# Skateboard Tricks and Topological Flips

Justus Carlisle  ,  Kyle Hammer  ,  Robert Hingtgen  and  Gabriel Martins

###### Abstract.

We study the motion of skateboard flip tricks by modeling them as continuous
curves in the group $SO(3)$ of special orthogonal matrices. We show that up to
continuous deformation there are only four flip tricks. The proof relies on an
analysis of the lift of such curves to the unit 3-sphere. We also derive
explicit formulas for a number of tricks and continuous deformations between
them.

###### Summary

  1. 1 Introduction
  2. 2 Formal Framework
  3. 3 Explicit Formulas
  4. 4 Quaternions and Rotations
  5. 5 Lifting Topological Flips to the 3-sphere
  6. 6 Proof of the Four Tricks Theorem
  7. 7 The Group Structure
  8. 8 Examples of homotopies
  9. 9 Stabilizing Rotations
  10. 10 Conclusion

##  1\. Introduction

Our work is inspired by the following question. Consider two different
skateboard flip tricks, call them $F_{1}$ and $F_{2}$. Is it possible for a
skater to perform a sequence of many tricks (say one hundred tricks for
example) such that every consecutive trick is very similar to each other, and
such that the first trick in the sequence is $F_{1}$ while the last trick is
$F_{2}$? If such a sequence exists we will say that the trick $F_{1}$ can be
continuously deformed into the trick $F_{2}$.

In order to study this problem we will model a flip trick as a continuous
curve in the group $SO(3)$ of special orthogonal matrices

$SO(3)=\\{R\in M_{3\times 3}(\mathbb{R})\mid R^{T}R=I_{3},\quad\det(R)=1\\}$

above $M_{3\times 3}(\mathbb{R})$ denotes the set of $3\times 3$ matrices with
real coefficients, $R^{T}$ denotes the transpose of the matrix $R$ and $I_{3}$
denotes the $3\times 3$ identity matrix.

More formally we will say that two tricks can be continuously deformed into
each other if the curves modeling them are _homotopic_.

Before establishing the formal framework, we would like to note that in our
study of this problem we will also consider non-physical tricks. In reality, a
skateboard trick would be modeled by a curve in $SO(3)$ of class $C^{2}$ which
satisfies _Euler’s equations_ of rigid body dynamics, or equivalently, by a
locally length-minimizing curve in $SO(3)$ with respect to a certain metric in
$SO(3)$ obtained from the inertia tensor of the skateboard, as described by
Arnold in [2, 3]. Other great references for concepts of geometric mechanics
can be found in [1, 5].

We will consider a broader class of curves in $SO(3)$. We require our curves
to be continuous, but not necessarily satisfy any particular differential
equation. We will call these curves _topological flips_ , as they are a
generalization of a physical flip trick. You could think of these tricks as
movements that could be realized by a skateboard equipped with perfect jet
propulsors, which would allow one to control their movement in the air
completely.

Allowing for this more general class of curves makes the problem a lot more
flexible and tractable. The disadvantage is that it does not provide a clear
answer for the more difficult problem regarding only physical tricks. We will
come back to this point later in the text.

##  2\. Formal Framework

We now define the resting position for the skateboard in 3-space. The origin
of our coordinate system will be placed at the center of mass of the
skateboard. The $z$-axis points upwards and is perpendicular to the plane
defined by the center of the board. The $y$-axis points to the front of the
board (also called the nose of the board), in that way the $yz$-plane divides
the skateboard into two equal parts. Finally, if you align your body with the
$z$-axis while facing the positive $y$-axis, the $x$-axis points to the right
side of the $y$-axis. The skater on the board will be facing the positive
$x$-axis with their right foot on the back of the board (also called the tail
of the board). We note that with this choice, the coordinate axes agree with
the principal axes of inertia of the skateboard, so that in these coordinates
the inertia matrix would be diagonalized.

We then consider an orthonormal frame $\vec{v}_{1},\vec{v}_{2},\vec{v}_{3}$
which is locked onto the skateboard and based at its center of mass. As the
skateboard moves about 3-space during a flip trick, this frame moves along
with it. We obtain in this way a time-dependent frame
$\vec{v}_{1}(t),\vec{v}_{2}(t),\vec{v}_{3}(t)$ which will describe the
rotational movement of the skateboard. We collect this frame into a $3\times
3$ matrix having the vectors in this moving frame as columns:

$R(t)=\begin{bmatrix}\uparrow&\uparrow&\uparrow\\\
\vec{v}_{1}(t)&\vec{v}_{2}(t)&\vec{v}_{3}(t)\\\
\downarrow&\downarrow&\downarrow\end{bmatrix}$

We will assume that the trick will be performed from time $t=0$ to $t=1$. At
time $t=0$ the skateboard is at its resting position and the frame is simply:

$\vec{v}_{1}=(1,0,0),\quad\vec{v}_{2}=(0,1,0),\quad\vec{v}_{3}=(0,0,1)$

in particular $R(0)=I_{3}$ is the $3\times 3$ identity matrix.

Since this frame is orthonormal at every instant $t\in[0,1]$ the matrix $R(t)$
is always orthogonal, that is $R(t)^{T}R(t)=I_{3}$. Notice that by taking
determinants in the previous equation we find that:

$\det(R(t))^{2}=\det\big{(}R(t)^{T}\big{)}\cdot\det\big{(}R(t)\big{)}=\det\left(R(t)^{T}R(t)\right)=\det(I_{3})=1$

which implies that $\det(R(t))=\pm 1$ for all $t\in[0,1]$. We will further
assume that the motion of the skateboard is continuous, that is, we will
assume that the function $R(t)$ is continuous in time. This means that every
component of the matrix $R(t)$, or equivalently every component of the vectors
$\vec{v}_{i}(t)$ in the frame, depend continuously on the time parameter $t$.
This implies that $\det(R(t))=1$ for all $t\in[0,1]$, since
$\det(R(0))=\det(I_{3})=1$.

We have found that the rotational motion of flip tricks can be described as a
continuous curve in the group $SO(3)$ of special orthogonal matrices. There is
however one last condition that these continuous curves must satisfy. At the
end of the flip trick the skateboard must land back on its wheels, aligned
with the $y$-axis as it started.

We can see that there are only two configurations allowed for $R(1)$, either
the skateboard lands back at its resting position, in which case $R(1)=I_{3}$,
or it lands in a reversed position where the front of the skateboard (its
nose) points towards the negative direction of the $y$-axis. This is the
landing position obtained for example after the skateboard performs a 180
degree rotation about the $z$-axis. In this case one can see that $R(1)$ must
be equal to the matrix $\mathcal{O}$ defined below:

$\mathcal{O}=\begin{bmatrix}-1&0&0\\\ 0&-1&0\\\ 0&0&1\end{bmatrix}$

We summarize this discussion with the following definition.

######  Definition 1.

A topological flip is a continuous curve $R:[0,1]\to SO(3)$ such that
$R(0)=I_{3}$ and $R(1)=I_{3}$ or $\mathcal{O}$.

We are now able to discuss when two topological flips $R(t)$ and $F(t)$ can be
continuously deformed into each other. Recall that a homotopy between $R(t)$
and $F(t)$ is a continuous function $H:[0,1]\times[0,1]\to SO(3)$ such that:

  1. i)

$H(0,t)=R(t)$ for all $t\in[0,1]$

  2. ii)

$H(1,t)=F(t)$ for all $t\in[0,1]$

  3. iii)

$H(s,0)=I_{3}$ for all $s\in[0,1]$

  4. iv)

$H(s,1)=I_{3}$ or $\mathcal{O}$ for all $s\in[0,1]$

Notice that for such a homotopy between $R(t)$ and $F(t)$ to exist it must be
the case that $R(1)=F(1)$. In the case that such a homotopy exists between two
topological flips we will say that these tricks are _homotopic_ and we will
denote $R(t)\approx F(t)$.

We notice that the relation $\approx$ defines an equivalence relation on the
set of topological flips. We call an equivalence class of this relation a
_homotopy class_ and we denote the homotopy class of a given topological flip
$R(t)$ by $[R(t)]$.

Our main result is the following.

######  Theorem 1 (Four Tricks Theorem).

There are exactly four homotopy classes of topological flips. The set of
homotopy classes has a natural group structure isomorphic to the cyclic group
$\mathbb{Z}/4\mathbb{Z}$.

In the next section, we will carefully describe a number of skateboard flip
tricks. The four homotopy classes can be represented by various different
choices of tricks. The most obvious obvious choice is perhaps the one
consisting of the iterations of a shove-it.

Below we list these four tricks with their corresponding residue class in
$\mathbb{Z}/4\mathbb{Z}$.

$\begin{array}[]{rcl}0&\leftrightarrow&\text{Ollie}\\\
1&\leftrightarrow&\text{180 Shove-it}\\\ 2&\leftrightarrow&\text{360 Shove-
it}\\\ 3&\leftrightarrow&\text{540 Shove-it}\end{array}$

Another interesting choice of representatives is given by a combination of the
shove-it and the kickflip.

$\begin{array}[]{rcl}0&\leftrightarrow&\text{Ollie}\\\
1&\leftrightarrow&\text{180 Shove-it}\\\ 2&\leftrightarrow&\text{Kickflip}\\\
3&\leftrightarrow&\text{Varial Kickflip}\end{array}$

##  3\. Explicit Formulas

We will now provide explicit formulas for a few standard skateboard tricks,
recall that we have chosen the domain of the curves modeling the topological
flips to be the interval $[0,1]$.

The first trick we will define is the ollie. It consists of a simple jump of
the skater. We will represent it with the constant topological flip
$O(t)=I_{3}$.

Next we analyze the trick called the shove-it (also known as a backside 180
shove-it). This trick consists of a 180 rotation of the skateboard about the
$z$-axis where the skater pushes the tail of the board backwards with their
back foot. This means that the rotation will follow a left-handed orientation.
Since this is a rotation about the $z$-axis we have that the last vector in
the orthonormal frame is stationary $\vec{v}_{3}(t)=(0,0,1)$, while the two
first vectors $\vec{v}_{1}(t)$ and $\vec{v}_{2}(t)$ will perform a 180 degree
rotation in the clockwise direction in the $xy$-plane. We will denote the
curve in $SO(3)$ representing the shove-it by $S(t)$, it has the form:

$S(t)=\begin{bmatrix}\cos(\pi t)&\sin(\pi t)&0\\\ -\sin(\pi t)&\cos(\pi
t)&0\\\ 0&0&1\end{bmatrix}$

Since the matrix operations of multiplication and inversion are continuous, we
are able to use them to derive formulas for new topological flips (which must
always be continuous curves in $SO(3)$). For example, we can now easily write
the formula for the trick called a 360 shove-it, which consists of a 360
degree rotation about the $z$-axis with a left-hand orientation. The curve in
$SO(3)$ representing the 360 shove-it can be obtained by squaring $S(t)$:

$S(t)^{2}=\begin{bmatrix}\cos(2\pi t)&\sin(2\pi t)&0\\\ -\sin(2\pi
t)&\cos(2\pi t)&0\\\ 0&0&1\end{bmatrix}$

Similarly a frontside shove-it is a trick which consists of a 180 degree
rotation of the skateboard about the $z$-axis, where in this case the skater
pushes the tail of the board forwards with their back foot. This means that
this rotation will follow a right-handed orientation. The curve in $SO(3)$
representing the frontside shove-it can be obtained by inverting $S(t)$:

$S(t)^{-1}=\begin{bmatrix}\cos(\pi t)&-\sin(\pi t)&0\\\ \sin(\pi t)&\cos(\pi
t)&0\\\ 0&0&1\end{bmatrix}$

In a similar fashion we can express a 360 frontside shove-it as $S(t)^{-2}$, a
540 shove-it as $S(t)^{3}$, a 540 frontside shove-it as $S(t)^{-3}$ and so on.

We now move our attention to a different axis of rotation and describe the
formula for the trick called a kickflip. In this trick the skater rotates
their skateboard by 360 degrees about the $y$ axis with a left-handed
orientation, that is the board spins counter-clockwise if viewed from the
back. The curve $K(t)$ in $SO(3)$ representing the kickflip has the form:

$K(t)=\begin{bmatrix}\cos(2\pi t)&0&-\sin(2\pi t)\\\ 0&1&0\\\ \sin(2\pi
t)&0&\cos(2\pi t)\end{bmatrix}$

In a double kickflip the skateboard rotates by 720 degrees about the $y$ axis
with a left-handed orientation, the associated curve is then $K(t)^{2}$. A
heelflip is a trick where the skateboard rotates by 360 degrees about the $y$
axis with a right-handed orientation, the associated curve is $K(t)^{-1}$.

All of the previous examples consist of rotations with constant angular
velocity about a principal axis of inertia of the skateboard, therefore they
are in fact physical flips in the sense that they satisfy Euler’s equations of
rigid body motion. The situation becomes more complicated once we combine
motions about two axes simultaneously.

Consider for example the trick called a varial kickflip. This trick’s motion
can be described as a simultaneous backside 180 shove-it together with a
kickflip. We may create a curve representing this trick by using the
multiplication operation and obtain $S(t)K(t)$. However, this representation
of a varial kickflip has a number of issues. Firstly, the notion of a
“simultaneous motion of a kickflip and backside shove-it” is not well defined,
this is because the multiplication operation in $SO(3)$ is not commutative.
For example the curve $K(t)S(t)$ can also be described as a simultaneous
kickflip and backside shove-it, however its movement does not resemble
anything that a skater might call a varial kickflip. Furthermore, the
topological flip $S(t)K(t)$ is not physical in the sense that it does not
satisfy Euler’s equation of rigid body motion (and in fact this is true no
matter the choice of moments of inertia for the skateboard). However visually
the curve $S(t)K(t)$ looks very similar to a physical varial kickflip and so
we will use it to model this trick. In the future we plan to study how close
the curves obtained by multiplication are to the physical curves, but this
question is beyond the scope of the present work.

This caveat having been clarified, we can apply this multiplication technique
to obtain formulas for many known tricks. A 360 flip consists of a
simultaneous 360 backside shove-it and kickflip, it can be described as
$S(t)^{2}K(t)$. A varial heelflip is a simultaneous heelflip and frontside
shove-it, it can be described by $S(t)^{-1}K(t)^{-1}$.

As a last example we will describe the hardflip. This is a simultaneous half
kickflip (a 180 degree rotation about the axis joining the tail to the nose of
the skateboard) and a 180 degree rotation about the $x$-axis in the right-hand
orientation. If we denote the rotation about the $x$-axis by:

$U(t)=\begin{bmatrix}1&0&0\\\ 0&\cos(\pi t)&-\sin(\pi t)\\\ 0&\sin(\pi
t)&\cos(\pi t)\end{bmatrix}$

then we may describe the hardflip as $U(t)K(t/2)$.

##  4\. Quaternions and Rotations

Recall that the quaternion numbers $\mathbb{H}$ can be represented in the form

$q=q_{0}+q_{1}\textbf{i}+q_{2}\textbf{j}+q_{3}\textbf{k}$

where the numbers $\textbf{i},\textbf{j}$ and k satisfy the relations:

$\textbf{i}^{2}=\textbf{j}^{2}=\textbf{k}^{2}=-1,\
\textbf{i}\textbf{j}\textbf{k}=-1$

The real number $q_{0}$ is called the scalar part (also the real part) of $q$
and the number $q_{1}\textbf{i}+q_{2}\textbf{j}+q_{3}\textbf{k}$ is called the
vector part (also the imaginary part) of $q$. We will often think of a
quaternion as a 4-vector $(q_{0},q_{1},q_{2},q_{3})$ and identify the vector
part of a quartenion with the 3-vector $(q_{1},q_{2},q_{3})$.

The conjugate $\overline{q}$ of a quaternion $q$ is given by switching the
sign of its vector part:

$\overline{q}=q_{0}-q_{1}\textbf{i}-q_{2}\textbf{j}-q_{3}\textbf{k}$

In general, conjugation reverses the order of quaternionic multiplication:

$\overline{pq}=\overline{q}\cdot\overline{p}$

As with complex numbers, the real and imaginary parts of a quaternion can be
written in terms of a quaternion and its conjugate,

$\text{Re}(q)=\frac{q+\overline{q}}{2},\hskip
28.45274pt\text{Im}(q)=\frac{q-\overline{q}}{2}.$

Another similarity between the complex and quaternion numbers is that we may
write the modulus of a quaternion as a product with its conjugate,

$q\overline{q}=\overline{q}q=q_{0}^{2}+q_{1}^{2}+q_{2}^{2}+q_{3}^{2}=|q|^{2}.$

as a consequence the inverse of a nonzero quaternion is given by

$q^{-1}=\frac{\overline{q}}{|q|^{2}}.$

Let us now consider the conjugation action of a fixed nonzero quaternion,
$q\neq 0$, on purely imaginary quaternions $\mathbf{v}$ given by
$q\mathbf{v}q^{-1}$. As $\mathbf{v}$ is a purely imaginary quaternion, we have
that $\overline{\mathbf{v}}=-\mathbf{v}$. When computing the conjugate of
$q\mathbf{v}q^{-1}$, we find that

$\overline{q\mathbf{v}q^{-1}}=\overline{q^{-1}}\cdot\overline{\mathbf{v}}\cdot\overline{q}=\frac{q}{|q|^{2}}(-\mathbf{v})\overline{q}=-q\mathbf{v}q^{-1}.$

With this it is clear that $\text{Re}(q\mathbf{v}q^{-1})=0$, that is
$q\mathbf{v}q^{-1}$ is itself a purely imaginary quaternion.

Furthermore, notice that

$\displaystyle q(\mathbf{u}+\mathbf{v})q^{-1}$
$\displaystyle=q\mathbf{u}q^{-1}+q\mathbf{v}q^{-1}$ $\displaystyle
q(\alpha\mathbf{v})q^{-1}$ $\displaystyle=\alpha q\mathbf{v}q^{-1}$

for every purely imaginary quaternions $\mathbf{u}$, $\mathbf{v}$ and scalar
$\alpha$. It follows that the conjugation action is real linear. By
identifying the purely imaginary quaternions with $\mathbb{R}^{3}$ we obtain a
map taking nonzero quaternions to linear maps on $\mathbb{R}^{3}$.

$\rho:\mathbb{H}-\\{0\\}\to L(\mathbb{R}^{3}),\ \
\rho(q)\mathbf{v}=q\mathbf{v}q^{-1}$

Notice that because $q^{-1}=\overline{q}/|q|$, the components of the matrix
associated to $\rho(q)$ can be expressed as a quadratic polynomial on $q$
divided by the function $|q|$, so that $\rho$ is a continuous map. Furthermore
$\rho(qp)=\rho(q)\rho(p)$, so that $\rho$ is a continuous group homomorphism.

A unit quaternion is a quaternion with unit length, i.e
$|q|^{2}=q\overline{q}=1$. Geometrically the set of unit quaternions is
identified with the unit 3-sphere $S^{3}$ inside of $\mathbb{R}^{4}$. Given a
unit quaternion $q$ and an imaginary quaternion $\mathbf{v}$, notice that:

$|\rho(q)\mathbf{v}|=|q\mathbf{v}q^{-1}|=|q||\mathbf{v}||q^{-1}|=|\mathbf{v}|$

Since $\rho(q)$ leaves the norm of vectors invariant, it must preserve the
inner product structure on $\mathbb{R}^{3}$ so that the matrix associated to
$\rho(q)$ is orthogonal. Furthermore since $\rho$ is continuous and
$\det(\rho(q))=\pm 1$ for $q\in S^{3}$, it must be the case that
$\det(\rho(q))=1$ for all $q$ in $S^{3}$, as $\rho(1)=I_{3}$. We may then
restrict $\rho$ to the unit 3-sphere and obtain a map $\rho:S^{3}\to SO(3)$.

Given a unit quaternion $q$, we may write $q=\cos\theta+\sin\theta\mathbf{u}$
for some unit imaginary quaternion $\mathbf{u}$. A somewhat long computation
allows us to write the following Rodriguez formula for a given imaginary
quaternion $\mathbf{v}$:

$\rho(q)\mathbf{v}=q\mathbf{v}q^{-1}=\cos(2\theta)\mathbf{v}_{\perp}+\sin(2\theta)(\mathbf{u}\times\mathbf{v})+\mathbf{v}_{\parallel}$

where $\mathbf{v}_{\parallel}=(\mathbf{v}\cdot\mathbf{u})\mathbf{u}$ and
$\mathbf{v}_{\perp}=\mathbf{v}-\mathbf{v}_{\parallel}$. This formula shows
that $\rho(q)$ can be described geometrically as a right-handed rotation of
angle $2\theta$ about the axis of rotation $\mathbf{u}$. Since every special
orthogonal transformation in 3-space is a rotation about a certain invariant
axis, we obtain the following:

######  Corollary 1.

The map $\rho:S^{3}\to SO(3)$ is surjective.

For the sake of completeness, we will also mention that in the coordinates
$q=q_{0}+q_{1}\textbf{i}+q_{2}\textbf{j}+q_{3}\textbf{k}$, the image of a unit
quaternion $q$ under the map $\rho$ is given by the following formula:

$\rho(q)=\left[\begin{array}[]{ccc}1-2(q_{2}^{2}+q_{3}^{2})&2(q_{1}q_{2}-q_{3}q_{0})&2(q_{1}q_{3}+q_{0}q_{2})\\\
2(q_{1}q_{2}+q_{3}q_{0})&1-2(q_{1}^{2}+q_{3}^{2})&2(q_{2}q_{3}-q_{1}q_{0})\\\
2(q_{1}q_{3}-q_{2}q_{0})&2(q_{2}q_{3}+q_{0}q_{1})&1-2(q_{1}^{2}+q_{2}^{2})\end{array}\right]$

##  5\. Lifting Topological Flips to the 3-sphere

In this section we will show that the map $\rho:S^{3}\to SO(3)$ is a covering
map. This will allow us to lift topological flips and homotopies between them
to the 3-sphere. We will then explore the simpler geometry of the sphere to
obtain a few explicit homotopies between different topological flips.

Recall that a map $\pi:X\to B$ between topological spaces is called a covering
map if for any point $b\in B$ there is an open neighborhood $V$ of $b$ such
that its preimage $\pi^{-1}(V)$ is a union of disjoint open sets
$\pi^{-1}(V)=\coprod_{\alpha\in A}U_{\alpha}$ and such that the restrictions
of $\pi$ to the open sets $U_{\alpha}$ denoted by
$\pi\mid_{U_{\alpha}}:U_{\alpha}\to V$ are homeomorphisms. The space $B$ is
called the base space and $X$ is called the covering space.

######  Proposition 1.

The map $\rho:S^{3}\to SO(3)$ is a covering map.

###### Proof.

We will first establish a small lemma.

######  Lemma 1.

Let $A$ be a matrix in $SO(3)$ such that $A=\rho(q)$. Then
$\rho^{-1}(A)=\\{q,-q\\}$.

###### Proof.

Notice that if $p\in S^{3}$ is a different quaternion such that $\rho(p)=A$,
then we know that for every imaginary quaternion $\mathbf{v}$ we have that:

$p\mathbf{v}p^{-1}=q\mathbf{v}q^{-1}$

which is equivalent to the equation $q^{-1}p\mathbf{v}=\mathbf{v}q^{-1}p$.
This means that the unit quaternion $q^{-1}p$ commutes with any imaginary
quaternion, and in fact it must commute with every quaternion since real
quaternions belong to the center of the quaternion algebra. Furthermore, since
the center of the quaternion algebra is in fact equal to the real quaternions,
we find that $q^{-1}p$ is real. Since $q^{-1}p$ is a unit quaternion we may
conclude that $q^{-1}p=\pm 1$, hence $p=\pm q$ which establishes the lemma. ∎

We now show that $\rho$ must be an open map. Since $S^{3}$ is compact and
$SO(3)$ is Hausdorff, $\rho$ is a closed map. Now given an open set $U\subset
S^{3}$, let:

$\widehat{U}=\\{q\in S^{3}\mid q\in U\text{ or }-q\in U\\}.$

Then notice that the surjectivity of $\rho$ allows us to write:

$SO(3)-\rho(U)=SO(3)-\rho(\widehat{U})=\rho(S^{3}-\widehat{U}).$

Since $\rho$ is a closed map, $\rho(S^{3}-\widehat{U})$ must be closed,
therefore $\rho(U)$ must be open.

Now let $A_{0}$ be any matrix in $SO(3)$ and let
$\rho^{-1}(A_{0})=\\{q_{0},-q_{0}\\}$. Consider the neighborhoods:

$\displaystyle U_{+}$ $\displaystyle=\\{q\in S^{3}\mid|q-q_{0}|<1\\}$
$\displaystyle U_{-}$ $\displaystyle=\\{q\in S^{3}\mid|q-(-q_{0})|<1\\}$

Notice that $q\in U_{+}\Rightarrow-q\in U_{-}$ and $q\in U_{-}\Rightarrow-q\in
U_{+}$. Let’s then define the open neighborhood $V=\rho(U_{+}\cup
U_{-})=\rho(U_{\pm})$ of $A_{0}$. Clearly we have that $\rho^{-1}(V)=U_{+}\cup
U_{-}$, and since $\rho$ is an open map, $V$ is an open subset of $SO(3)$. It
still remains to show that $U_{+}$ and $U_{-}$ are disjoint and that the
restrictions of $\rho$ to $U_{+}$ and $U_{-}$ are homeomorphisms.

We now show that $U_{+}$ and $U_{-}$ are disjoint. Suppose by contradiction
that there is a unit quaternion $q\in U_{+}\cap U_{-}$, then notice that:

$2=2|q|=|q-(-q)|=|q-q_{0}+q_{0}-(-q)|\leq|q-q_{0}|+|q_{0}-(-q)|<1+1=2$

which is a contradiction, so that $U_{+}$ and $U_{-}$ are disjoint.

Finally we consider the restrictions $\rho\mid_{U_{\pm}}:U_{\pm}\to V$ which
are clearly continuous bijective functions. Since $\rho:S^{3}\to SO(3)$ is an
open map, and since $U_{\pm}$ are open sets, we may conclude that the
restrictions $\rho\mid_{U_{\pm}}$ are also open maps and therefore
homeomorphisms, which finishes our proof. ∎

There are many interesting theorems that relate continuous curves in the base
space to continuous curves in the covering space. Let $\pi:X\to B$ be a
covering map and $c:[a,b]\to B$ be a continuous curve in the base space. A
lift of $c$ is a curve $\gamma:[a,b]\to X$ in the covering space such that
$\pi(\gamma(t))=c(t)$. A crucial theorem regarding lifts of curves is the
following (see [4] Theorem 1.7 (a)).

######  Theorem 2.

Let $c:[a,b]\to B$ be a continuous curve starting at a point $p$. Given a
choice of point $x\in\pi^{-1}(p)$, there exists a unique lift $\gamma:[a,b]\to
X$ of $c$ such that $\gamma(a)=x$.

This theorem allows us to relate topological flips to a certain family of
curves in the 3-sphere $S^{3}$. Since $\rho^{-1}(I_{3})=\\{1,-1\\}$ and
$\rho^{-1}(\mathcal{O})=\\{\textbf{k},-\textbf{k}\\}$ we may conclude that the
lift of a topological flip based at $1$ must be a continuous curve
$\gamma:[0,1]\to S^{3}$ such that $\gamma(0)=1$ and
$\gamma(1)\in\\{1,\textbf{k},-1,-\textbf{k}\\}$. We call such a curve in the
3-sphere a quaternionic flip. We obtain a one-to-one correspondence:

$\Big{\\{}\text{ Topological flips }\Big{\\}}\leftrightarrow\Big{\\{}\text{
Quaternionic flips }\Big{\\}}$

Notice that $\\{1,\mathbf{k},-1,-\mathbf{k}\\}$ is the subgroup of $S^{3}$
generated by $\mathbf{k}$, we will denote it by $\langle\mathbf{k}\rangle$.

Another interesting theorem concerns the lift of homotopies in the base $B$ to
homotopies in the covering space $X$ (see [4] Theorem 1.7 (b)).

######  Theorem 3.

Let $H:[0,1]\times[a,b]\to B$ be a homotopy in $B$ such that $H(s,a)=p$ for
all $s$ in $[0,1]$. Given a choice of point $x\in\pi^{-1}(p)$ there is a
unique homotopy $\tilde{H}:[0,1]\times[a,b]\to X$ such that
$\pi(\tilde{H}(s,t))=H(s,t)$ and $\tilde{H}(s,a)=x$ for all $s$ in $[0,1]$.

A consequence of this theorem is that we may obtain a one-to-one
correspondence between the set $[TF]$ of homotopy classes of topological flips
and the set $[QF]$ of homotopy classes of quaternionic flips.

$\Big{\\{}\text{ Topological flips
}\Big{\\}}\Big{/}\approx\;\leftrightarrow\;\Big{\\{}\text{ Quaternionic flips
}\Big{\\}}\Big{/}\approx$

We will denote this “lifting bijection” by $\lambda:[TF]\to[QF]$. With only a
few more results from algebraic topology we will soon be able to prove the
four tricks theorem.

Let’s now analyze the quaternionic lifts of the tricks described in section 3.
Recall that the shove-it $S(t)$ consists of a left-handed 180 degree rotation
of the skateboard about the $z$-axis. By using the Rodriguez formula we may
compute its quaternionic lift:

$\sigma(t)=\cos\left(\frac{\pi t}{2}\right)-\sin\left(\frac{\pi
t}{2}\right)\textbf{k}$

Since the $\mathbf{i}$ component of $\sigma$ vanishes, we may visualize the
curve $\sigma(t)$ by plotting its image in the 3-dimensional subspace
generated by $1$, $\mathbf{j}$ and $\mathbf{k}$. The intersection of the unit
3-sphere with this subspace will form a 2-dimensional sphere, see figure 1.
For aesthetic reasons we will use the coordinates $x-y\mathbf{j}-z\mathbf{k}$
in our plots. Please notice the negative signs in the $y$ and $z$ coordinates.

(a) $\sigma$ curve

(b) $\sigma^{2}$ curve

(c) $\sigma^{3}$ curve

Figure 1. Quaternionic lifts of the powers of a shove-it

We notice that $\sigma(1)=-\mathbf{k}$. The 360 shove-it can be lifted to the
curve:

$\sigma(t)^{2}=\cos\left(\pi t\right)-\sin\left(\pi t\right)\mathbf{k}$

The endpoint of this lift is $\sigma(1)^{2}=-1$. Similarly the endpoint of the
lift of a 540 shove-it is $\sigma(1)^{3}=\mathbf{k}$. We will show that
because these tricks exhaust the possibilities of the endpoints of
quaternionic lifts, they will (together with the ollie) exhaust the homotopy
classes of quaternionic flips. As a result their associated topological flips
will in turn exhaust the homotopy classes of topological flips, which is the
content of the main theorem.

(a) $\kappa$ curve

(b) $\kappa^{2}$ curve

(c) $\kappa^{-1}$ curve

Figure 2. Quaternionic lifts of the powers of a kickflip

Recall that the kickflip $K(t)$ consists of a left-handed 360 degree rotation
of the skateboard about the $y$-axis. Its quaternionic lift is:

$\kappa(t)=\cos(\pi t)-\sin(\pi t)\mathbf{j}.$

The lift of the double-kickflip can be described by $\kappa(t)^{2}=\cos(2\pi
t)-\sin(2\pi t)\mathbf{j}$. The heelflip, which is the kickflip with a
reversed rotation, that is, a right-handed 360 degree rotation about the
$y$-axis, has the quaternionic lift $\kappa(t)^{-1}=\cos(\pi t)+\sin(\pi
t)\mathbf{j}$.

These three curves all have vanishing $\mathbf{i}$ component, so we are able
to visualize them by using the same 3-dimensional subspace as before, see
figure 2.

The varial kickflip was modeled by $S(t)K(t)$. Because $\rho$ is a group
homomorphism, we deduce that its quaternionic lift is given by
$\sigma(t)\kappa(t)$. This curve does not remain in the 3-dimensional subspace
generated by $1$, $\mathbf{j}$ and $\mathbf{k}$, however the orthogonal
projection onto this subspace is injective when restricted to this curve, so
that we are still able to obtain a good visualization of this curve. We will
also renormalize the curve’s projection so that it remains on the surface of
the unit 2-sphere, see figure 3.

Figure 3. Renormalized projection of the quaternionic varial kickflip

Lastly we consider the hardflip. The right-handed 180 degree rotation about
the $x$-axis $U(t)$ has the quaternionic lift:

$\upsilon(t)=\cos\left(\frac{\pi t}{2}\right)+\sin\left(\frac{\pi
t}{2}\right)\mathbf{i}$

We then obtain that the quaternionic lift of the hardflip is
$\upsilon(t)\kappa(t/2)$.

##  6\. Proof of the Four Tricks Theorem

In order to prove the four tricks theorem it will suffice to show that $[QF]$
has cardinality equal to four. In fact we will show that the following map:

$\begin{array}[]{crcl}\text{ev}_{1}:&[QF]&\to&\langle\textbf{k}\rangle\\\
&\gamma&\mapsto&\gamma(1)\end{array}$

is a bijection. Furthermore we will later describe a geometrically defined
group structure on $[TF]$ which will make the composition
$\text{ev}_{1}\circ\lambda$ into a group homomorphism.

In order to do this, first recall that given a topological space $X$, and two
continuous curves $f$ and $g$ defined on the interval $[0,1]$ such that
$f(1)=g(0)$, we may define another continuous curve called their
_concatenation_ by:

$(f\\#g)(t)=\begin{cases}f(2t)\text{ for }t\in[0,1/2]\\\ g(2t-1)\text{ for
}t\in[1/2,1]\end{cases}$

Furthermore given a continuous curve $f$, we denote by $\bar{f}$ the curve
which traverses $f$ in the opposite direction, that is $\bar{f}(t)=f(1-t)$.

Recall that a loop $f$ in $X$ based at a point $p\in X$ is a continuous curve
defined on the interval $[0,1]$ such that $f(0)=f(1)=p$. A loop based at $p$
is contractible if it is homotopic to the constant loop at $p$.

######  Proposition 2.

Let $X$ be a topological space, $p$ and $q$ be two points in $X$, and let $f$
and $g$ be two continuous curves defined on $[0,1]$ from $p$ to $q$. Then $f$
and $g$ are homotopic if and only if the loop $f\\#\bar{g}$ is contractible.

###### Proof.

Suppose that $f$ and $g$ are homotopic, then $f\\#\bar{g}$ is homotopic to
$g\\#\bar{g}$ which is contractible via the homotopy:

$H(s,t)=\begin{cases}g(2st)\text{ for }t\in[0,1/2]\\\ g(2s(1-t))\text{ for
}t\in[1/2,1]\end{cases}$

Since $H(0,t)=p$ for all $t\in[0,1]$ and $H(1,t)=g\\#\bar{g}(t)$.

Conversely, suppose that $f\\#\bar{g}$ is contractible. Since $\bar{g}\\#g$ is
contractible we know that:

$[f]=[f\\#(\bar{g}\\#g)]=[(f\\#\bar{g})\\#g]=[g]$

∎

We may now prove the four tricks theorem by using the fact that in the
3-sphere every loop is contractible (that it $S^{3}$ is simply-connected).

###### Proof of the four tricks theorem.

We show that the map $\text{ev}_{1}$ is bijective. We start by showing that it
is injective. Let $f$ and $g$ be two quaternionic flips such that
$\text{ev}_{1}([f])=\text{ev}_{1}([g])$. Since $f(1)=g(1)$ we may consider the
loop $f\\#\bar{g}$. Because the 3-sphere is simply-connected this loop is
contractible, so that $[f]=[g]$.

In order to show that $\text{ev}_{1}$ is surjective, recall that the
quaternionic lift of the shove-it $S(t)$ is given by $\sigma(t)=\cos(\pi
t/2)-\sin(\pi t/2)\textbf{k}$, so that $\text{ev}_{1}([\sigma])=-\textbf{k}$.
Furthermore, given any integer $n\in\mathbb{Z}$ we may consider the
quaternionic flip $\sigma(t)^{n}$ and
$\text{ev}_{1}([\sigma^{n}])=(-\textbf{k})^{n}$, so that $\text{ev}_{1}$ is
surjective.

∎

##  7\. The Group Structure

We may define a group structure in the set $[TF]$ of homotopy classes of
topological flips in much the same way as one does with the fundamental group
of a topological space. The difference here is that if a topological flip $f$
lands at the configuration $\mathcal{O}$, that is if $f(1)=\mathcal{O}$, we
are not able to concatenate $f$ with another topological flip $g$, since $g$
must start at the configuration $I_{3}$. We may however concatenate the curve
$f$ with the trick $g_{\mathcal{O}}(t)=g(t)\cdot\mathcal{O}$ which is a curve
that models a trick which looks almost identical to $g$, but at the start of
the trick the skater has their back foot resting on the nose of the skateboard
instead of its tail (the tail would then point towards the positive $y$-axis).
If the skate is symmetric in the sense that its front side is exactly equal to
its back side, then the trick $g_{\mathcal{O}}$ would be visually
indistinguishable from $g$.

######  Definition 2.

Let $[f]$ and $[g]$ be homotopy classes of two topological flips, we define
their multiplication by:

$[f]\ast[g]=\begin{cases}[f\\#g]&\text{ \emph{if} }f(1)=I_{3}\\\
[f\\#g_{\mathcal{O}}]&\text{ \emph{if} }f(1)=\mathcal{O}\end{cases}$

Remark: If we allowed our flip tricks to start at the position $\mathcal{O}$
(with the back foot resting on the nose of the skateboard) we could use the
coset space $SO(3)/\langle\mathcal{O}\rangle$ as our configuration space
instead of $SO(3)$. We have not used that space in this text for two reasons.
First: its elements would no longer be represented by matrices but cosets
which is a bit inconvenient. Second: it does not have a natural group
structure (since $\langle\mathcal{O}\rangle$ is not a normal subgroup of
$SO(3)$), which would make defining more complex topological tricks more
cumbersome. There is an advantage of working in the space
$SO(3)/\langle\mathcal{O}\rangle$ however, in that case the topological flips
can be identified with loops in $SO(3)/\langle\mathcal{O}\rangle$ and the
group structure on the homotopy classes of topological flips is perhaps more
“standard”, as it would agree with the operation of the fundamental group
$\pi_{1}(SO(3)/\langle\mathcal{O}\rangle)$.

We will omit the proof that this group operation is well defined (it does not
depend on the choice of representative of the homotopy classes and satisfies
the axioms of a group), as it is very similar to the argument used for the
operation of the fundamental group of a topological space (see [4] Proposition
1.3).

Recall that $\lambda$ is the “lifting bijection” $\lambda:[TF]\to[QF]$. We now
define the map $\Lambda=\text{ev}_{1}\circ\lambda$:

$\begin{array}[]{crcl}\Lambda:&[TF]&\to&\langle\textbf{k}\rangle\end{array}$

We will now show that this map is compatible with the group structure that we
have defined on $[TF]$.

######  Proposition 3.

The map $\Lambda$ is a group isomorphism.

###### Proof.

We have already shown that both $\lambda$ and $\text{ev}_{1}$ are bijective,
so that $\Lambda$ is bijective. We must then show that $\Lambda$ is a group
homomorphism.

We have seen that any quaternionic flip is homotopic to a flip of the form
$\sigma(t)^{n}$ for some integer $n\in\mathbb{Z}$. Because $\lambda$ is
bijective, we conclude that any topological flip is homotopic to a flip of the
form $S(t)^{n}$. Since there are only four homotopy classes in total, it
suffices to prove that $[S(t)]^{n}=[S(t)^{n}]$ for integers $0\leq n\leq 3$.
This would imply that
$\Lambda([S(t)]^{n})=\Lambda([S(t)^{n}])=\text{ev}_{1}(\sigma(t)^{n})=(-\textbf{k})^{n}$,
so that $\Lambda$ will in fact be a group isomorphism.

The cases $n=0$ and $n=1$ are trivial. We then start by studying the case
$n=2$. We see that $[S]^{2}=[S\\#S_{\mathcal{O}}]$, as $S(1)=\mathcal{O}$.
Notice that for $t\in[0,1/2]$ we have:

$S\\#S_{\mathcal{O}}(t)=S(2t)=\begin{bmatrix}\cos(2\pi t)&\sin(2\pi t)&0\\\
-\sin(2\pi t)&\cos(2\pi t)&0\\\ 0&0&1\end{bmatrix}$

And for $t\in[1/2,1]$:

$\displaystyle S\\#S_{\mathcal{O}}(t)$
$\displaystyle=S_{\mathcal{O}}(2t-1)=S(2t-1)\cdot\mathcal{O}$
$\displaystyle=\begin{bmatrix}\cos(2\pi t-\pi)&\sin(2\pi t-\pi)&0\\\
-\sin(2\pi t-\pi)&\cos(2\pi t-\pi)&0\\\
0&0&1\end{bmatrix}\cdot\begin{bmatrix}-1&0&0\\\ 0&-1&0\\\ 0&0&1\end{bmatrix}$
$\displaystyle=\begin{bmatrix}-\cos(2\pi t)&-\sin(2\pi t)&0\\\ \sin(2\pi
t)&-\cos(2\pi t)&0\\\ 0&0&1\end{bmatrix}\cdot\begin{bmatrix}-1&0&0\\\
0&-1&0\\\ 0&0&1\end{bmatrix}$ $\displaystyle=\begin{bmatrix}\cos(2\pi
t)&\sin(2\pi t)&0\\\ -\sin(2\pi t)&\cos(2\pi t)&0\\\ 0&0&1\end{bmatrix}$

We conclude that for all $t\in[0,1]$ we have
$S\\#S_{\mathcal{O}}(t)=S(t)^{2}$, so that in particular at the level of
homotopy classes $[S]^{2}=[S\\#S_{\mathcal{O}}]=[S^{2}]$.

We now show that $[S(t)]^{3}=[S(t)^{3}]$. We notice that
$[S(t)]^{3}=[S(t)]^{2}\ast[S(t)]=[S^{2}\\#S(t)]$. We may “spread out” both the
360 shove-it $S^{2}$ and the shove-it $S$, so that they take place
simultaneously along the entire interval $[0,1]$, instead of only in one half
of the interval. We can then deform the curve $S^{2}\\#S(t)$ into the curve
$S^{3}(t)$.

Consider the homotopy:

$F(s,t)=\begin{cases}S^{2}\big{(}2t/(s+1)\big{)}&\text{ for }0\leq
t\leq(s+1)/2\\\ I_{3}&\text{ otherwise}\end{cases}$

When $s=0$ the curve $F(0,t)$ performs a 360 shove-it during the interval
$[0,1/2]$ and is constant at $I_{3}$ on the interval $[1/2,1]$, while when
$s=1$ the curve is simply equal to a 360 shove-it, $F(1,t)=S^{2}(t)$.

We may do something similar for the shove-it

$G(s,t)=\begin{cases}S\big{(}(2t+s-1)/(s+1)\big{)}&\text{ for }(1-s)/2\leq
t\leq 1\\\ I_{3}&\text{ otherwise}\end{cases}$

We then consider the homotopy $H(s,t)=F(s,t)\cdot G(s,t)$. We see that
$H(0,t)=S^{2}\\#S(t)$ and $H(1,t)=S(t)^{3}$, so that
$[S]^{3}=[S^{2}\\#S]=[S^{3}]$ which finishes the proof.

∎

We then obtain the simple corollary:

######  Corollary 2.

The group $[TF]$ is isomorphic to the cyclic group of order four
$\mathbb{Z}/4\mathbb{Z}$.

##  8\. Examples of homotopies

In this section we describe a few explicit homotopies between a number of
different tricks. Recall that the double kickflip is a trick where the
skateboard performs two full left-handed rotations about the y-axis:

$K(t)^{2}=\begin{bmatrix}\cos(4\pi t)&0&-\sin(4\pi t)\\\ 0&1&0\\\ \sin(4\pi
t)&0&\cos(4\pi t)\end{bmatrix}$

Its quaternionic lift is:

$\kappa^{2}(t)=\cos(2\pi t)-\sin(2\pi t)\textbf{j}$

Notice that since $\kappa^{2}(1)=1$ we know that $\kappa^{2}$ is homotopic to
the constant loop based at $1$. We may construct this homotopy geometrically
by noticing that the curve $\kappa^{2}(t)$ describes a great circle in the
3-sphere. We may deform it to the constant loop $O(t)=I_{3}$ by contracting
this great circle through the upper hemisphere of $S^{3}$. We then obtain the
following homotopy $H_{O\to K^{2}}:[0,1]\times[0,1]\to SO(3)$:

$H_{O\to K^{2}}(s,t)=\rho\big{(}1+s(\cos(2\pi t)-1),0,-s\sin(2\pi
t),-\sqrt{2s(1-s)(1-\cos(2\pi t))}\,\big{)}$

This homotopy remains in the 3-subspace generated by $1,\mathbf{j}$ and
$\mathbf{k}$ so we may visualize it as before. See figure 4.

Figure 4. Contraction of the double kickflip

Because the double kickflip $K^{2}(t)$ is homotopic to the constant loop
$O(t)$ we know that $2[K]=[K^{2}]=[O]=0$, and so $[K]=-[K]$. This means that
there is a deformation between the kickflip $K(t)$ and the reversed trick, the
heelflip $K(1-t)=K(t)^{-1}$. We have seen that their quaternionic lifts form
great semicircles in the plane generated by $1$ and $\mathbf{j}$. We can
deform one semicircle into the other by rotating them about the k axis, see
figure 5. We obtain the homotopy $H_{K\to K^{-1}}:[0,1]\times[0,1]\to SO(3)$:

$H_{K\to K^{-1}}(s,t)=\rho\big{(}\cos(\pi t),0,-\sin(\pi t)\cos(\pi
s),-\sin(\pi t)\sin(\pi s)\big{)}$

Figure 5. Deformation from a kickflip to a heelflip

Notice that in the deformation above as the parameter $s$ moves from $0$ to
$1/2$ we may find a homotopy between the kickflip and the 360 shove-it. The
quaternionic lift of both of these tricks form a great circle in the 3-sphere,
as the parameter $s$ increases, the axis of rotation $-\mathbf{j}$ of the
kickflip continuously moves towards the axis of rotation $-\mathbf{k}$ of the
360 shove-it. By rescaling the parameter $s$ so that it lies between $0$ and
$1$, we obtain the following homotopy:

$H_{K\to S^{2}}(s,t)=\rho\big{(}\cos(\pi t),0,-\sin(\pi t)\cos(\pi
s/2),-\sin(\pi t)\sin(\pi s/2)\big{)}$

As a last example we can use the multiplication in $SO(3)$ to deform the
varial kickflip $V(t)=S(t)K(t)$ into a 540 shove-it $S(t)^{3}$ by simply
multiplying the previous homotopy by $S(t)$. We obtain $H_{V\to
S^{3}}(s,t)=S(t)H_{K\to S^{2}}(s,t)$, see figure 6.

Figure 6. Deformation from a varial kickflip to a 540 shove-it

##  9\. Stabilizing Rotations

We can see that any topological flip can be deformed into a rotation about the
k axis with constant angular velocity. In this section we construct one more
interesting homotopy by exploring the geometry of the 3-sphere. This will be a
deformation between a somewhat generic topological flip and one which rotates
with constant angular velocity about a fixed axis of rotation. We then study
an example consisting of a shove-it with a wobbling axis of rotation and make
use of this homotopy to stabilize its axis and deform it into the standard
shove-it.

Consider a unit vector $\mathbf{u}$ in $\mathbb{R}^{3}$ and let
$P^{\perp}_{\mathbf{u}}$ be the 2-plane perpendicular to $\mathbf{u}$. We may
also identify $P^{\perp}_{\mathbf{u}}$ with the corresponding subset of the
imaginary quaternions, we then consider the circle
$C^{\perp}_{\mathbf{u}}=\rho(S^{3}\cap P^{\perp}_{\mathbf{u}})$ inside of
$SO(3)$. We also define the circle $C^{\parallel}_{\mathbf{u}}=\rho(S^{3}\cap
P^{\parallel}_{\mathbf{u}})$ where $P^{\parallel}_{\mathbf{u}}$ is the 2-plane
in $\mathbb{H}$ generated by $1$ and $\mathbf{u}$.

Finally we let $\mathbf{a},\mathbf{b}$ be an orthonormal basis of
$P^{\perp}_{\mathbf{u}}$ and consider the orthonormal basis
$\beta_{\mathbf{u}}=(\mathbf{a},\mathbf{b},\mathbf{u})$ of $\mathbb{R}^{3}$.

######  Proposition 4.

Let $A:[0,1]\to SO(3)$ be a continuous curve such that $A(0)$ and $A(1)$
belong to $C^{\parallel}_{\mathbf{u}}$, and suppose that $A$ does not
intersect the circle $C^{\perp}_{\mathbf{u}}$. There are numbers
$a\in\mathbb{R}$ and $\varphi\in[0,2\pi)$ such that $A(t)$ is homotopic to the
curve $S_{a,\varphi}(t)$ which in the basis $\beta_{\mathbf{u}}$ is given by:

$\Big{[}S_{a,\varphi}(t)\Big{]}_{\beta_{\mathbf{u}}}=\begin{bmatrix}\cos(at+\varphi)&-\sin(at+\varphi)&0\\\
\sin(at+\varphi)&\cos(at+\varphi)&0\\\ 0&0&1\end{bmatrix}$

###### Proof.

We will use the following coordinates for the quaternions:

$q=x_{0}+x_{1}\mathbf{a}+x_{2}\mathbf{b}+x_{3}\mathbf{u}$

Let $\Gamma^{\perp}_{\mathbf{u}}=S^{3}\cap P^{\perp}_{\mathbf{u}}$ and
$\Gamma^{\parallel}_{\mathbf{u}}=S^{3}\cap P^{\parallel}_{\mathbf{u}}$, so
that $\Gamma^{\perp}_{\mathbf{u}}=\rho^{-1}(C^{\perp}_{\mathbf{u}})$ and
$\Gamma^{\parallel}_{\mathbf{u}}=\rho^{-1}(C^{\parallel}_{\mathbf{u}})$. Using
our coordinates we may write:

$\Gamma^{\perp}_{\mathbf{u}}=\left\\{q\in S^{3}\mid
x_{0}^{2}+x_{3}^{2}=0\right\\}.$

Since over the open set $D_{\mathbf{u}}=S^{3}-\Gamma^{\perp}_{\mathbf{u}}$ of
$S^{3}$ we have the inequality $x_{0}^{2}+x_{3}^{2}\neq 0$, we are able to
define a homotopy $F:[0,1]\times D_{\mathbf{u}}\to D_{\mathbf{u}}$ given by

$F(s,q)=\left(\sqrt{n(s,q)}\cdot x_{0},\sqrt{1-s}\cdot x_{1},\sqrt{1-s}\cdot
x_{2},\sqrt{n(s,q)}\cdot x_{3}\right)$

where the function $n:[0,1]\times D_{\mathbf{u}}\to\mathbb{R}_{\geq 0}$ is
given by:

$n(s,q)=\frac{1-(1-s)(x_{1}^{2}+x_{2}^{2})}{x_{0}^{2}+x_{3}^{2}}.$

Denote $F_{s}(q)=F(s,q)$. Notice that when $s=0$ the function $F_{0}$ is the
identity map on $D_{\mathbf{u}}$, and when $s=1$ the function $F_{1}$ maps
$D_{\mathbf{u}}$ onto the circle $\Gamma^{\parallel}_{\mathbf{u}}$. Moreover,
for any value of the parameter $s$, the map $F_{s}$ leaves the points on the
circle $\Gamma^{\parallel}_{\mathbf{u}}$ fixed. The function $F$ is called a
_deformation retraction_ of $D_{\mathbf{u}}$ onto
$\Gamma^{\parallel}_{\mathbf{u}}$.

Now let $\alpha:[0,1]\to S^{3}$ be a quaternionic lift of $A$. We may define a
homotopy $\hat{F}:[0,1]\times[0,1]\to SO(3)$ by:

$\hat{F}(s,t)=\rho\circ F(s,\alpha(t)).$

This is a homotopy from $A(t)$ to a curve entirely contained in
$C^{\parallel}_{\mathbf{u}}$.

At this stage, we know that $\alpha(t)$ is homotopic to a curve of the form

$\alpha_{1}(t)=\cos(\theta(t))+\sin(\theta(t))\textbf{u}$

where $\theta(t):[0,1]\to\mathbb{R}$ is a continuous function. All that is
left for us to do is to deform this curve into one which has constant angular
velocity, that is, a curve where the rotation angle is linear in the time $t$.
We may accomplish this by using a convex combination:

$\gamma(s,t)=(1-s)\theta(t)+s(a_{0}t+\varphi_{0})$

where $a_{0}=\theta(1)-\theta(0)$ and $\varphi_{0}=\theta(0)$. We proceed by
defining the following homotopy $G:[0,1]\times[0,1]\to SO(3)$:

$G(s,t)=\rho\big{(}\cos(\gamma(s,t))+\sin(\gamma(s,t))\textbf{u}\big{)}$

We see that $G(1,t)$ is equal to the curve

$\cos(a_{0}t+\varphi_{0})+\sin(a_{0}t+\varphi_{0})\textbf{u}$

By concatenating the homotopies $\hat{F}$ and $G$, we find the homotopy
claimed in the proposition, where by the Rodriguez formula we must have
$a=a_{0}/2$ and $\varphi=\varphi_{0}/2$. ∎

We now discuss the example of a 360 shove-it with a wobbling axis of rotation.
The 360 shove-it rotates about the axis $-\mathbf{k}$ by an angle of 360
degrees with a right-handed orientation. We now consider a trick which is
close to the 360 shove-it, but whose axis of rotation wobbles about the
$z$-axis. We may consider for example the following formula for the axis of
rotation:

$\mathbf{k}_{a,\omega}(t)=a\cos(2\pi\omega t)\textbf{i}+a\sin(2\pi\omega
t)\textbf{j}-\sqrt{1-a^{2}}\textbf{k}$

where $\omega$ is a real number which determines the frequency of the wobble
and $0\leq a\leq 1$ determines its amplitude. When $a=0$ there is no wobble,
and when $a=1$ the resulting trick no longer bears any resemblance to a 360
shove-it. The resulting trick would then be:

$R(t)=\rho\big{(}\cos(\pi t)+\sin(\pi t)\mathbf{k}_{a,\omega}(t)\big{)}$

We may project the quaternionic curve onto the 3-subspace generated by $1$,
$\mathbf{j}$ and $\mathbf{k}$ as before, and obtain the visualization in
figure 7.

Figure 7. Renormalized projection of wobbling 360 shove-it

Let’s now stabilize the axis of rotation of this trick. We will simply apply
the homotopy $\hat{F}$ from the proposition in order to create a visualization
of that deformation. In this case, the axis $\mathbf{u}$ in the proposition is
$-\mathbf{k}$ and we may use the standard coordinates for the quaternions in
our computation. The quaternionic curve is $\alpha(t)=\cos(\pi t)+\sin(\pi
t)\mathbf{k}_{\omega,a}(t)$. We first compute the function
$n(s,t):=n(s,\alpha(t))$, we obtain:

$n(s,t)=1+\frac{s}{(1/a\sin(\pi t))^{2}-1}$

The homotopy is then given by:

$H(s,t)=\left(\sqrt{n(s,t)}\cdot\alpha_{0}(t),\sqrt{1-s}\cdot\alpha_{1}(t),\sqrt{1-s}\cdot\alpha_{2}(t),\sqrt{n(s,t)}\cdot\alpha_{3}(t)\right)$

We will not reparametrize the angle in order to obtain a curve with constant
angular velocity as this will not affect the image produced. We obtain figure
8.

Figure 8. Stabilizing the wobbling 360 shove-it

##  10\. Conclusion

We have ongoing work regarding the deformations of strictly physical
skateboard tricks. This involves an analysis of the deformation of closed
geodesics of the associated Riemannian metric on the Lie group $SO(3)$.

For a rigid body with enough symmetries this is a simpler task, when the
associated Riemannian metric is bi-invariant its geodesics correspond to
1-parameter subgroups of $SO(3)$, which can be studied through its Lie
algebra. However the asymmetry of the skateboard makes it so that the relevant
Riemannian metric in this case is not bi-invariant, so that a more careful
analysis of the closed geodesics becomes more difficult.

Finally we have developed a set of Python scripts which can generate rotation
matrices and animations for a number of skateboard tricks. You can find them
in the following GitHub repository:

https://github.com/holomorpheus/topological-flips

Acknowledgements

We would like to thank José Bravo for his contributions to the animation
scripts and Lance Johnson, Garrett Lent, Brenden Mucklow and Diego Vera for
our discussions on algebraic topology.

## References

  * [1] Abraham R.; Marsden J. E.: Foundations of Mechanics, American Mathematical Society, 2nd edition, (2008)
  * [2] Arnold, V.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Annales de l’Institut Fourier 16.1, (1966)
  * [3] Arnold, V.: Mathematical Methods of Classical Mechanics, Springer, (1989)
  * [4] Hatcher, A.: Algebraic Topology, Cambridge Univ. Press, (2001)
  * [5] Holm, D. D.; Schmah T.; Stoica C.: Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions, Oxford University Press (2009)


# Efficient Zero-shot Event Extraction with Context-Definition Alignment

Hongming Zhang, Wenlin Yao, Dong Yu
Tencent AI Lab, Bellevue, USA
{hongmzhang, wenlinyao<EMAIL_ADDRESS>
###### Abstract

Event extraction (EE) is the task of identifying interested event mentions
from text. Conventional efforts mainly focus on the supervised setting.
However, these supervised models cannot generalize to event types out of the
pre-defined ontology. To fill this gap, many efforts have been devoted to the
zero-shot EE problem. This paper follows the trend of modeling event-type
semantics but moves one step further. We argue that using the static embedding
of the event type name might not be enough because a single word could be
ambiguous, and we need a sentence to define the type semantics accurately. To
model the definition semantics, we use two separate transformer models to
project the contextualized event mentions and corresponding definitions into
the same embedding space and then minimize their embedding distance via
contrastive learning. On top of that, we also propose a warming phase to help
the model learn the minor difference between similar definitions. We name our
approach Zero-shot Event extraction with Definition (ZED). Experiments on the
MAVEN dataset show that our model significantly outperforms all previous zero-
shot EE methods with fast inference speed due to the disjoint design. Further
experiments also show that ZED can be easily applied to the few-shot setting
when the annotation is available and consistently outperforms baseline
supervised methods.

##  1 Introduction

Event extraction, the task of identifying event mentions from documents and
classifying them into pre-defined event types, is a fundamental NLP problem
Grishman et al. (2005). As a centric information extraction task, event
extraction is the foundation of a series of event-centric NLP applications
Chen et al. (2021) including event relation extraction Wang et al. (2020a),
event schema induction Li et al. (2020), and missing event prediction
Chaturvedi et al. (2017).

Figure 1: Zero-shot event extraction task demonstration. Given a corpus, the
goal is to identify all event mentions that fit the target event definitions
without using any annotation. The event definitions and corresponding mentions
are indicated in the same color.

Traditional event extraction efforts Wadden et al. (2019); Wang et al. (2019);
Lin et al. (2020) mostly focus on learning to identify and classify events
under a supervised learning setting, where a pre-defined event ontology and
large-scale expert annotations is available. However, the learned supervised
models cannot be easily applied to new event types out of the pre-defined
ontology, limiting these models’ usage in real applications.

Recently, large-scale pre-trained language models have demonstrated strong
semantics representation capabilities and motivated a series of works to
extract events in a zero-shot setting. For example, Du and Cardie (2020)
propose to manually design templates for each event type to convert the event
extraction problem into a question-answering (QA) task and then leverage QA
models to extract events. Following that, Lyu et al. (2021) propose to
verbalize candidate triggers and event types into hypothesis and premises and
leverage pre-trained textual entailment models to extract events. However, as
analyzed in Lyu et al. (2021), these models heavily rely on the template
design and often suffer from the domain-shifting problem between the original
training task and the new task. Moreover, as these models require jointly
encoding the event mentions and event types, the time complexity is $O(N*T)$,
where $N$ is the number of event mention candidates and $T$ is the number of
event types. Considering the low inference speed and high computation cost of
inference with a deep model, such complexity could be a massive burden for
real-time EE systems.

To avoid manually designing templates and to improve the inference efficiency,
another line of work Zhang et al. (2021) tries to leverage pre-trained
language representation models (i.e., BERT Devlin et al. (2019)) to acquire a
contextualized event type representation. The model can decouple the mention
and label representations during the inference time and predict the candidate
trigger to the most similar event type based on the cosine similarity. As a
result, this method could significantly reduce the inference time complexity
from $O(N*T)$ to $O(N+T)$. However, as the experiments show, using only the
label name might not lead to a good event-type representation because the
selected words could be ambiguous.

In this work, we follow the trend of representation learning Zhang et al.
(2021); Gao et al. (2021) and move forward from representing each event type
with a single name to a definition sentence. Specifically, we propose a three-
stage event representation learning framework. In the offline pre-training
phase, we leverage auto-extracted context-definition alignments to learn a
definition encoding model that can encode the contextualized mentions and
definitions into the same embedding space. In the second warming phase, we use
the target event types to retrieve hard negative examples to further polish
the model. In the end, we identify and classify event mentions based on the
cosine similarity between the mention representation and corresponding event-
type representations. As our system is a disjoint model, the inference time
complexity is also $O(N+T)$. Experiments on MAVEN Wang et al. (2020b), the
largest EE dataset to the best of our knowledge, show that ZED outperforms all
previous zero-shot approaches with high inference efficiency. Further
experiments show that ZED could also be applied to the supervised setting,
where it achieves comparable performance in the fully supervised setting and
consistently outperforms baseline supervised models in the data-scarce
learning settings. Specifically, with 10% of the training data, ZED could
achieve over 95% of the full performance. All the collected alignment data,
created definitions, and the code are available at:
https://github.com/tencent-ailab/ZED.

Figure 2: Overall framework of ZED. In the offline training phase, we train
the separate context and definition encoders with auto-extracted context-
definition alignment data. In the second warming phase, after knowing the
target event types, as no annotation is provided, we first retrieve similar
concepts from WordNet Miller (1998) and use corresponding alignment data to
polish the representation model. In the last inference phase, after encoding
all candidate event type definitions, for each candidate event mention, we
will encode it with the context encoder and determine whether it belongs to
one of the target event types based on the cosine similarity.

##  2 Related Works

In this section, we introduce related works about event extractions,
contrastive representation learning, and definition modeling.

###  2.1 Event Extraction

As a fundamental information extraction task Chen et al. (2021), event
extraction has attracted many efforts in the NLP community Sundheim (1992);
Grishman and Sundheim (1996); Riloff (1996); Grishman et al. (2005); Chen et
al. (2021); Hong et al. (2022). Recent success on the event extraction task
mostly relies on employing either symbolic features Ji and Grishman (2008);
Liao and Grishman (2010); Liu et al. (2016) or distributed features Chen et
al. (2015); Nguyen et al. (2016); Liu et al. (2018); Zhang et al. (2019);
Wadden et al. (2019); Lin et al. (2020) to learn supervised models with large-
scale high-quality annotations. However, the requirement of a pre-defined
ontology and corresponding annotations limits the application of these models
in real applications.

To address this issue and extract unseen event types, Huang et al. (2018)
propose a zero-shot event extraction task and use a transfer-learning
framework to apply the model trained with seen event types to unseen ones.
However, the prerequisite of their high performance is the similarity between
seen and unseen event types. Recently, with the fast development of large-
scale language models, several works Du and Cardie (2020); Lyu et al. (2021);
Zhang et al. (2021) propose to leverage the pre-trained models to encode the
label semantics either with templates or contextualized embeddings. In this
work, we follow the effort of using deep models to model the label semantics
but make a step further. Instead of directly using a pre-trained model, we
train a disjoint context-to-definition alignment encoding model, which can
effectively map the candidate event mentions and definitions into the same
embedding space and thus more accurately and efficiently extract events for
any arbitrarily defined event types.

###  2.2 Contrastive Representation Learning

The contrastive loss Chopra et al. (2005) is one of the most popular training
objectives for representation learning. The original contrastive loss and its
variations (e.g., triplet loss Schroff et al. (2015), lifted structured loss
Oh Song et al. (2016), N-pair loss Sohn (2016), and NCE loss Gutmann and
Hyvärinen (2010)) have been shown helpful for a series of vision applications
Radford et al. (2021). After being introduced to the NLP community, the
contrastive learning-based method also leads to the success of a series of
representation learning tasks such as sentence representation Gao et al.
(2021). Different from previous works, where the anchors and positive/negative
examples typically belong to the same category (e.g., image/sentence), we
propose to use the contextualized token representation as the anchor and event
type definition representations as the positive/negative examples to better
solve the zero-shot event extraction task. Moreover, motivated by the success
of the “pre-training+fine-tuning” paradigm, we propose a novel three-stage
representation learning framework.

###  2.3 Definition Modeling

Humans are capable of understanding new concepts by reading their glosses or
definitions. Thus, how to leverage the definitions and explanations from
dictionaries to help understand human language is a long-standing question in
the NLP community. Most of the previous efforts in this direction are working
on the word sense disambiguation task Luo et al. (2018); Huang et al. (2019);
Blevins and Zettlemoyer (2020); Kumar et al. (2019); Bevilacqua and Navigli
(2020); Yao et al. (2021); Su et al. (2022a, b). These models learn to map a
token into the correct pre-defined synset by either jointly or disjointly
encoding the tokens and definitions. Even though the setting of our model and
these WSD models are similar, identifying event mentions that satisfy an
arbitrary event type definition is a more challenging task Senel et al.
(2022). WSD aims to learn to distinguish the correct synset versus several
(typically less than 10) other pre-defined synsets, while our goal is to align
an event mention and the corresponding definition, where all other arbitrary
definitions are considered to be the negative candidates. To address the
engineering limitation that negative candidates exceed the GPU memory
limitation, we propose a coarse-to-fine negative sampling strategy to help
models learn the minor differences between similar definitions without
forgetting the big picture.

##  3 Task Definition

We define the zero-shot event extraction task as follows. Given a document in
the format of a sentence set ${\mathcal{S}}$ and event type set
${\mathcal{E}}$. Each event type $E\in{\mathcal{E}}$ is defined with a natural
sentence $d$. The task is to identify all mentions ${\mathcal{M}}_{E}$ in
${\mathcal{S}}$ that satisfy the definition of $E$ for each
$E\in{\mathcal{E}}$ without using direct annotations during the training
phase.

##  4 Model

We present the model framework in Figure 2. Motivated by the success of the
“pre-training + fine-tuning” learning paradigm, we propose to address the
zero-shot event extraction problem with a three-stage framework. Technical
details are as follows.

Mention in Context |  Definition
---|---
I love playing basketball. |  get pleasure from
Bob is studying computer science. |  be a student; follow a course of study; be enrolled at an institute of learning
I got promoted after many years of hard work |  give a promotion to or assign to a higher position
Table 1: Demonstration of collected context and definition alignments. Target
mentions are underlined.

###  4.1 Offline Pre-training

The offline pre-training step aims to train a decent definition encoder to map
the target mention representation and corresponding definitions into the same
embedding space. To achieve this goal, as no annotation is provided, we first
collect context-definition alignments and then train the encoder with a
contrastive learning loss.

####  4.1.1 Data Preparation

We select all verbal synsets from the WordNet ontology Miller (1998) to form
our open-world event definition set. In total, we collect 13,814 event
synsets. After that, to collect large-scale alignment data between context and
definitions, we apply the current state-of-the-art word sense disambiguation
model Yao et al. (2021) to the NYT corpus Sandhaus (2008) to align tokens in
NYT with their correct definitions. We randomly select 10 context instances
for each synset to speed up the training process. As a result, we collect 775K
context-definition alignments. Examples of extracted alignments are presented
in table 1.

####  4.1.2 Context-definition Alignment Encoding with Contrastive Learning

The goal of the context-definition alignment encoding is encoding the
contextualized representation of the target mention and the sentence
representation of the definition into the same embedding space and pushing
them to be closer to each other because they should have similar semantic
meanings. As this objective aligns well with the learning objective of the
contrastive learning framework, we follow the standard contrastive learning
framework Chopra et al. (2005). Specifically, we denote the pre-processed
context-definition alignment set as ${\mathcal{T}}$, where each instance
$(S,i,j,D)\in{\mathcal{T}}$ contains context sentence $S$, which is a list of
tokens $w_{1}^{S},w_{2}^{S},...,w_{n}^{S}$, target word starting position $i$,
target word ending position111Each word could have multiple tokens because we
follow the standard tokenization of BERT Devlin et al. (2019). $j$, and a
definition sentence $D$, which is also a list of tokens
$w_{1}^{D},w_{2}^{D},...,w_{m}^{D}$. We follow the standard approach to get
the contextualized word representation as the mean pooling of all sub-token
representations:

${\bf e}_{S,i,j}=\frac{\sum_{i\leq k\leq j}{\bf e}_{k}}{j-i+1},$ (1)

where ${\bf e}_{k}$ is the contextualized representation of token $k$ produced
by a transformer baseline language model (e.g., BERT Devlin et al. (2019)).
For the sentence encoding, we choose to use the average representation of all
tokens as follows:

${\bf d}_{D}=\frac{\sum_{1\leq k\leq m}FFN({\bf e}_{k})}{m},$ (2)

where $FFN$ represents a two-layer feed-forward neural network and ${\bf
e}_{k}$ is the token representation of token $w_{k}$.

Following the contrastive learning framework, during this step, we optimize
the marginal ranking loss222We chose ranking loss over entropy loss mainly
because the alignment data we used for the pre-training is automatically
collected, and the training signal may contain noise. . Assume that the set of
randomly sampled negative definitions is ${\mathcal{D}}^{\prime}$, for each
$D^{\prime}\in{\mathcal{D}}^{\prime}$, we could follow equation 2 to compute
its representation as ${\bf d}_{D^{\prime}}$. For each instance
$(S,i,j,D)\in{\mathcal{T}}$ and a randomly sampled negative definition set
${\mathcal{D}}^{\prime}$, we minimize the following marginal ranking loss:

$\frac{\sum_{D^{\prime}\in{\mathcal{D}}^{\prime}}max(0,\epsilon-(cos({\bf
e}_{S,i,j},{\bf d}_{D})-cos({\bf e}_{S,i,j},{\bf
d}_{D^{\prime}})))}{\|{\mathcal{D}}^{\prime}\|},$ (3)

where $max$ means the maximum operation, $cos$ is the cosine similarity, and
$\epsilon$ is the margin.

###  4.2 Query-specific Warming

After the pre-training phase, the model briefly understands how to project the
contextualized event mentions and corresponding definitions into similar
positions in the embeddings. However, its capability of distinguishing similar
definitions is still limited because the previous negative sampling strategy
does not encourage such capabilities. To address this issue, we introduce an
additional warming phase to help models learn the minor difference between
similar definitions.

Similar to how human beings understand new concepts by recalling relevant
knowledge, we also retrieve relevant knowledge from ${\mathcal{T}}$ to further
fine-tune the model. Specifically, assume that the set of interested event
definitions is $\hat{{\mathcal{D}}}$, for each
$\hat{D}\in\hat{{\mathcal{D}}}$, we first retrieve the most similar definition
$\tilde{D}$ from the original definition set ${\mathcal{D}}$ by:

$\tilde{D}=\arg\max_{D\in{\mathcal{D}}}sim(PLM(D),PLM(\hat{D})),$ (4)

where $sim$ is the similarity measurement and $PLM$ represents the encoding
with a pre-trained language model. In our experiment, we select cosine
similarity as the similarity measurement and average contextualized token
embedding encoded with BERT-base Devlin et al. (2019) as the encoding. But
other techniques could also be applied.

We thus denote the set of all retrieved relevant definitions as
$\tilde{{\mathcal{D}}}$ and select a subset $\tilde{{\mathcal{I}}}$ of
${\mathcal{I}}$ such that all definitions in $\tilde{{\mathcal{I}}}$ belong to
$\tilde{{\mathcal{D}}}$ to further fine-tune the model. After generating all
the data, we will fine-tune all models following the loss function in Equation
3.

###  4.3 Inference

During the inference, we compute the representation for each candidate event
mention in and target event type descriptions. After that, for each candidate
mention, we compute its cosine similarity with all the target event-type
representations. If the largest similarity is larger than a threshold $t$,
this mention is identified and labeled as the most similar event type. Assume
that the size of all candidate mentions and target event types are $N$ and
$T$, respectively. Compared with previous zero-shot models that rely on the
joint encoding of the candidate mention and target event types Du and Cardie
(2020); Lyu et al. (2021); Yao et al. (2021), we successfully reduce the
computation complexity from $O(N*T)$ to $O(N+T)$. A numerical evaluation of
the computation efficiency is shown in Section 6.2.

Model | Identification | Identification+Classification
---|---|---
P | R | F1 | P | R | F1
Chance Performance | 19.33 | 20.05 | 19.68 | 0.11 | 0.11 | 0.11
Most Popular Event Type | 18.59 | 19.45 | 19.01 | 0.74 | 0.77 | 0.75
QA Du and Cardie (2020) | 19.76 | 45.18 | 27.49 | 4.19 | 9.58 | 5.83
TE Lyu et al. (2021) | 20.20 | 32.83 | 25.01 | 4.59 | 7.46 | 5.68
WSD Yao et al. (2021) | 24.66 | 80.52 | 37.76 | 5.36 | 17.51 | 8.21
CLE Zhang et al. (2021) | 55.07 | 14.63 | 23.00 | 42.99 | 11.34 | 17.95
ZED | 59.37 | 42.28 | 49.39 | 39.63 | 28.22 | 32.96
Table 2: Zero-shot Event identification and classification results on MAVEN
Wang et al. (2020b), which has 168 event types. Best F1 performances are
indicated with bold font.

##  5 Experiments

This section introduces experiment details, including the selected baseline
methods, experiment datasets, and implementation details.

###  5.1 Baseline Methods

In the past two years, the community has been devoting significant effort to
solving the zero-shot event extraction problem with different approaches.
Specifically, we select the following best-performing models as our baselines.

  1. 1.

Pre-trained Question Answering Models Du and Cardie (2020) (QA): Most NLP
tasks can be converted into a QA format and event extraction is not an
exception. Motivated by this, Du and Cardie (2020) propose to design a
question template for each target event type and directly ask a QA model to
answer whether a mention is the target event.

  2. 2.

Pre-trained Textual Entailment Models Lyu et al. (2021) (TE): Motivated by the
QA approach, Lyu et al. (2021) explore the possibility of utilizing a pre-
trained textual entailment (TE) model to automatically extract events.
Specifically, for each target event type, Lyu et al. (2021) manually design a
template to convert it into a hypothesis, treat the target event mention as
the premise, and ask the TE model whether the target event mention can entail
an event type.

  3. 3.

Word Sense Disambiguation Models Yao et al. (2021) (WSD): Prior WSD works also
heavily rely on the correctly modeling of the definitions, so conceptually
they could also be applied to the event extraction task following our setup.
There are mainly two key differences between our work and Yao et al. (2021):
(1) Yao et al. (2021) encode the context and definition jointly while our
model encodes them separately; (2) Yao et al. (2021) aim at modeling the minor
difference between different synsets of the same word while our work aims at
modeling general definition semantics.

  4. 4.

Contextualized Label Embedding Zhang et al. (2021) (CLE): The last baseline we
compare with is the contextualized label representation. Specifically, for
each target event type, Zhang et al. (2021) generate a contextualized label
representation by putting the label name back into contexts and directly
extracting events based on the similarity between the mention representation
and event type representations.

Besides these baselines, we also present the “Chance” performance, where a
mention is randomly selected following the percentage of gold mentions and
randomly assigned an event type, and the “Most Popular Event Type”
performance, where a mention is also randomly selected following the
percentage of gold mentions and is always predicted to be the most popular
event type.

###  5.2 Evaluation Dataset

We select MAVEN Wang et al. (2020b) as the evaluation dataset due to its
large-scale and balanced distribution. Specifically, MAVEN contains 186 unique
event types selected from FrameNet Baker et al. (1998) and 118,732 annotated
event mentions, which is almost two magnitudes larger than the previous
datasets such as ACE Grishman et al. (2005). Moreover, MAVEN provides the
official event mention candidates to evaluate the mention understanding
capability of all event extraction models more fairly. As the original dataset
only provides the event name in the format of a phrase (e.g.,
“Body_movement”), we directly use definitions from Wordnet as the
description333For event names that have multiple synsets or not covered by
WordNet, we manually select the most accurate description from WordNet..
Examples of the event types and corresponding definitions are presented in
Appendix Table 5.

###  5.3 Implementation Details

For baseline models, we conduct experiments with officially released code,
hyperparameters, templates, and pre-trained models. For ZED, we use two
separate encoders for the context and definition encoding. Both of them are
initialized with BERT-base Devlin et al. (2019). As no training set is needed
in the zero-shot setting and the test set of MAVEN is not publicly available,
we report the performance on the dev set. Specifically, we set the margin to
0.2 for the marginal ranking loss and set the number of negative examples to
2. The selection threshold at the inference phrase is set to be 0.7. We train
the model with ten epochs for both the pre-training and warming phrases. We
directly evaluate the last checkpoint to simulate the real application, where
no dev set is available. All models are trained with Tesla P40 with batch size
16. The pre-training and warming phrases will take around 200 and 3 hours on a
single GPU, respectively, but we could speed it up with multiple GPUs.

| F1 (I) | F1 (I+C)
---|---|---
ZED | 49.39 | 32.96
\- Warming | 47.86 (-1.53) | 21.89 (-11.07)
\- Strong Negative Sampling | 48.91 (-0.48) | 29.20 (-3.76)
Table 3: Ablation study. “I” and “C” represent the identification and
classification, respectively.

##  6 Zero-shot Performance

The zero-shot performance of all models is presented in Table 2, from which we
can make the following observations:

  1. 1.

All models significantly outperform the naive baselines even though they do
not use any annotations. This observation shows that current deep models can
indeed learn rich semantics that could generalize outside of their original
training goal.

  2. 2.

The overall performance of pre-trained QA, TE, and WSD models is not
satisfying because they suffer from domain shifting. For example, even though
current deep-model-driven QA models have surpassed human performance on
several leaderboards, they are still not ready to be used as a general QA
model for solving tasks that require deep understanding, such as zero-shot
event extraction.

  3. 3.

Compared with other methods, Contextualized label embedding achieves lower
identification F1 but higher classification accuracy, which aligns with the
original observation in Zhang et al. (2021). The reason behind this is that
due to the cone property of the BERT representation (i.e., most of the token
representations of BERT are grouped in a small region), it is tough to
determine the cosine similarity boundary of whether an event mention fits a
specific event type. As a result, even though CLE could accurately identify
high-confident mentions, it cannot handle boundary ones very well.

  4. 4.

Compared with baseline methods, ZED could perform better on both the
identification and classification tasks. The main reason is that we are using
definitions to model the label semantics, which is more accurate than a single
word.

###  6.1 Ablation Study

Figure 3: Effect of training instance number per definition. (Zero-shot
performance on the Identification+Classification F1 is reported.)

From the ablation study results in Table 3, we can see that if we remove the
warming phase, the model’s performance will drop on both the identification
and classification, especially the classification step. This aligns well with
our assumption that the model can learn to model the general definition
semantics after the pre-training step but cannot distinguish minor differences
very well. The performance drop of removing the strong negative sampling
module indicates that strong negatives are crucial for the success of
representation learning, which aligns well with previous observations Clark et
al. (2020).

Besides those ablation studies, we also show the impact of the pre-training
data scale in Figure 3. As expected, the more data we use, the better
performance we will get. However, the performance gain after 10 instances per
synset is limited. As a result, we select 10 instances for each synset as the
pre-training data for training efficiency.

###  6.2 Inference Efficiency

Model | Identification | Identification+Classification
---|---|---
P | R | F1 | P | R | F1
DMBERT | 73.37 | 87.82 | 79.95 | 61.20 | 73.25 | 66.69
BERT+CRF | 75.19 | 81.80 | 78.35 | 64.40 | 70.28 | 67.21
ZED + Supervision | 82.48 | 80.76 | 81.61 | 67.87 | 66.46 | 67.16
Table 4: Model Performance with full annotations, Best F1 performances are
indicated with the bold font.

We present the inference speed of all evaluated models in Figure 4. As ZED
adopts a disjoint encoding design, we successfully reduce the computation
complexity from $O(N*T)$ to $O(N+T)$, where $N$ is the number of event
mentions and $T$ number of event types. On Maven, which has 168 different
event types, ZED could speed up the inference efficiency by almost two
magnitudes.

##  7 Warming with Gold Annotation

Figure 4: Inference Speed of all zero-shot models. For a fair comparison, we
evaluate all models with the same GPU and use batch size 1. As our model is
smaller than baseline models, we could use a larger batch size in real
applications to further boost efficiency.

ZED can also be adapted to a fully supervised or few-shot learning setting
when the annotation is available. Specifically, during the warming phase of
our model, we can replace the auto-retrieved examples with the annotated ones
and fine-tune the model. In this section, we follow the benchmark paper Wang
et al. (2020b) to compare with the recent language model-driven
baselines444Several other recent works (e.g., OneIE Lin et al. (2020)) further
improves the performance on event extraction by utilizing the constraints
between trigger and arguments. However, as such information is not available
in MAVEN and extracting arguments is beyond the research scope of this paper,
we cannot compare with them.: DMBERT Wang et al. (2019) and BERT Devlin et al.
(2019) \+ CRF Lafferty et al. (2001), which also achieved the previous best
performance. Please refer to the original papers for technical details of
these baseline models. We implement all models with the officially released
code555https://github.com/THU-KEG/MAVEN-dataset and report the average
performance of five trials on the development set. All models are trained for
ten epochs, and the final model is evaluated. Like the zero-shot setting, we
also report the micro precision, recall, and F1 for both the “identification”
and “identification+classification” settings. All hyper-parameters are based
on the officially released code.

Figure 5: Model performance with limited annotation.

Results in Table 4 show that with the help of the pre-training step, our model
can outperform all previous supervised models on the identification task and
comparable performance on the classification task. This makes sense because a
carefully designed deep model could learn to identify and classify event
mentions well with the large-scale annotation provided by MAVEN.

However, we argue that such a large-scale annotation is often expensive in
terms of money and time. The data-scarce learning setting might be more
applicable in real applications. Thus, we also test the performance of these
supervised settings under the data-scarce learning setting. Specifically, we
randomly select 1% to 10% of the training sentences to be sampled from the
training data and report the performances in Figure 5. Our model can
constantly outperform baseline models with a small number of annotations.
Especially when only 1% of the data is available, we only have 7.07 mentions
per event type, ZED could achieve over 50 F1. With 10% of the training data,
ZED could achieve over 95% of full supervised performance. These observations
show that our framework could be applied to broader applications where limited
or enough annotations are available besides the zero-shot setting. Besides
that, another interesting finding is that even though “BERT+CRF” could
outperform “DMBERT” slightly when enough annotation is available, which is
consistent with the observations in Wang et al. (2020b), its performance is
worse under the data-scarce setting. This observation indicates that using CRF
might not be the optimal option when the annotation scale is limited.

##  8 Conclusion

This paper proposes a novel zero-shot event extraction framework ZED. Given a
set of interested event types in the format of definitions, ZED could
automatically extract all the event mentions that fit the definitions from raw
documents much better than previous methods. Experiments show that the
proposed warming phase and the mixed strong negative examples sampling
strategies contribute to the success of ZED. Additional experiments also show
that ZED could be applied to the supervised setting. Thanks to the pre-
training phase, it could achieve good performance under both the fully
supervised and data-scarce settings.

## Acknowledgements

We thank anonymous reviewers for their insightful comments and suggestions.

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##  Appendix A MAVEN Ontology Demonstration

Name |  Definition
---|---
Manufacturing |  make or cause to be or to become
Achieve |  to gain with effort
Communication |  express in words
Employment |  engage or hire for work
Process_start |  take the first step or steps in carrying out an action
Theft |  take without the owner’s consent
Legal_rulings |  pronounce a sentence in a court of law
Influence |  have an effect upon
Give_up |  give up, such as power, as of monarchs and emperors, or duties and obligations
Catastrophe |  a violent weather condition
Table 5: Representative MAVEN event types and associated definitions. All used
definitions will be released with the code.