# Deep Temporal Contrastive Clustering

Ying Zhong<EMAIL_ADDRESS>Dong Huang<EMAIL_ADDRESS>Chang-Dong Wang<EMAIL_ADDRESS>College of Mathematics and
Informatics, South China Agricultural University, China  School of Computer
Science and Engineering, Sun Yat-sen University, China  Guangdong Key
Laboratory of Information Security Technology, China

###### Abstract

Recently the deep learning has shown its advantage in representation learning
and clustering for time series data. Despite the considerable progress, the
existing deep time series clustering approaches mostly seek to train the deep
neural network by some instance reconstruction based or cluster distribution
based objective, which, however, lack the ability to exploit the sample-wise
(or augmentation-wise) contrastive information or even the higher-level (e.g.,
cluster-level) contrastiveness for learning discriminative and clustering-
friendly representations. In light of this, this paper presents a deep
temporal contrastive clustering (DTCC) approach, which for the first time, to
our knowledge, incorporates the contrastive learning paradigm into the deep
time series clustering research. Specifically, with two parallel views
generated from the original time series and their augmentations, we utilize
two identical auto-encoders to learn the corresponding representations, and in
the meantime perform the cluster distribution learning by incorporating a
$k$-means objective. Further, two levels of contrastive learning are
simultaneously enforced to capture the instance-level and cluster-level
contrastive information, respectively. With the reconstruction loss of the
auto-encoder, the cluster distribution loss, and the two levels of contrastive
losses jointly optimized, the network architecture is trained in a self-
supervised manner and the clustering result can thereby be obtained.
Experiments on a variety of time series datasets demonstrate the superiority
of our DTCC approach over the state-of-the-art.

###### keywords:

Time series , Data clustering , Deep clustering , Contrastive learning , Deep
learning

##  1 Introduction

Time series data widely exist in many areas of science and engineering, such
as financial data [1], robot sensor data [2], earthquake data [3], and genomic
data [4]. The rapid emergence of time series data has brought new challenges
to the current clustering research, where the traditional clustering
algorithms are mostly designed for static data and lack the ability to well
capture the temporal information [5, 6, 7, 8, 9].

In recent years, the deep learning has shown its promising ability in
representation learning and clustering for time series data. As one of the
earliest deep time series clustering works, Madiraju [10] proposed a deep
temporal clustering (DTC) model, which utilizes an auto-encoder and a temporal
clustering layer to learn the clustering assignment. The clustering layer is
designed to minimize the Kullback-Leibler (KL) divergence between the
predicted distribution and a target distribution [10]. Further, Ma et al. [11]
incorporated the $k$-means objective and the temporal reconstruction into a
sequence to sequence (seq2seq) model for learning the cluster-specific
representations. The representations learned via the temporal auto-encoder are
utilized to form the cluster structure with the guidance of the $k$-means
objective.

Despite the recent advances, these deep time series clustering methods [10,
11] typically learn the representations by considering instance reconstruction
or cluster distributions, yet cannot well exploit the sample-wise (or
augmentation-wise) information for learning more discriminative
representations. More recently, the contrastive learning has attracted
significant attention, especially for the image clustering task, which is able
to effectively train the deep neural network by leveraging data augmentations
and their pair-wise relationships in a self-supervised manner. For example, Li
et al. [12] incorporated the contrastive learning into the deep image
clustering, and presented the contrastive clustering (CC) method by exploiting
the instance-level and cluster-level contrastive learning simultaneously. Li
et al. [13] presented an unsupervised representation learning method called
prototypical contrastive learning (PCL), which not only obtains the low-level
representations for the instance discrimination task, but also encodes the
semantic structures acquired by the clustering process into the learned
embedding space. Peng et al. [14] proposed the ContrastiveCrop method, which
is able to generate better crops for the Siamese representation learning and
further designs a center-suppressed sampling strategy to enlarge the variance
of crops. However, these contrastive learning based deep clustering methods
[12, 13, 14] are mostly designed for the image clustering task, which cannot
well handle the temporal information and thus are not suitable for time series
clustering. Though considerable progress has been made, it is still an open
problem how to effectively and jointly incorporate the contrastive learning
and the cluster structure learning into the deep time series clustering
framework.

Figure 1: The overall architecture of our DTCC framework

In light of this, this paper proposes a novel deep clustering approach termed
deep temporal contrastive clustering (DTCC), which for the first time, to the
best of our knowledge, introduces the contrastive learning into the deep time
series clustering research. As illustrated in Fig. 1, the data augmentation
(randomly selected from a family of augmentations) is performed on each time
series sample, leading to two parallel views, namely, the original view
(corresponding to the original samples) and the augmented view (corresponding
to the augmented samples). Then, two identical temporal auto-encoders are
utilized to learn representations for the two views, respectively. To obtain
clustering-friendly representations, the soft $k$-means objective is exploited
for the cluster distribution learning. Further, two levels of contrastive
learning are enforced to jointly leverage the instance-level and cluster-level
contrastiveness. With the temporal instance reconstruction (via auto-encoder),
the cluster distribution learning, and the dual contrastive learning
seamlessly integrated into a unified objective, the overall network can be
effectively trained in a self-supervised manner and the time series clustering
result can thereby be obtained. Experimental results on a variety of time
series datasets demonstrate the superiority of the proposed DTCC approach.

For clarity, the contributions of this work are summarized as follows:

  * 1.

This paper is the first attempt (to our knowledge) to incorporate the
contrastive learning into the deep time series clustering process.

  * 2.

A novel time series clustering approach termed DTCC is proposed, which is
capable of enforcing the temporal instance reconstruction, the cluster
distribution learning, and the dual contrastive learning simultaneously.

  * 3.

Extensive experiments are conducted on ten time series datasets, which confirm
the superior clustering performance of our DTCC approach.

The rest of the paper is organized as follows. The related works on time
series clustering are reviewed in Section 2. The overall framework of DTCC is
described in Section 3. Section 4 reports the experimental results. Finally,
Section 5 concludes this paper.

##  2 Related work

Time series clustering has been an important research topic in time series
analysis. In this section, we first review the traditional time series
clustering methods, which can be mainly divided into two categories, namely,
the raw data based methods and the feature based methods. Then we proceed to
review the recent advances in the deep time series clustering methods.

The raw data based methods are typically applied to the original time series
features, which achieve the clustering result by exploiting some modified
distance or similarity function (e.g. via scaling and distortion). Petitjean
et al. [15] utilized the dynamic time warping (DTW) metric and performed
clustering via the computation of the average of a set of sequences. Yang and
Leskovec [16] proposed the $k$-Spectral Centroid($k$-SC) method, which
explores the temporal dynamics by using a similarity metric that is invariant
to scaling and shifting. Paparrizos and Gravano [17] proposed a $k$-Shape
method for preserving the shapes of time series, which can effectively compute
the centroids of time series under the scaling and shift invariances.

The feature based methods attempt to extract feature representations of time
series for the clustering purpose, which can reduce the impact of noise or
outliers when compared with the raw data based methods. Guo et al. [18]
converted the raw time series data into feature vectors of lower dimensions by
independent component analysis (ICA), and then employed a modified $k$-means
clustering algorithm upon the extracted feature vectors. Zakaria et al. [19]
exploited a concept called shapelets to obtain the local patterns in time
series for the clustering task. Yang et al. [20] performed time series
clustering by taking advantage of pseudo-labels, which adopts the local
learning for selecting discriminative features.

In recent years, the deep learning has shown it advantageous ability in
simultaneous representation learning and clustering for the time series data,
and several deep time series clustering methods have been devised. For
example, Madiraju [10] proposed the deep temporal clustering (DTC) method,
which utilizes an auto-encoder to reduce the temporal dimension and
incorporates a temporal clustering layer for learning the clustering
assignment. Ma et al. [11] utilized a seq2seq model to reconstruct the time
series and integrated the $k$-means objectives into the model to guide the
representation learning and clustering. Further, they introduced a fake-sample
generation strategy, and designed an auxiliary classification task to enhance
the discriminative ability of the encoder [11]. Anand et al.[21] dealt with
time series clustering based on feature transformation and feature learning,
where each 1-D time-series is transformed into a 2-D image sample and the
feature set is obtained by utilizing a pre-trained convolutional neural
network on the transformed search space. Ma et al. [22] developed a time
series clustering framework called self-supervised time series clustering
network (STCN) by jointly optimizing the feature extraction and clustering. To
capture the temporal dynamics and maintain the local structures of time
series, a recurrent neural network (RNN) in the feature extraction module is
exploited to perform the one-step time series prediction that acts as the
reconstruction of input data. Then, model-based dynamic features are obtained
by the output layer of the RNN and further fed into a self-supervised
clustering module to achieve the predict labels [22].

##  3 Proposed Framework

In this paper, we present a deep clustering approach termed DTCC for time
series data, which incorporates the time series reconstruction (via auto-
encoder), the cluster distribution learning, and the dual contrastive learning
in a unified framework. In the following, Section 3.1 first describes the
overall framework of our DTCC approach. Then the time series reconstruction,
the cluster distribution learning, and the dual contrastive learning are
depicted in Sections 3.2, 3.3, and 3.4, respectively. Finally, the overall
loss function of DTCC is presented in Section 3.5.

###  3.1 Framework Overview

The overall framework of the proposed DTCC approach is illustrated in Fig. 1.
In the network architecture of DTCC, we utilize two identical temporal auto-
encoders to reconstruct the input time series data in two different views. the
cluster distribution learning module is designed to enforce clustering-
friendly representation learning. Further, the dual contrastive learning
module is incorporated to learn discriminative representations via
simultaneously instance-level and cluster-level contrastive learning.

Specifically, besides the original time series, we construct an augmented view
by performing time-series-specific augmentations on the input data. Then, the
original sample and the augmented sample are respectively encoded into latent
temporal representations via an encoder, which will be exploited in the
following clustering task. The encoder is instantiated by a multi-layer
dilated recurrent neural network (RNN). The clustering-friendly
representations are obtained by concatenating the last hidden state output of
each layer in the multi-layer dilated RNN [Chang2017]. Thereby, two identical
encoder-decoder (auto-encoder) networks are trained via the temporal
reconstruction loss, while two soft $k$-means objectives are utilized for the
cluster distribution learning.

With the cluster distribution learning module that deploys two $k$-means
objectives (on the original view and the augmented view, respective) to guide
the latent temporal representation learning, the dual contrastive learning
module further jointly enforces the instance-level contrastive learning and
the cluster-level contrastive learning. With the simultaneous modeling of the
temporal reconstruction, the cluster distribution learning, and the dual
contrastive learning, the overall network of DTCC can be trained in a self-
supervised manner and thus the clustering result can be learned.

###  3.2 Temporal Auto-encoder Reconstruction with Parallel Views

Given a time series dataset
$\mathcal{X}_{data}=\\{{x}_{1},\ldots,{x}_{i},\ldots,{x}_{N}\\}$ with $N$
samples (or instances), where the $i$-th time series sample
${x}_{i}\in{R}^{m}$ is denoted as
${x}_{i}=({x}_{i,1},{x}_{i,2},\ldots,{x}_{i,m})$ and ${x}_{i,t}$ is the
corresponding data at the $t$-th time step. Upon the original time series
sample ${x}_{i}$, we perform a time series data augmentation $T^{a}$, which is
randomly selected from a family of augmentations $\mathcal{T}$, to obtain an
augmented sample $x_{i}^{a}=T(x_{i})$. Note that the original sample ${x}_{i}$
and the augmented sample ${x}_{i}^{a}$ are fed to two different views of two
identical auto-encoders, respectively.

In the auto-encoder, we employ a bidirectional multi-layer dilated RNN as the
encoder for capturing the dynamics and multi-scale characteristics of time
series, and utilize a single-layer RNN as the decoder. Specifically, the
encoder is denoted as $G_{e}:{x}_{i}\rightarrow z_{i}$ for the original view,
$G_{e}:{x}_{i}^{a}\rightarrow z_{i}^{a}$) for the augmented view, which non-
linearly map the input time series (of both views) to the latent space
$\mathcal{Z}$, where ${z}_{i}\in\mathcal{Z}$ and ${z}_{i}^{a}\in\mathcal{Z}$
denote the learned latent representations of the original sample and the
augmented sample, respectively. Also, we denote the decoders in the original
view and the augmented view as $G_{d}:{z}_{i}\rightarrow\hat{x}_{i}$ and
$G_{d}:{z}_{i}^{a}\rightarrow\hat{x}_{i}^{a}$, respectively. Here, $z_{i}\in
R^{d}$ (or $z_{i}^{a}\in R^{d}$) is the $d$-dimensional latent representation
of an input time series $x_{i}$ (or ${x}_{i}^{a}$). Formally, the encoding
process can be denoted as

$\displaystyle z_{i}=G_{e}\left({x}_{i}\right),$ (1) $\displaystyle
z_{i}^{a}=G_{e}\left({x}_{i}^{a}\right).$ (2)

Specifically, $z_{i}$(or $z_{i}^{a}$) is the concatenation of the last hidden
state of each layer of the multi-layer Dilated RNN. Thereafter, $z_{i}$ and
$z_{i}^{a}$ are fed to two decoder networks for reconstructing the input data,
respectively.

For the decoding task, we obtain the output of the two decoders as
$\hat{{x}}_{{i}}$ and $\hat{{x}}_{{i}}^{{a}}$, where $\hat{{x}_{{i}}}\in
R^{m}$ and $\hat{{x}}_{{i}}^{{a}}\in R^{m}$ are respectively given by

$\displaystyle\hat{{x}}_{{i}}=G_{d}\left({z}_{{i}}\right),$ (3)
$\displaystyle\hat{{x}}_{{i}}^{{a}}=G_{d}\left({z}_{{i}}^{{a}}\right).$ (4)

Let $n$ denote the number of samples in a mini-batch. Then the mean square
error (MSE) is used to compute the reconstruction losses of the two parallel
views (i.e., the original view and the augmented view), that is

$\displaystyle\mathcal{L}_{{recon-
org}}=\frac{1}{n}\sum_{i=1}^{n}\left\|{x}_{{i}}-\hat{{x}}_{{i}}\right\|_{2}^{2},$
(5) $\displaystyle\mathcal{L}_{{recon-
aug}}=\frac{1}{n}\sum_{i=1}^{n}\left\|{x}_{{i}}^{{a}}-\hat{{x}}_{{i}}^{{a}}\right\|_{2}^{2}.$
(6)

Thereby, the joint reconstruction loss of both views can be represented as

$\displaystyle\mathcal{L}_{{reconsruction}}=\mathcal{L}_{{recon-
org}}+\mathcal{L}_{{recon-aug}}$ (7)

###  3.3 Cluster Distribution Learning

Though the auto-encoder with the reconstruction losses can learn some
informative features from the original time series and its augmented time
series, yet it only considers the information reconstruction of each sample,
which cannot reflect capture the higher-level (such as the cluster-level)
information for tackling the clustering task. In light of this, in this
section, we proceed to incorporate the cluster distribution learning in the
network architecture.

With the feature representation matrix $Z\in R^{d\times n}$, we seek to
minimize a soft $k$-means objective, which can be rewritten as a trace
maximization problem associated the Gram matrices ${Z}^{T}{Z}$, and has an
optimal local-minima-free global solution [23]. Formally, the $k$-means
objective can be converted (via spectral relaxation) into the following form,
that is

$\displaystyle\mathcal{L}_{km-{org}}=\operatorname{Tr}\left({Z}^{{T}}{Z}\right)-\operatorname{Tr}\left({Q}^{{T}}{Z}^{{T}}{Z}{Q}\right)$
(8)

where $\operatorname{Tr}(\cdot)$ is the matrix trace, and ${Q}\in R^{n\times
k}$ is the cluster indicator matrix. With $Z$ fixed, the $k$-means based
clustering distribution loss (for the original view) can be transformed into a
trace maximization problem, that is

$\displaystyle\max_{{Q}}\operatorname{Tr}\left({Q}^{{T}}{Z}^{{T}}{Z}{Q}\right),\text{
s.t. }{Q}^{{T}}{Q}={I}$ (9)

For the augmented view, the clustering distribution loss can be calculated as
follows:

$\displaystyle\mathcal{L}_{km-{aug}}=\operatorname{Tr}\left({(Z^{a})}^{{T}}Z^{a}\right)-\operatorname{Tr}\left({(Q^{a})}^{{T}}{(Z^{a})}^{{T}}{Z^{a}}{Q^{a}}\right)$
(10)

Similarly, it can be transformed into a trace maximization problem, that is

$\displaystyle\max_{{Q^{a}}}\operatorname{Tr}\left({(Q^{a})}^{{T}}{(Z^{a})}^{{T}}{Z^{a}}{Q^{a}}\right),\text{
s.t. }{(Q^{a})}^{{T}}{Q^{a}}={I}$ (11)

Thereby, the cluster distribution loss for both views can be written as

$\displaystyle\mathcal{L}_{cd}=\frac{1}{2}(\mathcal{L}_{km-{org}}+\mathcal{L}_{km-{aug}}).$
(12)

In this paper, the cluster distribution learning is coupled with instance
reconstruction and contrastive learning. A regularization term $\lambda$ is
introduced for learning feature representations with tight clustering
structures. Thereby, the objective function can be represented as

$\displaystyle\min_{{Z},{Q},{Z^{a}},{Q^{a}}}J({Z})+\lambda\mathcal{L}_{cd},$
(13) $\displaystyle\text{ s.t. }{Q}^{{T}}{Q}={I},{(Q^{a})}^{{T}}{Q^{a}}={I}$

where $J({Z})$ is the sum of the two reconstruction losses and the two
contrastive losses. In the following, the contrastive losses (for contrastive
learning) will be described in Section 3.4.

For the original view, the representation ${Z}$ is dynamically learned by the
auto-encoder as well as the training process of the network in DTCC, where
${Z}$ and ${Q}$ are jointly optimized. The same network update as above is
performed for the augmented version. Thus, we focus on the network updating of
the original view, and divide it into the following two steps.

First, with ${Q}$ fixed, the standard stochastic gradient descent (SGD) is
adopted to update ${Z}$, with the gradient given as $\nabla
J({Z})+\lambda{Z}\left({I}-{Q}{Q}^{{T}}\right)$.

Second, with ${Z}$ fixed, according to the Ky Fan theorem, we update ${Q}$ by
calculating the $k$-truncated singular value decomposition (SVD) of $Z$. In
this paper, it is updated every five epochs so as to avoid potential
instability. Thereby, the cluster distribution learning via the $k$-means
objective can guide the temporal representation learning with better cluster
structures.

###  3.4 Dual Contrastive Learning

In the previous section, the cluster distribution learning is performed on the
original view and the augmented view separately. In this section, we proceed
to bridge the original view and the augmented view via two levels of
contrastive learning. In the following, the instance-level contrastive
learning (via the instance contrastive loss) and the cluster-level contrastive
learning (via the cluster contrastive loss) will be described in Sections
3.4.1 and 3.4.2, respectively.

####  3.4.1 Instance Contrastive Loss

The instance-level contrastive learning aims to maximize the similarity
between the learned representations of an original sample and its augmented
sample, while minimizing the similarity between the representations of
different samples. Specifically, DTCC takes the representations of the same
input time series sample from the two views as the positive sample pairs, and
the representations of different input time series samples as the negative
sample pairs.

Let $z_{i}$ and $z_{i}^{a}$ denote the representations for the $i$-th time
series sample from the original view and the augmented view, respectively. The
instance contrastive loss for $z_{i}$ can be written as

$\displaystyle{\ell}_{z_{i}}=-\log\frac{\exp\left(sim\left(z_{i},z_{i}^{a}\right)/\tau_{I}\right)}{\sum_{j=1}^{n}\left[\exp\left(sim\left(z_{i},z_{j}\right)/\tau_{I}\right)+\mathbbm{1}_{[i\neq
j]}\exp\left(sim\left(z_{i},z_{j}^{a}\right)/\tau_{I}\right)\right]}$ (14)

where $\tau_{I}$ is the instance-wise temperature parameter, $n$ is the batch
size, $\mathbbm{1}$ is the indicator function, and
$sim(u,v)={u}^{T}{v}/\|{u}\|\|{v}\|$ is the dot product between
$\ell_{2}$-normalized $u$ and $v$. The instance contrastive loss for
${z_{i}^{a}}$ can be similarly defined. Therefore, the instance-wise
contrastive loss (for the $n$ samples in a mini-batch) with both view
considered can be formulated as

$\displaystyle\mathcal{L}_{{instance}}=\frac{1}{2n}\sum_{i=1}^{n}({\ell}_{z_{i}}+{\ell}_{z_{i}^{a}})$
(15)

####  3.4.2 Cluster Contrastive Loss

Besides the instance-level contrastive learning, we further enforce the
cluster-level contrastive learning, which constrains the _consistency_ between
the cluster distributions of the two views. According to Eq.(8) and Eq.(10),
let $Q\in R^{n\times k}$ and $Q^{a}\in R^{n\times k}$ denote the cluster
indicator matrices for a mini-batch from the two views, where $k$ is the
number of clusters. Note that $Q$ and $Q^{a}$ are obtained by computing the
$k$-truncated SVD of $Z$ and $Z^{a}$, respectively.

Let $q_{i}$ be the $i$-th column of $Q$, which can be seen as a representation
of the $i$-th cluster (with its $j$-th entry indicating the probability that
the $j$-th sample belong to this cluster). To enable the cluster-level
contrastive learning, the two clusters $q_{i}$ with $q_{i}^{a}$ from two
different views are regarded as a positive cluster pair
$\\{q_{i},q_{i}^{a}\\}$, where $q_{i}^{a}$ denotes the $i$-th column of
$Q^{a}$. Meanwhile, the remaining $(2k-2)$ pairs within the same mini-batch
are regarded as negative cluster pairs. Thereby, the cluster contrastive loss
for cluster $q_{i}$ can be formulated as

$\displaystyle\ell_{q_{i}}=-\log\frac{\exp(sim(q_{i},q_{i}^{a})/\tau_{C})}{\sum_{j=1}^{k}[\exp(sim(q_{i},q_{j})/\tau_{C})+\mathbbm{1}_{[i\neq
j]}\exp(sim(q_{i},q_{j}^{a})/\tau_{C})]}$ (16)

where $\tau_{C}$ is the cluster-wise temperature parameter. Here, we utilize
the cosine similarity to measure the similarity between cluster pairs, that is

$\displaystyle
sim\left(q_{i},q_{i}^{a}\right)=\frac{\left(q_{i}\right)^{\top}\left(q_{i}^{a}\right)}{\left\|q_{i}\right\|\left\|q_{i}^{a}\right\|}$
(17)

Then, with $k$ clusters on each view, the cluster-wise contrastive loss for
the totally $2k$ clusters on the two views can be defined as

$\displaystyle\begin{split}\mathcal{L}_{cluster}=\frac{1}{2k}\sum_{i=1}^{k}({\ell}_{q_{i}}+\ell_{q_{i}^{a}})-{{H}(q)}\end{split}$
(18)

where $H(q)=-\sum_{i=1}^{k}\left[P\left(q_{i}\right)\log
P\left(q_{i}\right)+P\left(q_{i}^{a}\right)\log
P\left(q_{i}^{a}\right)\right]$ is the entropy of the cluster-assignment
probability, $P\left(q_{i}\right)=\frac{1}{n}\sum_{j=1}^{n}q_{ji}$,
$P\left(q_{i}^{a}\right)=\frac{1}{n}\sum_{j=1}^{n}q_{ji}^{a}$, and $q_{ji}$
and $q_{ji}^{a}$ are respectively the $(j,i)$-th entry of $Q$ and $Q^{a}$. The
entropy term is incorporated to avoid the degenerate solution that most
instances are assigned to the same cluster.

###  3.5 Overall Loss Function

Finally, with the temporal reconstruction via auto-encoders, the cluster
distribution learning, and the the dual contrastive learning, the overall loss
function of our DTCC framework is formulated as

$\displaystyle\begin{split}\mathcal{L}_{DTCC}=\mathcal{L}_{{reconstruction}}+\mathcal{L}_{{instance}}+\mathcal{L}_{{cluster}}+\lambda\mathcal{L}_{cd}.\end{split}$
(19)

Thereby, the network architecture of DTCC can be trained in a self-supervised
manner. For clarity, the overall process of the proposed DTCC method is
summarized in Algorithm 1.

Input: Data set: $\mathcal{X}_{data}$; Number of clusters: $k$; Batch size:
$n$; Alternate: $T$; Training epochs: $E$.

Output: The clustering result $Q$

1 for  _epoch= $1$ to $E$_ do

2 Sample a mini-batch of time series $\\{x_{i}\\}_{i=1}^{n}$ from
$\mathcal{X}_{data}$.

3

4 For a mini-batch of time series $\\{x_{i}\\}_{i=1}^{n}$, generate the
augmented samples.

5

6 Update the latent representations
$\left\\{{z}_{i}=G_{e}\left(x_{i}\right)\right\\}_{i=1}^{n}$ and
$\left\\{{z}_{i}^{a}=G_{e}\left({x}_{i}^{a}\right)\right\\}_{i=1}^{n}$ by
minizing the overall loss $\mathcal{L}_{DTCC}$ with SGD.

7 if  _$epoch\%T=0$_ then

8 Update $\\{q_{i}\\}_{i=1}^{n}$ using the closed-form solution of Eq.(9)

9 Update$\\{q_{i}^{a}\\}_{i=1}^{n}$ using the closed-form solution of Eq.(11)

10

11 end if

12

13 end for

Obtain the clustering result Q.

Algorithm 1 Deep Temporal Contrastive Clustering (DTCC)

##  4 Experiments

###  4.1 Experimental Settings and Datasets

In this paper, two identical encoders with bidirectional multi-layer Dilated
RNN are utilized to capture the multi-scale features of the time series data,
and a single-layer RNN is utilized as the decoder. The number of units in each
layer is set to
$\left[n_{1},n_{2},n_{3}\right]\subset\\{[100,50,50],[50,30,30]\\}$. The
number of hidden dimensions is $\left(n_{1}+n_{2}+n_{3}\right)\times 2$. In
our experiments, we set the fixed number of layers in the network and the
number of each dilation layer to 3 and 1, and 4 and 16, respectively. The
decoder treats the final state of the encoder as the initial state and makes
iterative predictions. The batch size is one half of the sample size. Our
experiments are conducted on the Tensorflow platform using a GeForce GTX 1030
4G GPU. The Adam optimizer is applied with an initial learning rate of 5e-3.
To evaluate the clustering performance, two evaluation metrics are adopted,
namely, the normalized mutual information (NMI) [24] and the Rand index (RI)
[25].

In the experiments, ten real-world time series datasets from the UCR Archive
[26] are used, namely, Beef, dist.phal.outl.agegroup, ECG200, ECGFiveDays,
Meat, MoteStrain, OSULeaf, Plane, Prox.phal.outl.ageGroup, and Prox.phal.TW.
The detailed statistics of the datasets are given in Table 1.

Table 1: Statistics of the used time series datasets Dataset | Train | Test | Length | classes
---|---|---|---|---
Beef | 30 | 30 | 471 | 5
dist.phal.outl.agegroup | 139 | 400 | 81 | 3
ECG200 | 100 | 100 | 97 | 2
ECGFiveDays | 23 | 86 1 | 137 | 2
Meat | 60 | 60 | 449 | 3
MoteStrain | 20 | 1252 | 85 | 2
OSULeaf | 200 | 242 | 428 | 6
Plane | 105 | 105 | 145 | 7
Prox.phal.outl.ageGroup | 400 | 205 | 81 | 3
Prox.phal.TW | 205 | 400 | 81 | 6

###  4.2 Compared with State-of-the-Art Methods

In the experiments, we compare our DTCC method against twelve deep or non-deep
clustering methods, which are introduced below.

  1. 1.

$k$-means [27] : Perform the $k$-means clustering on the entire time series.

  2. 2.

UDFS [20]: The unsupervised discriminative feature selection method exploits
the manifold structure, the local discriminative information, and the feature
correlations for selecting discriminative features.

  3. 3.

NDFS [28]: The nonnegative discriminant feature selection (NDFS) selects the
discriminative features by exploiting the clustering labels acquired by
spectral clustering and the feature selection matrix.

  4. 4.

RUFS [29]: The robust unsupervised feature selection (RUFS) utilized the local
learning regularized robust nonnegative matrix factorization to peform the
feature learning.

  5. 5.

RSFS [30]: The robust spectral learning for unsupervised feature selection
(RSFS) method jointly improves the robustness of graph embedding and sparse
spectral regression.

  6. 6.

KSC [16]: The $k$-spectral centroid (KSC) clustering method performs
clustering by identifying the cluster centroids with the time series
similarity measure.

  7. 7.

$k$-DBA) [15] : The $k$-DBA method utilizes the $k$-means and the dynamic time
warping distance to acquire the clustering result.

  8. 8.

u-shapelet [19]: The u-shapelet method performs time series clustering by
exploiting some local patterns and ignoring the rest of the data.

  9. 9.

DTC [10]: The deep temporal clustering (DTC) method naturally integrates the
dimensionality reduction and the temporal clustering into an end-to-end
unsupervised learning model.

  10. 10.

DEC [31]: The deep embedding clustering (DEC) method can jointly learn the
feature representations and the cluster assignments by optimizing a KL-
divergence based clustering objective.

  11. 11.

DTCR [11]: The deep temporal clustering representation (DTCR) method
incorporates the auto-encoder reconstruction and a $k$-means objective into
the seq2seq model for learning cluster-specific temporal representations.

  12. 12.

CRLI [32]: The clustering representation learning on incomplete time-series
data (CRLI) method simultaneously optimizes the imputation and the clustering
process in a unified deep learning framework.

Table 2: The NMI scores of different clustering methods on the ten time series datasets Dataset | K-means | UDFS | NDFS | RUFS | RSFS | KSC | KDBA
---|---|---|---|---|---|---|---
Beef | 0.2925 | 0.2718 | 0.3647 | 0.3799 | 0.3597 | 0.3828 | 0.3340
dist.phal.outl.agegroup | 0.1880 | 0.3262 | 0.1943 | 0.2762 | 0.3548 | 0.3331 | 0.4261
ECG200 | 0.1403 | 0.1854 | 0.1403 | 0.2668 | 0.2918 | 0.1403 | 0.1886
ECGFiveDays | 0.0002 | 0.0600 | 0.1296 | 0.0352 | 0.1760 | 0.0682 | 0.1983
Meat | 0.2510 | 0.2832 | 0.2416 | 0.1943 | 0.3016 | 0.2846 | 0.3661
MoteStrain | 0.0551 | 0.1187 | 0.1919 | 0.1264 | 0.2373 | 0.3002 | 0.0970
OSULeaf | 0.0208 | 0.0200 | 0.0552 | 0.0246 | 0.0463 | 0.0421 | 0.0327
Plane | 0.8598 | 0.8046 | 0.8414 | 0.8675 | 0.8736 | 0.9218 | 0.8784
Prox.phal.outl.ageGroup | 0.0635 | 0.0182 | 0.0830 | 0.0726 | 0.0938 | 0.0682 | 0.0377
Prox.phal.TW | 0.0082 | 0.0308 | 0.2215 | 0.1187 | 0.0809 | 0.1919 | 0.2167
Avg. NMI | 0.1879 | 0.2119 | 0.2464 | 0.2362 | 0.2816 | 0.2732 | 0.2776
Dataset | u-shapelet | DTC | DEC | DTCR | CRLI | DTCC
Beef | 0.3413 | 0.2751 | 0.2463 | 0.4484 | 0.4570 | [0.4753]
dist.phal.outl.agegroup | 0.2577 | 0.3406 | 0.4405 | 0.3957 | 0.3080 | [0.4596]
ECG200 | 0.1323 | 0.0918 | 0.1885 | 0.1805 | 0.2064 | [0.3009]
ECGFiveDays | 0.1498 | 0.0022 | 0.0178 | 0.0900 | 0.1875 | [0.3623]
Meat | 0.2716 | 0.2250 | 0.5176 | 0.3987 | [0.6608] | 0.6099
MoteStrain | 0.0082 | 0.0094 | 0.3867 | 0.3218 | 0.2748 | [0.4653]
OSULeaf | 0.0203 | 0.2201 | 0.2141 | 0.1953 | 0.2071 | [0.2354]
Plane | [1.0000] | 0.8678 | 0.8947 | 0.9049 | [1.0000] | 0.9252
Prox.phal.outl.ageGroup | 0.0332 | 0.4153 | 0.2500 | 0.4878 | 0.5227 | [0.5317]
Prox.phal.TW | 0.0107 | 0.6199 | 0.5864 | 0.5190 | 0.5709 | [0.6213]
Avg. NMI | 0.2225 | 0.3067 | 0.3743 | 0.3942 | 0.4395 | [0.4987]

Figure 2: The number of times being ranked in top 1 or top 2 on the ten datasets (w.r.t. the results in Table 2). Table 3: The RI scores of different clustering methods on the ten time series datasets Dataset | K-means | UDFS | NDFS | RUFS | RSFS | KSC | KDBA
---|---|---|---|---|---|---|---
Beef | 0.6713 | 0.6759 | 0.7034 | 0.7149 | 0.6975 | 0.7057 | 0.6713
dist.phal.outl.agegroup | 0.6171 | 0.6531 | 0.6239 | 0.6252 | 0.6539 | 0.6535 | 0.6750
ECG200 | 0.6315 | 0.6533 | 0.6315 | [0.7018] | 0.6916 | 0.6315 | 0.6018
ECGFiveDays | 0.4783 | 0.5020 | 0.5573 | 0.5020 | 0.5953 | 0.5257 | 0.5573
Meat | 0.6595 | 0.6483 | 0.6635 | 0.6578 | 0.6657 | 0.6723 | 0.6816
MoteStrain | 0.4947 | 0.5579 | 0.6053 | 0.5579 | 0.6168 | 0.6632 | 0.4789
OSULeaf | 0.5615 | 0.5372 | 0.5622 | 0.5497 | 0.5665 | 0.5714 | 0.5541
Plane | 0.9081 | 0.8949 | 0.8954 | 0.9220 | 0.9314 | 0.9603 | 0.9225
Prox.phal.outl.ageGroup | 0.5288 | 0.4997 | 0.5463 | 0.5780 | 0.5384 | 0.5305 | 0.5192
Prox.phal.TW | 0.4789 | 0.4947 | 0.6053 | 0.5579 | 0.5211 | 0.6053 | 0.5211
Avg. RI | 0.6030 | 0.6117 | 0.6394 | 0.6367 | 0.6478 | 0.6519 | 0.6183
Dataset | u-shapelet | DTC | DEC | DTCR | CRLI | DTCC
Beef | 0.6966 | 0.6345 | 0.5954 | 0.7452 | [0.7655] | 0.7633
dist.phal.outl.agegroup | 0.6273 | [0.7812] | 0.7785 | 0.7417 | 0.6520 | 0.7516
ECG200 | 0.5758 | 0.6018 | 0.6422 | 0.6085 | 0.6315 | 0.6733
ECGFiveDays | 0.5968 | 0.5016 | 0.5103 | 0.5419 | 0.6237 | [0.6581]
Meat | 0.6742 | 0.3220 | 0.6475 | 0.6934 | [0.8305] | 0.7200
MoteStrain | 0.4789 | 0.5062 | 0.7435 | 0.6956 | 0.6792 | [0.7740]
OSULeaf | 0.5525 | 0.7329 | 0.7484 | 0.7497 | 0.7476 | [0.7546]
Plane | [1.0000] | 0.9040 | 0.9447 | 0.9404 | [1.0000] | 0.9373
Prox.phal.outl.ageGroup | 0.5206 | 0.7430 | 0.4263 | 0.8023 | 0.8020 | [0.8091]
Prox.phal.TW | 0.4789 | 0.8380 | 0.8189 | [0.8617] | 0.8439 | 0.8606
Avg. RI | 0.6202 | 0.6565 | 0.6856 | 0.7380 | 0.7576 | [0.7729]

Figure 3: The number of times being ranked in top 1 or top 2 on the ten
datasets (w.r.t. the results in Table 3).

Specifically, the clustering performances (w.r.t. NMI and RI) of our DTCC
method and the baseline methods are reported in Tables 2 and 3, respectively.
In terms of NMI, as shown in Table 2, the proposed DTCC method achieves the
best clustering performance on the Beef, dist.phal.outl.agegroup, ECG200,
ECGFiveDays, MoteStrain, OSULeaf, Prox.phal.TW datasets, and Prox.phal.TW
datasets. On the other datasets, DTCC is still among the best two clustering
methods w.r.t. the NMI scores. The average NMI scores (across the ten
datasets) of different methods are also provided in Table 2. Our DTCC obtains
the average NMI score of 0.4987, which significantly outperforms the second
best method, whose average NMI score is only 0.4395. In terms of RI, similar
advantages of DTCC can also be observed. Our DTCC method obtains the average
RI score of 0.7729, which is substantially higher than the average score of
0.7576 obtained by the second best method.

Table 4: Ablation analysis on the dual contrastive losses in DTCC (w.r.t. NMI). ICL: instance contrastive loss. CCL: cluster contrastive loss. ICL |  CCL | Datasets
---|---|---
Beef | dist.phal.outl.agegroup | ECG200 | ECGFiveDays | Meat
✕ | ✓ | 0.4211 | 0.4096 | 0.1140 | 0.1291 | 0.1683
✓ | ✕ | 0.4753 | 0.4229 | 0.2436 | 0.2966 | 0.3574
✕ | ✕ | 0.4284 | 0.4170 | 0.1126 | 0.0730 | 0.2049
✓ | ✓ | 0.4753 | 0.4596 | 0.3009 | 0.3623 | 0.6099
| MoteStrain | OSULeaf | Plane | Prox.phal.outl.ageGroup | Prox.phal.TW
✕ | ✓ | 0.4040 | 0.1487 | 0.7751 | 0.5310 | 0.5334
✓ | ✕ | 0.4086 | 0.2435 | 0.8570 | 0.5246 | 0.5328
✕ | ✕ | 0.3751 | 0.1794 | 0.8342 | 0.5271 | 0.5365
✓ | ✓ | 0.4653 | 0.2354 | 0.9252 | 0.5317 | 0.6213
Table 5: Ablation analysis on the dual contrastive losses in DTCC (w.r.t.RI). ICL: instance contrastive loss. CCL: cluster contrastive loss. ICL |  CCL | Datasets
---|---|---
Beef | dist.phal.outl.agegroup | ECG200 | ECGFiveDays | Meat
✕ | ✓ | 0.7367 | 0.7505 | 0.6085 | 0.5777 | 0.6332
✓ | ✕ | 0.7593 | 0.7445 | 0.6724 | 0.6903 | 0.6954
✕ | ✕ | 0.7423 | 0.7419 | 0.5819 | 0.5127 | 0.6376
✓ | ✓ | 0.7632 | 0.7516 | 0.6733 | 0.6851 | 0.7200
| MoteStrain | OSULeaf | Plane | Prox.phal.outl.ageGroup | Prox.phal.TW
✕ | ✓ | 0.7426 | 0.7375 | 0.9155 | 0.8042 | 0.7927
✓ | ✕ | 0.7457 | 0.7575 | 0.9396 | 0.8059 | 0.8612
✕ | ✕ | 0.7213 | 0.7417 | 0.9283 | 0.8042 | 0.7969
✓ | ✓ | 0.7740 | 0.7546 | 0.9373 | 0.8091 | 0.8606

Furthermore, Figs. 2 and 3 illustrate the number of times of different
clustering methods to be ranked in top 1 or top 2 in terms of NMI and RI,
respectively. As shown in Figs. 2 and 3, our DTCC method is ranked in the top
2 on all of the ten datasets in terms of NMI, and on seven out of the ten
datasets in terms of RI, which confirm the highly competitive clustering
performance of the proposed DTCC method in comparison with the state-of-the-
art methods.

###  4.3 Ablation Study

In this section, the ablation study is conducted. Specifically, the influence
of the two contrastive losses are evaluated in Section 4.3.1, and the
influence of the cluster distribution loss is tested in Section 4.3.2.

####  4.3.1 Influence of the Two Contrastive Losses

In DTCC, two levels of constrastive learning are enforced, corresponding to
two types of contrastive losses, i.e., the instance contrastive loss and the
cluster contrastive loss. In this section, we evaluate the influence of these
two contrastive losses in DTCC.

As shown in Table 4, our DTCC method with the two types of contrastive losses
outperforms or significantly outperforms the variant with no contrastive loss
(in terms of NMI). When comparing DTCC with both contrastive losses against
the variant with only one of them, the joint use of both contrastive losses
exhibits overall better NMI score than using only one level of
contrastiveness. Similarity results can also be seen in Table 5, where the
advantage of using the dual contrastive losses has also been verified on most
of the benchmark datasets.

Table 6: Ablation analysis on the cluster distribution losses in DTCC (w.r.t. NMI). $\mathcal{L}_{km-{org}}$ | $\mathcal{L}_{km-{aug}}$ | Datasets
---|---|---
Beef | dist.phal.outl.agegroup | ECG200 | ECGFiveDays | Meat
✕ | ✓ | 0.4507 | 0.4163 | 0.2187 | 0.1510 | 0.3921
✓ | ✕ | 0.4710 | 0.4370 | 0.2905 | 0.1482 | 0.4424
✕ | ✕ | 0.4606 | 0.4555 | 0.2042 | 0.2183 | 0.4224
✓ | ✓ | 0.4753 | 0.4596 | 0.3009 | 0.3623 | 0.6099
| MoteStrain | OSULeaf | Plane | Prox.phal.outl.ageGroup | Prox.phal.TW
✕ | ✓ | 0.3669 | 0.1940 | 0.8200 | 0.5093 | 0.5459
✓ | ✕ | 0.3864 | 0.2308 | 0.8404 | 0.5225 | 0.5594
✕ | ✕ | 0.3826 | 0.2072 | 0.8485 | 0.5127 | 0.5331
✓ | ✓ | 0.4653 | 0.2354 | 0.9252 | 0.5317 | 0.6213
Table 7: Ablation analysis on the cluster distribution losses in DTCC (w.r.t. RI). $\mathcal{L}_{km-{org}}$ | $\mathcal{L}_{km-{aug}}$ | Datasets
---|---|---
Beef | dist.phal.outl.agegroup | ECG200 | ECGFiveDays | Meat
✕ | ✓ | 0.7610 | 0.7382 | 0.6687 | 0.6003 | 0.7102
✓ | ✕ | 0.7485 | 0.7513 | 0.7040 | 0.5982 | 0.7127
✕ | ✕ | 0.7553 | 0.7545 | 0.6568 | 0.6430 | 0.7242
✓ | ✓ | 0.7632 | 0.7516 | 0.6733 | 0.6851 | 0.7200
| MoteStrain | OSULeaf | Plane | Prox.phal.outl.ageGroup | Prox.phal.TW
✕ | ✓ | 0.7169 | 0.7492 | 0.9264 | 0.8003 | 0.8003
✓ | ✕ | 0.7426 | 0.7525 | 0.9455 | 0.8051 | 0.8665
✕ | ✕ | 0.7393 | 0.7546 | 0.9505 | 0.8014 | 0.8564
✓ | ✓ | 0.7740 | 0.7546 | 0.9373 | 0.8091 | 0.8606

####  4.3.2 Influence of the Cluster Distribution Losses

In this section, we evaluate the influence of the cluster distribution losses
in DTCC. Note that in the proposed DTCC method, the cluster distribution
losses are enforced on both of the original view and the augmented view. As
shown in Tables 6 and 7, the proposed DTCC method using both cluster
distribution losses outperforms the variants with zero or one cluster
distribution loss on most of the datasets, which demonstrate the benefit of
enforcing the cluster distribution learning on both the original and augmented
views in DTCC.

##  5 Conclusion

In this paper, we propose a novel deep clustering approach termed DTCC for
time series data, which integrates the temporal reconstruction via auto-
encoders, the cluster distribution learning, and the dual contrastive learning
into a unified framework. Particular, two parallel views (with two identical
temporal auto-encoders) are formulated, where the first view corresponds to
the original samples and the second view corresponds to the augmented samples.
To strengthen the cluster structures in the representations learned via the
auto-encoder, we incorporate the cluster structure learning with a soft
$k$-means objective on both the original and augmented views. Further, the
dual contrastive learning is integrated to learn more discriminative features
by exploiting the sample-wise and augmentation-wise relationships as well as
the global cluster-wise contrastiveness, which is able to enforce the
instance-level contrastive learning and the cluster-level contrastive learning
simultaneously. By modeling the auto-encoder reconstruction loss, the cluster
distribution loss, and the dual contrastive losses into an overall objective,
the network architecture can be trained in a self-supervised manner and the
final clustering can thus be obtained for the time series data. Experiments on
multiple time series datasets have confirmed the superiority of the proposed
DTCC approach over the state-of-the-art time series clustering approaches.

## Acknowledgments

This work was supported by the NSFC (61976097 & 62276277) and the Natural
Science Foundation of Guangdong Province (2021A1515012203).

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# Generalized Yang Poisson models on canonical phase space

Tea Martinić Bilać  Faculty of Science, University of Split, Rudjera Boškovića
33, 21000 Split, Croatia
E-mail<EMAIL_ADDRESS>Stjepan Meljanac  Rudjer Bošković Institute,
Theoretical Physics Division, Bijenička c. 54, HR 10002 Zagreb, Croatia
E-mail<EMAIL_ADDRESS>Salvatore Mignemi  Dipartimento di Matematica,
Università di Cagliari via Ospedale 72, 09124 Cagliari, Italy and INFN,
Sezione di Cagliari Cittadella Universitaria, 09042 Monserrato, Italy
E-mail<EMAIL_ADDRESS>
###### Abstract

We discuss the generalized Yang-Poisson models. We costruct generalizations of
the Yang Poisson algebra related to $so(1,5)$ algebra discussed in [14]. The
exact realizations of this generalized algebra on canonical phase space are
presented and the corresponding differential equations are solved in simple
case. Futhermore, we discuss a Poisson algebras related to $so(3,3$) and
$so(2,4)$ algebras.

##  1 Yang Poisson model

The Yang model of noncommutative geometry on a curved background spacetime was
proposed in [16] and is a generalization of noncommutative geometry first
introduced by Snyder [15]. This model is defined in terms of a 15-parameter
algebra isomorphic to $so(1,5)$ containing the generators of the Lorentz
algebra with the coordinates of phase spaces, and was generalized in [4].
Furthermore, Snyder-de Sitter or triply special relativity model, where the
symmetry is realized nonlinearly, was proposed in [1, 3, 5, 10].

Realizations of the Yang model have been discussed in [6, 7, 11, 12, 13]. They
cannot be obtained in closed analytic form, but only as power series in
coupling constants. In the classical limit, where quantum commutators are
replaced by Poisson brackets, the algebra has a simpler structure and no
ordering problems arise. The corresponding realizations of this Yang-Poisson
model can be obtained in an exact form [14].

The Yang Poisson model is generated with a Poisson algebra containing usual
Lorentz agebra of generators $M_{\mu\nu}$, with its standard action on phase
space parametrized by $\hat{x}_{\mu}$ and $\hat{p}_{\mu}$,

$\left\\{M_{\mu\nu},M_{\rho\sigma}\right\\}=\left(\eta_{\mu\rho}M_{\nu\sigma}-\eta_{\mu\sigma}M_{\nu\rho}-\eta_{\nu\rho}M_{\mu\sigma}+\eta_{\nu\sigma}M_{\mu\rho}\right),$
(1.1)
$\left\\{M_{\mu\nu},\hat{x}_{\lambda}\right\\}=\left(\eta_{\mu\lambda}\hat{x}_{\nu}-\eta_{\nu\lambda}\hat{x}_{\mu}\right),\quad\left\\{M_{\mu\nu},\hat{p}_{\lambda}\right\\}=\left(\eta_{\mu\lambda}\hat{p}_{\nu}-\eta_{\nu\lambda}\hat{p}_{\mu}\right)$
(1.2)
$\left\\{\hat{x}_{\nu},\hat{x}_{\nu}\right\\}=\beta^{2}M_{\mu\nu},\quad\left\\{\hat{p}_{\nu},\hat{p}_{\nu}\right\\}=\alpha^{2}M_{\mu\nu},$
(1.3) $\left\\{\hat{x}_{\mu},\hat{p}_{\nu}\right\\}=\eta_{\mu\nu}h,$ (1.4)
$\left\\{h,\hat{x}_{\mu}\right\\}=\beta^{2}\hat{p}_{\mu},\quad\left\\{h,\hat{p}_{\mu}\right\\}=-\alpha^{2}\hat{x}_{\mu},$
(1.5) $\left\\{M_{\mu\nu},h\right\\}=0,$ (1.6)

where $\left\\{,\right\\}$ are classical Poisson brackets,
$\mu,\nu=0,\ldots,n-1$, $\;\eta_{\mu\nu}=diag(-1,1,\ldots,1)$ is the Minkowski
metric and $\alpha$ and $\beta$ are real parameters. Generators denoted by
$A,B,C\;$ of this Poisson algebra satisfy Jacobi relations of type

$\left\\{A,\left\\{B,C\right\\}\right\\}+\left\\{B,\left\\{C,A\right\\}\right\\}+\left\\{C,\left\\{A,B\right\\}\right\\}=0.$
(1.7)

We look for realizations of $\hat{x}_{\mu}$ and $\hat{p}_{\mu}$ on a phase
space with coordinates $x_{\mu}$ and momenta $p_{\mu}$ satisfying the
canonical algebra

$\left\\{x_{\mu},x_{\nu}\right\\}=\left\\{p_{\mu},p_{\nu}\right\\}=0,\quad\left\\{x_{\mu},p_{\nu}\right\\}=\eta_{\mu\nu}.$
(1.8)

Realizations of the Yang Poisson algebra on a canonical phase space of
coordinates $x_{\mu}$ and $p_{\mu}$, with the Lorentz algebra generators
realized as $M_{\mu\nu}=x_{\mu}p_{\nu}-x_{\nu}p_{\mu}$ were analyzed and
discussed in [14]. Special solution for realizations of $\hat{x}_{\mu}$ and
$\hat{p}_{\mu}$ are given as

$\hat{x}_{\mu}=x_{\mu}\sqrt{1-\beta^{2}p^{2}+\phi_{1}(z)},$ (1.9)
$\hat{p}_{\mu}=p_{\mu}\sqrt{1-\alpha^{2}x^{2}+\phi_{2}(z)},$ (1.10)

where

$\phi_{1}\phi_{2}+\phi_{1}+\phi_{2}=z^{2},\quad z=\alpha\beta\left(xp\right)$
(1.11)

and

$h=\sqrt{\left(1-\beta^{2}p^{2}+\phi_{1}(z)\right)\left(1-\alpha^{2}x^{2}+\phi_{2}(z)\right)}.$
(1.12)

Generally, it holds that in terms of original variables $x_{\mu},p_{\mu}$,

$h=\sqrt{1-\alpha^{2}\hat{x}^{2}-\beta^{2}\hat{p}^{2}-\frac{\alpha^{2}\beta^{2}}{2}M^{2}}.$
(1.13)

This result for $h$ is universal and generally valid in the Yang Poisson model
[14]. The most general realizations are obtained using the group of
automorphisms applied to the special solution defined with $\phi_{2}(z)=0$
[14].

In the present paper, we shall consider generalized Yang-Poisson models and
their related models, as well as their exact realizations. In section (2) we
costruct generalizations of the Yang Poisson algebra (1.1)-(1.6). We construct
realizations of this algebra and solve the corresponding differential
equations, presenting the general results in section (3). Furthermore, we
discuss related Poisson algebras in section (4).

##  2 Generalized Yang Poisson models

The most general new generators linear in $\hat{x}_{\mu},\;\hat{p}_{\mu}\;$
and $M_{\mu\nu}$ are

$\tilde{X}_{\mu}=A\left(\cos\varphi\hat{x}_{\mu}+\frac{\beta}{\alpha}\sin\varphi\hat{p}_{\mu}\right)+\beta
a_{\nu}M_{\mu\nu},$ (2.1)
$\tilde{P}_{\mu}=B\left(\cos\psi\hat{p}_{\mu}+\frac{\alpha}{\beta}\sin\psi\hat{x}_{\mu}\right)+\alpha
b_{\nu}M_{\mu\nu},$ (2.2)

where $M_{\mu\nu}=x_{\mu}p_{\nu}-x_{\nu}p_{\mu}$ is unchanged and
$A,B,\varphi,\psi,a_{\mu},b_{\mu}\;$ are dimensionless parameters with $AB\neq
0$.
The inverse transformations are

$\hat{x}_{\mu}=\frac{A^{-1}\alpha\cos\psi\left(\tilde{X}_{\mu}-\beta
a_{\nu}M_{\mu\nu}\right)-B^{-1}\beta\sin\varphi\left(\tilde{P}_{\mu}-\alpha
b_{\nu}M_{\mu\nu}\right)}{\alpha\cos\left(\varphi+\psi\right)},$ (2.3)
$\hat{p}_{\mu}=\frac{B^{-1}\beta\cos\varphi\left(\tilde{P}_{\mu}-\alpha
b_{\nu}M_{\mu\nu}\right)-A^{-1}\alpha\sin\psi\left(\tilde{X}_{\mu}-\beta
a_{\nu}M_{\mu\nu}\right)}{\beta\cos\left(\varphi+\psi\right)}.$ (2.4)

The new generators $\tilde{X}_{\mu}\;$ and $\;\tilde{P}_{\mu}\;$ generate a
new class of Poisson algebras isomorphic to the initial Yang Poisson algebra.
The new algebra generated by $\tilde{X}_{\mu}\;\tilde{P}_{\mu},M_{\mu\nu}\;$
and $\;\tilde{h}$ is given by

$\left\\{\tilde{X}_{\mu},\tilde{X}_{\nu}\right\\}=\left(\beta^{2}\tilde{A}M_{\mu\nu}+\beta\left(a_{\mu}\tilde{X}_{\nu}-a_{\nu}\tilde{X}_{\mu}\right)\right),$
(2.5)
$\left\\{\tilde{P}_{\mu},\tilde{P}_{\nu}\right\\}=\left(\alpha^{2}\tilde{B}M_{\mu\nu}+\alpha\left(b_{\mu}\tilde{P}_{\nu}-b_{\nu}\tilde{P}_{\mu}\right)\right),$
(2.6)
$\left\\{\tilde{X}_{\mu},\tilde{P}_{\nu}\right\\}=\left(\eta_{\mu\nu}\tilde{h}+\alpha
b_{\mu}\tilde{X}_{\nu}-\beta a_{\nu}\tilde{P}_{\mu}+\alpha\beta
AB\tilde{\rho}M_{\mu\nu}\right),$ (2.7)
$\left\\{M_{\mu\nu},\tilde{X}_{\lambda}\right\\}=\left(\eta_{\mu\lambda}\tilde{X}_{\nu}-\eta_{\nu\lambda}\tilde{X}_{\mu}+\beta\left(a_{\mu}M_{\lambda\nu}-a_{\nu}M_{\lambda\mu}\right)\right),$
(2.8)
$\left\\{M_{\mu\nu},\tilde{P}_{\lambda}\right\\}=\left(\eta_{\mu\lambda}\tilde{P}_{\nu}-\eta_{\nu\lambda}\tilde{P}_{\mu}+\alpha\left(b_{\mu}M_{\lambda\nu}-b_{\nu}M_{\lambda\mu}\right)\right),$
(2.9)
$\left\\{M_{\mu\nu},\tilde{h}\right\\}=\alpha\left(b_{\nu}\tilde{X}_{\mu}-b_{\mu}\tilde{X}_{\nu}\right)-\beta\left(a_{\nu}\tilde{P}_{\mu}-a_{\mu}\tilde{P}_{\nu}\right),$
(2.10)
$\left\\{\tilde{h},\tilde{X}_{\mu}\right\\}=\beta^{2}\tilde{A}\tilde{P}_{\mu}-\alpha\beta
AB\tilde{\rho}\tilde{X}_{\mu}-\beta a_{\mu}\tilde{h},$ (2.11)
$\left\\{\tilde{h},\tilde{P}_{\mu}\right\\}=-\left(\alpha^{2}\tilde{B}\tilde{X}_{\mu}-\alpha\beta
AB\tilde{\rho}\tilde{P}_{\mu}-\alpha b_{\mu}\tilde{h}\right),$ (2.12)

where

$\tilde{h}=AB\cos(\varphi+\psi)h+\beta a\tilde{P}-\alpha
b\tilde{X}-\alpha\beta a_{\mu}b_{\nu}M_{\mu\nu},$ (2.13)

and

$\tilde{\rho}=\sin(\varphi+\psi)+\frac{ab}{AB},\quad\tilde{A}=A^{2}+a^{2}\quad\text{and}\quad\tilde{B}=B^{2}+b^{2}.$
(2.14)

The algebra (2.5)-(2.14) is invariant under Born duality [2],
$\alpha\leftrightarrow\beta,\;a_{i}\rightarrow-b_{i},\;b_{i}\rightarrow
a_{i},\;\tilde{A}\leftrightarrow\tilde{B},\;\tilde{\rho}\leftrightarrow-\tilde{\rho},\;\tilde{X}_{\mu}\rightarrow-\tilde{P}_{\mu},\;\tilde{P}_{\mu}\rightarrow\tilde{X}_{\mu},\;M_{\mu\nu}\leftrightarrow
M_{\mu\nu},\;\tilde{h}\leftrightarrow\tilde{h}.$

In the following we cosider a case where $a_{\mu}=b_{\mu}=0$. In this case the
Poisson algebra (2.5)-(2.14) corresponds to classical limit of Khruschev
Leznov algebra [4], where quantum commutators are replaced by Poisson
brackets. Let us denote $\tilde{\alpha}^{2}=\alpha^{2}\tilde{B}\;$ and
$\tilde{\beta}^{2}=\beta^{2}\tilde{A}$. Then we have

$\tilde{h}=AB\cos(\varphi+\psi)\sqrt{1-\tilde{\alpha}^{2}\tilde{X}^{2}-\tilde{\beta}^{2}\tilde{P}^{2}+2\tilde{\rho}\tilde{\alpha}\tilde{\beta}\tilde{X}\tilde{P}-\frac{\tilde{\alpha}^{2}\tilde{\beta}^{2}}{2A^{2}B^{2}}M^{2}},$
(2.15)

where

$\tilde{\rho}=\sin(\varphi+\psi),\quad\tilde{A}=A^{2}\quad\text{and}\quad\tilde{B}=B^{2}.$

Using realizations of $\hat{x}_{\mu}\;$ and $\;\hat{p}_{\mu}$ from section (1)
((1.9) and (1.10)) we can write realization of
$\tilde{X}_{\mu},\;\tilde{P}_{\mu}$ in terms of $x_{\mu}\;$ and $p_{\mu}$

$\tilde{X}_{\mu}=x_{\mu}A\cos(\varphi)\sqrt{1-\frac{\tilde{\beta}^{2}}{A^{2}}p^{2}+\varphi_{1}(z)}+\frac{\tilde{\beta}}{\tilde{\alpha}}p_{\mu}B\sin(\varphi)\sqrt{1-\frac{\tilde{\alpha}^{2}}{B^{2}}x^{2}+\varphi_{2}(z)},$
(2.16)
$\tilde{P}_{\mu}=p_{\mu}B\cos(\psi)\sqrt{1-\frac{\tilde{\alpha}^{2}}{B^{2}}x^{2}+\varphi_{2}(z)}+\frac{\tilde{\alpha}}{\tilde{\beta}}x_{\mu}A\sin(\varphi)\sqrt{1-\frac{\tilde{\beta}^{2}}{A^{2}}p^{2}+\varphi_{1}(z)}.$
(2.17)

Generalizing (2.16) and (2.17), we get the general ansatz for
$\tilde{X}_{\mu}\;$ and $\tilde{P}_{\mu}$

$\tilde{X}_{\mu}=x_{\mu}f+\frac{\tilde{\beta}}{\tilde{\alpha}}p_{\mu}g$ (2.18)

and

$\tilde{P}_{\mu}=p_{\mu}\tilde{f}+\frac{\tilde{\alpha}}{\tilde{\beta}}x_{\mu}\tilde{g},$
(2.19)

where $f,g,\tilde{f},\tilde{g}\;$ are functions of $u,v,z$ and
$u=\tilde{\beta}^{2}p^{2},\;v=\tilde{\alpha}^{2}x^{2},\;z=\frac{\tilde{\alpha}\tilde{\beta}}{AB}\left(xp\right)$.
From
$\left\\{\tilde{X}_{\mu},\tilde{X}_{\nu}\right\\}=\tilde{\beta^{2}}M_{\mu\nu}$
we get

$\displaystyle-2f\frac{\partial f}{\partial u}-2g\frac{\partial g}{\partial
v}+4z\left(\frac{\partial f}{\partial v}\frac{\partial g}{\partial
u}-\frac{\partial f}{\partial u}\frac{\partial g}{\partial
v}\right)+2v\left(\frac{\partial f}{\partial v}\frac{\partial g}{\partial
z}-\frac{\partial f}{\partial z}\frac{\partial g}{\partial v}\right)$
$\displaystyle+2u\left(\frac{\partial f}{\partial z}\frac{\partial g}{\partial
u}-\frac{\partial f}{\partial u}\frac{\partial g}{\partial
z}\right)+f\frac{\partial g}{\partial z}+g\frac{\partial f}{\partial z}=1$
(2.20)

and from
$\left\\{\tilde{P}_{\mu},\tilde{P}_{\nu}\right\\}=\tilde{\alpha^{2}}M_{\mu\nu}$
it follows

$\displaystyle-2\tilde{f}\frac{\partial\tilde{f}}{\partial
v}-2\tilde{g}\frac{\partial\tilde{g}}{\partial
u}+4z\left(\frac{\partial\tilde{f}}{\partial
u}\frac{\partial\tilde{g}}{\partial v}-\frac{\partial\tilde{f}}{\partial
v}\frac{\partial\tilde{g}}{\partial
u}\right)+2v\left(\frac{\partial\tilde{f}}{\partial
z}\frac{\partial\tilde{g}}{\partial v}-\frac{\partial\tilde{f}}{\partial
v}\frac{\partial\tilde{g}}{\partial z}\right)$
$\displaystyle+2u\left(\frac{\partial\tilde{f}}{\partial
u}\frac{\partial\tilde{g}}{\partial z}-\frac{\partial\tilde{f}}{\partial
z}\frac{\partial\tilde{g}}{\partial
u}\right)+\tilde{f}\frac{\partial\tilde{g}}{\partial
z}+\tilde{g}\frac{\partial\tilde{f}}{\partial z}=1.$ (2.21)

The relation
$\left\\{\tilde{X}_{\mu},\tilde{P}_{\nu}\right\\}=\eta_{\mu\nu}\tilde{h}+\tilde{\alpha}\tilde{\beta}\tilde{\rho}M_{\mu\nu}$
yields following five equations:

$f\tilde{f}-g\tilde{g}=\tilde{h},$ (2.22) $\displaystyle
2\tilde{f}\frac{\partial f}{\partial v}-2g\frac{\partial\tilde{g}}{\partial
v}+4z\left(\frac{\partial f}{\partial v}\frac{\partial\tilde{g}}{\partial
u}-\frac{\partial f}{\partial u}\frac{\partial\tilde{g}}{\partial
v}\right)+2v\left(\frac{\partial f}{\partial
v}\frac{\partial\tilde{g}}{\partial z}-\frac{\partial f}{\partial
z}\frac{\partial\tilde{g}}{\partial v}\right)$
$\displaystyle+2u\left(\frac{\partial f}{\partial
z}\frac{\partial\tilde{g}}{\partial u}-\frac{\partial f}{\partial
u}\frac{\partial\tilde{g}}{\partial
z}\right)+f\frac{\partial\tilde{g}}{\partial z}-\tilde{g}\frac{\partial
f}{\partial z}=0.$ (2.23) $\displaystyle-2f\frac{\partial\tilde{f}}{\partial
u}-2\tilde{g}\frac{\partial g}{\partial
u}+4z\left(\frac{\partial\tilde{f}}{\partial u}\frac{\partial g}{\partial
v}-\frac{\partial\tilde{f}}{\partial v}\frac{\partial g}{\partial
u}\right)+2v\left(\frac{\partial\tilde{f}}{\partial z}\frac{\partial
g}{\partial v}-\frac{\partial\tilde{f}}{\partial v}\frac{\partial g}{\partial
z}\right)$ $\displaystyle+2u\left(\frac{\partial\tilde{f}}{\partial
u}\frac{\partial g}{\partial z}-\frac{\partial\tilde{f}}{\partial
z}\frac{\partial g}{\partial u}\right)+\tilde{f}\frac{\partial g}{\partial
z}-g\frac{\partial\tilde{f}}{\partial z}=0.$ (2.24)
$\displaystyle-2\tilde{g}\frac{\partial f}{\partial
u}-2g\frac{\partial\tilde{f}}{\partial v}+4z\left(\frac{\partial f}{\partial
v}\frac{\partial\tilde{f}}{\partial u}-\frac{\partial f}{\partial
u}\frac{\partial\tilde{f}}{\partial v}\right)+2v\left(\frac{\partial
f}{\partial v}\frac{\partial\tilde{f}}{\partial z}-\frac{\partial f}{\partial
z}\frac{\partial\tilde{f}}{\partial v}\right)$
$\displaystyle+2u\left(\frac{\partial f}{\partial
z}\frac{\partial\tilde{f}}{\partial u}-\frac{\partial f}{\partial
u}\frac{\partial\tilde{f}}{\partial
z}\right)+f\frac{\partial\tilde{f}}{\partial z}+\tilde{f}\frac{\partial
f}{\partial z}=\tilde{\rho}.$ (2.25) $\displaystyle
2f\frac{\partial\tilde{g}}{\partial u}+2\tilde{f}\frac{\partial g}{\partial
v}+4z\left(\frac{\partial g}{\partial v}\frac{\partial\tilde{g}}{\partial
u}-\frac{\partial g}{\partial u}\frac{\partial\tilde{g}}{\partial
v}\right)+2v\left(\frac{\partial g}{\partial
v}\frac{\partial\tilde{g}}{\partial z}-\frac{\partial g}{\partial
z}\frac{\partial\tilde{g}}{\partial v}\right)$
$\displaystyle+2u\left(\frac{\partial g}{\partial
z}\frac{\partial\tilde{g}}{\partial u}-\frac{\partial g}{\partial
u}\frac{\partial\tilde{g}}{\partial
z}\right)-g\frac{\partial\tilde{g}}{\partial z}-\tilde{g}\frac{\partial
g}{\partial z}=-\tilde{\rho}.$ (2.26)

Note that from $\eqref{1.3}$ and $\eqref{1.4}$ and comparing them with (2.18)
and (2.19) it follows that

$f=A\cos(\varphi)\sqrt{1-\frac{u}{A^{2}}+\varphi_{1}(z)},\quad
g=B\sin(\varphi)\sqrt{1-\frac{v}{B^{2}}+\varphi_{2}(z)}$ (2.27)

and

$\tilde{f}=B\cos(\psi)\sqrt{1-\frac{v}{B^{2}}+\varphi_{2}(z)},\quad\tilde{g}=A\sin(\varphi)\sqrt{1-\frac{u}{A^{2}}+\varphi_{1}(z)}.$
(2.28)

We have checked that functions $f,g,\tilde{f},\tilde{g}$ in (2.27) and (2.28)
are solutions of seven differential equations (2.20)-(2.26). Also, it is easy
to verify that special solutions from section (1) ((1.9) and (1.10)) satisfy
differential equations (2.20)-(2.26).

In particular, for $\tilde{f}=\sqrt{1-v},\;\tilde{g}=0\;$ and $g\equiv g(v)$
from equations (2.20)-(2.26) we find that

$f=\sqrt{\left(1-\tilde{\rho}^{2}\right)\left(1-u+z^{2}\right)},\quad
g=\tilde{\rho}\sqrt{1-v}$

and

$h=\sqrt{\left(1-\tilde{\rho}^{2}\right)\left(1-u-v+uv+z^{2}-vz^{2}\right)}.$

In case where $h=\tilde{h}$ we have

$AB\cos\left(\varphi+\psi\right)=1\quad\text{and}\quad
1-\tilde{\rho}^{2}=\frac{1}{A^{2}B^{2}}$

which implies that

$\tilde{h}=\sqrt{1-\tilde{\alpha}^{2}\tilde{X}^{2}-\tilde{\beta}^{2}\tilde{P}^{2}+2\tilde{\rho}\tilde{\alpha}\tilde{\beta}\tilde{X}\tilde{P}-\frac{1-\tilde{\rho}^{2}}{2}\tilde{\alpha}^{2}\tilde{\beta}^{2}M^{2}}.$

In the case when $\tilde{\rho}=1$ we find that
$\cos\left(\varphi+\psi\right)=0$ which implies $\tilde{h}=0$ and then from
(2.11) and (2.12) we have
$\;\tilde{X}_{\mu}=\frac{\tilde{\beta}}{\tilde{\alpha}}\tilde{P}_{\mu}$.
Simirarly, in the case when $\tilde{\rho}=-1,\;$ we have $\tilde{h}=0$ and
$\;\tilde{X}_{\mu}=-\frac{\tilde{\beta}}{\tilde{\alpha}}\tilde{P}_{\mu}$.

##  3 General solution

###### Proposition 3.1

If $\hat{\mathcal{L}}$ is an operator acting on the deformed phase space
defined on $\hat{x}_{\mu}$ and $\hat{p}_{\mu}$ as

$\hat{\mathcal{L}}\left(\hat{x}_{\mu}\right)=\left\\{\frac{1}{\alpha\beta}h,\hat{x}_{\mu}\right\\}=\frac{\beta}{\alpha}\hat{p}_{\mu},$
(3.1)
$\hat{\mathcal{L}}\left(\hat{p}_{\mu}\right)=\left\\{\frac{1}{\alpha\beta}h,\hat{p}_{\mu}\right\\}=-\frac{\alpha}{\beta}\hat{x}_{\mu},$
(3.2)

where $h$ is defined in (1.12) and (1.13), then it holds

$\left(e^{\varphi\hat{\mathcal{L}}}\right)\left(\hat{x}_{\mu}\right)=\hat{x}_{\mu}\cos{\varphi}+\frac{\beta}{\alpha}\hat{p}_{\mu}\sin{\varphi}$
(3.3)

and

$\left(e^{-\psi\hat{\mathcal{L}}}\right)\left(\hat{p}_{\mu}\right)=\hat{p}_{\mu}\cos{\psi}+\frac{\alpha}{\beta}\hat{x}_{\mu}\sin{\psi}$
(3.4)

Proof. First, we have that

$\left(e^{\varphi\hat{\mathcal{L}}}\right)\left(\hat{x}_{\mu}\right)=\hat{x}_{\mu}+\frac{\varphi}{\alpha\beta}\left\\{h,\hat{x}_{\mu}\right\\}+\frac{1}{2!}\left(\frac{\varphi}{\alpha\beta}\right)^{2}\left\\{h,\left\\{h,\hat{x}_{\mu}\right\\}\right\\}+\cdots+\frac{1}{n!}\left(\frac{\varphi}{\alpha\beta}\right)^{n}\underset{n\quad\text{times}}{\underbrace{\left\\{h,\ldots,\left\\{h,\left\\{h,\hat{x}_{\mu}\right\\}\right\\}\ldots\right\\}}}+\cdots$
(3.5)

By induction on $n$ and using (1.5) we prove relations

$\underset{n=2k\quad\text{times}}{\underbrace{\left\\{h,\ldots,\left\\{h,\left\\{h,\hat{x}_{\mu}\right\\}\right\\}\ldots\right\\}}}=\left(-1\right)^{k}\alpha^{2k}\beta^{2k}\hat{x}_{\mu},\quad
k=1,2,\ldots$ (3.6)

and

$\underset{n=2k+1\quad\text{times}}{\underbrace{\left\\{h,\ldots,\left\\{h,\left\\{h,\hat{x}_{\mu}\right\\}\right\\}\ldots\right\\}}}=\left(-1\right)^{k}\alpha^{2k}\beta^{2k+2}\hat{p}_{\mu},\quad
k=0,1,\ldots$ (3.7)

For the case when $n=2k$, it is easy to see, using (1.5) that for $k=1$ we get

$\left\\{h,\left\\{h,\hat{x}_{\mu}\right\\}\right\\}=-\alpha^{2}\beta^{2}\hat{x}_{\mu}.$
(3.8)

Assume that the relation (3.6) holds for some $k>1$. Then by the induction
assumption and using (1.5) we have

$\displaystyle\underset{n=2k+2\quad\text{times}}{\underbrace{\left\\{h,\ldots,\left\\{h,\left\\{h,\hat{x}_{\mu}\right\\}\right\\}\ldots\right\\}}}$
$\displaystyle=\left(-1\right)^{k}\alpha^{2k}\beta^{2k}\left\\{h,\left\\{h,\hat{x}_{\mu}\right\\}\right\\}$
$\displaystyle=\left(-1\right)^{k+1}\alpha^{2\left(k+1\right)}\beta^{2\left(k+1\right)}\hat{x}_{\mu}.$

Simirarly, by induction on $n$ and using (1.5) we prove that relation (3.7)
holds. Now, inserting relations (3.6) and (3.7) in (3.5) we finally prove that
(3.3) holds. Also, in a similar way we prove (3.4). $\blacksquare$

Let us define an operator $\mathcal{L}_{G}$ acting on a classical phase space
defined on $x_{\mu}$ and $p_{\mu}$ as

$\mathcal{L}_{G}\left(f\right)=\left\\{G,f\right\\},$ (3.9)

where $G(x,p)$ and $f(x,p)$ are functions on classical phase space.
Furthermore, we define

$\hat{x}_{\mu}^{(0)}=\sqrt{1-\beta^{2}p^{2}}x_{\mu},$ (3.10)
$\hat{p}_{\mu}^{(0)}=\sqrt{1-\alpha^{2}x^{2}}p_{\mu}$ (3.11)

and then construct an operator $\mathcal{O}$ such that

$\left\\{\mathcal{O}\left(\hat{x}_{\mu}^{(0)}\right),\hat{p}_{\nu}^{(0)}\right\\}=\eta_{\mu\nu}h,$
(3.12)

where $h$ is given in (1.12). The general structure of an operator
$\mathcal{O}$ acting on $\hat{x}_{\mu}^{(0)}$ is

$\mathcal{O}\left(\hat{x}_{\mu}^{(0)}\right)=\left(e^{\mathcal{L}_{G}}\right)\left(\hat{x}_{\mu}^{(0)}\right)=\hat{x}_{\mu}^{(0)}+\left\\{G,\hat{x}_{\mu}^{(0)}\right\\}+\frac{1}{2!}\left\\{G,\left\\{G,\hat{x}_{\mu}^{(0)}\right\\}\right\\}+\frac{1}{3!}\left\\{G\left\\{G,\left\\{G,\hat{x}_{\mu}^{(0)}\right\\}\right\\}\right\\}+\cdots$
(3.13)

Solving pertrbatively (3.12) a uniqe solution was found for $G$ in [14],

$G=\sum_{n=1}^{\infty}\alpha^{2n}\beta^{2n}g_{2n},$ (3.14)

where

$g_{2n}=\frac{\left(-1\right)^{n}\cdot\left(xp\right)^{2n+1}}{2n\cdot\left(2n+1\right)}.$
(3.15)

The summation of the equation (3.14) for $G$ gives an exact result,

$G=\frac{1}{\alpha\beta}\left(z\left(1-\frac{1}{2}\ln\left(1+z^{2}\right)\right)-\arctan
z\right).$ (3.16)

If we fix
$\;\tilde{P}_{\mu}^{(0)}=\hat{p}_{\mu}^{(0)}=\sqrt{1-\alpha^{2}x^{2}}p_{\mu}\;$
then we obtain the corresponding

$\tilde{X}_{\mu}^{(1)}=A\left(e^{\varphi\hat{\mathcal{L}}}\circ
e^{\mathcal{L}_{G}}\right)\left(\hat{x}_{\mu}^{(0)}\right)=A\left(e^{\varphi\hat{\mathcal{L}}}\circ
e^{\mathcal{L}_{G}}\right)\left(\sqrt{1-\beta^{2}p^{2}}x_{\mu}\right)$ (3.17)

and

$h^{(1,0)}=\sqrt{\left(1-\beta^{2}p^{2}+z^{2}\right)\left(1-\alpha^{2}x^{2}\right)}$
(3.18)

with the property

$\left\\{\tilde{X}_{\mu}^{(1)},\tilde{P}_{\mu}^{(0)}\right\\}=\eta_{\mu\nu}A\cos\left(\varphi\right)h^{(1,0)}+A\sin\left(\varphi\right)\alpha\beta
M_{\mu\nu}.$ (3.19)

The composition of mappings $\;e^{\varphi\hat{\mathcal{L}}}\circ
e^{\mathcal{L}_{G}}\;$ can be calculated pertubatively applying the BCH
formula.

###### Proposition 3.2

Application of the BCH formula to a Poisson algebra with the classical Poisson
brackets is

$e^{\mathcal{L}_{A}}\circ e^{\mathcal{L}_{B}}=e^{\mathcal{L}_{C}},$ (3.20)

where
$C=A+B+\frac{1}{2}\left\\{A,B\right\\}+\frac{1}{12}\left\\{A,\left\\{A,B\right\\}\right\\}-\frac{1}{12}\left\\{B,\left\\{A,B\right\\}\right\\}+\cdots$

Proof. From the BCH formula we have

$e^{\mathcal{L}_{A}}\circ
e^{\mathcal{L}_{B}}=e^{\left(\mathcal{L}_{A}+L_{B}+\frac{1}{2}\left[\mathcal{L}_{A},\mathcal{L}_{B}\right]+\frac{1}{12}\left[\mathcal{L}_{A},\left[\mathcal{L}_{A},\mathcal{L}_{B}\right]\right]-\frac{1}{12}\left[\mathcal{L}_{B},\left[\mathcal{L}_{A},\mathcal{L}_{B}\right]\right]+\cdots\right)}$
(3.21)

which implies that in order to prove (3.20) is sufficient to prove that the
relation

$\left[\mathcal{L}_{A_{1}},\ldots\left[\mathcal{L}_{A_{n-1}},\left[\mathcal{L}_{A_{n}},\mathcal{L}_{B}\right]\right]\ldots\right]=\mathcal{L}_{\left(\left\\{A_{1},\ldots\left\\{A_{n-1},\left\\{A_{n},B\right\\}\right\\}\ldots\right\\}\right)}$
(3.22)

holds. We prove (3.22) by induction on $n$. For $n=1$, using the Jacobi
identity we get

$\displaystyle\left(\left[\mathcal{L}_{A_{1}},\mathcal{L}_{B}\right]\right)\left(F(x,p)\right)$
$\displaystyle=\left(\mathcal{L}_{A_{1}}\mathcal{L}_{B}-\mathcal{L}_{B}\mathcal{L}_{A_{1}}\right)\left(F(x,p)\right)$
$\displaystyle=\left\\{A_{1},\left\\{B,F\right\\}\right\\}-\left\\{B,\left\\{A_{1},F\right\\}\right\\}$
$\displaystyle=\left\\{\left\\{A_{1},B\right\\},F\right\\}=\mathcal{L}_{\left(\left\\{A_{1},B\right\\}\right)}\left(F(x,p)\right).$

Let us assume that the relation (3.22) holds for some $n>1$. Then by the
induction assumption and using the Jacobi identity we have

$\displaystyle\left(\left[\mathcal{L}_{A_{1}},\ldots\left[\mathcal{L}_{A_{n}},\left[\mathcal{L}_{A_{n+1}},\mathcal{L}_{B}\right]\right]\ldots\right]\right)\left(F(x,p)\right)$
$\displaystyle=\left(A_{1}\left[\mathcal{L}_{A_{2}},\ldots\left[\mathcal{L}_{A_{n}},\left[\mathcal{L}_{A_{n+1}},\mathcal{L}_{B}\right]\right]\ldots\right]-\left[\mathcal{L}_{A_{2}},\ldots\left[\mathcal{L}_{A_{n}},\left[\mathcal{L}_{A_{n+1}},\mathcal{L}_{B}\right]\right]\ldots\right]A_{1}\right)\left(F(x,p)\right)$
$\displaystyle=\left\\{A_{1},\left\\{\left\\{A_{2},\ldots\left\\{A_{n},\left\\{A_{n+1},B\right\\}\right\\}\ldots\right\\},F\right\\}\right\\}-\left\\{\left\\{A_{2},\ldots\left\\{A_{n},\left\\{A_{n+1},B\right\\}\right\\}\ldots\right\\},\left\\{A_{1},F\right\\}\right\\}$
$\displaystyle=\left\\{\left\\{A_{1},\left\\{A_{2},\ldots\left\\{A_{n},\left\\{A_{n+1},B\right\\}\right\\}\ldots\right\\}\right\\},F\right\\}=\mathcal{L}_{\left(\left\\{A_{1},\ldots\left\\{A_{n},\left\\{A_{n+1},B\right\\}\right\\}\ldots\right\\}\right)}\left(F(x,p)\right).$

$\blacksquare$

Let us denote $e^{\mathcal{L}_{\tilde{G}}}=e^{\varphi\hat{\mathcal{L}}}\circ
e^{\mathcal{L}_{G}}$. Then in the first order we get
$\mathcal{L}_{\tilde{G}}=\varphi\hat{\mathcal{L}}+\mathcal{L}_{G}+\frac{1}{2}\varphi\left\\{\hat{\mathcal{L}},\mathcal{L}_{G}\right\\}\pm\cdots$.
The most general realizations of $\tilde{X}_{\mu},\tilde{P}_{\mu}$ and $h$ are
obtained using the group of automorphisms applied to the special solution
$\tilde{X}_{\mu}^{(1)},\tilde{P}_{\mu}^{(0)}$ and $h^{(1,0)}$, namely
$\tilde{X}_{\mu}=O_{F}\left(\tilde{X}_{\mu}^{(1)}\right),\;\tilde{P}_{\mu}=O_{F}\left(\tilde{P}_{\mu}^{(0)}\right)$
and $h=O_{F}\left(h^{(1,0)}\right)$, where $O_{F}=e^{\mathcal{L}_{F}}$ and $F$
is arbitrary function of $\alpha^{2}x^{2},\;\beta^{2}p^{2}$ and $z$.
Alternatively we can write

$\tilde{X}_{\mu}=A\left(e^{\varphi\hat{\mathcal{L}}}\circ
e^{\mathcal{L}_{F}}\right)\left(\hat{x}_{\mu}^{(1)}\right)=A\left(e^{\varphi\hat{\mathcal{L}}}\circ
e^{\mathcal{L}_{F}}\right)\left(\sqrt{1-\beta^{2}p^{2}+z^{2}}x_{\mu}\right)$
(3.23)

and

$\tilde{P}_{\mu}=B\left(e^{-\psi\hat{\mathcal{L}}}\circ
e^{\mathcal{L}_{F}}\right)\left(\hat{p}_{\mu}^{(0)}\right)=B\left(e^{-\psi\hat{\mathcal{L}}}\circ
e^{\mathcal{L}_{F}}\right)\left(\sqrt{1-\alpha^{2}x^{2}}p_{\mu}\right).$
(3.24)

These solutions satisfy the above seven differential equations (2.20)-(2.26).

###### Remark 3.1

Note that the Poisson brackets are covariant under action of
$e^{\mathcal{L}_{F}}$ i.e they satisfy

$\left\\{\left(e^{\mathcal{L}_{F}}\right)(f),\left(e^{\mathcal{L}_{F}}\right)(g)\right\\}=\left(e^{\mathcal{L}_{F}}\right)\left(\left\\{f,\;g\right\\}\right)$
(3.25)

for any function $F,f,g$. Also
$\;\left(e^{\mathcal{L}_{F}}\right)\left(M_{\mu\nu}\right)=M_{\mu\nu}\;$ if
$\;\left\\{F,M_{\mu\nu}\right\\}=0\;$ and
$\;\left(e^{\mathcal{L}_{F}}\right)\left(\eta_{\mu\nu}\right)=\eta_{\mu\nu}\;$.

###### Remark 3.2

Using realizations of the Snyder model [8] we can write the corresponding
realizations of the Snyder Poisson model. From the results that were found in
[8] we get

$\hat{x}_{\mu}=\left(e^{\mathcal{L}_{G}}\right)\left(x_{\mu}+\beta^{2}(x\cdot
p)p_{\mu}\right)=x_{\mu}\varphi_{1}(u)+\beta^{2}(x\cdot
p)p_{\mu}\varphi_{2}(u)$

and

$\hat{p}_{\mu}=\left(e^{\mathcal{L}_{G}}\right)\left(p_{\mu}\right)=p_{\mu}\frac{1}{\varphi_{1}(u)},$

where

$G=(x\cdot
p)F(u),\quad\varphi_{2}(u)=\frac{1+\dot{\varphi}_{1}(u)\varphi_{1}(u)}{\varphi_{1}(u)-2u\dot{\varphi}_{1}(u)},\quad\dot{\varphi}_{1}=\frac{d\varphi_{1}(u)}{du}\quad\text{and}\quad
u=\beta^{2}p^{2}.$

##  4 Related Poisson models

Here we introduce related Poisson algebras generalizing (1.3), (1.4) and
(1.5):

$\left\\{\hat{x}_{\nu},\hat{x}_{\nu}\right\\}=\epsilon_{1}\beta^{2}M_{\mu\nu},\quad\left\\{\hat{p}_{\nu},\hat{p}_{\nu}\right\\}=\epsilon_{2}\alpha^{2}M_{\mu\nu},$
(4.1) $\left\\{\hat{x}_{\mu},\hat{p}_{\nu}\right\\}=\eta_{\mu\nu}h,$ (4.2)
$\left\\{h,\hat{x}_{\mu}\right\\}=\epsilon_{1}\beta^{2}\hat{p}_{\mu},\quad\left\\{h,\hat{p}_{\mu}\right\\}=-\epsilon_{2}\alpha^{2}\hat{x}_{\mu},$
(4.3)

where $\epsilon_{1}^{2}=\epsilon_{2}^{2}=1$. The Yang-Poisson model in section
(1) corresponds to the $\epsilon_{1}=\epsilon_{2}=1$ case.

Now, let us consider the case when $\mathbf{\epsilon_{1}=\epsilon_{2}=-1}$. A
special solution for realizations of $\hat{x}_{\mu}$ and $\hat{p}_{\mu}$ in
this case is given as

$\hat{x}_{\mu}=x_{\mu}\sqrt{1+\beta^{2}p^{2}+\phi_{1}(z)},$ (4.4)
$\hat{p}_{\mu}=p_{\mu}\sqrt{1+\alpha^{2}x^{2}+\phi_{2}(z)},$ (4.5)

where

$\phi_{1}\phi_{2}+\phi_{1}+\phi_{2}=z^{2},\quad z=\alpha\beta\left(xp\right)$
(4.6)

and

$h=\sqrt{\left(1+\beta^{2}p^{2}+\phi_{1}(z)\right)\left(1+\alpha^{2}x^{2}+\phi_{2}(z)\right)}.$
(4.7)

The new generators $\tilde{X}_{\mu}\;$ and $\tilde{P}_{\mu}$ linear in
$\hat{x}_{\mu},\;\hat{p}_{\mu}\;$ and $M_{\mu\nu}$ are given in (2.1) and
(2.2). Also, a new algebra generated by
$\tilde{X}_{\mu}\;\tilde{P}_{\mu},M_{\mu\nu}\;$ and $\;\tilde{h}$ is given in
(2.5)-(2.13), where

$\tilde{\rho}=-\sin(\varphi+\psi)+\frac{ab}{AB},\quad\tilde{A}=-A^{2}+a^{2}\quad\text{and}\quad\tilde{B}=-B^{2}+b^{2}.$
(4.8)

In the case when $\mathbf{\epsilon_{1}=1}$ and $\mathbf{\epsilon_{2}=-1}$
special solution is given by

$\hat{x}_{\mu}=x_{\mu}\sqrt{1-\beta^{2}p^{2}+\phi_{1}(z)},$ (4.9)
$\hat{p}_{\mu}=p_{\mu}\sqrt{1+\alpha^{2}x^{2}+\phi_{2}(z)},$ (4.10)

where

$\phi_{1}\phi_{2}+\phi_{1}+\phi_{2}=-z^{2},\quad z=\alpha\beta\left(xp\right)$
(4.11)

and

$h=\sqrt{\left(1-\beta^{2}p^{2}+\phi_{1}(z)\right)\left(1+\alpha^{2}x^{2}+\phi_{2}(z)\right)}.$
(4.12)

The new generators $\tilde{X}_{\mu}\;$ and $\tilde{P}_{\mu}$ are given by

$\tilde{X}_{\mu}=A\left(\cosh\varphi\hat{x}_{\mu}+\frac{\beta}{\alpha}\sinh\varphi\hat{p}_{\mu}\right)+\beta
a_{\nu}M_{\mu\nu},$ (4.13)
$\tilde{P}_{\mu}=B\left(\cosh\psi\hat{p}_{\mu}+\frac{\alpha}{\beta}\sinh\psi\hat{x}_{\mu}\right)+\alpha
b_{\nu}M_{\mu\nu},$ (4.14)

where $M_{\mu\nu}=x_{\mu}p_{\nu}-x_{\nu}p_{\mu}$ and
$A,B,\varphi,\psi,a_{\mu},b_{\mu}\;$ are dimensionless parameters with $AB\neq
0$. New generators $\tilde{X}_{\mu}\;$ and $\tilde{P}_{\mu}$ together with
$M_{\mu\nu}\;$ and $\;\tilde{h}$ generate a new algebra, isomorphic to the
initial Yang Poisson algebra, which is given in (2.5)-(2.12), where

$\tilde{h}=AB\cosh(\psi-\varphi)h+\beta a\tilde{P}-\alpha
b\tilde{X}-\alpha\beta a_{\mu}b_{\nu}M_{\mu\nu},$ (4.15)

and

$\tilde{\rho}=\sinh(\psi-\varphi)+\frac{ab}{AB},\quad\tilde{A}=A^{2}+a^{2}\quad\text{and}\quad\tilde{B}=-B^{2}+b^{2}.$
(4.16)

Finally, in the last case when $\mathbf{\epsilon_{1}=-1}\;$ and
$\mathbf{\epsilon_{2}=1}$ a special solution is given by

$\hat{x}_{\mu}=x_{\mu}\sqrt{1+\beta^{2}p^{2}+\phi_{1}(z)},$ (4.17)
$\hat{p}_{\mu}=p_{\mu}\sqrt{1-\alpha^{2}x^{2}+\phi_{2}(z)},$ (4.18)

where

$\phi_{1}\phi_{2}+\phi_{1}+\phi_{2}=-z^{2},\quad z=\alpha\beta\left(xp\right)$
(4.19)

and

$h=\sqrt{\left(1+\beta^{2}p^{2}+\phi_{1}(z)\right)\left(1-\alpha^{2}x^{2}+\phi_{2}(z)\right)}.$
(4.20)

The new generators $\tilde{X}_{\mu}\;$ and $\tilde{P}_{\mu}$ are given in
(4.13) and (4.14) and a new algebra generated by them together with
$M_{\mu\nu}$ and $\tilde{h}$ is given in (2.5)-(2.12), where

$\tilde{h}=AB\cosh(\psi-\varphi)h+\beta a\tilde{P}-\alpha
b\tilde{X}-\alpha\beta a_{\mu}b_{\nu}M_{\mu\nu},$ (4.21)

and

$\tilde{\rho}=-\sinh(\psi-\varphi)+\frac{ab}{AB},\quad\tilde{A}=-A^{2}+a^{2}\quad\text{and}\quad\tilde{B}=B^{2}+b^{2}.$
(4.22)

##  5 Discussion

We have obtained the exact realizations of the generalized Yang-Poisson
algebra on a canonical phase space related to the $so(1,5)$ algebra. These
realizations are simpler than in the quantum case. The results obtained in
this paper can be considered as a limit of the quantum formalism for
$\;\hbar\rightarrow 0,\;$ presenting a classical approximation of quantum
realizations. Futhermore, we have discussed the Poisson algebras related to
$so(3,3)$ and $so(2,4)$ algebras. The generalized Yang models are examples of
noncommutative geometry on a background spacetime of constant curvature and
possess duality between position and momentum manifolds.

In the present paper a canonical phase space is defined with coordinates
$x_{\mu}$, momenta $p_{\mu}$ and Lorentz generators
$M_{\mu\nu}=x_{\mu}p_{\nu}-x_{\nu}p_{\mu}$. However it is possible to define
the extended coordinates $x_{\mu\nu}$ with corresponding momenta $p_{\mu\nu}$
and to interpret Lorentz generators $M_{\mu\nu}$ as extended coordinates
$x_{\mu\nu}$ and $\tilde{h}$ as additional scalar coordinate. Then algebra
(2.5) – (2.14) is generated with 15 coordinates. The corresponding
realizations can be obtained from quantum realizations of the generalized Yang
models, presented in [6, 9].

### Acknowledgement

S.Mignemi thanks Gruppo Nazionale di Fisica Matematica for support.

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