can lead the representations to neglect meaningful
dynamic features contained in the data, and (ii) incorporating the
dissimilarities between positive pairs results in the representation of the
dynamic features and, consequently, an improvement in the performance of
downstream tasks such as AFib identification.

##  5 Conclusion

In this paper, we have presented DEBS, the first SSL method that incorporates
dissimilarities between positive pairs. We have shown that by incorporating
this new objective function, the representations capture not only the static
nature of the data but also the dynamic features. It leads to a significant
improvement of over 10% when evaluated on AFib classification in dynamic ECG
time series data. More generally, the approach presented in this paper may
open new perspectives for improved SSL methods for handling temporal data.

## References

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  *[SSL]: Self-supervised Learning
  *[DEBS]: Distilled Encoding Beyond Similarities
  *[ECG]: electrocardiagram
  *[AFib]: Atrial Fibrillation
  *[TF-C]: Time-Frequency Consistency
  *[PCLR]: Patient Contrastive Learning
  *[BYOL]: Bootstrap Your Own Latent
  *[EMA]: exponential moving average
  *[PAR]: Pondered Average Representation
  *[ViT]: Vision Transformer
  *[MLP]: Multilayer Perceptron
  *[ReLU]: rectified linear unit
  *[SHHS]: Sleep Heart Health Study
  *[SVC]:  Support Vector Classificatier
  *[MIT-ARR]: MIT-BIH Arrhythmia Database
  *[MIT-AFIB]: MIT-BIH Atrial Fibrillation Database
  *[CINC2017]: Computing in Cardiology Challenge 2017


# CIRCE at SemEval-2020 Task 1:
Ensembling Context-Free and Context-Dependent Word Representations

Martin Pömsl
Osnabrück University
<EMAIL_ADDRESS>

&Roman Lyapin
Cogent Labs Inc.
<EMAIL_ADDRESS>
Work done during internship at Cogent Labs Inc.

###### Abstract

This paper describes the winning contribution to SemEval-2020 Task 1:
Unsupervised Lexical Semantic Change Detection (Subtask 2) handed in by team
UG Student Intern. We present an ensemble model that makes predictions based
on context-free and context-dependent word representations. The key findings
are that (1) context-free word representations are a powerful and robust
baseline, (2) a sentence classification objective can be used to obtain useful
context-dependent word representations, and (3) combining those
representations increases performance on some datasets while decreasing
performance on others.

##  1 Introduction

SemEval-2020 Task 1 poses an evaluation framework for unsupervised Lexical
Semantic Change Detection (LSCD). Its two subtasks operate on a set of non-
parallel corpus pairs from two different time periods and are evaluated
against human annotations for semantic change of a subset of words. Subtask 1
requires binary classification of whether or not the meaning of the words has
changed. Subtask 2 requires ranking the words by degree of lexical semantic
change and is evaluated in Spearman’s rank-order correlation coefficient
$\uprho$ [Schlechtweg et al. (2020]. This paper primarily addresses Subtask 2.

One of the most successful methods for predicting the degree of semantic
change of words is comparing context-free (static) semantic vector spaces.
Such models seperately induce word vectors for all words in two corpora (e.g.
with Word2Vec [Mikolov et al. (2013a]), align the resulting vector spaces and
take the distance of the word vectors as a measure of semantic change
[Schlechtweg et al. (2019].

With recent advances in language model pretraining, it is now possible to
extract context-dependent (contextualized) word representations for each use
of a word in the two corpora by using a language model (e.g. BERT [Devlin et
al. (2019]) as feature extractor. The distance between these context-dependent
word representations can then be taken as a measure of semantic change
[Giulianelli (2019].

In this paper, we present a model based on context-free word representations,
a model based on context-dependent word representations and an ensemble model
that combines their predictions. We obtain context-free representations by
following the methodology of the best model reported by ?), aligning SGNS
vectors. For context-dependent representations, we finetune BERT with a
sentence classification objective - predicting the time period of sentences -
and extract internal representations for all words from the finetuned BERT
model. We show that this classification finetuning can both produce useful
word representations and provide an indicator for how to parameterize the
ensemble.

In the evaluation phase of SemEval-2020 Task 1, the context-free model ranked
first out of 128 contributions to Subtask 2. The ensemble model performed
better in one language, but significantly worse in another, causing it to be
ranked fifth among all contributions. The context-dependent model ranked 69th,
suggesting that it can sometimes add information to the context-free model,
but is largely inadequate on its own. In the results section, we analyze the
submission experiments and show that the usefulness of the context-dependent
representations is linked to the BERT classification accuracy. Code and
predictions for all models are available publicly at
https://github.com/mpoemsl/circe.

00footnotetext:  This work is licensed under a Creative Commons Attribution
4.0 International License.
License details: http://creativecommons.org/licenses/by/4.0/.

##  2 Related Work

###  2.1 LSCD with Context-Free Word Representations

Approaches that rely on the comparison of context-free word representations in
diachronic corpora have a long history in lexical semantic change detection.
The word vectors are usually either explicitly derived from co-occurrence
statistics or implicitly through the use of neural methods [Tahmasebi et al.
(2018]. Context-free models often follow a three-step scheme: Representing
words in semantic vectors, aligning the resulting vector spaces and comparing
relevant word vectors.

?) use Skip-Gram, the neural word embedding method introduced by ?) to
represent words over multiple diachronic corpora and compare the
representations. A similar method that includes an alignment step was used by
?).

###  2.2 LSCD with Context-Dependent Word Representations

Context-dependent word representations assign a semantic vector to each word-
use within the context of its sentence, rather than to each unique word. One
way to get context-dependent word representations is using a pretrained neural
language model as a feature extractor [Peters et al. (2018]. Such
representations have proven useful for a wide range of NLP tasks [Liu et al.
(2019] and are becoming increasingly popular for LSCD. Like context-free
models, context-dependent models usually follow a three-step scheme:
Extracting semantic vectors for each use of the relevant words, clustering the
resulting semantic vectors and comparing the mean clusters with a distance
metric.

?) use context-dependent representations derived from a neural language model
as the basis for word sense tracking. ?) clusters the resulting word-use
representations into usage type distributions, which can then be compared.

##  3 Available Datasets

A complete dataset for lexical semantic change ranking as defined by ?)
consists of two lemmatized corpora from different time periods ($t_{1}$ and
$t_{2}$) and a corresponding testset. This testset contains gold ranks for a
subset of words as annotated by human experts. Predictions are made for all
target words on the basis of the two corpora and evaluated against the true
ranks in the testset.

In the development experiments, we validate our approach by following ?) in
evaluating our models on the diachronic testset DURel [Schlechtweg et al.
(2018] (de-durel) and the synchronic testset SURel [Hätty et al. (2019] (de-
surel) in combination with the corresponding corpora. It should be noted that
de-surel presents domain-specific rather than time-specific meaning
differences, but since diachronic and synchronic lexical semantic change
detection are closely related, it is still a useful development dataset.

In the submission experiments, we compute our submission for the evaluation
phase based on the diachronic datasets provided in SemEval-2020 Task 1
[Schlechtweg et al. (2020], which consist of an English (en-semeval), German
(de-semeval), Latin (ln-semeval) and Swedish (sw-semeval) dataset. Results are
averaged over submission datasets. An overview of all datasets can be found in
Table 1.

|  |  Language |  Total Sentences |  Annotated Words |  Time Period $t_{1}$ |  Time Period $t_{2}$
---|---|---|---|---|---|---
Development | de-durel |  German |  $3.2\cdot 10^{6}$ |  19 |  1750 - 1799 |  1850 - 1899
de-surel |  German |  $0.9\cdot 10^{6}$ |  21 |  - |  -
Submission | en-semeval |  English |  $0.6\cdot 10^{6}$ |  37 |  1810 - 1860 |  1960 - 2010
de-semeval |  German |  $6.1\cdot 10^{6}$ |  48 |  1800 - 1899 |  1946 - 1990
ln-semeval |  Latin |  $0.6\cdot 10^{6}$ |  40 |  -200 - 0 |  0 - 2000
sw-semeval |  Swedish |  $8.5\cdot 10^{6}$ |  31 |  1790 - 1830 |  1895 - 1903
Table 1: Properties of datasets used in development and submission
experiments.

##  4 System Overview

We present three models for SemEval-2020 Task 1 Subtask 2: A context-free
model, a context-dependent model and the ensemble model CIRCE, which stands
for Classification-Informed Representation Comparison Ensemble. For Subtask 1,
we binarize the CIRCE rank predictions by naively assuming that the upper half
of ranks has changed while the lower half has not.

###  4.1 Context-Free Model

The context-free model is structured analogously to the best performing model
reported by ?). We adopt the use of word vectors generated by Skip-Gram with
Negative Sampling (SGNS) [Mikolov et al. (2013b] as context-free word
representations. Similarly, we follow their use of orthogonal Procrustes
analysis [Schönemann (1966] to align the embeddings.

However, we diverge from the best performing model reported by ?) in that we
employ Euclidean distance rather than cosine distance to compare the aligned
representations, since this metric achieved more robust results in development
experiments.

###  4.2 Context-Dependent Model

The context-dependent model follows a similar outline to the one described in
?), with a few key changes. We adopt the use of context-dependent
representations derived from the masked language model BERT [Devlin et al.
(2019]. We also recognize the need for domain-adaptive finetuning of the
pretrained BERT model as described by ?).

However, instead of the standard language modelling objective, we use a
sentence time classification objective. This is motivated by the assumption
that a successful time classifier for sentences must learn time-specific word
features that are useful for measuring lexical semantic change.

In order to reduce the number of model parameters, we do not cluster the
resulting word-use representations, but instead directly compare all
representations of one relevant word $W_{t_{1}}$ at time $t_{1}$ and
$W_{t_{2}}$ at time $t_{2}$ using a Mean Pairwise Euclidean (MPE) distance
metric:

$d_{MPE}(W_{t_{1}},W_{t_{2}})=\frac{1}{|W_{t_{1}}|\cdot|W_{t_{2}}|}\cdot\sum_{w_{t_{1}}\in
W_{t_{1}}}\sum_{w_{t_{2}}\in W_{t_{2}}}\lVert w_{t_{1}}-w_{t_{2}}\rVert$

###  4.3 CIRCE

We propose the ensemble model CIRCE to combine the predictions of context-free
and context-dependent models. The CIRCE rank prediction $r_{CIRCE}$ is
generated from the context-free rank $r_{CF}$ and the context-dependent rank
$r_{CD}$ through linear combination with a single parameter
$\theta\in\mathopen{[}0.0,1.0\mathclose{]}$:

$r_{CIRCE}=\theta\cdot r_{CD}+(1-\theta)\cdot r_{CF}$

Development experiments confirmed that there often are values for $\theta$
that cause CIRCE to perform better than both the context-free and the context-
dependent model. This indicates that both kinds of representations contain
unique time-specific features that can be exploited in order to predict the
degree of lexical semantic change.

We employ a simple heuristic to calculate $\theta$ in an unsupervised setting
such as SemEval-2020 Task 1. Development experiments suggested that the
usefulness of context-dependent representations as measured by the optimal
value for
$\theta_{optimal}=\operatorname*{argmax}_{\theta\in\mathopen{[}0.0,1.0\mathclose{]}}\rho(r_{CIRCE},r_{true})$
roughly correlates with the time classification accuracy of the BERT model
after finetuning. Consequently, we predict $\theta$ at test time with a simple
linear mapping from the set of realistic BERT classification accuracies
$acc_{BERT}\in\mathopen{[}0.5,1.0\mathclose{]}$ to the set of valid CIRCE
weights $\theta\in\mathopen{[}0.0,1.0\mathclose{]}$:

$\theta_{CIRCE}=2\cdot(acc_{BERT}-0.5)$

##  5 Experimental Setup

###  5.1 Preprocessing

In the context-free model, we follow ?) in removing words with frequencies
below a threshold. We use $\frac{|S|}{5\cdot 10^{4}}$ for this threshold,
where $|S|$ is the number of sentences in the corpus. In order to preserve
information, we skip this step for datasets with fewer than $10^{6}$ sentences
in total.

In the context-dependent model, we create a balanced binary time
classification dataset from the two diachronic corpora. We employ a train-test
split of $0.8$ / $0.2$ for classification finetuning and evaluation.
Furthermore, we create a version of the classification dataset in which words
that occur only in one corpus are replaced by the [MASK]-token. The intention
behind this preprocessing step is to avoid the learning of rule-based features
that are not useful for LSCD such as the memorization of unique words.

###  5.2 Implementations

We use the Word2Vec111https://github.com/danielfrg/word2vec SGNS [Mikolov et
al. (2013b] implementation to create word vectors and
VecMap222https://github.com/artetxem/vecmap [Artetxe et al. (2018] to align
them. We use the Transformers333https://github.com/huggingface/transformers
library by HuggingFace [Wolf et al. (2019] to finetune BERT and to extract
context-dependent word representations.

###  5.3 Parameters

In the context-free model, we create SGNS vectors of dimensionality 300 with
window size 10 and negative sample 1. Following ?), we length-normalize and
mean-center the representations when applying orthogonal Procrustes analysis.

In the context-dependent model, we use the pretrained BERT model bert-base-
german-cased in development experiments and bert-base-multilingual-cased in
the SemEval submission experiments. We finetune the model with a sequence
classification head for one epoch at a learning rate of $4\cdot 10^{-5}$ with
a warm-up step ratio of $0.05$. We extract a context-dependent representation
vector of size 768 for each word-use by feeding the whole sentence and taking
the mean over the corresponding tokens in the activations of the last hidden
layer of BERT.

##  6 Results

###  6.1 LSCD Performance

|  |  Context-Free |  CIRCE |  Context-Dependent
---|---|---|---|---
Development | de-durel |  0.7263 |  0.8018 |  0.4157
de-surel |  0.5802 |  0.7251 |  0.7089
Submission | en-semeval |  0.4221 |  0.2465 |  0.0816
de-semeval |  0.7253 |  0.7253 |  0.2075
ln-semeval |  0.4124 |  0.4637 |  0.4439
sw-semeval |  0.5467 |  0.5467 |  0.0416
Table 2: Evaluation phase results for Subtask 2 in development and submission
experiments in Spearman’s $\uprho$. For each dataset all results with the
maximum value are underlined.

As Table 2 shows, the context-free model reliably scores well on all datasets
for Subtask 2, while the context-dependent model only punctually matches its
performance. In cases where both models achieve good results (e.g. de-surel,
ln-semeval), the ensemble model CIRCE is able to exceed both predictions.

In the evaluation phase results for Subtask 2, the context-free submission
ranks first with a mean correlation of 0.527 over all submission datasets,
while the CIRCE submission ranks fifth at a mean correlation of 0.495. The
context-dependent submission ranks 69th at a mean correlation of 0.194.

In the evaluation phase results for Subtask 1, the binarized CIRCE predictions
ranks fifth as well with a mean accuracy of 0.639. On en-semeval, it achieved
an accuracy of 0.568, on de-semeval 0.728, on ln-semeval 0.550 and on sw-
semeval 0.710.

###  6.2 Time Classification Performance

| Time Clf. Accuracy | CIRCE Weight $\theta$ | LSCD Spearman’s $\uprho$
---|---|---|---
|  |  $\theta_{CIRCE}$ |  $\theta_{optimal}$ |  $\theta_{CIRCE}$ |  $\theta_{optimal}$
de-durel |  0.59 |  0.18 |  0.34 |  0.8018 |  0.8772
de-surel |  0.95 |  0.90 |  0.92 |  0.7251 |  0.7264
en-semeval |  0.82 |  0.64 |  0.08 |  0.2465 |  0.4274
de-semeval |  0.50 |  0.00 |  0.08 |  0.7253 |  0.7345
ln-semeval |  0.73 |  0.46 |  0.74 |  0.4637 |  0.4986
sw-semeval |  0.50 |  0.00 |  0.16 |  0.5467 |  0.5726
Table 3: Time classification accuracies (independent of $\theta$) as well as
evaluation phase ($\theta_{CIRCE}$) and post-evaluation phase
($\theta_{optimal}$) CIRCE weights and Subtask 2 performances in Spearman’s
$\uprho$.

As Table 3 shows, time classification accuracy of BERT after finetuning varies
widely across different datasets. Most remarkably, BERT entirely fails to
optimize the time classification objective on de-semeval and sw-semeval, while
it achieves exceptionally good results on de-surel.

In general, the classification accuracy seems to be a good predictor for the
optimal weight - the submission weight $\theta_{CIRCE}$ calculated with the
linear mapping is often within $\pm 0.20$ of the optimal weight
$\theta_{optimal}$, and consequently the submission correlations are often
close to the optimal correlations.

One notable exception is en-semeval. The high classification accuracy leads to
a predicted $\theta_{CIRCE}$ of 0.64, while $\theta_{optimal}$ is located at
0.08. As a result, CIRCE falls short of its potential in the results for
Subtask 2 and ranks below the context-free model despite gains on ln-semeval,
where CIRCE improves by $0.05$ upon the context-free and by $0.02$ upon the
context-dependent model.

###  6.3 Error Analysis

The failure to achieve a significant time classification accuracy for de-
semeval and sw-semeval is curious, but might simply be due to the properties
of the datasets. Predicting the time period of a given sentence can be a
difficult task even for humans, depending on several factors such as the
occurrence of unique words (which are masked in the classification dataset)
and the distinctiveness of grammatical structures.

The mismatch of predicted submission weight $\theta_{CIRCE}$ and optimal
weight $\theta_{optimal}$ in the case of en-semeval is a more severe error,
since it challenges the notion that classification accuracy is a good
indicator for representation usefulness. Without extensive experiments, it is
difficult to determine the cause of this outlier. However, one contributing
factor might be the masking preprocessing step. While the distribution of
labels in the BERT classification dataset is balanced, the distribution of
[MASK]-tokens is not, since there is often one time period that has more words
unique to it than the other. This makes it possible for BERT to learn
[MASK]-specific rather than time-specific features during finetuning, which
would cause the representations to be useful for time classification but not
for LSCD.

In line with this, repeating the submission experiments without masking in the
post-evaluation phase causes the classification accuracy on en-semeval to drop
and the predicted submission weight to come within $\pm 0.20$ of the optimal
weight for all datasets. This modification boosts the overall mean correlation
of CIRCE on the submission datasets to 0.545, which exceeds the performance of
all systems for Subtask 2 in the evaluation phase. However, without further
experiments on other datasets, the results of this modified model cannot be
considered conclusive.

##  7 Conclusion

We presented a context-free model, a context-dependent model and an ensemble
model for SemEval-2020 Task 1 Subtask 2. We showed that while the context-free
model outperformed all other systems during the evaluation phase, ensembling
its predictions with those of the context-dependent model leads to increased
performance on the development datasets (de-surel, de-durel) and one
submission dataset (ln-semeval) at the cost of decreased performance on
another submission dataset (en-semeval).

The submission experiments have made it clear that, although its performance
in the development experiments was competitive, the context-dependent model on
its own is not a reliable predictor of lexical semantic change. However, it is
worth pointing out that the context-dependent representations in all cases
contain at least some information that is useful for LSCD, since the optimal
CIRCE weight is greater than zero in all experiments. Still, the marginal
improvements accomplished with this information do not justify the significant
computational effort of finetuning BERT for one epoch.

In further research, it would be interesting to validate the methods described
in this paper on additional datasets. In particular, it could be worthwhile to
empirically explore the link between classification accuracy and
representation usefulness. If the relation were to hold up for other domains
and models with a lower computational complexity than BERT, representations
obtained through self-supervised classification training could be used on a
whole range of other unsupervised tasks.

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# Minimal Timelike Surfaces in the Lorentz-Minkowski 3-space and their
Canonical Parameters

Ognian Kassabov and Velichka Milousheva  Institute of Mathematics and
Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str. bl. 8, 1113,
Sofia, Bulgaria<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We study minimal timelike surfaces in $\mathbb{R}^{3}_{1}$ using a special
Weierstrass-type formula in terms of holomorphic functions defined in the
algebra of the double (split-complex) numbers. We present a method of
obtaining an equation of a minimal timelike surface in terms of canonical
parameters, which play a role similar to the role of the natural parameters of
curves in $\mathbb{R}^{3}$. Having one holomorphic function that generates a
minimal timelike surface, we find all holomorphic functions that generate the
same surface. In this way we give a correspondence between a minimal timelike
surface and a class of holomorphic functions. As an application, we prove that
the Enneper surfaces are the only minimal timelike surfaces in
$\mathbb{R}^{3}_{1}$ with polynomial parametrization of degree 3 in isothermal
parameters.

###### Key words and phrases:

Timelike surfaces, canonical parameters, Weierstrass formula

2020 Mathematics Subject Classification: 53A10; 53B30; 53C50

##  1\. Introduction

The study of minimal surfaces is one of the main topics in classical
differential geometry which goes back to the 18th century. Lagrange initiated
in 1760 the study of minimal surfaces in Euclidean 3-space and found the
minimal surface equation when he looked for a necessary condition for
minimizing the area functional. He showed that a minimal surface parametrized
as a graphic $x=(u,v,\varphi(u,v))$ satisfies the following equation, known
nowadays as the Lagrange’s equation, 1762:

$(1+\varphi_{v}^{2})\varphi_{uu}-2\varphi_{u}\varphi_{v}\varphi_{uv}+(1+\varphi_{u}^{2})\varphi_{vv}=0.$

The link between curvature and minimal surfaces was made by Meusnier in 1776
who proved that the Lagrange’s equation implies that the mean curvature is
zero everywhere on a minimal surface. Usually, minimal surfaces are defined as
surfaces with zero mean curvature, but they are also characterized as surfaces
of minimal surface area for given boundary conditions, as a critical point of
the area functional, or as a graphic of the solution of a differential
equation.

The Weierstrass representation formula (1866) describes minimal surfaces in
terms of two holomorphic functions $f(z)$ and $g(z)$ as follows [18]:

$\Psi(z)=\Re\int_{z_{0}}^{z}\,\left(\frac{1}{2}f(z)(1-g^{2}(z)),\frac{i}{2}f(z)(1+g^{2}(z)),f(z)g(z)\right)\,dz.$

The theory of minimal surfaces in real space forms have been attracting the
attention of many mathematicians for more than two centuries and have inspired
many authors to study minimal surfaces in other ambient spaces. In the last
years, great attention is paid to Lorentz surfaces in pseudo-Euclidean spaces,
since pseudo-Riemannian geometry has many important applications in Physics,
especially in problems related to General Relativity.

However, the local geometry of surfaces in the Lorentz-Minkowski space
$\mathbb{R}^{3}_{1}$ is much more complicated than that in the Euclidean space
$\mathbb{R}^{3}$, since in $\mathbb{R}^{3}_{1}$ the vectors have different
casual characters (spacelike, timelike or lightlike), which yield more cases
to be considered. One could consider spacelike, timelike or lightlike surfaces
in $\mathbb{R}^{3}_{1}$. In the present paper we consider minimal timelike
surfaces. Although in the timelike case the minimal surfaces neither maximize
nor minimize surfaces area, they have many geometric properties similar to
minimal surfaces in the Euclidean space $\mathbb{R}^{3}$. For example,
Weierstrass representation formula was introduced by M. Magid in [16]. While
the classical Weierstrass representation theorem uses a relationship between
holomorphic functions and solutions of certain elliptic PDEs, in the case of
minimal timelike surfaces in $\mathbb{R}^{3}_{1}$ the problem of finding a
Weierstrass type representation is related to solving a certain system of
hyperbolic PDEs. This shows that the difference between the pseudo-Riemannian
and Riemannian case is as between hyperbolic and elliptic PDEs. In [16], M.
Magid obtained local results. A global version of the Weierstrass
representation theorem for Lorentz surfaces in $\mathbb{R}^{3}_{1}$ is given
by J. Konderak in [14]. After the work of Konderak, many researches have
studied spacelike and timelike surfaces in a 3-dimensional Lorentz space via
Weierstrass-type formulas, see e.g. [2], [15], [17].

In the Euclidean space, the classical Björling problem, proposed by Björling
in 1844, is related to the construction of a minimal surface in
$\mathbb{R}^{3}$ containing a prescribed analytic strip. The Björling problem
for timelike surfaces in the Lorentz-Minkowski spaces $\mathbb{R}^{3}_{1}$ was
solved in [1], where a representation formula was obtained by use of split-
complex numbers and natural split-complex extensions. The split-complex
numbers are also known as Lorentz numbers, para-complex, double or hyperbolic
numbers, and play a role similar to that played by the ordinary complex
numbers in the spacelike case. The algebra of Lorentz numbers is often used in
the study of timelike surfaces, see for instance [4], [5], [12].

In differential geometry, in the study of surfaces in both Euclidean and
pseudo-Euclidean geometry, it is appropriate to use different types of special
parameters: isothermal, principal, asymptotic, geodesic, etc., in different
particular cases. In this paper, we deal with a special kind of parameters,
which play a role similar to the role of the natural parameters of a curve
which are essential in the theory of curves in $\mathbb{R}^{3}$. Note that two
important invariants – the curvature and the torsion of a regular curve,
parametrized by natural parameters, determine the curve up a motion in the
space.

Actually, natural parameters of a surface are not known in the general case,
even in the Euclidean space $\mathbb{R}^{3}$. However, such parameters were
introduced for the Weingarten surfaces which form a wide class of surfaces in
$\mathbb{R}^{3}$ [10]. Some results in [10] were specialized in [6] for the
class of minimal surfaces (i.e. surfaces with vanishing mean curvature) in
$\mathbb{R}^{3}$ and in this case the special parameters are called canonical
principal parameters. They are determined up to renumbering, sign and additive
constants. The normal curvature, expressed in terms of canonical principal
parameters, satisfies a special partial differential equation determining the
surface up to a motion in $\mathbb{R}^{3}$. Note that a surface may be
parametrized in quite different ways and it is not easy to say whether or not
two parametric equations define one and the same surface. The normal curvature
of minimal surfaces gives a simple method to answer this question: one may
simply calculate the normal curvatures of two minimal surfaces in terms of
canonical principal parameters and then compare the results. The problem is
that usually a minimal surface is defined in arbitrary (not canonical)
parameters. An effective method to solve this problem was proposed in [13].

Similar questions may be considered for surfaces with vanishing mean curvature
in the Lorentz-Minkowski space $\mathbb{R}^{3}_{1}$. An important tool to
investigate the minimal surfaces in $\mathbb{R}^{3}$ are the complex numbers
and the complex functions (on the field of complex numbers), see e.g. [11].
These functions are also convenient in the study of spacelike surfaces in
$\mathbb{R}^{3}_{1}$ with vanishing mean curvature – the maximal surfaces.

In the present paper, we are interested in minimal timelike surfaces in
$\mathbb{R}^{3}_{1}$, so we apply the theory of functions on the algebra of
double numbers. We use also the special Weierstrass formula of G. Ganchev,
proposed in [7], where a new approach to timelike surfaces is given and a
determination of a minimal timelike surface via canonical parameters is
established. We find a method of obtaining an equation of a minimal timelike
surface in canonical parameters.

Having one holomorphic function (defined on a domain of the plane of double
numbers) that generates a minimal timelike surface, we find all holomorphic
functions that generate the same surface. Thus we obtain a correspondence
between a minimal timelike surface and a class of holomorphic functions. As an
application, we prove that the Enneper surfaces are the only minimal timelike
surfaces in $\mathbb{R}^{3}_{1}$ with polynomial parametrization of degree 3
in isothermal parameters.

##  2\. Preliminaries

We deal with the 3-dimensional Lorentz-Minkowski space $\mathbb{R}^{3}_{1}$,
endowed with the standard flat metric of signature $(2,1)$:

$\langle x,y\rangle=-x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}\ .$

The considerations in this paper are local and all functions are supposed to
be of class $C^{\infty}$.

A regular surface $S$ in $\mathbb{R}^{3}_{1}$ is said to be:

\- timelike, if the restriction of $\langle.,.\rangle$ to each tangent space
of $S$ is indefinite;

\- spacelike, if the restriction of $\langle.,.\rangle$ to each tangent space
of $S$ is positive definite;

\- lightlike, if the restriction of $\langle.,.\rangle$ to each tangent space
of $S$ is degenerate.

In what follows, we suppose that the surface $S$ is timelike and is defined by
the parametric equation

${\bf x=x}(u,v)=(x_{1}(u,v),x_{2}(u,v),x_{3}(u,v)),\qquad(u,v)\in
U\subset\mathbb{R}^{2}.$

We denote the derivatives of the vector function ${\bf x=x}(u,v)$ by

${\bf x}_{u}=\frac{\partial\bf x}{\partial u},\qquad{\bf
x}_{v}=\frac{\partial\bf x}{\partial v},\qquad{\bf
x}_{uv}=\frac{\partial^{2}\bf x}{\partial u\partial v},\ ...$

The coefficients of the first fundamental form are given by

$E=\langle{\bf x}_{u},{\bf x}_{u}\rangle,\qquad F=\langle{\bf x}_{u},{\bf
x}_{v}\rangle,\qquad G=\langle{\bf x}_{v},{\bf x}_{v}\rangle.$

Denote by ${\bf U}$ the unit normal to the surface, i.e.

${\bf U}=\frac{{\bf x}_{u}\times{\bf x}_{v}}{|{\bf x}_{u}\times{\bf x}_{v}|}.$

Then, the coefficients of the second fundamental form are

$L=\langle{\bf U},{\bf x}_{uu}\rangle,\qquad M=\langle{\bf U},{\bf
x}_{uv}\rangle,\qquad N=\langle{\bf U},{\bf x}_{vv}\rangle.$

The Gauss curvarture and the mean curvature of $S$ are defined respectively by

$K=\frac{LN-M^{2}}{EG-F^{2}},\qquad H=\frac{EN-2FM+GL}{2(EG-F^{2})}.$

The surface $S$ is said to be minimal if the mean curvature vanishes
identically.

Probably, the most used parameters of a surface are the isothermal parameters.
The coefficients of the first fundamental form of a timelike surface in
$\mathbb{R}^{3}_{1}$ parametrized in terms of isothermal parameters satisfy
$E=-G$, $F=0$. In the study of minimal timelike surfaces in
$\mathbb{R}^{3}_{1}$ we use the algebra $\mathbb{D}$ of the double numbers,
which is determined in the following way: $\mathbb{D}=\\{a+\mathrm{j}b:\
a,b\in\mathbb{R},\ \mathrm{j}^{2}=1\\}$, where $\mathrm{j}$ commutes with the
elements of $\mathbb{R}$. For the element $z=a+\mathrm{j}b$ of $\mathbb{D}$ we
have $|z|^{2}=z\bar{z}=(a+\mathrm{j}b)(a-\mathrm{j}b)=a^{2}-b^{2}$. This shows
that $\mathbb{D}$ is the hyperbolic analogue of the algebra of complex numbers
$\mathbb{C}$ and reflects the Lorentz geometry. The algebra of double numbers
is used essentially in paper [9] in the study of the Lorentz surfaces in
$\mathbb{R}^{4}_{2}$.

Let $f(z)$ and $g(z)$ be two holomorphic functions defined in a domain of
$\mathbb{D}$. Consider the Weierstrass curve, defined by

$\Psi(z)=\int_{z_{0}}^{z}\,\left(-\frac{1}{2}\,f(z)(1+g^{2}(z)),\frac{\mathrm{j}}{2}\,f(z)(1-g^{2}(z)),f(z)g(z)\right)\,dz\
.$

The real and the ”imaginary” parts ${\bf x}(u,v)$ and ${\bf y}(u,v)$ define
two minimal timelike surfaces of Gauss curvature $K<0$ and $-K$, respectively.

For example, with the functions $f(z)=1,\ g(z)=z$ we obtain the classical
Enneper surface of negative Gauss curvature

${\bf
x}(u,v)=\left(-\frac{u}{6}(u^{2}+3v^{2}+3),-\frac{v}{6}(3u^{2}+v^{2}-3),\frac{1}{2}(u^{2}+v^{2})\right)$

as the real part of the Weierstrass curve and the classical Enneper surface of
positive Gauss curvature

${\bf
y}(u,v)=\left(-\frac{v}{6}(3u^{2}+v^{2}+3),-\frac{u}{6}(u^{2}+3v^{2}-3),uv\right)$

as the ”imaginary” part of the Weierstrass curve.

Enneper surface with $K<0$

Enneper surface with $K>0$

Figure 1. Enneper surfaces

Conversely, every minimal timelike surface can be obtained at least locally in
this way. Note, however, that a minimal timelike surface can be generated via
the Weierstrass formula by different pairs of holomorphic functions on the
algebra of double numbers.

In his study of minimal timelike surfaces in $\mathbb{R}^{3}_{1}$, G. Ganchev
[7] specialized the Weierstrass formula by introducing special isothermal
parameters, called canonical. If the surface is parametrized with respect to
canonical parameters, then the coefficients of the first and second
fundamental forms are

(2.1) $\begin{array}[]{l}\displaystyle-E=G=\frac{1}{\sqrt{-K}}>0,\qquad
F=0,\vspace{1mm}\\\ \displaystyle L=-1,\qquad M=0,\qquad N=-1,\end{array}$

in the case of minimal surfaces of negative Gauss curvature $K$, and

(2.2) $\begin{array}[]{l}\displaystyle E=-G=\frac{1}{\sqrt{K}}>0,\qquad
F=0,\\\ \displaystyle L=0,\qquad M=1,\qquad N=0,\end{array}$

in the case of minimal surfaces of positive Gauss curvature $K$. Note that,
because of (2.1) the parametric lines are principal in the case of minimal
surfaces with $K<0$. Analogously, because of (2.2) the parametric lines are
asymptotic in the case of minimal surfaces with $K>0$. The idea of Ganchev
leads to the special Weierstrass curve

(2.3)
$\Phi(z)=\int_{z_{0}}^{z}\,\left(-\frac{1}{2}\,\frac{1+g^{2}(z)}{g^{\prime}(z)},\frac{\mathrm{j}}{2}\,\frac{1-g^{2}(z)}{g^{\prime}(z)},\frac{g(z)}{g^{\prime}(z)}\right)\,dz\
.$

The real (resp. the ”imaginary”) part of this curve is a minimal timelike
surface with canonical parametrization and negative (resp. positive) Gauss
curvature. We shall use also the following theorem:

Theorem A [7]. If a timelike surface in $\mathbb{R}^{3}_{1}$ with non-
vanishing Gauss curvature is parametrized by canonical parameters $(u,v)$,
then the Gauss curvature $K$ satisfies the equation

$(\ln\sqrt{-K})_{uu}-(\ln\sqrt{-K})_{vv}=2\sqrt{-K},$

in the case $K<0$, and

$(\ln\sqrt{K})_{uu}-(\ln\sqrt{K})_{vv}=2\sqrt{K},$

in the case $K>0$. Conversely, for any solution $K(u,v)$ of any of these
equations there exists a unique (up to position in the space) minimal timelike
surface of Gauss curvature $K(u,v)$, $(u,v)$ being canonical parameters.

The canonical parameters $(u,v)$ are determined uniquely up to the following
transformations [7]:

(2.4) $\begin{array}[]{l}\vspace{2mm}u=\varepsilon\bar{u}+A\\\
\vspace{2mm}v=\varepsilon\bar{v}+B\end{array}\qquad\varepsilon=\pm 1\ ,\
A=const,\ B=const.$

The idea of canonical parameters is further developed for the class of
timelike surfaces in $\mathbb{R}^{n}_{1}$, see [8].

##  3\. Transformation of the isothermal parameters to canonical ones

Suppose the surface $S$ is defined as the real part of the Weierstrass curve

(3.1)
$\Psi(z)=\int_{z_{0}}^{z}\,\left(-\frac{1}{2}\,f(z)(1+g^{2}(z)),\frac{\mathrm{j}}{2}\,f(z)(1-g^{2}(z)),f(z)g(z)\right)\,dz\
.$

We look for a transformation $z=z(w)$ such that the curve $\Phi(w)=\Psi(z(w))$
has the form

$\Phi(w)=\int_{w_{0}}^{w}\,\left(-\frac{1}{2}\,\frac{1+\tilde{g}^{2}(w)}{\tilde{g}^{\prime}(w)},\frac{\mathrm{j}}{2}\,\frac{(1-\tilde{g}^{2}(w))}{\tilde{g}^{\prime}(w)},\frac{\tilde{g}(w)}{\tilde{g}^{\prime}(w)}\right)\,dw$

for some holomorphic function $\tilde{g}(w)$. The real part of this curve will
be a canonical representation of the given surface $S$. The equality
$\Psi(z(w))=\Phi(w)$ implies
$\Psi^{\prime}(z(w))z^{\prime}(w)=\Phi^{\prime}(w)$. Hence, it is easy to
derive

(3.2) $f(z(w))z^{\prime}(w)=\frac{1}{\tilde{g}^{\prime}(w)}\ ,\qquad\qquad
g(z(w))=\tilde{g}(w)\ .$

The last equality implies

$\tilde{g}^{\prime}(w)=g^{\prime}(z(w))z^{\prime}(w)$

and using the first equality of (3.2) we obtain

(3.3) $(z^{\prime}(w))^{2}=\frac{1}{f(z(w))g^{\prime}(z(w))}\ .$

Now, we know also the function $\tilde{g}(w)=g(z(w))$ that generates the
surface in canonical parameters.

Similar considerations can be done in the case the surface is defined as the
”imaginary” part of the Weierstrass curve determined by (3.1).

So, we can state the following result:

######  Theorem 3.1.

Let the minimal timelike surface $S$ be defined by the real or ”imaginary”
part of (3.1). Any solution to differential equation (3.3) defines a
transformation of the isothermal parameters of $S$ to canonical ones.
Moreover, the function $\tilde{g}(w)$ that defines $S$ via formula (2.3) is
given by $\tilde{g}(w)=g(z(w))$.

As a consequence, we may obtain also relations (2.4) between two different
pairs of canonical parameters.

As an application of Theorem 3.1, consider the minimal surfaces $S$ (of
negative Gauss curvature) generated by the functions

$f(z)=a\ ,\qquad g(z)=bz+c$

via the Weierstrass formula, $a,b,c$ being double numbers, $a\neq 0$, $b\neq
0$, $ab\neq 0$. Equation (3.3) takes the form

$(z^{\prime}(w))^{2}=\frac{1}{ab}$

and has the following solution

$z(w)=\pm\frac{w}{\sqrt{a}\sqrt{b}}+const.$

According to (2.4) and Theorem 3.1, we may replace $z$ in $g(z)$ with
$\displaystyle{\frac{z}{\sqrt{a}\sqrt{b}}-\frac{c}{b}}$ and we will obtain a
parametrization of the surface $S$ in canonical parameters via formula (2.3)
and the function

$\tilde{g}(z)=g\left(\frac{z}{\sqrt{a}\sqrt{b}}-\frac{c}{b}\right)=\frac{\sqrt{b}}{\sqrt{a}}\,z\
.$

The Gauss curvature is given by the following formula:

$K=-\frac{16\left|\frac{b}{a}\right|^{2}}{\left(1-\left|\frac{b}{a}\right|(u^{2}-v^{2})\right)^{4}}\
.$

This result shows that the surface $S_{0}$ of negative Gauss curvature
generated via the Weierstrass formula by the functions

$f(z)=\frac{|a|}{|b|}\ ,\qquad g(z)=z,$

has the same Gauss curvature in canonical parameters. Hence, due to Theorem A
we may identify $S$ with $S_{0}$. On the other hand, the Weierstrass formula
implies that $S_{0}$ is homothetic to the standard timelike Enneper surface
($a=b=1$) with $K<0$.

So, as in the case of minimal surfaces in the Euclidean space [3], we have

######  Corollary 3.2.

The minimal timelike surface generated by the pair of linear functions
$f(z)=a,\,g(z)=bz+c$ via the Weierstrass formula coincides with the Enneper
surface up to position in the space and homothety.

##  4\. Holomorphic functions generating a minimal timelike surface

As we said before, a minimal timelike surface is generated by different pairs
of holomorphic functions via the Weierstrass formula. For example, the Enneper
surface of negative (resp. positive) Gauss curvature is the real (resp.
”imaginary”) part of the curve defined via the Weierstrass formula by the pair
of functions

(4.1) $f(z)=1;\qquad g(z)=z,$

but also by

(4.2) $f(z)=e^{z};\qquad g(z)=e^{z},$

and, of course, by many others. So, the following natural question arises:
under what conditions do two pairs of holomophic functions give rise to one
and the same minimal timelike surface via the Weierstrass representation? It
is not difficult to prove the following:

######  Proposition 4.1.

Suppose the pairs $(\tilde{f}(z),\tilde{g}(z))$ and $(f(w),g(w))$ generate two
minimal timelike surfaces via the Weierstrass formula. Then, these surfaces
coincide (up to translation) if and only if there exists a function $w=w(z)$,
such that

$\tilde{g}(z)=g(w(z))\qquad{\rm and}\qquad\tilde{f}(z)=f(w(z))w^{\prime}(z)\
.$

For the two pairs (4.1) and (4.2), that generate the Enneper surface, the
function $w(z)=e^{z}$ transfers the first pair into the second one.

Similarly, the following question related to formula (2.3) arises: what is the
relation between the functions that generate a minimal timelike surface in
canonical parameters? A result in this direction is given by the following
theorem.

######  Theorem 4.2.

Let the holomorphic function $g(z)$ (defined on a domain of $\mathbb{D}$)
generate a minimal timelike surface $S$ in canonical parameters, i.e. via
formula (2.3). Then, for an arbitrary real number $\varphi$ and an arbitrary
double number $\alpha$, by the transformations

(4.3) $\tilde{g}(z)=\pm e^{\varphi
j}\frac{\alpha+g(z)}{1+\bar{\alpha}g(z)};\qquad\qquad\tilde{g}(z)=\pm
e^{\varphi j}\frac{1}{f(z)},$

we obtain the same (up to position in the space) surface in canonical
parameters. Conversely, any function that generates (up to position) the
surface $S$ in canonical parameters may be obtained in this way.

Proof. Let us consider the first transformation. Denote by $S$ the minimal
timelike surface of negative Gauss curvature, generated via formula (2.3) by
the function $g(z)$ and let $\Psi(z)$ be the corresponding curve. Analogously,
we define $\widetilde{S}$ and $\widetilde{\Psi}(z)$.

We may prove that $S$ and $\widetilde{S}$ coincide (up to position) by a
direct computation of their Gauss curvatures using the formula

$K=-\frac{16|g^{\prime}|^{4}}{(1-|g|^{2})^{4}}$

and applying Theorem A. Now we give another proof, thus clarifying the
relation between $S$, $\widetilde{S}$ and transformation (4.3). We have

$\Psi^{\prime}(z)=\left(-\frac{1+g^{2}(z)}{2g^{\prime}(z)},\mathrm{j}\frac{1-g^{2}(z)}{2g^{\prime}(z)},\frac{g(z)}{g^{\prime}(z)}\right)\
,$
$\widetilde{\Psi}^{\prime}(z)=\left(-\frac{1+\widetilde{g}^{2}(z)}{2\widetilde{g}^{\prime}(z)},\mathrm{j}\frac{1-\widetilde{g}^{2}(z)}{2\widetilde{g}^{\prime}(z)},\frac{\widetilde{g}(z)}{\widetilde{g}^{\prime}(z)}\right)\
.$

Let $\alpha=a+\mathrm{j}b$, $a,b\in\mathbb{R}$. Define the SO(1,2)-matrices

$A=\left(\begin{array}[]{ccc}\cosh\varphi&\sinh\varphi&0\\\
\sinh\varphi&\cosh\varphi&0\\\ 0&0&1\end{array}\right);\qquad\qquad
B=\left(\begin{array}[]{ccc}\frac{1+a^{2}+b^{2}}{1-a^{2}+b^{2}}&\frac{-2ab}{1-a^{2}+b^{2}}&\frac{-2a}{1-a^{2}+b^{2}}\vspace{.1cm}\\\
\frac{2ab}{1-a^{2}+b^{2}}&\frac{1-a^{2}-b^{2}}{1-a^{2}+b^{2}}&\frac{-2b}{1-a^{2}+b^{2}}\vspace{.1cm}\\\
\frac{-2a}{1-a^{2}+b^{2}}&\frac{2b}{1-a^{2}+b^{2}}&\frac{1+a^{2}-b^{2}}{1-a^{2}+b^{2}}\end{array}\right).$

A straightforward verification shows that

$A\,B\,\Psi^{\prime}(z)=\widetilde{\Psi}^{\prime}(z)\ .$

The last equality implies that up to translation

$A\,B\,{\bf x}(u,v)=\tilde{\bf x}(u,v)\ .$

Hence, the considered transformation of type (4.3) of the function $g(z)$
corresponds to a motion of the surface $S$.

Conversely, it is clear that any surface that coincides (up to position) with
$S$ may be obtained from $S$ using as above two $SO(1,2)$ matrices and a
translation.

∎

As an application of Theorem 4.2, we may prove that any polynomial minimal
surfaces $S$ which has polynomial parametrization of degree 3 in isothermal
coordinates is (up to position and homothety) an Enneper surface. Namely, we
have the following result:

######  Theorem 4.3.

Let the minimal timelike surface $S$ of negative Gauss curvature has
polynomial parametrization of degree 3 in isothermal parameters. Then, up to
position in space and homothety, $S$ is (a part of) the Enneper surface of
negative curvature.

Proof. Suppose that the surface is defined by

$S\ :\ \ {\bf x}={\bf x}(u,v).$

Similarly to the case of surfaces in the Euclidean space (see e.g. section
22.4 in [11]), $S$ is the real part of the Weierstrass curve obtained by
substituting the double number variables $\displaystyle{\frac{z}{2}}$ and
$\displaystyle{\frac{z}{2\mathrm{j}}}$ formally in the places of the real
variables $u,v$:

$\Psi(z)=2{\bf x}\left(\frac{z}{2},\frac{z}{2\mathrm{j}}\right)-{\bf x}(0,0).$

Using a translation (if necessary) we may assume that ${\bf x}(0,0)=0$. Since
the curve $\Psi(z)$ is a cubic polynomial, then

$\Psi^{\prime}(z)=(\phi_{1}(z),\phi_{2}(z),\phi_{3}(z))=\left(-\frac{f(z)}{2}(1+g^{2}(z)),\frac{\mathrm{j}}{2}f(z)(1-g^{2}(z)),f(z)g(z)\right)\,dz\
$

for some functions $f(z)$ and $g(z)$ and the functions $\phi_{i}(z)$,
$i=1,2,3$ are polynomials of degrees at most 2. Moreover, at least one of them
is of degree exactly 2. Hence, the same is true for the following three
functions

(4.4) $f(z)=-\phi_{1}+\mathrm{j}\phi_{2};\qquad
f(z)g^{2}(z)=-\phi_{1}-\mathrm{j}\phi_{2};\qquad f(z)g(z)=\phi_{3}\ .$

So, $f(z)$ is a polynomial of degree at most 2. From the third equality of
(4.4) we have $g=\displaystyle{\frac{\phi_{3}}{f}}$. Since $\phi_{3}$ and $f$
are polynomials, we may write $g(z)$ in the form

$g(z)=\frac{P(z)}{Q(z)}\ ,$

where the polynomials $P(z)$ and $Q(z)$ have no common zeros. If we assume
that $Q(z)$ is a constant, then $g(z)$ is a polynomial and having in mind
(4.4), we get $f(z)=const$, $g(z)=cz+d$. If we assume that $Q(z)$ is not a
constant, then from the second equality of (4.4), we get

$f(z)\frac{P^{2}(z)}{Q^{2}(z)}=-\phi_{1}-j\phi_{2},$

which is a polynomial and hence

$f(z)=\pm(az+b)^{2}=\pm Q^{2}(z)\ .$

Up to symmetry of the surface, we assume that

$f(z)=(az+b)^{2},\qquad Q(z)=az+b\ .$

Now, $\psi_{3}=f(z)g(z)=(az+b)P(z)$ and since it is of degree at most 2, then
$P(z)$ is of degree at most 1, i.e. $P(z)=cz+d$.

Hence, we conclude that, up to homothety of the surface, $f(z)$ and $g(z)$
have the form

$f(z)=(az+b)^{2};\qquad g(z)=\frac{cz+d}{az+b}\ .$

On the other hand, the Enneper surface is generated in canonical parameters by
the pair of functions $f_{1}(z)=1$, $g_{1}(z)=z$ and due to Theorem 4.2 also
by the functions

$g_{2}(z)=e^{\varphi\mathrm{j}}\frac{\alpha+z}{1+\bar{\alpha}z};\qquad\qquad
f_{2}(z)=\frac{1}{g_{2}^{\prime}(z)}=e^{-\varphi\mathrm{j}}\frac{(1+\bar{\alpha}z)^{2}}{|\alpha|^{2}-1}.$

We can change the parameter $z$ by

$\frac{(a\alpha e^{\varphi\mathrm{j}}-c)z+\alpha
be^{\varphi\mathrm{j}}-d}{(\bar{\alpha}c-ae^{\varphi\mathrm{j}})z+\bar{\alpha}d-be^{\varphi\mathrm{j}}}$

and then the generating functions take the form:

$f_{3}(z)=\frac{e^{\varphi\mathrm{j}}(1-|\alpha|^{2})(az+b)^{2}}{\Big{(}(\bar{\alpha}c-ae^{\varphi\mathrm{j}})z+\bar{\alpha}d-be^{\varphi\mathrm{j}}\Big{)}^{2}};\qquad\qquad
g_{3}(z)=\frac{cz+d}{az+b}.$

Of course, in the last expressions the parameters are not canonical. Note that
we may choose arbitrary $\varphi$ and $\alpha$, so we put

$e^{\varphi\mathrm{j}}=\frac{\frac{c^{2}}{(bc-
ad)^{2}}}{\left|\frac{c^{2}}{(bc-
ad)^{2}}\right|};\qquad\qquad\bar{\alpha}=\frac{ae^{\varphi\mathrm{j}}}{c}\ .$

Then, $f_{3}(z)$ becomes proportional (with real coefficient) to $f(z)$, thus
proving the assertion. Note that $bc-ad$ can not be zero (if we assume that
$bc-ad=0$, then the surface is planar which is not our case).

∎

######  Remark 4.1.

The same proposition holds for minimal timelike surfaces of positive Gauss
curvature.

Acknowledgments: The authors are partially supported by the National Science
Fund, Ministry of Education and Science of Bulgaria under contract
KP-06-N52/3.

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