# Jensen-type inequalities for convex and $m$-convex functions via fractional
calculus

Quintana, Yamilet
Universidad Carlos III de Madrid, Departamento de Matemáticas
Avenida de la Universidad 30, 28911 Leganés, Madrid, Spain
<EMAIL_ADDRESS>
Rodríguez, José M.
Universidad Carlos III de Madrid, Departamento de Matemáticas
Avenida de la Universidad 30, 28911 Leganés, Madrid, Spain
<EMAIL_ADDRESS>
Sigarreta Almira, José M.
Universidad Autónoma de Guerrero, Centro Acapulco
CP 39610, Acapulco de Juárez, Guerrero, México
<EMAIL_ADDRESS>
###### Abstract

Inequalities play an important role in pure and applied mathematics. In
particular, Jensen’s inequality, one of the most famous inequalities, plays a
main role in the study of the existence and uniqueness of initial and boundary
value problems for differential equations. In this work we prove some new
Jensen-type inequalities for $m$-convex functions, and we apply them to
generalized Riemann-Liouville-type integral operators. It is remarkable that,
if we consider $m=1$, we obtain new inequalities for convex functions.

AMS Subject Classification (2010):  26A33, 26A51, 26D15

Key words and phrases: Jensen-type inequalities, convex functions, $m$-convex
functions, fractional derivatives and integrals, fractional integral
inequalities.

##  1 Introduction

Integral inequalities are used in countless mathematical problems such as
approximation theory and spectral analysis, statistical analysis and the
theory of distributions. Studies involving integral inequalities play an
important role in several areas of science and engineering.

In recent years there has been a growing interest in the study of many
classical inequalities applied to integral operators associated with different
types of fractional derivatives, since integral inequalities and their
applications play a vital role in the theory of differential equations and
applied mathematics. Some of the inequalities studied are Gronwall, Chebyshev,
Hermite-Hadamard-type, Ostrowski-type, Grüss-type, Hardy-type, Gagliardo-
Nirenberg-type, reverse Minkowski and reverse Hölder inequalities (see, e.g.,
[2, 5, 11, 12, 14, 15, 16, 17, 18]).

In this work we obtain new Jensen-type inequalities for convex and $m$-convex
functions, and we apply them to the generalized Riemann-Liouville-type
integral operators defined in [1], which include most of known Riemann-
Liouville-type integral operators.

##  2 Preliminaries

One of the first operators that can be called fractional is the Riemann-
Liouville fractional derivative of order $\alpha\in\mathbb{C}$, with
$Re(\alpha)>0$, defined as follows (see [4]).

###### Definition 1

Let $a<b$ and $f\in L^{1}((a,b);\mathbb{R})$. The _right and left side
Riemann-Liouville fractional integrals of order_ $\alpha$, with
$Re(\alpha)>0$, are defined, respectively, by

$^{RL}\\!{{J}_{{a}^{+}}^{\alpha}}f(t)=\frac{1}{\Gamma(\alpha)}\int_{a}^{t}{{(t-s)}^{\alpha-1}f(s)\,ds},$
(1)

and

$^{RL}\\!{{J}_{{b}^{-}}^{\alpha}}f(t)=\frac{1}{\Gamma(\alpha)}\int_{t}^{b}{{(s-t)}^{\alpha-1}f(s)\,ds},$
(2)

with $t\in(a,b)$.

When $\alpha\in(0,1)$, their corresponding _Riemann-Liouville fractional
derivatives_ are given by

$\displaystyle\big{(}{{}^{RL}\\!D_{a^{+}}^{\alpha}}f\big{)}(t)$
$\displaystyle=\frac{d}{dt}\left({{}^{RL}\\!{{J}_{{a}^{+}}^{1-\alpha}}f(t)}\right)=\frac{1}{\Gamma(1-\alpha)}\,\frac{d}{dt}\int_{a}^{t}{{\frac{f(s)}{(t-s)^{\alpha}}}\,ds},$
$\displaystyle\big{(}{{}^{RL}\\!D_{b^{-}}^{\alpha}}f\big{)}(t)$
$\displaystyle=-\frac{d}{dt}\left({{}^{RL}\\!{{J}_{{b}^{-}}^{1-\alpha}}f(t)}\right)=-\frac{1}{\Gamma(1-\alpha)}\,\frac{d}{dt}\int_{t}^{b}{{\frac{f(s)}{(s-t)^{\alpha}}}\,ds}.$

Other definitions of fractional operators are the following ones.

###### Definition 2

Let $a<b$ and $f\in L^{1}((a,b);\mathbb{R})$. The _right and left side
Hadamard fractional integrals of order_ $\alpha$, with Re($\alpha)>0$, are
defined, respectively, by

${{H}_{{a}^{+}}^{\alpha}}f(t)=\frac{1}{\Gamma(\alpha)}\int_{a}^{t}{{\Big{(}\log\frac{t}{s}\,\Big{)}}^{\alpha-1}\frac{f(s)}{s}\,ds},$
(3)

and

${{H}_{{b}^{-}}^{\alpha}}f(t)=\frac{1}{\Gamma(\alpha)}\int_{t}^{b}{{\Big{(}\log\frac{s}{t}\,\Big{)}}^{\alpha-1}\frac{f(s)}{s}\,ds},$
(4)

with $t\in(a,b)$.

When $\alpha\in(0,1)$, _Hadamard fractional derivatives_ are given by the
following expressions:

$\displaystyle{\big{(}{}^{H}\\!{D_{a^{+}}^{\alpha}}}f\big{)}(t)$
$\displaystyle=t\,\frac{d}{dt}\big{(}{{H}_{{a}^{+}}^{1-\alpha}}f(t)\big{)}=\frac{1}{\Gamma(1-\alpha)}\,t\,\frac{d}{dt}\int_{a}^{t}{{\Big{(}\log\frac{t}{s}\Big{)}}^{-\alpha}\frac{f(s)}{s}\,ds},$
$\displaystyle{\big{(}{}^{H}\\!{D_{b^{-}}^{\alpha}}}f\big{)}(t)$
$\displaystyle=-t\,\frac{d}{dt}\big{(}{{{H}_{{b}^{-}}^{1-\alpha}}f(t)}\big{)}=\frac{-1}{\Gamma(1-\alpha)}\,t\,\frac{d}{dt}\int_{t}^{b}{{\Big{(}\log\frac{s}{t}\Big{)}}^{-\alpha}\frac{f(s)}{s}\,ds},$

with $t\in(a,b)$.

###### Definition 3

Let $0<a<b$, $g:[a,b]\rightarrow\mathbb{R}$ an increasing positive function on
$(a,b]$ with continuous derivative on $(a,b)$, $f:[a,b]\rightarrow\mathbb{R}$
an integrable function, and $\alpha\in(0,1)$ a fixed real number. The right
and left side fractional integrals in [7] of order $\alpha$ of $f$ with
respect to $g$ are defined, respectively, by

$I_{g,a^{+}}^{\alpha}f(t)=\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\frac{g^{\prime}(s)f(s)}{{\big{(}g(t)-g(s)\big{)}}^{1-\alpha}}\,ds,$
(5)

and

$I_{g,b^{-}}^{\alpha}f(t)=\frac{1}{\Gamma(\alpha)}\int_{t}^{b}\frac{g^{\prime}(s)f(s)}{{\big{(}g(s)-g(t)\big{)}}^{1-\alpha}}\,ds,$
(6)

with $t\in(a,b)$.

There are other definitions of integral operators in the global case, but they
are slight modifications of the previous ones.

##  3 General fractional integral of Riemann-Liouville type

Now, we give the definition of a general fractional integral in [1].

###### Definition 4

Let $a<b$ and $\alpha\in\mathbb{R}^{+}$. Let $g:[a,b]\rightarrow\mathbb{R}$ be
a positive function on $(a,b]$ with continuous positive derivative on $(a,b)$,
and $G:[0,g(b)-g(a)]\times(0,\infty)\rightarrow\mathbb{R}$ a continuous
function which is positive on $(0,g(b)-g(a)]\times(0,\infty)$. Let us define
the function $T:[a,b]\times[a,b]\times(0,\infty)\rightarrow\mathbb{R}$ by

$T(t,s,\alpha)=\frac{G\big{(}|g(t)-g(s)|,\alpha\big{)}}{g^{\prime}(s)}\,.$

The _right and left integral operators_ , denoted respectively by
$J_{T,a^{+}}^{\alpha}$ and $J_{T,b^{-}}^{\alpha}$, are defined for each
measurable function $f$ on $[a,b]$ as

$J_{T,a^{+}}^{\alpha}f(t)=\int_{a}^{t}\frac{f(s)}{T(t,s,\alpha)}\,ds,$ (7)
$J_{T,b^{-}}^{\alpha}f(t)=\int_{t}^{b}\frac{f(s)}{T(t,s,\alpha)}\,ds,$ (8)

with $t\in[a,b]$.

We say that $f\in L_{T}^{1}[a,b]$ if
$J_{T,a^{+}}^{\alpha}|f|(t),J_{T,b^{-}}^{\alpha}|f|(t)<\infty$ for every
$t\in[a,b]$.

Note that these operators generalize the integral operators in Definitions 1,
2 and 3:

$(A)$ If we choose

$g(t)=t,\quad G(x,\alpha)=\Gamma(\alpha)\,x^{1-\alpha},\quad
T(t,s,\alpha)=\Gamma(\alpha)\,|t-s|^{1-\alpha},$

then $J_{T,a^{+}}^{\alpha}$ and $J_{T,b^{-}}^{\alpha}$ are the right and left
Riemann-Liouville fractional integrals ${}^{RL}\\!{{J}_{{a}^{+}}^{\alpha}}$
and ${}^{RL}\\!{{J}_{{b}^{-}}^{\alpha}}$ in (1) and (2), respectively. Its
corresponding right and left Riemann-Liouville fractional derivatives are

$\displaystyle\big{(}{{}^{RL}\\!D_{a^{+}}^{\alpha}}f\big{)}(t)=\frac{d}{dt}\left({{}^{RL}\\!{{J}_{{a}^{+}}^{1-\alpha}}f(t)}\right),\quad\big{(}{{}^{RL}\\!D_{b^{-}}^{\alpha}}f\big{)}(t)=-\frac{d}{dt}\left({{}^{RL}\\!{{J}_{{b}^{-}}^{1-\alpha}}f(t)}\right).$

$(B)$ If we choose

$g(t)=\log t,\quad G(x,\alpha)=\Gamma(\alpha)\,x^{1-\alpha},\quad
T(t,s,\alpha)=\Gamma(\alpha)\,t\,\Big{|}\log\frac{t}{s}\Big{|}^{1-\alpha},$

then $J_{T,a^{+}}^{\alpha}$ and $J_{T,b^{-}}^{\alpha}$ are the right and left
Hadamard fractional integrals ${H}_{{a}^{+}}^{\alpha}$ and
${H}_{{b}^{-}}^{\alpha}$ in (3) and (4), respectively. Its corresponding right
and left Hadamard fractional derivatives are

$\displaystyle{\big{(}{}^{H}\\!{D_{a^{+}}^{\alpha}}}f\big{)}(t)=t\,\frac{d}{dt}\big{(}{{H}_{{a}^{+}}^{1-\alpha}}f(t)\big{)},\quad{\big{(}{}^{H}\\!{D_{b^{-}}^{\alpha}}}f\big{)}(t)=-t\,\frac{d}{dt}\big{(}{{{H}_{{b}^{-}}^{1-\alpha}}f(t)}\big{)}.$

$(C)$ If we choose a function $g$ with the properties in Definition 4 and

$G(x,\alpha)=\Gamma(\alpha)\,x^{1-\alpha},\quad
T(t,s,\alpha)=\Gamma(\alpha)\,\frac{\,|\,g(t)-g(s)|^{1-\alpha}}{g^{\prime}(s)}\,,$

then $J_{T,a^{+}}^{\alpha}$ and $J_{T,b^{-}}^{\alpha}$ are the right and left
fractional integrals $I_{g,a^{+}}^{\alpha}$ and $I_{g,b^{-}}^{\alpha}$ in (5)
and (6), respectively.

###### Definition 5

Let $a<b$ and $\alpha\in\mathbb{R}^{+}$. Let $g:[a,b]\rightarrow\mathbb{R}$ be
a positive function on $(a,b]$ with continuous positive derivative on $(a,b)$,
and $G:[0,g(b)-g(a)]\times(0,\infty)\rightarrow\mathbb{R}$ a continuous
function which is positive on $(0,g(b)-g(a)]\times(0,\infty)$. For each
function $f\in L_{T}^{1}[a,b]$, its _right and left generalized derivative of
order_ $\alpha$ are defined, respectively, by

$\displaystyle D_{T,a^{+}}^{\alpha}f(t)$
$\displaystyle=\frac{1}{g^{\prime}(t)}\,\frac{d}{dt}\left({J}_{T,a^{+}}^{1-\alpha}f(t)\right),$
(9) $\displaystyle D_{T,b^{-}}^{\alpha}f(t)$
$\displaystyle=\frac{-1}{g^{\prime}(t)}\,\frac{d}{dt}\left({J}_{T,b^{-}}^{1-\alpha}f(t)\right).$

for each $t\in(a,b)$.

Note that if we choose

$g(t)=t,\quad G(x,\alpha)=\Gamma(\alpha)\,x^{1-\alpha},\quad
T(t,s,\alpha)=\Gamma(\alpha)\,|t-s|^{1-\alpha},$

then $D_{T,a^{+}}^{\alpha}f(t)=\,^{RL}\\!{{D}_{a^{+}}^{\alpha}}f(t)$ and
$D_{T,b^{-}}^{\alpha}f(t)=\,^{RL}\\!{{D}_{b^{-}}^{\alpha}}f(t)$. Also, we can
obtain Hadamard and others fractional derivatives as particular cases of this
generalized derivative.

##  4 Jensen-type inequalities for $m$-convex functions

The property of $m$-convexity for functions on $[0,b]$, $b>0$ was introduced
in [19] as an intermediate property between the usual convexity and starshaped
property. Since then many properties, especially inequalities, have been
obtained for them (cf. [3, 8, 9, 13]). One of the classical integral
inequalities frequently studied in this setting is Jensen’s inequality, which
relates the value of a convex function of an integral to the integral of the
convex function. It was proved in 1906 [6], and it can be stated as follows:

Let $\mu$ be a probability measure on the space $X$. If $f:X\rightarrow(a,b)$
is $\mu$-integrable and $\varphi$ is a convex function on $(a,b)$, then

$\varphi\left(\int_{X}f\,d\mu\right)\leq\int_{X}\varphi\circ f\,d\mu\,.$

###### Definition 6

Let $I\subseteq\mathbb{R}$ be an interval containing the zero, and let
$m\in(0,1]$. A function $\varphi:I\rightarrow\mathbb{R}$ is said to be
$m$-convex if the inequality

$\varphi(tx+m(1-t)y)\leq t\varphi(x)+m(1-t)\varphi(y),$ (10)

holds for every pair of points $x,y\in I$ and every coefficient $t\in[0,1]$.

If $m\in(0,1)$, then the hypothesis $0\in I$ guarantees that $tx+m(1-t)y\in
I$.

It is clear that taking $m=1$ in Definition 6 we recover the concept of
classical convex functions on $I$. Note that in this case it is not necessary
the hypothesis $0\in I$, since $tx+(1-t)y\in I$ for every $x,y\in I$.

Note that if we choose the coefficient $t=0$ in (10), we get the inequality
$\varphi(my)\leq m\varphi(y)$.

Also, Definition 6 is equivalent to

$\varphi(mtx+(1-t)y)\leq mt\varphi(x)+(1-t)\varphi(y),$ (11)

for all $x,y\in I$ and $t\in[0,1]$.

The following discrete Jensen-type inequality for $m$-convex functions was
established in [13, Theorem 3.2]:

###### Theorem 7

Let $I\subseteq\mathbb{R}$ be an interval containing the zero, and let
$\sum_{k=1}^{n}w_{k}x_{k}$ be a convex combination of points $x_{k}\in I$ with
coefficients $w_{k}\in[0,1]$. If $\varphi$ is an $m$-convex function on $I$,
with $m\in(0,1]$, then

$\varphi\Big{(}m\sum_{k=1}^{n}w_{k}x_{k}\Big{)}\leq
m\sum_{k=1}^{n}w_{k}\varphi(x_{k}).$ (12)

This inequality is a discrete version of the following one for continuous
$m$-convex functions [13, Corollary 4.2]:

###### Theorem 8

Let $\mu$ be a probability measure on the space $X$. If $I\subseteq\mathbb{R}$
is an interval containing the zero, $f:X\rightarrow I$ is $\mu$-integrable and
$\varphi$ is a continuous $m$-convex function on $I$, with $m\in(0,1]$, then

$\varphi\left(m\int_{X}f\,d\mu\right)\leq m\int_{X}\varphi\circ f\,d\mu\,.$
(13)

The following discrete Jensen-type inequality for convex functions appears in
[10, Theorem 1.2]:

###### Theorem 9

Let $x_{1}\leq x_{2}\leq\dots\leq x_{n}$ and let $\\{w_{k}\\}_{k=1}^{n}\\!$ be
positive weights whose sum is $1$. If $\varphi$ is a convex function on
$[x_{1},x_{n}]$, then

$\varphi\Big{(}x_{1}+x_{n}-\sum_{k=1}^{n}w_{k}x_{k}\Big{)}\leq\varphi(x_{1})+\varphi(x_{n})-\sum_{k=1}^{n}w_{k}\varphi(x_{k}).$

Our purpose is to prove continuous versions of the above discrete inequality
in the setting of $m$-convexity (see Theorems 14 and 15). Before stating such
a result, we require some properties of the $m$-convex functions.

###### Lemma 10

Let $I\subseteq\mathbb{R}$ be an interval containing the zero, and let
$\varphi$ be an $m$-convex function on $I$ with $m\in(0,1]$. For
$\\{x_{k}\\}_{k=1}^{n}\subset I\\!$ such that $x_{1}\leq x_{2}\leq\cdots\leq
x_{n}$, the following inequalities hold:

$\varphi\left(x_{1}+mx_{n}-mx_{k}\right)\leq\varphi\left(x_{1}\right)+m\varphi\left(x_{n}\right)-\varphi\left(mx_{k}\right),\quad
1\leq k\leq n.$ (14)

Proof. Let us consider $y_{k}=x_{1}+mx_{n}-mx_{k}$. Then
$x_{1}+mx_{n}=y_{k}+mx_{k}$ and so, the pairs $x_{1},mx_{n}$ and
$y_{k},mx_{k}$ have the same mid-point. Since $x_{1}\leq y_{k}$ and
$mx_{k}\leq mx_{n}$, we have $x_{1}\leq y_{k},mx_{k}\leq mx_{n}$ and there
exists $\lambda\in[0,1]$ such that

$\displaystyle mx_{k}=\lambda x_{1}+m(1-\lambda)x_{n},$ $\displaystyle
y_{k}=(1-\lambda)x_{1}+m\lambda x_{n},$

for $1\leq k\leq n$. From Definition 6 and its equivalent form (11) we get

$\displaystyle\varphi\left(y_{k}\right)$ $\displaystyle\leq
m\lambda\varphi\left(x_{n}\right)+(1-\lambda)\varphi\left(x_{1}\right)$
$\displaystyle=\varphi\left(x_{1}\right)+m\varphi\left(x_{n}\right)-\big{[}\lambda\varphi\left(x_{1}\right)+m(1-\lambda)\varphi\left(x_{n}\right)\big{]}$
$\displaystyle\leq\varphi\left(x_{1}\right)+m\varphi\left(x_{n}\right)-\varphi\left(\lambda
x_{1}+m(1-\lambda)x_{n}\right)$
$\displaystyle=\varphi\left(x_{1}\right)+m\varphi\left(x_{n}\right)-\varphi\left(mx_{k}\right),$

and (14) follows.

Note that since $m\in(0,1]$, the hypothesis $0\in I$ guarantees that
$mx_{k}\in I$.

The following two results generalize Theorem 9 in the setting of
$m$-convexity.

###### Theorem 11

Let $I\subseteq\mathbb{R}$ be an interval containing the zero, let
$\\{x_{k}\\}_{k=1}^{n}\subset I$ with $x_{1}\leq x_{2}\leq\dots\leq x_{n}$ and
let $\\{w_{k}\\}_{k=1}^{n}\\!$ be positive weights whose sum is $1$. If
$\varphi$ is an $m$-convex function on $I$, with $m\in(0,1]$, then

$\varphi\Big{(}mx_{1}+m^{2}x_{n}-m^{2}\sum_{k=1}^{n}w_{k}x_{k}\Big{)}\leq
m\varphi(x_{1})+m^{2}\varphi(x_{n})-m\sum_{k=1}^{n}w_{k}\varphi(mx_{k}).$ (15)

###### Remark 12

Theorem 9 gives that if $m=1$, the inequality in Theorem 11 also holds if we
remove the hypothesis $0\in I$.

Proof. First, note that

$x_{1}+mx_{n}-m\sum_{k=1}^{n}w_{k}x_{k}=\sum_{k=1}^{n}w_{k}(x_{1}+mx_{n}-mx_{k}),$

and thus

$mx_{1}+m^{2}x_{n}-m^{2}\sum_{k=1}^{n}w_{k}x_{k}=m\sum_{k=1}^{n}w_{k}(x_{1}+mx_{n}-mx_{k}).$

Then it follows from (12) and (14) that

$\displaystyle\varphi\Big{(}mx_{1}+m^{2}x_{n}-m^{2}\sum_{k=1}^{n}w_{k}x_{k}\Big{)}$
$\displaystyle=\varphi\Big{(}m\sum_{k=1}^{n}w_{k}(x_{1}+mx_{n}-mx_{k})\Big{)}$
$\displaystyle\leq
m\sum_{k=1}^{n}w_{k}\varphi\left(x_{1}+mx_{n}-mx_{k}\right)$
$\displaystyle\leq
m\sum_{k=1}^{n}w_{k}\left(\varphi(x_{1})+m\varphi(x_{n})-\varphi(mx_{k})\right)$
$\displaystyle=m\varphi(x_{1})+m^{2}\varphi(x_{n})-m\sum_{k=1}^{n}w_{k}\varphi(mx_{k}),$

and this concludes the proof of the inequality.

Let us check that the hypothesis $0\in I$ guarantees that
$mx_{1}+m^{2}x_{n}-m^{2}\sum_{k=1}^{n}w_{k}x_{k}\in I$:

Assume that $I=[a,b]$. Then

$\displaystyle mx_{1}+m^{2}x_{n}-m^{2}\sum_{k=1}^{n}w_{k}x_{k}$
$\displaystyle\geq mx_{1}+m^{2}x_{n}-m^{2}\sum_{k=1}^{n}w_{k}x_{n}$
$\displaystyle=mx_{1}\geq\min\\{0,x_{1}\\}\geq a.$

Also,

$\displaystyle mx_{1}+m^{2}x_{n}-m^{2}\sum_{k=1}^{n}w_{k}x_{k}$
$\displaystyle\leq mx_{1}+m^{2}x_{n}-m^{2}\sum_{k=1}^{n}w_{k}x_{1}$
$\displaystyle=mx_{1}-m^{2}x_{1}+m^{2}x_{n}.$

If $x_{1}\leq 0$, then $mx_{1}-m^{2}x_{1}\leq 0$ and so,

$\displaystyle mx_{1}+m^{2}x_{n}-m^{2}\sum_{k=1}^{n}w_{k}x_{k}$
$\displaystyle\leq mx_{1}-m^{2}x_{1}+m^{2}x_{n}\leq
m^{2}x_{n}\leq\max\\{0,x_{n}\\}\leq b.$

Assume now that $x_{1}>0$. Let us consider the function
$v(t)=tx_{1}-t^{2}x_{1}+t^{2}x_{n}$. Since
$v^{\prime}(t)=x_{1}+2t(x_{n}-x_{1})>0$ and $m\in(0,1]$, we have

$\displaystyle v(m)\leq v(1)$ $\displaystyle=x_{n}\leq b,$ $\displaystyle
mx_{1}+m^{2}x_{n}-m^{2}\sum_{k=1}^{n}w_{k}x_{k}$ $\displaystyle\leq
mx_{1}-m^{2}x_{1}+m^{2}x_{n}=v(m)\leq b.$

If $I$ is not a closed interval $[a,b]$, a similar argument gives the result.

If $\varphi$ is a continuous $m$-convex function, we can obtain the following
improvement of Theorem 11.

###### Theorem 13

Let $a\leq 0\leq b$, let $\\{y_{k}\\}_{k=1}^{n}\subset[a,b]$ and let
$\\{w_{k}\\}_{k=1}^{n}\\!$ be positive weights whose sum is $1$. If $\varphi$
is a continuous $m$-convex function on $[a,b]$, with $m\in(0,1]$, then

$\varphi\Big{(}ma+m^{2}b-m^{2}\sum_{k=1}^{n}w_{k}y_{k}\Big{)}\leq
m\varphi(a)+m^{2}\varphi(b)-m\sum_{k=1}^{n}w_{k}\varphi(my_{k}).$ (16)

Proof. If we consider $0<\varepsilon<1$, $y_{0}=a$, $y_{n+1}=b$,
$w_{k}^{\prime}=(1-\varepsilon)w_{k}$ $(1\leq k\leq n)$,
$w_{0}^{\prime}=\varepsilon/2$ and $w_{n+1}^{\prime}=\varepsilon/2$, then
$\sum_{k=0}^{n+1}w_{k}^{\prime}=1$ and Theorem 11 gives

$\displaystyle\varphi\Big{(}ma$
$\displaystyle+m^{2}b-\frac{m^{2}\varepsilon}{2}\,a-\frac{m^{2}\varepsilon}{2}\,b-m^{2}\sum_{k=1}^{n}(1-\varepsilon)w_{k}y_{k}\Big{)}$
$\displaystyle\leq
m\varphi(a)+m^{2}\varphi(b)-\frac{m\varepsilon}{2}\,\varphi(ma)-\frac{m\varepsilon}{2}\,\varphi(mb)-m\sum_{k=1}^{n}(1-\varepsilon)w_{k}\varphi(my_{k}).$

Since $\varphi$ is a continuous function on $[a,b]$, if we take
$\varepsilon\to 0^{+}$, we obtain (16).

Next, we present a continuous version of the above discrete inequality.

###### Theorem 14

Let $\mu$ be a probability measure on the space $X$ and $a\leq 0\leq b$ real
constants. If $f:X\rightarrow[a,b]$ is a measurable function and $\varphi$ is
a continuous $m$-convex function on $[a,b]$, with $m\in(0,1]$, then $f$ and
$\varphi(mf)$ are $\mu$-integrable functions and

$\varphi\Big{(}ma+m^{2}b-m^{2}\int_{X}f\,d\mu\Big{)}\leq
m\varphi(a)+m^{2}\varphi(b)-m\int_{X}\varphi(mf)\,d\mu.$ (17)

If $m=1$, this inequality also holds if we remove the hypothesis $0\in[a,b]$.

Proof. Since $a\leq f\leq b$ and $\varphi$ is a continuous function on
$[a,b]$, we have that $f$ and $\varphi(mf)$ are bounded measurable functions
on $X$. And using that $\mu$ is a probability measure on $X$, we conclude that
$f$ and $\varphi(mf)$ are $\mu$-integrable functions.

For each $n\geq 1$ and $0\leq k\leq 2^{n}$, let us consider the sets

$E_{n,k}=\big{\\{}x\in X:\,a+k2^{-n}(b-a)\leq
f(x)<a+(k+1)2^{-n}(b-a)\big{\\}}.$

Since $f$ is a measurable function satisfying $a\leq f\leq b$, we have that
$\\{E_{n,k}\\}_{k=0}^{2^{n}}$ are pairwise disjoint measurable sets and
$X=\cup_{k=0}^{2^{n}}E_{n,k}$ for each $n$. Thus,

$\sum_{k=0}^{2^{n}}\mu(E_{n,k})=1$

for each $n$.

Since $f$ is a measurable function satisfying $a\leq f\leq b$ and
$\\{E_{n,k}\\}_{k=0}^{2^{n}}$ is a partition of $X$, the sequence of simple
functions

$f_{n}=\sum_{k=0}^{2^{n}}\big{(}a+k2^{-n}(b-a)\big{)}\chi_{E_{n,k}}$

satisfies $a\leq f_{n}\leq b$ and $f-2^{-n}(b-a)<f_{n}\leq f$ for every $n$
and so,

$\lim_{n\to\infty}f_{n}=f.$

Note that

$\int_{X}f_{n}\,d\mu=\sum_{k=0}^{2^{n}}\big{(}a+k2^{-n}(b-a)\big{)}\mu(E_{n,k}).$

Since $\\{E_{n,k}\\}_{k=0}^{2^{n}}$ is a partition of $X$, we have

$\displaystyle\varphi(mf)$
$\displaystyle=\sum_{k=0}^{2^{n}}\varphi\big{(}ma+mk2^{-n}(b-a)\big{)}\chi_{E_{n,k}},$
$\displaystyle\int_{X}\varphi(mf)\,d\mu$
$\displaystyle=\sum_{k=0}^{2^{n}}\varphi\big{(}ma+mk2^{-n}(b-a)\big{)}\mu(E_{n,k}).$

Hence, Theorem 13 gives

$\varphi\Big{(}ma+m^{2}b-m^{2}\int_{X}f_{n}\,d\mu\Big{)}\leq
m\varphi(a)+m^{2}\varphi(b)-m\int_{X}\varphi(mf)\,d\mu.$ (18)

If $m=1$, Theorem 9 gives the above inequality without the hypothesis
$0\in[a,b]$.

Since $a\leq f_{n}\leq b$ for every $n$, $\mu$ is a finite measure and
$\displaystyle{\lim_{n\to\infty}f_{n}=f}$, dominated convergence theorem gives

$\lim_{n\to\infty}\int_{X}f_{n}\,d\mu=\int_{X}f\,d\mu.$

If $m\in(0,1)$, then the hypothesis $0\in[a,b]$ guarantees that
$ma+m^{2}b-m^{2}\int_{X}f_{n}\,d\mu\in[a,b]$:

Since $a\leq 0\leq b$, we have

$\displaystyle ma+m^{2}b-m^{2}\int_{X}f_{n}\,d\mu$ $\displaystyle\leq
ma+m^{2}b-m^{2}a=m(1-m)a+m^{2}b\leq m^{2}b\leq b,$ $\displaystyle
ma+m^{2}b-m^{2}\int_{X}f_{n}\,d\mu$ $\displaystyle\geq ma+m^{2}b-m^{2}b=ma\geq
a.$

If $m=1$, then

$\displaystyle a+b-\int_{X}f_{n}\,d\mu$ $\displaystyle\leq a+b-a=b,$
$\displaystyle a+b-\int_{X}f_{n}\,d\mu$ $\displaystyle\geq a+b-b=a,$

and so, we do not need the hypothesis $0\in[a,b]$.

Since

$a\leq ma+m^{2}b-m^{2}\int_{X}f_{n}\,d\mu\leq b$

for every $n$ and $m\in(0,1]$, and $\varphi$ is a continuous function on
$[a,b]$,

$\lim_{n\to\infty}\varphi\Big{(}ma+m^{2}b-m^{2}\int_{X}f_{n}\,d\mu\Big{)}=\varphi\Big{(}ma+m^{2}b-m^{2}\int_{X}f\,d\mu\Big{)}.$

Since $a\leq mf_{n}\leq b$ for every $n$,
$\displaystyle{\lim_{n\to\infty}f_{n}=f}$ and $\varphi$ is a continuous
function on $[a,b]$,
$\displaystyle{\lim_{n\to\infty}\varphi(mf_{n})=\varphi(mf)}$.

Again, from the continuity of $\varphi$ on $[a,b]$, there exists a positive
constant $K$ with $|\varphi|\leq K$ on $[a,b]$ and so, $|\varphi(mf_{n})|\leq
K$ for every $n$.

In view of the finiteness of $\mu$, dominated convergence theorem guarantees
that

$\lim_{n\to\infty}\int_{X}\varphi(mf)\,d\mu=\int_{X}\varphi(mf)\,d\mu\,.$

Combining the foregoing facts with (18), we obtain (17).

If $m=1$, it is possible to improve Theorem 14, by removing the hypothesis of
continuity.

###### Theorem 15

Let $\mu$ be a probability measure on the space $X$ and $a\leq b$ real
constants. If $f:X\rightarrow[a,b]$ is a measurable function and $\varphi$ is
a convex function on $[a,b]$, then $f$ and $\varphi\circ f$ are
$\mu$-integrable functions and

$\varphi\Big{(}a+b-\int_{X}f\,d\mu\Big{)}\leq\varphi(a)+\varphi(b)-\int_{X}\varphi\circ
f\,d\mu.$

Proof. Since $\varphi$ is a convex function on $[a,b]$, $\varphi$ is
continuous on $(a,b)$ and there exist the limits

$\lim_{s\to a^{+}}\varphi(s),\qquad\lim_{s\to b^{-}}\varphi(s).$

Define $\varphi^{*}$ as follows

$\varphi^{*}(t)=\begin{cases}\varphi(t)\quad&\text{if }\,t\in(a,b),\\\
\lim_{s\to a^{+}}\varphi(s)\quad&\text{if }\,t=a,\\\ \lim_{s\to
b^{-}}\varphi(s)\quad&\text{if }\,t=b.\end{cases}$

Hence, $\varphi^{*}$ is a continuous convex function on $[a,b]$, and by
Theorem 14 with $m=1$, we have

$\varphi^{*}\Big{(}a+b-\int_{X}f\,d\mu\Big{)}\leq\varphi^{*}(a)+\varphi^{*}(b)-\int_{X}\varphi^{*}\circ
f\,d\mu.$

Assume that $f=a$ $\mu$-a.e. or $f=b$ $\mu$-a.e.; in the first case,

$\displaystyle\varphi\Big{(}a+b-\int_{X}f\,d\mu\Big{)}$
$\displaystyle=\varphi(a+b-a)=\varphi(b)$
$\displaystyle=\varphi(a)+\varphi(b)-\varphi(a)=\varphi(a)+\varphi(b)-\int_{X}\varphi\circ
f\,d\mu\,;$

in the second case,

$\displaystyle\varphi\Big{(}a+b-\int_{X}f\,d\mu\Big{)}$
$\displaystyle=\varphi(a+b-b)=\varphi(a)$
$\displaystyle=\varphi(a)+\varphi(b)-\varphi(b)=\varphi(a)+\varphi(b)-\int_{X}\varphi\circ
f\,d\mu\,.$

Otherwise, $\displaystyle{a<\int_{X}f\,d\mu<b}$ and
$\displaystyle{a<a+b-\int_{X}f\,d\mu<b}$. Consequently,

$\varphi\Big{(}a+b-\int_{X}f\,d\mu\Big{)}=\varphi^{*}\Big{(}a+b-\int_{X}f\,d\mu\Big{)}.$

If we define

$\Delta_{a}=\varphi(a)-\varphi^{*}(a)\geq
0,\qquad\Delta_{b}=\varphi(b)-\varphi^{*}(b)\geq 0,$

then

$\displaystyle\varphi$
$\displaystyle=\varphi^{*}+\Delta_{a}\,\chi_{\\{a\\}}+\Delta_{b}\,\chi_{\\{b\\}},$
$\displaystyle\varphi\circ f$ $\displaystyle=\varphi^{*}\circ
f+\Delta_{a}\,\chi_{\\{f=a\\}}+\Delta_{b}\,\chi_{\\{f=b\\}},$

where $\chi_{A}$ is the function with value $1$ on the set $A$ and $0$
otherwise (i.e., the characteristic function of $A$). Hence,

$\int_{X}\varphi\circ f\,d\mu=\int_{X}\varphi^{*}\circ
f\,d\mu+\Delta_{a}\,\mu\big{(}\\{f=a\\}\big{)}+\Delta_{b}\,\mu\big{(}\\{f=b\\}\big{)},$

and we have

$\displaystyle\varphi\Big{(}a+b-\int_{X}f\,d\mu\Big{)}=\varphi^{*}\Big{(}a+b-\int_{X}f\,d\mu\Big{)}$
$\displaystyle\quad\leq\varphi^{*}(a)+\varphi^{*}(b)-\int_{X}\varphi^{*}\circ
f\,d\mu$
$\displaystyle\quad=\varphi(a)-\Delta_{a}+\varphi(b)-\Delta_{b}-\int_{X}\varphi\circ
f\,d\mu+\Delta_{a}\,\mu\big{(}\\{f=a\\}\big{)}+\Delta_{b}\,\mu\big{(}\\{f=b\\}\big{)}$
$\displaystyle\quad=\varphi(a)+\varphi(b)-\int_{X}\varphi\circ
f\,d\mu-\Delta_{a}\big{[}1-\mu\big{(}\\{f=a\\}\big{)}\big{]}-\Delta_{b}\big{[}1-\mu\big{(}\\{f=b\\}\big{)}\big{]}$
$\displaystyle\quad\leq\varphi(a)+\varphi(b)-\int_{X}\varphi\circ f\,d\mu.$

Theorem 15 has the following direct consequence.

###### Corollary 16

Let $a\leq b$, let $\\{y_{k}\\}_{k=1}^{n}\subset[a,b]$ and let
$\\{w_{k}\\}_{k=1}^{n}\\!$ be positive weights whose sum is $1$. If $\varphi$
is a convex function on $[a,b]$, then

$\varphi\Big{(}a+b-\sum_{k=1}^{n}w_{k}y_{k}\Big{)}\leq\varphi(a)+\varphi(b)-\sum_{k=1}^{n}w_{k}\varphi(y_{k}).$

Note that Theorem 15 provides a kind of converse of the classical Jensen’s
inequality for convex functions.

###### Proposition 17

Let $\mu$ be a probability measure on the space $X$ and $a\leq b$ real
constants. If $f:X\rightarrow[a,b]$ is a measurable function and $\varphi$ is
a convex function on $[a,b]$, then $f$ and $\varphi\circ f$ are
$\mu$-integrable functions and

$\varphi\Big{(}\int_{X}f\,d\mu\Big{)}\leq\int_{X}\varphi\circ
f\,d\mu\leq\varphi(a)+\varphi(b)-\varphi\Big{(}a+b-\int_{X}f\,d\mu\Big{)}.$

Theorems 15 and 14 have, respectively, the following direct consequences for
general fractional integrals of Riemann-Liouville type.

###### Proposition 18

Let $c<d$ and $a\leq b$ be real constants. If $f:[c,d]\rightarrow[a,b]$ is a
measurable function, $\varphi$ is a convex function on $[a,b]$, and

$\mathbb{T}(\alpha)=\int_{c}^{d}\frac{1}{T(d,s,\alpha)}\,ds=\int_{0}^{g(d)-g(c)}\frac{dx}{G(x,\alpha)}<\infty,$

then
$f(s)/T\big{(}d,s,\alpha\big{)},\varphi(f(s))/T\big{(}d,s,\alpha\big{)}\in
L^{1}[c,d]$ and

$\varphi\Big{(}a+b-\frac{1}{\mathbb{T}(\alpha)}\int_{c}^{d}\frac{f(s)}{T\big{(}d,s,\alpha\big{)}}\,ds\Big{)}\leq\varphi(a)+\varphi(b)-\frac{1}{\mathbb{T}(\alpha)}\int_{c}^{d}\frac{\varphi(f(s))}{T\big{(}d,s,\alpha\big{)}}\,ds.$

###### Proposition 19

Let $c<d$ and $a\leq 0\leq b$ be real constants. If $f:[c,d]\rightarrow[a,b]$
is a measurable function, $\varphi$ is a continuous $m$-convex function on
$[a,b]$, with $m\in(0,1]$, and

$\mathbb{T}(\alpha)=\int_{c}^{d}\frac{1}{T(d,s,\alpha)}\,ds=\int_{0}^{g(d)-g(c)}\frac{dx}{G(x,\alpha)}<\infty,$

then
$f(s)/T\big{(}d,s,\alpha\big{)},\varphi(mf(s))/T\big{(}d,s,\alpha\big{)}\in
L^{1}[c,d]$ and

$\varphi\Big{(}ma+m^{2}b-\frac{m^{2}}{\mathbb{T}(\alpha)}\\!\\!\int_{c}^{d}\\!\\!\frac{f(s)}{T\big{(}d,s,\alpha\big{)}}\,ds\Big{)}\leq
m\varphi(a)+m^{2}\varphi(b)-\frac{m}{\mathbb{T}(\alpha)}\\!\\!\int_{c}^{d}\\!\frac{\varphi(mf(s))}{T\big{(}d,s,\alpha\big{)}}\,ds.$

## Acknowledgments

The research of Yamilet Quintana, José M. Rodríguez and José M. Sigarreta is
supported by a grant from Agencia Estatal de Investigación
(PID2019-106433GB-I00 / AEI / 10.13039/501100011033), Spain. The research of
José M. Rodríguez and Yamilet Quintana is supported by the Madrid Government
(Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the
line of Excellence of University Professors (EPUC3M23), and in the context of
the V PRICIT (Regional Programme of Research and Technological Innovation).

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# NBA2Vec: Dense Feature Representations of NBA Players

Webster Guan1, Nauman Javed2, and Peter Lu3 1Department of Chemical
Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA.
<EMAIL_ADDRESS>Health Sciences and Technology, Cambridge, MA, USA.
<EMAIL_ADDRESS>of Physics, Massachusetts Institute of Technology,
Cambridge, MA, USA<EMAIL_ADDRESS>
###### Abstract

Understanding a player’s performance in a basketball game requires an
evaluation of the player in the context of their teammates and the opposing
lineup. Here, we present NBA2Vec, a neural network model based on Word2Vec [1]
which extracts dense feature representations of each player by predicting play
outcomes without the use of hand-crafted heuristics or aggregate statistical
measures. Specifically, our model aimed to predict the outcome of a possession
given both the offensive and defensive players on the court. By training on
over 3.5 million plays involving 1551 distinct players, our model was able to
achieve a 0.3 K-L divergence with respect to the empirical play-by-play
distribution. The resulting embedding space is consistent with general
classifications of player position and style, and the embedding dimensions
correlated at a significant level with traditional box score metrics. Finally,
we demonstrate that NBA2Vec accurately predicts the outcomes to various 2017
NBA Playoffs series, and shows potential in determining optimal lineup match-
ups. Future applications of NBA2Vec embeddings to characterize players’ style
may revolutionize predictive models for player acquisition and coaching
decisions that maximize team success.

##  I INTRODUCTION

Successful coaches construct optimal lineups for given situations in
basketball games based on a deep understanding of each player’s play-style,
strengths, and weaknesses in the context of all other players on the court.
Studying the distribution of contexts and their outcomes in which a player
takes part may provide insights into aspects of player’s performance and play
style that are not otherwise reflected in traditional basketball statistics.
While much of basketball analytics relies on the use of hand-crafted advanced
statistics (e.g. Wins Above Replacement and Offensive/Defensive rating) and
aggregate statistics (e.g. FG%, assists), they tend to not capture these
contextual influences and effects not present in box scores. Models capable of
characterizing players based on these contextual factors would offer greater
insight into individual player performance and play-style, and may shed light
on how to construct optimal lineups for specific situations. Constructing such
frameworks may be possible given the wealth of play-by-play game data and
recent advances in machine learning and natural language processing (NLP)
algorithms.

In particular, the problem of generating accurate representations of players
in sports analytics is analogous to the problem of word embeddings in NLP.
Word embedding models aim to create real-valued vector embeddings of words
that encode complex semantic structure. Word2Vec is a class of word embedding
models that extract useful features of words given the sentences, known as the
“context,” in which the words are commonly used [1]. This allows Word2Vec to
be trained in an unsupervised way on a large corpus of written works and
extract meaningful relationships. Once trained, the word embeddings can then
be applied to a variety of other tasks as a pretrained initial transformation
in manner analogous to transfer learning.

In the training phase, the way in which the context of each word is used can
be different; in particular, Word2Vec uses either a continuous bag-of-words
(CBOW) model, or a skip-gram model. The skip-gram method (Figure 1a) takes the
target word as input to a neural network’s hidden layer(s) and attempts to
predict the words that immediately surround the target word in a given
sentence. On the other hand, the CBOW method (Figure 1b) takes the words
around a target word as input, predicting the target word as output. The
result of training these models is a dense vectorial word representation that
captures the meaning of each word. For example, Word2Vec finds more similarity
between the words “king” and “queen” than between “king” and “vindication.”
The ability of word embeddings to accurately capture the relationship and
analogies among words is shown by Word2Vec arithmetic: for instance, Paris $-$
France $+$ Italy $=$ Rome.

Figure 1: (a) Skip-gram word embedding model. The embedding is extracted from
the hidden layer after training to a target of n context words. v represents
the vocabulary size, or length of the word vectors, while h represents the
embedding length. (b) CBOW model. A given word’s embedding is computed by
averaging the hidden layer representations of all contexts in which it
appears.

The success of word embeddings in NLP has inspired its recent application in
sports analytics to characterizing batter and pitcher performance and play
style in baseball [2]. In that study, a neural network was trained using
pitcher-batter matchups as inputs and the outcome of each at-bat (e.g. single,
home run, flyout to right field) as outputs to create the player embeddings.
The author was able to successfully visualize macroscopic differences in
embedding clusters (e.g. right-handed vs. left-handed vs. switch hitters,
power hitters vs. on-base hitters) and model previously unseen at-bat
matchups, suggesting that the word embedding concept may be feasible and
promising for creating player representations.

In this study, we applied this concept to extract representations of different
NBA players by producing a embeddings of every player, which we term NBA2Vec.
Similar to [2], the embedding for each player was generated by training a
neural network aimed at predicting the outcome of each possession given the
ten players on the court. Unlike in [2], which takes advantage of the mostly
one-on-one nature of baseball dynamics, we used all players on the court to
ensure accurate modeling of holistic relationships and dynamics between
players on the same and different teams. This increased the complexity of the
network, and due to the n-body nature of the problem, required the network to
exhibit permutation invariance. Unlike previous attempts to generate NBA
player representations [3] using purely high level, aggregate statistics and
hand-picked features (e.g. number of shots taken in the paint, FG%, FT%,
assists, etc.), our embedding approach learns directly from raw play-by-play
data by automatically generating rich features which account for the “context”
that affects a player’s style and statistics. The latent features encoded in
these player embeddings can shed light on both the play style and
effectiveness of different types of players, and can be used as inputs for
further downstream processing and prediction tasks.

##  II METHODS

###  II-A Data Sets and Preprocessing

We used play-by-play and players-on-court data provided by the NBA, which
featured over 9 million distinct plays, with 1551 distinct players taking the
court in these plays. To create the input to the network, we denoted each
player with an index from 0 to 1550. For the outputs to the network, we needed
to encode the possible outcomes of each play. In order to encourage learning,
we only considered key outcomes, omitting rebounds and defensive plays. We
chose to use 23 distinct outcomes, some examples of which are included in
Table 1. The provided raw play-by-play data was more specific on outcomes of
plays, but we grouped many of these plays (e.g. “driving layup shot, dunk
shot, reverse dunk shot, hook shot” were all considered “close-up shots”)
together for simplicity. This preprocessing resulted in 4.5 million plays, of
which we used 3.7 million for a training set and the remainder as a validation
set. We used the Pandas library to preprocess the data [4].

TABLE 1: Types of play outcomes. Play Index | Play Outcome
---|---
0 | Mid-range jump shot made
1 | Mid-range jump shot missed
2 | Mid-range jump shot made + 1 free throw made
3 | Mid-range jump shot made + 1 free throw missed
4 | Close-range shot made
5 | Close-range shot missed
6 | Close-range shot made + 1 free throw made
7 | Close-range shot made + 1 free throw missed
8 | 0/1 FT made
9 | 1/1 FT made
10 | 0/2 FT made
11 | 1/2 FT made
12 | 2/2 FT made
13 | 0/3 FT made
14 | 1/3 FT made
15 | 2/3 FT made
16 | 3/3 FT made
17 | 3PT shot made
18 | 3PT shot missed
19 | 3PT shot made + 1 free throw made
20 | 3PT shot made + 1 free throw missed
21 | Turnover
22 | Foul

###  II-B NBA2Vec: Player Embedding Model

To train informative embeddings for players in the NBA, we created a neural
network architecture that predicts the distribution of play outcomes given a
particular offensive and defensive lineup (Figure 2). For each play, we first
embed the 10 players on the court using an 8 dimensional shared player
embedding. We then separately average the 5 offensive and 5 defensive player
embedding vectors. These two mean player embeddings (i.e. an offensive and a
defensive lineup embedding) are concatenated and fed through one additional
hidden layer of size 128 with a ReLU activation before outputting 23 outcome
scores. Applying a softmax activation to the scores produces probabilities
that we interpret as the distribution of play outcomes (Table 1). The entire
network is trained end-to-end with a cross entropy loss function that
stochastically minimizes the K–L divergence between the true play outcomes
from the data and the predicted distribution from the model. This model was
built and trained using the PyTorch framework [5].

###  II-C Validation and Post-processing

####  II-C1 Validation of NBA2Vec

To evaluate the efficacy of the NBA2Vec model used to generate the embeddings,
we characterized the difference between the predicted and empirical
distributions of play outcomes. The Kullback–Leibler (K–L) divergence was used
as the metric to compare the distributions. K–L divergence is given by

$D_{KL}\left(p\middle\|q\right)=\sum_{i=1}^{N}{p(x_{i})\log{\frac{p(x_{i})}{q(x_{i})}}}.$
(1)

This measures the number of encoded bits lost when modeling a target
distribution $p(x)$ (in this case, the empirical distribution) with some
approximate distribution $q(x)$ (in this case, our predictive model). Thus, a
low K–L divergence value ($D_{KL}\approx 0$) implies a good approximation of
the target distribution, while a larger value ($D_{KL}\gg 0$) implies poor
approximation.

Due to the large number of unique lineup-matchup combinations, some of which
do not appear enough for a proper empirical distribution to be generated, we
decided to only look at K–L divergences for lineup-matchup combinations with
more than 15 plays. This analysis was performed on the last 25 games of the
data set (corresponding to the last 25 playoff games in the 2018 NBA playoffs,
and 5102 plays).

####  II-C2 Embedding Analysis

After training the model, we extracted the shared embedding layer and use
dimensionality reduction, clustering, and visualization methods to explore the
learned player embeddings. In particular, we used t-SNE—a dimensionality
reduction method based on local neighbor structure—to visualize our 8
dimensional embeddings in 2 dimensions [6]. We also used 2D principal
component analysis (PCA) for dimensionality reduction before performing
k-means clustering. PCA is a statistical method that uses orthogonal
decomposition to transform a set of observations of correlated variables into
a set of observations of uncorrelated variables, or principal components. Each
successive principal component explains the maximal variance in the data while
remaining orthogonal to the preceding component. K-means clustering is a
simple clustering method that aims to partition n observations into k
clusters, where each observation belongs to the cluster with the nearest mean.
Our approaches are further described in section III-A. These dimensionality
and clustering techniques were conducted using implementations from the
Scikit-learn library [7].

####  II-C3 Exploring Lineup Combinations

To further explore the macroscopic predictive nature allowed by these
embeddings, we used the neural network model to predict the outcomes of games
based on each team’s most frequent 5-player lineup. For each pair of teams,
the model would output the distribution of possible play outcomes. We would
then sample these distributions to determine which plays would occur in a
given game, and based on this, predict a game score. Assuming 100 possessions
for each team and that no substitutions are ever made, we ran the model on
various playoff series match-ups from the 2016-17 season, simulating 1000
best-of-7 series between each pair of teams. Certain playoff series were not
simulated because the most frequent lineups contained players that were not
among the raw data’s most common 500 players.

Figure 2: NBA2Vec model architecture. We have $v=1551$ players, which are
mapped to $h=8$ dimensional embeddings. After the $n=5$ offensive and
defensive player embeddings are separately averaged and then concatenated,
they are fed through an $i=128$ hidden layer with a ReLU activation. The final
output layer with a softmax activation predicts a probability distribution
over $o=23$ play outcomes.

##  III RESULTS AND DISCUSSION

###  III-A Embedding Analysis

In order to better understand the generated player representations, we used
t-SNE (described in section II-C2) to visualize the 8 dimensional feature
vector in 2D. As depicted in the 2D t-SNE plot (described in section II-C2) 3,
we see that the centers and guards separate nicely, while forwards (yellow)
are scattered throughout both groups. This is consistent with our intuition
about the roles and play-styles of these classes of players—guards and centers
fulfill very distinct roles while forwards can be more center-like or guard-
like and fulfill multiple roles (i.e. LeBron James). Despite t-SNE’s utility
in preserving high dimensional local structure, it is not effective at
preserving the global structure of data, which along with the high
dimensionality of our player representations, may help explain why it is
difficult to visualize clear separation between different groups of players.

To further characterize the learned embeddings, we k-means clustered 8 scaled
and centered embedded dimensions. We decided to select 3 clusters for our
initial analysis after observing a decreased rate of decline in the variance
as a function of the number of clusters, as seen in the elbow plot in Figure
4. In Figure 5 we see that k-means yields 3 distinct clusters, which roughly
group players with similar roles/playing styles. For example, the yellow
cluster seems to correspond to guards, the green cluster with
centers/forwards, and the red cluster with forwards. Note that because of the
high dimensional nature of our player representations and the fact that we are
projecting players onto two dimensions, the euclidean distance between points
on the shown plot is not entirely representative of player similarity.

The observed clusters also seem to suggest that as opposed to fitting neatly
within the traditional 5 positions of basketball, players actually perform
diverse roles that would place them in multiple categories. As seen in figure
5, each of the clusters comprises multiple groups of players—clusters 1 and 2
comprise positions 1-5, while cluster 2 corresponds mostly to
centers/forwards. Roughly, this may reflect that successful players are rarely
one trick ponies—centers must be able to shoot, and point guards must be able
to score. In general, we can observe that the learned embeddings roughly
correspond to general basketball intuition. However, the embeddings are also
capturing player characteristics that may not be entirely reflected in
traditional metrics such as box score.

Exploring the structure of the embedded space by calculating the nearest
neighbors by distance for various players further validates the learned player
representations. For example, Chris Paul, a canonical point guard, has nearest
neighbors including other point guards such as Steve Nash, Jose Calderon, and
Jason Terry. Shaquille O’Neal, a classic big man, has nearest neighbors
including centers such as Dwight Howard, Roy Hibbert, Tiago Splitter, and Rudy
Gobert.

Next, we calculated Pearson’s correlation between the top two PCA dimensions
and player metrics including minute adjusted rates for field goals made, three
pointers, assists, rebounds, and plus—minus, as shown in figure 7. Rudimentary
analysis revealed that PCA dimension 1 correlated at a significant level
(corrected $\alpha=5\times 10^{-4}$) with rebounds, assists, and three
pointers, while PCA dimension 2 correlated at a significant level with
rebounds and assists (corrected $\alpha=5\times 10^{-4}$). For both
dimensions, the noted correlations remained significant even with Bonferroni
adjustment.

Our exploratory analysis of the embeddings reveals that the learned player
representations encode meaningful information that correspond roughly to our
intuition about various players. Through a rudimentary analysis, the embedded
dimensions seem to correspond to a complex combination of player
characteristics as well as real player performance metrics.

Figure 3: Two-dimensional t-SNE visualization of player representations,
colored by position. (G = guards, C = centers, F = forwards, G-F = guard-
forwards, F-C = forward-centers). Figure 4: Elbow method showing variance in
data as a function of the number of clusters in order to optimize $k$. Figure
5: Points projected on to first two principal components, colored by clusters
identified by k-means clustering with $k=3$. Figure 6: Different types of
players comprising each identified cluster in Figure 5.  Figure 7:
Correlations and p-values of different metrics with the two top PCA dimensions
for each player. Each raw metric is summed for each player and normalized by
the total number of minutes played. dim1 = PCA dimension 1, dim2 = PCA
dimension 2, fg = field goal. Reported p-values are not Bonferroni corrected
but, after correction, remain significant at $\alpha=5\times 10^{-4}$ with 5
comparisons.

###  III-B Validation of NBA2Vec

Validation of the NBA2Vec network was performed on plays in the final 25 games
of the data set, and a mean K–L divergence of $0.301\pm 0.162$ was achieved.
Some example predicted vs. empirical distributions of play outcomes are shown
in Figure 9, showing that the model is able to closely approximate the target
distribution.

We also wanted to determine the minimum number of plays needed to create an
accurate empirical distribution that can be modeled by the predictive network.
Plotting the K–L divergence vs. number of plays used in the empirical
distribution (Figure 8), we can estimate the minimum number of plays to reach
the minimum K–L divergence to be around 30.

Figure 8: K–L divergence vs. Number of plays used in empirical distribution.
The K–L divergence reaches a minimum plateau after about 30 plays.

Figure 9: Empirical vs. predicted distributions for various lineup matchups.
The outcome corresponding to each play index can be retrieved from Table 1.

###  III-C Exploring Lineup Combinations

The results (Table 2) show that even with some crude assumptions, the winner
of any given 7-game series and the average margin of victory can be
approximated using NBA2Vec embeddings and this neural network model. More
accurate game outcomes would require more precise sequence modeling of game-
by-game dynamics instead of our current play-by-play treatment; however, this
demonstrates the potential of NBA2Vec embeddings and the play outcome
predictive network.

TABLE 2: Results of 1000 simulations of 7-game series of various 2017 playoff matchups vs. ground truth. | Team 1 | Team 2 | Series Score | Margin | Team 1 Game Win %
---|---|---|---|---|---
Simulation | Cavaliers | Warriors | 1.69 vs. 4 | $-8.85$ | 29.8%
Truth | Cavaliers | Warriors | 1 vs. 4 | $-6.8$ | 20.0%
Simulation | Jazz | Clippers |  4 vs. 3.77 | $+0.61$ | 51.5%
Truth | Jazz | Clippers |  4 vs. 3 | $+1.14$ | 57.1%
Simulation | Rockets | Thunder |  4 vs. 2.05 | $+6.63$ | 66.1%
Truth | Rockets | Thunder |  4 vs. 1 | $+8.6$ | 80.0%
Simulation | Warriors | Trailblazers |  4 vs. 1.06 | $+13.3$ | 79.0%
Truth | Warriors | Trailblazers |  4 vs. 0 | $+18$ | 100%
Simulation | Spurs | Rockets |  4 vs. 3.96 | $-0.04$ | 50.3%
Truth | Spurs | Rockets |  4 vs. 2 | $+5$ | 66.7%
Simulation | Warriors | Jazz |  4 vs. 1.51 | $+10.0$ | 72.5%
Truth | Warriors | Jazz |  4 vs. 0 | $+15$ | 100%
Simulation | Wizards | Hawks |  4 vs. 3.26 | $+1.92$ | 55.1%
Truth | Wizards | Hawks |  4 vs. 2 | $+1.17$ | 66.7%
Simulation | Celtics | Wizards |  4 vs. 3.53 | $+1.11$ | 53.1%
Truth | Celtics | Wizards |  4 vs. 3 | $+1$ | 57.1%
Simulation | Celtics | Cavaliers | 3.49 vs. 4 | $-1.62$ | 46.6%
Truth | Celtics | Cavaliers | 1 vs. 4 | $-20$ | 20.0%

While serviceable as a predictor of game and series outcomes, there is also
potential use for NBA2Vec as a lineup optimizer. Given an opposing lineup,
this model facilitates selection of a corresponding optimally matched lineup
for both offense and defense. This optimization can be accomplished by
sampling the model’s predicted distribution many times for a given pair of
lineups. As an example, we optimize a lineup to face the Golden State
Warriors’ “death lineup” (Stephen Curry, Klay Thompson, Andre Iguodala, Kevin
Durant, and Draymond Green) for the Houston Rockets, where we fix the first
four players (James Harden, Chris Paul, Eric Gordon, Clint Capela) and vary
the fifth. From this analysis, we can predict the Rockets’ best possible 5th
man, and also compare his performance to that of previous starting forward
Trevor Ariza. As the simulated win percentages show (Table 3), the ideal 5th
man that is currently on the Rockets roster—and is also among the data’s 500
most common players—for combating this Warriors lineup is Nene. Compared to
Trevor Ariza, only Nene is predicted to add more value. Interestingly,
offseason acquisition and superstar Carmelo Anthony is projected to add
slightly less value for the Rockets when facing the Warriors than Trevor Ariza
(Table 3).

TABLE 3: Rockets’ lineups vs. Warriors for varying “5th man” Player | Win % | Avg. Margin of Victory ($\pm 1\sigma$)
---|---|---
Trevor Ariza | 34.4 % | $-6.69\pm 17$
Nene | 37.3% | $-5.3\pm 17$
Carmelo Anthony | 33.4% | $-7.37\pm 17.1$
Gerald Green | 31.1% | $-8.47\pm 17.1$
Joe Johnson | 30.7% | $-8.1\pm 16.8$
Brandon Knight | 28.5% | $-9.22\pm 16.7$
Michael Carter-Williams | 27.4% | $-9.58\pm 16.6$

##  IV CONCLUSIONS AND FUTURE WORK

In this study, we have demonstrated that NBA2Vec embeddings can be applied to
micro-scale prediction tasks: in particular, predicting the play-by-play
outcome distribution given players on the court. Moreover, we have also
demonstrated that NBA2Vec also shows potential towards macro-scale prediction
tasks, such as in identification of optimal lineups and prediction of game
outcomes. In future applications, this could be extrapolated to predictive
algorithms for projections of each team’s win-loss record given the players on
its roster.

In addition to applications in predictive tasks, we have shown that the
generated NBA2Vec embeddings are able to reveal underlying features of players
without using aggregate statistics such as points, FG%, and assists.
Clustering on the embeddings generally groups players in agreement with their
position and our priors about their play style/characteristics. Furthermore,
the embeddings in part reflect traditional performance metrics, as we are able
to show that they correlate at a significant level with box score statistics
including rebounds, assists, and field goal rate. Given enough G-League and
NCAA training data, player embeddings for potential recruits could also be
generated. By examining the nearest neighbor embeddings in the NBA player
space, the recruit’s “equivalent NBA player” representation could aid scouts
in characterizing him and how he would contribute to a given NBA roster.

There are various improvements that can be made to potentially extract better
player embeddings. Instead of training to predict a singular outcome to every
play, a more complex model would train to predict a series of outcomes to each
play (e.g. missed shot followed by defensive rebound, shooting foul followed
by 2/2 free throws made). To further increase the richness of the embeddings,
the network could also be modified to predict the player who commits each
action. Finally, with the appropriate player tracking data, a recurrent neural
network could be used to take as input time series of player spatial positions
and attempt to predict play outcomes and later player spatial positions.
Similar to the embeddings generated in this study, these improvements would
use only raw data to capture each player’s features and “identity.”
Ultimately, we envision a future for basketball analytics in which player
embeddings allow for unprecedented levels of player characterization, driving
predictive models that revolutionize the way front office and coaching
decisions are made.

##  V ACKNOWLEDGEMENTS

We would like to thank the NBA for organizing the 2018 NBA Hackathon and
providing the data for this analysis. We would also like to extend our thanks
to Caitlin Chen for her generosity during our trip.

## References

  * [1] T. Mikolov, K. Chen, G. Corrado, and J. Dean, “Efficient estimation of word representations in vector space,” arXiv preprint arXiv:1301.3781, 2013\.
  * [2] M. A. Alcorn, “(batter$\mid$pitcher)2vec: Statistic-free talent modeling with neural player embeddings,” MIT Sloan Sports Analytics Conference, 2018\.
  * [3] D. Lutz, “A cluster analysis of nba players,” in Proceedings of the MIT Sloan Sports Analytics Conference, 2012.
  * [4] W. McKinney et al., “Data structures for statistical computing in python,” in Proceedings of the 9th Python in Science Conference, vol. 445, pp. 51–56, Austin, TX, 2010.
  * [5] A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in PyTorch,” in NIPS-W, 2017.
  * [6] L. v. d. Maaten and G. Hinton, “Visualizing data using t-SNE,” Journal of Machine Learning Research, vol. 9, no. Nov, pp. 2579–2605, 2008.
  * [7] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay, “Scikit-learn: Machine learning in Python,” Journal of Machine Learning Research, vol. 12, pp. 2825–2830, 2011.


††thanks: These authros contributed equally††thanks: These authros contributed
equally††thanks: These authros contributed equally††thanks: These authros
contributed equally††thanks: These authros contributed equally††thanks:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Integrated Optical Vortex Microcomb

Bo Chen  State Key Laboratory of Optoelectronic Materials and Technologies,
School of Physics, School of Electronics and Information Technology, Sun Yat-
sen University, Guangzhou, 510275, China  Yueguang Zhou  DTU Electro,
Technical University of Denmark, DK-2800 Lyngby, Denmark  Peinian Huang  State
Key Laboratory of Optoelectronic Materials and Technologies, School of
Physics, School of Electronics and Information Technology, Sun Yat-sen
University, Guangzhou, 510275, China  Chaochao Ye  DTU Electro, Technical
University of Denmark, DK-2800 Lyngby, Denmark  Qian Cao  School of Optical-
Electrical and Computer Engineering,University of Shanghai for Science and
Technology, Shanghai, China  Yang Liu  DTU Electro, Technical University of
Denmark, DK-2800 Lyngby, Denmark  Chanju Kim  DTU Electro, Technical
University of Denmark, DK-2800 Lyngby, Denmark  Kresten Yvind  DTU Electro,
Technical University of Denmark, DK-2800 Lyngby, Denmark  Jin Li  CAS Key
Laboratory of Quantum Information, University of Science and Technology of
China, Hefei, 230026, China  Chunhua Dong  CAS Key Laboratory of Quantum
Information, University of Science and Technology of China, Hefei, 230026,
China  Songnian Fu  Institute of Advanced Photonics Technology, School of
Information Engineering, Guangdong University of Technology, Guangzhou,
510006, China  Qiwen Zhan  School of Optical-Electrical and Computer
Engineering,University of Shanghai for Science and Technology, Shanghai, China
Xuehua Wang  State Key Laboratory of Optoelectronic Materials and
Technologies, School of Physics, School of Electronics and Information
Technology, Sun Yat-sen University, Guangzhou, 510275, China  Minhao Pu  DTU
Electro, Technical University of Denmark, DK-2800 Lyngby, Denmark  Jin Liu
State Key Laboratory of Optoelectronic Materials and Technologies, School of
Physics, School of Electronics and Information Technology, Sun Yat-sen
University, Guangzhou, 510275, China

###### Abstract

The explorations of physical degrees of freedom with infinite
dimensionalities, such as orbital angular momentum and frequency of light,
have profoundly reshaped the landscape of modern optics with representative
photonic functional devices including optical vortex emitters
erhard2018twisted ; ni2021multidimensional ; he2022towards  and frequency
combs kippenberg2018dissipative ; picque2019frequency ; diddams2020optical .
In nanophotonics, whisper gallery mode microresonators naturally support
orbital angular momentum of light matsko2005whispering  and have been
demonstrated as on-chip emitters of monochromatic optical vortices
cai2012integrated . On the other hand, whisper gallery mode microresonators
serve as a highly-efficient nonlinear optical platform for producing light at
different frequencies - i.e., microcombs chang2022integrated . Here, we
interlace the optical vortices and microcombs by demonstrating an optical
vortex comb on an III-V integrated nonlinear microresonator. The angular-
grating-dressed nonlinear microring simultaneously emits up to 50 orbital
angular momentum modes that are each spectrally addressed to the frequency
components ((longitudinal whispering gallery modes) of the generated
microcomb. We further show that the integrated vortex comb with varied orbital
angular momenta distributed across different frequencies exhibits
unprecedented advantages in the synthesis of spatio-temporal optical pulses
with time-varying orbital angular momenta. This work may immediately boost the
development of integrated nonlinear/quantum photonics for exploring
fundamental optical physics and advancing photonic quantum technology.

Photons with helical phase fronts can carry unlimited, but quantized amount of
orbital angular momentum (OAM) allen1992orbital  and therefore serve as an
enabling technology with applications across fundamental physicshe2022towards
, optical communications wang2012terabit  and quantum photonics
erhard2018twisted . Circular optical resonators such as microdisks or
microrings sustain whisper gallery modes (WGMs) naturally carrying OAM. The
introduction of periodic angular gratings to the WGM resonator collectively
scatters the WGMs to the free space, leading to on-chip emissions of optical
vortices cai2012integrated . Compared to other on-chip solutions of vortex
beam generators, e.g., metasurfaces ren2019metasurface ; sroor2020high  and
fork gratings zhou2019ultra , microresonator vortex emitters benefit from
strong light-matter interactions due to the prolonged photon lifetimes and
enhanced light intensities. For example, spontaneous and stimulated emissions
can be dramatically enhanced by the cavity quantum electrodynamics effect,
facilitating the realization of highly-efficient OAM single-photon sources
chen2021bright ; ma2022chip  and low-threshold vortex microlasers
miao2016orbital ; hayenga2019direct ; zhang2020tunable . Hitherto, optical
vortex emitters based on either passive or active microrings are operated
monochromatically. On the other hand, integrated microresonators in nonlinear
materials enable Kerr comb generation (KCG), revolutionizing applications in
timekeeping, telecommunication, chemical sensing, distance ranging, etc
gaeta2019photonic ; chang2022integrated . Unlike the OAM emitters, the
microcombs rely on high-quality ($Q$) factor microresonators with an extremely
low cavity roundtrip loss pfeiffer2018ultra ; puckett2021422 . Though
microresonators support OAM and KCG, the developments of microresonator-based
vortex emitters and comb generators have been detached from each other due to
their contradicting requirements regarding cavity structure scattering (the
former relies on light scattering while the latter requires good suppression
of light scattering). Here, we demonstrate a conceptually new class of
nanophotonic devices, vortex microcombs, on an integrated AlGaAs photonic
platform pu2016efficient ; chang2020ultra . A delicate resonator design
balances the light confinement and vortex emission, where the scattering
(emission) loss induced by angular gratings is compensated by the ultra-high
nonlinearity of AlGaAs-on-insulator (AlGaAsOI) waveguides pu2016efficient . We
show that such a vortex microcomb is capable of emitting a massive assembly of
spectrally multiplexed optical vortices, each translated from the
corresponding spectral component of the microcomb. The special configuration
of varied OAMs distributed over different frequencies promises unprecedented
advantages in the generations of spatiotemporal beams with unique dynamic
features zhao2020dynamic ; chen2022synthesizing . In particular, self-torque
pulses with time-varying OAMs rego2019generation ; chen2022synthesizing  are
accessible from our device with low energy consumption, small device
footprint, engineerable OAM configuration and wide spectral coverage.

Figure 1: Schematic of generating self-torque pulses from the vortex
microcomb. The angular-grating-addressed nonlinear microring emits multiple
optical vortices with each OAM mode (right side) in a different frequency
(left side). A dispersive medium is employed to generate spatio-temporal
optical pulses with varying OAMs.

Our devices consist of a high-$Q$ microring with inner sidewall gratings made
from the emerging AlGaAsOI platform exhibiting tight mode confinements and
strong optical nonlinearities for low-threshold microcomb
generationspu2016efficient ; chang2020ultra . The vortex microcomb [Fig. 1] is
driven by a continuous wave (CW) pump that initializes optical parametric
oscillation (OPO) and cascaded four-wave mixing (FWM) between the adjacent
WGMs of the microring, resulting in the generation of microcombs with
equidistant sidebands. Each comb line corresponds to a WGM with an azimuthal
number $m$, where the angular gratings establish a modal relationship between
the emitted optical vortices and the WGMs cai2012integrated . The generated
comb lines - WGMs - are consequently ejected upwards to the free space, which
forms multiple optical vortices with topological charges of $l=m-N$ ($N$ is
the number of periods of the angular grating). By passing the vortex comb
through a dispersive medium, self-torque pulses with time-varying OAMs can be
generated, as elaborated in Fig. (4).

The device was fabricated on the AlGaAsOI platform [see Fig. S1 of SM]. Fig.
2a presents scanning electron microscope (SEM) images of a
25-$\mu\rm{m}$-radius microring coupled with a bus waveguide. Fig. 2b shows
the zoomed coupling region, where the angular gratings can be clearly
identified. To mitigate the scattering loss of the fundamental TE00 mode, a
multi-mode waveguide (380 nm $\times$ 750 nm) exhibiting a normal dispersion
is used [see Fig. S2 of SM]. The mode profile is superimposed in the cross-
section SEM image of the microring in Fig. 2c. The grating protrusion is
designed to 30 nm to achieve appreciable efficiencies for vortice ejection and
high-$Q$s for comb generation lu2022highly . The transmission spectra for
different resonances of the microring are presented in Fig. 2(d-g), where
mode-splittings are presented for all the resonances. For resonances with non-
zero topological charges puckett2021422 (see Fig. 2(e-g)), only small mode-
splittings ($\textless$1 GHz) are induced by the counter-propagating mode
coupling due to the unavoidable fabrication imperfections. One can observe a
pronounced mode splitting of 9.3 GHz in Fig. 2d, which is an inherent
characteristic feature for the zero-order ($l=0$) cavity resonance of the
vortex emittercai2012integrated ; rego2019generation . The zero-order mode
splitting is dependent on the mode-coupling strength and can be tailored by
controlling the grating protrusion, which could be employed to generate low-
noise phase-locking comb states and tuning the free spectral ranges (FSRs) of
the comb states for engineering the self-torque pulses liu2014investigation ;
karpov2019dynamics , see Fig. S(3) of SM.

Figure 2: Characterizations of the AlGaAs microring. (a) SEM image of the
fabricated device. (b) Zoom-in of white dashed line highlighted area in (a),
showing inner sidewall angular gratings. (c) SEM image of the cross-section of
microring with $\rm{TE_{00}}$ mode superimposed. (d-f) Transmission spectra of
the resonances corresponding to (d) $l=0$, (e) $l=-1$, and (f) $l=-2$ OAM
modes. (g) Transmission spectrum of the resonance with the highest intrinsic
$Q$ of 410,000. The waveguide cross-section dimension: 380$\times$750 nm2 and
angular grating feature size 30 nm.

The experimental setup for characterizing the vortex comb is schematically
shown in Fig. 3a. Fig. 3b presents the full comb spectrum spanning from 1480
nm to 1680 nm. All the generated comb lines were ejected to the free space by
the embedded angular gratings and further spatially separated by an off-chip
transmission grating to resolve different OAM modes. We imaged the far-field
patterns of each optical vortex from $l=-4$ to $l=4$ with a high-sensitivity
CCD mounted on a movable stage. The direct imaging of the far-field emissions
resulted in clockwise (CW) and counterclockwise (CCW) interference patterns
[Fig. 3c] which can be faithfully reproduced by numerical simulations with an
internal dipole source placed in the microring, as shown in Fig. 3d [see more
discussions in Fig. S(4,5) of SM]. Such interferometric behaviors have been
readily observed in the optical vortices from both low-$Q$ GaAs chen2021bright
and high-$Q$ SiN microrings lu2022highly . The OAM nature was further
confirmed by interfering the upwards-emitted photons with a reference beam
transmitted from the waveguide output, as presented in Fig. 3e. Before the
interference, a quarter waveplate was employed to project vectorial reference
beams with topological charge $l$ to the circular polarization basis, forming
left-handed polarized beams with $l’=\lvert l-1\rvert$. The measured
interference patterns therefore exhibited spiral arms equal to $\lvert
l-1\rvert$, as confirmed by the numerical simulations in Fig. 3f (see Fig. S6
of SM). For the vortex beams with large topological charges, e.g., $l=-5$ to
$l=–13$, the CW and CCW interference patterns looked complicated due to the
superpositions of azimuthal and axial components of the cavity field (see Fig.
S7 of SM), which make it challenging to quantitatively compare all the
features between experiments and simulations, as presented in Fig.3(g,h). In
addition, spiral arms for high-order vortex beams become vague [Fig. 3(d,e)],
preventing clear identifications of the topological charges. To reliably
extract the high-order topological charges of the vortex beams, we projected
the CW/CCW intedrference patterns to the linear polarization basis, forming
clear standing-wave patterns with number of the anti-modes equal to
$l^{\prime}=2\lvert l-1\rvert$ [see Fig. S(7) of SM], as presented in Fig.
3(i,j). We use such a method to successfully quantify the topological charge
up to 25, as shown in Fig. S(8) of SM. The emitted vortices feature
simultaneous high-degrees of OAM mode purity and ejection efficiency [see Fig.
S(9,10) of SM], which is consistent with the recent study on high-$Q$
microring OAM emitters lu2022highly .

Figure 3: Characterizations of the vortex frequency microcomb. (a) Schematic
of the experimental setup. A tunable CW laser is amplified by an EDFA, and its
polarization state is controlled by a FPC before coupling to a waveguide via a
lensed fiber. The transmitted light from waveguide output is collected by the
other lensed fiber and sent to either an OSA or to the interference path. The
ejected optical vortices are either directly imaged on a CCD or combined with
a reference beam and then imaged on a CCD after spatial separations by a
transmission grating. FPC: fiber polarization controller. EDFA: Erbium-Doped
Fiber Amplifier. BS: beam splitter. OSA: optical spectrum analyzer. (b)
Spectrum of the microcomb measured by OSA. The light blue area denotes the
comb lines with OAM charge from 4 to -4, the light red area represents the
comb lines with OAM charge from -5 to -13. (c) Measured and simulated (d) far-
field patterns without the reference beam for $l$ from 4 to -4. (e) Measured
and simulated (f) far-field patterns with the reference beam for $l$ from 4 to
-4. (g) Measured and (h) simulated far-field patterns for $l$ from -5 to -13.
(i) Measured and simulated (j) far-field patterns projected to linear
polarization basis for $l$ from -5 to -13.

The vortex microcombs may immediately be employed to explore optical physics
and advance quantum technology. In particular, sculpturing light in the
spatiotemporal domain with increasingly complex topological structures
represents the most cutting-edge frontier for both fundamental physics and
applied optics. Some of the recent examples are spatiotemporal optical vortex
chong2020generation , toroidal vortex wan2022toroidal , and toroidal pulse
zdagkas2022observation . One particular class of spatiotemporal wavepacket
that is directly accessible with our vortex microcomb platform is the self-
torque beams, i.e., optical pulses with time-dependent OAMs. The self-torque
pulses so far have only been created in extremely-ultraviolet regime by using
high-harmonic generation (HHG) processes of near-infrared (NIR) ultrashort
pulses rego2019generation . We, now, show, in Fig. 4, self-torque optical
pulses can be versatilely synthesized with the vortex microcombs. Here, we
simulate vortex microcombs operated in the fundamental mode-locking and
harmonic mode-locking states which could be engineered by how the microcomb is
initialized with the pumping cole2017soliton ; karpov2019dynamics . Fig.
4(a,d) show the simulated comb spectra together with the intensity and phase
profile for each mode. For both mode-locking states, we simulate 7 comb lines
with radial polarization zhu2014spin . For the fundamental mode-locking state,
the topological charge $l$ carried by the microcomb varies from -3 to +3 in an
interval of 1. For the harmonic mode-locking, $l$ varies from -6 to +6 in an
interval of 2. With the spectral phase properly managed [see discussion of
Fig. S(11,12)], the output of the vortex microcomb forms a self-torque pulse
carrying time-dependent OAM. Figure 4(b,e) show 3D intensity iso-surface plots
of the self-torque pulses. Fig. 4(c,f) plot the modal decompositions of the
self-torque pulses in time. Comparing with the HHG approach, generation of
self-torque space-time pulses by using vortex microcombs offers significant
advantages. First, the vortex combs only require a CW laser instead of high-
power ultrafast pulses. Second, the topological configurations of the OAM can
be versatilely engineered by tuning FSR of the comb states, which is not
possible with the HHG approach. Third, the frequency of the self-torque beams
from vortex microcombs can be straightforwardly expanded from visible to NIR
by employing different material platforms. Finally, our integrated devices are
much more compact and suitable for large-scale production using modern
nanofabrication technology. In quantum optics, high-dimensional OAM entangled
quantum states of light can be created from the vortex microcombs with
potential applications in stronger violations of local
realismdada2011experimental ; hu2022high  and generating high-dimensional
multiphoton entanglement for future quantum technologies .

Figure 4: Synthesis of self-torque pulses using vortex microcombs. (a), (d):
Simulated microcomb structure for fundamental mode-locking (FSR = 500 GHz) and
harmonic mode-locking (FSR = 1000 GHz). Total of 7 comb lines are simulated
with their topological charge varying from -3 to +3 and from -6 to +6. (b),
(e): Space-time intensity profile of the microcomb output. The output self-
torque pulse carries time-varying OAM. The plot shows the iso-surface for the
wavepacket at 80/60/30$\%$ of its peak intensity. The red/blue color
corresponds to OAM with a positive/negative sign. (c), (f): Modal
decomposition of the self-torque pulse in the time domain.

We have demonstrated a conceptually new nanophotonic device by fusing two
separately developing fields, optical vortices and microcombs. Our device can
emit up to 50 OAM modes, each translated from corresponding Kerr comb lines
generated by an AlGaAs microring. We further present the unprecedented
advantages of vortex microcombs in the generations of self-torque pulses.
III-V compound semiconductors employed in this work feature both second- and
third-order nonlinearities and engineerable bandgaps, which are highly
beneficial for integrated nonlinear photonics. Furthermore, the direct bandgap
of AlGaAs provides opportunities of monolithic integrations with on-chip
lasers and detectors. Our method of generating vortex microcombs can be easily
implemented in other material platforms such as SiN, SiC, AlN and LiNbO3 etc,
providing exciting opportunities to integrated photonics for harnessing
structured light matter interactions.

Acknowledgements This research is supported by National Key R&D Program of
China (2021YFA1400800), the National Natural Science Foundation of China
(62035017) and Guangdong Special Support Program (2019JC05X397). We thank
Siyuan Yu and Jie Liu for loaning the equipments.

Author contributions J. L conceived the project. B. C, Y. G. Z, and Q. C
performed to the numerical simulations. Y. G. Z, C. C. Y and Y. L fabricated
the devices. B. C, P. N. H, J. Li Y. L and Y. G. Z built the setup and
characterized the devices. B. C., P. N, H, C. L, L. Y, Y. G. Z, C. J. K, Q. C,
Q. W. Z, M. H. P and J. L analyzed the data. M. H. P and J. L wrote the
manuscript with inputs from all authors. C. H. D, K. Y, Q. W. Z, X. H. W, M.
H. P and J. L supervised the project.

Competing interests The authors declare no competing financial interests.

Data and materials availability The data sets will be available upon
reasonable requests.

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