# Barycentric coordinate neighbourhoods in Riemannian manifolds

Ramsay Dyer<EMAIL_ADDRESS>Gert Vegter Johann Bernoulli Institute,
Rijksuniversiteit<EMAIL_ADDRESS>Mathijs Wintraecken INRIA Sophia
<EMAIL_ADDRESS>
###### Abstract

We quantify conditions that ensure that a signed measure on a Riemannian
manifold has a well defined centre of mass. We then use this result to
quantify the extent of a neighbourhood on which the Riemannian barycentric
coordinates of a set of $n+1$ points on an $n$-manifold provide a true
coordinate chart, i.e., the barycentric coordinates provide a diffeomorphism
between a neighbourhood of a Euclidean simplex, and a neighbourhood containing
the points on the manifold.

#### Keywords.

Riemannian centre of mass, Karcher means, barycentric coordinates, Riemannian
manifold, Riemannian simplices

###### Contents

  1. 1 Background and notation
    1. 1.1 Convexity
    2. 1.2 Riemannian centre of mass
    3. 1.3 Comparison theorems
    4. 1.4 Signed measures
  2. 2 Center of mass of signed measures
    1. 2.1 Ensuring the convexity of $\mathcal{E}_{\mu}$
    2. 2.2 The gradient of $\mathcal{E}_{\mu}$ on the boundary of a ball
  3. 3 Barycentric coordinate neighbourhoods
    1. 3.1 Simplifying the setting
    2. 3.2 The extent of barycentric coordinate neighbourhoods
    3. 3.3 Riemannian simplices that share a facet

The focus of this work is on identifying conditions under which barycentric
coordinates will parameterise a neighbourhood of a simplex in a Riemannian
manifold, and specifically on quantifying the extent of the permissible
neighbourhood. Recall that if
$\sigma=\\{p_{0},\ldots,p_{n}\\}\subset\mathbb{E}^{n}$ defines a
_nondegenerate_ simplex in Euclidean space111We consider $\mathbb{E}^{n}$ to
have an affine structure, and Euclidean distance function, but no canonical
origin or coordinate system., i.e.,
$\operatorname{aff}(\sigma)=\mathbb{E}^{n}$, where
$\operatorname{aff}(\sigma)$ is the affine hull of $\sigma$, then
$\mathbb{E}^{n}$ can be parameterised by the barycentric coordinate functions;
any point $u\in\mathbb{E}^{n}$ is uniquely defined by a set of $(n+1)$
barycentric coordinates, $(\lambda_{i})$ that satisfy $\sum\lambda_{i}=1$. The
point $u$ corresponding to $(\lambda_{i})$ can be identified as the point that
minimises the function

$x\mapsto\tfrac{1}{2}\sum_{i=0}^{n}\lambda_{i}d_{\mathbb{E}^{n}}(x,p_{i})^{2},$
(1)

where $d_{\mathbb{E}^{n}}(x,y)$ is the Euclidean distance function. In this
way the barycentric coordinates define the familiar affine functions that
satisfy $\lambda_{i}(p_{j})=\delta_{ij}$.

We can view a given set of barycentric coordinates
$\lambda=(\lambda_{0},\ldots,\lambda_{n})$ as a point in $\mathbb{R}^{n+1}$.
The set $\boldsymbol{\Delta}^{\\!n}$ of all points in $\mathbb{R}^{n+1}$ with
non-negative coefficients that sum to $1$ is called the _standard Euclidean
$n$-simplex_. Thus the minimisation of the function (1) defines a _barycentric
coordinate map_ from the the plane
$\\{\lambda\in\mathbb{R}^{n+1}\mid\sum\lambda_{i}=1\\}$ to $\mathbb{E}^{n}$
that brings the standard Euclidean simplex to the Euclidean simplex
$\boldsymbol{\sigma}_{\mathbb{E}^{n}}=\operatorname{conv}(\sigma)\subset\mathbb{E}^{n}$.

This same technique can be used to define barycentric coordinate
neighbourhoods in a Riemannian manifold $M$. If $\sigma=\\{p_{i}\\}\subset M$,
we consider the function (1) except that the Euclidean distance
$d_{\mathbb{E}^{n}}$ is replaced by $d_{M}$, the intrinsic metric of the
manifold. The first problem is to identify conditions under which this
function admits a unique minimum. This is a particular case of a more general
problem of identifying conditions under which the _Riemannian centre of mass_
of a measure is well defined. There are several demonstrations of the
existence and uniqueness of Riemannian centres of mass.

The standard reference for the subject is the excellent exposition by Karcher
[Kar77] (see also the recent note outlining the earlier history of the subject
[Kar14]). He uses Jacobi field estimates to demonstrate that if the support of
a measure $\mu$ is contained in a sufficiently small convex ball, then the
_energy function_ $x\mapsto\tfrac{1}{2}\int d_{M}(x,y)^{2}\,d\mu(y)$ is
convex, and since the gradient is pointing outward on the boundary of the
ball, there is a unique minimum in the ball’s interior.

Kendall [Ken90] gave a different proof based on the concept of “convex
geometry” which arose in earlier work on Dirichlet problems. Kendal was able
to relax Karcher’s bounds on the size of the ball that must contain the
support of the measure.

Groisser [Gro04] gives a proof based on a contraction mapping argument, using
a map defined from the gradient of the above mentioned energy function. This
yields a constructive algorithm for finding the centre of mass, but the
constraint on the extent of the admissible neighbourhood is more restrictive
than Karcher’s.

These works all assume an unsigned measure, so they do not apply to situations
where negative weights are involved. The existence and uniqueness of the
Riemannian centre of mass with non-negative weights at least allows us to
define a “filled in” geometric _Riemannian simplex_ associated to a finite set
of points with small diameter. If $\sigma\subset M$ is a set of $n+1$ points
in a Riemannian $n$-manifold, and $\sigma$ is contained in a sufficiently
small convex ball $B$, then we can define the Riemannian simplex
$\boldsymbol{\sigma}_{\\!\\!M}$ as the image of the standard simplex
$\boldsymbol{\Delta}^{\\!n}$ under the barycentric coordinate map.

In order to make the concept of Riemannian simplices useful for triangulating
manifolds, we need to ensure that the barycentric coordinate map is not only
well-defined, but also an embedding. When this is the case, we say that the
Riemannian simplex is _nondegenerate_. We recently established conditions,
based on the local curvatures of the manifold and the quality of the point set
$\sigma$, that ensure that the barycentric coordinate map is an embedding
[DVW15].

Although we used the result on nondegenerate Riemannian simplices to establish
criteria for ensuring that a simplicial complex triangulates a Riemannian
manifold, we did not establish some simple properties that we expect from
geometric simplices. Specifically, if $\boldsymbol{\sigma}_{\\!\\!M}$ and
$\boldsymbol{\tau}_{\\!M}$ are two Riemannian $n$-simplices in an
$n$-manifold, and they have $n$ vertices in common, and agree with a local
orientation of the manifold, then is
$\boldsymbol{\tau}_{\\!M}\cap\boldsymbol{\sigma}_{\\!\\!M}$ exactly the
Riemannian simplex that is the common facet? Of course it is impossible to
ensure a triangulation by Riemannian simplices without ensuring this property,
but in [DVW15] we did not show this property for two isolated simplices.

This is one of the motivations for establishing the barycentric coordinate
neighbourhoods that are the focus of the current work. If the barycentric
neighbourhood defined by $\boldsymbol{\sigma}_{\\!\\!M}$ encompasses
$\boldsymbol{\tau}_{\\!M}$ and vice versa, then the above problem is easily
resolved, as discussed in Section 3.3.

In order to establish such barycentric coordinate neighbourhoods we must first
ensure that the barycentric coordinate map is well defined when the weights
may be negative. To this end we give an alternate proof of the existence and
uniqueness of Riemannian centres of mass. The result, Theorem 3, can
accommodate signed measures, i.e., mass distributions that include negative
weights. This is the main technical contribution of this paper.

A demonstration of the existence and uniqueness of Riemannian centres of mass
for signed measures has already been published by Sander [San16, Theorem
3.19]. He was motivated by a desire to describe higher order interpolation on
Riemannian manifolds, which needs to accommodate negative interpolation
weights. Sander’s proof is modelled on Groisser’s proof for unsigned measures
[Gro04]. The result is stated in terms of several parameters governing the
neighbourhood of validity, and they are bounded in an intricate manner. We are
specifically interested in obtaining an explicit bound on the extent of the
admissible neighbourhood in terms of primitive properties of the measure and
bounds on the sectional curvatures. Such an explicit bound is not easily
obtained from Sander’s result.

We provide an elementary proof of the existence and uniqueness of Riemannian
centres of mass for signed measures that is in the spirit of Karcher’s
demonstration for the unsigned case. However, instead of using Jacobi field
estimates to demonstrate the convexity of the energy function, we rely on a
series expansion of the cosine rule in constant curvature spaces. At the
expense of this brute force calculation, we obtain a concise and elementary
demonstration. Furthermore, our bounds on the admissible neighbourhood are
explicit, and they can be reduced to Karcher’s bounds in the case where the
measure is unsigned. This is especially evident in the formulation of the
result expressed in Corollary 8.

In Section 1 we briefly review the tools we will use for our exposition,
including a recap of the main ideas in defining Riemannian centres of mass and
barycentric coordinates. We demonstrate our main theorem on Riemannian centres
of mass of signed measures in Section 2, and Section 3 is devoted to
exploiting this for the development of barycentric coordinate neighbourhoods
that motivates this work.

In order to obtain true coordinate neighbourhoods we need to ensure not only
that the barycentric coordinate map is well defined, but also that it is an
embedding. We observe in Section 3 that we can use the arguments in [DVW15]
verbatim since they did not rely on the sign of the weights, only on the
convexity of the energy function, but this is established as part of the
demonstration of Theorem 3. We obtain Theorem 9 and Proposition 11 which
describe the extent of admissible barycentric coordinate neighbourhoods.
Finally, in Section 3.3 we make the observation that suitably constrained
compatibly oriented Riemannian $n$-simplices that share $n$-vertices intersect
only on their shared facet.

##  1 Background and notation

In this work $M$ refers to a $C^{\infty}$ Riemannian manifold (without
boundary) of dimension $n$. The adjective _smooth_ is synonymous with
$C^{\infty}$. A function defined on a closed set $A\subset M$ is _smooth_ if
it can be extended to a smooth function on an open neighbourhood of $A$. The
distance between $x,y\in M$ is denoted $d_{M}(x,y)$, and $B_{M}(c,r)=\\{x\in
M\mid d_{M}(c,x)<r\\}$ is the open geodesic ball of radius $r$ centred at
$c\in M$. The topological closure of a set $B\subset M$ is denoted
$\overline{B}$. A geodesic is _minimising_ if it is the shortest path between
any two of its points (all geodesics are locally minimising). A (geodesic)
_segment_ is the trace of a minimising geodesic. We sometimes abuse notation
and terminology by identifying a geodesic with its trace.

The injectivity radius $\iota_{M}$ is the supremum of the distances $r$ such
that any two points $x,y\in M$ with $d_{M}(x,y)<r$ have a unique minimising
geodesic connecting them. This number is positive if $M$ is compact. For any
$x\in M$, the _exponential map_ $\exp_{x}\colon T_{x}M\to M$ maps a
vector222In general $\exp_{x}$ may not be defined on all of $T_{x}M$, unless
$M$ is _complete_ , but this detail is not a concern for us. $v$ in the
tangent space $T_{x}M$ to the point $\gamma(\left|v\right|)\in M$, where
$\gamma$ is the unique unit-speed geodesic emenating from $x$ with
$\gamma^{\prime}(0)=v/\left|v\right|$, and $\left|v\right|=\left\langle
v,v\right\rangle^{1/2}$ is the norm defined by the Riemannian metric. If
$r<\iota_{M}$, then $\exp_{x}$ is a diffeomorphism (onto its image) when
restricted to $B_{T_{x}M}(x,r)$. Identifying $T_{x}M$ with $\mathbb{R}^{n}$,
this yields a local coordinate system called _Riemann normal coordinates_.

For any $x\in M$, and any 2-dimensional subspace $H\subseteq T_{x}M$ we
associate a _sectional curvature_ , which is the Gaussian curvature at $x$ of
the surface $\exp_{x}(H)$. We denote by $K$ the sectional curvature function,
but we are concerned only with bounds on $K$, never individual values.

###  1.1 Convexity

A set $B\subseteq M$ is _convex_ if for all $x,y\in B$ there is a unique
minimising geodesic in $M$ connecting $x$ to $y$, and this geodesic is
contained in $B$. If ${\Lambda_{u}}$ is an upper bound on the sectional
curvatures of $M$, then for any point $c\in M$ the ball
$\overline{B}_{M}(c,r)$ is convex if
$r<\min\bigl{\\{}\frac{\iota_{M}}{2},\frac{\pi}{2\sqrt{{\Lambda_{u}}}}\bigr{\\}}$,
where we set $\frac{\pi}{2\sqrt{{\Lambda_{u}}}}=\infty$ if ${\Lambda_{u}}\leq
0$ (see [Cha06, Thm. IX.6.1]).

A function $f\colon A\to\mathbb{R}$, with $A\subseteq M$ is _convex_ if for
any $p\in A$ and geodesic $\gamma$ with $\gamma(0)=p$, we have
$\frac{d^{2}}{dt^{2}}f(\gamma(t))|_{t=0}\geq 0$. We say $f$ is _strictly
convex_ if this inequality is strict. If $A$ is compact and convex with
nonempty interior, and $f$ is strictly convex with $\operatorname{grad}f$
pointing outwards on $\partial{A}$ (i.e., for any $x\in\partial{A}$,
$y\in\operatorname{int}(A)$, and $\gamma$ a minimising geodesic from $x$ to
$y$, we have
$\left\langle\gamma^{\prime}(0),\operatorname{grad}f\right\rangle<0$), then
$f$ has a unique minimum in the interior of $A$. (The gradient assumption
precludes a minimum on the boundary, and the existence of multiple minima
contradicts strict convexity via the same argument as in the Euclidean case.)

###  1.2 Riemannian centre of mass

Let $\mu$ be an unsigned measure whose support is contained within a convex
geodesic ball $B_{\rho}\subseteq M$ with radius $\rho$. Consider the energy
function $\mathcal{E}_{\mu}\colon\overline{B}_{\rho}\to\mathbb{R}$ defined by

$\mathcal{E}_{\mu}(x)=\frac{1}{2}\int d_{M}(x,y)^{2}\,d\mu(y),$ (2)

where $d\mu(y)$ indcates that the integration is with respect to the variable
$y$, and the domain of integration is understood to be $M$, or equivalently
$B_{\rho}$, since it contains the support of $\mu$. Karcher [Kar77, Theorem
1.2] showed that $\mathcal{E}_{\mu}$ has a unique minimum on $B_{\rho}$,
provided that, if there are positive sectional curvatures in $B_{\rho}$, the
radius satisfies $\rho\leq\frac{\pi}{4\sqrt{\kappa}}$, where $\kappa$ is an
upper bound on the sectional curvature in $B_{\rho}$. This unique minimum is
called the _Riemannian centre of mass_ of $\mu$.

Since we will be working within these same constraints, it is convenient to
define

$\rho_{0}=\min\left\\{\frac{\iota_{M}}{2},\frac{\pi}{4\sqrt{{\Lambda_{u}}}}\right\\},$
(3)

where ${\Lambda_{u}}$ is an upper bound on the sectional curvatures of $M$,
and we understand $\frac{1}{\sqrt{{\Lambda_{u}}}}=\infty$ if
${\Lambda_{u}}\leq 0$. Then $\rho<\rho_{0}$ is sufficient to enforce the
convexity condition and the curvature sensitive bound on the radius.

In the case of a discrete measure concentrated on a finite set of points
$\sigma=\\{p_{0},\ldots,p_{k}\\}\subset B_{\rho}$, then instead of $\mu$ we
use $\lambda$, with $\lambda_{i}=\lambda(p_{i})$, and when $\sum\lambda_{i}=1$
this defines the _barycentric coordinates_ of the centre of mass. In other
words, the unique point $x\in B_{\rho}$ that has barycentric coordinates
$\lambda$ with respect to $\sigma$ is the point that minimises

$\mathcal{E}_{\lambda}(x)=\frac{1}{2}\sum_{i}\lambda_{i}d_{M}(x,p_{i})^{2}.$
(4)

The “filled-in” _Riemannian simplex_ $\boldsymbol{\sigma}_{\\!\\!M}$ defined
by $\sigma$ is defined to be the set of points in $B_{\rho}$ defined by all
nonnegative barycentric coordinates.

###  1.3 Comparison theorems

We denote by $\mathbb{H}_{\kappa}=\mathbb{H}_{\kappa}^{n}$ the simply
connected space of dimension $n$ whose sectional curvatures are all equal to
the constant $\kappa$. A comparison theorem provides inequalities relating
geometry on a manifold $M$ to that on an appropriate $\mathbb{H}_{\kappa}$.
The comparison theorems we state here are equivalent corollaries of the Rauch
comparison theorem [BK81, 6.4.1] and can be found in [Kar89, BK81, 6.4.3,
6.4.4], and many other standard references.

The first comparison theorem captures the notion that geodesics diverge more
quickly in low curvature than in high curvature. A _hinge_ in $M$ consists of
two geodesic segments of lengths $\ell_{1}$ and $\ell_{2}$ meeting at a common
endpoint at an angle $\alpha$. A _comparison hinge_ is a hinge in
$\mathbb{H}_{\kappa}$ with the same corresponding side lengths and angle.
Three points $w,x,y$ in a convex ball in $M$ define a hinge with the angle
$\alpha$ at $x$ being the angle between the unique geodesic segments
connecting $w$ to $x$ and $x$ to $y$.

######  Lemma 1 (Hinge comparison).

Assume the sectional curvature $K$ in $M$ is bounded by ${\Lambda_{\ell}}\leq
K\leq{\Lambda_{u}}$, and let $B_{\rho}\subseteq M$ be a convex ball with
radius $\rho\leq\rho_{0}$ (as defined in (3)). Suppose $w,x,y\in B_{\rho}$
define a hinge with angle $\alpha$ at $x$, and let
$\bar{w},\bar{x},\bar{y}\in\mathbb{H}_{{\Lambda_{\ell}}}$ and
$\tilde{w},\tilde{x},\tilde{y}\in\mathbb{H}_{{\Lambda_{u}}}$ define comparison
hinges.

Then

$d_{\mathbb{H}_{{\Lambda_{u}}}}(\tilde{w},\tilde{y})\leq d_{M}(w,y)\leq
d_{\mathbb{H}_{{\Lambda_{\ell}}}}(\bar{w},\bar{y}).$

A _geodesic triangle_ $T$ is a collection of three points (vertices) in $M$,
together with geodesic segments between them. A _comparison triangle_
$T_{\kappa}$ is a geodesic triangle in $\mathbb{H}_{\kappa}$ with the same
edge lengths. Thus a vertex, edge, or angle in $T$ has a corresponding vertex,
edge, or angle in $T_{\kappa}$.

The second comparison theorem says that triangles are fatter in spaces with
higher curvature.

######  Lemma 2 (Angle comparison).

Assume the sectional curvature $K$ in $M$ is bounded by ${\Lambda_{\ell}}\leq
K\leq{\Lambda_{u}}$, and let $T$ be a geodesic triangle contained in a convex
ball $B_{\rho}\subseteq M$, with radius $\rho\leq\rho_{0}$ (as defined in
(3)).

Then comparison triangles $T_{{\Lambda_{\ell}}}$ and $T_{{\Lambda_{u}}}$ exist
in $\mathbb{H}_{{\Lambda_{\ell}}}$ and $\mathbb{H}_{{\Lambda_{u}}}$. For any
angle $\alpha$ in $T$, the corresponding angles in $T_{{\Lambda_{\ell}}}$ and
$T_{{\Lambda_{u}}}$ satisfy

$\alpha_{{\Lambda_{\ell}}}\leq\alpha\leq\alpha_{{\Lambda_{u}}}.$

###  1.4 Signed measures

In this work we are only concerned with finite (i.e., bounded) Borel measures.
We will be considering signed measures $\mu$ on $M$. The Jordan decomposition
theorem [Hal74, §29], states that there are unique unsigned measures $\mu_{+}$
and $\mu_{-}$ such that $\mu=\mu_{+}-\mu_{-}$, and for our purposes we can
take this as a definition of a signed measure.

If $\nu$ is an unsigned measure on $M$, then the _support_ of $\nu$ is the
closed set

$\operatorname{supp}(\nu)=\\{p\in M\mid\nu(U)>0\text{ for all open }U\subseteq
M\text{ with }p\in U\\}.$

The support of a signed measure $\mu$ is defined by
$\operatorname{supp}(\mu)=\operatorname{supp}(\mu_{+})\cup\operatorname{supp}(\mu_{-})$.

##  2 Center of mass of signed measures

We consider a signed measure $\mu$ with support contained in a convex ball
$B_{\rho}\subset M$. In this section we establish constraints on $\rho$ in
terms of $\mu$ and bounds on the sectional curvatures that guarantee two
properties which together are sufficient to guarantee that $\mathcal{E}_{\mu}$
has a unique minimum in $B_{\rho}$. In Section 2.1 we show when
$\mathcal{E}_{\mu}$ is guaranteed to be strictly convex, and in Section 2.2 we
find conditions that ensure that the gradient of $\mathcal{E}_{\mu}$ is
pointing outwards on $\partial{B_{\rho}}$. This results in the main theorem of
this section:

######  Theorem 3 (Centre of mass of signed measures).

Let $M$ be a manifold whose sectional curvature $K$ is bounded by
${\Lambda_{\ell}}\leq K\leq{\Lambda_{u}}$, and let $\mu$ be a signed measure
on $M$, whose support is contained in a geodesic ball $\overline{B}_{M}(c,r)$.
Suppose $B_{\rho}=B_{M}(c,\rho)$, with $r<\rho<\rho_{0}$ as defined in (3).

Then $\mathcal{E}_{\mu}\colon\overline{B}_{\rho}\to\mathbb{R}$ (Equation (2))
has a unique minimum in $B_{\rho}$ if

$(\rho-r)\mu_{+}(M)-(\rho+r)\mu_{-}(M)>0,$

and

$\displaystyle\frac{\vartheta_{u}}{\tan\vartheta_{u}}\mu_{+}(M)-\mu_{-}(M)$
$\displaystyle>0$ $\displaystyle\text{if
}0\leq{\Lambda_{\ell}}\leq{\Lambda_{u}},$
$\displaystyle\frac{\vartheta_{u}}{\tan\vartheta_{u}}\mu_{+}(M)-\frac{\vartheta_{\ell}}{\tanh\vartheta_{\ell}}\mu_{-}(M)$
$\displaystyle>0$ $\displaystyle\text{if }{\Lambda_{\ell}}\leq
0\leq{\Lambda_{u}},$
$\displaystyle\mu_{+}(M)-\frac{\vartheta_{\ell}}{\tanh\vartheta_{\ell}}\mu_{-}(M)$
$\displaystyle>0$ $\displaystyle\text{if
}{\Lambda_{\ell}}\leq{\Lambda_{u}}\leq 0,$

where $\vartheta_{\ell}=2\rho\sqrt{\left|{\Lambda_{\ell}}\right|}$, and
$\vartheta_{u}=2\rho\sqrt{\left|{\Lambda_{u}}\right|}$.

###  2.1 Ensuring the convexity of $\mathcal{E}_{\mu}$

Our demonstration of the convexity of $\mathcal{E}_{\mu}$ relies on an
asymptotic expansion of the squared distance function in constant curvature
spaces. We consider points $x,y\in\mathbb{H}_{\kappa}$, and for any point $w$
close to $x$ we express the distance $d_{\mathbb{H}_{\kappa}}(w,y)$ in terms
of $d_{\mathbb{H}_{\kappa}}(x,y)$ and terms involving powers of
$\delta=d_{\mathbb{H}_{\kappa}}(x,w)$.

######  Lemma 4.

Suppose $B_{\rho}\subset\mathbb{H}_{\kappa}$ is a geodesic ball of radius
$\rho$, and $w,x,y\in B_{\rho}$ define a hinge with angle $\alpha$ at $x$, and
$d_{\mathbb{H}_{\kappa}}(x,w)=\delta$. If $\kappa>0$, assume
$\rho<\frac{\pi}{4\sqrt{\kappa}}$.

Then

$d_{\mathbb{H}_{\kappa}}(w,y)^{2}=\left|\exp_{x}^{-1}(w)-\exp_{x}^{-1}(y)\right|^{2}+(f_{\kappa}(\alpha,x,y)-1)\delta^{2}+O(\delta^{3}),$

where

$f_{\kappa}(\alpha,x,y)=\begin{cases}\cos^{2}\alpha+\frac{\vartheta}{\tan\vartheta}\sin^{2}\alpha&\text{if
}\kappa\geq 0,\\\
\cos^{2}\alpha+\frac{\vartheta}{\tanh\vartheta}\sin^{2}\alpha&\text{if
}\kappa\leq 0,\end{cases}$

and $\vartheta=d_{\mathbb{H}_{\kappa}}(x,y)\sqrt{\left|\kappa\right|}$.

###### Proof.

The result is obtained directly from a series expansion of the cosine rule for
a space of constant curvature. Letting $a=d_{\mathbb{H}_{\kappa}}(w,y)$, and
$c=d_{\mathbb{H}_{\kappa}}(x,y)=\left|\exp_{x}^{-1}(y)\right|$, the cosine
rules are (see e.g., [Ber87, §§18.6.8, 19.3.1]):

$\displaystyle\cos(a\sqrt{\kappa})$
$\displaystyle=\cos(c\sqrt{\kappa})\cos(\delta\sqrt{\kappa})+\sin(c\sqrt{\kappa})\sin(\delta\sqrt{\kappa})\cos\alpha$
$\displaystyle\text{if }\kappa>0,$ $\displaystyle a^{2}$
$\displaystyle=c^{2}+\delta^{2}-2c\delta\cos\alpha$ $\displaystyle\text{if
}\kappa=0,$ $\displaystyle\cosh(a\sqrt{-\kappa})$
$\displaystyle=\cosh(c\sqrt{-\kappa})\cosh(\delta\sqrt{-\kappa})-\sinh(c\sqrt{-\kappa})\sinh(\delta\sqrt{-\kappa})\cos\alpha$
$\displaystyle\text{if }\kappa<0.$

With the aid of a computer algebra system, we compute the series expansion
with respect to $\delta$ and find

$\left(\kappa^{-1/2}\arccos\left(\cos(c\sqrt{\kappa})\cos(\delta\sqrt{\kappa})+\sin(c\sqrt{\kappa})\sin(\delta\sqrt{\kappa})\cos\alpha\right)\right)^{2}\\\
=c^{2}-2c\delta\cos\alpha+\left(\cos^{2}\alpha+\frac{c\sqrt{\kappa}}{\tan(c\sqrt{\kappa})}\sin^{2}\alpha\right)\delta^{2}+O(\delta^{3}),$

when $\kappa>0$ and $c\sqrt{\kappa}$ and $\delta\sqrt{\kappa}$ are less than
$\frac{\pi}{2}$, and

$\left((-\kappa)^{-1/2}\operatorname{arccosh}\left(\cosh(c\sqrt{-\kappa})\cosh(\delta\sqrt{-\kappa})+\sinh(c\sqrt{-\kappa})\sinh(\delta\sqrt{-\kappa})\cos\alpha\right)\right)^{2}\\\
=c^{2}-2c\delta\cos\alpha+\left(\cos^{2}\alpha+\frac{c\sqrt{-\kappa}}{\tanh(c\sqrt{-\kappa})}\sin^{2}\alpha\right)\delta^{2}+O(\delta^{3}),$

when $\kappa<0$.

Since the Euclidean cosine rule in this notation yields

$\left|\exp_{x}^{-1}(w)-\exp_{x}^{-1}(y)\right|^{2}=c^{2}-2c\delta\cos\alpha+\delta^{2},$

the result follows.

Now let $M$ be an arbitrary Riemannian manifold with sectional curvature $K$
bounded by ${\Lambda_{\ell}}\leq K\leq{\Lambda_{u}}$, and let $B_{\rho}$ be a
geodesic ball of radius $\rho<\rho_{0}$, as defined in (3). We consider a
hinge defined by $w,x,y\in B_{\rho}$, and let
$\bar{w},\bar{x},\bar{y}\in\mathbb{H}_{{\Lambda_{\ell}}}$ and
$\tilde{w},\tilde{x},\tilde{y}\in\mathbb{H}_{{\Lambda_{u}}}$ define comparison
hinges. Then the hinge comparison theorem (Lemma 1) says

$d_{\mathbb{H}_{{\Lambda_{u}}}}(\tilde{w},\tilde{y})^{2}\leq
d_{M}(w,y)^{2}\leq d_{\mathbb{H}_{{\Lambda_{\ell}}}}(\bar{w},\bar{y})^{2}.$
(5)

Consider now that $w=w(t)$ is a point on a unit-speed geodesic with $w(0)=x$,
so that $d_{M}(x,w(t))=\left|\exp_{x}^{-1}(w(t))\right|=\pm t$, and associate
the angle $\alpha$ with the positive values of $t$, i.e., $\left\langle
w^{\prime}(0),\exp_{x}^{-1}(y)\right\rangle=\left|\exp_{x}^{-1}(y)\right|\cos\alpha$.
Employing Lemma 4 on the bounds (5), and expanding the squared norm using the
inner product, we find

$d_{M}(w(t),y)^{2}=\left|\exp_{x}^{-1}(y)\right|^{2}-2t\left|\exp_{x}^{-1}(y)\right|\cos\alpha+O(t^{2}).$

From this we conclude that

$\frac{d}{dt}\mathcal{E}_{\mu}(w(t))|_{t=0}=-\int\left\langle
w^{\prime}(0),\exp_{x}^{-1}(y)\right\rangle\,d\mu(y),$

and since the geodesic $w$ through $x$ was arbitrary, and

$\frac{d}{dt}\mathcal{E}_{\mu}(w(t))|_{t=0}=\left\langle
w^{\prime}(0),\operatorname{grad}\mathcal{E}_{\mu}(x)\right\rangle,$

by definition of the gradient, we see that

$\operatorname{grad}\mathcal{E}_{\mu}(x)=-\int\exp_{x}^{-1}(y)\,d\mu(y).$ (6)

We use the same technique to get a lower bound on the second derivative. _If
$\mu$ were an unsigned measure_, then the lower bound in (5), would yield

$\frac{d^{2}}{dt^{2}}\mathcal{E}_{\mu}(w(t))|_{t=0}\geq\int
f_{\mathbb{H}_{{\Lambda_{u}}}}(\alpha,x,y)\,d\mu(y).$ (7)

Our bound
$d_{\mathbb{H}_{{\Lambda_{u}}}}(\tilde{x},\tilde{y})=d_{M}(x,y)<2\rho_{0}\leq\frac{\pi}{2\sqrt{{\Lambda_{u}}}}$
ensures that $f_{\mathbb{H}_{{\Lambda_{u}}}}>0$ when $x$ is distinct from $y$.
It follows that if $\mu$ is an unsigned measure, then $\mathcal{E}_{\mu}$ is
strictly convex in $B_{\rho}$.

However, for the case of a signed measure that interests us here, more work is
required. Using the Jordan decomposition (Section 1.4), our lower bound now
becomes

$\frac{d^{2}}{dt^{2}}\mathcal{E}_{\mu}(w(t))|_{t=0}\geq\int
f_{\mathbb{H}_{{\Lambda_{u}}}}(\alpha,x,y)\,d\mu_{+}(y)-\int
f_{\mathbb{H}_{{\Lambda_{\ell}}}}(\alpha,x,y)\,d\mu_{-}(y),$ (8)

where we have used both of the inequalities in (5). We wish to ensure that
this bound is strictly positive.

Observe that for $\vartheta\in[0,\frac{\pi}{2})$, the function
$\vartheta/\tan\vartheta$ is monotonically decreasing, and
$\vartheta/\tanh\vartheta$ is monotonically increasing for $\vartheta\geq 0$.
Let $z=d_{M}(x,y)$. Writing
$\vartheta_{\kappa}(z)=z\sqrt{\left|\kappa\right|}$ for the variable that
appears in the definition of $f_{\kappa}$, we have

$\displaystyle\frac{\vartheta_{\kappa}(2\rho)}{\tan\vartheta_{\kappa}(2\rho)}$
$\displaystyle<\frac{\vartheta_{\kappa}(z)}{\tan\vartheta_{\kappa}(z)}\leq 1$
$\displaystyle\text{if }\kappa\geq 0,$ $\displaystyle 1$
$\displaystyle\leq\frac{\vartheta_{\kappa}(z)}{\tanh\vartheta_{\kappa}(z)}<\frac{\vartheta_{\kappa}(2\rho)}{\tanh\vartheta_{\kappa}(2\rho)}$
$\displaystyle\text{if }\kappa\leq 0.$

Finally, by defining
$\vartheta_{\ell}=2\rho\sqrt{\left|{\Lambda_{\ell}}\right|}$ and
$\vartheta_{u}=2\rho\sqrt{\left|{\Lambda_{u}}\right|}$, and expanding
$f_{\mathbb{H}_{{\Lambda_{u}}}}$ and $f_{\mathbb{H}_{{\Lambda_{\ell}}}}$ in
(8) using these inequalities, we obtain the convexity property of
$\mathcal{E}_{\mu}$ desired for Theorem 3:

######  Lemma 5.

Let $M$ be a Riemannian manifold with sectional curvature $K$ bounded by
${\Lambda_{\ell}}\leq K\leq{\Lambda_{u}}$. Let $B_{\rho}\subseteq M$ be a
geodesic ball of radius $\rho<\rho_{0}$, and $\mu$ a signed measure with
support in $B_{\rho}$.

The function $\mathcal{E}_{\mu}\colon\overline{B}_{\rho}\to\mathbb{R}$ of
Equation (2) is strictly convex provided

$\displaystyle\frac{\vartheta_{u}}{\tan\vartheta_{u}}\mu_{+}(M)-\mu_{-}(M)$
$\displaystyle>0$ $\displaystyle\text{if
}0\leq{\Lambda_{\ell}}\leq{\Lambda_{u}},$
$\displaystyle\frac{\vartheta_{u}}{\tan\vartheta_{u}}\mu_{+}(M)-\frac{\vartheta_{\ell}}{\tanh\vartheta_{\ell}}\mu_{-}(M)$
$\displaystyle>0$ $\displaystyle\text{if }{\Lambda_{\ell}}\leq
0\leq{\Lambda_{u}},$
$\displaystyle\mu_{+}(M)-\frac{\vartheta_{\ell}}{\tanh\vartheta_{\ell}}\mu_{-}(M)$
$\displaystyle>0$ $\displaystyle\text{if
}{\Lambda_{\ell}}\leq{\Lambda_{u}}\leq 0,$

where $\vartheta_{\ell}=2\rho\sqrt{\left|{\Lambda_{\ell}}\right|}$, and
$\vartheta_{u}=2\rho\sqrt{\left|{\Lambda_{u}}\right|}$.

######  Remark.

Equation (7) encompasses the bound on the second derivative announced by
Karcher [Kar77, (1.2.2) and (1.2.3)] in the following sense. If $\mu$ is an
unsigned measure with $\mu(M)=1$, then if ${\Lambda_{u}}<0$, (7) may be
expanded as

$\frac{d^{2}}{dt^{2}}\mathcal{E}_{\mu}(w(t))|_{t=0}\geq\int\left(\cos^{2}\alpha+\frac{\vartheta}{\tanh\vartheta}\sin^{2}\alpha\right)\,d\mu(y)\geq\cos^{2}\alpha+\frac{2\rho\sqrt{\Lambda}}{\tanh(2\rho\sqrt{\Lambda})}\sin^{2}\alpha\geq
1,$

and if ${\Lambda_{u}}>0$, then

$\frac{d^{2}}{dt^{2}}\mathcal{E}_{\mu}(w(t))|_{t=0}\geq\int\left(\cos^{2}\alpha+\frac{\vartheta}{\tan\vartheta}\sin^{2}\alpha\right)\,d\mu(y)\geq\cos^{2}\alpha+\frac{2\rho\sqrt{\Lambda}}{\tan(2\rho\sqrt{\Lambda})}\sin^{2}\alpha\geq\frac{2\rho\sqrt{\Lambda}}{\tan(2\rho\sqrt{\Lambda})}.$

The right hand side in each case corresponds to Karcher’s bound.

###  2.2 The gradient of $\mathcal{E}_{\mu}$ on the boundary of a ball

We have a signed measure $\mu$ with support in $\overline{B}_{M}(c,r)$, and as
usual the sectional curvatures on $M$ satisfy the upper bound ${\Lambda_{u}}$.
We consider a convex ball $B_{\rho}=B_{M}(c,\rho)\supset\overline{B}_{M}(c,r)$
and we wish to find conditions on $\mu$ and $\rho$ such that
$\mathcal{E}_{\mu}\colon\overline{B}_{\rho}\to\mathbb{R}$ has gradient
pointing outward on $\partial{B_{\rho}}$. To be specific, let $N$ be an
outward pointing unit normal vector field on $\partial{B_{\rho}}$. Then
$\operatorname{grad}\mathcal{E}_{\mu}$ is _pointing outward_ on
$\partial{B_{\rho}}$ if $\left\langle
N(x),\operatorname{grad}\mathcal{E}_{\mu}(x)\right\rangle>0$ for all
$x\in\partial{B_{\rho}}$.

Using the Jordan decomposition in (6), and defining

$X(x)=-\int\exp_{x}^{-1}(y)\,d\mu_{+}(y),\qquad
Z(x)=-\int\exp_{x}^{-1}(y)\,d\mu_{-}(y).$

we have $\operatorname{grad}\mathcal{E}_{\mu}=X-Z$, with $\left\langle
N,X\right\rangle>0$ and $\left\langle N,Z\right\rangle>0$.

It is convenient to introduce $R=\rho-r>0$. For the “inward pointing” part of
the gradient, we have $\left\langle N,Z\right\rangle\leq(2r+R)\mu_{-}(M)$,
because any given point in $\overline{B}_{M}(c,r)$ lies at most $2r+R$ from a
point on $\partial B_{\rho}$.

The “outward pointing” part of the gradient satisfies the bound $\left\langle
N,X\right\rangle\geq R\mu_{+}(M)$. This bound is not as easy to prove as the
bound on the inward pointing part of the gradient. We wish to show that if
$x\in\partial{B_{\rho}}$ and $y\in\partial{B_{M}(c,r)}$, then

$\left\langle-\exp_{x}^{-1}(y),N(x)\right\rangle=\left|\exp_{x}^{-1}(y)\right|\cos\varphi\geq
R,$ (9)

where $\varphi$ is the angle between $-N(x)$ and $\exp_{x}^{-1}(y)$. In other
words, we want to show that the image of $\overline{B}_{M}(c,r)$ under
$\exp_{x}^{-1}$ is contained in the half-space $\\{v\in T_{x}M\mid\left\langle
v,-N(x)\right\rangle\geq R\\}$.

Let $b=\left|\exp_{x}^{-1}(y)\right|$ be the length of the geodesic segment
between $x$ and $y$. We show that, if $b\leq R/\cos\varphi$, the only geodesic
triangle with edge lengths $r$, $r+R$, $b$, vertices $c$, $x$, $y$, and angle
$\varphi$ at $x$, as in Figure 1, is the trivial triangle with $\varphi=0$.

Observe that $\sin(as)\leq a\sin s$ for $a\geq 1$ and $0\leq as\leq\pi/2$, and
this inequality is strict if $s>0$ and $a>1$. This can be seen by
differentiating with respect to $s$ and observing that $\cos as\leq\cos s$
under the same conditions.

$b$$c$$x$$z$$\varphi$$r$$R$$r$$y$

Figure 1: Sketch for the argument that $b\cos\varphi\geq R=\rho-r$. This is a
schematic view in Riemann normal coordinates whose origin coincides with the
centre of the concentric geodesic balls $B_{M}(c,r)$ and $B_{M}(c,\rho)$.

Returning to the geodesic triangle, the angle comparison theorem (Lemma 2)
implies that $\cos\varphi$ is bounded from below by
$\cos\varphi_{\Lambda_{u}}$, where $\varphi_{\Lambda_{u}}$ denotes the angle
corresponding to $\varphi$ for the comparison triangle in the space of
curvature ${\Lambda_{u}}$. Without loss of generality we can assume that
${\Lambda_{u}}$ is positive (for our purposes here, nothing is gained by
requiring ${\Lambda_{u}}$ to be a _least_ upper bound on the sectional
curvatures).

We have $b\geq R$, by the triangle inequality. Using our comparison triangle,
and letting $\tilde{r}=r\sqrt{{\Lambda_{u}}}$,
$\tilde{R}=R\sqrt{{\Lambda_{u}}}$, and $\tilde{b}=b\sqrt{{\Lambda_{u}}}$, the
cosine rule for spaces of positive curvature now yields

$\displaystyle\cos\tilde{r}$
$\displaystyle=\cos(\tilde{r}+\tilde{R})\cos(\tilde{b})+\sin(\tilde{r}+\tilde{R})\sin(\tilde{b})\cos\varphi_{\Lambda_{u}}$
$\displaystyle\leq\cos(\tilde{r}+\tilde{R})\cos(\tilde{b})+\sin(\tilde{r}+\tilde{R})\sin(\tilde{R}/\cos\varphi_{\Lambda_{u}})\cos\varphi_{\Lambda_{u}}$
$\displaystyle\leq\cos(\tilde{r}+\tilde{R})\cos(\tilde{b})+\sin(\tilde{r}+\tilde{R})\sin(\tilde{R})$
$\displaystyle\leq\cos(\tilde{r}+\tilde{R})\cos(\tilde{R})+\sin(\tilde{r}+\tilde{R})\sin(\tilde{R})$
$\displaystyle=\cos\tilde{r},$

and since the inequalities are strict unless $\varphi_{\Lambda_{u}}=0$, we
must have $\varphi=0$. This confirms Equation (9), and we obtain our desired
result:

######  Lemma 6.

Let $\mu$ be a signed measure on $M$ with support in $\overline{B}_{M}(c,r)$,
which is contained in a concentric convex geodesic ball
$B_{\rho}=B_{M}(c,\rho)$. The gradient of $\mathcal{E}_{\mu}$ is pointing
outward on $\partial{B_{\rho}}$ if

$(\rho-r)\mu_{+}(M)-(\rho+r)\mu_{-}(M)>0.$

##  3 Barycentric coordinate neighbourhoods

As mentioned in Section 1.2, the centre of mass construction allows us to
define barycentric coordinates on Riemannian manifolds. The signed measure
$\mu$ is replaced with a discrete signed measure $\lambda$ that assigns
“weights” to the points in a set $\sigma$ of $n+1$ points contained in a
sufficiently small ball $B_{\rho}\subseteq M$. The associated Riemannian
simplex $\boldsymbol{\sigma}_{\\!\\!M}$ is the set of points with nonnegative
barycentric coordinates. Theorem 3 allows us to define a barycentric
coordinates that extend to an open neighbourhood containing
$\boldsymbol{\sigma}_{\\!\\!M}$.

In this section we quantify the extent of these barycentric coordinate
neighbourhoods. To facilitate this we first make some mild simplifying
assumptions to the setting of Theorem 3, arriving at Corollary 8, which
provides transparent upper and lower bounds on $\rho$ in terms of the total
negative mass, and allows for easy comparison with Karcher’s result [Kar77,
Theorem 1.2]. Then in Section 3.2 we use these simplified bounds to quantify
the extent of well-defined barycentric coordinate neighbourhoods. The size of
these neighbourhoods depends on a measure of the _quality_ of $\sigma$, as
discussed in Section 3.2.

Our main result, Theorem 9, quantifies the size of a ball in the parameter
domain, $B\subset\mathbb{R}^{n}$, on which the barycentric coordinate map
$b\colon B\to B_{\rho}\subseteq M$ is well defined. By earlier work [DVW15] we
are ensured that the barycentric coordinate map is not only well defined, but
is in fact an embedding on this domain. We also estimate the size of a
geodesic ball that is contained in the barycentric coordinate neighbourhood
$b(B)$ (Proposition 11). In Section 3.3 we exploit barycentric coordinate
neighbourhoods to show that Riemannian simplices behave like their Euclidean
counterparts: simplices with the same orientation that share a facet intersect
only in that facet.

###  3.1 Simplifying the setting

Our goal now is to simplify the constraints imposed in Theorem 3, albeit at
the expense of weakening the statement. We will assume $\mu(M)=1$; this
normalisation was not required for Theorem 3, although we did assume
$\mu_{-}(M),\mu_{+}(M)<\infty$. We will also replace ${\Lambda_{\ell}}$ and
${\Lambda_{u}}$ with a single bound $\Lambda$ on the absolute value of the
sectional curvatures. Before exploiting these assumptions, we first replace
the bounds of Lemma 5 with constraints linear in $\rho$.

######  Lemma 7.

We have

$\displaystyle 1-\frac{2\vartheta}{\pi}$
$\displaystyle\leq\frac{\vartheta}{\tan\vartheta}$ $\displaystyle\text{for
}\vartheta\in[0,\tfrac{\pi}{2}),$ $\displaystyle 1+\vartheta$
$\displaystyle\geq\frac{\vartheta}{\tanh\vartheta}$ $\displaystyle\text{for
}\vartheta\geq 0.$

###### Proof.

By taking the second derivatives, and observing that
$\frac{\vartheta}{\tan\vartheta}\geq 1$ and
$\frac{\vartheta}{\tanh\vartheta}\leq 1$ in the stated ranges, we find that
$\frac{\vartheta}{\tan\vartheta}$ is concave and
$\frac{\vartheta}{\tanh\vartheta}$ is convex in their ranges. The inequalities
now follow from Jensen’s inequality, i.e., by comparing the function to a
secant line.

Revisiting Lemma 5 with the assumption that ${\Lambda_{\ell}}=-\Lambda$ and
${\Lambda_{u}}=\Lambda>0$, the convexity of $\mathcal{E}_{\mu}$ is ensured if
the middle inequality is satisfied. Using Lemma 7, and then expanding
$\vartheta_{\ell}=\vartheta_{u}=2\rho\sqrt{\Lambda}$ we get that
$\mathcal{E}_{\mu}$ is convex if

$\left(1-\frac{4\rho\sqrt{\Lambda}}{\pi}\right)\mu_{+}(M)-(1+2\rho\sqrt{\Lambda})\mu_{-}(M)>0.$

Now using the assumption that $\mu(M)=\mu_{+}(M)-\mu_{-}(M)=1$, we obtain a
convenient simplified version of Theorem 3.

######  Corollary 8 (Centre of mass of signed measures).

Suppose the sectional curvature $K$ of $M$ is bounded by
$\left|K\right|\leq\Lambda$, and $\mu$ is a signed measure on $M$ with
$\mu(M)=1$ and support contained in a geodesic ball $\overline{B}_{M}(c,r)$.
If $B_{\rho}=B_{M}(c,\rho)$ with $r<\rho<\rho_{0}$ as defined in (3), then
$\mathcal{E}_{\mu}\colon\overline{B}_{\rho}\to\mathbb{R}$ (Equation (2)) has a
unique minimum in $B_{\rho}$ if

$\displaystyle\rho$ $\displaystyle>(1+2\mu_{-}(M))r,$ (10) and
$\displaystyle\rho$
$\displaystyle<\frac{\pi}{4\sqrt{\Lambda}}\left(1+C\mu_{-}(M)\right)^{-1},$
(11)

where $C=\left(1+\frac{\pi}{2}\right)$.

If $\mu$ is an unsigned measure, the conditions of Corollary 8 reduce to the
same conditions required in Karcher’s theorem [Kar77, Thm. 1.2].

###  3.2 The extent of barycentric coordinate neighbourhoods

We consider a set $\sigma$ of $n+1$ points contained in a geodesic ball
$B_{\rho}\subseteq M$ of radius $\rho<\rho_{0}$ ($\rho$ will be further
constrained below). We will require that there exists a Euclidean $n$-simplex
$\tilde{\sigma}\subset\mathbb{E}^{n}$ whose vertices are in correspondence
with the points of $\sigma$ and whose edge lengths are equal to the geodesic
distances between the corresponding points in $\sigma$. This in itself is a
constraint on $\sigma$ (see [DVW15, §5.1]).

Let $(\lambda_{i})$ be the barycentric coordinate functions associated with
$\tilde{\sigma}$. We define $\lambda_{+}\colon\mathbb{E}^{n}\to\mathbb{R}$ to
be the function that gives the sum of the positive barycentric coordinates,
and $\lambda_{-}$ to be the sum of the negative coordinates, so
$\lambda_{+}+\lambda_{-}=1$ is constant. Given any fixed $u\in\mathbb{E}^{n}$,
the barycentric coordinates $(\lambda_{i}(u))$ define a discrete measure on
$M$ (a weight for each point of $\sigma$) via the correspondence between
$\sigma$ and the vertices of $\tilde{\sigma}$. In this context,
$\left|\lambda_{-}(u)\right|$ plays the role of $\mu_{-}(M)$ in Corollary 8.

We would like to find a domain $B\subset\mathbb{E}^{n}$, with
$\tilde{\sigma}\subset B$, such that the constraints of Corollary 8 are
satisfied for any $u\in B$. So we are interested in finding a bound on
$\left|\lambda_{-}\right|$.

We choose an arbitrary vertex $\tilde{v}\in\tilde{\sigma}$, and we will find a
radius $\tilde{r}$ such that for all points $u$ in
$B=B_{\mathbb{E}^{n}}(\tilde{v},\tilde{r})$, this goal is attained.

Let $a$ be a lower bound on the altitudes of $\tilde{\sigma}$. The magnitude
of the gradient of each barycentric coordinate function $\lambda_{i}$ is
bounded by $a^{-1}$. Therefore $-\lambda_{i}<\tilde{r}/a$ on $B$. Since at
most $n$ coordinate functions can be negative at any point in $B$, we find

$\left|\lambda_{-}\right|<\frac{n\tilde{r}}{a}=\frac{\tilde{r}}{Lt},$ (12)

where $L$ is a strict upper bound on the edge lengths, and $t=a/nL$ is the
_thickness_ of $\tilde{\sigma}$.

We choose $r=L$ as the radius of a ball containing $\sigma\subset M$.
Observing that $\left|\lambda_{-}\right|=\lambda_{+}-1$ and exploiting (12) we
find that (10) is satisfied if

$\rho>L+\frac{2\tilde{r}}{t}.$ (13)

Similarly, (11) may be replaced with

$\rho<\frac{\pi}{4\sqrt{\Lambda}}\left(1+C\frac{\tilde{r}}{Lt}\right)^{-1},$
(14)

where $C=1+\pi/2$.

In order for there to exist a $\rho$ satisfying both of these constraints, we
require

$\Big{(}L+\frac{2\tilde{r}}{t}\Big{)}\Big{(}1+C\frac{\tilde{r}}{Lt}\Big{)}<\frac{\pi}{4\sqrt{\Lambda}},$

or

$\frac{2C}{Lt^{2}}\tilde{r}^{2}+\frac{2+C}{t}\tilde{r}+\Big{(}L-\frac{\pi}{4\sqrt{\Lambda}}\Big{)}<0.$

We require $\tilde{r}$ to be strictly smaller than the largest root:

$\tilde{r}<\frac{Lt(2+C)}{4C}\bigg{(}{-}1+\bigg{(}1+\frac{8C}{(2+C)L}\Big{(}\frac{\pi}{4\sqrt{\Lambda}}-L\Big{)}\bigg{)}^{1/2}\bigg{)}.$
(15)

Since $\frac{4}{9}<\frac{2+C}{4C}<\frac{1}{2}$, Equation (15) is satisfied
when

$\tilde{r}\leq\frac{Lt}{3}\bigg{(}{-}1+\sqrt{-3+\frac{\pi}{4L\sqrt{\Lambda}}}\bigg{)}.$
(16)

Observe that this bound is positive provided
$L\leq\frac{1}{6\sqrt{\Lambda}}<\frac{\pi}{16\sqrt{\Lambda}}$. Also, the bound
(16) is maximised when
$L=\frac{\pi}{48\sqrt{\Lambda}}\sim\frac{1}{15\sqrt{\Lambda}}$.

We want to ensure that the entire simplex $\tilde{\sigma}$ is contained in the
domain of the barycentric coordinates. Therefore, using a scale parameter
$s\geq 1$, we set $\tilde{r}=sL$ in (16) and find that $L$ must satisfy

$L\leq\frac{\pi}{4\sqrt{\Lambda}}\Big{(}\frac{9s^{2}}{t^{2}}+\frac{6s}{t}+4\Big{)}^{-1}.$

Since $t<1$, and $s\geq 1$ this is satisfied when

$L\leq\frac{t^{2}}{25s^{2}\sqrt{\Lambda}}.$ (17)

This is the criterion bounding $\tilde{r}$ that we sought.

Thus when (17) is satisfied, the barycentric coordinate map $b\colon
B_{\mathbb{E}^{n}}(\tilde{v},sL)\to B_{\rho}$ is well defined, provided $\rho$
satisfies (13) and (14), with $\tilde{r}=sL$; the existence of such a $\rho$
is assured. But we can say more. It has been shown [DVW15, Prop. 29] that the
differential of the barycentric coordinate map is nondegenerate provided
$L\leq t/(3\sqrt{\Lambda})$, which is obviously satisfied when (17) is.
Although the argument was made in the context of positive weights, this fact
was not used. What is required is that the energy functional (4) that defines
the barycentric coordinate map be convex, but we still have that in our
current context. Thus we obtain our main result:

######  Theorem 9 (Barycentric coordinate neighbourhoods).

Let $M$ be an $n$-dimensional Riemannian manifold with sectional curvature $K$
bounded by $\left|K\right|\leq\Lambda$, and let $\sigma\subset M$ be a set of
$n+1$ points such that $d_{M}(p,q)<L$ for any $p,q\in\sigma$.

Given a scale parameter $s\geq 1$, if $\sigma$ defines a Euclidean simplex
$\tilde{\sigma}\subset\mathbb{E}^{n}$ with the same edge lengths, and with
thickness $t$ satisfying

$t^{2}\geq 25s^{2}L\sqrt{\Lambda},$

then, the barycentric coordinate map $b\colon
B_{\mathbb{E}^{n}}(\tilde{v},sL)\to B_{\rho}$, where $\tilde{v}$ is any vertex
of $\tilde{\sigma}$, is well defined, and is in fact an embedding. The ball
$B_{\rho}$ is centred on the vertex of $\sigma$ corresponding to $\tilde{v}$,
and its radius $\rho$ must lie in the nonempty interval given by

$\left(1+\frac{2s}{t}\right)L<\rho<\frac{\pi}{4\sqrt{\Lambda}}\left(1+C\frac{s}{t}\right)^{-1}\\!,$

where $C=1+\pi/2$.

Although the image of $B_{\mathbb{E}^{n}}(\tilde{v},sL)$ under the barycentric
coordinate map $b$ is guaranteed to be contained in $B_{\rho}$ for any
sufficiently large $\rho$, it may be also useful to know the size of a
geodesic ball that is contained in $b(B_{\mathbb{E}^{n}}(\tilde{v},sL))$. The
metric distortion of the barycentric coordinate map has already been
calculated [DVW15, §5.2], and although this argument was made in the context
of non-negative weights, this fact was not exploited, and the argument carries
verbatim to the case of interest here. The argument does require that $\rho$
be bounded by $\rho\leq\frac{t}{6\sqrt{\Lambda}}$ but this is satisfied if the
conditions of Theorem 9 are satisfied.

######  Lemma 10 ([DVW15, §5.2]).

Given the assumptions of Theorem 9, the barycentric coordinate map $b\colon
B_{\mathbb{E}^{n}}(\tilde{v},sL)\to B_{\rho}$ satisfies

$\left|d_{M}(b(u),b(w))-\left|u-w\right|\right|\leq\frac{50\Lambda\rho^{2}}{t^{2}}\left|u-w\right|,$

for any $u,w\in B_{\mathbb{E}^{n}}(\tilde{v},sL)$.

There is no reason to choose a $\rho$ larger than necessary; with the
assumptions of Theorem 9, we can choose
$\rho^{2}=\left(1+\frac{2s}{t}\right)^{2}L^{2}+\varepsilon^{2}\leq\frac{3^{2}t^{2}}{25^{2}s^{2}\Lambda}$,
where $\varepsilon$ is an arbitrarily small distance. Employing this bound in
Lemma 10, and choosing a point
$w\in\partial{B_{\mathbb{E}^{n}}(\tilde{v},sL)}$, we find

$d_{M}(b(\tilde{v}),b(w))\geq\left(1-\frac{18}{25s^{2}}\right)sL.$

We desire that the image of $\tilde{\sigma}$ be contained in this ball.
Demanding $\left(s-\frac{18}{25s}\right)L\geq L$, we find this will be
satisfied provided $s\geq\frac{3}{2}$, and we have the following addendum to
Theorem 9:

######  Proposition 11.

Assume that the conditions of Theorem 9 are satisfied, and that $v\in\sigma$
is the vertex corresponding to $\tilde{v}\in\tilde{\sigma}$. Then the image of
$B_{\mathbb{E}^{n}}(\tilde{v},sL)$ under the barycentric coordinate map
contains the geodesic ball $B_{M}(v,\bar{r})$, where

$\bar{r}=\left(s-\frac{18}{25s}\right)L.$

If $s\geq\frac{3}{2}$, then $\bar{r}\geq L$, and so $B_{M}(v,\bar{r})$
contains $\tilde{\sigma}$.

###  3.3 Riemannian simplices that share a facet

Suppose $\boldsymbol{\sigma}_{\\!\\!M}\subset M$ is a Riemannian $n$-simplex
defined by $\sigma=\\{p,p_{0},\ldots,p_{n-1}\\}$. The _facet_ of
$\boldsymbol{\sigma}_{\\!\\!M}$ opposite $p$ is the Riemannian $(n-1)$-simplex
$\boldsymbol{\eta}_{M}$ defined by $\eta=\\{p_{0},\ldots,p_{n-1}\\}$, i.e.,
the set of points in $\boldsymbol{\sigma}_{\\!\\!M}$ where the barycentric
coordinate associated with $p$ is $0$. Suppose now that
$\boldsymbol{\tau}_{\\!M}\subset M$ is another Riemannian simplex defined by
$\tau=\\{p_{0},q,p_{1},\ldots,p_{n-1}\\}$, so that $\boldsymbol{\tau}_{\\!M}$
shares the facet $\boldsymbol{\eta}_{M}$ with $\boldsymbol{\sigma}_{\\!\\!M}$.

Suppose that both $\boldsymbol{\sigma}_{\\!\\!M}$ and
$\boldsymbol{\tau}_{\\!M}$ satisfy the conditions of Theorem 9, with scale
factor $s=3/2$, and with $L$ being a common strict upper bound on their edge
lengths. Then they are nondegenerate, and we can define their orientation in
$B_{\rho}$ (see [DVW15, §3.3]). If they have the same orientation, then they
can be represented by Euclidean simplices
$\tilde{\sigma}=\\{\tilde{p},\tilde{p}_{0},\ldots,\tilde{p}_{n-1}\\}$ and
$\tilde{\tau}=\\{\tilde{p}_{0},\tilde{q},\tilde{p}_{1},\ldots,\tilde{p}_{n-1}\\}$
that share a common facet, $\tilde{\eta}=\tilde{\sigma}\cap\tilde{\tau}$, such
that $\tilde{p}$ and $\tilde{q}$ lie on opposite sides of the hyperplane
$\operatorname{aff}(\tilde{\eta})$. Thus
$\boldsymbol{\tilde{\tau}}_{\mathbb{E}^{n}}=\operatorname{conv}(\tilde{\tau})$
lies in the region where the barycentric coordinate associated with
$p\in\tilde{\sigma}$ is nonpositive, and similarly,
$\boldsymbol{\tilde{\sigma}}_{\mathbb{E}^{n}}=\operatorname{conv}(\tilde{\sigma})$
lies in the region where the barycentric coordinate associated with
$\tilde{q}\in\tilde{\tau}$ is nonpositive, and
$\operatorname{aff}(\tilde{\eta})$ is the common 0 level set of both of these
barycentric coordniate functions.

The barycentric coordinate map for $\sigma$ is an embedding of
$B_{\mathbb{E}^{n}}(\tilde{p}_{0},sL)$ that maps
$\boldsymbol{\tilde{\sigma}}_{\mathbb{E}^{n}}$ to
$\boldsymbol{\sigma}_{\\!\\!M}$, preserving the associated barycentric
coordinates, and similarly the barycentric coordinate map for $\tau$ maps
$\boldsymbol{\tilde{\tau}}_{\mathbb{E}^{n}}$ to $\boldsymbol{\tau}_{\\!M}$.
These maps agree on $\operatorname{aff}(\tilde{\eta})$, and by our choice of
scale factor $s$, Proposition 11 ensures that $B_{M}(p_{0},L)$ is contained in
the image of both maps. Since the $(n-1)$-submanifold that is the image of
$\operatorname{aff}(\tilde{\eta})$ separates this ball into two components
according to the sign of the barycentric coordinate of $p$ (and likewise of
$q$), it follows that
$\boldsymbol{\sigma}_{\\!\\!M}\cap\boldsymbol{\tau}_{\\!M}=\boldsymbol{\eta}_{M}$.

#### Acknowledgements

Theorem 3 appeared in Wintraecken’s PhD thesis [Win15, Theorem 3.4.9]. The
current exposition benefited from the comments and suggestions of the jury. In
particular we thank John Sullivan for valuable feedback on this work. This
research has been partially supported by the 7th Framework Programme for
Research of the European Commission, under FET-Open grant number 255827 (CGL
Computational Geometry Learning). Partial support has been provided by the
Advanced Grant of the European Research Council GUDHI (Geometric Understanding
in Higher Dimensions).

## References

  * [Ber87] M. Berger.  Geometry II.  Universitext. Springer-Verlag, 1987.
  * [BK81] P. Buser and H. Karcher.  Gromov’s almost flat manifolds.  Number 81 in Astérisque. Société mathématique de France, 1981.
  * [Cha06] I. Chavel.  Riemannian Geometry: A Modern Introduction.  Number 98 in Cambridge studies in advanced mathematics. Cambridge University Press, 2006.
  * [DVW15] R. Dyer, G. Vegter, and M. Wintraecken.  Riemannian simplices and triangulations.  Geometriae Dedicata, 179(1):91–138, 2015.
  * [Gro04] D. Groisser.  Newton’s method, zeroes of vector fields, and the Riemannian center of mass.  Adv. in Appl. Math., 33(1):95–135, 2004.
  * [Hal74] Paul R Halmos.  Measure theory.  Number 18 in GTM. Springer, 1974.
  * [Kar77] H. Karcher.  Riemannian center of mass and mollifier smoothing.  Communications on Pure and Applied Mathematics, 30:509–541, 1977\.
  * [Kar89] H. Karcher.  Riemannian comparison constructions.  In S.S. Chern, editor, Global Differential Geometry, pages 170–222. The mathematical association of America, 1989.
  * [Kar14] H. Karcher.  Riemannian Centre of Mass and so called karcher mean.  Historical Note 1407.2087, arXiv, 2014.
  * [Ken90] W.S. Kendall.  Probability, convexity, and harmonic maps with small image I: Uniqueness and fine existence.  Procedings of the London Mathematical society, s3-61 (Issue 2):371–406, 1990.
  * [San16] O. Sander.  Geodesic finite elements of higher order.  IMA Journal of Numerical Analysis, 36(1):238–266, 2016.
  * [Win15] M.H.M.J. Wintraecken.  Ambient and intrinsic triangulations and topological methods in cosmology.  PhD thesis, Rijksuniversiteit Groningen, 2015.  https://www.rug.nl/research/portal/files/23072604/Complete_thesis.pdf.


#  Coordinated Beamforming in Quantized Massive MIMO Systems with Per-Antenna
Constraints

Yunseong Cho, Graduate Student Member, IEEE, Jinseok Choi, Member, IEEE,
and Brian L. Evans, Fellow, IEEE Y. Cho and B. L. Evans are with the Wireless
Networking and Communication Group (WNCG), Dept. of Electrical and Computer
Engineering, The University of Texas at Austin, Austin, TX 78712 USA. (e-mail:
<EMAIL_ADDRESS>bevans@ece.utexas.edu). J. Choi is with Dept. of Electrical
and Computer Engineering, Ulsan National Institute of Science and Technology
(UNIST), Ulsan, South Korea (e-mail:jinseokchoi@unist.ac.kr).

###### Abstract

In this work, we present a solution for coordinated beamforming for large-
scale downlink (DL) communication systems with low-resolution data converters
when employing a per-antenna power constraint that limits the maximum antenna
power to alleviate hardware cost. To this end, we formulate and solve the
antenna power minimax problem for the coarsely quantized DL system with target
signal-to-interference-plus-noise ratio requirements. We show that the
associated Lagrangian dual with uncertain noise covariance matrices achieves
zero duality gap and that the dual solution can be used to obtain the primal
DL solution. Using strong duality, we propose an iterative algorithm to
determine the optimal dual solution, which is used to compute the optimal DL
beamformer. We further update the noise covariance matrices using the optimal
DL solution with an associated subgradient and perform projection onto the
feasible domain. Through simulation, we evaluate the proposed method in
maximum antenna power consumption and peak-to-average power ratio which are
directly related to hardware efficiency.

###### Index Terms:

Transmit power minimax problem, beamforming, low-resolution quantizers, per-
antenna power constraint.

##  I Introduction

Massive multiple-input-multiple-output (MIMO) systems have drawn attention for
fifth-generation wireless communication systems [1]. However, a large number
of analog-to-digital converters (ADCs) and digital-to-analog converters (DACs)
connected to the antennas require prohibitively high power consumption.
Accordingly, employing low-resolution quantizers has gathered momentum as a
power-efficient solution [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The design of
multicell systems should take into account intra-cell and inter-cell
interferers as well as the quantization error. Moreover, for a realistic
deployment, a per-antenna power constraint that limits the power of each
antenna is desirable by allowing the system to operate with more energy-
efficient power amplifiers [12, 13].

In recent years, low-resolution converters have been introduced in many
communication systems and their corresponding algorithms [2, 3, 4, 5, 8, 7, 9,
10, 11]. The authors in [4] employed artificial noise to precisely learn the
likelihood probabilities for maximum likelihood detection under one-bit ADCs.
In [5], computation of soft metrics with one-bit observations and its
application to channel coding were presented. A hybrid MIMO architecture with
resolution-adaptive ADCs for millimeter wave communication was developed in
[6]. Linear estimation and model-based deep neural networks approach were
combined in [9] for data detection with one-bit ADCs. Considering different
traffic load requirements of IoT devices, the authors in [10] developed a
grant-free access scheme for massive MIMO systems with mixed-ADC access points
(APs).

Figure 1: Multicell multiuser-MIMO configuration with low-resolution ADCs and
DACs at the base station (BS).

Coordinated multipoint (CoMP) has been explored to increase throughput,
coverage and reliability [14, 15, 13, 8]. In [14], CoMP beamforming (BF) and
power control (PC) in uplink (UL) were presented. Considering downlink (DL) as
a virtual UL, UL-DL CoMP BF and PC were further proposed in [15] in a
distributed manner using local measurements. The authors in [13] showed
Lagrangian-based duality for multiuser MIMO systems and proposed a distributed
method to lower computational load on users and BSs. CoMP BF and PC were also
studied for coarsely quanitized massive MIMO systems, and their closed-form
solution and extension to a wideband scenario were investigated [8]. The
synergy between massive MIMO and CoMP is studied with higher throughput [16].

In this paper, we design CoMP-based BF with the per-antenna power constraint
in coarsely quantized large-scale MIMO systems. We first formulate the DL
antenna power minimax problem with individual signal-to-interference-plus-
noise-ratio (SINR) constraints. We then derive the Lagrangian dual of the DL
problem which can be considered as a virtual UL problem with uncertain noise
covariance matrices. By transforming the DL problem to a strictly feasible
second-order cone program (SOCP), we derive strong duality between the DL
problem and its dual. Leveraging strong duality, we propose an iterative
algorithm to solve the dual in a distributed manner for fixed noise covariance
matrices. The solutions of the dual are used to obtain an optimal DL BF. The
optimal DL solutions update the noise covariance matrices which are then
projected onto the feasible constraint set.

Notation: $\bf{A}$ is a matrix and $\bf{a}$ is a column vector.
$\mathbf{A}^{H}$ and $\mathbf{A}^{T}$ denote conjugate transpose and
transpose. $[{\bf A}]_{i,:}$ and $\mathbf{a}_{i}$ indicate the $i$th row and
column vectors of $\bf A$. We denote $a_{i,j}$ as the $\\{i,j\\}$th element of
$\bf A$ and $a_{i}$ as the $i$th element of $\bf a$.
$\mathcal{CN}(\mu,\sigma^{2})$ is a complex Gaussian distribution with mean
$\mu$ and variance $\sigma^{2}$. The diagonal matrix $\rm diag(\bf A)$ has
$\\{a_{i,i}\\}$ as its diagonal entries, and $\rm diag(\bf a)$ or $\rm
diag({\bf a}^{T})$ has $\\{a_{i}\\}$ as its diagonal entries. A block diagonal
matrix is presented as ${\rm blkdiag}({\bf A}_{1},\dots,{\bf A}_{N})$. ${\bf
I}_{N}$ is a $N\times N$ identity matrix and ${\bf 1}_{N}$ is a $N\times 1$
ones vector. $\otimes$ represents Kronecker product.

##  II System Model

We consider a multicell multiuser-MIMO network with $N_{c}$ cells, $N_{u}$
single-antenna users per cell. BSi represents the BS with $N_{b}$ antennas in
cell $i$ that serves $N_{u}$ dedicated users. We assume that the BSs
distributed over $N_{c}$ cells cooperate and employ low-resolution DACs.

###  II-A DL System Models with Low-Resolution DACs

Let ${\bf s}_{i}\\!\in\\!\mbox{$\mathbb{C}$}^{N_{u}}$ denote the symbols for
$N_{u}$ users in cell $i$ and ${\bf W}_{i}\\!=\\!\left[{\bf
w}_{i,1},\ldots,{\bf
w}_{i,N_{c}}\right]\\!\in\\!\mbox{$\mathbb{C}$}^{N_{b}\times N_{u}}$ collects
the associated precoders. We define the precoded signals as ${\bf x}_{i}={\bf
W}_{i}{\bf s}_{i}$. Upon generating ${\bf x}_{i}$, the signal is treated by
the low-resolution DACs with $b$ quantization bits. For analytic tractability,
we adopt the AQNM [8, 17] that delivers a linear approximation of the
quantization process derived from assuming a scalar minimum-mean-squared-error
(MMSE) quantizer. Under the AQNM, the quantized signal is modeled as

$\displaystyle{\bf x}_{{\rm q},i}=\alpha{\bf x}_{i}+{\bf q}_{i}$ (1)

where ${\bf q}_{i}\sim\mbox{$\mathcal{C}$}\mbox{$\mathcal{N}$}({\bf 0},{\bf
C}_{{\bf q}_{i}{\bf q}_{i}})$ is the quantization noise vector at BSi with its
covariance matrix of [8]

$\displaystyle{\bf C}_{{\bf q}_{i}{\bf q}_{i}}=\alpha\beta{\rm
diag}\big{(}{\bf W}_{i}{\bf W}_{i}^{H}\big{)}.$ (2)

The quantization gain $\alpha$ is defined as $\alpha\\!=\\!1-\beta$ where
$\beta$’s are listed in Table 1 in [18] for $b\leq 5$ assuming ${\bf
s}_{i}\\!\sim\\!\mathcal{CN}({\bf 0}_{N_{u}},{\bf I}_{N_{u}}),\forall
i\in\\{1,\ldots,N_{c}\\}$.

Due to the broadcast channel, users in cell $i$ receive signals from all BSs.
By defining ${\bf H}_{i,j}$ as the channel between BSi and users in cell $j$,
the received signal at user $u$ in cell $i$ is

$\displaystyle y_{i,u}\\!=\\!\ $ $\displaystyle\alpha{\bf
h}_{i,i,u}^{H}\\!{\bf w}_{i,u}{s}_{i,u}$
$\displaystyle+\alpha\\!\\!\\!\\!\\!\\!\sum_{(j,v)\neq(i,u)}^{(N_{c},N_{u})}\\!\\!\\!\\!{\bf
h}_{j,i,v}^{H}\\!{\bf w}_{j,v}{s}_{j,v}\\!+\\!\sum_{j=1}^{N_{c}}{\bf
h}_{j,i,u}^{H}{\bf q}_{j}\\!+\\!n_{i,u}$ (3)

where the DL channel between BSj and user $u$ in cell $i$ is ${\bf
h}^{H}_{j,i,u}$, and $n_{i,u}\sim\mathcal{CN}(0,1)$. Based on (2) and (3) the
DL SINR for user $u$ in cell $i$ is expressed as

$\displaystyle\Gamma_{i,u}=\frac{\alpha^{2}|{\bf h}_{i,i,u}^{H}{\bf
w}_{i,u}|^{2}}{\alpha^{2}\sum_{(j,v)\neq(i,u)}^{N_{c},N_{u}}\\!|{\bf
h}_{j,i,u}^{H}{\bf w}_{j,v}|^{2}+{\rm Q}_{i,u}+\\!\sigma^{2}}$ (4)

with the quantization noise terms of

$\displaystyle{\rm Q}_{i,u}=\sum_{j=1}^{N_{c}}{\bf h}_{j,i,u}^{H}{\bf C}_{{\bf
q}_{j},{\bf q}_{j}}{\bf h}_{j,i,u}.$ (5)

Finally, the DL transmit power minimization problem with per-antenna power
constraints is formulated as

$\displaystyle\mathop{{\text{minimize}}}_{{\bf w}_{i,u},\ p_{\rm 0}}$
$\displaystyle\;\;p_{\rm 0}$ (6) subject to
$\displaystyle\;\;\Gamma_{i,u}\geq\gamma_{i,u}$ (7) $\displaystyle\
\Big{[}\mbox{$\mathbb{E}$}[{\bf x}_{{\rm q},i}{\bf x}_{{\rm
q},i}^{H}]\Big{]}_{m,m}\leq p_{\rm 0}$ (8) $\displaystyle\;\;\forall i,u,m.$

Since directly solving the above problem is difficult, we first derive a
Lagrangian dual and solve the problem in the dual space with a more efficient
solver.

###  II-B Lagrangian Dual Problem

Let us collect Lagrangian multipliers as ${\boldsymbol{\Lambda}}={\rm
blkdiag}\big{(}{\boldsymbol{\Lambda}}_{1},\dots,{\boldsymbol{\Lambda}}_{N_{c}}\big{)}$
where ${\boldsymbol{\Lambda}}_{i}={\rm
diag}\big{(}\lambda_{i,1},\dots,\lambda_{i,N_{u}}\big{)}$. In Theorem 1, a
virtual UL problem is derived as a dual problem of (6).

######  Theorem 1 (Duality).

The Lagrangian dual problem of the DL problem in (6) is equivalent to

$\displaystyle\max_{{\bf D}_{i}}\min_{\lambda_{i,u}}$
$\displaystyle\;\;\sum_{i,u}^{N_{c},N_{u}}\lambda_{i,u}\sigma^{2}$ (9) subject
to $\displaystyle\;\;\max_{{\bf f}_{i,u}}\hat{\Gamma}_{i,u}\geq\gamma_{i,u},$
(10) $\displaystyle\;\;{\bf D}_{i}\succeq 0,\ {\bf
D}_{i}\in\mbox{$\mathbb{R}$}^{N_{b}\times N_{b}}:\text{\rm diagonal},$ (11)
$\displaystyle\;\;{\rm tr}({\bf D}_{i})\leq N_{b}\quad\forall i,u$ (12)

with UL SINR of

$\displaystyle\hat{\Gamma}_{i,u}\\!=\\!\frac{\alpha^{2}\lambda_{i,u}|{\bf
f}_{i,u}^{H}{\bf h}_{i,i,u}|^{2}}{{\bf f}_{i,u}^{H}{\bf Z}_{i,u}{\bf
f}_{i,u}}$ (13)

where

$\displaystyle{\bf Z}_{i,u}=$
$\displaystyle\alpha^{2}\\!\\!\\!\\!\\!\\!\\!\sum_{(j,v)\neq(i,u)}\\!\\!\\!\\!\\!\\!\\!\lambda_{j,v}{\bf
h}_{i,j,v}{\bf h}_{i,j,v}^{H}$ $\displaystyle+\alpha^{2}{{\bf
D}_{i}}+\alpha\beta{\rm diag}({\bf H}_{i}{\boldsymbol{\Lambda}}{\bf
H}_{i}^{H}\\!+\\!{\bf D}_{i}).$ (14)

###### Proof.

Considering ${\bf f}_{i,u}$ as a combiner for user $u$ in cell $i$, we let
${\bf f}_{i,u}$ be the MMSE equalizer as

$\displaystyle{\bf f}_{i,u}=$ $\displaystyle{\bf Z}_{i,u}^{-1}{\bf
h}_{i,i,u}.$ (15)

With the MMSE combiner that maximizes $\hat{\Gamma}_{i,u}$, (10) is simplified
as

$\displaystyle\max_{{{\bf D}}_{i}}\min_{\lambda_{i,u}}$
$\displaystyle\;\;\sum_{i,u}^{N_{c},N_{u}}\lambda_{i,u}\sigma^{2}$ (16)
subject to $\displaystyle\;\;{\bf
K}_{i}({\boldsymbol{\Lambda}})\preceq\alpha\left(1+\frac{1}{\gamma_{i,u}}\right)\lambda_{i,u}{\bf
h}_{i,i,u}{\bf h}_{i,i,u}^{H},$ (17) $\displaystyle\;\;{\bf D}_{i}\succeq 0,\
{\bf D}_{i}\in\mbox{$\mathbb{R}$}^{N_{b}\times N_{b}}:\text{\rm diagonal},$
$\displaystyle\;\;{\rm tr}({\bf D}_{i})\leq N_{b}\quad\forall i,u$

where

$\displaystyle{\bf K}_{i}({\boldsymbol{\Lambda}})\\!=\\!{\bf
D}_{i}\\!+\\!\alpha\\!\sum_{j,v}\\!\lambda_{j,v}{\bf h}_{i,j,v}{\bf
h}_{i,j,v}^{H}\\!+\\!\beta{\rm diag}\Big{(}{\bf
H}_{i}{\boldsymbol{\Lambda}}{\bf H}_{i}^{H}\Big{)},$ (18)

which is the covariance matrix of the received signal at BSi.

Now, we show that (16) is equivalent to the Lagrangian dual of (6). The per-
antenna constraint in (8) is rewritten as

$\displaystyle\Big{[}\mbox{$\mathbb{E}$}[{\bf x}_{{\rm q},i}{\bf x}_{{\rm
q},i}^{H}]\Big{]}_{m,m}=\left[\alpha{\bf W}_{i}{\bf W}_{i}^{H}\right]_{m,m}.$
(19)

We replace the objective function in (6) with $N_{c}N_{b}p_{\rm 0}$ that does
not affect the problem. With Lagrangian multipliers $\lambda_{i,u}$ and
$\mu_{i,m}$, the Lagrangian of (6) is then given as

$\displaystyle{\mbox{$\mathcal{L}$}}\\!=\\!$ $\displaystyle\,N_{c}N_{b}p_{\rm
0}\\!-\\!\\!\sum_{i,u}\\!\lambda_{i,u}\Bigg{(}\\!\frac{\alpha^{2}|{\bf
w}_{i,u}^{H}{\bf
h}_{i,i,u}|^{2}}{\gamma_{i,u}}\\!-\alpha^{2}\\!\\!\\!\\!\\!\\!\\!\\!\sum_{(j,v)\neq(i,u)}^{N_{c},N_{u}}\\!\\!\\!\\!\\!\\!\\!|{\bf
h}_{j,i,u}^{H}{\bf w}_{j,v}|^{2}$ $\displaystyle\\!-\\!{\rm
Q}_{i,u}\\!-\\!\sigma^{2}\\!\Bigg{)}+\sum_{i,m}\mu_{i,m}\left(\left[\alpha{\bf
W}_{i}{\bf W}_{i}^{H}\right]_{m,m}-p_{\rm 0}\right).$ (20)

First, from the proof of Theorem 2 in [8], we have
$\sum_{i,u}\lambda_{i,u}{\rm Q}_{i,u}$ in (20) rewritten as

$\displaystyle\sum_{i,u}\lambda_{i,u}{\rm
Q}_{i,u}=\alpha\beta\sum_{i,u}\\!{\bf w}_{i,u}^{H}{\rm diag}\left({\bf
H}_{i}{\boldsymbol{\Lambda}}{\bf H}_{i}^{H}\\!\right)\\!{\bf w}_{i,u}.$ (21)

Next, we let ${\bf D}_{i}={\rm diag}(\mu_{i,1},\dots,\mu_{i,N_{b}})$. We can
cast $\sum_{i,m}\mu_{i,m}\left[\alpha{\bf W}_{i}{\bf W}_{i}^{H}\right]_{m,m}$
in (20) to

$\displaystyle\sum_{i,m}\mu_{i,m}\bigg{[}\sum_{u}{\bf w}_{i,u}{\bf
w}_{i,u}^{H}\bigg{]}_{m,m}\\!$
$\displaystyle=\sum_{i,m}\sum_{u}w^{*}_{i,u,m}\mu_{i,m}w_{i,u,m}$
$\displaystyle=\sum_{i,u}{\bf w}^{H}_{i,u}{\bf D}_{i}{\bf w}_{i,u}$ (22)

where $w_{i,u,m}$ denotes $m$th element of ${\bf w}_{i,u}$. Lastly,
$\sum_{i,m}\mu_{i,m}p_{\rm 0}$ can be rewritten as

$\displaystyle p_{\rm 0}\sum_{i,m}\mu_{i,m}=p_{\rm 0}\sum_{i=1}^{N_{c}}{\rm
tr}({\bf D}_{i}).$ (23)

Applying (21), (22), and (23) to the Lagrangian in (20), we finally have the
reformed Lagrangian as

$\displaystyle{\mbox{$\mathcal{L}$}}\\!=\\!$
$\displaystyle\sum_{i,u}\lambda_{i,u}\sigma^{2}\\!-\\!p_{\rm
0}\\!\sum_{i}\\!\big{[}{\rm tr}({\bf D}_{i})\\!-\\!N_{b}\big{]}\\!$ (24)
$\displaystyle+\\!\sum_{i,u}\\!{\bf w}_{i,u}^{H}\\!\bigg{(}\alpha{\bf
D}_{i}\\!-\\!\alpha^{2}\\!\left(\\!1\\!+\\!\frac{1}{\gamma_{i,u}}\\!\right)\\!\lambda_{i,u}{\bf
h}_{i,i,u}{\bf h}_{i,i,u}^{H}$
$\displaystyle+\alpha^{2}\\!\sum_{j,v}\lambda_{j,v}{\bf h}_{i,j,v}{\bf
h}_{i,j,v}^{H}\\!+\\!\alpha\beta{\rm diag}\\!\left({\bf
H}_{i}\,{\boldsymbol{\Lambda}}\,{\bf H}_{i}^{H}\\!\right)\\!\bigg{)}{\bf
w}_{i,u}.$

Let the dual objective function be $g({\bf D}_{i},\lambda_{i,u})=\min_{{\bf
w}_{i,u},p_{\rm 0}}\mathcal{L}$. We then need ${\rm tr}({\bf D}_{i})\leq
N_{b}$ and ${\bf
K}_{i}({\boldsymbol{\Lambda}})\succeq\alpha\big{(}1+1/{\gamma_{i,u}}\big{)}\lambda_{i,u}{\bf
h}_{i,i,u}{\bf h}_{i,i,u}^{H}$, where ${\bf K}_{i}({\boldsymbol{\Lambda}})$ is
in (18). Consequently, the Lagrangian dual problem of (6) becomes

$\displaystyle\max_{{{\bf D}}_{i}}\max_{\lambda_{i,u}}$
$\displaystyle\;\;\sum_{i,u}^{N_{c},N_{u}}\lambda_{i,u}\sigma^{2}$ (25)
subject to $\displaystyle\;\;{\bf
K}_{i}({\boldsymbol{\Lambda}})\succeq\alpha\bigg{(}1+\frac{1}{\gamma_{i,u}}\bigg{)}\lambda_{i,u}{\bf
h}_{i,i,u}{\bf h}_{i,i,u}^{H},$ (26) $\displaystyle\;\;{\bf D}_{i}\succeq 0,\
{\bf D}_{i}\in\mbox{$\mathbb{R}$}^{N_{b}\times N_{b}}:\text{\rm diagonal},$
$\displaystyle\;\;{\rm tr}({\bf D}_{i})\leq N_{b}\quad\forall i,u$

We note that the differences between the problem in (16) and in (25) are the
reversed objectives with respect to $\lambda_{i,u}$ (i.e., $\min$ vs. $\max$)
and reversed SINR inequalities in (17) and (26). Since the problems in (16)
and (25) have optimal solutions when the SINR constraints are active, the
solutions for the problems are indeed equivalent to each other with the active
SINR constraints. ∎

We remark that $\hat{\Gamma}_{i,u}$ in (13) can be interpreted as the SINR of
user $u$ in cell $i$ for the UL system with low-resolution ADCs, i.e., ${\bf
f}_{i,u}$ is a combiner for user $u$ in cell $i$, $\lambda_{i,u}$ is transmit
power for user $u$ in cell $i$, ${\bf C}_{i}$ is an aggregated quantization
noise of BSi after quantization, and ${\bf D}_{i}$ is a diagonal matrix of
noise variances at the antennas of BSi with uncertain noise covariance in UL
direction. Accordingly, the Lagrangian dual problem is considered to be an
antenna power minimax problem with noise variance constraints for a virtual UL
system with low-resolution ADCs at the BSs.

######  Corollary 1 (Strong Duality).

Zero duality gap exists between the DL formulation and its associated dual.

###### Proof.

The primal DL problem is rewritten as

$\displaystyle\min_{{{\bf W}},P_{o}}p_{o}$ (27) $\displaystyle{\rm s.t.}\
\Gamma_{i,u}\geq\gamma_{i,u},\quad\forall i,u$ (28)
$\displaystyle\left[\alpha{\bf W}_{i}{\bf W}_{i}^{H}\right]_{m,m}\leq p_{o}$
(29)

Let ${\bf W}_{\rm BD}\\!=\\!{\rm blkdiag}({\bf W}_{1},\dots,{\bf W}_{N_{c}})$,
$\tilde{{\bf W}}_{\rm BD}\\!=\\!{\rm blkdiag}(({\bf I}_{N_{b}}\otimes{\bf
W}_{1}),\dots,({\bf I}_{N_{b}}\otimes{\bf W}_{N_{c}}))$, ${\bf
E}_{j,i,u}\\!=\\!{\rm diag}({\bf h}_{j,i,u}{\bf h}_{j,i,u}^{H})$, and
$\mbox{$\mathbb{E}$}_{i,u}\\!=\\!{\rm vec}({\bf E}^{1/2}_{1,i,u},\ldots,{\bf
E}^{1/2}_{N_{c},i,u})$. The SINR constraints in (28) can be rewritten as

$\displaystyle\alpha^{2}\bigg{(}1+\frac{1}{\gamma_{i,u}}\bigg{)}|{\bf
w}_{i,u}^{H}{\bf h}_{i,i,u}|^{2}$ (30)
$\displaystyle\geq\left\|\begin{matrix}\alpha{\bf W}_{\rm BD}^{H}{\rm
vec}({\bf h}_{1,i,u},\dots,{\bf h}_{N_{c},i,u})\\\
\sqrt{\alpha\beta}\tilde{{\bf W}}_{\rm BD}\mbox{$\mathbb{E}$}_{i,u}\\\
\sigma\end{matrix}\right\|^{2}$ (31)

for all $i$, $u$, and $k$. In addition, the per-antenna constraint in (29) is
rewritten as

$\displaystyle\left[\alpha{\bf W}_{i}{\bf W}_{i}^{H}\right]_{m,m}=\alpha\|{\bf
e}_{m}^{H}{\bf W}_{i}\|^{2}$ (32)

for all $m$ which is a convex constraint. Accordingly, we eventually have the
standard SOCP form. Next, (6) is strictly feasible because, for a given
solution ${\bf W}$, it can be scaled by a factor of $c>1$ satisfying the
constraints. Thus, strong duality holds between (6) and (9). ∎

######  Corollary 2.

An optimal DL precoder forms a linear relationship with the UL MMSE receiver,
i.e., ${\bf w}_{i,u}=\sqrt{{\tau}_{i,u}}{\bf f}_{i,u}\;\forall i,u$. Here,
${\tau}_{i,u}$ is derived from solving
${\boldsymbol{\tau}}={\boldsymbol{\Sigma}}^{-1}{\bf 1}$, where ${\bf 1}$ is a
$N_{u}N_{c}\times 1$ column vector,
${\boldsymbol{\tau}}=[{\boldsymbol{\tau}}_{1}^{T},{\boldsymbol{\tau}}_{2}^{T},\cdots,{\boldsymbol{\tau}}_{N_{c}}^{T}]^{T}$
and
${\boldsymbol{\tau}}_{i}^{T}=[{\tau}_{i,1},{\tau}_{i,2},\cdots,{\tau}_{i,N_{u}}]^{T}$,
and ${\boldsymbol{\Sigma}}$ is defined as

${\boldsymbol{\Sigma}}=\begin{pmatrix}{\boldsymbol{\Sigma}}_{1,1}&{\boldsymbol{\Sigma}}_{1,2}&\cdots&{\boldsymbol{\Sigma}}_{1,N_{c}}\\\
{\boldsymbol{\Sigma}}_{2,1}&{\boldsymbol{\Sigma}}_{2,2}&\cdots&{\boldsymbol{\Sigma}}_{2,N_{c}}\\\
\vdots&\vdots&\ddots&\vdots\\\
{\boldsymbol{\Sigma}}_{N_{c},1}&{\boldsymbol{\Sigma}}_{N_{c},2}&\cdots&{\boldsymbol{\Sigma}}_{N_{c},N_{c}}\end{pmatrix},$
(33)

and

$\displaystyle[{\boldsymbol{\Sigma}}_{i,j}]_{u,v}$
$\displaystyle=\begin{cases}\frac{\alpha^{2}}{\gamma_{i,u}}|{\bf
h}_{i,i,u}^{H}{\bf f}_{i,u}|^{2}\\!-\alpha(1\\!-\\!\alpha){\bf
f}_{i,u}^{H}{\rm diag}\big{(}{\bf h}_{i,i,u}{\bf h}_{i,i,u}^{H}\\!\big{)}{\bf
f}_{i,u}\\\ \text{if }i=j\text{, }u=v,\\\ -\alpha^{2}|{\bf h}_{j,i,u}^{H}{\bf
f}_{j,v}|^{2}\\!-\alpha(1\\!-\\!\alpha){\bf f}_{j,v}^{H}{\rm diag}\big{(}{\bf
h}_{j,i,u}{\bf h}_{j,i,u}^{H}\\!\big{)}{\bf f}_{j,v}\\\
\text{otherwise.}\end{cases}$

###### Proof.

Starting from the Lagrangian in (24), we find the derivative of the Lagrangian
regarding ${\bf w}_{i,u}$ as

$\displaystyle 2\bigg{(}\alpha{\bf
D}_{i}\\!-\\!\alpha^{2}\\!\left(\\!1\\!+\\!\frac{1}{\gamma_{i,u}}\\!\right)\\!\lambda_{i,u}{\bf
h}_{i,i,u}{\bf h}_{i,i,u}^{H}$ (34)
$\displaystyle+\alpha^{2}\\!\sum_{j,v}\lambda_{j,v}{\bf h}_{i,j,v}{\bf
h}_{i,j,v}^{H}+\\!\alpha(1-\alpha){\rm diag}\\!\left({\bf
H}_{i}\,{\boldsymbol{\Lambda}}\,{\bf H}_{i}^{H}\\!\right)\\!\bigg{)}{\bf
w}_{i,u}.$

We then set the derivative to zero, and solve it for ${\bf w}_{i,u}$ as

$\displaystyle{\bf
w}_{i,u}=\bigg{(}\\!\alpha^{2}\\!\\!\\!\\!\\!\sum_{(j,v)\neq(i,u)}\\!\\!\\!\lambda_{j,v}{\bf
h}_{i,j,v}{\bf h}_{i,j,v}^{H}\\!$ $\displaystyle+\\!\alpha(1\\!-\\!\alpha){\rm
diag}\\!\left({\bf H}_{i}{\boldsymbol{\Lambda}}{\bf
H}_{i}^{H}\\!\right)\\!+\\!\alpha{\bf
D}_{i}\\!\bigg{)}^{\\!-1}\\!\\!\frac{\alpha^{2}}{\gamma_{i,u}}\\!\lambda_{i,u}{\bf
h}_{i,i,u}{\bf h}_{i,i,u}^{H}{\bf w}_{i,u}$
$\displaystyle=\frac{\alpha^{2}}{\gamma_{i,u}}\lambda_{i,u}{\bf
h}_{i,i,u}^{H}{\bf w}_{i,u}{\bf f}_{i,u}$

where ${\bf f}_{i,u}$ is in (15) and the last equality is valid because ${\bf
h}_{i,i,u}^{H}{\bf w}_{i,u}$ is a scalar. Accordingly, we can justify the form
of ${\bf w}_{i,u}=\sqrt{\tau_{i,u}}{\bf f}_{i,u}$ with properly designed
$\tau_{i,u}$.

To satisfy the KKT stationarity condition with the DL constraint in (7),
$\Gamma_{i,u}$ has to meet the target SINR constraint with equality. Since the
DL precoder is deeply embedded in the quantization noise term, i.e., ${\rm
Q}_{i,u}$, in the DL SINR expression, we first simplify ${\rm Q}_{i,u}$. Let
us define $\mu_{i^{\prime},u^{\prime}}$ where $\mu_{i^{\prime},u^{\prime}}=1$
if $i^{\prime}=i$ and $u^{\prime}=u$, and $\mu_{i^{\prime},u^{\prime}}=0$
otherwise. To compose the DL constraint in a tractable form, we can rewrite
the quantization error term in (4) as

$\displaystyle{\rm Q}_{i,u}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\\!\alpha\beta\sum_{j=1}^{N_{c}}{\bf
h}_{j,i,u}^{H}{\rm
diag}\big{(}\mbox{${\bf{W}}$}_{j}\mbox{${\bf{W}}$}_{j}^{H}\big{)}{\bf
h}_{j,i,u}\\!$
$\displaystyle=\alpha\beta\sum_{i^{\prime},u^{\prime},j}\mu_{i^{\prime},u^{\prime}}{\bf
h}_{j,i^{\prime},u^{\prime}}^{H}{\rm
diag}\big{(}\mbox{${\bf{W}}$}_{j}\mbox{${\bf{W}}$}_{j}^{H}\big{)}{\bf
h}_{j,i^{\prime},u^{\prime}}$ $\displaystyle=\alpha\beta\sum_{j,v}{\bf
w}_{j,v}^{H}{\rm diag}\left({\bf H}_{j}\,\mbox{${\bf{M}}$}\,{\bf
H}_{j}^{H}\right)\\!{\bf w}_{j,v}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\alpha\beta\sum_{j,v}{\bf
w}_{j,v}^{H}{\rm diag}\left({\bf h}_{j,i,u}{\bf
h}_{j,i,u}^{H}\\!{}^{H}\\!\right)\\!{\bf w}_{j,v}^{H},$ (35)

where $(a)$ is obtained by plugging (2) into (5). We define ${\bf
M}_{i}\\!=\\!{\rm diag}(\mu_{i,1},\dots,\mu_{i,N_{u}})$ and ${{\bf
M}}\\!=\\!{\rm blkdiag}({{\bf M}}_{1},\dots,{{\bf M}}_{N_{c}})$. By the
definition of ${\bf M}$, $(b)$ is from ${\bf H}_{j}\,\mbox{${\bf{M}}$}\,{\bf
H}_{j}^{H}={\bf h}_{j,i,u}{\bf h}_{j,i,u}^{H}$.

Accordingly, we can create the DL SINR constraints with equality as follows:

$\displaystyle\sigma^{2}$ $\displaystyle=\frac{\alpha^{2}}{\gamma_{i,u}}|{\bf
h}_{i,i,u}^{H}{\bf
w}_{i,u}|^{2}-\alpha^{2}\\!\\!\\!\\!\sum_{(j,v)\neq(i,u)}^{N_{c},N_{u}}\\!|{\bf
h}_{j,i,u}^{H}{\bf w}_{j,v}|^{2}-{\rm Q}_{i,u}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\frac{\alpha^{2}}{\gamma_{i,u}}|{\bf
h}_{i,i,u}^{H}{\bf
f}_{i,u}|^{2}\tau_{i,u}-\alpha^{2}\\!\\!\\!\\!\\!\\!\\!\sum_{(j,v)\neq(i,u)}\\!\\!\\!\\!\\!{|{\bf
h}_{j,i,u}{\bf f}_{j,v}^{H}|^{2}}\tau_{j,v}\\!$
$\displaystyle\quad-\\!\alpha\beta\\!\sum_{j,v}\\!{\bf w}_{j,v}^{H}{\rm
diag}\\!\left({\bf h}_{j,i,u}{\bf h}_{j,i,u}^{H}\\!{}^{H}\right)\\!{\bf
w}_{j,v}^{H},$ (36)

for all $i$, $u$ where $(a)$ is from (35) and ${\bf
w}_{i,u}=\sqrt{\tau_{i,u}}{\bf f}_{i,u}$. Combining (36) for all $i,u$ gives a
system of linear equations as $\sigma^{2}{\bf
1}={\boldsymbol{\Sigma}}{\boldsymbol{\tau}}$, thereby having $\tau_{i,u}$
through ${\boldsymbol{\tau}}=\sigma^{2}{\boldsymbol{\Sigma}}^{-1}{\bf 1}$. ∎

###  II-C Distributed Iterative Algorithm

In this subsection, we characterize solutions by exploiting strong duality and
further adopt the fixed-point iteration with a subgradient projection method
[13].

######  Corollary 3.

For a fixed ${\bf D}_{i}$, the optimal power for the uplink total power
minimization problem in (9) is given as

$\displaystyle\lambda_{i,u}=\frac{1}{\alpha\left(1+\frac{1}{\gamma_{i,u}}\right){\bf
h}_{i,i,u}^{H}{\bf K}_{i}^{-1}({\boldsymbol{\Lambda}}){\bf h}_{i,i,u}},$ (37)

where ${\bf K}_{i}({\boldsymbol{\Lambda}})$ is defined in (18).

###### Proof.

Setting (34) to zero and solving it for $\lambda_{i,u}$ produce (37). The
solution of $\lambda_{i,u}$ then satisfies the stationary condition, and we
further observe that the UL SINR constraint in (8) is active at the solution
satisfying the complementary slackness condition. Therefore, (37) is optimal
solution of the virtual UL problem. ∎

The UL power control solution from Corollary 3 is fundamentally designed for
the subproblem of (9) written as

$\displaystyle f\\!\left({\bf D}_{i}\right)=$
$\displaystyle\min_{\lambda_{i,u}}\;\;\sum_{i,u}^{N_{c},N_{u}}\lambda_{i,u}\sigma^{2}$
(38) $\displaystyle{\text{ \rm subject to}}\;\;\max_{{\bf
f}_{i,u}}\hat{\Gamma}_{i,u}\geq\gamma_{i,u}\;\;\forall i,u,$

which is the inner optimization on $\lambda_{i,u}$ of $g({\bf
D}_{i},\lambda_{i,u})$ when MMSE filter in (15) is used for ${\bf f}_{i,u}$.
However, the solutions do not guarantee a global optimum over the entire
feasible candidates of ${\bf D}_{i}$ in (11)-(12). Therefore, as a second
stage, the external loop on ${\bf D}_{i}$ is needed. We then use a projected
subgradient ascend method to maximize the objective function while satisfying
the constraint on ${\bf D}_{i}$. Note that $f\\!\left({\bf D}_{i}\right)$ is
concave in ${\bf D}_{i}$ since the Lagrangian function in
$\eqref{eq:lagrangian}$ is affine in $\lambda$ and $\mu$, and the infimum of
affine functions is still concave.

######  Corollary 4 (Subgradient).

${\rm diag}\left(\sum_{u}{\bf w}_{i,u}{\bf w}_{i,u}^{H}\right)$ is a
subgradient of (38) in updating ${\bf D}_{i}$.

###### Proof.

Using strong duality between UL and DL, the UL subproblem $f\\!\left({\bf
D}_{i}\right)$ in (38) for a fixed ${\bf D}_{i}$ is equivalent to

$\displaystyle f\\!\left({\bf D}_{i}\right)=$ $\displaystyle\min_{{\bf
w}_{i,u}}\;\;\sum_{u}^{N_{u}}{\bf w}_{i,u}^{H}{\bf D}_{i}{\bf w}_{i,u}$ (39)
$\displaystyle{\text{ \rm subject
to}}\;\;\Gamma_{i,u}\geq\gamma_{i,u}\;\;\forall i,u,$

based on the proof of Theorem 1. We introduce two arbitrary diagonal
covariance matrices ${\bf D}_{i}$ and ${\bf D}_{i}^{\prime}$ whose associated
optimal beamformer is ${\bf w}_{i,u}$ and ${\bf w}_{i,u}^{\prime}$,
respectively. We then derive a subgradient as follows:

$\displaystyle f\\!\left({\bf D}_{i}^{\prime}\right)-f\\!\left({\bf
D}_{i}\right)=\sum_{u}{\bf w}_{i,u}^{{}^{\prime}H}{\bf D}_{i}^{\prime}{\bf
w}_{i,u}^{\prime}-\sum_{u}{\bf w}_{i,u}^{H}{\bf D}_{i}{\bf w}_{i,u}$ (40)
$\displaystyle\qquad\qquad\qquad\;\;\leq\sum_{u}{\bf w}_{i,u}^{H}{\bf
D}_{i}^{\prime}{\bf w}_{i,u}-\sum_{u}{\bf w}_{i,u}^{H}{\bf D}_{i}{\bf
w}_{i,u}$ (41) $\displaystyle\qquad\qquad={\rm tr}\left({\rm
diag}\left(\sum_{u}{\bf w}_{i,u}{\bf w}_{i,u}^{H}\right)\left({\bf
D}_{i}^{\prime}-{\bf D}_{i}\right)\right).$ (42)

By the definition of subgradients, the multiplier in front of ${\bf
D}_{i}^{\prime}-{\bf D}_{i}$ becomes a subgradient of $f({\bf D}_{i})$. ∎

We finally plug the subproblem into the entire DL problem in (9). Recall that
the main target is to maximize $f({\bf D}_{i})$ while satisfying the
constraints on ${\bf D}_{i}$. After obtaining a converged solution of (38) for
a fixed ${\bf D}_{i}$, we take a step in the direction of a positive
subgradient. We further project the updated ${\bf D}_{i}$ onto the feasible
set on ${\bf D}_{i}$ in (11) and (12) because the updated ${\bf D}_{i}$
probably violates the feasible domain. The complete algorithm is summarized in
Algorithm 1.

1 Initialize $\lambda_{i,u}$, $\forall i,u$ and ${\bf D}_{i}^{(0)}$, $\forall
i$.

2 while  _${\bf D}_{i}^{(n)}$ ’s do not converge_ do

3 Iteratively update $\lambda_{i,u}$ until converges as
$\displaystyle\lambda_{i,u}=\frac{1}{\alpha\Big{(}1+\frac{1}{\gamma_{i,u}}\Big{)}{\bf
h}_{i,i,u}^{H}\left[{\bf K}_{i}^{(n)}({\boldsymbol{\Lambda}})\right]^{-1}{\bf
h}_{i,i,u}},$ for all $i,u$ where ${\bf K}_{i}^{(n)}({\boldsymbol{\Lambda}})$
is updated according to (18) using $\lambda_{i,u}$ and ${\bf D}_{i}^{(n)}$.

4 Find the UL MMSE equalizer ${\bf f}_{i,u}$ in (15) with ${\bf D}_{i}^{(n)}$
and the converged $\lambda_{i,u}$

5 Compute the DL precoder ${\bf w}_{i,u}$ from Corollary 2.

6 for  _$i=1$ to $N_{c}$_ do

7 ${\bf D}_{i}^{(n+1)}\\!\leftarrow\\!{\bf D}_{i}^{(n)}\\!+\\!\eta\;{\rm
diag}\left(\sum_{u}{\bf w}_{i,u}{\bf w}_{i,u}^{H}\right)$.

8 Project ${\bf D}_{i}^{(n+1)}$ onto the feasible set (11)-(12) until
converges as

9 ${\bf D}_{i}^{(n+1)}\\!\leftarrow\\!{\bf D}_{i}^{(n+1)}\\!\\!-\\!\frac{\rm
max\left(0,{\rm tr}\left({\bf D}_{i}^{(n+1)}\right)-N_{b}\right)}{\left\|{\bf
1}_{N_{b}}\right\|^{2}}{\bf 1}_{N_{b}}$

10 $n\leftarrow n+1$

return __ ${\bf w}_{i,u}$ for all $i,u$.

Algorithm 1 Joint Power-Minimizing Transmission with Per-Antenna Constraints

##  III Simulation Results

We evaluate the derived results of the proposed quantization-aware iterative
CoMP algorithm with per-antenna constraints (Q-iCoMP-PA) against the
quantization-aware iterative CoMP algorithm (Q-iCoMP) in [8]. The former is
based on the antenna power minimax problem while the latter focuses on the
total transmit power minimization problem.

Each BS is in the center of own hexagonal cell and BSs operate beside each
other. We assume that the small scale fading of each channel follows Rayleigh
fading with zero mean and unit variance. For large scale fading, we use the
log-distance pathloss model in [19]. The distance between adjacent BSs is $2\
\rm km$. The minimum distance between BS and user is $100\ \rm m$. Considering
a $2.4\ \rm GHz$ carrier frequency with $10\ \rm MHz$ bandwidth, we use $8.7\
\rm dB$ lognormal shadowing variance and $5\ \rm dB$ noise figure. For
simplicity, we assume an equal target SINR for all users, i.e.,
$\gamma_{i,u}=\gamma$ for all $i,u$.

Figure 2: Maximum transmit antenna power vs. target SINR for the network with
$N_{b}=32$ antennas per BS, $N_{c}=4$ cells, $N_{u}=2$ users per cell, and
$b\in\\{2,3,\infty\\}$ bits using 20 channel realizations per target SINR
value.

Fig. 2 shows maximum transmit antenna power in DL direction across all
$N_{c}N_{b}$ transmit antennas for given target SINRs. We consider a
communication configuration with $N_{b}=32$ BS antennas, $N_{c}=4$ cells, and
$N_{u}=2$ users per cell. We test both infinite-resolution and low-resolution
converters, i.e., $b\\!\in\\!\\{2,3,\infty\\}$. When using infinite-resolution
ADCs and DACs, we have slightly lower peak power compared to one with 3-bit
data converters, however the gap between $b=2$, $b=3$, and $b=\infty$ cases is
marginal on both Q-iCoMP and Q-iCoMP-PA by properly incorporating the coarse
quantization error into the design of beamformers. With multi-cell
coordination, both Q-iCoMP and Q-iCoMP-PA do not suffer from implausible power
consumption and undesirable divergence. However, based on the primal problem
of Q-iCoMP-PA, the proposed algorithm can limit the maximum transmit power
providing around 2 dB gain over the regular Q-iCoMP.

Figure 3:  Empirical CDF of the transmit power of the BS antennas in the
network with $N_{b}=32$ antennas per BS, $N_{c}=5$ cells, $N_{u}=2$ users per
cell, $b=3$ quantization bits, and $\gamma=2\ \rm dB$ target SINR using one
channel realization. Empirical CDF gives the probability of an antenna
operating at or below the given transmit power.

Fig. 3 shows the cumulative density function (CDF) of the transmit power of
all antennas, i.e., $N_{b}N_{c}$ antennas, considering one channel
realization. We employ a network with $N_{b}=32$ BS antennas, $N_{c}=5$ cells,
$N_{u}=2$ users per cell, and $b=3$ quantization bits. From the figure, the
proposed algorithm reveals two main advantages: 1) maximum transmit antenna
power; and 2) operating range. Since the CDF is plotted over all antennas, the
rightmost point of the CDF represents the maximum transmit power. When
comparing the rightmost point of two methods, Q-iCoMP-PA achieves more than
$2\ \rm dB$ gain over Q-iCoMP, which corresponds to the main purpose of the
proposed method. Also, the CDF gives the operating range which is defined as
the gap between the leftmost and rightmost points of the CDF. Q-iCoMP-PA works
with a much narrower operating range compared with Q-iCoMP, thereby
increaseing efficiency of power-related components such as power amplifier.

TABLE I: Comparison of Peak-to-average power ratio (PAPR).

$\gamma$ | Methods
---|---
[dB] | Q-iCoMP | Q-iCoMP-PA
2 | 3.81 dB | 2.16 dB
-3 | 3.77 dB | 2.09 dB

$N_{b}=32$ antennas per BS, $N_{c}=4$ cells, $N_{u}=2$ users per cell, and
$b=3$ bits.

In Table. I, we further simulate the peak-to-average power ratio (PAPR) which
is directly related to the efficiency of power amplifiers. We consider a
network configuration with $N_{b}=16$ BS antennas, $N_{c}\in\\{2,3\\}$ cells,
$N_{u}=2$ users per cell, and $b=3$ bits over different constraint SINRs. With
two cells in the network, the Q-iCoMP-PA achieves significant reduction over
the Q-iCoMP, showing more than 1.6 dB gain on average. Therefore, Q-iCoMP-PA
is more favorable for mobile communication systems by properly providing
multi-cell coordination and limiting the peak power of antennas.

##  IV Conclusion

In this paper, we investigated the CoMP solution for a multicell configuration
with low-resolution data converters when employing per-antenna power
constraints for more practical deployment. Considering the coarse quantization
error, we derived the antenna power minimax problem and effective UL problem
with uncertain noise covariance as dual problem, and further proved zero
duality gap between two problems. Leveraging strong duality, we proposed the
iterative algorithm that finds the optimal dual solution for a fixed
covariance and used the solution to compute the optimal DL beamformer. We
further update the UL noise covariance using the optimal DL solution with the
projected subgradient descent method. In simulation, the proposed Q-iCoMP-PA
achieves significant gain over Q-iCoMP in terms of maximum antenna power,
operating range, and PAPR, thereby improving hardware efficiency. Therefore,
we can emphasize the need to limit the antenna power when deploying multicell
massive MIMO communication systems with low-resolution data converters.

## References

  * [1] T. L. Marzetta, “Noncooperative Cellular Wireless with Unlimited Numbers of Base Station Antennas,” _IEEE Trans. on Wireless Commun._ , vol. 9, no. 11, pp. 3590–3600, 2010.
  * [2] J. Choi, J. Mo, and R. W. Heath, “Near maximum-likelihood detector and channel estimator for uplink multiuser massive MIMO systems with one-bit ADCs,” _IEEE Trans. Comm._ , vol. 64, no. 5, pp. 2005–2018, Mar. 2016.
  * [3] C. Studer and G. Durisi, “Quantized massive mu-mimo-ofdm uplink,” _IEEE Trans. on Commun._ , vol. 64, no. 6, pp. 2387–2399, 2016.
  * [4] J. Choi, Y. Cho, B. L. Evans, and A. Gatherer, “Robust learning-based ML detection for massive MIMO systems with one-bit quantized signals,” in _IEEE Global Commun. Conf._ , 2019, pp. 1–6.
  * [5] Y. Cho and S.-N. Hong, “One-bit successive-cancellation soft-output (OSS) detector for uplink MU-MIMO systems with one-bit ADCs,” _IEEE Access_ , vol. 7, pp. 27 172–27 182, 2019.
  * [6] J. Choi, B. L. Evans, and A. Gatherer, “Resolution-Adaptive Hybrid MIMO Architectures for Millimeter Wave Communications,” _IEEE Trans. on Signal Process._ , vol. PP, no. 99, pp. 1–1, 2017.
  * [7] J. Choi, G. Lee, and B. L. Evans, “Two-Stage Analog Combining in Hybrid Beamforming Systems With Low-Resolution ADCs,” _IEEE Trans. on Signal Process._ , vol. 67, no. 9, pp. 2410–2425, 2019.
  * [8] J. Choi, Y. Cho, and B. L. Evans, “Quantized massive mimo systems with multicell coordinated beamforming and power control,” _IEEE Transactions on Communications_ , vol. 69, no. 2, pp. 946–961, 2021.
  * [9] L. V. Nguyen, A. Lee Swindlehurst, and D. H. N. Nguyena, “Linear and deep neural network-based receivers for massive mimo systems with one-bit adcs,” _IEEE Trans. on Wireless Commun._ , pp. 1–1, 2021.
  * [10] J. Yuan, Q. He, M. Matthaiou, T. Q. S. Quek, and S. Jin, “Toward massive connectivity for iot in mixed-adc distributed massive mimo,” _IEEE Internet of Things Journal_ , vol. 7, no. 3, pp. 1841–1856, 2020.
  * [11] S. Park, Y. Cho, and S. Hong, “Construction of 1-bit transmit signal vectors for downlink mu-miso systems: Qam constellations,” _IEEE Transactions on Vehicular Technology_ , vol. 70, no. 10, pp. 10 065–10 076, 2021.
  * [12] W. Yu and T. Lan, “Transmitter optimization for the multi-antenna downlink with per-antenna power constraints,” _IEEE Trans. on Signal process._ , vol. 55, no. 6, pp. 2646–2660, 2007.
  * [13] H. Dahrouj and W. Yu, “Coordinated beamforming for the multicell multi-antenna wireless system,” _IEEE Trans. on Wireless Commun._ , vol. 9, no. 5, pp. 1748–1759, 2010.
  * [14] F. Rashid-Farrokhi, L. Tassiulas, and K. R. Liu, “Joint optimal power control and beamforming in wireless networks using antenna arrays,” _IEEE Trans. on Commun._ , vol. 46, no. 10, pp. 1313–1324, 1998.
  * [15] F. Rashid-Farrokhi, K. R. Liu, and L. Tassiulas, “Transmit beamforming and power control for cellular wireless systems,” _IEEE J. on Sel. Areas in Commun._ , vol. 16, no. 8, pp. 1437–1450, 1998.
  * [16] V. Jungnickel, K. Manolakis, W. Zirwas, B. Panzner, V. Braun, M. Lossow, M. Sternad, R. Apelfröjd, and T. Svensson, “The role of small cells, coordinated multipoint, and massive MIMO in 5G,” _IEEE Commun. Mag._ , vol. 52, no. 5, pp. 44–51, 2014.
  * [17] O. Orhan, E. Erkip, and S. Rangan, “Low power analog-to-digital conversion in millimeter wave systems: Impact of resolution and bandwidth on performance,” in _IEEE Info. Theory and App. Work._ , Feb. 2015, pp. 191–198.
  * [18] L. Fan, S. Jin, C.-K. Wen, and H. Zhang, “Uplink achievable rate for massive MIMO systems with low-resolution ADC,” _IEEE Comm. Letters_ , vol. 19, no. 12, pp. 2186–2189, Oct. 2015.
  * [19] V. Erceg, L. J. Greenstein, S. Y. Tjandra, S. R. Parkoff, A. Gupta, B. Kulic, A. A. Julius, and R. Bianchi, “An empirically based path loss model for wireless channels in suburban environments,” _IEEE J. on Sel. Areas in Commun._ , vol. 17, no. 7, pp. 1205–1211, 1999.