# Families of Equiangular Lines and Lattices

Jacques Martinet<EMAIL_ADDRESS>
###### Abstract.

We construct Euclidean lattices whose sets of minimal vectors support some
large equiangular families of lines, using notably reduction modulo $2$ of
lattices. We also consider some related problems, and answer a question raised
by Greaves ([G], Subsection 1.3.2).

###### Key words and phrases:

Euclidean lattices, equiangular families, reduction modulo $2$

###### 2000 Mathematics Subject Classification:

11H55, 11H71

## Introduction

In this note we consider constructions of sets of equiangular lines in a
Euclidean space $E$ derived from Euclidean lattices. We postpone to Section 1
the necessary prerequisites on Euclidean lattices together with the statement
of various general results, stating here only the following theorem:

######  Theorem 0.1.

Let ${\Lambda}$ be an even (integral) lattice in a Euclidean space $E$, of
minimum $m$, and let $x_{0}\in{\Lambda}$ of norm (the square of the length)
$2m\\!-\\!2$. Then the set vectors of norm $2m+2$ in ${\Lambda}$ orthogonal to
$x_{0}$ and congruent to $x_{0}$ modulo $2$ supports an equiangular family of
lines of common angle $\arccos\frac{1}{m+1}$.

In the rest of the introduction we solely recall some basic facts and
questions concerning families of equiangular lines. Since the publication in
1973 of Lehmens-Seidel seminal paper [L-S], there has been a large literature
on the subject, among which I would like to quote the collective work [JTYZZ]
(Ann. Math., 2021). We refer to Greaves’ survey [G] for the results mentioned
below without a reference. We shall in particular note that for a set of a
given rank $n$, the numbers of lines $t=n$, $n+1$, $2n$ and $\frac{n(n+1)}{2}$
play somewhat special rôles.

Consider a set $X$ of $t$ lines of rank $n$ with common angle
${\theta}\in[0,\frac{\pi}{2}]$, with which we associate the parameter
$\alpha=\arccos{\theta}$, and assume that $t>n$. We then have
${\theta}\neq\frac{\pi}{2}$, thus $0<\alpha<1$. Equip each line with a
norm-$1$ vector, and consider their Gram matrix $G(X)$, with entries $1$ on
its diagonal and $\pm\alpha$ off its diagonal. The matrix

$S(X):=\frac{1}{\alpha}(I_{n}-G(X))$

is a Seidel matrix for $X$. It depends on the ordering and orientations of the
lines in $X$, but its spectrum solely depends on $X$, and in particular its
smallest eigenvalue ${\lambda}$, equal to $-\frac{1}{\alpha}<-1$. It is
immediate that if all $\pm\alpha$ are $+\alpha$ (resp. $-\alpha$), we have
${\lambda}=-1$, which must be excluded (resp, ${\lambda}=-(t-1)$, $t=n+1$ and
$\alpha=\frac{1}{n}$): in other words, for an acute (resp. obtuse) equiangular
family of vectors, we have $t\leq n$ (resp. $t\leq n+1$, with
$\alpha=\frac{1}{n}$ if equality holds). Also we see that $t\geq n+1$ implies
that the angle may take only finitely many values.

The absolute bound (Gerzen’s theorem) is the inequality
$t\leq\frac{n(n+1)}{2}$, which holds because the orthogonal projections to the
line of $X$ are independent symmetric automorphisms.

The importance of the value $2n$ (or $2n+1$) comes from Neuman’s theorem: if
$t>2n$, $\frac{1}{\alpha}$ is an odd integer; compare Theorem 0.1.

In general one considers the maximum number of lines in an equiangular set
without taking into account their rank: in terms of the definition below, one
considers values or estimations for $t_{n}$, not for $t^{\prime}_{n}$.

######  Definition 0.2.

Let $t_{n}$ be the maximal cardinality of an equiangular set contained in an
$n$-dimensional space (resp. spanning an $n$-dimensional space). [Thus we have
$t^{\prime}_{n}\leq t_{n}=\max_{k\leq n}\,t_{k}$.]

The exact values of $t_{n}$ are known up to $n=17$. As we shall see, we have
$t^{\prime}_{n}=t_{n}$ for $n\leq 7$ and $14\leq n\leq 17$, but in the range
$7<n<14$, we have the strict inequality $t^{\prime}_{n}<t_{n}$. It is also
known (cf. Neuman’s theorem above) that $t_{n}$ is strictly larger than $2n$
except if $n=2$, $3$ $4$, $5$ or $14$. This suggests the following double
question:

######  Question 0.3.

For which dimensions $n$ is it true that $t^{\prime}_{n}=t_{n}$? that
$t^{\prime}_{n}>2n$?

A neighbour question closer to the subject of this note is:

######  Question 0.4.

For which dimensions is the bound $t^{\prime}_{n}$ attained on the set of
minimal vectors of a Euclidean lattice ?

I have put emphasis on $t^{\prime}_{n}$ rather than on $t_{n}$ because I
believe that $t^{\prime}_{n}$ presents irregularities as a function of $n$
which throw some light on individual properties of dimensions. In this respect
it is worth considering the absolute bound $\frac{n(n+1)}{2}$. This is known
to be attained for $n=2,3,7,23$, but on no other dimensions. These dimensions
are somewhat special, related to the existence of the regular hexagon and
icosahedron for $n=2,3$, and to the lattices ${\mathbb{E}}_{8}$ and Leech’s
${\Lambda}_{24}$ of dimension $n+1$ for $n=7,23$.
[This can be related to the double transitivity of the groups $S_{3}$,
$A_{5}$, $\operatorname{O}_{7}(2)\sim S_{6}(2)$ and $\operatorname{Co}_{3}$ in
degrees $3,6,28$ and $276$, respectively. For $n=3$, one identifies the action
of ${\mathbb{A}}_{5}$ on the diagonals of a regular icosahedron as its action
on its $5$-Sylow subgroups; lattices for $n=2,7,23$ will be described later.]

Recent works have shown the special behaviour of the “magic” dimensions $8$
and $24$ for kissing number and sphere packing problems. This suggest that the
absolute bound could be strict in all other dimensions. The following question
of maximality could be considered in place of it. Consider a configuration $X$
of equiangular lines which is maximal in the $n$-dimensional space its spans.
Say that $X$ is universally maximal if $X$ remains maximal when embedded in
dimension $n+1$ (or any larger dimension). The four dimensions above are
universally maximal: trivially for $n=2$, because
$t^{\prime}_{n}=t_{n}=t_{n+1}$ for $n=3,7,23$.

######  Problem 0.5.

For which dimensions $n$ does there exist a universally maximal configuration
of $t^{\prime}_{n}$ lines?

Here are two more results we shall need later. First the relative bound (see
[G], Th. 1.9): if $t\leq\frac{1}{a^{2}}$, then
$t\leq\frac{n(1-\alpha^{2})}{1-n\alpha^{2}}$.

Next ([JTYZZ]): if $\alpha=\frac{1}{2k-1}$ ($k\geq 2$), then we have
$t^{\prime}_{n}=\lfloor\frac{k(n-1)}{k-1}\rfloor$ for $n$ large enough.

In Section 1 we explain how to apply the theory of lattices modulo $2$ as
developed in [Ma1] and [Ma2] to the construction of equiangular systems of
lines. In Section 2 we consider root lattices (certain lattices of minimum
$2$) and their duals, and lattices of minimum 3 derived from them, and in
Section 3, lattices of minimum $4$ and $5$.

Acknowledgments. The author thanks Bill Allombert for his help in working with
PARI.GP.

##  1\. Basic Results

Let $E$ be a Euclidean space of dimension $n$, and ${\Lambda}$ be a (full)
lattice in $E$. We denote by $m$ its minimum, by $S$ the set of its minimal
vectors and set $s=\frac{1}{2}\lvert S\rvert$. More generally, given $r>0$,
let

$S_{r}=\\{x\in{\Lambda}\mid x\cdot x=r\\}$ and let $s_{r}=\frac{1}{2}\lvert
S_{r}\rvert$ ;

thus, $S=S_{m}$ and $s=s_{m}$. We also set $N(x)=x\cdot x$ (the norm of $x$,
the square of the usual $\|x\|$). We say that ${\Lambda}$ is integral if all
scalar products on ${\Lambda}$ are integral, and then even if all norms are
even, odd otherwise. Note that ${\Lambda}$ is integral if and only if it is
contained in its dual ${\Lambda}^{*}=\\{x\in
E\mid\forall\,y\in{\Lambda},\,x\cdot y\in{\mathbb{Z}}\\}$.

Consider vectors $x,y\in{\Lambda}$, with $x\neq 0$, $y\neq 0$ and $y\neq\pm
x$, and such that $y\equiv x\\!\\!\mod 2$. Set

$e=\frac{y-x}{2}\,{\text{ and }}\,f=\frac{y+x}{2}$ .

Then $e,f$ are nonzero, the set $\\{\pm e,\pm f\\}$ only depends on $\\{\pm
x,\pm y\\}$, and we have

$x=-e+f\,{\text{ and }}\,y=e+f$ .

Moreover, in an equality $\pm y=\pm x+2u$, $u$ is one of $\pm e$ or $\pm f$.

######  Proposition 1.1.

([Ma1], § 1) With the notation above, we have

  1. (1)

$N(x)+N(y)\geq 4m$.

  2. (2)

If ${\Lambda}$ is integral, $N(y)\equiv N(x)\\!\\!\mod 4$.

  3. (3)

If $(N(x)+N(y)=4m$, then $e$ and $f$ are minimal and $x\cdot y=0$.

###### Proof.

(1) We have

$N(x)+N(y)=\frac{1}{2}\big{(}N(y-x)+N(y+x)\big{)}=2\big{(}N(e)+N(f)\big{)}\geq
4m$

since $e$ and $f$ are nonzero.

(2) We then have

$N(y)-N(x)=N(e+f)-N(-e+f)=4e\cdot f\equiv 0\\!\\!\mod 4$ .

(3) We then have $N(e)+N(f)=2m$, hence $N(e)=N(f)=m$ and $x\cdot
y=N(f)-N(e)=0$. ∎

Proposition 1.1 shows that a list of representatives of classes modulo $2$ of
smallest possible norms must include vectors of norm $N\leq 2m$, and that the
corresponding classes then include a unique pair $\pm x$ if $N<2m$, and at
most $n$ such pairs if $N=2m$. Among known examples are the celebrated
lattices ${\mathbb{E}}_{8}$ and ${\Lambda}_{24}$.

######  Proposition 1.2.

Let $m^{\prime}\geq m$ and $m^{\prime\prime}\geq m^{\prime}$, and let
$x,y,y^{\prime}\in{\Lambda}$ such that $N(x)=m^{\prime}$,
$N(y)=N(y^{\prime})=m^{\prime\prime}$, $y\equiv y^{\prime}\equiv x\mod 2$, and
$y^{\prime}\neq\pm y$, Then $\lvert y\cdot y^{\prime}\rvert\leq
m^{\prime\prime}-2m$.

###### Proof.

Write as in Proposition 1.1

$x=-e+f,y=e+f\,{\text{ and
}}\,x=-e^{\prime}+f^{\prime},y^{\prime}=e^{\prime}+f^{\prime}$ .

Calculating $N(y-y^{\prime})$ in two ways we obtain the equalities

$2m^{\prime\prime}-2y\cdot y^{\prime}=4N(e-e^{\prime})\geq 4m$

(since $e^{\prime}=e$ implies $y^{\prime}=y$), hence

$y\cdot y^{\prime}=m^{\prime\prime}-2N(e-e^{\prime})\leq m^{\prime\prime}-2m$
,

since $N(e\pm e^{\prime})\geq m$.

Calculating similarly $N(y+y^{\prime})$, we obtain

$2m^{\prime\prime}-2y\cdot y^{\prime}=4N(-2x+f+f^{\prime})\geq 4m$

(since $x=f+f^{\prime}$ implies $y^{\prime}=-y$), hence

$y\cdot y^{\prime}=m^{\prime\prime}-2N(e-e^{\prime})\leq m^{\prime\prime}-2m$
.

∎

######  Theorem 1.3.

Assume that ${\Lambda}$ is integral of even minimum. Let $x_{0}\in{\Lambda}$
of norm $2m-2$. Consider the set ${\mathcal{E}}(x_{0},2m+2)=\\{\pm
y_{1},\dots,\pm y_{k}\\}$ of those vectors of norm $2m+2$ congruent to $x_{0}$
modulo $2$. Then the $y_{i}$ support an equiangular family of lines of angle
$\arccos\frac{1}{m+1}$ and rank $\leq n-1$. [By an abuse of language, we
accept sets of less than three lines.]

###### Proof.

We have $\lvert y_{i}\cdot y_{j}\rvert\leq 2$ by Proposition 1.2. Next by the
calculation done in the proof of this proposition we have $y_{i}\cdot
y_{j}=m^{\prime\prime}-N(e-e^{\prime})$ with $m^{\prime\prime}=2m+2$, and
since $N(e)=N(e^{\prime})=m$, we have $N(e-e^{\prime})=2m-2e\cdot e^{\prime}$,
which implies

$y_{i}\cdot y_{j}=-2\,m+2+4\,e\cdot e^{\prime}\equiv 2\mod 4$ ,

hence $y_{i}\cdot y_{j}=\pm 2$.

The bound for the rank comes from Proposition 1.1, (4). ∎

######  Remark 1.4.

  1. (1)

The restriction to even minima is essential. Indeed, if $m$ is odd, we may
apply Proposition 1.1 to the even sublattice ${\Lambda}_{ev}$ of ${\Lambda}$,
of minimum $m_{ev}\geq m+1$. Since $m^{\prime}+m^{\prime\prime}$ is strictly
smaller than $4(m+1)$, ${\mathcal{E}}(x_{0},2m+2)$ is then empty.

  2. (2)

We may also clearly restrict ourselves to irreducible lattices.

The proposition below, extracted from Section 5 of [Ma2], shows that the set
of equiangular lines constructed in Theorem 1.3 can be realized as the set of
minimal vectors of an integral (relative) lattice of minimum $m+1$.

######  Proposition 1.5.

Let $x_{0}\in{\Lambda}$ of norm $m^{\prime}<2m$, let ${\mathcal{C}}$ be its
class modulo $2$, let $L_{0}={\mathcal{C}}\cup 2{\Lambda}$ and let
$L=L_{0}\cap x_{0}^{\perp}$. Set $m^{\prime\prime}=4m-m^{\prime}$, and assume
that ${\mathcal{E}}(x_{0},m^{\prime\prime})$ is not empty. Then $L$ is a
lattice with invariants

$\dim L=n-1,\ \min L=m^{\prime\prime},\ {\text{ and }}\
S(L)={\mathcal{E}}(x_{0},m^{\prime\prime})$ .

###### Proof.

We have ${\mathcal{C}}=x_{0}+2{\Lambda}$ and $2x_{0}\in 2{\Lambda}$, hence
$L_{0}$ is a lattice (containing $2{\Lambda}$ to index $2$). By Proposition
1.1, the first minimum of the norm on ${\mathcal{C}}$ is $m^{\prime}$,
attained uniquely at $\pm x_{0}$, and since ${\mathcal{E}}$ is not empty, its
second minimum is $m"$, attained exactly on ${\mathcal{E}}$. Since $\min
2{\Lambda}=4m>m^{\prime\prime}$, these are the first two minima of $L_{0}$,
and since ${\mathcal{E}}$ is orthogonal to $x_{0}$, we have $\min
L=m^{\prime\prime}$ and $S(L)={\mathcal{E}}$.
[The calculation of $\det(L)$ is carried out in [Ma2], Section 5.] ∎

######  Remark 1.6.

The vectors in ${\Lambda}^{\prime}$ are sums of minimal vectors. This shows
that its even part that ${\Lambda}^{\prime}_{ev}$ is generated by sums
$e+e^{\prime}$, $e,e^{\prime}\in S({\Lambda}^{\prime})$. Easy calculations
then show that on ${\Lambda}^{\prime}_{ev}$, all scalar products are even and
all norms are divisible by $4$. Hence
$\frac{1}{\sqrt{2}}{\Lambda}^{\prime}_{ev}$ is an even (integral) lattice.

From an algorithmic viewpoint, listing the $y_{i}$ in Theorem 1.3 from vectors
of norm $2m+2$ can need lengthy computations. The following proposition allows
us to complete this list using only minimal vectors. This also makes easy the
calculation of the rank of ${\mathcal{E}}$. In the following proposition we
keep the notation of 1.3, setting moreover

{$S_{0}=\\{x\in S\mid x_{0}\cdot x=m-1$ .}

######  Proposition 1.7.

The map

$x\mapsto x_{0}-2x\,:\,S_{0}\to{\Lambda}$

induces a bijection of $S_{0}$ onto ${\mathcal{E}}$, and we have
$\operatorname{rk}{\mathcal{E}}=\operatorname{rk}S_{0}-1$.

###### Proof.

The first assertion follows from the calculation of $x_{0}\cdot\frac{y-x}{2}$
and $N(\frac{y-x}{2})$ for $y\in S_{2m+2}$.

Let $r=\operatorname{rk}{\mathcal{E}}$, let $y_{1},\dots,y_{r}$ be $r$
independent vectors in ${\mathcal{E}}$, and let $x_{i}=\frac{x_{0}-y_{i}}{2}$.
We have

$\langle y_{1},...,y_{r}\rangle=\langle
y_{1},y_{2}-y_{1},\dots,y_{r}-y_{1}\rangle=\langle
y_{1},x_{2}-x_{1},\dots,x_{r}-x_{1}\rangle$ .

Now $\langle S_{0}\rangle$ contains the $x_{0}-x_{i}$ (because
$-y_{i}=x_{0}-2(x_{0}-x_{i})$) and $x_{0}-2x_{1}$ ($=y_{1}$). Hence $\langle
S_{0}\rangle=\langle x_{0},y_{1},\dots,y_{r}\rangle$ , and since the $y_{i}$
but not $x_{0}$ are orthogonal to $x_{0}$, we have
$\operatorname{rk}S_{0}=r+1$. ∎

Let $v$ be a nonzero vector in $E$ and let $H:=({\mathbb{R}}v)^{\perp}$. The
orthogonal projection to $H$ (or along $v$) of $x\in E$ is

$p(x)=x-\frac{v\cdot x}{v\cdot v}\,v$ .

Scalar products and norms of projections are given by the formulae

$None$ $p(x)\cdot p(y)=x\cdot y-\frac{(v\cdot x)(v\cdot y)}{N(v)}{\text{ and
}}N(p(x))=N(x)-\frac{(v\cdot x)^{2}}{N(v)}\,.$

Note that for $x\in S$ and $x\neq\pm v$, we have $\lvert v\cdot
x\rvert\leq\frac{N(v)}{2}$, since we have $N(v\pm x)\geq m$, hence $\mp
2(v\cdot x)\leq N(v)$.

Consider a lattice ${\Lambda}$. Then $p({\Lambda})$ is a lattice if and only
if $v\in{\Lambda}$, and we may assume that $v$ is primitive (because $p$ only
depends on the line ${\mathbb{R}}v$). We can then construct bases
$(v_{1},\dots,v_{n})$ for ${\Lambda}$ with $v_{1}=v$, so that
$(p(v_{2}),\dots,p(v_{n})$ is a basis for $p({\Lambda})$.

In the setting of Theorem 1.3 projections along $x_{0}$ preserve the norms on
${\mathcal{E}}$, and shall be used to construct lattices whose minimal vectors
support equiangular families of lines. However I cannot give a priori a
general procedure to find minimal vectors in projections. Note that if $\lvert
v\cdot x\rvert$,takes values $\alpha_{1}<\dots<\alpha_{t}$ on
$S({\Lambda})\smallsetminus\\{\pm v\\}$, the corresponding norms of the
projections occur in the reverse order, and $\alpha_{t}=\frac{N(v)}{2}$ needs
$N(v)<4m$ by $(\boldsymbol{\ast})$. We shall consider the projections of the
$x\in S({\Lambda})$ such that $v\cdot x=\pm\alpha_{t}$, in particular when
$N(v)=2m-2$ and $\alpha_{t}=\frac{N(v)}{2}$, since we then obtain equiangular
families of line by applying Theorem 1.3. It will often happen that
$S(p({\Lambda}))$ is the subset of $p(S({\Lambda}))$ of the $p(x)$ with
$v\cdot x=\frac{N(v)}{2}$. This will be checked for all lattices we shall
construct in the forthcoming sections.

We state below a proposition which gives us some precisions on projections.
Give $v$ we consider a partition $S^{+}\cup S^{-}$ of $S$ for which $x\in
S^{+}$ needs $v\cdot x\geq 0$ (the choice of $x\in S^{+}$ among $\pm x$ is
well-defined only when $v\cdot x\neq 0$). Using formulae $(\ast$), we easily
prove:

######  Proposition 1.8.

Consider a lattice ${\Lambda}$ of minimum $m$, a vector $v\in{\Lambda}$ of
norm $2m-2$ and the orthogonal projection $p$ along $v$, and assume that
$S(p({\Lambda})$ is contained in $p(S({\Lambda}))$. Let $x$ and $y\neq\pm x$
in $S^{+}$, not colinear with $v$. Then we have

$\frac{m-3}{4}\leq x\cdot y\leq\frac{3m-1}{4}$ .

In particular if $m=2$ (resp. $m=4$) we have $x\cdot y\in\\{0,1\\}$ (resp.
$x\cdot y\in\\{1,2\\}$. ∎

##  2\. Root Lattices and their duals

Root lattices are integral lattices generated by norm-$2$ vectors. Theses are
orthogonal sums of irreducible lattices, isometric to exactly one of
${\mathbb{A}}_{n}$ ($n\geq 1)$, ${\mathbb{D}}_{n}$ ($n\geq 4$) or
${\mathbb{E}}_{n}$ ($n=6,7,8$, the definition of which we recall below; see
[Ma], Chapter 4 for details.

Inside the lattice ${\mathbb{R}}^{n+1}$ (resp. ${\mathbb{R}}^{n}$), equipped
with its canonical basis $({\varepsilon}_{0},\dots,{\varepsilon}_{n})$ (resp.
$({\varepsilon}_{1},\dots,{\varepsilon}_{n})$, Let

${\mathbb{A}}_{n}=\\{x\in{\mathbb{Z}}^{n+1}\mid\sum_{i=0}^{i=n}x_{i}=0\\}{\text{
and }}{\mathbb{D}}_{n}=\\{x\in{\mathbb{Z}}^{n}\mid\sum_{i=1}^{i=n}x_{i}\equiv
0\mod 0\\}$ .

In ${\mathbb{R}}^{8}$ let
$e=\frac{{\varepsilon}_{1}+\dots+{\varepsilon}_{8}}{2}$, set
${\mathbb{E}}_{8}={\mathbb{D}}_{8}\cup({\mathbb{D}}_{8}+e)$, and define
${\mathbb{E}}_{7}$ and ${\mathbb{E}}_{8}$ by successive sections orthogonal to
${\varepsilon}_{7}-{\varepsilon}_{8}$ and
${\varepsilon}_{6}-{\varepsilon}_{7}$.

For a lattice ${\Lambda}$ in the list above the automorphism group acts
transitively on its set of minimal vectors, among which we may choose
arbitrarily $x_{0}$ when applying Theorem 1.3. Let us choose
$x_{0}={\varepsilon}_{0}-{\varepsilon}_{1}$,
${\varepsilon}_{1}-{\varepsilon}_{2}$ and $e$ when
${\Lambda}={\mathbb{A}}_{n}$, ${\mathbb{D}}_{n}$ and ${\mathbb{E}}_{n}$,
respectively.

For ${\Lambda}={\mathbb{A}}_{n}$, we have

$S_{0}=\\{-{\varepsilon}_{0}+{\varepsilon}_{i},{\varepsilon}_{1}-{\varepsilon}_{i},\,i\geq
2\\}\ {\text{ and }}\
\pm{\mathcal{E}}=\\{{\varepsilon}_{0}+{\varepsilon}_{1}-2{\varepsilon}_{i},\,i\geq
2\\}$ .

For ${\Lambda}={\mathbb{D}}_{n}$, we have

$S_{0}=\\{-{\varepsilon}_{1}\pm{\varepsilon}_{i},{\varepsilon}_{2}\pm{\varepsilon}_{i},\,i\geq
3\\}\ {\text{ and }}\
\pm{\mathcal{E}}=\\{{\varepsilon}_{1}+{\varepsilon}_{2}\pm{\varepsilon}_{i},\,i\geq
3\\}$.

For ${\Lambda}={\mathbb{E}}_{8}$, $S_{0}$ consists of $28\times 2=56$ vectors,
the ${\varepsilon}_{i}+{\varepsilon}_{j},\,1\leq i<j\leq 8$ and the $28$
vectors obtained by negating $6$ basis vectors in $e$. The $28$ vectors in
${\mathcal{E}}$ (up to sign) are then obtain by permutations of

$\frac{3{\varepsilon}_{1}+3{\varepsilon}_{2}-{\varepsilon}_{3}-{\varepsilon}_{4}-{\varepsilon}_{5}-{\varepsilon}_{6}-{\varepsilon}_{7}-{\varepsilon}_{8}}{2}$
.

The results for ${\mathbb{E}}_{7}$ and ${\mathbb{E}}_{6}$ are obtained by
sections of ${\mathbb{E}}_{8}$. Alternatively we could also have used the sets
of minimal vectors in projections of
${\mathbb{A}}_{n},{\mathbb{D}}_{n},{\mathbb{E}}_{n}$.

Summarizing, we obtain:

######  Proposition 2.1.

Let ${\Lambda}$ be an irreducible root lattice of dimension $n$. Then the
equiangular family of Theorem 1.3 is of rank $r=n-1$ and contains $t$ lines
according to the following data:

${\Lambda}={\mathbb{A}}_{n}$ : $t=r$ ; ${\Lambda}={\mathbb{D}}_{n}$ : $t=2r-2$
;

${\Lambda}={\mathbb{E}}_{8}$: $t=28$ ; ${\Lambda}={\mathbb{E}}_{7}$ : $t=16$ ;
${\Lambda}={\mathbb{E}}_{6}$: $t=10$.

[${\mathbb{D}}_{n+1}$ provides the asymptotic bound for $\alpha=\frac{1}{3}$,
valid for all $n\geq 15$.] ∎

As for the duals, we may discard ${\mathbb{D}}_{4}^{*}\sim{\mathbb{D}}_{4}$,
$E_{8}^{*}=E_{8}$ and ${\mathbb{D}}_{n}^{*},\,n\geq 5$ (because
$S(D_{n}^{*})=S({\mathbb{Z}}^{n})$). The set $S({\mathbb{A}}_{n}^{*})$
supports the equiangular set of vectors with common angle
$-\arccos\frac{1}{n}$, and $S({\mathbb{E}}_{7}^{*})$ is is the projection of
${\mathcal{E}}$ attached to ${\mathbb{E}}_{8}$. The lattice
${\mathbb{E}}_{6}^{*}$ does not apply directly to equiangular systems of
lines.
[The function $\lvert x\cdot y\rvert$ takes two values on non-proportional
$x,y\in S({\mathbb{E}}_{6}^{*})$. We shall return to such lattices in [Ma3] in
connection with the theory of strongly regular graphs (with
$S({\mathbb{E}}_{6}^{*})$ we can recover the Schläfli graph, attached to the
system of lines on a non-singular cubic surface).]

The values of $t_{n}$ are known up to $n=17$ (see [G]). For $n=2$ to $7$,
$t_{n}$ is equal to $3,6,6,10$ and $16$, respectively, and we have $t_{n}=28$
for $7\leq n\leq 14$.

######  Proposition 2.2.

  1. (1)

We have $t^{\prime}_{n}=t_{n}$ for $2\leq n\leq 7$, and except for $n=3$,
$t^{\prime}_{n}$ is attained on the set of minimal vectors of a lattice.

  2. (2)

For $n=3$, the maximum number of lines defined by a lattice is $4$, attained
uniquely on the configuration of $S({\mathbb{A}}_{3}^{*})$.

  3. (3)

For $8\leq n\leq 13$, $t^{\prime}_{n}$ is strictly smaller than $t_{n}$.

###### Proof.

(1) This is clear for $n=2$ and $3$. For $n=4,5,6,7$, consider the projections
of ${\mathbb{D}}_{5}$, ${\mathbb{E}}_{6}$, ${\mathbb{E}}_{7}$ and
${\mathbb{E}}_{8}$ (Proposition 2.2).

(2) This results from the classification of minimal classes for $n=3$ (see
[Ma], Theorem 9.2): one checks that a lattice with $s\geq 4$ must have a
system of minimal vectors containing $e_{1},e_{2},e_{3},e_{1}+e_{2}+e_{3}$
with equal scalar products $e_{i}\cdot e_{j}$.

(3) Let $n$ as above, Assume that there exists an $n$-dimensional system of
$s=28$ equiangular lines. Let ${\theta}$ be their common angle and let
$\alpha=\arccos{\theta}$. We then have $s>2n$, so that by Neumann’s theorem
([G], Theorem 1.16), we have $\alpha=\frac{1}{m}$ for some odd integer $m\geq
3$. If $m\geq 5$, the relative bound $s\leq\frac{1-\alpha^{2}}{1-s\alpha^{2}}$
(see [G], Theorem 1.9) implies $s\leq 2n$, a contradiction. Hence we have
$m=3$, and a theorem of Kao-Yu’s ([K-Y]) implies $n\leq 27$. (If $n\leq 14$, a
system of $28$ equiangular lines with angle $\arccos\frac{1}{3}$ comes from
dimension $7$.) ∎

An other way to construct lattices of minimum $3$ consists in viewing them as
lattices containing to index $2$ their even sublattice. For further use we
consider more generally lattices of minimum $m\geq 3$ odd. To state the
results below we introduce some notation. Given a lattice $L$ and $a>0$, we
denote by ${}^{a}L$ the group $L$ equipped with the scalar product $a\,(x\cdot
y)$. (Thus we have $a\,(x\cdot y)\simeq\sqrt{a}\,L$.) Given an integral
lattice $L$, $L_{ev}$ denotes the even part of $L$, i.e., the set of $x\in L$
having an even norm. If $L$ is odd, we have $[L:L_{ev}]=2$ and $L=\langle
L_{ev},e\rangle$ where $e\in L$ is any vector of odd norm.

######  Proposition 2.3.

Let ${\Lambda}$ be a lattice of dimension $n\geq 2$ and of odd minimum $m\geq
3$, generated by its minimal vectors. Then $L:={}^{1/2}{\Lambda}_{ev}$ is an
even lattice of minimum $m^{\prime}\geq\frac{m}{2}$, and either we have
$s({\Lambda})\leq n$, or $L$ has a basis of vectors of norm $m-1$.

###### Proof.

First note that ${\Lambda}_{ev}$ is generated by the sums $x+y$, $x,y\in
S({\Lambda})$, with $y\neq x$ since $2x=(x+y)+(x-y)$. We have $N(x+y)=2(m\pm
1)\equiv 0\mod 4$, which shows that $L$ is even.

Choose a half-system $e_{1},\dots,e_{s}$ of minimal vectors of ${\Lambda}$
such that $e_{1}\cdot e_{i}=+1$ for $i\geq 2$ and that $e_{1},\dots,e_{n}$ are
independent, and set $e^{\prime}_{i}=e_{1}-e_{i},i\geq 2$. Suppose first that
all $e_{i}\cdot e_{j},i<j$ are equal to $+1$ and that $s>n$. Then we may write
$x:=e_{n+1}$ as a ${\mathbb{Q}}$-linear combination of $e_{1},\dots,e_{n}$,
say, $x=\sum{\lambda}_{k}\,e_{k}$. Calculating $x\cdot e_{i}$ for $i\leq n$,
we obtain

$1=x\cdot e_{i}=m{\lambda}_{i}+\sum_{i\neq
j}{\lambda}_{j}=(m-1){\lambda}_{i}+\sum_{i}\,{\lambda}_{j}$ ,

which implies that the ${\lambda}_{i}$ have the common value
${\lambda}=\frac{1}{n+m-1}$.

From $x={\lambda}\sum e_{i}$ we deduce that $m=x\cdot x={\lambda}\,n$, which
implies

$\frac{1}{n+m-1}=\frac{m}{n}\ \Leftrightarrow\ (m+n)\,(n-1)=0$ ,

a contradiction.

[Lattices with $e_{i}\cdot e_{j}=+1$ and $s=n$ exist, and are unique up to
isometry. They can be represented by the Gram matrix $M$ with entities
$M_{i,i}=m-1$ and $M_{i,j}=\frac{m-1}{2}$ if $j\neq i$, except $M_{1,1}=m+1$
and $M_{1,2}=M_{2,1}=0$.]

Otherwise, let $i>1$ and $j>i$ such that $e_{i}\cdot e_{j}=-1$. If $i>n$,
write $e_{i}$ as a $Q$=linear combination of $e_{k},k\leq n$. There must be at
least three nonzero components, so that exchanging $i$ with some
$i^{\prime}\leq n$, me may assume that $i\in[2,n]$, that me may then exchange
with $2$. We thus reduce ourselves to the case when $i=2$, and then we may
similarly assume that $j=3$. Finally we check that
$e_{2}+e_{3},e^{\prime}_{2},\dots,e^{\prime}_{n}$ is a basis for
${\Lambda}_{ev}$, which completes the proof of the dichotomy between the two
cases above. ∎

Note that given $L$ we can reconstruct ${\Lambda}$ from $L$ by the formula

${\Lambda}={}^{2}L_{1}\text{ with }L_{1}=\langle L,\frac{v}{2}\rangle\text{
and }v\in L\text{ of norm }2m$ .

Using Proposition 2.3 together with the classification of root lattices we now
prove:

######  Theorem 2.4.

Let ${\Lambda}$ be an $n$-dimensional lattice of minimum $3$ whose set $S$ of
minimal vectors supports an equiangular family of lines of rank $n$. Then
except if $n=5,6$ or $7$, the maximal number $s_{n}$ of lines is equal to
$2(n-1)$, and except if $n=2$, attained uniquely on the projection of
${\mathbb{D}}_{n+1}$ described in Proposition 2.1.

###### Proof.

The first case of Proposition 2.3 needs $n\geq 2(n-1)$, hence $n=2$.

Let now $n\geq 3$. Then ${\Lambda}$ is of the form $\langle
L,\frac{v}{2}\rangle$ where ${\Lambda}$ is a root lattice and $v$ a norm-$6$
vector which is not congruent modulo $2$ to a shorter vector. This condition
eliminates ${\mathbb{E}}_{6}$ and ${\mathbb{E}}_{8}$, and implies $n\geq 5$
(resp. $n\geq 6)$ if ${\Lambda}={\mathbb{A}}_{n}$ (resp. ${\mathbb{D}}_{n}$),
and because $S$ must be of rank $n$, $n=5$ (resp. $6$). Finally we are left
with ${\mathbb{A}}_{5}$, ${\mathbb{D}}_{6}$ and ${\mathbb{E}}_{7}$, which
accounts for the exceptional dimensions $5,6,7$.

From now on we assume that $L=L_{1}\perp L_{2}$ and $v=\frac{x+y}{2}$, $x\in
L_{1}$, $y\in L_{2}$. The vectors congruent to $\frac{v}{2}$ modulo $l$ are of
the form $\frac{x+2z,y+2t}{2}$ with z,t in $L$. We have

$N(x+2z)=N(x)\Leftrightarrow x\cdot z+N(z)=0$ ,

and if $N(z)\geq N(x)$ (a condition which is automatic if $N(x)=2$), this
needs $z=-x$.

We may assume that $N(x)=2$ and $N(y)=4$, and since $x^{\prime}=x+2z$ reduces
to $x^{\prime}=\pm x$, $\dim S=n$ needs $\dim L_{1}=1$, i.e.,
$L_{1}={\mathbb{A}}_{1}$, and then either
$L\simeq{\mathbb{A}}_{1}\perp{\mathbb{A}}_{1}\perp{\mathbb{A}}_{1}$ or $L2$ is
irreducible. The former case accounts for the theorem in dimension $3$. We are
now left with $L={\mathbb{A}}_{1}+{\mathbb{A}}_{k},k\geq 3$,
$L={\mathbb{A}}_{1}+{\mathbb{D}}_{k},k\geq 4$ and
$L={\mathbb{A}}_{1}+{\mathbb{E}}_{k},k=6,7,8$ ($n=k+1$), the convenient $y+2t$
not $\pm y$ being obtained with $N(t)=2$ and $y\cdot t=-2$.

If $L={\mathbb{A}}_{1}\perp{\mathbb{A}}_{k}$ we may take
$y={\varepsilon}_{0}+{\varepsilon}_{1}-{\varepsilon}_{2}-{\varepsilon}_{3}$
and must take $k=3$. We obtain $s=6$, which accounts for the theorem in
dimension $4$.

If $L={\mathbb{A}}_{1}+{\mathbb{D}}_{k}$ we may take (a)
$y=2{\varepsilon}_{1}$ or (b)
$y={\varepsilon}_{1}+{\varepsilon}_{2}+{\varepsilon}_{3}+{\varepsilon}_{4}$
(in the same orbit if $n=4$). Then $y+2t$ is one of $\pm{\varepsilon}_{i}$
($2(n-1)$ solutions, $\operatorname{rk}\,S=n$) or
$\pm{\varepsilon}_{1}\pm\dots$ with an even number of minus signs ($8$
solutions, $\operatorname{rk}\,S=5$).

If $L={\mathbb{A}}_{1}\perp{\mathbb{E}}_{n-1}$, we perform direct
calculations. The maximal value of $\operatorname{rk}\,S$ is then $6,7$ and
$9$ for $n=7,8$ and $9$, respectively. In the latter cases $S$ generates a
sublattice of index $2$ in ${\Lambda}$, isometric to the lattice constructed
with ${\mathbb{A}}_{1}\perp{\mathbb{D}}_{8}$¿

We have proved that fir $n\neq 2,5,6,7$, the maximum of $s$ is $2(n-1)$,
attained on a unique lattice (up to isometry) generated by its minimal
vectors. This completes the proof of Theorem 2.4. ∎

[As a Gram matrix for the projection of ${\mathbb{D}}_{n+1}$ we may choose
$A=(a_{i,j})$ with entries $3$ on the diagonal and $1$ off the diagonal,
except $a_{1,2}=a_{2,1}=-1$.]

It might well be that the methods of [K-Y] could prove better upper bounds
than $t^{\prime}_{n}\leq 27$ for some $n$. In the other direction, we have the
lower bounds $t^{\prime}_{n}\geq 2(n-1)$ coming from ${\mathbb{D}}_{n+1}$, and
we shall see in Section 3 that we have $t^{\prime}_{13}\geq 26$, attained with
a system of angle $\arccos\frac{1}{5}$. Experimentation based on Proposition
2.3 applied to various “classical” lattices of minimum $4$ in dimensions $8$
to $12$ always produced lattices with $s<2(n-1)$. This supports the following
conjecture:

######  Conjecture 2.5.

For $8\leq n\leq 13$ we have $t^{\prime}_{n}\leq 2n$, and
$t^{\prime}_{13}=26$.

The upper bound $2n$ allows that some $t^{\prime}_{n}$ may be attained on
angles not of the form $\arccos\frac{1}{m}$, $m$ odd, like in dimension $3$.

##  3\. Lattices of minimum 4 and 5

In this section we concentrate on lattices of minimum $5$ in which pairs of
non-proportional minimal vectors have scalar product $\pm 1$. In dimensions
$n\in[15$–$23]$ (and in some higher dimension) examples show that we have the
strict inequality $t^{\prime}_{n}>2n$,so that the highest values of
$t^{\prime}_{n}$ are attains on sets of angle $\arccos\frac{1}{5}$, whence the
special interest of these lattices.

Most of the lattices we shall consider (though not all) have been constructed
using projections of a lattice of minimum $4$ along a norm-$6$ vector, and
using descending chains of cross-sections. We first consider such an
exception.

###  3.1. A 13-Dimensional Lattice

This is the lattice $L$ denoted by $C2\times\operatorname{PSL}(2,25):C2$ in
[WebN-S] (after its automorphism group). It has $s=26$, and defines the
equiangular family referred to in Conjecture 2.5. It has a unique (up to
isometry) $14$-dimensional extension $\overline{L}$ with $s=28$, the value of
$t_{14}$, which extends uniquely wxactly up to dimension $19$ under the
condition that $s$ be as large as possible. We recover these lattices as the
unique sequence of densest sections, for which Gram matrices denoted by
$\operatorname{Qa}\\!n$ can be read in [WebM], file Min5.GP.
[The even sublattice $L_{ev}$ of $L$ deserves a remark: this is the best known
example (found by Conway and Sloane, [C-S]) of a lattice in its dimension
having a large “Bergé-Martinet invariant” ${\gamma}^{\prime}$, defined by
${\gamma}^{\prime}({\Lambda})=\big{(}{\gamma}({\Lambda}){\gamma}({\Lambda}^{*})\big{)}^{1/2}$.]

###  3.2. Lattices from the Leech Lattice

There is a unique orbit of norm-$6$ vectors in the Leech lattice
${\Lambda}_{24}$, so that Theorem 1.3 defines a unique lattice (up to
isomorphism). Its set of minimal vectors consists in $276$ pairs $\pm x$,
which are the projections along the corresponding norm-$6$ vector of vectors
$x\in S({\Lambda}_{24}$ such that $v\cdot x=\pm 3$. This configuration is
known as the Witt design. Its automorphism group is
$2\times\operatorname{Co}_{3}$.

Consequently, for any pair $(L,v)$ of a relative lattice
$L\subset{\Lambda}_{24}$ of minimum $4$ and of a norm-$6$ vector $v\in L$,
provided that there exists an $x\in S(L)$ with $v\cdot x=\pm 3$, projection
along $v$ defines a lattice whose set of minimal vectors supports an
equiangular family contained in the Witt design.

We have considered projections of various lattices contained in the Leech
lattice in dimensions $14-24$ taken from [WebM], file Lambda.gp. The best
results were obtained using Conway-Sloane’s laminated lattices ${\Lambda}_{n}$
or successive sections of such lattices. The file Min5.gp contains series of
Gram matrices $\operatorname{Qb}\\!n$, $16\geq n\geq 8$ and
$\operatorname{Qc}\\!n$, $23\geq n\geq 8$, starting with Gram matrices for
projections of ${\Lambda}_{17}$ and ${\Lambda}_{24}$, respectively.

The table below displays the known values of $t_{n}$ (taken from [GSY] and the
largest values found on lattices for equiangular families, using
$\operatorname{Qa}\\!n$ and $\operatorname{Qb}\\!n$ for $n=14$,
$\operatorname{Qb}\\!n$ for $n=15,16$, and $\operatorname{Qc}\\!n$ for $17\leq
n\leq 23$.

Table for dimensions $14$ – $23$

$n$ | $14$ | $15$ | $16$ | $17$ | $18$ | $19$ | $20$ | $21$ | $22$ | $23$
---|---|---|---|---|---|---|---|---|---|---
$t_{n}$ | $28$ | $36$ | $40$ | $48$ | $57-59$ | $72-74$ | $90-94$ | $126$ | $176$ | $276$
$lat\geq$ | $28$ | $36$ | $38$ | $48$ | $56$ | $72$ | $90$ | $126$ | $176$ | $276$

Inspection of the second line of this table clearly shows that we have
$t^{\prime}_{n}=t_{n}$ for all $n\in[14,23]$, and inspection of the third line
shows that this values are attained on minimaL vectors of lattices for $n=14$,
$15$, $17$ and $21$–$23$, As for the remaining values of $n$ I conjecture:

######  Conjecture 3.1.

For $n=16$, $18$, $19$ and $20$, the largest number of lines produced by the
minimal vectors of a lattice is $38$, $56$, $72$ and $90$, respectively.

We now prove two complements, first for $n=14$, then for $n=18$, the latter
answering a question raised in [G], Subsection 1.3.2.

######  Proposition 3.2.

The sets of equiangular lines afforded by $S(\operatorname{Qa}_{14})$ and
$S(\operatorname{Qb}_{14})$ are not isometric.

###### Proof.

The construction of $S(\operatorname{Qa}_{14})$ from
$S(\operatorname{Qa}_{13})$ shows that removing two convenient lines from
$S(\operatorname{Qa}_{14})$ produces a set of rank $13$ (indeed, in a unique
way), whereas one must remove at least four lines from
$S(\operatorname{Qb}14))$ to obtain a set of rank $\leq 13$. ∎

In [GSY] the authors construct four sets of $57$ vectors which, when rescaled
to norm $5$ have pairwise scalar products $\pm 1$. I have checked that in all
examples the sublattice they generate in ${\mathbb{Z}}^{18}$ has a minimum
$m\leq 4$ (and contains norm $5$ vectors with pairwise scalar products not
$\pm 1$). Since any subset of norm $5$ vectors in $\operatorname{Qa}_{23}$
generate a lattice of minimum $5$ in its span, we have:

######  Proposition 3.3.

The four equiangular systems above are not contained in the Witt design. ∎

###  3.3. Beyond dimension 23

The arguments used to prove Proposition 2.2, (3) (relying on results of [K-Y]
together with the relative bound) prove:

######  Proposition 3.4.

In the range $24\leq n\leq 41$, $t^{\prime}_{n}$ is strictly smaller than
$t_{n}$ (equal to $276$). ∎

In analogy with what was observed in Section 2 for angle $\arccos\frac{1}{3}$,
we expect that $\arccos\frac{1}{5}$ should play a major rôle in dimensions,
say, $24$ to $50$. However, whereas lattices contained in the Leech lattice
constitute a rich source of even lattices of minimum $4$, our knowledge of
such lattices in larger dimensions is poor. I have carried out some
experimentation (far from being exhaustive) on the even sublattices of
unimodular lattices of minimum $3$, using the Bacher-Venkov classification in
dimensions $27$ and $28$ (see the file unimod23to28.gp.gp in[WebM]). Using
projections of the even part of the lattice denoted there by $o27b1$ we found
lattices with $(n,s)=(26,82)$ and $(25,108)$, and then $(n,s)=(24,100)$ by
cross-sections. Gram matrices can be downloaded from Part 4 of Min5.gp.

Also no infinite series of lattices having a fixed minimum $m\geq 4$ are known
(by contrast with minima $m=2$ or $3$ for which we can use root lattices).
This leaves wide open the following question:

######  Question 3.5.

Do there exist for each odd $m\geq 5$ infinite series $L_{n}$ of lattices with
minimum $m$ and pairwise scalar products $\pm 1$ on $S(L_{n})$ such that
$s\sim\frac{m+1}{m-1}\,n$ for $n\to\infty\,$? (If not the exact bound
$\lfloor\frac{m+1}{m-1}\,(n-1)\rfloor$ of [JTYZZ].)

I have also considered angles $\arccos\frac{1}{7}$, using projections of
lattices of minimum $6$. The only example which deserves to be mentioned is
due to G. Nebe ([Ne], answering a question of mine), the proof of which relies
on the theory of modular forms: applying Theorem 1.3 to an even, unimodular
lattice of minimum $6$ yields an equiangular family of $100$ lines.

## References

  * [C-S] J.H. Conway, N.J.A. Sloane, On Lattices Equivalent to Their!la Duals, J. Number Theory 48 (1994), 373–382.
  * [G] G. Greaves, Equiangular lines, University of Montreal (August 2022).
  * [GSY] G. Greaves, J. Syatriadi, P. Yatsyna, Equiangular lines in Euclidean spaces: dimensions 17 and 18, Combinatorica 41 (2021), 839-–872; preprint at arXiv:2104.04330v1 [math.CO], 9 Apr 2021.
  * [JTYZZ] Z.Jiang, J. Tidor, Y. Yao, S. Zhang, Y. Zhao, Equiangular Lines with a Fixed Angle, Ann. Math. 194 (2021), 729–743; see also arXiv [math.CO], 28 Feb 2022.
  * [K-Y] W.-J. Kao, W.-H. Yu, Four-point semidefinite bound for equiangular lines, preprint at arXiv:2203.05828 [math.CO], 11 Mar 2022.
  * [L-S] P.W.H. Lemmens and J.J. Seidel, Equiangular lines, J. Algebra 24 (1973), 494–512.
  * [Ma] J. Martinet, Perfect Lattices in Euclidean Spaces, Grundlehren 327, Springer-Verlag, Heidelberg (2003).
  * [Ma1] J. Martinet, Reduction Modulo $2$ and $3$ of Euclidean Lattices, J. Algebra 251 (2002), 864–887.
  * [Ma2] J. Martinet, Reduction Modulo $2$ and $3$ of Euclidean Lattices, II, Contemporary Math. 344 (2004), 265–273.
  * [Ma3] J. Martinet, Lattices, Graphs and Groups, in preparation.
  * [Ne] G.Nebe, Mail to the author, January 25th, 2024.
  * [PARI] PARI/GP, a computer algebra system designed for fast computations in number theory, url http://pari.math.u-bordeaux.fr/ .
  * [WebM] J. Martinet, A Catalogue of Perfect Lattices,
https://jamartin.perso.math.cnrs.fr/Lattices/index.html.

  * [WebN-S] G. Nebe, N.J.A. Sloane, A Catalogue of Lattices,
https://www.math.rwth-aachen.de/ Gabriele.Nebe/LATTICES/.


# Learning Personas from Dialogue with
Attentive Memory Networks

Eric Chu
MIT Media Lab
<EMAIL_ADDRESS>
&Prashanth Vijayaraghavan 11footnotemark: 1
MIT Media Lab
<EMAIL_ADDRESS>
&Deb Roy
MIT Media Lab
<EMAIL_ADDRESS>
The first two authors contributed equally to this work.

###### Abstract

The ability to infer persona from dialogue can have applications in areas
ranging from computational narrative analysis to personalized dialogue
generation. We introduce neural models to learn persona embeddings in a
supervised character trope classification task. The models encode dialogue
snippets from IMDB into representations that can capture the various
categories of film characters. The best-performing models use a multi-level
attention mechanism over a set of utterances. We also utilize prior knowledge
in the form of textual descriptions of the different tropes. We apply the
learned embeddings to find similar characters across different movies, and
cluster movies according to the distribution of the embeddings. The use of
short conversational text as input, and the ability to learn from prior
knowledge using memory, suggests these methods could be applied to other
domains.

##  1 Introduction

Individual personality plays a deep and pervasive role in shaping social life.
Research indicates that it can relate to the professional and personal
relationships we develop Barrick and Mount (1993), Shaver and Brennan (1992),
the technological interfaces we prefer Nass and Lee (2000), the behavior we
exhibit on social media networks Selfhout et al. (2010), and the political
stances we take Jost et al. (2009).

With increasing advances in human-machine dialogue systems, and widespread use
of social media in which people express themselves via short text messages,
there is growing interest in systems that have an ability to understand
different personality types. Automated personality analysis based on short
text analysis could open up a range of potential applications, such as
dialogue agents that sense personality in order to generate more interesting
and varied conversations.

We define persona as a person’s social role, which can be categorized
according to their conversations, beliefs, and actions. To learn personas, we
start with the character tropes data provided in the CMU Movie Summary Corpus
by Bamman et al. (2014). It consists of 72 manually identified commonly
occurring character archetypes and examples of each. In the character trope
classification task, we predict the character trope based on a batch of
dialogue snippets.

In their original work, the authors use Wikipedia plot summaries to learn
latent variable models that provide a clustering from words to topics and
topics to personas – their persona clusterings were then evaluated by
measuring similarity to the ground-truth character trope clusters. We asked
the question – could personas also be inferred through dialogue? Because we
use quotes as a primary input and not plot summaries, we believe our model is
extensible to areas such as dialogue generation and conversational analysis.

Character Trope | Character | Movie
---|---|---
Corrupt corporate executive | Les Grossman | Tropic Thunder
Retired outlaw | Butch Cassidy | Butch Cassidy and the Sundance Kid
Lovable rogue | Wolverine | X-Men
Table 1: Example tropes and characters

Our contributions are:

  1. 1.

Data collection of IMDB quotes and character trope descriptions for characters
from the CMU Movie Summary Corpus.

  2. 2.

Models that greatly outperform the baseline model in the character trope
classification task. Our experiments show the importance of multi-level
attention over words in dialogue, and over a set of dialogue snippets.

  3. 3.

We also examine how prior knowledge in the form of textual descriptions of the
persona categories may be used. We find that a ‘Knowledge-Store’ memory
initialized with descriptions of the tropes is particularly useful. This
ability may allow these models to be used more flexibly in new domains and
with different persona categories.

##  2 Related Work

Prior to data-driven approaches, personalities were largely measured by asking
people questions and assigning traits according to some fixed set of
dimensions, such as the Big Five traits of openness, conscientiousness,
extraversion, agreeability, and neuroticism Tupes and Christal (1992).
Computational approaches have since advanced to infer these personalities
based on observable behaviors such as the actions people take and the language
they use Golbeck et al. (2011).

Our work builds on recent advances in neural networks that have been used for
natural language processing tasks such as reading comprehension Sukhbaatar et
al. (2015) and dialogue modeling and generation Vinyals and Le (2015); Li et
al. (2016); Shang et al. (2015). This includes the growing literature in
attention mechanisms and memory networks Bahdanau et al. (2014); Sukhbaatar et
al. (2015); Kumar et al. (2016).

The ability to infer and model personality has applications in storytelling
agents, dialogue systems, and psychometric analysis. In particular,
personality-infused agents can help “chit-chat” bots avoid repetitive and
uninteresting utterances Walker et al. (1997); Mairesse and Walker (2007); Li
et al. (2016); Zhang et al. (2018). The more recent neural models do so by
conditioning on a ‘persona’ embedding – our model could help produce those
embeddings.

Finally, in the field of literary analysis, graphical models have been
proposed for learning character personas in novels Flekova and Gurevych
(2015); Srivastava et al. (2016), folktales Valls-Vargas et al. (2014), and
movies Bamman et al. (2014). However, these models often use more structured
inputs than dialogue to learn personas.

##  3 Datasets

Characters in movies can often be categorized into archetypal roles and
personalities. To understand the relationship between dialogue and personas,
we utilized three different datasets for our models: (a) the Movie Character
Trope dataset, (b) the IMDB Dialogue Dataset, and (c) the Character Trope
Description Dataset. We collected the IMDB Dialogue and Trope Description
datasets, and these datasets are made publicly available
111https://pralav.github.io/emnlp_personas/.

###  3.1 Character Tropes Dataset

The CMU Movie Summary dataset provides tropes commonly occurring in stories
and media Bamman et al. (2014). There are a total of 72 tropes, which span 433
characters and 384 movies. Each trope contains between 1 and 25 characters,
with a median of 6 characters per trope. Tropes and canonical examples are
shown in Table 1.

###  3.2 IMDB Dialogue Snippet Dataset

To obtain the utterances spoken by the characters, we crawled the IMDB Quotes
page for each movie. Though not every single utterance spoken by the character
may be available, as the quotes are submitted by IMDB users, many quotes from
most of the characters are typically found, especially for the famous
characters found in the Character Tropes dataset. The distribution of quotes
per trope is displayed in Figure 1. Our models were trained on 13,874 quotes
and validated and tested on a set of 1,734 quotes each.

We refer to each IMDB quote as a (contextualized) dialogue snippet, as each
quote can contain several lines between multiple characters, as well as
italicized text giving context to what might be happening when the quote took
place. Figure 2 show a typical dialogue snippet. 70.3% of the quotes are
multi-turn exchanges, with a mean of 3.34 turns per multi-turn exchange. While
the character’s own lines alone can be highly indicative of the trope, our
models show that accounting for context and the other characters’ lines and
context improves performance. The context, for instance, can give clues to
typical scenes and actions that are associated with certain tropes, while the
other characters’ lines give further detail into the relationship between the
character and his or her environment.

Figure 1: Number of IMDB dialogue snippets per trope Figure 2: Example IMDB
dialogue snippet containing multiple characters and context.

###  3.3 Character Trope Description Dataset

We also incorporate descriptions of each of the character tropes by using the
corresponding descriptions scraped from TVTropes222http://tvtropes.org. Each
description contains several paragraphs describing typical characteristics,
actions, personalities, etc. about the trope. As we demonstrate in our
experiments, the use of these descriptions improves classification
performance. This could allow our model to be applied more flexibly beyond the
movie character tropes – as one example, we could store descriptions of
personalities based on the Big Five traits in our Knowledge-Store memory.

##  4 Problem Formulation

Our goal is to train a model that can take a batch of dialogue snippets from
the IMDB dataset and predict the character trope.

Formally, let $N_{P}$ be the total number of character tropes in the character
tropes dataset. Each character $C$ is associated with a corresponding ground-
truth trope category $P$. Let $S=(D,E,O)$ be a dialog snippet associated with
a character $C$, where $D=[w_{D_{1}},w_{D_{2}}...,w_{D_{T}}]$ refers to the
character’s own lines, $E=[w_{E_{1}},w_{E_{2}}...,w_{E_{T}}]$ is the
contextual information and $O=[w_{O_{1}},w_{O_{2}}...,w_{O_{T}}]$ denotes the
other characters’ lines. We define all three components of $S$ to have fixed
sequence length $T$ and pad when necessary. Let $N_{S}$ be the total number of
dialogue snippets for a trope. We sample a set of $N_{diag}$ (where
$N_{diag}\ll N_{S}$) snippets from $N_{S}$ snippets related to the trope as
inputs to our model.

##  5 Attentive Memory Network

The Attentive Memory Network consists of two major components: (a) Attentive
Encoders, and (b) a Knowledge-Store Memory Module. Figure 3 outlines the
overall model. We describe the components in the following sections.

Figure 3: Illustration of the Attentive Memory Network. The network takes
dialogue snippets as input and predicts its associated character trope. In
this example, dialogue snippets associated with the character trope “Bruiser
with a Soft Corner” is given as input to the model.

###  5.1 Attentive Encoders

Not every piece of dialogue may be reflective of a latent persona. In order to
learn to ignore words and dialogue snippets that are not informative about the
trope we use a multi-level attentive encoder that operates at (a) the
individual snippet level, and (b) across multiple snippets.

#### Attentive Snippet Encoder

The snippet encoder extracts features from a single dialogue snippet $S$, with
attention over the words in the snippet. A snippet $S=(D,E,O)$ is fed to the
encoder to extract features from each of these textual inputs and encode them
into an embedding space. We use a recurrent neural network as our encoder,
explained in detail in Section 5.1.1. In order to capture the trope-reflective
words from the input text, we augment our model with a self-attention layer
which scores each word in the given text for its relevance. Section 5.1.2
explains how the attention weights are computed. The output of this encoder is
an encoded snippet embedding $S^{e}=(D^{e},E^{e},O^{e})$.

#### Attentive Inter-Snippet Encoder

As shown in Figure 3, the $N_{diag}$ snippet embeddings $S^{e}$ from the
snippet encoder are fed to our inter-snippet encoder. This encoder captures
inter-snippet relationship using recurrence over the $N_{diag}$ snippet
embeddings for a given trope and determines their importance. Some of the
dialogue snippets may not be informative about the trope, and the model learns
to assign low attention scores to such snippets. The resulting attended
summary vector from this phase is the persona representation $z$, defined as:

$\begin{array}[]{c}z=\gamma_{D}^{s}D^{s}+\gamma_{E}^{s}E^{s}+\gamma_{O}^{s}O^{s}\\\
\gamma_{D}^{s}+\gamma_{E}^{s}+\gamma_{O}^{s}=1\end{array}$ (1)

where $\gamma_{D}^{s},\gamma_{E}^{s},\gamma_{O}^{s}$ are learnable weight
parameters. $D^{s},E^{s},O^{s}$ refers to summary vectors of the $N_{diag}$
character’s lines, contextual information, and other characters’ lines,
respectively. In Section 7, we experiment with models that have
$\gamma_{E}^{s}$ and $\gamma_{O}^{s}$ set to 0 to understand how the
contextual information and other characters’ lines contribute to the overall
performance.

####  5.1.1 Encoder

Given an input sequence $(x_{1},x_{2},...,x_{T})$, we use a recurrent neural
network to encode the sequence into hidden states $(h_{1},h_{2},...,h_{T})$.
In our experiments, we use a gated recurrent network (GRU) Chung et al. (2014)
over LSTMs Hochreiter and Schmidhuber (1997) because the latter is more
computationally expensive. We use bidirectional GRUs and concatenate our
forward and backwards hidden states to get $\overleftrightarrow{h_{t}}$ for
$t=1,...,T$.

####  5.1.2 Attention

We define an attention mechanism $Attn$ that computes $s$ from the resultant
hidden states $\overleftrightarrow{h_{t}}$ of a GRU by learning to generate
weights $\alpha_{t}$. This can be interpreted as the relative importance given
to a hidden state $h_{t}$ to form an overall summary vector for the sequence.
Formally, we define it as:

$\displaystyle a_{t}=f_{attn}(h_{t})$ (2)
$\displaystyle\alpha_{t}=softmax(a_{t})$ (3) $\displaystyle
s=\sum_{t=1}^{T}{\alpha_{t}h_{t}}$ (4)

where $f_{attn}$ is a two layer fully connected network in which the first
layer projects $h_{t}\in{\rm I\\!R}^{d_{h}}$ to an attention hidden space
$g_{t}\in{\rm I\\!R}^{d_{a}}$, and the second layer produces a relevance score
for every hidden state at timestep $t$.

###  5.2 Memory Modules

Our model consists of a read-only ‘Knowledge-Store’ memory, and we also test a
recent read-write memory. External memories have been shown to help on natural
language processing tasks Sukhbaatar et al. (2015); Kumar et al. (2016);
Kaiser and Nachum (2017), and we find similar improvements in learning
capability.

####  5.2.1 Knowledge-Store Memory

The main motivation behind the Knowledge-Store memory module is to incorporate
prior domain knowledge. In our work, this knowledge refers to the trope
descriptions described in Section 3.3.

Related works have initialized their memory networks with positional encoding
using word embeddings Sukhbaatar et al. (2015); Kumar et al. (2016); Miller et
al. (2016). To incorporate the descriptions, we represent them with skip
thought vectors Kiros et al. (2015) and use them to initialize the memory keys
$K_{M}\in{\rm I\\!R}^{N_{P}\times d_{K}}$, where $N_{P}$ is the number of
tropes, and $d_{K}$ is set to the size of embedded trope description $R^{D}$,
i.e. $d_{K}=||R^{D}||$.

The values in the memory represent learnable embeddings of corresponding trope
categories $V_{M}\in{\rm I\\!R}^{N_{P}\times d_{V}}$, where $d_{V}$ is the
size of the trope category embeddings. The network learns to use the persona
representation $z$ from the encoder phase to find relevant matches in the
memory. This corresponds to calculating similarities between $z$ and the keys
$K_{M}$. Formally, this is calculated as:

$\displaystyle z_{M}=f_{z}(z)$ (5) $\displaystyle
p_{M}^{(i)}=softmax(z_{M}\cdot K_{M}[i])$ (6) $\displaystyle\forall
i\in\\{1,..,N_{P}\\}$

where $f_{z}:{\rm I\\!R}^{d_{h}}\mapsto{\rm I\\!R}^{d_{K}}$ is a fully-
connected layer that projects the persona representation in the space of
memory keys $K_{M}$. Based on the match probabilities $p_{M}^{(i)}$, the
values $V_{M}$ are weighted and cumulatively added to the original persona
representation as:

$r_{out}=\sum_{i=1}^{N_{P}}{p_{M}^{(i)}\cdot V_{M}[i]}$ (7)

We iteratively combine our mapped persona representation $z_{M}$ with
information from the memory $r_{out}$. The above process is repeated $n_{hop}$
times. The memory mapped persona representation $z_{M}$ is updated as follows:

$z_{M}^{{hop}}=f_{r}(z_{M}^{{hop}-1})+r_{out}$ (8)

where $z_{M}^{0}=z_{M}$, and $f_{r}:{\rm I\\!R}^{d_{V}}\mapsto{\rm
I\\!R}^{d_{K}}$ is a fully-connected layer. Finally, we transform the
resulting $z_{M}^{n_{hop}}$ using another fully-connected layer,
$f_{out}\in{\rm I\\!R}^{d_{K}}\mapsto{\rm I\\!R}^{d_{h}}$, via:

$\hat{z}_{M}=f_{out}(z_{M}^{n_{hop}})$ (9)

####  5.2.2 Read-Write Memory

We also tested a Read-Write Memory following Kaiser et. al Kaiser and Nachum
(2017), which was originally designed to remember rare events. In our case,
these ‘rare’ events might be key dialogue snippets that are particularly
indicative of latent persona. It consists of keys, which are activations of a
specific layer of model, i.e. the persona representation $z$, and values,
which are the ground-truth labels, i.e. the trope categories. Over time, it is
able to facilitate predictions based on past data with similar activations
stored in the memory. For every new example, the network writes to memory for
future look up. A memory with memory size $N_{M}$ is defined as:

$M=(K_{N_{M}\times d_{H}},V_{N_{M}},A_{N_{M}})$ (10)

##### Memory Read

We use the persona embedding $z$ as a query to the memory. We calculate the
cosine similarities between $z$ and the keys in $M$, take the softmax on the
top-k neighbors, and compute a weighted embedding $\hat{z}_{M}$ using those
scores.

##### Memory Write

We update the memory in a similar fashion to the original work by Kaiser and
Nachum (2017), which takes into account the maximum age of items as stored in
$A_{N_{M}}$.

##  6 Objective Losses

To train our model, we utilize the different objective losses described below.

###  6.1 Classification Loss

We calculate the probability of a character belonging to a particular trope
category $P$ through Equation 11, where $f_{P}:{\rm I\\!R}^{d_{h}}\mapsto{\rm
I\\!R}^{N_{P}}$ is a fully-connected layer, and $z$ is the persona
representation produced by the multi-level attentive encoders described in
Equation 1. We then optimize the categorical cross-entropy loss between the
predicted and true tropes as in Equation 12, where $N_{P}$ is the total number
of tropes, $q_{j}$ is the predicted distribution that the input character
fulls under trope $j$, and $p_{j}\in\\{0,1\\}$ denotes the ground-truth of
whether the input snippets come from characters from the $j^{th}$ trope.

$\displaystyle q=softmax(f_{P}(z))$ (11) $\displaystyle
J_{CE}=\sum_{j=1}^{N_{P}}{-p_{j}\log(q_{j})}$ (12)

###  6.2 Trope Description Triplet Loss

In addition to using trope descriptions to initialize the Knowledge-Store
Memory, we also test learning from the trope descriptions through a triplet
loss Hoffer and Ailon (2015). We again use the skip thought vectors to
represent the descriptions. Specifically, we want to maximize the similarity
of representations obtained from dialogue snippets with their corresponding
description, and minimize their similarity with negative examples. We
implement this as:

$\displaystyle R^{P}=f_{D}(z)$ (13) $\displaystyle
J_{T}=max(0,s(R^{P},R^{D}_{n})-s(R^{P},R^{D}_{p})+\alpha_{T})$ (14)

where $f_{D}:{\rm I\\!R}^{d_{h}}\mapsto{\rm I\\!R}^{||R^{D}||}$ is a fully-
connected layer. The triplet ranking loss is then Equation 14, where
$\alpha_{T}$ is a learnable margin parameter and $s(\cdot,\cdot)$ denotes the
similarity between trope embeddings ($R^{P}$), positive ($R^{D}_{p}$) and
negative ($R^{D}_{n}$) trope descriptions.

#### Trope Description Triplet Loss with Memory Module

If a memory module is used, we compute a new triplet loss in place of the one
described in Equation 14. Models that use a memory module should learn a
representation $\hat{z}_{M}$, based on either the prior knowledge stored in
the memory (as in Knowledge-Store memory) or the top-$k$ key matches (as in
Read-Write memory), that is similar to the representation of the trope
descriptions.

This is achieved by replacing the persona embedding $z$ in Equation 13 with
the memory output $\hat{z}_{M}$ as shown in Equation 15, where $f_{D_{M}}:{\rm
I\\!R}^{d_{h}}\mapsto{\rm I\\!R}^{||R^{D}||}$ is a fully-connected layer. To
compute the new loss, we combine the representations obtained from Equations
13 and 15 through a learnable parameter $\gamma$ that determines the
importance of each representation. Finally, we utilize this combined
representation $\hat{R}^{P}$ to calculate the loss as shown in Equation 17.

$\displaystyle R^{P}_{M}=f_{D_{M}}(\hat{z}_{M})$ (15)
$\displaystyle\hat{R}^{P}=\gamma R^{P}+(1-\gamma)R_{M}^{P}$ (16)
$\displaystyle
J_{MT}=max(0,s(\hat{R}^{P},R^{D}_{n})-s(\hat{R}^{P},R^{D}_{p})+\alpha_{MT})$
(17)

###  6.3 Read-Write Memory Losses

When the Read-Write memory is used, we use two extra loss functions. The first
is a Memory Ranking Loss $J_{MR}$ as done in Kaiser and Nachum (2017), which
learns based on whether a query with the persona embedding $z$ returns nearest
neighbors with the correct trope. The second is a Memory Classification Loss
$J_{MCE}$ that uses the values returned by the memory to predict the trope.
The full details for both are found in Supplementary Section A.

###  6.4 Overall Loss

We combine the above losses through:

$\begin{array}[]{c}J=\beta_{CE}\cdot J_{CE}\\\ +\;\beta_{T}\cdot\hat{J}_{T}\\\
+\;\beta_{MR}\cdot J_{MR}+\;\beta_{MCE}\cdot J_{MCE}\\\ \\\
\hat{J}_{T}=\begin{cases}J_{MT}&\text{if memory module is used }\\\
J_{T}&\text{otherwise}.\end{cases}\end{array}$ (18)

where $\beta=[\beta_{CE},\beta_{MCE},,\beta_{T},\beta_{MR}]$ are learnable
weights such that $\sum_{i}\beta_{i}=1$. Depending on which variant of the
model is being used, the list $\beta$ is modified to contain only relevant
losses. For example, when the Knowledge-Store memory is used, we set
$\beta_{MR}=\beta_{MCE}=0$ and $\beta$ is modified to
$\beta=[\beta_{CE},\beta_{T}]$. We discuss different variants of our model in
the next section.

Model | Accuracy | Precision | Recall | F1
---|---|---|---|---
baseline_char | $\numprint{0.286057692308}$ | $\numprint{0.538461538462}$ | $\numprint{0.286153846154}$ | $\numprint{0.339230769231}$
baseline_3 | $\numprint{0.287259615385}$ | $\numprint{0.552307692308}$ | $\numprint{0.288461538462}$ | $\numprint{0.349230769231}$
attn_char | $\numprint{0.629807692308}$ | $\numprint{0.586153846154}$ | $\numprint{0.627692307692}$ | $\numprint{0.6}$
attn_3 | $\numprint{0.629807692308}$ | $\numprint{0.59}$ | $\numprint{0.63}$ | $\numprint{0.603076923077}$
attn_3_tropetrip | $\numprint{0.615384615385}$ | $\numprint{0.566153846154}$ | $\numprint{0.614615384615}$ | $\numprint{0.583076923077}$
attn_3_tropetrip-500 | $\numprint{0.644230769231}$ | $\numprint{0.600769230769}$ | $\numprint{0.642307692308}$ | $\numprint{0.614615384615}$
attn_3_ks-mem | $\numprint{0.677884615385}$ | $\numprint{0.647692307692}$ | $\numprint{0.676153846154}$ | $\numprint{0.656923076923}$
attn_3_rw-mem | $\numprint{0.634615384615}$ | $\numprint{0.598461538462}$ | $\numprint{0.634615384615}$ | $\numprint{0.610769230769}$
attn_3_tropetrip_ks-mem | $\numprint{0.663461538462}$ | $\numprint{0.628461538462}$ | $\numprint{0.662307692308}$ | $\numprint{0.639230769231}$
attn_3_tropetrip-500_ks-mem | $\numprint{0.653846153846}$ | $\numprint{0.617692307692}$ | $\numprint{0.652307692308}$ | $\numprint{0.629230769231}$
attn_3_tropetrip_rw-mem | $\numprint{0.649038461538}$ | $\numprint{0.601538461538}$ | $\numprint{0.647692307692}$ | $\numprint{0.616923076923}$
attn_3_tropetrip-500_rw-mem | $\numprint{0.644230769231}$ | $\numprint{0.607692307692}$ | $\numprint{0.643846153846}$ | $\numprint{0.619230769231}$
attn_3_tropetrip-500_ks-mem_ndialog16 | $\numprint{0.740384615385}$ | $\numprint{0.706923076923}$ | $\numprint{0.740769230769}$ | $\numprint{0.717692307692}$
attn_3_tropetrip-500_ks-mem_ndialog32 | $\numprint{0.75}$ | $\numprint{0.75}$ | $\numprint{0.75}$ | $\numprint{0.75}$
attn_3_tropetrip_ks-mem_ndialog16 | $\numprint{0.740384615385}$ | $\numprint{0.721538461538}$ | $\numprint{0.740769230769}$ | $\numprint{0.727692307692}$
attn_3_tropetrip_ks-mem_ndialog32 | $\numprint{0.730769230769}$ | $\numprint{0.711538461538}$ | $\numprint{0.730769230769}$ | $\numprint{0.718461538462}$
Table 2:  Experimental results. Details and analysis are given in Section 7.1.
The best performing results in each block are bolded. The first block examines
the baseline model vs. the attention model, as well as use of different
inputs. The second block uses the triplet loss, and the third block uses our
memory modules. The fourth block combines the triplet loss and memory module,
which the fifth block extends to larger $N_{diag}$.

##  7 Experiments

We experimented with combinations of our various modules and losses. The
experimental results and ablation studies are described in the following
sections, and the experimental details are described in Supplementary Section
B. The different model permutation names in Table 2, e.g.
“attn_3_tropetrip_ks-mem_ndialog16”, are defined as follows:

  * •

baseline vs attn: The ‘baseline’ model uses only one dialogue snippet $S$ to
predict the trope, i.e. $N_{diag}=1$. Hence, the inter-snippet encoder is not
used. The ‘attn’ model operates on $N_{diag}$ dialogue snippets using the
inter-snippet encoder to assign an attention score for each snippet $S_{i}$.

  * •

char vs. 3: To measure the importance of context and other characters’ lines,
we have two variants – ‘char’ uses only the character’s lines, while ‘3’ uses
the character’s lines, other character’s lines, and all context lines.
Formally, in ‘char’ mode, we set $\gamma_{E}^{s}$ and $\gamma_{O}^{s}$ to 0 in
Equation 1. In ‘attn’ mode, $(\gamma_{E}^{s},\gamma_{O}^{s},\gamma_{D}^{s})$
are learned by the model.

  * •

tropetrip: The presence of ’tropetrip’ indicates that the triplet loss on the
trope descriptions was used. If ‘-500’ is appended to ‘tropetrip’, then the
4800-dimensional skip embeddings representing the descriptions in Equations 15
and 17 are projected to 500 dimensions using a fully connected layer.

  * •

ks-mem vs. rw-mem: ‘ks-mem’ refers to the Knowledge-Store memory, and ‘rw-mem’
refers to the Read-Write memory.

  * •

ndialog: The number of dialogue snippets $N_{diag}$ used as input for the
attention models. Any attention model without the explicit $N_{diag}$ listed
uses $N_{diag}=8$.

###  7.1 Ablation Results

Baseline vs. Attention Model. The attention model shows a large improvement
over the baseline models. This matches our intuition that not every quote is
strongly indicative of character trope. Some may be largely expository or
‘chit-chat’ pieces of dialogue. Example attention scores are shown in Section
7.2.

Though our experiments showed marginal improvement between using the ‘char’
data and the ‘3’ data, we found that using all 3 inputs had greater
performance for models with the triplet loss and read-only memory. This is
likely because the others’ lines and context capture more of the social
dynamics and situations that are described in the trope descriptions.
Subsequent results are shown only for the ‘attn_3’ models.

Figure 4: Attention scores for a batch of dialogues for the “byronic hero”
trope 444TVtropes.org defines a byronic hero as “Sometimes an Anti-Hero,
others an Anti-Villain, or even Just a Villain, Byronic heroes are charismatic
characters with strong passions and ideals, but who are nonetheless deeply
flawed individuals….”.

Trope Description Triplet Loss. Adding the trope description loss alone
provided relatively small gains in performance, though we see greater gains
when combined with memory. While both use the descriptions, perhaps the
Knowledge Store memory matches an embedding against all the tropes, whereas
the trope triplet loss is only provided information from one positive and one
negative example.

Memory Modules. The Knowledge-Store memory in particular was helpful.
Initialized with the trope descriptions, this memory can ‘sharpen’ queries
toward one of the tropes. The Read-Write memory had smaller gains in
performance. It may be that more data is required to take advantage of the
write capabilities.

Combined Trope Description Triplet Loss and Memory Modules. Using the triplet
loss with memory modules led to greater performance when compared to the
$attn\\_3$ model, but the performance sits around the use of either triplet
only or memory only. However, when we increase the $N_{diag}$ to 16 or 32, we
find a jump in performance. This is likely the case because the model has both
increased learning capacity and a larger sample of data at every batch, which
means at least some of the $N_{diag}$ quotes should be informative about the
trope.

###  7.2 Attention Scores

Because the inter-snippet encoder provides such a large gain in performance
compared to the baseline model, we provide an example illustrating the weights
placed on a batch of $N_{diag}$ snippets. Figure 4 shows the attention scores
for the character’s lines in the “byronic hero” trope. Matching what we might
expect for an antihero personality, we find the top weighted line to be full
of confidence and heroic bluster, while the middle lines hint at the
characters’ personal turmoil. We also find the lowly weighted sixth and
seventh lines to be largely uninformative (e.g. “I heard things.”), and the
last line to be perhaps too pessimistic and negative for a hero, even a
byronic one.

###  7.3 Purity scores of character clusters

Finally, we measure our ability to recover the trope ‘clusters’ (with one
trope being a cluster of its characters) with our embeddings through the
purity score used in Bamman et al. (2014). Equation 19 measures the amount of
overlap between two clusterings, where $N$ is the total number of characters,
$g_{i}$ is the $i$-ith ground truth cluster, and $c_{j}$ is the $j$-th
predicted cluster.

$\text{Purity}=\frac{1}{N}\sum_{i}\text{max}_{j}|g_{i}\cap c_{j}|$ (19)

We use a simple agglomerative clustering method on our embeddings with a
parameter $k$ for the number of clusters. The methods in Bamman et al. (2014)
contain a similar hyper-parameter for the number of persona clusters. We note
that the metrics are not completely comparable because not every character in
the original dataset was found on IMDB. The results are shown in Table 3. It
might be expected that our model perform better because we use the character
tropes themselves as training data. However, dialogue may be noisier than the
movie summary data; their better performing Persona Regression (PR) model also
uses useful metadata features such as the movie genre and character gender. We
simply note that our scores are comparable or higher.

$k$ | PR | DP | AMN
---|---|---|---
25 | 42.9 | 39.63 | 48.4
50 | 36.5 | 31.0 | 48.1
100 | 30.3 | 24.4 | 45.2
Table 3: Cluster purity scores. $k$ is the number of clusters, PR and DP are
the Persona Regression and Dirichlet Persona models from Bamman et al. (2014),
and AMN is our attention memory network.

##  8 Application: Narrative Analysis

We collected IMDB quotes for the top 250 movies on IMDB. For every character,
we calculated a character embedding by taking the average embedding produced
by passing all the dialogues through our model. We then calculated movie
embeddings by taking the weighted sum of all the character embeddings in the
movie, with the weight as the percentage of quotes they had in the movie. By
computing distances between pairs of character or movie embeddings, we could
potentially unearth notable similarities. We note some of the interesting
clusters below.

###  8.1 Clustering Characters

  * •

Grumpy old men: Carl Fredricksen (Up); Walk Kowalski (Gran Torino)

  * •

Shady tricksters, crooks, well versed in deceit: Ugarte (Casablanca); Eames
(Inception)

  * •

Intrepid heroes, adventurers: Indiana Jones (Indiana Jones and the Last
Crusade); Nemo (Finding Nemo); Murph (Interstellar)

###  8.2 Clustering Movies

  * •

Epics, historical tales: Amadeus, Ben-Hur

  * •

Tortured individuals, dark, violent: Donnie Darko, Taxi Driver, Inception, The
Prestige

  * •

Gangsters, excess: Scarface, Goodfellas, Reservoir Dogs, The Departed, Wolf of
Wall Street

##  9 Conclusion

We used the character trope classification task as a test bed for learning
personas from dialogue. Our experiments demonstrate that the use of a multi-
level attention mechanism greatly outperforms a baseline GRU model. We were
also able to leverage prior knowledge in the form of textual descriptions of
the trope. In particular, using these descriptions to initialize our
Knowledge-Store memory helped improved performance. Because we use short text
and can leverage domain knowledge, we believe future work could use our models
for applications such as personalized dialogue systems.

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##  Appendix A Read-Write Memory Losses

###  A.1 Memory Ranking Loss $J_{MR}$

We want the model to learn to make efficient matches with the memory keys in
order to facilitate look up on past data. To do this, we find a positive and
negative neighbor after computing the $k$ nearest neighbors
$(n_{1},...,n_{k})$ by finding the smallest index $p$ such that $V[p]=P$ and
$n$ such that $V[n]\neq P$ respectively. We define the memory ranking loss as:

$J_{MR}=max(0,s(z,K[n])-s(z,K[p])+\alpha_{MR})$ (20)

where $\alpha_{MR}$ is a learnable margin parameter and $s(\cdot,\cdot)$
denotes the similarity between persona embeddings ($z$), key representations
of positive ($K[p]$) and negative ($K[n]$) neighbors. The above equation is
consistent with the memory loss defined in the original work Kaiser and Nachum
(2017).

###  A.2 Memory Classification Loss $J_{MCE}$

The Read-Write memory returns $\hat{z}_{M}$ and values
$v_{M}=V[n_{i}]\;\forall i\in\\{n_{1},..,n_{k}\\}$. The probability of the
given input dialogues belonging to a particular persona category $P$ is
computed using values $v_{M}$ returned from the memory via:

$q^{M}=softmax(f_{{P}_{M}}(v_{M}))$ (21)

where $f_{{P}_{M}}:{\rm I\\!R}^{k}\mapsto{\rm I\\!R}^{N_{P}}$ is a fully-
connected layer. We replace the $q_{j}$ with $q_{j}^{M}$ in Equation 12 and
calculate the categorical cross entropy to get $J_{MCE}$.

##  Appendix B Experimental details

The vocabulary size was set to 20000. We used a GRU hidden size of 200, a word
embedding size of 300, and the word embedding lookup was initialized with
GLoVe Pennington et al. (2014). For the Read-Write memory module, we used k=8
when calculating the nearest neighbors and a memory size of 150. Our models
were trained using Adam Kingma and Ba (2014).