# The square integrable representations on generalized Weyl- Heisenberg groups

Fatemeh Esmaeelzadeh 1Department of Mathematics, Bojnourd Branch, Islamic Azad
University, Bojnourd, Iran.
<EMAIL_ADDRESS>
###### Abstract.

This paper presents the square integrable representations of generalized Weyl-
Heisenberg group. We investigate the quasi regular representation of
generalized Weyl-Heisenberg group. Moreover, we obtain a concrete form for
admissible vector of this representation . Finally, we provide some examples
to support our technical considerations.

###### Key words and phrases:

generalized Weyl-Heisenberg group, square integrable, admissible wavelet.

###### 2010 Mathematics Subject Classification:

Primary 43A15, Secondary

##  1\. Introduction

Wavelet transform has rich theoretical structures and is extremely useful as
tools for building signal transforms, adapted to various signal geometries,
quantum mechanics, etc. Continuous wavelet transform admits a generalization
to locally compact groups. Such a unified approach seems to be useful, since
it emphasizes on a clear way to basic features of continuous wavelet transform
and includes all important cases for applications [2, 3, 5]. It should be
mentioned that, the Weyl Heisenberg group plays a significant designations in
various aspects of the connections between the classical harmonic analysis and
concrete applications of numerical harmonic analysis.

This paper contains $4$ sections. Section 2 includes the definition of semi-
direct product of two locally compact groups and generalized Weyl Heisenberg
group. In Section 3, we study the square integrable representations on the
generalized Weyl Heisenberg group and then we obtain the necessary and
sufficient conditions for admissible wavelet on this group. In Section 4, some
examples are proved as application of our results.

##  2\. Preliminaries and notation

Let $H$ and $K$ be two locally compact groups with the identity elements
$e_{H}$ and $e_{K}$, respectively and let $\tau:H\rightarrow Aut(K)$ be a
homomorphism such that the map $(h,k)\mapsto\tau_{h}(k)$ is continuous from
$H\times K$ onto $K$, where $H\times K$ equips with the product topology. The
semi- direct product topological group $G_{\tau}=H\times_{\tau}K$ is the
locally compact topological space $H\times K$ under the product topology, with
the group operations:

$(h_{1},k_{1})\times_{\tau}(h_{2},k_{2})=(h_{1}h_{2},k_{1}\tau_{h_{1}}(k_{2}),$
$(h,k)^{-1}=(h^{-1},\tau_{h^{-1}}(k^{-1})).$

It is worth to note that $K_{1}=\\{(e_{H},k);k\in K\\}$ is a closed normal
subgroup and $H_{1}=\\{(h,e_{K});h\in H\\}$ is a closed subgroup of $G_{\tau}$
such that $G_{\tau}=HK$ . Moreover, the left Haar measure of the locally
compact group $G_{\tau}$ is

$d\mu_{G_{\tau}}(h,k)=\delta_{H}(h)d\mu_{H}(h)d\mu_{K}(k),$

in which $d\mu_{H},d\mu_{K}$ are the left Haar measures on $H$ and $K$,
respectively and $\delta_{H}:H\rightarrow(0,\infty)$ is a positive continuous
homomorphism that satisfies

$d\mu_{K}(k)=\delta_{H}(h)d\mu(\tau_{h}(k)),$

for $h\in H,k\in K$. Moreover, the modular function $\Delta_{G_{\tau}}$ is

$\Delta_{G_{\tau}}=\delta_{H}(h)\Delta_{H}(h)\Delta_{K}(k),$

where $\Delta_{H},\Delta_{K}$ are the modular functions of $H,K$,
respectively.
When $K$ is also abelian, one can define $\hat{\tau}:H\rightarrow
Aut(\hat{K})$ via $h\mapsto\hat{\tau_{h}}$ where

$\hat{\tau_{h}}(\omega)=\omega\circ\tau_{h^{-1}},$

for all $\omega\in{\hat{K}}$. We usually denote $\omega\circ\tau_{h^{-1}}$ by
$\omega_{h}$. With this notation, it is easy to see

$\omega_{h_{1}h_{2}}=(\omega_{h_{2}})_{h_{1}},$

where $h_{1},h_{2}\in H$ and $\omega\in{\hat{K}}$. The semi-direct product
$G_{\hat{\tau}}=H\times_{\hat{\tau}}\hat{K}$ is a locally compact group with
the left Haar measure

$d\mu_{\hat{G}}(h,\omega)=\delta_{H}(h)^{-1}d\mu_{H}(h)d\mu_{\hat{K}}(\omega),$

where $d\mu_{\hat{K}}$ is the Haar measure on $\hat{K}$. Also, for all $h\in
H$,

$d\mu_{\hat{K}}(\omega_{h})=\delta_{H}(h)d\mu_{\hat{K}}(\omega),$

for $\omega\in{\hat{K}},$ (see more details in [4, 1, 3].)
Let $G_{\tau}=H\times_{\tau}K$, and define $\theta:G_{\tau}\rightarrow
Aut(\hat{K}\times\mathbb{T})$ via

$(h,k)\mapsto\theta_{(h,k)}(\omega,z)=(\hat{\tau_{h}}(\omega),\hat{\tau_{h}}(\omega)(k)z)=(\omega_{h},\omega_{h}(k)z),$

for all $(h,k)\in{H\times_{\tau}K}$ and
$(\omega,z)\in{\hat{K}\times\mathbb{T}}.$ The mapping $\theta$ is a continuous
homomorphism. Thus the semi-direct prodoct

$G_{\tau}\times_{\theta}(\hat{K}\times\mathbb{T})=(H\times_{\tau}K)\times_{\theta}(\hat{K}\times\mathbb{T}),$

is a locally compact group and it is called the generalized Weyl Heisenberg
group associated with the semi direct product group
$G_{\tau}=H\times_{\tau}K$, and denoted by $\mathbb{H}(G_{\tau})$. It is easy
to see that the group operations of $\mathbb{H}(G_{\tau})$ are

$(h_{1},k_{1},\omega_{1},z_{1}).(h_{2},k_{2},\omega_{2},z_{2})=(h_{1}h_{2},k_{1}\tau_{h_{1}}(k_{2}),\omega_{1}{\omega_{2}}_{h_{1}},{\omega_{2}}_{h_{1}}(k)z_{1}z_{2}),$
$(h_{1},k_{1},\omega_{1},z_{1})^{-1}=(h_{1}^{-1},\tau_{h_{1}}^{-1}(k^{-1}),\bar{\omega}_{h_{1}^{-1}},\bar{\omega}_{h_{1}^{-1}}(\tau_{h_{1}}^{-1}(k^{-1}))z^{-1}),$

for
$(h_{1},k_{1},\omega_{1},z_{1}),(h_{2},k_{2},\omega_{2},z_{2})\in{\mathbb{H}(G_{\tau})}$
(see [4]) and the left Haar measure of $\mathbb{H}(G_{\tau})$ is:

$d\mu_{\mathbb{H}(G_{\tau})}(h,k,\omega,z)=d\mu_{H}(h)d\mu_{K}(k)d\mu_{\hat{K}}(\omega)d\mu_{\mathbb{T}}(z).$

##  3\. The square integrable representation of $\mathbb{H}(G_{\tau})$

Throughout this section, we assume that $H$ and $K$ are locally compact
topological groups and that $K$ is abelian, too. We denote the left Haar
measures of $H$ and $K$ by $d\mu_{H},d\mu_{K}$, respectively. Suppose that
$h\mapsto\tau_{h}$ from $H$ to $Aut(K)$ is a homomorphism such that
$(h,k)\mapsto\tau_{h}(k)$ from $H\times K$ into $K$ is continuous.
$G_{\tau}=H\times_{\tau}K$ is the semi-direct product of $H$ and $K$ that is a
locally compact topology group with the left Haar measure
$d\mu_{G_{\tau}}(h,k)=\delta_{H}(h)d\mu_{H}((h)d\mu_{K}(k)$, where
$\delta_{H}:H\mapsto(0,\infty)$ is a continuous homomorphism. Consider the
homomorphism $\theta:G_{\tau}\rightarrow Aut(\hat{K}\times\mathbb{T})$ is
defined by

$((h,k),(\omega,z))\mapsto\theta_{(h,k)}(\omega,z),$

where
$\theta_{(h,k)}(\omega,z)=(\omega\circ\tau_{h^{-1}},\omega\circ\tau_{h^{-1}}(k).z).$
This makes
$\mathbb{H}(G_{\tau})=G_{\tau}\times_{\theta}(\hat{K}\times\mathbb{T})$ a
locally compact topological group where $\mathbb{H}(G_{\tau})$ is equipped
with the product topology and the group operations as

$(h_{1},k_{1},\omega_{1},z_{1}).(h_{2},k_{2},\omega_{2},z_{2})=(h_{1}h_{2},k_{1}\tau_{h_{1}}(k_{2}),\omega_{1}{\omega_{2}}_{h_{1}},{\omega_{2}}_{h_{1}}(k)z_{1}z_{2}),$
$(h_{1},k_{1},\omega_{1},z_{1})^{-1}=(h_{1}^{-1},\tau_{h_{1}}^{-1}(k^{-1}),\bar{\omega}_{h_{1}^{-1}},\bar{\omega}_{h_{1}^{-1}}(\tau_{h_{1}}^{-1}(k^{-1}))z^{-1}),$

for
$(h_{1},k_{1},\omega_{1},z_{1}),(h_{2},k_{2},\omega_{2},z_{2})\in{\mathbb{H}(G_{\tau})}.$
The left Haar measure of $\mathbb{H}(G_{\tau})$ is

$d\mu_{\mathbb{H}(G_{\tau})}(h,k,\omega,z)=d\mu_{H}(h)d\mu_{K}(k)d\mu_{\hat{K}}(\omega)d\mu_{\mathbb{T}}(z).$

Now, we are going to define a square integrable representation on
$\mathbb{H}(G_{\tau})$. With the above notations define
$\pi:\mathbb{H}(G_{\tau})\rightarrow U(L^{2}(\hat{K}))$ by

(3.1)
$\pi(h,k,\omega,z)f(\xi)=\delta_{H}^{-1/2}(h)z\xi(k)\overline{\omega(k)}f((\xi\overline{\omega})_{h^{-1}}),$

then $\pi$ is a homomorphism. Indeed,
$\begin{array}[]{lll}\pi\left((h_{1},k_{1},\omega_{1},z_{1})(h_{2},k_{2},\omega_{2},z_{2})\left)f(\xi)=\pi(h_{1}h_{2},k_{1}\tau_{h_{1}}(k_{2}),\omega_{1}\right(\omega_{2}\right)_{h_{1}},(\omega_{2})_{h_{1}}(k_{1})z_{1}z_{2})f(\xi)\\\\[4.30554pt]
=\delta_{H}^{-1/2}(h_{1}h_{2})(\omega_{2})_{h_{1}}(k_{1})z_{1}z_{2}\xi(k_{1}\tau_{h_{1}}(k_{2}))\overline{\omega_{1}(\omega_{2})_{h_{1}}}(k_{1}\tau_{h_{1}}(k_{2}))f((\xi\overline{\omega_{1}(\omega_{2})_{h_{1}}})_{(h_{1}h_{2})^{-1}}\\\\[4.30554pt]
=\delta_{H}^{-1/2}(h_{1}h_{2})(\omega_{2})_{h_{1}}(k_{1})z_{1}z_{2}\xi(k_{1})\xi_{h_{1}^{-1}}(k_{2})\overline{\omega_{1}(k_{1})}\overline{(\omega_{1})_{h_{1}^{-1}}(k_{2})}\overline{\omega_{2}(k_{2})}f(\xi_{h_{2}^{-1}h_{1}^{-1}}\overline{(\omega_{1})}_{h_{2}^{-1}h_{1}^{-1}}\overline{(\omega_{2})}_{h_{2}^{-1}})\end{array}.$
Also,
$\begin{array}[]{lll}\pi(h_{1},k_{1},\omega_{1},z_{1})\pi(h_{2},k_{2},\omega_{2},z_{2})f(\xi)\\\\[4.30554pt]
=\delta_{H}^{-1/2}(h_{1})z_{1}\xi(k_{1})\overline{\omega_{1}}(k_{1})\pi(h_{2},k_{2},\omega_{2},z_{2})f((\xi\overline{\omega_{1}})_{h_{1}^{-1}}\\\\[4.30554pt]
=\delta_{H}^{-1/2}(h_{1})\delta_{H}^{-1/2}(h_{2})z_{1}z_{2}\xi(k_{1})\overline{\omega_{1}}(k_{1})\overline{\omega_{2}}(k_{2})(\xi\overline{\omega_{1}})_{h_{1}^{-1}}(k_{2})f((\xi\overline{\omega_{1}})_{h_{1}^{-1}}(\overline{\omega_{2}})_{h_{2}^{-1}})\\\\[4.30554pt]
=\delta_{H}^{-1/2}(h_{1}h_{2})z_{1}z_{2}\xi(k_{1})\xi_{h_{1}^{-1}}(k_{2})\overline{\omega_{1}(k_{1})}\overline{(\omega_{1})_{h_{1}^{-1}}(k_{2})}\overline{\omega_{2}(k_{2})}f(\xi_{h_{2}^{-1}h_{1}^{-1}}\overline{(\omega_{1})}_{h_{2}^{-1}h_{1}^{-1}}\overline{(\omega_{2})}_{h_{2}^{-1}}).\end{array}$.

Moreover, $\pi$ is unitary. In fact we have,
$\begin{array}[]{rcl}\|\pi(h,k,\omega,z)f\|^{2}_{2}&=&\int_{\hat{K}}|\pi(h,k,\omega,z)f(\xi)|^{2}d\mu_{\hat{K}}(\xi)\\\\[4.30554pt]
&=&\int_{\hat{K}}\delta_{H}^{-1}(h)|f((\xi\overline{\omega})_{h^{-1}}|^{2}d\mu_{\hat{K}}(\xi)\\\\[4.30554pt]
&=&\int_{\hat{K}}\delta_{H}^{-1}(h)|f((\xi)_{h^{-1}}|^{2}d\mu_{\hat{K}}(\xi)\\\\[4.30554pt]
&=&\int_{\hat{K}}f((\xi)|^{2}d\mu_{\hat{K}}(\xi)\\\\[4.30554pt]
&=&\|f\|_{2}^{2}.\end{array}$.
And it is easy to check that $\pi$ is continuous and onto. So, $\pi$ is a
continuous unitary representation of group $\mathbb{H}(G_{\tau})$ to the
Hilbert space $L^{2}(\hat{K})$. In the sequel, we show that $\pi$ is
irreducible when $H$ is compact. Furthermore, it is also shown that $\pi$ is
square integrable if and only if $H$ is compact. Note that when $H$ is a
compact group, we normalize the Haar measure $\mu_{H}$ such that
$\mu_{H}(H)=1$.

######  Theorem 3.1.

Let
$\mathbb{H}(G_{\tau})=(H\times_{\tau}K)\times_{\theta}(\hat{K}\times\mathbb{T})$
where $H$ is a locally compact group and $K$ is a locally compact abelian
group. Then for $\varphi,\psi$ in $L^{2}(\hat{K})$,

(3.2)
$\int_{\mathbb{H}(G_{\tau})}|\prec\varphi,\pi(h,k,\omega,z)\psi\succ|^{2}d\mu_{\mathbb{H}(G_{\tau})}(h,k,\omega,z)=\|\varphi\|_{2}^{2}\|\psi\|_{2}^{2}.$

if and only if $H$ is compact.

###### Proof.

For $\varphi,\psi$ in $L^{2}(\hat{K})$ we first consider the following
observations:
$\begin{array}[]{lll}\int_{\mathbb{H}(G_{\tau})}|\prec\varphi,\pi(h,k,\omega,z)\psi\succ|^{2}d\mu_{\mathbb{H}(G_{\tau})}(h,k,\omega,z)\\\\[4.30554pt]
=\int_{\mathbb{H}(G_{\tau})}|\int_{\hat{K}}\varphi(\xi)\overline{\pi(h,k,\omega,z)\psi(\xi)}d\mu_{\hat{K}}(\xi)|^{2}d\mu_{\mathbb{H}(G_{\tau})}(h,k,\omega,z)\\\\[4.30554pt]
=\int_{\mathbb{H}(G_{\tau})}|\int_{\hat{K}}\varphi(\xi)\delta_{H}^{-1/2}(h)\overline{z}\overline{\xi(k)}\omega(k)\overline{\psi(\xi\overline{\omega})_{h^{-1}}}d\mu_{\hat{K}}(\xi)|^{2}d\mu_{\mathbb{H}(G_{\tau})}(h,k,\omega,z)\\\\[4.30554pt]
=\int_{\mathbb{H}(G_{\tau})}|\int_{\hat{K}}\varphi(\xi\omega)\delta_{H}^{-1/2}(h)\overline{z}\overline{\xi(k)}\overline{\psi(\xi)_{h^{-1}}}d\mu_{\hat{K}}(\xi)|^{2}d\mu_{\mathbb{H}(G_{\tau})}(h,k,\omega,z)\\\\[4.30554pt]
=\int_{\mathbb{H}(G_{\tau})}|\int_{\hat{K}}R_{\omega}\varphi(\xi)\delta_{H}^{-1/2}(h)\overline{z}\overline{\xi(k)}\overline{\psi(\xi\circ\tau_{h})}d\mu_{\hat{K}}(\xi)|^{2}d\mu_{\mathbb{H}(G_{\tau})}(h,k,\omega,z)\\\\[4.30554pt]
=\int_{\mathbb{H}(G_{\tau})}|\int_{\hat{K}}R_{\omega}\varphi(\xi\circ\tau_{h^{-1}})\delta_{H}^{-1/2}(h)\overline{z}\overline{\xi\circ\tau_{h^{-1}}(k)}\overline{\psi(\xi)}d\mu_{\hat{K}}(\xi_{h})|^{2}d\mu_{\mathbb{H}(G_{\tau})}(h,k,\omega,z)\\\\[4.30554pt]
=\int_{\mathbb{H}(G_{\tau})}|\int_{\hat{K}}R_{\omega}\varphi(\xi\circ\tau_{h^{-1}})\delta_{H}^{1/2}(h)\overline{z}\overline{\xi(\tau_{h^{-1}}(k))}\overline{\psi(\xi)}d\mu_{\hat{K}}(\xi)|^{2}d\mu_{\mathbb{H}(G_{\tau})}(h,k,\omega,z)\\\\[4.30554pt]
=\int_{\mathbb{H}(G_{\tau})}\delta_{H}(h)|\int_{\hat{K}}(R_{\omega}\varphi(.\circ\tau_{h^{-1}}).\overline{\psi})(\xi)\overline{\xi(\tau_{h^{-1}}(k))}d\mu_{\hat{K}}(\xi)|^{2}d\mu_{\mathbb{H}(G_{\tau})}(h,k,\omega,z)\\\\[4.30554pt]
=\int_{\mathbb{H}(G_{\tau})}\delta_{H}(h)\widehat{|(R_{\omega}\varphi(.\circ\tau_{h^{-1}}).\overline{\psi})}(\tau_{h^{-1}}(k))|^{2}d\mu_{\mathbb{H}(G_{\tau})}(h,k,\omega,z)\\\\[4.30554pt]
=\int_{H}\delta_{H}(h)\int_{\hat{K}}\int_{K}|\widehat{(R_{\omega}\varphi(.\circ\tau_{h^{-1}}).\overline{\psi})}(\tau_{h^{-1}}(k))|^{2}d\mu_{K}(k)d\mu_{\hat{K}}(\omega)d\mu_{H}(h)\\\\[4.30554pt]
=\int_{H}\int_{\hat{K}}\int_{K}|\widehat{(R_{\omega}\varphi(.\circ\tau_{h^{-1}}).\overline{\psi})}(k)|^{2}d\mu_{K}(k)d\mu_{\hat{K}}(\omega)d\mu_{H}(h)\\\\[4.30554pt]
=\int_{H}\int_{\hat{K}}\int_{\hat{K}}|(R_{\omega}\varphi(.\circ\tau_{h^{-1}}).\overline{\psi})(\xi)|^{2}d\mu_{\hat{K}}(\xi)d\mu_{\hat{K}}(\omega)d\mu_{H}(h)\\\\[4.30554pt]
=\int_{H}\int_{\hat{K}}\int_{\hat{K}}|R_{\omega}\varphi(\xi\circ\tau_{h^{-1}}).\overline{\psi}(\xi)|^{2}d\mu_{\hat{K}}(\xi)d\mu_{\hat{K}}(\omega)d\mu_{H}(h)\\\\[4.30554pt]
=\int_{H}\int_{\hat{K}}\int_{\hat{K}}\delta_{H}(h)|R_{\omega}\varphi(\xi).\overline{\psi}(\xi\circ\tau_{h})|^{2}d\mu_{\hat{K}}(\xi)d\mu_{\hat{K}}(\omega)d\mu_{H}(h)\\\\[4.30554pt]
=\int_{H}\int_{\hat{K}}\|\varphi\|_{2}^{2}\delta_{H}(h)|\overline{\psi}(\xi\circ\tau_{h})|^{2}d\mu_{\hat{K}}(\xi)d\mu_{H}(h)\\\\[4.30554pt]
=\|\varphi\|_{2}^{2}\|\psi\|_{2}^{2}\mu_{H}(H)\end{array}$.
Now, if $H$ is compact, then $\mu_{H}(H)=1$. So, (3.2) holds. Conversely, if
(3.2) holds,the above observation implies that $\mu_{H}(H)=1$ . So, we can
conclude that $H$ is compact. ∎

######  Corollary 3.2.

With notation as above, the representation $\pi$ of $\mathbb{H}(G_{\tau})$ on
$L^{2}(\hat{K})$ is irreducible if $H$ is compact.

###### Proof.

If $H$ is compact, then (3.2) in Theorem 3.1 holds. Now, suppose that $M$ is a
closed subspace of the Hilbert space $L^{2}(\hat{K})$ that is invariant under
$\pi$. Then for any $\varphi\in M$ we have,

$\\{\pi(h,k,\omega,z)\varphi;\ \
(h,k,\omega,z)\in{\mathbb{H}(G_{\tau})}\\}\subseteq M.$

Let $\psi\in{L^{2}(\hat{K})}$ be orthogonal to $M$, that is
$\prec\psi,\pi(h,k,\omega,z)\varphi\succ=0,$ for all
$(h,k,\omega,z)\in{\mathbb{H}(G_{\tau})}$. Thus by (3.2),
$\|\varphi\|_{2}\|\psi\|_{2}=0$, and hence $\psi=0$. So, $M^{\perp}=\\{0\\}$,
that is, $M=L^{2}(\hat{K})$. Namely, $\pi$ is irreducible. ∎

We remind the reader that, an irreducible representation $\pi$ of
$\mathbb{H}(G_{\tau})$ on $L^{2}(\hat{K})$ is called square integrable if
there exists a non zero element $\psi$ in $L^{2}(\hat{K})$ such that

(3.3) $\prec\pi(.,.,.,.)\psi,f\succ\in{L^{2}({\mathbb{H}(G_{\tau})})},$

for all $f\in{L^{2}(\hat{K})}$. A unit vector $\psi$ satisfying (3.3) is said
to be an admissible wavelet for $\pi$, and the constant

$c_{\psi}=\int_{\mathbb{H}(G_{\tau})}|\prec\pi(h,k,\omega,z)\psi,\psi\succ|^{2}d\mu_{\mathbb{H}(G_{\tau})},$

is called the wavelet constant associated to the admissible wavelet $\psi$.
Also, for the wavelet vector $\psi$, the continuous wavelet transform is
defined by

$W_{\psi}f(h,k,\omega,z)=\prec f,\pi(h,k,\omega,z)\psi\succ.$

It is easy to see that $(h,k,\omega,z)\mapsto W_{\psi}f(h,k,\omega,z)$ is a
continuous function on $\mathbb{H}(G_{\tau}).$ Moreover, $W_{\psi}$
intertwines $\pi$ and the left regular representation on
$\mathbb{H}(G_{\tau}).$

######  Corollary 3.3.

The representation $\pi$ of the $GWH$ group
$\mathbb{H}(G_{\tau})=(H\times_{\tau}K)\times_{\theta}(\hat{K}\times\mathbb{T})$
on $L^{2}(\hat{K})$ is square integrable if and only if $H$ is compact.

###### Proof.

If $H$ is compact, then by Theorem 3.1 and Corollary 3.2, $\pi$ is square
integrable. For the inverse, if $\pi$ is square integrable, then there exists
a non zero element $\varphi\in L^{2}(\hat{K})$ such that

$\prec\pi(.,.,.,.)\varphi,\psi\succ\in{L^{2}({\mathbb{H}(G_{\tau})})},$

for all $\psi\in{L^{2}(\hat{K})}$. On the other hand,

$\int_{\mathbb{H}(G_{\tau})}|\prec\varphi,\pi(h,k,\omega,z)\psi\succ|^{2}d\mu_{\mathbb{H}(G_{\tau})}(h,k,\omega,z)=\|\varphi\|_{2}^{2}\|\psi\|_{2}^{2}\mu_{H}(H).$

So $\mu_{H}(H)<\infty$. That is $H$ is compact. ∎

######  Remark 3.4.

There is another irreducible representation of $\mathbb{H}(G_{\tau})$ on
Hilbert space $L^{2}(K)$. Indeed, consider

$\tilde{\pi}:\mathbb{H}(G_{\tau})\rightarrow U(L^{2}(K)),\ \ \ \
\tilde{\pi}(h,k,\omega,z)f(k^{\prime})=\delta_{H}(h)^{1/2}z\omega(k^{\prime})f(\tau_{h^{-1}}(k^{\prime}k)),$

for all $(h,k,\omega,z)\in{\mathbb{H}(G_{\tau})},f\in{L^{2}(K)}$.
$\tilde{\pi}$ is homomorphism and unitary. In fact we have
$\begin{array}[]{lll}\tilde{\pi}((h_{1},k_{1},\omega_{1},z_{1})(h_{2},k_{2},\omega_{2},z_{2}))f(k^{\prime})\\\\[4.30554pt]
=\tilde{\pi}(h_{1}h_{2},k_{1}\tau_{h_{1}}(k_{2}),\omega_{1}(\omega_{2})_{h_{1}},(\omega_{2})_{h_{1}}(k_{1})z_{1}z_{2})f(k^{\prime})\\\\[4.30554pt]
=\delta_{H}^{1/2}(h_{1}h_{2})(\omega_{2})_{h_{1}}(k_{1})z_{1}z_{2}\omega_{1}(\omega_{2})_{h_{1}}(k^{\prime})f(\tau_{(h_{1}h_{2})^{-1}}(k^{\prime}(k_{1}\tau_{h}{}_{1}(k_{2}))\\\\[4.30554pt]
=\delta_{H}^{1/2}(h_{1}h_{2})z_{1}z_{2}\omega_{1}(k^{\prime})(\omega_{2})_{h_{1}}(k^{\prime}k_{1})f(\tau_{h_{2}^{-1}h_{1}^{-1}}(k^{\prime}k_{1}\tau_{h_{1}}(k_{2}))\\\\[4.30554pt]
=\delta_{H}^{1/2}(h_{1}h_{2})z_{1}z_{2}\omega_{1}(k^{\prime})\omega_{2})_{h_{1}}(k^{\prime}k_{1})f(\tau_{h_{2}^{-1}h_{1}^{-1}}(k^{\prime}k_{1})\tau_{h_{2}^{-1}}(k_{2})),\end{array}$
and
$\begin{array}[]{lll}\tilde{\pi}(h_{1},k_{1},\omega_{1},z_{1})\tilde{\pi}(h_{2},k_{2},\omega_{2},z_{2})f(k^{\prime})\\\\[4.30554pt]
=\delta_{H}^{1/2}(h_{1})z_{1}\omega_{1}(k^{\prime})\tilde{\pi}(h_{2},k_{2},\omega_{2},z_{2})f(\tau_{h_{1}^{-1}}(k^{\prime}k_{1}))\\\\[4.30554pt]
=\delta_{H}^{1/2}(h_{1})\delta_{H}^{1/2}(h_{2})z_{1}z_{2}\omega_{1}(k^{\prime})\omega_{2}(\tau_{h_{1}^{-1}}(k^{\prime}k_{1}))f(\tau_{h_{2}^{-1}}(\tau_{h_{1}^{-1}}(k^{\prime}k_{1})k_{2})\\\\[4.30554pt]
=\delta_{H}^{1/2}(h_{1}h_{2})z_{1}z_{2}\omega_{1}(k^{\prime})(\omega_{2})_{h_{1}}(k^{\prime}k_{1})f(\tau_{h_{2}^{-1}h_{1}^{-1}}(k^{\prime}k_{1})\tau_{h_{2}^{-1}}(k_{2})).\end{array}$.
Also,
$\begin{array}[]{rcl}\|\tilde{\pi}(h,k,\omega,z)f\|^{2}_{2}&=&\int_{K}|\tilde{\pi}(h,k,\omega,z)f(k^{\prime})|^{2}d\mu_{K}(k^{\prime})\\\\[4.30554pt]
&=&\int_{K}\delta_{H}(h)|f(\tau_{h^{-1}}(k^{\prime}k))|^{2}d\mu_{K}(k^{\prime})\\\\[4.30554pt]
&=&\int_{K}\delta_{H}(h)|f(k^{\prime})|^{2}d\mu_{K}(\tau_{h}(k^{\prime}))\\\\[4.30554pt]
&=&\int_{K}|f(k^{\prime})|^{2}d\mu_{K}(k^{\prime})\\\\[4.30554pt]
&=&\|f\|_{2}^{2}.\end{array}$.
Using the Plancherel theorem, $\pi,\tilde{\pi}$ are unitarily equivalent. So,
$\tilde{\pi}$ is square integrable if and only if $\pi$ is square integrable.

######  Remark 3.5.

The inverse of Corollary 3.2 does not hold, generally. An obvious example is
when $H$ is a non compact group and $K$ is the trivial group $\\{e\\}$. Then
the representation $\pi:\mathbb{H}(H\times_{\tau}\\{e\\})\rightarrow
U(\mathbb{C})$ is an irreducible representation. Here we give a non trivial
example in which $\pi$ is an irreducible representation, but $H$ is not
compact. Let $H=\mathbb{R}^{+},K=\mathbb{R}$. Define the representation $\pi$
of $\mathbb{H}(\mathbb{R}^{+}\times_{\tau}\mathbb{R})$ as follows:

$\pi:\mathbb{H}(\mathbb{R}^{+}\times_{\tau}\mathbb{R})\rightarrow
U(L^{2}(\mathbb{R}));\ \ \ \ \pi(a,x,\omega,z)f(\xi)=a^{1/2}ze^{2\pi
ix(\xi-\omega)}f((\xi\bar{\omega)}_{a^{-1}}),$

in which $(\xi\bar{\omega)}_{a^{-1}}=(\xi\bar{\omega)}\circ\tau_{a}$,
$\tau_{a}(x)=a.x$ and $\delta_{H}(a)=a^{-1}$. This representation is
irreducible. Indeed, let $M$ be a closed invariant subspace of
$L^{2}(\mathbb{R})$ under $\pi$. Then for any $f\in{M}$, we have
$\pi(h,k,\omega,z)f\in M$. Consider $0\neq g\in{M^{\perp}}$, so that $\prec
g,\pi(h,k,\omega,z)f\succ=0$. Then

$0=\int_{\mathbb{R}}g(\xi)e^{-2\pi
ix\xi}\bar{f}((\xi\bar{\omega})_{a^{-1}})d\xi=\\\
\int_{\mathbb{R}}g(\xi_{a}\omega)e^{-2\pi ix\xi_{a}\omega}\bar{f}(\xi)d\xi.$

Thus, $g(\xi_{a}\omega)\bar{f}(\xi)=0$, for almost all $\xi\in\mathbb{R}$.
Suppose that $\bar{f}(\xi)\neq 0$, for all $\xi$ in a set $A$ with positive
measure. Then for all $\xi\in A,\ \ g(\xi_{a}\omega)=0$, for all
$\omega\in{\mathbb{R}},a\in{\mathbb{R}^{+}}$. Thus $g=0$. This is a
contradiction. So, $\pi$ is an irreducible representation, but $H$ is not
compact.

In the sequel, we define the quasi regular representation and we obtain a
concrete form for an admissible vector. Note that $\mathbb{H}(G_{\tau})$ acts
on the Hilbert space $L^{2}(\hat{K}\times\mathbb{T})$ and this action induces
the quasi regular representation $\\{\rho,L^{2}(\hat{K}\times\mathbb{T})\\}$
as follows:

(3.4)
$\rho:(H\times_{\tau}K)\times_{\theta}(\hat{K}\times\mathbb{T})\rightarrow
U(L^{2}(\hat{K}\times\mathbb{T})),$

where

$\displaystyle\rho(h,k,\omega,z)f(\xi,t)$ $\displaystyle=$
$\displaystyle\delta_{H\times_{\tau}K}^{1/2}(h,k)f(\theta_{(h,k)^{-1}}(\xi,t)(\omega,z)^{-1})$
$\displaystyle=$
$\displaystyle\delta_{H}^{-1/2}(h)f(\theta_{(h^{-1},\tau_{h^{-1}}({k^{-1}})}(\xi\bar{\omega},tz^{-1}))$
$\displaystyle=$
$\displaystyle\delta_{H}^{-1/2}(h)f((\xi\bar{\omega})_{h^{-1}},(\xi\bar{\omega})_{h^{-1}}(\tau_{h^{-1}}(k^{-1})).tz^{-1}).$

Note that $\delta_{H\times_{\tau}K}(h,k)=\delta_{H}(h)^{-1}.$ (see Corollary
3.3 in [4])
A type of the Fourier transform of the quasi regular representation $\rho$
obtains as follows:
$\begin{array}[]{lll}\widehat{\rho(h,k,\omega,z)f}(k^{\prime},n^{\prime})\\\\[4.30554pt]
=\int_{\hat{K}\times\mathbb{T}}\rho(h,k,\omega,z)f(\xi,t)\overline{(k^{\prime},n^{\prime})(\xi,t)}d\mu_{\hat{K}}(\xi)d\mu_{\mathbb{T}}(t)\\\\[4.30554pt]
=\delta_{H}(h)^{-1/2}\int_{\hat{K}\times\mathbb{T}}f((\xi\overline{\omega})_{h^{-1}},(\xi\overline{\omega})_{h^{-1}}(\tau_{h^{-1}}(k^{-1}))tz^{-1})\overline{\xi(k^{\prime})}\overline{t^{n^{\prime}}}d\mu_{\hat{K}}(\xi)d\mu_{\mathbb{T}}(t)\\\\[4.30554pt]
=\delta(h)^{-1/2}\overline{z^{n^{\prime}}}\int_{\hat{K}\times\mathbb{T}}f(\xi)_{h^{-1}},\xi_{h^{-1}}(\tau_{h^{-1}}(k^{-1}))t)\overline{\omega}(k^{\prime})\bar{\xi(k^{\prime})}\overline{t^{n^{\prime}}}d\mu_{\hat{K}}(\xi)d\mu_{\mathbb{T}}(t)\\\\[4.30554pt]
=\delta_{H}(h)^{-1/2}\bar{z^{n^{\prime}}}\bar{\omega}(k^{\prime})\int_{\hat{K}\times\mathbb{T}}f(\theta_{(h^{-1},\tau_{h^{-1}})}(\xi,t)\bar{\xi(k^{\prime})}\bar{t^{n^{\prime}}}d\mu_{\hat{K}}(\xi)d\mu_{\mathbb{T}}(t)\\\\[4.30554pt]
=\delta_{H}(h)^{-1/2}\bar{z^{n^{\prime}}}\bar{\omega}(k^{\prime})\int_{\hat{K}\times\mathbb{T}}f\circ\theta_{(h,k)^{-1}}(\xi,t)\overline{(k^{\prime},n^{\prime})(\xi,t)}d\mu_{\hat{K}}(\xi)d\mu_{\mathbb{T}}(t)\\\\[4.30554pt]
=\delta_{H}(h)^{-1/2}\bar{z^{n^{\prime}}}\bar{\omega}(k^{\prime})\widehat{(f\circ\theta_{(h,k)^{-1}})}(k^{\prime},n^{\prime}),\end{array}$

for all
$(k^{\prime},n^{\prime})\in{K\times\mathbb{Z}}=\widehat{(\hat{K}\times\mathbb{T})}.$
So,

(3.5)
$\widehat{\rho(h,k,\omega,z)f}(k^{\prime},n^{\prime})=\delta_{H}(h)^{-1/2}\bar{z^{n^{\prime}}}\bar{\omega}(k^{\prime})\widehat{(f\circ\theta_{(h,k)^{-1}})}(k^{\prime},n^{\prime}).$

######  Theorem 3.6.

With the notation as above, let $\rho$ be the quasi regular representation on
$\mathbb{H}(G_{\tau}),$ and $\psi,f\in{L^{2}(\hat{K}\times\mathbb{T})}$.

If $\psi$ is a wavelet vector, then

$W_{\psi}f(h,k,\omega,z)=\delta_{H}^{-1/2}(h)\int_{K}\sum_{n^{\prime}\in{\mathbb{Z}}}\hat{f}(k^{\prime},n^{\prime})z^{n^{\prime}}\omega(k^{\prime})\overline{\widehat{{(\psi\circ\theta)}}}_{(h,k)^{-1}}(k^{\prime},n^{\prime})d\mu_{K}(k^{\prime}).$

The vector $\psi$ is wavelet if

$\int_{H\times_{\tau}K}|\hat{\psi}(k^{\prime},n^{\prime})\circ\theta_{(h,k)^{-1}}|^{2}d\mu_{H\times
K}(h,k)<\infty.$

###### Proof.

For $(k^{\prime},n^{\prime})\in{K\times\mathbb{Z}},$

By the Plancherel’s theorem and (3.5), we have

$\displaystyle W_{\psi}f(h,k,\omega,z)$ $\displaystyle=$ $\displaystyle\prec
f,\rho(h,k,\omega,z)\psi\succ$ $\displaystyle=$
$\displaystyle\prec\hat{f},\widehat{\rho(h,k,\omega,z)\psi}\succ$
$\displaystyle=$
$\displaystyle\delta_{H}^{-1/2}(h)\int_{K}\sum_{n^{\prime}\in{\mathbb{Z}}}\hat{f}(k^{\prime},n^{\prime})z^{n^{\prime}}\omega(k^{\prime})\overline{\widehat{{(\psi\circ\theta)}}}_{(h,k)^{-1}}(k^{\prime},n^{\prime})d\mu_{K}(k^{\prime}).$

By applying the part (i), for $f\in{L^{2}(\hat{K}\times\mathbb{T})}$, we get
$\begin{array}[]{lll}\int_{\hat{K}\times\mathbb{T}}|W_{\psi}f(h,k,\omega,z)|^{2}d\mu_{\hat{K}\times\mathbb{T}}(\omega,z)\\\\[4.30554pt]
=\int_{\hat{K}\times\mathbb{T}}W_{\psi}f(h,k,\omega,z)\overline{W_{\psi}f(h,k,\omega,z)}d\mu_{\hat{K}\times\mathbb{T}}(\omega,z)\\\\[4.30554pt]
=\delta_{H}^{-1}(h)\int_{\hat{K}\times\mathbb{T}}[(\int_{K}\sum_{n^{\prime}\in{\mathbb{Z}}}\hat{f}(k^{\prime},n^{\prime})z^{n^{\prime}}\omega(k^{\prime})\overline{\widehat{{(\psi\circ\theta)}}}_{(h,k)^{-1}}(k^{\prime},n^{\prime})d\mu_{K}(k^{\prime}))\\\
\times(\overline{\int_{K}\sum_{n^{\prime\prime}\in{\mathbb{Z}}}\hat{f}(k^{\prime},n^{\prime})z^{n^{\prime\prime}}\omega(k^{\prime\prime})\overline{\widehat{{(\psi\circ\theta)}}}_{(h,k)^{-1}}(k^{\prime\prime},n^{\prime\prime})}d\mu_{K}(k^{\prime\prime}))]\\\\[4.30554pt]
=\delta_{H}^{-1}(h)\int_{\hat{K}\times\mathbb{T}}|\hat{F}(\omega,z)|^{2}d\mu_{\hat{K}\times\mathbb{T}}\\\\[4.30554pt]
=\delta_{H}^{-1}(h)\int_{\hat{K}\times\mathbb{T}}|F(k^{\prime},n^{\prime})|^{2}d\mu_{{K}\times\mathbb{Z}}\\\\[4.30554pt]
=\delta_{H}^{-1}(h)\int_{K}\sum_{n^{\prime}\in{\mathbb{Z}}}|\hat{f}(k^{\prime},n^{\prime})|^{2}|\widehat{{(\psi\circ\theta)}}(k^{\prime},n^{\prime})|^{2}d\mu_{K}(k^{\prime}),\end{array}$

where $\hat{F}=\hat{f}\hat{(\psi\circ\theta)}\in{L^{1}(K\times\mathbb{Z}}).$
It is easy to see that

$\widehat{(\psi\circ\theta)}((k^{\prime},n^{\prime})=\delta_{H}^{-1}(h)\hat{\psi}(k^{\prime},n^{\prime})\circ\theta_{(h,k)^{-1}}.$

Then

(3.6)
$\int_{\hat{K}\times\mathbb{T}}|W_{\psi}f(h,k,\omega,z)|^{2}d\mu_{\hat{K}\times\mathbb{T}}(\omega,z)=\delta_{H}^{-1}(h)\int_{K}\sum_{n^{\prime}\in{\mathbb{Z}}}|\hat{f}(k^{\prime},n^{\prime})|^{2}|\widehat{(\psi(k^{\prime},n^{\prime})}\circ\theta_{(h,k)^{-1}}|^{2}d\mu_{K}(k^{\prime}).$

Now, by using (3.6) we have
$\begin{array}[]{lll}\|W_{\psi}f\|_{2}^{2}&=&\int_{\mathbb{H}(G_{\tau})}|W_{\psi}f(h,k,\omega,z)|^{2}d\mu_{\mathbb{H}(G_{\tau})}(h,k,\omega,z)\\\\[4.30554pt]
&=&\int_{H\times_{\tau}K}\int_{\hat{K}\times\mathbb{T}}|W_{\psi}f(h,k,\omega,z)|^{2}\delta_{H}^{-1}(h)d\mu_{\hat{K}\times\mathbb{T}}(\omega,z)d\mu_{H\times_{\tau}K}(h,k)\\\\[4.30554pt]
&=&\int_{H\times_{\tau}K}\int_{K}\sum_{n^{\prime}\in{\mathbb{Z}}}|\hat{f}(k^{\prime},n^{\prime})|^{2}|\widehat{(\psi(k^{\prime},n^{\prime})}\circ\theta_{(h,k)^{-1}}|^{2}d\mu_{K}(k^{\prime})d\mu_{H\times_{\tau}K}(h,k)\\\\[4.30554pt]
&=&\|f\|_{2}^{2}\int_{H\times_{\tau}K}|\widehat{(\psi(k^{\prime},n^{\prime})}\circ\theta_{(h,k)^{-1}}|^{2}d\mu_{H\times_{\tau}K}(h,k),\end{array}$
and then the proof of part $(ii)$ is complete. ∎

##  4\. Examples and applications

######  Example 4.1.

Let $K$ be an abelian locally compact group and $H=\\{e\\}$ (the trivial
group). In this case the generalized weyl Heisenberg group
$\mathbb{H}(G_{\tau})$ coincides with the standard weyl Heisenberg group
$G:=K\times_{\theta}(\hat{K}\times\mathbb{T})$. In this case the square
integrable representation of $G=K\times_{\theta}(\hat{K}\times\mathbb{T})$ on
$L^{2}(\hat{K})$ is as follows:

(4.1)
$\pi(k,\omega,z)f(\xi)=z\xi(k)\overline{\omega(k)}f(\xi\overline{\omega}).$

######  Example 4.2.

Let $E(n)$ be the Euclidean group which is the semi-direct product of
$So(n)\times_{\tau}\mathbb{R}^{n}$ where the continuous homomorphism
$\tau:So(n)\rightarrow Aut(\mathbb{R}^{n})$ given by
$\sigma\mapsto\tau_{\sigma}$ via $\tau_{\sigma}(x)=\sigma x$, for all
$x\in{\mathbb{R}^{n}}$. The group operation for $E(n)$ is

$(\sigma_{1},x_{1})\times_{\tau}(\sigma_{2},x_{2})=(\sigma_{1}\sigma_{2},x_{1}+\sigma_{1}x_{2}).$

Consider the continuous homomorphism $\hat{\tau}:So(n)\rightarrow
Aut(\mathbb{R}^{n})$ via $\sigma\mapsto\hat{\tau_{\sigma}}$ which is given by
$\hat{\tau_{\sigma}}((\omega)=\omega_{\sigma}=\omega\circ\tau_{\sigma^{-1}}$.
Thus the generalized Weyl Heisenberg group of $E(n)$, is the set
$\mathbb{H}(E(n))=(So(n)\times_{\tau}\mathbb{R}^{n})\times_{\theta}(\mathbb{R}^{n}\times\mathbb{T})$
with the group operation

$(\sigma_{1},x_{1},\omega_{1},z_{1})(\sigma_{2},x_{2},\omega_{2},z_{2})=(\sigma_{\sigma_{2}},x_{1}+\sigma_{1}x_{2},\omega_{1}(\omega_{2})_{\sigma_{1}},(\omega_{2})_{\sigma_{1}}(x_{1})z_{1}z_{2}),$

for all
$(\sigma_{1},x_{1},\omega_{1},z_{1})(\sigma_{2},x_{2},\omega_{2},z_{2})\in{\mathbb{H}(E(n))}$
and with the product topology. Then the square integrable representation $\pi$
of $\mathbb{H}(E(n))$ onto $L^{2}(\mathbb{R}^{n})$ is

$\pi(\sigma,x,\omega,z)f(\xi)=e^{2\pi
ix(\xi-\omega)}f((\xi-\omega)_{\sigma^{-1}}).$

Note that $H$ is compact and $\delta_{H}(h)=1$.

######  Example 4.3.

Let
$\mathbb{H}(\mathbb{R}^{n})=\mathbb{R}^{n}\times_{\theta}(\mathbb{R}^{n}\times\mathbb{T})$
be the classical Heisenberg group on $\mathbb{R}^{n},$ in which the continuous
homomorphism $x\mapsto\theta_{x}$ from $\mathbb{R}^{n}$ into
$Aut(\mathbb{R}^{n}\times\mathbb{T})$ is defined by
$\theta_{x}(y,z)=(y,ze^{2\pi ix.y}).$ Then the square integrable
representation $\pi$ of $\mathbb{H}(\mathbb{R}^{n})$ onto
$L^{2}(\mathbb{R}^{n})$ is

$\pi(x,\omega,z)f(\xi)=z.e^{2\pi ix(\xi-\omega)}f(\xi-\omega).$

## References

  * [1] A.A. Arefijamaal, R.A. Kamyabi Gol,  A Characterization of Square Integrable Representations Associated with CWT, J. Sci. Islam. Repub. Iran., Vol 18, No. 2, 159-166, 2007.
  * [2] R. Fabec, G. Ólafsson  The Continuous Wavelet Transform and Symmetric Spaces, Acta. Appl. Math., Vol. 77, No. 1, 41–69, 2003.
  * [3] H. Fuehr, M. Mayer, Continuous wavelet transforms from semidirect products: Cyclic representations and Plancherel measure, J. Fourier Anal. Appl., Vol. 8, 375-398, 2002.
  * [4] A. Ghaani Farashahi, Generalized Weyl-Heisenberg group, Anal. Math. Phys., Vol.4, 187-197, 2014.
  * [5] M.W. Wong, Wavelet Transform and Localization Operator, Verlag, Basel- Boston- Berlin, 2002.


# Human mobility generation with physics-enhanced network

Zongyuan Huang<EMAIL_ADDRESS>Shanghai Jiao Tong UniversityShanghaiChina
,  Shengyuan Xu<EMAIL_ADDRESS>Shanghai Jiao Tong
UniversityShanghaiChina ,  Menghan Wang<EMAIL_ADDRESS>eBay
Inc.ShanghaiChina ,  Hansi Wu<EMAIL_ADDRESS>eBay Inc.ShanghaiChina ,
Yanyan Xu<EMAIL_ADDRESS>Shanghai Jiao Tong UniversityShanghaiChina and
Yaohui Jin<EMAIL_ADDRESS>Shanghai Jiao Tong UniversityShanghaiChina

###### Abstract.

Modeling human mobility helps to understand how people are accessing resources
and physically contacting with each other in cities, and thus contributes to
various applications such as urban planning, epidemic control, and location-
based advertisement. Next location prediction is one decisive task in
individual human mobility modeling and is usually viewed as sequence modeling,
solved with Markov or RNN-based methods. However, the existing models paid
little attention to the logic of individual travel decisions and the
reproducibility of the collective behavior of population. To this end, we
propose a Causal and Spatial-constrained Long and Short-term Learner (CSLSL)
for next location prediction. CSLSL utilizes a causal structure based on
multi-task learning to explicitly model the
”when$\rightarrow$what$\rightarrow$where”, a.k.a.
”time$\rightarrow$activity$\rightarrow$location” decision logic. We next
propose a spatial-constrained loss function as an auxiliary task, to ensure
the consistency between the predicted and actual spatial distribution of
travelers’ destinations. Moreover, CSLSL adopts modules named Long and Short-
term Capturer (LSC) to learn the transition regularities across different time
spans. Extensive experiments on three real-world datasets show promising
performance improvements of CSLSL over baselines and confirm the effectiveness
of introducing the causality and consistency constraints. The implementation
is available at https://github.com/urbanmobility/CSLSL.

Next Location Prediction; Human Mobility; Causality; Spatial Consistency;
Multi-task Network

##  1\. Introduction

Human mobility modeling aims to explore the regularities and patterns of human
behavior (Gonzalez et al., 2008; Barbosa et al., 2018) and plays a significant
role in numerous applications, such as urban planning (Xu et al., 2021),
travel demand management (Çolak et al., 2016; Xu et al., 2018), health risk
assessment (Xu et al., 2019), epidemic spreading modeling and control (Arenas
et al., 2020; Jia et al., 2020), and so on. In the big data era, the
accessibility to GPS, mobile phone records, and location-based social networks
(LBSNs) provides an unprecedented chance to understand and model human
mobility (Barbosa et al., 2018; Luca et al., 2020).

In the research community of human mobility, physicists focus on statistical
analysis from a macroscopic perspective and have summarized empirical rules
(Barbosa et al., 2018). For example, they found that, truncating the power law
distribution can well fit the displacement distribution (Gonzalez et al.,
2008); despite the significant differences in the travel patterns, a majority
of users’ mobility behaviors are predictable (Song et al., 2010). Computer
scientists, on the other hand, prefer to model the transition regularities
from location sequences, using Markov models (Rendle et al., 2010), recurrent
neural networks (RNNs) (Feng et al., 2018), etc. In summary, statistical
physics study the collective behavior at population level, while deep learning
methods emphasize modeling individual travel trajectories. Thus, we can expect
that integrating physical domain knowledge into a deep learning model
encourages the model to pay attention to group behaviors and promotes the
performance of deep learning models at population level.

Here we place our emphasis on next location prediction, a vital task in human
mobility modeling at individual level. A body of work leverages machine
learning methods to tackle this problem due to the sequential nature of
mobility behavior. A common thread of these studies is to efficiently capture
behavior patterns from sparse data (Luca et al., 2020; Lian et al., 2020; Guo
et al., 2020; Luo et al., 2021). Traditional methods mainly adopt Markov
chains to model transition probability matrices across locations, along with
techniques like factorization (Rendle et al., 2010; Cheng et al., 2013; He et
al., 2016) or metric embedding (Feng et al., 2015). In recent years, deep
learning methods are gaining increasing attention in next location prediction
as the recurrent neural network (RNN) presents its capability to capture
sequential dependency. To model multi-scale spatio-temporal periodicity,
researchers designed attention or gate mechanisms and introduced time and
distance interval information (Feng et al., 2018; Manotumruksa et al., 2018;
Zhao et al., 2019; Sun et al., 2020; Yang et al., 2020). Also a few studies
incorporate semantic information such as location categories to cope with the
data sparsity (He et al., 2017; Yu et al., 2020; Guo et al., 2020). However,
methods that capture dependencies only from location sequences are difficult
to fully fit complex human travel behaviors, especially with sparse data.

To tackle this challenge, we seek to integrate physical knowledge into deep
learning methods to enhance the capability of human mobility prediction.
Specifically, we propose two physical constraints. The first one is summarized
as ”when$\rightarrow$what$\rightarrow$where” causal relationship. ”When”,
”what”, and ”where” are the three core elements of human travel behavior and
the dependencies between them can explain the motivation of location transfer.
For example, as shown in Figure 1(a), people have specific demands at
different times, causing the shifts between locations. Considering causal
dependencies enables more comprehensive modeling of human mobility. The second
constraint is the macro-statistical characteristics reflecting group behavior.
Figure 1(b) illustrates the deviation of the modeled displacement distribution
via LSTM from the true distribution in New York City and Tokyo, suggesting
LSTM is more likely to focus on shorter trips with higher frequency. Ensuring
the consistency between the model output and the macro-statistical
characteristics is expected to improve the model’s capability to fit travel
behavior. We summarize these two constraints as causality and consistency
constraints and incorporate them into deep learning models.

Figure 1. Illustrations for causality and spatial consistency in human
mobility. (a) An example to explain the
”when$\rightarrow$what$\rightarrow$where” decision logic. (b) The deviation of
the modeled displacement distribution via LSTM from the ground truth (GT) in
New York City and Tokyo.

To this end, we propose a Causal and Spatial-constrained Long and Short-term
Learner (CSLSL), a model integrating the decision logic and the consistency
constraints of human mobility modeling. To model the
”when$\rightarrow$what$\rightarrow$where” decision logic, we introduce a
causal structure in CSLSL. Based on a multi-task learning, the causal
structure utilizes three similar network branches to model the regularities of
time, activity, and location, respectively. In line with the
”when$\rightarrow$what$\rightarrow$where” logic, we explicitly build
connections between the three branches in the causal structure. As for
consistency, we exploratively propose a spatial-constrained loss to reduce the
distance between the predicted and actual locations, and indirectly ensure the
consistency of the spatial density distribution. In addition, we adopt a Long
and Short-term Capturer (LSC) to learn the transition patterns of different
time spans. There are two units in LSC that focus on long-term and short-term
regularities respectively.

The main contributions of this work are summarized as follows:

  * •

We propose CSLSL to integrate the travel decision logic and the macro-
statistical consistency for human mobility modeling. To our best knowledge,
CSLSL is the first model to learn the causality and consistency constraints
for next location prediction.

  * •

We introduce a causal structure that can capture not only the separate
regularities of time, activity, and location, but also the
”when$\rightarrow$what$\rightarrow$where” causal dependencies. In this way,
CSLSL models more essential travel logic in addition to sequence
relationships.

  * •

To ensure the consistency in spatial distribution, we propose a spatial-
constrained loss to reduce the gap between the predicted and actual
destinations.

  * •

We evaluate CSLSL on three real-world datasets to confirm the performance
improvements. We also conduct ablation studies and visualization analyses of
results such as displacement distribution to demonstrate the effectiveness of
our design.

##  2\. Related Work

###  2.1. Next Location Prediction

Here we classify the approaches to the next location prediction problem into
two categories: traditional and deep learning methods. Traditional methods
mainly apply Markov chain (MC) and focus on constructing a better location
transition probability matrix (Rendle et al., 2010; Cheng et al., 2013; Feng
et al., 2015; Zhao et al., 2016). For instance, factorized personalized Markov
chain (FPMC) combines the matrix factorization technique with Markov chains to
learn users’ personalized transition matrices (Rendle et al., 2010). The
limitation of the MC-based methods lies in the difficulty in capturing long-
term and high-order regularity (Guo et al., 2020; Luo et al., 2021).

Deep learning methods have advantages of learning dense representation and
complex dependency. Recently, RNN-based methods show promising performance in
mining sequential information. A popular scheme of deep learning methods is
incorporating time and distance intervals to assist the model in learning the
spatio-temporal regularities of human mobility. Specifically, these methods
integrate spatio-temporal information into hidden state transition (Liu et
al., 2016), gate mechanisms (Kong and Wu, 2018; Manotumruksa et al., 2018;
Zhao et al., 2019), or self-attention mechanisms (Lian et al., 2020; Luo et
al., 2021), and exploit spatio-temporal contexts in a implicit manner. To
leverage the spatio-temporal contexts, researchers explicitly used spatial and
temporal factors as attention weights to select the historical hidden states
(Yang et al., 2020; Zhao et al., 2020). Another scheme emphasizes the long-
term patterns of human behavior, such as DeepMove (Feng et al., 2018) and
LSTPM (Sun et al., 2020). They introduce two different components to model
long-term and short-term preferences respectively. There is also another
scheme that utilizes semantic information such as location categories to
improve the performance of location prediction (Yu et al., 2020; Guo et al.,
2020; Wang et al., 2021). However, methods that focus on modeling location
transfer patterns in sequences cannot effectively capture complex human
decision logic. In our work, we propose a causal structure to explicitly
capture the ”when$\rightarrow$what$\rightarrow$where” decision logic.

###  2.2. Time- or Activity-jointed Location Prediction

The methods that jointly predict time or activity learn knowledge from related
tasks to improve the prediction performance of location. RMTPP (Du et al.,
2016) combines RNN and temporal point process (TPP) to jointly model the time
and location information. He et al. (He et al., 2017) proposed a two-fold
approach that predicts category with Bayesian Personalized Ranking (BPR)
technique and then predicts the category-based location. Krishna et al.
(Krishna et al., 2018) utilized two distinct LSTM networks to predict
activities and durations. DeepJMT (Chen et al., 2020) fuses spatio-temporal
information and social context to predict time and location with a
hierarchical RNN and TPP technique. Sun et al. (Sun and Kim, 2021) proposed a
hybrid LSTM and a sequential LSTM with a self-attention mechanism to jointly
model location and travel time. The limitation of these approaches is that
they attempt to implicitly and passively learn the correlation between time,
category, and location information, but this relationship is explicit and can
be directly exploited. In contrast, CSLSL explicitly models the causal
dependencies between time, category, and location information through two
structural designs.

###  2.3. Statistical Physics-informed human mobility modeling

Explicitly integrating knowledge of statistical physics contributes to guiding
model optimization and improving the performance of machine learning methods.
On the task of trajectory generation, researchers introduced knowledge of
statistical physics to constrain the macroscopic performance of their models,
such as the individual trajectory generation model TimeGeo (Jiang et al.,
2016) and DITRAS (Pappalardo and Simini, 2018), and flow generation model
DeepGravity (Simini et al., 2021). Unlike the trajectory generation task, only
a limited amount of work on individual mobility prediction incorporates
knowledge of statistical physics. Zhao et al. (Zhao et al., 2020) integrated
domain knowledge, specifically a power-law decay for distances and an
exponential decay for time intervals, into an attention mechanism to adjust
the impact of historical information on current prediction. However, deep
learning-based methods focus more on fitting individual behavior while
neglecting group behavior constraints described by macro-statistical
characteristics, for example, the regional attractiveness of city blocks. The
spatial distribution of model prediction results should be consistent with the
actual statistical distribution. Predicted locations closer to the actual
location are more expected. Toward this, we propose a spatial-constrained loss
function to narrow the distance between the predicted and actual locations,
thereby ensuring the consistency of the spatial distribution.

##  3\. Problem Formulation

A person’s travel behavior can be represented as a sequence of locations,
associated with the timestamps and her user ID. In LBSN datasets, each
location is also associated with its functional category to support the
analysis of user’s activity. Let
$\mathcal{U}=\\{u_{1},\dots,u_{|\mathcal{U}|}\\}$,
$\mathcal{L}=\\{l_{1},\dots,l_{|\mathcal{L}|}\\}$ and
$\mathcal{C}=\\{c_{1},\dots,c_{|\mathcal{C}|}\\}$ denote a set of users,
locations and functional categories, respectively. Each location $l_{i}$ is
associated with its category and geographical coordinate
$(c_{i},lat_{i},lon_{i})$.

Definition 1 (Record). Record $r$ is a 3-tuple $(u_{i},l_{j},t_{k})$,
representing that the user $u_{i}$ visited location $l_{j}$ at time $t_{k}$,
where $u_{i}\in\mathcal{U},l_{j}\in\mathcal{L}$.

Definition 2 (Individual Trajectory). A person’s trajectory is defined as a
record sequence $\mathcal{R}=\\{r_{1},r_{2},\dots,r_{|\mathcal{R}|}\\}$, which
consists of the person’s all records arranged in chronological order. Note
that the time interval between two consecutive records is heterogeneous due to
the irregular travel behavior.

Definition 3 (Session). Session $S$ is a subsequence of records in a time
slot. One user’s trajectory $\mathcal{R}$ can be split into a series of
sessions with various strategies. For example, DeepMove adopts a specific time
interval between two consecutive records to split the trajectory (Feng et al.,
2018). Other strategies segment users’ trajectories using a fixed number of
records (Yang et al., 2020; Luo et al., 2021) or a meaningful time window such
as days, weeks, etc (Sun et al., 2020). We define the session where the
prediction target is located as the short-term session $S_{p}$ and the
previous historical sessions as long-term sessions
$\\{S_{q}\\},q\in\\{1,\dots,p-1\\}$.

The location prediction problem is formulated as: given a record sequence of a
user $\mathcal{R}_{t-1}=\\{r_{1},\dots,r_{t-1}\\}$, the goal is to predict
where the user $u$ is most likely to go in her next trip. We use $\hat{l}_{t}$
to denote the predicted next location. Note that the timestamp of the next
trip $t$ is also unknown.

##  4\. Methodology

In this section, we first analyze the causality and consistency constraints in
human mobility modeling, and then elaborately introduce the design of the
proposed model, Casual and Spatial-constrained Long and Short-term Learner
(CSLSL). The architecture of CSLSL is illustrated in Figure 2. It mainly
consists of two parts, an embedding part for learning the representations of
arrival time, category, and location, from users’ recent and historical
records; and the second part for learning the regularities of mobility
behavior in a multi-task learning based causal module and making predictions.

Figure 2. The architecture of the proposed CSLSL model.

###  4.1. Causality and Consistency Constraints

A common practice for next location prediction is to discover similar
subsequences or location transition relationships from historical records.
This is accomplished by integrating the context information such as distance
or time intervals into attention or RNN-centered framework (Zhao et al., 2019;
Yang et al., 2020; Sun et al., 2020; Luo et al., 2021). We can formulate such
mainstream scheme as $P(\hat{l}_{t}|\mathcal{R}_{t-1})$, where
$\mathcal{R}_{t-1}$ is the historical record sequence. Another scheme adopts
multi-task learning techniques to jointly predict next location with time or
activity (Chen et al., 2020; Sun and Kim, 2021), formulated as
$P(\hat{l}_{t},\hat{c}_{t},\hat{t}|\mathcal{R}_{t-1})=P(\hat{l}_{t}|\mathcal{R}_{t-1})P(\hat{c}_{t}|\mathcal{R}_{t-1})P(\hat{t}|\mathcal{R}_{t-1})$,
where we assume that the location category can approximate the type of
activity. Although these two schemes combine contextual information to capture
hidden regularities of location transition, they ignore the causal
dependencies in the context information.

As aforementioned, we regard ”when”, ”what” and ”where” as three crucial
elements to describe human mobility (Zhang et al., 2021; Pacheco et al.,
2022). ”When” refers to the time the trip takes place, e.g. ”midday”. ”What”
tells about the activities people participate in and also answers the reasons
for the trip, such as ”having lunch”. ”Where” is the destination of the trip,
like ”steakhouse”. Periodic activities exist in human mobility and occur at
specific times and places, such as going to work in the morning and going to a
restaurant for lunch, which reveals the correlation between the three
elements. When we mention a specific timestamp, we have various activity
choices. But we are accustomed to doing certain activities at certain times,
such as going to the gym in the evening. Similarly, one activity (category)
corresponds to multiple locations (POIs), while one location ID only
corresponds to one activity, also reflected in the dataset. Moreover, our
target is location prediction, thus location should be the final subtask to
leverage the predicted time and activity information. Therefore, we summarize
a ”when$\rightarrow$what$\rightarrow$where”, a.k.a.
”time$\rightarrow$activity$\rightarrow$location” causal relationship, which is
in line with the coarse-to-fine logic of the human decision. The proposed
scheme can be formulated as:

(1)
$P(\hat{l}_{t},\hat{c}_{t},\hat{t}|\mathcal{R}_{t-1})=P(\hat{l}_{t}|\hat{c}_{t},\hat{t},\mathcal{R}_{t-1})P(\hat{c}_{t}|\hat{t},\mathcal{R}_{t-1})P(\hat{t}|\mathcal{R}_{t-1}).$

The scheme explicitly models the dependencies between time, activity, and
location, and alleviates the difficulty of location prediction. For example,
people are accustomed to going to restaurants at midday instead of bar, that
is,
$P(\hat{c}_{t}=restaurant|\hat{t}=midday,\mathcal{R}_{t-1})>P(\hat{c}_{t}=bar|\hat{t}=midday,\mathcal{R}_{t-1})$.
Each individual has her personalized
$P(\hat{l}_{t}|\hat{c}_{t}=restaurant,\hat{t}=midday,\mathcal{R}_{t-1})$, and
the casual constrained location distribution is easier to learn than
$P(\hat{l}_{t}|\mathcal{R}_{t-1})$. In CSLSL, we introduce a causal structure
to implement the scheme. In experiments, we also demonstrate that
”time-¿acitivity-¿location” outperforms ”activity-¿time-¿location” and further
discuss the reason.

On the other side, integrating physical knowledge provides more information
and prior constraints to guide the optimization of deep learning models
(Karniadakis et al., 2021; Willard et al., 2020). In human mobility modeling,
one can expect that properly introducing the physical laws and domain
knowledge would narrow down the gap between the output of deep learning-based
approaches and the observed macro-statistical characteristics of human
behavior. Due to the difficulty in applying statistical constraints in the
training of deep learning models, here we consider the geographic spatial
consistency in an indirect way. Specifically, we devise a loss function to
constrain the distance between the predicted and actual locations. That is,
the closer the predicted location is to the ground truth, the smaller loss we
have. By this way, we can indirectly ensure the consistency of the
displacement distribution and the consistency of the spatial distribution of
travelers’ destinations.

###  4.2. Long and Short-term Capturer

Human travel behavior has long and short-cycle repetitive patterns, such as
going to work every day and going to the supermarket once a week. Inspired by
DeepMove (Feng et al., 2018) and LSTPM (Sun et al., 2020), we devise a Long
and Short-term Capturer (LSC) to learn the behavioral patterns in different
observation cycles. In the whole framework shown in Figure 2, we apply three
LSCs to model the time, activity, and location sequences, respectively.

Let
$\boldsymbol{e}^{l}\in\mathbb{R}^{d^{l}},\boldsymbol{e}^{c}\in\mathbb{R}^{d^{c}},\boldsymbol{e}^{t}\in\mathbb{R}^{d^{t}}$
and $\boldsymbol{e}^{u}\in\mathbb{R}^{d^{u}}$ denote the embedded
representation of location, category, time and user, respectively. Given a
historical record sequence $\mathcal{R}$, CSLSL embeds each record as
$(\boldsymbol{e}^{l},\boldsymbol{e}^{c},\boldsymbol{e}^{t},\boldsymbol{e}^{u})$
in hidden spaces. Note that we first convert the continuous timestamp as the
hour in a day $t^{h}$ and the day in a week $t^{d}$ to present the daily and
weekly periodicity. By doing so, we have
$\boldsymbol{e}^{t}=\boldsymbol{e}^{t^{h}}\oplus\boldsymbol{e}^{t^{d}}$. Then
these representations in a record are concatenated together,
$\boldsymbol{e}^{r}=\boldsymbol{e}^{l}\oplus\boldsymbol{e}^{c}\oplus\boldsymbol{e}^{t}\oplus\boldsymbol{e}^{u}$.
We next split each user’s record sequence into multiple sessions with a
certain time window, like days or weeks. The records in short-term session and
long-term sessions are represented as
$\widetilde{\boldsymbol{e}}^{r}_{p}=\\{\boldsymbol{e}^{r}_{1},\dots,\boldsymbol{e}^{r}_{t-1}\\}$
and
$\\{\widetilde{\boldsymbol{e}}^{r}_{q}\\}=\\{\widetilde{\boldsymbol{e}}^{r}_{1},\dots,\widetilde{\boldsymbol{e}}^{r}_{p-1}\\}$,
respectively.

Figure 3. The illustration of the LSC module.

Our proposed LSC consists of two capturers to learn the transition
regularities in the short-term and long-term sessions, respectively, as shown
in Figure 3. We formulate LSC as:

(2)
$\boldsymbol{h}_{t}=LSC(\widetilde{\boldsymbol{e}}^{r}_{p},\\{\widetilde{\boldsymbol{e}}^{r}_{q}\\},\boldsymbol{h}_{0}),$

where $\boldsymbol{h}_{0}$ is the initial hidden state. In the LSC structure,
the short-term capturer takes the hidden state $h_{|S_{p-1}|}$ of the long-
term capturer as the initial hidden state to combine the historical
information. Because GRU is simple but efficient in modeling temporal data, we
apply a layer of GRU in both of the capturers:

(3) $\boldsymbol{h}_{i}=GRU(\boldsymbol{e}^{r}_{i-1},\boldsymbol{h}_{i-1}),$

where $i\in\\{1,2,\dots,|S_{p-1}|\\}$ for long-term capturer and
$i\in\\{1,2,\dots,t\\}$ for short-term capturer.

###  4.3. Causal Structure

To model the ”time$\to$activity$\rightarrow$location” logic relationship
discussed in subsection 4.1, we introduce a causal structure based on multi-
task learning techniques. As illustrated in Figure 2, we utilize three
branches with the same architecture to model the change patterns of time,
activity, and location, respectively. Specifically, in each branch, we convey
the same record representations to the LSC module and then transfer the output
hidden states to the predictor. To explicitly model the summarized causal
relation in human travel behavior, we next design two paths for information
transfer between various tasks. The first path lies between two LSC modules,
passing on the task-specific hidden states. The second path lies between two
predictors. In this path, the predicted result of the upstream task is
processed by the converter and then conveyed to the downstream task. Here we
use the fully connected layer as the predictor (P) and converter(C). That is
$\boldsymbol{y}=Linear(\boldsymbol{x})=\boldsymbol{W}\boldsymbol{x}+\boldsymbol{b}$.

Mathematically, the branch of ”time” is formulated as:

(4)
$\boldsymbol{h}^{t}_{t}=LSC(\widetilde{\boldsymbol{e}}^{r}_{p},\\{\widetilde{\boldsymbol{e}}^{r}_{q}\\},0),$
(5) $\hat{t}=Linear^{(P^{t})}(\boldsymbol{h}^{t}_{t}),$

where $\boldsymbol{W}^{(P^{t})}\in\mathbb{R}^{1\times|\boldsymbol{h}^{t}|}$,
$\boldsymbol{h}^{t}_{t}$ is the hidden state of the next time, and $\hat{t}$
is the predicted time. As the downstream task of ”time” in causal structure,
the branch of ”activity” can be formulated as:

(6)
$\boldsymbol{h}^{c}_{t}=LSC(\widetilde{\boldsymbol{e}}^{r}_{p},\\{\widetilde{\boldsymbol{e}}^{r}_{q}\\},\boldsymbol{h}^{t}_{t}),$
(7) $\boldsymbol{c}_{t}=Linear^{(P^{c})}(\boldsymbol{h}^{c}_{t}\oplus
Linear^{(C^{t})}(\hat{t})),$ (8) $\hat{c}_{t}=argmax(\boldsymbol{c}_{t}),$

where $\boldsymbol{W}^{(C^{t})}\in\mathbb{R}^{|\boldsymbol{e}^{t}|\times
1},\boldsymbol{W}^{(P^{c})}\in\mathbb{R}^{|\mathcal{C}|\times(|\boldsymbol{h}^{c}|+|\boldsymbol{e}^{t}|)}$,
$\boldsymbol{h}^{c}_{t}$ is the hidden state of the next activity, and
$\hat{c}_{t}$ is the predicted activity. Eventually, we can formulate the
branch of ”location” as:

(9)
$\boldsymbol{h}^{l}_{t}=LSC(\widetilde{\boldsymbol{e}}^{r}_{p},\\{\widetilde{\boldsymbol{e}}^{r}_{q}\\},\boldsymbol{h}^{c}_{t}),$
(10) $\boldsymbol{l}_{t}=Linear^{(P^{l})}(\boldsymbol{h}^{l}_{t}\oplus
Linear^{(C^{c})}(\boldsymbol{c}_{t})),$ (11)
$\hat{l}_{t}=argmax(\boldsymbol{l}_{t}),$

where
$\boldsymbol{W}^{(C^{c})}\in\mathbb{R}^{|\boldsymbol{e}^{c}|\times|\mathcal{C}|},\boldsymbol{W}^{(P^{l})}\in\mathbb{R}^{|\mathcal{L}|\times(|\boldsymbol{h}^{l}|+|\boldsymbol{e}^{c}|)}$,
$\boldsymbol{h}^{l}_{t}$ is the hidden state of the next location,
$\boldsymbol{l}_{t}$ is the distribution of predicted location, and
$\hat{l}_{t}$ is the predicted location.

Figure 4. Structure variation. (a) Long and Short-term Learner (LSL); (b)
Share-Bottom LSL (SBLSL); (c) Multi-Experts LSL (MELSL); (d) Separate LSL
(SLSL); (e) Hierarchical LSL (HLSL). Note the gate networks of MELSL are not
drawn for simplicity.

###  4.4. Spatial-constrained Loss Function

As discussed in subsection 4.1, for seeking an alignment of spatial
distribution of destinations, we propose a spatial-constrained loss function
to shorten the distance from the predicted location to the ground truth at
individual level. The distance constraint can be regarded as a self-supervised
auxiliary task, integrating the geographical information and restricting the
candidate set for better next location prediction. The existing methods
introduce the distance constraints in a regression subtask, directly
predicting the geographical locations of destinations (Xue et al., 2021).
However, in the classification scheme, we must query the coordinates of
location IDs to calculate their distance. This operation is not derivable. We
get inspiration from REINFORCE (Williams, 1992), which introduces the reward
in the loss function to train a policy network, and also consider the distance
error as a coefficient to weight the cross-entropy between ground truth and
the predicted location ID with their physical distance. The spatial-
constrained loss function is defined as:

(12) $L_{s}=-\sum^{N}_{i=1}distance(\hat{l}_{t,i},l_{t,i})\cdot
log(\sigma(\boldsymbol{l}_{t,j})),$

where $N$ is the total number of records and $\sigma$ is the softmax function.

We next employ MAE loss for time prediction and cross entropy loss for
category and location prediction. Thus we have
$L_{t}=MAE(\hat{t},t)=\sum^{N}_{i=1}|\hat{t}_{i}-t_{i}|$ and
$L_{*}=CrossEntropy(*)=-\sum^{N}_{i=1}log(\sigma(\boldsymbol{*_{t,i}})),*\in\\{c,l\\}$.
Thus, the total loss function can be written as

(13) $L_{total}=L_{l}+\lambda_{t}L_{t}+\lambda_{c}L_{c}+\lambda_{s}L_{s},$

where $\lambda_{t},\lambda_{c},\lambda_{s}$ are weights for their loss
functions.

###  4.5. Structure Comparison

There are various strategies for task combination in multi-task learning, such
as share-bottom structure (Ruder, 2017; Qin et al., 2020), hierarchical
structure (Sanh et al., 2019; Chen et al., 2020), and multi-expert structure
(Ma et al., 2018; Qin et al., 2020). Inspired by these structures, we propose
five variants, as shown in Figure 4, to demonstrate the advantages of our
causal structure. Note that these variants use the same basic components as
CSLSL, such as GRUs and fully connected layers.

Long and Short-term Learner (LSL) is a basic approach with only one branch to
predict location. To jointly predict ”time”, ”activity”, and ”location”,
Share-Bottom LSL (SBLSL) introduces two additional predictors that share the
same bottom LSC module with the original one. Multi-Experts LSL (MELSL) is an
advanced version of SBLSL, with a similar structure to Mixture of Sequential
Expert (MoSE) (Qin et al., 2020). MELSL employs several GRUs as experts to
focus on different aspects of sequence dependencies and gate networks to
combine relevant aspects for each task.

Unlike the share-bottom structure, Separate LSL (SLSL) employs a separate
branch for each task and the only shared information between each task is the
same record representations. Considering the dependencies between tasks,
Hierarchical LSL (HLSL) concatenates the record embedding and the output
hidden state of the upstream task as the input of the downstream task. Thus
the equation 3 changes to:

(14)
$\boldsymbol{h}^{k}_{i}=GRU(\boldsymbol{e}^{r}_{i-1}\oplus\boldsymbol{h}^{k-1}_{i},\boldsymbol{h}^{k}_{i-1}),$

where $\boldsymbol{h}^{k}_{i}$ is the hidden state of the $k$-th task at
$i$-th time step, and the equation 2 changes to:

(15)
$\boldsymbol{h}^{k}_{t}=LSC([\widetilde{\boldsymbol{e}}^{r}_{p},\widetilde{\boldsymbol{h}}^{k-1}_{p}],\\{[\widetilde{\boldsymbol{e}}^{r}_{q},\widetilde{\boldsymbol{h}}^{k-1}_{q}]\\},0),$

where
$[\widetilde{\boldsymbol{e}}^{r}_{p},\widetilde{\boldsymbol{h}}^{k-1}_{p}]=\\{\boldsymbol{e}^{r}_{1}\oplus\boldsymbol{h}^{k-1}_{1},\dots,\boldsymbol{e}^{r}_{t-1}\oplus\boldsymbol{h}^{k-1}_{t-1}\\}$.

##  5\. Experiments

###  5.1. Data Description

We leverage three publicly available check-in datasets in the experiments: two
datasets from Foursquare (Yang et al., 2014) in New York (NYC) and Tokyo (TKY)
and one dataset from Gowalla (Cho et al., 2011) in Dallas. Data in NYC and TKY
were collected from 3 April 2012 to 16 February 2013, and data in Dallas was
collected from 4 October 2016 to 30 April 2017. The number of users,
locations, records in three datasets are summarized in Table 1, where
$|\mathcal{*}|$ denotes the number of $*$. The number of location categories
$|\mathcal{C}|$ in NYC and TKY are 400 and 385, while Dallas does not contain
category information.

To prepare data for baselines and proposed models, we first filter out both
users and locations with fewer than 10 records, in line with previous work
(Feng et al., 2018; Sun et al., 2020). We then merge the consecutive records
with the same user and location on the same day. The statistic information of
the raw and processed data are depicted in Table 1. After pre-processing, the
number of categories for NYC and TKY are reduced to 308 and 286. For CSLSL and
its variations, we split trajectories into sessions according to week due to
the data sparsity. In addition, we require that each session contains at least
two records and a user contains at least five sessions to guarantee a
training/testing split of 8/2, following (Feng et al., 2018). All baselines
have their own further data preparation strategies and the model-specific
dataset information is also shown in Table 1. It’s noteworthy that, LSTPM (Sun
et al., 2020) requires at least three records in each session and Flashback
(Yang et al., 2020) limits the minimum records of each user to 100. These
practices filter out more sparse data and reduce the challenge of prediction.
Moreover, GETNext requires category information as input, thus it cannot work
on dataset Dallas.

Table 1. Statistic information of raw and processed data.

| NYC | TKY | Dallas
---|---|---|---
| $|\mathcal{U}|$ | $|\mathcal{L}|$ | $|\mathcal{R}|$ | $|\mathcal{U}|$ | $|\mathcal{L}|$ | $|\mathcal{R}|$ | $|\mathcal{U}|$ | $|\mathcal{L}|$ | $|\mathcal{R}|$
Raw | 1,083 | 38,333 | 227,428 | 2,293 | 61,858 | 573,703 | 5,894 | 5,767 | 167,016
Processed | 1,083 | 4,638 | 139,183 | 2,293 | 7,222 | 427,746 | 2,412 | 5,642 | 146,117
FPMC-D | 1,083 | 4,638 | 94,168 | 2,293 | 7,222 | 334,561 | 2,412 | 5,642 | 100,743
FPMC-W | 1,083 | 4,638 | 92,260 | 2,293 | 7,222 | 303,854 | 2,412 | 5,642 | 122,419
DeepMove | 1,061 | 4,627 | 111,968 | 2,284 | 7,206 | 333,215 | 1,193 | 5,346 | 93,911
Flashback | 439 | 4,093 | 96,771 | 1,451 | 6,998 | 366,604 | 340 | 5,434 | 82,287
LSTPM | 579 | 4,128 | 62,511 | 1,866 | 7,007 | 273,880 | 587 | 5,169 | 55,784
GeoSAN | 1,073 | 4,611 | 138,229 | 2,289 | 7,209 | 427,157 | 2,300 | 5,357 | 142,980
STAN | 1,000 | 4,503 | 82,783 | 2,200 | 7,047 | 201,831 | 2,400 | 5,615 | 95,656
GETNext | 1,066 | 4,621 | 131,920 | 2,280 | 7,200 | 414,993 | - | - | -
Ours | 1,065 | 4,635 | 133,944 | 2,280 | 7,204 | 422,432 | 1,357 | 5,428 | 120,531

Table 2. Performance comparison between baselines, CSLSL, its variants and
ablations on three real-world datasets.

| NYC | TKY | Dallas
---|---|---|---
| Category | Location | Category | Location | Location
| R@1 | R@5 | R@10 | R@1 | R@5 | R@10 | R@1 | R@5 | R@10 | R@1 | R@5 | R@10 | R@1 | R@5 | R@10
FPMC-D | 0.217 | 0.510 | 0.631 | 0.173 | 0.424 | 0.526 | 0.462 | 0.623 | 0.688 | 0.176 | 0.408 | 0.503 | 0.086 | 0.204 | 0.261
FPMC-W | 0.196 | 0.394 | 0.482 | 0.162 | 0.323 | 0.382 | 0.428 | 0.513 | 0.557 | 0.117 | 0.245 | 0.313 | 0.055 | 0.125 | 0.161
DeepMove | 0.244 | 0.499 | 0.586 | 0.198 | 0.407 | 0.470 | 0.398 | 0.567 | 0.626 | 0.161 | 0.331 | 0.400 | 0.089 | 0.185 | 0.229
Flashback* | 0.280 | 0.618 | 0.737 | 0.223 | 0.521 | 0.639 | 0.479 | 0.730 | 0.801 | 0.207 | 0.486 | 0.583 | 0.090 | 0.199 | 0.256
LSTPM* | 0.335 | 0.659 | 0.765 | 0.267 | 0.560 | 0.662 | 0.461 | 0.727 | 0.797 | 0.231 | 0.457 | 0.543 | 0.123 | 0.247 | 0.316
GeoSAN | 0.193 | 0.433 | 0.602 | 0.166 | 0.430 | 0.584 | 0.317 | 0.551 | 0.694 | 0.158 | 0.392 | 0.528 | 0.078 | 0.186 | 0.265
STAN | 0.218 | 0.480 | 0.591 | 0.192 | 0.411 | 0.494 | 0.376 | 0.575 | 0.668 | 0.167 | 0.388 | 0.478 | 0.074 | 0.153 | 0.196
GETNext | 0.303 | 0.646 | 0.749 | 0.246 | 0.536 | 0.622 | 0.452 | 0.758 | 0.844 | 0.216 | 0.456 | 0.550 | - | - | -
LSL | 0.288 | 0.587 | 0.684 | 0.242 | 0.506 | 0.589 | 0.446 | 0.697 | 0.762 | 0.225 | 0.462 | 0.548 | 0.101 | 0.203 | 0.254
SBLSL | 0.290 | 0.595 | 0.682 | 0.242 | 0.488 | 0.571 | 0.446 | 0.769 | 0.851 | 0.229 | 0.460 | 0.545 | 0.107 | 0.194 | 0.234
MELSL | 0.275 | 0.586 | 0.662 | 0.227 | 0.432 | 0.526 | 0.439 | 0.648 | 0.745 | 0.211 | 0.419 | 0.475 | 0.078 | 0.121 | 0.164
SLSL | 0.281 | 0.611 | 0.722 | 0.253 | 0.534 | 0.632 | 0.409 | 0.731 | 0.825 | 0.233 | 0.458 | 0.559 | 0.114 | 0.220 | 0.272
HLSL | 0.296 | 0.628 | 0.734 | 0.256 | 0.536 | 0.625 | 0.441 | 0.760 | 0.847 | 0.232 | 0.472 | 0.562 | 0.115 | 0.230 | 0.284
CLSL-ctl | 0.315 | 0.647 | 0.745 | 0.257 | 0.543 | 0.636 | 0.472 | 0.780 | 0.861 | 0.227 | 0.472 | 0.560 | - | - | -
CLSL | 0.322 | 0.658 | 0.747 | 0.261 | 0.553 | 0.643 | 0.459 | 0.782 | 0.864 | 0.238 | 0.476 | 0.567 | 0.120 | 0.229 | 0.282
CSLSL-c | 0.315 | 0.638 | 0.757 | 0.247 | 0.546 | 0.643 | 0.450 | 0.775 | 0.858 | 0.230 | 0.463 | 0.551 | - | - | -
CSLSL-t | 0.319 | 0.648 | 0.749 | 0.264 | 0.556 | 0.652 | 0.479 | 0.740 | 0.829 | 0.233 | 0.478 | 0.568 | 0.118 | 0.231 | 0.284
CSLSL | 0.327 | 0.661 | 0.759 | 0.268 | 0.568 | 0.656 | 0.488 | 0.801 | 0.875 | 0.240 | 0.488 | 0.580 | 0.126 | 0.243 | 0.297

  * *

Flashback and LSTPM filter out much more sparse users in their data
preparation, reducing the challenge of prediction.

###  5.2. Baselines and Settings

Baselines. We compare CSLSL with the state-of-the-art baselines:

  * •

FPMC (Rendle et al., 2010) is a Markov-based model that uses factorization to
learn individual transition matrices.

  * •

DeepMove (Feng et al., 2018) adopts an attention mechanism to learn long-term
preference and an GRU module to capture short-term preference.

  * •

Flashback (Yang et al., 2020) uses spatio-temporal distances as attention
weights to search the historical hidden states for current prediction.

  * •

LSTPM (Sun et al., 2020) considers temporal similarity and distance factor to
model long-term preferences and geographical relevance to model short-term
preferences.

  * •

GeoSAN (Lian et al., 2020) designs a geography encoder to implicitly capture
spatial proximity and introduces a loss function based on importance sampling
to better use the informative negative samples.

  * •

STAN (Luo et al., 2021) introduces a two-layer attention architecture with
spatio-temporal relation matrices to explicitly capture the spatio-temporal
correlations.

  * •

GETNext (Yang et al., 2022) utilizes a GCN to integrate collective movement
patterns and a transformer encoder to capture transition regularities.
Besides, it introduces location categories as inputs and prediction targets.

Settings. For convincingly comparing these baselines with our CSLSL, we
collected the open-source codes released by the authors and attempted to find
the optimal hyper-parameters in the experiments. It’s worth noting that most
of the baselines only predict next location, without category and time of
visitation. Thus, we match the predicted location ID to its category for
comparison and exclude the performance comparison of time prediction. Besides,
we split users’ trajectories by day and week for FPMC model, referred to as
FPMC-D and FPMC-W, respectively. For CSLSL, the dimensions of representation
vectors
$\boldsymbol{e}^{l},\boldsymbol{e}^{c},\boldsymbol{e}^{t^{h}},\boldsymbol{e}^{t^{d}}$
and $\boldsymbol{e}^{u}$ are set to 200, 100, 10, 20, and 20 for all datasets.
The dimension of the hidden state in all GRUs is set to 600. We use the Adam
optimizer with the learning rate of 0.0001, and $\lambda_{t}$, $\lambda_{c}$,
and $\lambda_{s}$ are set to $10$.

Metrics. In the next location prediction task, what we care about is whether
the actual location is in the top $N$ of our predictions, $N=\\{1,5,10\\}$.
$Recall@N$ is the most commonly used metric and is equal to $Accuracy@N$
because we don’t have false positive (FP) and true negative (TN). The
definition of $Recall@N$ is

(16)
$Recall@N=\frac{1}{|\mathcal{U}|}\sum_{u\in\mathcal{U}}\frac{|\mathcal{L}_{u}^{T}\cap\mathcal{L}_{u}^{P}|}{|\mathcal{L}_{u}^{T}|},$

where $\mathcal{L}_{u}^{T}$ and $\mathcal{L}_{u}^{P}$ are the target and top
$N$ prediction location sets, respectively.

###  5.3. Performance Comparison with Baselines

The experimental results are averaged over 10 independent runs and shown in
Table 2. For each city, the results are presented in three pieces,
representing the results of baselines (line 1-8), variants (line 9-14), and
ablations (line 15-18), respectively. The best performance in each column is
highlighted in bold text and the second best one is underlined. For NYC and
TKY, we present the predicted results for categories and locations, while for
Dallas, we only show the location prediction results due to the lack of
category information.

From the experiment results, we can observe that the proposed CSLSL shows
promising performances compared with baselines. In terms of $Recall@1$ in
location prediction, CSLSL achieves $27\%$, $37\%$, and $43\%$ averaged
performance improvements over these deep learning baselines in three datasets.
For $Recall@1$ in category prediction, the improvements are $34\%$ and $23\%$
in NYC and TKY, respectively. Considering $Recall@\\{5,10\\}$ in location
prediction, CSLSL achieves similar performances with LSTPM and Flashback,
which filter more than $46\%,18\%,$ and $57\%$ of sparse users than we do on
three datasets shown in Table 1. These similar performances in the more
challenging dataset settings can also reflect the superiority of our model.
Disregarding these two models, CSLSL still obtains over $20\%$ averaged
improvements than the rest of deep learning baselines. Moreover, CSLSL has
improved by 8.9% and 11.1% in Recall@1 in NYC and TKY compared to GetNext,
which has similar dataset statistics to ours. The poor performances of all
models on the Dallas dataset may be due to the data sparseness. Even so, CSLSL
can still utilize the time-location relationship and the spatial constraints
to achieve performance gains.

Among the baselines, we observe that the overall performance of location
prediction on TKY is lower than NYC. This is probably because the TKY dataset
has a larger number of users and locations than NYC, increasing the difficulty
of mobility predicting. However, CSLSL obtains more performance improvement on
TKY than NYC for location prediction compared with baselines. For instance,
the performance of CSLSL improves 8.9% on NYC compared with GETNext, while the
improvement is 11.1% on TKY. On the other side, the category predictions for
all models have higher accuracy on TKY than on NYC. We may conclude that the
larger performance improvement on TKY than NYC mainly owes to the proper
modeling of the dependencies.

###  5.4. Performance Comparison with Variants

To fairly demonstrate the effectiveness of the proposed causal structure, here
we develop an ablated version of CSLSL via dropping the spatial-constraint
loss, namely CLSL, and compare it with the 5 variants discussed in the
subsection 4.5. Moreover, we also consider the ”what$\to$when$\to$where”
relationship, thus we change the order of these three branches in CLSL from
”time$\to$category$\to$location” to ”category$\to$time$\to$location” and this
variant is named CLSL-ctl. We present the results of the variants and CLSL in
the second and third pieces of Table 2. The category prediction results of LSL
are obtained in the same way as the baselines.

Compared with LSL, SBLSL has a similar performance of location prediction and
slightly improved performance of category prediction, suggesting that the
shared bottom of SBLSL has indeed learned the category transfer regularities.
However, these learned regularities make no contribution to the location
prediction. Besides, the performance of MELSL is weaker than LSL and SBLSL,
which may be because MELSL does not clarify the relationship between tasks and
its experts cannot find suitable optimization directions. The better
performance of SLSL than SBLSL indicates that the separate modules to learn
transition relationships are better than the shared one. HLSL achieves the
best performance in the second pieces of Table 2, suggesting that there are
dependencies between time, category, and location, and that capturing the
dependencies facilitates location prediction.

The performances of these variants are weaker than CLSL, suggesting that
although these variants utilize temporal and categorical information, they
cannot effectively and autonomously capture the dependencies between time,
category, and location. In contrast, the causal structure explicitly captures
the dependencies between tasks in two ways, thereby fully exploiting their
dependencies to improve performance. Moreover, the better performance of CLSL
than CLSL-ctl is in line with expectations, because location has stronger
dependencies with category than time and category information can bring more
performance gains for location prediction. Therefore, our proposed causal
structure explicitly models ”time$\to$category$\to$location” rather than
”category$\to$time$\to$location”.

###  5.5. Ablation Study

We also conduct ablation studies to examine the contributions of different
components in CSLSL. The ablated models include:

  * •

LSL: the version that only keep the location branch.

  * •

CLSL: the version that removes the spatial-constraint loss.

  * •

CSLSL-t: the version that removes the time branch.

  * •

CSLSL-c: the version that removes the category branch.

As shown in Table 2, we can find that CSLSL-t achieves better performance than
CSLSL-c, indicating the ”category$\to$location” relationship has stronger
dependency constraints than ”time$\to$location”. This result is also
consistent with what we discussed in the subsection 5.4. The best performance
of the complete CSLSL demonstrates the significance of the entire
”time$\to$category$\to$location” decision logic. Comparing the performance of
CLSL and CSLSL, we can confirm that the spatial-constraint loss function have
a positive impact on the performance improvement. Moreover, LSL achieves
decent performance compared with baselines, probably because it leverages
category information and the LSC module is capable of capturing the long-term
and short-term preferences.

###  5.6. Results Visualization Analysis

We conduct result visualization analysis to further understand the
effectiveness of the causal structure and the spatial-constrained loss. For
the causal structure, we compare the category and the location prediction
results of GETNext and CSLSL, as shown in Figure 5. We can observe that for
CSLSL, the records with successfully predicted both categories and locations
on NYC and TKY account for 21% and 20% of all records. Compared with GETNext,
CSLSL successfully predicted 10% and 18% more locations with more accurately
predicted categories on NYC and TKY, respectively. This intuitively indicates
that the causal structure can enhance the location prediction with more
accurate category prediction results. Interestingly, the location can be
predicted correctly with an unsuccessfully predicted category. This is because
the category information is introduced as additional auxiliary information
without imposing mandatory constraints on the location prediction.

Figure 5. Effect analysis of the causal structure. Figure 6. Distance
distribution between the predicted and target location. Figure 7. (a)
Comparison of the predicted displacement with the ground truth (GT). (b)
Prediction error comparison of regional attractiveness.

Regarding to the spatial-constrained loss, we examine whether the distances
between predicted and actual locations are successfully constrained, and
compare CSLSL with four baselines. As shown in Figure 6, the results show that
the predicted locations of CSLSL is closer to the actual locations, which
indicates that the proposed loss can successfully constrain the distance
errors. Furthermore, we inspect the constraining effect of the proposed loss
on spatial consistency. Figure 7(a) shows the comparison of the predicted
displacement with the ground truth. It can be seen that the predicted
displacement of CSLSL is closer to the true distribution. This is because the
constraint between the predicted and actual locations can indirectly ensure
the consistency of the predicted displacements and the ground truth. Figure
7(b) shows the prediction error of regional attractiveness. We divided the
geographic regions into square grids with side length of 500m, and count the
difference between the predicted and actual visits in each grid. As presented
in Figure 7(b), CSLSL has a smaller prediction error of regional
attractiveness, suggesting that the proposed loss successfully constrains the
spatial consistency.

###  5.7. Sensitivity Analysis

Figure 8. Analysis of parameter sensitivity.

We perform sensitivity analysis on dataset NYC to examine how the performance
of CSLSL is affected by $\lambda_{*}$, $*\in\\{t,c,s\\}$. We first vary
$\lambda_{t}$ and $\lambda_{c}$ to analyze the effect of time and category
prediction subtasks with a fixed $\lambda_{s}=1$. Then we fix $\lambda_{t}$
and $\lambda_{c}$ and vary $\lambda_{s}$ to observe the impact of spatial-
constrained auxiliary tasks. $Recall@1$ is chosen as the evaluation metric and
the results of location prediction are averaged of three runs, shown in Figure
8.

From Figure 8 (a), we can observe that the performance of location prediction
is more sensitive to $\lambda_{c}$ than $\lambda_{t}$, reflecting that the
accurate category prediction exerts more influence on the location prediction
accuracy, which is also consistent with our proposed decision logic. In
addition, the best performance is obtained with $\lambda_{t}=5$ and
$\lambda_{c}=10$ when $\lambda_{s}=1$. Figure 8 (b) shows that CSLSL reaches a
more stable accuracy on NYC when $\lambda_{s}=5.0$, while the average accuracy
is higher when $\lambda_{s}=10.0$. In summary, CSLSL is robust to these
parameters and does not suffer from large performance fluctuations with
parameter changes.

##  6\. Conclusion

In this work, we propose a Causal and Spatial-constrained Long and Short-Term
Learner (CSLSL) to incorporate the individual travel decision logic and the
group consistency for next location prediction. In CSLSL, we introduce a
causal structure based on multi-task learning to explicitly capture the
”when$\rightarrow$what$\rightarrow$where” decision logic and enhance location
prediction by fully exploiting the temporal and categorized information. We
further propose a simple but effective spatial-constrained loss function that
acts as a self-supervised auxiliary task to incorporate geographical
information and indirectly ensure spatial consistency. We conduct extensive
experiments to confirm the effectiveness of the design. In the future, we will
assess the scalability of CSLSL in dense mobility datasets with large number
of users.

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