11institutetext: Dustin G. Mixon 22institutetext: The Ohio State University,
Columbus, OH, 22email<EMAIL_ADDRESS>33institutetext: Tom Needham
44institutetext: Florida State University, Tallahassee, FL, 44email:
<EMAIL_ADDRESS>55institutetext: Clayton Shonkwiler 66institutetext: Colorado
State University, Fort Collins, CO, 66email<EMAIL_ADDRESS>77institutetext: Soledad Villar 88institutetext: Johns Hopkins University,
Baltimore, MD, 88email<EMAIL_ADDRESS>
# Three proofs of the Benedetto–Fickus theorem

Dustin G. Mixon  Tom Needham  Clayton Shonkwiler  Soledad Villar

###### Abstract

In 2003, Benedetto and Fickus introduced a vivid intuition for an objective
function called the frame potential, whose global minimizers are fundamental
objects known today as unit norm tight frames. Their main result was that the
frame potential exhibits no spurious local minimizers, suggesting local
optimization as an approach to construct these objects. Local optimization has
since become the workhorse of cutting-edge signal processing and machine
learning, and accordingly, the community has identified a variety of
techniques to study optimization landscapes. This chapter applies some of
these techniques to obtain three modern proofs of the Benedetto–Fickus
theorem.

##  1 Introduction

Frame theory is the study of overcomplete systems of vectors in a Hilbert
space. Much of the last 20 years of research in finite frame theory has
concerned various modifications of the following fundamental object: (the
sequence of column vectors of) $Z=[z_{1}\cdots z_{n}]\in\mathbb{C}^{d\times
n}$ is a unit norm tight frame if

  * •

$\|z_{j}\|^{2}=1$ for every $j\in[n]:=\\{1,\ldots,n\\}$, and

  * •

$ZZ^{*}$ is a multiple of the $d\times d$ identity matrix.

The multiple is necessarily the redundancy $\frac{n}{d}$ of $Z$ since
$\operatorname{tr}(ZZ^{*})=\operatorname{tr}(Z^{*}Z)=n$. Given a $d\times n$
unit norm tight frame, it necessarily holds that

$n\geq\operatorname{rank}(Z)=\operatorname{rank}(ZZ^{*})=\operatorname{rank}(\tfrac{n}{d}I_{d})=d.$

The unit norm tight frames with $n=d$ are precisely the orthonormal bases for
$\mathbb{C}^{d}$. When $n>d$, a unit norm tight frame $Z\in\mathbb{C}^{d\times
n}$ provides both an encoding $y:=Z^{*}x$ of data $x\in\mathbb{C}^{d}$ that is
robust to various types of errors GoyalKK:01 ; CasazzaK:03 ; HolmesP:04
(thanks in part to the redundancy $\frac{n}{d}>1$) and a painless
reconstruction formula

$x=\frac{d}{n}ZZ^{*}x=\frac{d}{n}Zy$

(thanks to the resemblance $ZZ^{*}=\frac{n}{d}I_{d}$ to orthonormal bases).

Historically, frames were introduced in the setting of infinite-dimensional
Hilbert space by Duffin and Schaeffer DuffinS:52  in 1952 in the development
of non-harmonic Fourier series. In 1986, Daucechies, Grossmann, and Meyer
DaubechiesGY:86  further developed the theory of frames in the context of
signal processing. In the time since, finite-dimensional treatments of the
subject have proved interesting in their own right.

Today, certain unit norm tight frames are known to emerge as optimal frames
under various objectives, for example, when hunting for $n$ points on the unit
sphere in $\mathbb{C}^{d}$ for which the absolute values of the inner products
between pairs of points are uniformly as small as possible Welch:74 ;
StrohmerH:03 ; FickusM:15 . In this chapter, we focus on an objective for
which the $d\times n$ unit norm tight frames are precisely the global
minimizers. Let $\operatorname{S}(d,n)$ denote the set of matrices
$Z\in\mathbb{C}^{d\times n}$ for which every column of $Z$ has unit norm. (We
use $\operatorname{S}$ since each column resides in the sphere.) The frame
potential $\operatorname{FP}\colon\operatorname{S}(d,n)\to\mathbb{R}$ is
defined by

$\operatorname{FP}(Z):=\|Z^{*}Z\|_{F}^{2},$

where $\|\cdot\|_{F}$ denotes the Frobenius matrix norm.

######  Proposition 1 (Welch Welch:74 )

For every $Z\in\operatorname{S}(d,n)$, it holds that
$\operatorname{FP}(Z)\geq\frac{n^{2}}{d}$, with equality precisely when $Z$ is
a unit norm tight frame.

###### Proof

Rearrange the inequality

$0\leq\|ZZ^{*}-\tfrac{n}{d}I_{d}\|_{F}^{2}=\|ZZ^{*}\|_{F}^{2}-2\langle
ZZ^{*},\tfrac{n}{d}I_{d}\rangle+\|\tfrac{n}{d}I_{d}\|_{F}^{2}=\operatorname{FP}(Z)-\tfrac{n^{2}}{d},$

and observe that equality holds precisely when $ZZ^{*}=\frac{n}{d}I_{d}$. ∎

We note that when $n<d$, Welch’s bound is dominated by the simpler bound

$\operatorname{FP}(Z)\geq n,$

and equality is achieved precisely when the columns of $Z$ are orthonormal.

In BenedettoF:03 , Benedetto and Fickus introduced the idea of locally
minimizing the frame potential, much like how point charges confined to a
sphere will dynamically repel each other in order to locally minimize a
Coulomb potential. For context, there are several longstanding open problems
that concern minimizing a given potential of a point configuration on the
sphere, to include the Thomson problem Thomson:04 , the Tammes problem
Tammes:30 , and Smale’s 7th problem Smale:98 . The difficulty with these
problems stems from what appear to be exponentially many spurious local
minimizers, i.e., local minimizers that are not global minimizers. Perhaps
surprisingly, the frame potential does not share this defect:

######  Proposition 2 (Benedetto–Fickus theorem BenedettoF:03 )

For each $d,n\in\mathbb{N}$, the frame potential
$\operatorname{FP}\colon\operatorname{S}(d,n)\to\mathbb{R}$ has no spurious
local minimizers.

We offer two comments about the results in BenedettoF:03 . First, Benedetto
and Fickus consider both the real and complex cases. We isolate the complex
case here because it appears to be more amenable to analysis by a variety of
interesting techniques. Throughout, we discuss to what extent certain
arguments can be transferred to the real case. Second, even if we were to
include the real case, Benedetto and Fickus prove a bit more than what is
contained in the Benedetto–Fickus theorem. Specifically, they use Lagrange
multipliers to characterize the critical points of the frame potential, and
then they find a descent direction for every critical point that is neither
orthonormal (in the case where $n\leq d$) nor a unit norm tight frame (in the
case where $n\geq d$). Note that by compactness, a global minimum necessarily
exists, which in turn is a critical point with no descent direction, and by
the above, this point must be a unit norm tight frame whenever $n\geq d$. By
this argument, Benedetto and Fickus provide a clever implicit proof of the
existence of unit norm tight frames and the achievability of Welch’s bound.

Regardless, we isolate the Benedetto–Fickus theorem as the main result of
BenedettoF:03 . One reason is that this feature of the frame potential clearly
distinguishes it from other potentials such as the Coulomb potential. Second,
there are much shorter explicit proofs of the existence of unit norm tight
frames (even in the real case): for each $d,n\in\mathbb{N}$ with $n\geq d$, it
suffices to select $d$ appropriate rows of the $n\times n$ real discrete
Fourier transform matrix and scale the resulting columns to have unit norm.
Indeed, the frame potential is unusual in how easy it is to characterize its
global minimizers. Finally, our statement of the Benedetto–Fickus theorem
highlights the relevance of BenedettoF:03  to modern trends in signal
processing and machine learning. In particular, much of the last decade of
research has centered around analyzing non-convex landscapes for optimization
in various settings, such as orthogonal tensor decomposition GeHJY:15 , the
Burer–Monteiro method BoumalVB:16 , dictionary learning SunQW:16 , matrix
completion GeLM:16 , phase retrieval SunQW:18 , and neural network training
LaurentB:18 . In all of these settings, a primary goal has been to establish
the absence of spurious local minimizers so as to explain the empirical
performance of local optimization.

The recent literature has introduced many techniques to our community, and in
this chapter, we use some of them to prove three different strengthenings of
the Benedetto–Fickus theorem. The proof in the next section is closest to the
original, though we use a stronger Lagrange multiplier result to make the
proof more routine, and we use Wirtinger derivatives to simplify the necessary
calculations. In Section 3, we apply a general observation that a convex
function of an open map has no spurious local minimizers, and we obtain such a
factorization of the frame potential using the theory of eigensteps developed
in Cahill:13 . Finally, Section 4 leverages ideas from geometric invariant
theory to show that locally minimizing the frame potential by gradient flow
sends almost every initialization to a unit norm tight frame.

##  2 Lagrange multipliers and Wirtinger derivatives

Lagrange multipliers provide a standard approach for optimization subject to
equality constraints. What follows is a fundamental result in this setting:

######  Proposition 3 (Lagrange multiplier theorem, Proposition 3.1.1 in
Bertsekas:99 )

Given twice continuously differentiable functions
$f,h_{1},\ldots,h_{m}\colon\mathbb{R}^{n}\to\mathbb{R}$, consider

$S:=\\{x\in\mathbb{R}^{n}:h_{1}(x)=\cdots=h_{m}(x)=0\\},$

and suppose $x_{0}\in S$ is a local minimum of the restriction of $f$ to $S$
with linearly independent constraint gradients $\\{\nabla
h_{i}(x_{0})\\}_{i\in[m]}$. Then there exists a unique choice of Lagrange
multipliers $\lambda_{1},\ldots,\lambda_{m}\in\mathbb{R}$ such that

$\nabla f(x_{0})+\sum_{i\in[m]}\lambda_{i}\nabla h_{i}(x_{0})=0,\quad
y^{\top}\bigg{(}\nabla^{2}f(x_{0})+\sum_{i\in[m]}\lambda_{i}\nabla^{2}h_{i}(x_{0})\bigg{)}y\geq
0$ (1)

for all $y\in(\operatorname{span}\\{\nabla
h_{i}(x_{0})\\}_{i\in[m]})^{\perp}$.

This motivates a couple of definitions. In the setting of Proposition 3, we
say $x_{0}\in S$ is a (first-order) critical point if the constraint gradients
$\\{\nabla h_{i}(x_{0})\\}_{i\in[m]}$ are linearly dependent or there exist
Lagrange multipliers $\lambda_{1},\ldots,\lambda_{m}\in\mathbb{R}$ such that

$\nabla f(x_{0})+\sum_{i\in[m]}\lambda_{i}\nabla h_{i}(x_{0})=0,$

and we say $x_{0}\in S$ is a second-order critical point if $\\{\nabla
h_{i}(x_{0})\\}_{i\in[m]}$ are linearly dependent or there exist
$\lambda_{1},\ldots,\lambda_{m}\in\mathbb{R}$ such that (1) holds for all
$y\in(\operatorname{span}\\{\nabla h_{i}(x_{0})\\}_{i\in[m]})^{\perp}$. By
Proposition 3, every local minimizer is a second-order critical point, and by
relaxation, every second-order critical point is a first-order critical point.
(Indeed, we include the dependence of constraint gradients in the definition
of critical points so that this implication holds, but as we will see, such
degeneracy never arises in our setting.)

In this section, we simplify the original proof of the Benedetto–Fickus
theorem to obtain a proof of the following strengthening:

###### Theorem 2.1

For each $d,n\in\mathbb{N}$, the second-order critical points of the frame
potential $\operatorname{FP}\colon\operatorname{S}(d,n)\to\mathbb{R}$ are
precisely the global minimizers of the frame potential, namely, the $d\times
n$ matrices with orthonormal columns when $n\leq d$, and the $d\times n$ unit
norm tight frames when $n\geq d$.

While we will prove Theorem 2.1 in the complex setting, the same proof works
(and is even simpler) in the real setting. In our setting, we are optimizing
the restriction of a function $f\colon\mathbb{C}^{n}\to\mathbb{R}$, and so to
find critical points, we are inclined to take
$g\colon\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{C}^{n}$ defined by
$g(x,y):=x+\mathrm{i}y$ and then compute the gradient of the composition
$f\circ g$. Working with this composition can be cumbersome, but as we will
see, Wirtinger derivatives nicely encode the desired gradient information in
terms of functions over the native space $\mathbb{C}^{n}$. This formalism drew
widespread attention from the signal processing community when Candès, Li, and
Soltanolkotabi used it to analyze the phase retrieval problem in CandesLS:15 .
The Wirtinger derivative $\frac{\partial
f}{\partial\overline{z}_{i}}\colon\mathbb{C}^{n}\to\mathbb{C}$ is defined by

$\frac{\partial
f}{\partial\overline{z}_{i}}:=\frac{1}{2}\bigg{(}\frac{\partial(f\circ
g)}{\partial x_{i}}+\mathrm{i}\frac{\partial(f\circ g)}{\partial
y_{i}}\bigg{)}\circ g^{-1},$

which in turn determines the Wirtinger gradient $\frac{\partial
f}{\partial\overline{z}}\colon\mathbb{C}^{n}\to\mathbb{C}^{n}$ defined by

$\frac{\partial f}{\partial\overline{z}}:=\bigg{(}\frac{\partial
f}{\partial\overline{z}_{1}},\ldots,\frac{\partial
f}{\partial\overline{z}_{n}}\bigg{)}.$

For example, given $f\colon\mathbb{C}^{n}\to\mathbb{R}$ defined by
$f(z):=\|z\|^{2}$, we have $(f\circ
g)(x,y)=\sum_{j\in[n]}(x_{j}^{2}+y_{j}^{2})$, and so $\frac{\partial(f\circ
g)}{\partial x_{i}}(x,y)=2x_{i}$, $\frac{\partial(f\circ g)}{\partial
y_{i}}(x,y)=2y_{i}$, and $\frac{\partial f}{\partial\overline{z}}(z)=z$. Next,
the gradient $\nabla f\colon\mathbb{C}^{n}\to\mathbb{C}^{n}$ is defined in
terms of the best real linear approximation of $f$ at the input $z$:

$f(z+\Delta z)=f(z)+\operatorname{Re}\langle\nabla f(z),\Delta
z\rangle+\text{h.o.t.}$

Since $g$ is a real linear isometry, it follows that $\nabla(f\circ
g)=g^{-1}\circ\nabla f\circ g$, meaning $\nabla f$ is the notion of gradient
that is relevant to optimization. One may verify that

$\nabla f=2\frac{\partial f}{\partial\overline{z}}.$

For example, for $f(z):=\|z\|^{2}$, we have $\nabla f(z)=2z$, which nicely
resembles the gradient expression when $z$ is real. The most attractive aspect
of Wirtinger derivatives is that if $f(z)$ is a polynomial in $z$ and
$\overline{z}$, then $\frac{\partial f}{\partial\overline{z}_{i}}(z)$ may be
computed by formally differentiating in $\overline{z}_{i}$ as if everything
else (including $z_{i}$) is constant. For example, for
$f(z):=\|z\|^{2}=\sum_{j\in[n]}z_{j}\overline{z}_{j}$, we have $\frac{\partial
f}{\partial\overline{z}_{i}}(z)=z_{i}$, matching our previous calculation.

###### Lemma 1

Consider the extension $\operatorname{EP}\colon\mathbb{C}^{d\times
n}\to\mathbb{R}$ of the frame potential defined by
$\operatorname{EP}(Z):=\|Z^{*}Z\|_{F}^{2}$, and for each $j\in[n]$, consider
the constraint $h_{j}\colon\mathbb{C}^{d\times n}\to\mathbb{R}$ defined by
$h_{j}(Z):=\|Ze_{j}\|^{2}-1$, where $e_{j}$ denotes the $j$th standard basis
element in $\mathbb{C}^{n}$. Then for $Z=[z_{1}\cdots z_{n}]$, it holds that

$\nabla\operatorname{EP}(Z)=4ZZ^{*}Z,\qquad\nabla h_{j}(Z)=2z_{j}e_{j}^{*}.$

###### Proof

For $\operatorname{EP}$, we expand

$\displaystyle\operatorname{EP}(Z)=\|Z^{*}Z\|_{F}^{2}=\sum_{i\in[n]}\sum_{j\in[n]}|(Z^{*}Z)_{ij}|^{2}$
$\displaystyle=\sum_{i\in[n]}\sum_{j\in[n]}\sum_{k\in[d]}\sum_{\ell\in[d]}\overline{Z}_{ki}Z_{kj}Z_{\ell
i}\overline{Z}_{\ell j},$

and formally differentiate

$\displaystyle\frac{\partial\operatorname{EP}}{\partial\overline{Z}_{pq}}(Z)$
$\displaystyle=\sum_{i\in[n]}\sum_{j\in[n]}\sum_{k\in[d]}\sum_{\ell\in[d]}(\delta_{pk}\delta_{qi}Z_{kj}Z_{\ell
i}\overline{Z}_{\ell j}+\overline{Z}_{ki}Z_{kj}Z_{\ell
i}\delta_{p\ell}\delta_{qj})$
$\displaystyle=\sum_{j\in[n]}\sum_{\ell\in[d]}Z_{pj}Z_{\ell
q}\overline{Z}_{\ell
j}+\sum_{i\in[n]}\sum_{k\in[d]}\overline{Z}_{ki}Z_{kq}Z_{pi}$
$\displaystyle=\sum_{j\in[n]}\sum_{\ell\in[d]}Z_{pj}(Z^{*})_{j\ell}Z_{\ell
q}+\sum_{i\in[n]}\sum_{k\in[d]}Z_{pi}(Z^{*})_{ik}Z_{kq}=(2ZZ^{*}Z)_{pq},$

and then double the Wirtinger gradient to obtain the gradient. For $h_{j}$, we
similarly expand and formally differentiate

$h_{j}(Z)=\sum_{i\in[d]}Z_{ij}\overline{Z}_{ij}-1,\quad\frac{\partial
h_{j}}{\partial\overline{Z}_{pq}}(Z)=\sum_{i\in[d]}Z_{ij}\delta_{pi}\delta_{qj}=Z_{pj}\delta_{qj}=(z_{j}e_{j}^{*})_{pq},$

and then double the Wirtinger gradient to obtain the gradient. ∎

###### Proposition 4

For each $d,n\in\mathbb{N}$, the first-order critical points of the frame
potential $\operatorname{FP}\colon\operatorname{S}(d,n)\to\mathbb{R}$ are
precisely the matrices $Z\in\operatorname{S}(d,n)$ for which every column of
$Z$ is an eigenvector of $ZZ^{*}$.

###### Proof

For every $Z\in\operatorname{S}(d,n)$, it holds that $\\{\nabla
h_{j}(Z)\\}_{j\in[n]}$ are nonzero matrices with disjoint support, meaning
they are linearly independent. Thus, $Z$ is a first-order critical point if
and only if $\nabla\operatorname{EP}(Z)$ is in the span of $\\{\nabla
h_{j}(Z)\\}_{j\in[n]}$, if and only if for every $j\in[n]$, the $j$th column
of $\nabla\operatorname{EP}(Z)$ is a scalar multiple of the $j$th column of
$\nabla h_{j}(Z)$. The result follows.

Now that we have characterized the first-order critical points of the frame
potential, we isolate a useful feature of the non-minimizing critical points.
To enunciate this feature, we need another definition: We say
$Z\in\operatorname{S}(d,n)$ is a unit norm tight frame for a subspace
$S\subseteq\mathbb{C}^{d}$ if $ZZ^{*}$ is a multiple of the orthogonal
projection matrix with image $S$. In particular, the multiple equals
$n/\operatorname{dim}(S)$ by a trace argument.

###### Corollary 1

Fix $d,n\in\mathbb{N}$, and suppose $Z\in\operatorname{S}(d,n)$ is a first-
order critical point of the frame potential
$\operatorname{FP}\colon\operatorname{S}(d,n)\to\mathbb{R}$ that is not a
global minimizer. Then there exist $d^{\prime},n^{\prime}\in\mathbb{N}$
satisfying

$d^{\prime}<d,\qquad\frac{n^{\prime}}{d^{\prime}}>\max\bigg{\\{}\frac{n}{d},1\bigg{\\}}$

such that the maximum eigenvalue of $ZZ^{*}$ has a $d^{\prime}$-dimensional
eigenspace $E$ that contains exactly $n^{\prime}$ columns of $Z$, which in
turn form a unit norm tight frame for $E$.

###### Proof

Let $\Lambda$ denote the set of eigenvalues of $ZZ^{*}$, and let $d^{\prime}$
denote the multiplicity of $\max\Lambda$ as an eigenvalue of $ZZ^{*}$. Since
$Z$ is not a unit norm tight frame by assumption, it holds that
$d^{\prime}<d$. Next, let $P_{\lambda}$ denote the orthogonal projection
matrix onto the eigenspace of $ZZ^{*}$ corresponding to $\lambda\in\Lambda$.
Then the spectral theorem gives

$\sum_{\lambda\in\Lambda}\lambda\frac{\operatorname{tr}P_{\lambda}}{d}=\frac{1}{d}\operatorname{tr}\bigg{(}\sum_{\lambda\in\Lambda}\lambda
P_{\lambda}\bigg{)}=\frac{1}{d}\operatorname{tr}(ZZ^{*})=\frac{1}{d}\operatorname{tr}(Z^{*}Z)=\frac{n}{d},$

i.e., $\frac{n}{d}$ is a weighted average of the eigenvalues of $ZZ^{*}$.
Since $Z$ is not a unit norm tight frame by assumption, there are at least two
distinct eigenvalues, and so $\max\Lambda>\frac{n}{d}$. In the case where
$n<d$, the analogous weighted average of eigenvalues of $Z^{*}Z$ equals $1$,
and so $\max\Lambda>1$ follows from the assumption that the columns of $Z$ are
not orthonormal. Overall, we have

$d^{\prime}<d,\qquad\max\Lambda>\max\bigg{\\{}\frac{n}{d},1\bigg{\\}}.$

Next, consider the assignment $f\colon[n]\to\Lambda$ afforded by Proposition
4. Then the spectral theorem gives

$\sum_{\lambda\in\Lambda}\sum_{j\in
f^{-1}(\lambda)}z_{j}z_{j}^{*}=\sum_{j\in[n]}z_{j}z_{j}^{*}=ZZ^{*}=\sum_{\lambda\in\Lambda}\lambda
P_{\lambda}.$

Since distinct eigenspaces are orthogonal, then for each
$\lambda^{\prime}\in\Lambda$, we have

$\sum_{j\in
f^{-1}(\lambda^{\prime})}z_{j}z_{j}^{*}=P_{\lambda^{\prime}}\sum_{\lambda\in\Lambda}\sum_{j\in
f^{-1}(\lambda)}z_{j}z_{j}^{*}=P_{\lambda^{\prime}}\sum_{\lambda\in\Lambda}\lambda
P_{\lambda}=\lambda^{\prime}P_{\lambda^{\prime}},$

i.e., $\\{z_{j}z_{j}^{*}\\}_{j\in f^{-1}(\lambda^{\prime})}$ is a unit norm
tight frame for the eigenspace corresponding to $\lambda^{\prime}$. Taking
$n^{\prime}:=|f^{-1}(\max\Lambda)|$ gives
$\max\Lambda=\frac{n^{\prime}}{d^{\prime}}$, and the result follows. ∎

Next, we will characterize the second-order critical points of the frame
potential. To accomplish this, we first recall how Wirtinger calculus captures
Hessian information. The Wirtinger derivative $\frac{\partial f}{\partial
z_{i}}\colon\mathbb{C}^{n}\to\mathbb{C}$ is defined by

$\frac{\partial f}{\partial z_{i}}:=\frac{1}{2}\bigg{(}\frac{\partial(f\circ
g)}{\partial x_{i}}-\mathrm{i}\frac{\partial(f\circ g)}{\partial
y_{i}}\bigg{)}\circ g^{-1},$

and the Wirtinger Hessians $\frac{\partial^{2}f}{\partial
z^{2}},\frac{\partial^{2}f}{\partial
z\partial\overline{z}},\frac{\partial^{2}f}{\partial\overline{z}\partial
z},\frac{\partial^{2}f}{\partial\overline{z}^{2}}\colon\mathbb{C}^{n}\to\mathbb{C}^{n\times
n}$ are defined by

$\bigg{(}\frac{\partial^{2}f}{\partial
z\partial\overline{z}}(u)\bigg{)}_{ij}:=\frac{\partial}{\partial
z_{i}}\bigg{(}\frac{\partial f}{\partial\overline{z}_{j}}\bigg{)}(u),$

and similarly for the other Wirtinger Hessians. The following result is
essentially contained in the monograph Kreutz:09 , but the proof is
straightforward, so we provide a sketch for completeness:

###### Proposition 5

Define $g\colon\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{C}^{n}$ by
$g(x,y):=x+\mathrm{i}y$ and
$h\colon\mathbb{C}^{n}\to\mathbb{C}^{n}\times\mathbb{C}^{n}$ by
$h(z):=(z,\overline{z})$. Consider any $f\colon\mathbb{C}^{n}\to\mathbb{R}$
such that $f\circ g$ is twice continuously differentiable. For every
$u,v\in\mathbb{R}^{n}\times\mathbb{R}^{n}$, it holds that

$v^{\top}\nabla^{2}(f\circ g)(u)v=(h\circ
g)(v)^{\top}\left[\begin{array}[]{cc}\frac{\partial^{2}f}{\partial
z^{2}}(g(u))&\frac{\partial^{2}f}{\partial z\partial\overline{z}}(g(u))\\\
\frac{\partial^{2}f}{\partial\overline{z}\partial
z}(g(u))&\frac{\partial^{2}f}{\partial\overline{z}^{2}}(g(u))\end{array}\right](h\circ
g)(v).$

######  Proof (sketch)

By a standard abuse of notation, we write

$\frac{\partial}{\partial x_{i}}=\frac{\partial}{\partial
z_{i}}+\frac{\partial}{\partial\overline{z}_{i}},\qquad\frac{\partial}{\partial
y_{i}}=\mathrm{i}\bigg{(}\frac{\partial}{\partial
z_{i}}-\frac{\partial}{\partial\overline{z}_{i}}\bigg{)}.$

Then, for example,

$\frac{\partial^{2}}{\partial y_{i}\partial
x_{j}}=\mathrm{i}\bigg{(}\frac{\partial}{\partial
z_{i}}-\frac{\partial}{\partial\overline{z}_{i}}\bigg{)}\bigg{(}\frac{\partial}{\partial
z_{j}}+\frac{\partial}{\partial\overline{z}_{j}}\bigg{)},$

and expanding implies

$\nabla_{yx}^{2}=\mathrm{i}\bigg{(}\frac{\partial^{2}}{\partial
z^{2}}+\frac{\partial^{2}}{\partial
z\partial\overline{z}}-\frac{\partial^{2}}{\partial\overline{z}\partial
z}-\frac{\partial^{2}}{\partial\overline{z}^{2}}\bigg{)}.$

The same approach delivers similar expressions for $\nabla_{xx}^{2}$,
$\nabla_{xy}^{2}$, and $\nabla_{yy}^{2}$. The desired result can then be
verified by expanding the left-hand side in terms of these expressions and
rearranging to obtain the right-hand side. ∎

Proposition 5 will help us identify the second-order critical points of the
frame potential. To this end, the following lemma summarizes a straightforward
calculation:

###### Lemma 2

For $\operatorname{EP},h_{1},\ldots,h_{n}\colon\mathbb{C}^{d\times
n}\to\mathbb{R}$ defined in Lemma 1, it holds that

$\frac{\partial^{2}\operatorname{EP}}{\partial Z_{pq}\partial
Z_{rs}}(Z)=2\overline{Z}_{rq}\overline{Z}_{ps},\qquad\frac{\partial^{2}\operatorname{EP}}{\partial\overline{Z}_{pq}\partial\overline{Z}_{rs}}(Z)=2Z_{rq}Z_{ps},$
$\frac{\partial^{2}\operatorname{EP}}{\partial
Z_{pq}\partial\overline{Z}_{rs}}(Z)=2\bigg{(}\delta_{pr}(Z^{*}Z)_{qs}+\delta_{qs}(ZZ^{*})_{rp}\bigg{)},$
$\frac{\partial^{2}h_{j}}{\partial Z_{pq}\partial
Z_{rs}}(Z)=\frac{\partial^{2}h_{j}}{\partial\overline{Z}_{pq}\partial\overline{Z}_{rs}}(Z)=0,\qquad\frac{\partial^{2}h_{j}}{\partial
Z_{pq}\partial\overline{Z}_{rs}}(Z)=\delta_{pr}\delta_{qj}\delta_{sj}.$

We may now prove the main result of this section:

######  Proof (of Theorem 2.1)

One containment is trivial: every global minimizer is necessarily a local
minimizer, which is necessarily a second-order critical point. For the other
direction, we show that every point $Z\in\operatorname{S}(d,n)$ that is not a
global minimizer is necessarily not a second-order critical point. Since every
second-order critical point is necessarily a first-order critical point, the
implication trivially holds if $Z$ is not a first-order critical point.
Suppose $Z=[z_{1}\cdots z_{n}]$ is a first-order critical point that is not a
global minimizer. The corresponding Lagrange multipliers
$\\{\lambda_{j}\\}_{j\in[n]}$ are the unique solution to

$ZZ^{*}z_{j}=-\frac{1}{2}\lambda_{j}z_{j}$

for $j\in[n]$. Consider the map $L\colon\mathbb{C}^{d\times n}\to\mathbb{R}$
defined by

$L(Z):=\operatorname{EP}(Z)+\sum_{j\in[n]}\lambda_{j}h_{j}(Z).$

To show that $Z$ is not a second-order critical point, Proposition 5 gives
that it suffices to find $Y\in\mathbb{C}^{d\times n}$ such that

$\operatorname{Re}\operatorname{tr}(\nabla h_{j}(Z)^{*}Y)=0$

for every $j\in[n]$ and

$\displaystyle\sum_{p\in[d]}\sum_{q\in[n]}\sum_{r\in[d]}\sum_{s\in[n]}\bigg{(}\frac{\partial^{2}L(Z)}{\partial
Z_{pq}\partial Z_{rs}}Y_{pq}Y_{rs}+\frac{\partial^{2}L(Z)}{\partial
Z_{pq}\partial\overline{Z}_{rs}}Y_{pq}\overline{Y}_{rs}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\frac{\partial^{2}L(Z)}{\partial\overline{Z}_{pq}\partial
Z_{rs}}\overline{Y}_{pq}Y_{rs}+\frac{\partial^{2}L(Z)}{\partial\overline{Z}_{pq}\partial\overline{Z}_{rs}}\overline{Y}_{pq}\overline{Y}_{rs}\bigg{)}<0.$

To enunciate our choice for $Y$, let $J\subseteq[n]$ denote the index set of
the $n^{\prime}$ columns of $Z$ implicated by Corollary 1. Then Corollary 1
gives that $\\{z_{j}\\}_{j\in J}$ spans a $d^{\prime}$-dimensional eigenspace
$E$ of $ZZ^{*}$ with $d^{\prime}<\min\\{d,n^{\prime}\\}$. Since
$d^{\prime}<d$, there is another eigenspace $E^{\prime}$ of $ZZ^{*}$, and we
select $u\in E^{\prime}$ with $\|u\|^{2}=1$. Since $d^{\prime}<n^{\prime}$,
then $\\{z_{j}\\}_{j\in J}$ is linearly dependent, and we select
$v\in\mathbb{C}^{n}$ with support contained in $J$ such that $Zv=0$ and
$\|v\|^{2}=1$. We take $Y:=uv^{*}$, and it remains to verify that $Y$
satisfies the above conditions. First, Lemma 1 gives

$\operatorname{Re}\operatorname{tr}(\nabla
h_{j}(Z)^{*}Y)=\operatorname{Re}\operatorname{tr}(2e_{j}z_{j}^{*}uv^{*})=2\operatorname{Re}(v^{*}e_{j}z_{j}^{*}u)=0,$

where the last step follows from the facts that $v^{*}e_{j}=0$ whenever
$j\not\in J$ and $z_{j}^{*}u=0$ whenever $j\in J$. Next, Lemma 2 gives

$\displaystyle\sum_{p\in[d]}\sum_{q\in[n]}\sum_{r\in[d]}\sum_{s\in[n]}\frac{\partial^{2}\operatorname{EP}(Z)}{\partial
Z_{pq}\partial Z_{rs}}Y_{pq}Y_{rs}$
$\displaystyle=\sum_{p\in[d]}\sum_{q\in[n]}\sum_{r\in[d]}\sum_{s\in[n]}2\overline{Z}_{rq}\overline{Z}_{ps}u_{p}\overline{v}_{q}u_{r}\overline{v}_{s}$
$\displaystyle=2(v^{*}Z^{*}u)^{2}=0,$

where the last step follows from the fact that $v$ and $Z^{*}u$ have disjoint
support. We similarly have

$\sum_{p\in[d]}\sum_{q\in[n]}\sum_{r\in[d]}\sum_{s\in[n]}\frac{\partial^{2}\operatorname{EP}(Z)}{\partial\overline{Z}_{pq}\partial\overline{Z}_{rs}}\overline{Y}_{pq}\overline{Y}_{rs}=0.$

Considering Lemma 2, it remains to show that

$\operatorname{Re}\bigg{(}\sum_{p\in[d]}\sum_{q\in[n]}\sum_{r\in[d]}\sum_{s\in[n]}\frac{\partial^{2}L(Z)}{\partial
Z_{pq}\partial\overline{Z}_{rs}}Y_{pq}\overline{Y}_{rs}\bigg{)}<0.$ (2)

To this end, let $\alpha$ and $\beta$ denote the eigenvalues corresponding to
eigenspaces $E$ and $E^{\prime}$, respectively. Notably, Corollary 1 gives
$\alpha>\beta$, and this will be the source of our desired inequality. Lemma 2
gives

$\displaystyle\sum_{p\in[d]}\sum_{q\in[n]}\sum_{r\in[d]}\sum_{s\in[n]}\frac{\partial^{2}\operatorname{EP}(Z)}{\partial
Z_{pq}\partial\overline{Z}_{rs}}Y_{pq}\overline{Y}_{rs}$
$\displaystyle=\sum_{p\in[d]}\sum_{q\in[n]}\sum_{r\in[d]}\sum_{s\in[n]}2\bigg{(}\delta_{pr}(Z^{*}Z)_{qs}+\delta_{qs}(ZZ^{*})_{rp}\bigg{)}u_{p}\overline{v}_{q}\overline{u}_{r}v_{s}$
$\displaystyle=2\sum_{q\in[n]}\sum_{r\in[d]}\sum_{s\in[n]}(Z^{*}Z)_{qs}u_{r}\overline{v}_{q}\overline{u}_{r}v_{s}+2\sum_{p\in[d]}\sum_{r\in[d]}\sum_{s\in[n]}(ZZ^{*})_{rp}u_{p}\overline{v}_{s}\overline{u}_{r}v_{s}$
$\displaystyle=2\|Zv\|^{2}\|u\|^{2}+2\|Z^{*}u\|^{2}\|v\|^{2}=2\beta,$ (3)

where the last step uses the facts that $Zv=0$, $\|v\|^{2}=1$, and
$\|Z^{*}u\|^{2}=u^{*}ZZ^{*}u=\beta$. Next, Lemma 2 gives

$\displaystyle\sum_{p\in[d]}\sum_{q\in[n]}\sum_{r\in[d]}\sum_{s\in[n]}\frac{\partial^{2}h_{j}(Z)}{\partial
Z_{pq}\partial\overline{Z}_{rs}}Y_{pq}\overline{Y}_{rs}$
$\displaystyle=\sum_{p\in[d]}\sum_{q\in[n]}\sum_{r\in[d]}\sum_{s\in[n]}\delta_{pr}\delta_{qj}\delta_{sj}u_{p}\overline{v}_{q}\overline{u}_{r}v_{s}$
$\displaystyle=\|u\|^{2}|v_{j}|^{2}=|v_{j}|^{2}.$ (4)

We combine (3) and (4) to get

$\sum_{p\in[d]}\sum_{q\in[n]}\sum_{r\in[d]}\sum_{s\in[n]}\frac{\partial^{2}L(Z)}{\partial
Z_{pq}\partial\overline{Z}_{rs}}Y_{pq}\overline{Y}_{rs}=2\beta+\sum_{j\in[n]}\lambda_{j}|v_{j}|^{2}=2(\beta-\alpha)<0,$

where the last equality follows from the facts that $\lambda_{j}=-2\alpha$
whenever $j\in J$, $v_{j}=0$ whenever $j\not\in J$, and $\|v\|^{2}=1$. This
implies the desired inequality (2). ∎

##  3 Convexity and topology

Given a self-adjoint matrix $A\in\mathbb{C}^{d\times d}$, let
$\lambda(A)\in\mathbb{R}^{d}$ denote the vector of eigenvalues of $A$ sorted
in monotonically decreasing order. This section proves the following
strengthening of the Benedetto–Fickus theorem:

###### Theorem 3.1

Given a convex function $f\colon\mathbb{R}^{d}\to\mathbb{R}$, consider
$\operatorname{FP}_{f}\colon\operatorname{S}(d,n)\to\mathbb{R}$ defined by
$\operatorname{FP}_{f}(Z):=f(\lambda(ZZ^{*}))$. Then $\operatorname{FP}_{f}$
has no spurious local minimizers.

Notably, if $f(\cdot)=\|\cdot\|^{2}$, then $\operatorname{FP}_{f}$ is the
frame potential. We will prove Theorem 3.1 using ideas from both convexity and
topology. Recall that given topological spaces $(X,\tau_{X})$ and
$(Y,\tau_{Y})$, we say $f\colon X\to Y$ is open at $x\in X$ if for every
$U\in\tau_{X}$ such that $x\in U$, there exists $V\in\tau_{Y}$ such that
$f(x)\in V\subseteq f(U)$. Furthermore, $f$ is open if for every $x\in X$, $f$
is open at $x$. Throughout this section, $\operatorname{int}(U)$ and
$\operatorname{cl}(U)$ denote the topological interior and closure of $U$,
respectively. The following observation appears to originate with the recent
preprint NouiehedR:18 :

###### Lemma 3

Given a topological space $X$, a convex subset $Y$ of a real topological
vector space, an open map $g\colon X\to Y$ in the subspace topology for $Y$,
and a convex function $f\colon Y\to\mathbb{R}$, it holds that $f\circ g$ has
no spurious local minimizers.

###### Proof

Suppose $x_{0}\in X$ is a local minimizer of $f\circ g$, i.e., there exists
$U\in\tau_{X}$ with $x_{0}\in U$ such that $(f\circ g)(x)\geq(f\circ
g)(x_{0})$ for every $x\in U$. Since $g$ is open, it follows that
$V:=g(U)\in\tau_{Y}$ contains $g(x_{0})$ and has the property that $f(y)\geq
f(g(x_{0}))$ for every $y\in V$. That is, $g(x_{0})$ is a local minimizer of
$f$. Since $f$ is convex, $g(x_{0})$ is necessarily a global minimizer of $f$,
meaning $f(y)\geq f(g(x_{0}))$ for every $y\in Y$. Then $f(g(x))\geq
f(g(x_{0}))$ for every $x\in X$, i.e., $x_{0}$ is a global minimizer of
$f\circ g$.

As such, to prove Theorem 3.1, it suffices to factor $\operatorname{FP}_{f}$
as the composition of an open map with a convex function. To proceed, we need
a definition. Given $x,y\in\mathbb{R}^{d}$, we say $x$ interlaces $y$ and
write $x\sqsupseteq y$ if

$x_{1}\geq y_{1}\geq x_{2}\geq y_{2}\geq\cdots\geq x_{d}\geq y_{d}.$

Let $\operatorname{E}(d,n)$ denote the polytope of matrices
$[\lambda_{1}\cdots\lambda_{n}]\in\mathbb{R}^{d\times n}$ such that

$\lambda_{n}\sqsupseteq\cdots\sqsupseteq\lambda_{1}\sqsupseteq
0\qquad\text{and}\qquad\sum_{i\in[d]}(\lambda_{j})_{i}=j\quad\forall j\in[n].$

For example, one may verify that

$\frac{1}{4}\left[\begin{array}[]{ccccc}4&6&7&8&9\\\ 0&2&4&6&7\\\
0&0&1&2&4\end{array}\right]\in\operatorname{E}(3,5).$

Throughout this section, we use $\Lambda$ to denote a member of
$\operatorname{E}(d,n)$.

Define $\Sigma\colon\operatorname{S}(d,n)\to\operatorname{E}(d,n)$ so that the
$j$th column of $\Sigma([z_{1}\cdots z_{n}])$ is the vector of eigenvalues of
$\sum_{i\in[j]}z_{i}z_{i}^{*}$ sorted in monotonically decreasing order. The
fact that $\Sigma(Z)\in\operatorname{E}(d,n)$ for every
$Z\in\operatorname{S}(d,n)$ essentially follows from Cauchy’s interlacing
theorem and is established in the proof of the ($\Rightarrow$) direction of
Theorem 2 in Cahill:13 . In words, $\Sigma(Z)$ is known as the eigensteps of
$Z$. (We use $\Sigma$ since it looks like an E for eigen and sounds like an S
for steps.) Observe the relationship between eigensteps and Gelfand–Tsetlin
polytopes DeLoeraM:04 . Eigensteps were introduced in Cahill:13  to explicitly
construct the set of unit norm tight frames, and were later used to establish
the path-connectivity of this set CahillMS:17 .

Considering $\lambda(ZZ^{*})=\Sigma(Z)e_{d}$, then $\operatorname{FP}_{f}$ is
the composition of $Z\mapsto\Sigma(Z)$ with the convex function
$\Lambda\mapsto f(\Lambda e_{d})$. Thus, we would like to show that the
eigensteps map $\Sigma$ is open. The following lemma summarizes our approach
to this end:

###### Lemma 4

Given topological spaces $X$ and $Y$, consider a point $x\in X$ and a map
$f\colon X\to Y$ that is continuous at $x$. Suppose there exists
$W\in\tau_{Y}$ such that $f(x)\in W$ and a continuous map $g\colon W\to X$
such that

$(g\circ f)(x)=x\qquad\text{and}\qquad f\circ g=\operatorname{id}_{W}.$

Then $f$ is open at $x$.

###### Proof

Given any $U\in\tau_{X}$ such that $x\in U$, take $V:=g^{-1}(U\cap
f^{-1}(W))$. Then $V\in\tau_{Y}$ since $W$ is open, $f(x)\in W$, $f$ is
continuous at $x$, $U$ is open, and $g$ is continuous. Next, since $x\in U$
and $f(x)\in W$, we have $x\in U\cap f^{-1}(W)$, and so $g(f(x))=(g\circ
f)(x)=x$ gives $f(x)\in g^{-1}(U\cap f^{-1}(W))=V$. It remains to verify that
$V\subseteq f(U)$. Select any $v\in V=g^{-1}(U\cap f^{-1}(W))\subseteq W$.
Then $v=(f\circ g)(v)=f(g(v))\in f(U\cap f^{-1}(W))\subseteq f(U)$. ∎

Due to the nature of the eigensteps map, our construction of $g$ uses a bit of
matrix analysis. We start with a couple of lemmas:

###### Lemma 5

Given a self-adjoint matrix $A\in\mathbb{C}^{d\times d}$ and eigenvalue
$\lambda$ of multiplicity $m$, let $P_{\lambda}\in\mathbb{C}^{d\times d}$
denote orthogonal projection onto the corresponding eigenspace. For
$z\in\mathbb{C}^{d}$ with $P_{\lambda}z\neq 0$, the multiplicity of $\lambda$
as an eigenvalue of $A+zz^{*}$ is $m-1$.

###### Proof

Let $\\{v_{i}\\}_{i\in[n]}$ denote any orthonormal basis of eigenvectors of
$A$ with corresponding eigenvalues $\\{\lambda_{i}\\}_{i\in[n]}$. We apply the
matrix determinant lemma to express the characteristic polynomial of
$A+zz^{*}$:

$\displaystyle p_{A+zz^{*}}(t)=\operatorname{det}((tI-A)-zz^{*})$
$\displaystyle=\operatorname{det}(tI-A)\cdot(1-z^{*}(tI-A)^{-1}z)$
$\displaystyle=\bigg{(}\prod_{i\in[n]}(t-\lambda_{i})\bigg{)}\bigg{(}1-\sum_{j\in[n]}\frac{|\langle
v_{j},z\rangle|^{2}}{t-\lambda_{j}}\bigg{)}$
$\displaystyle=\prod_{i\in[n]}(t-\lambda_{i})-\sum_{j\in[n]}|\langle
v_{j},z\rangle|^{2}\prod_{\begin{subarray}{c}i\in[n]\\\ i\neq
j\end{subarray}}(t-\lambda_{j}).$

We rewrite this expression in terms of the multiplicity $m(\lambda^{\prime})$
of each eigenvalue $\lambda^{\prime}$ of $A$ to find polynomials $q(t)$ and
$r(t)$ with $q(\lambda)\neq 0\neq r(\lambda)$ such that

$\displaystyle p_{A+zz^{*}}(t)$
$\displaystyle=\prod_{\lambda^{\prime}}(t-\lambda^{\prime})^{m(\lambda^{\prime})}-\sum_{\lambda^{\prime}}\|P_{\lambda^{\prime}}z\|^{2}(t-\lambda^{\prime})^{m(\lambda^{\prime})-1}\prod_{\lambda^{\prime\prime}\neq\lambda^{\prime}}(t-\lambda^{\prime\prime})^{m(\lambda^{\prime\prime})}$
$\displaystyle=(t-\lambda)^{m}q(t)-\|P_{\lambda}z\|^{2}(t-\lambda)^{m-1}r(t).$

Since $\|P_{\lambda}z\|^{2}\neq 0$ by assumption, the result follows. ∎

###### Lemma 6

The preimage $\Sigma^{-1}(\operatorname{int}(\operatorname{E}(d,n)))$ is dense
in $\operatorname{S}(d,n)$.

###### Proof

Given arbitrary $Z=[z_{1}\cdots z_{n}]\in\operatorname{S}(d,n)$ and
$\epsilon>0$, we identify a matrix
$\tilde{Z}=[\tilde{z}_{1}\cdots\tilde{z}_{n}]\in\Sigma^{-1}(\operatorname{int}(\operatorname{E}(d,n)))$
such that $\|\tilde{Z}-Z\|_{F}<\epsilon$. Put $\tilde{z}_{1}:=z_{1}$, and then
for each $j\in[n-1]$, select any unit vector $\tilde{z}_{j+1}$ in the open
$2$-ball of radius $\epsilon/n$ about $z_{j+1}$ that avoids the orthogonal
complements of the eigenspaces of
$\sum_{i\in[j]}\tilde{z}_{i}\tilde{z}_{i}^{*}$. Then
$\Sigma(\tilde{Z})\in\operatorname{int}(\operatorname{E}(d,n))$ by Lemma 5 and
$\|\tilde{Z}-Z\|_{F}<\epsilon$ by the triangle inequality. ∎

Let $\mathbb{T}^{k}\times\operatorname{U}(d-k)$ denote the group of matrices
in $\operatorname{U}(d)$ of the form

$\left[\begin{array}[]{cc}D&0\\\ 0&U\end{array}\right]$

with diagonal $D\in\operatorname{U}(k)$, and consider the following group of
$n$-tuples of matrices:

$\operatorname{U}(d,n):=\operatorname{U}(d)\times(\mathbb{T}^{1}\times\operatorname{U}(d-1))\times\cdots\times(\mathbb{T}^{d-1}\times\operatorname{U}(1))\times(\mathbb{T}^{d})^{n-d}.$

(We use $\operatorname{U}$ since each component of the tuple is unitary.) The
following lemma describes how this set allows us to map back from
$\operatorname{E}(d,n)$ to $\operatorname{S}(d,n)$. This portion of our proof
appears to break in the real case since the orthogonal group
$\operatorname{O}(1)$ is discrete; it would be interesting if the real case
could be treated differently.

###### Lemma 7

There exists a continuous map
$\Phi\colon\operatorname{E}(d,n)\times\operatorname{U}(d,n)\to\operatorname{S}(d,n)$
such that

  * (a)

for every $Z\in\operatorname{S}(d,n)$, there exists
$U\in\operatorname{U}(d,n)$ such that $\Phi(\Sigma(Z),U)=Z$, and

  * (b)

for every $\Lambda\in\operatorname{E}(d,n)$ and $U\in\operatorname{U}(d,n)$,
it holds that $\Sigma(\Phi(\Lambda,U))=\Lambda$.

###### Proof

We will define $\Phi$ explicitly on
$\operatorname{int}(\operatorname{E}(d,n))\times\operatorname{U}(d,n)$ and
then extend continuously to the full domain. For each
$\Lambda=[\lambda_{1}\cdots\lambda_{n}]\in\operatorname{int}(\operatorname{E}(d,n))$
and $j\in[n]$, denote $r_{j}:=\min\\{j+1,d\\}$, define
$v_{j},w_{j}\in\mathbb{R}^{r_{j}}$ by

$(v_{j})_{i}:=\sqrt{-\frac{\displaystyle\prod_{i^{\prime}\in[r_{j}]}((\lambda_{j})_{i}-(\lambda_{j+1})_{i^{\prime}})}{\displaystyle\prod_{\begin{subarray}{c}i^{\prime}\in[r_{j}]\\\
i^{\prime}\neq
i\end{subarray}}((\lambda_{j})_{i}-(\lambda_{j})_{i^{\prime}})}},\quad(w_{j})_{i}:=\sqrt{\frac{\displaystyle\prod_{i^{\prime}\in[r_{j}]}((\lambda_{j+1})_{i}-(\lambda_{j})_{i^{\prime}})}{\displaystyle\prod_{\begin{subarray}{c}i^{\prime}\in[r_{j}]\\\
i^{\prime}\neq
i\end{subarray}}((\lambda_{j+1})_{i}-(\lambda_{j+1})_{i^{\prime}})}},$

and define $W_{j}\in\operatorname{U}(r_{j})$ by

$(W_{j})_{i,i^{\prime}}:=\frac{(v_{j})_{i}(w_{j})_{i^{\prime}}}{(\lambda_{j+1})_{i^{\prime}}-(\lambda_{j})_{i}}.$

By Theorem 7 in Cahill:13 , $\Sigma^{-1}(\Lambda)$ equals the set of matrices
$[z_{1}\cdots z_{n}]\in\mathbb{C}^{d\times n}$ obtained from all
$(U_{1},V_{1},\ldots,V_{n-1})\in\operatorname{U}(d,n)$ by taking
$z_{1}:=U_{1}e_{1}$ and applying the following iteration for $j\in[n-1]$:

$z_{j+1}:=U_{j}V_{j}\left[\begin{array}[]{c}v_{j}\\\
0\end{array}\right],\qquad
U_{j+1}:=U_{j}V_{j}\left[\begin{array}[]{cc}W_{j}&0\\\
0&I_{d-r_{j}}\end{array}\right].$

This process defines a map
$\Phi_{0}\colon\operatorname{int}(\operatorname{E}(d,n))\times\operatorname{U}(d,n)\to\operatorname{S}(d,n)$
such that

  * (a’)

for every $Z\in\Sigma^{-1}(\operatorname{int}(\operatorname{E}(d,n)))$, there
exists $U\in\operatorname{U}(d,n)$ such that $\Phi_{0}(\Sigma(Z),U)=Z$, and

  * (b’)

for every $\Lambda\in\operatorname{int}(\operatorname{E}(d,n))$ and
$U\in\operatorname{U}(d,n)$, it holds that
$\Sigma(\Phi_{0}(\Lambda,U))=\Lambda$.

Observe that
$R\colon\Lambda\mapsto(\\{(v_{j})_{i}^{2}\\}_{i,j},\\{(W_{j})_{i,i^{\prime}}^{2}\\}_{j,i,i^{\prime}})$
is a rational map with bounded domain, and furthermore, $R$ is bounded since
$\Phi_{0}$ is bounded. It follows that $R$ is Lipschitz, and so $\Phi_{0}$ is
uniformly continuous since $\Phi_{0}(\Lambda,U)$ is a polynomial in
$(\sqrt{R(\Lambda)},U)$ with bounded domain. In addition,
$\operatorname{S}(d,n)$ is a complete metric space, and so there exists a
unique continuous extension $\Phi$ of $\Phi_{0}$ to all of
$\operatorname{E}(d,n)\times\operatorname{U}(d,n)$.

We now verify properties (a) and (b). For (a), take any
$Z\in\operatorname{S}(d,n)$. By Lemma 6, there exists a sequence
$\\{Z_{i}\\}_{i=1}^{\infty}$ in $\operatorname{S}(d,n)$ such that
$\Sigma(Z_{i})\in\operatorname{int}(\operatorname{E}(d,n))$ for every $i$ and
$Z_{i}\to Z$. By (a’), it holds that for each $i$, there exists
$U_{i}\in\operatorname{U}(d,n)$ such that $\Phi(\Sigma(Z_{i}),U_{i})=Z_{i}$.
By compactness, there is a subsequence of $\\{U_{i}\\}_{i=1}^{\infty}$ with
some limit point $U\in\operatorname{U}(d,n)$. By continuity, it follows that
$\Phi(\Sigma(Z),U)=Z$. For (b), take any $\Lambda\in\operatorname{E}(d,n)$ and
$U\in\operatorname{U}(d,n)$. There exists a sequence
$\\{\Lambda_{i}\\}_{i=1}^{\infty}$ in
$\operatorname{int}(\operatorname{E}(d,n))$ such that $\Lambda_{i}\to\Lambda$.
By (b’), it holds that $\Sigma(\Phi(\Lambda_{i},U))=\Lambda_{i}$ for each $i$.
Since $\Sigma$ and $\Phi$ are continuous, taking limits of both sides gives
$\Sigma(\Phi(\Lambda,U))=\Lambda$.

###### Lemma 8

The eigensteps map $\Sigma\colon\operatorname{S}(d,n)\to\operatorname{E}(d,n)$
is open in the standard subspace topologies for
$\operatorname{S}(d,n)\subseteq\mathbb{C}^{d\times n}$ and
$\operatorname{E}(d,n)\subseteq\mathbb{R}^{d\times n}$.

###### Proof

Take any $Z\in\operatorname{S}(d,n)$. By Lemma 7(a), there exists
$U\in\operatorname{U}(d,n)$ such that $\Phi(\Sigma(Z),U)=Z$. We may apply
Lemma 4 with

$f:=\Sigma,\qquad W:=\operatorname{E}(d,n),\qquad
g\colon\Lambda\mapsto\Phi(\Lambda,U).$

Indeed, Lemma 7(a) gives $(g\circ f)(Z)=\Phi(\Sigma(Z),U)=Z$ and Lemma 7(b)
gives $(f\circ g)(\Lambda)=\Sigma(\Phi(\Lambda,U))=\Lambda$ for every
$\Lambda\in\operatorname{E}(d,n)$. Thus, $\Sigma$ is open at $Z$. Since $Z$
was arbitrary, it follows that $\Sigma$ is open. ∎

We may now prove the main result of this section:

######  Proof (of Theorem 3.1)

By Lemma 8, the eigensteps map $\Sigma$ is open. As such,
$\operatorname{FP}_{f}$ is the composition of the open map $Z\mapsto\Sigma(Z)$
with the convex map $\Lambda\mapsto f(\Lambda e_{d})$, and so the result
follows from Lemma 3. ∎

##  4 Geometric invariant theory

The recent literature showcases an industry of using geometric methods to
prove fundamental results in frame theory NeedhamS:21 ; NeedhamS:arXiv1 ;
NeedhamS:arXiv2 . This section represents another step in this direction. We
say $Z\in\mathbb{C}^{d\times n}$ is full spark if every $d\times d$ submatrix
of $Z$ is invertible AlexeevCM:12 . We will prove the following strengthening
of the Benedetto–Fickus theorem in the special case where $n>d$:

###### Theorem 4.1

Fix $d,n\in\mathbb{N}$ with $n>d$, and consider the gradient flow
$F\colon\operatorname{S}(d,n)\times[0,\infty)\to\operatorname{S}(d,n)$ defined
by the differential equation

$F(Z_{0},0)=Z_{0},\qquad\frac{d}{dt}F(Z_{0},t)=-\operatorname{grad}\operatorname{FP}(F(Z_{0},t)),$

where $\operatorname{grad}$ denotes Riemannian gradient. If $Z_{0}$ is full
spark, then $\lim_{t\to\infty}F(Z_{0},t)$ is a unit norm tight frame.

Given a critical point that is not a unit norm tight frame, almost every
member of any neighborhood of the critical point is full spark, which by
Theorem 4.1 is sent by gradient flow to a unit norm tight frame. The absence
of a basin of attraction indicates that the critical point is not a local
minimizer. Since the frame potential is real analytic (in fact, polynomial),
this can be made rigorous with a standard application of the Łojasiewicz
inequality (à la AbsilK:06 ), thereby proving the Benedetto–Fickus theorem.
Furthermore, Theorem 4.1 easily extends to the real case since the real
submanifold of $\operatorname{S}(d,n)$ is invariant under gradient flow. To
prove Theorem 4.1, we identify a property P such that

  * •

every full spark member of $\operatorname{S}(d,n)$ satisfies P, and

  * •

gradient flow (and its limit) preserves P, but

  * •

no non-minimizing critical point of the frame potential satisfies P.

It turns out that this property originated from the geometric invariant theory
developed by Mumford MumfordFK:94  using ideas from Hilbert Hilbert:93 ; see
Thomas:06  for a survey. We start by introducing a few terms in some level of
generality.

Let $G$ be a reductive group acting linearly on a finite-dimensional complex
vector space $V$, such as $\operatorname{GL}(V)$, $\operatorname{SL}(V)$, or
$\operatorname{O}(V)$. A nonzero vector $v\in V$ is said to be unstable under
the action of $G$ if the closure $\operatorname{cl}(G\cdot v)$ of the orbit of
$v$ contains $0$, and otherwise $v$ is semi-stable under the action of $G$.

###### Proposition 6

Given a nonzero vector $v\in V$ that is semi-stable under the action of $G$,
it holds that every member of $\operatorname{cl}(G\cdot v)$ is also semi-
stable under the action of $G$.

###### Proof

We prove the contrapositive. Suppose there exists a nonzero
$w\in\operatorname{cl}(G\cdot v)$ that is unstable under the action of $G$.
Then there exist sequences $\\{g_{i}\\}_{i=1}^{\infty}$ and
$\\{h_{j}\\}_{j=1}^{\infty}$ in $G$ such that $g_{i}\cdot v\to w$ and
$h_{j}\cdot w\to 0$. For each $j$, the continuity of the linear action of
$h_{j}$ on $V$ implies that

$h_{j}\cdot w=h_{j}\cdot\lim_{i\to\infty}(g_{i}\cdot
v)=\lim_{i\to\infty}(h_{j}g_{i}\cdot v)\in\operatorname{cl}(G\cdot v),$

and so $0=\lim_{j\to\infty}(h_{j}\cdot w)\in\operatorname{cl}(G\cdot v)$.
Thus, $v$ is unstable under the action of $G$. ∎

We are interested in the special case where $G=\operatorname{SL}(d)$ acts on
$V=(\mathbb{C}^{d})^{\otimes n}$ by

$g\cdot(z_{1}\otimes\cdots\otimes z_{n}):=gz_{1}\otimes\cdots\otimes gz_{n}$

and extending linearly. We pass to this space by “tensoring up columns” with
$\tau\colon\mathbb{C}^{d\times n}\to(\mathbb{C}^{d})^{\otimes n}$ defined by
$[z_{1}\cdots z_{n}]\mapsto z_{1}\otimes\cdots\otimes z_{n}$. Then
$Z\in\operatorname{S}(d,n)$ satisfies the previously mentioned property P if
$\tau(Z)$ is semi-stable under the action of $\operatorname{SL}(d)$.

###### Lemma 9

Fix $d,n\in\mathbb{N}$ with $n>d$.

  * (a)

For every full spark $Z\in\operatorname{S}(d,n)$, it holds that $\tau(Z)$ is
semi-stable under the action of $\operatorname{SL}(d)$.

  * (b)

For every non-minimizing critical point $Z\in\operatorname{S}(d,n)$ of the
frame potential, it holds that $\tau(Z)$ is unstable under the action of
$\operatorname{SL}(d)$.

Assuming the lemma for now, we proceed with the proof of Theorem 4.1.

######  Proof (sketch of Theorem 4.1)

For every $Z\in\operatorname{S}(d,n)$, Lemma 1 gives the Euclidean gradient
$\nabla\operatorname{EP}(Z)=4ZZ^{*}Z$, and so the Riemannian gradient
$\operatorname{grad}\operatorname{FP}(Z)$ is the matrix obtained by
normalizing the columns of $ZZ^{*}Z$. Consider the action of
$\operatorname{GL}(d)\times(\mathbb{C}^{\times})^{n}$ on $\mathbb{C}^{d\times
n}$ defined by $(g,\\{a_{i}\\}_{i\in[n]})\cdot
X:=gX\operatorname{diag}\\{a_{i}\\}_{i\in[n]}^{-1}$. Then
$\operatorname{grad}\operatorname{FP}(Z)$ is in the tangent space at $Z$ of
the orbit $(\operatorname{GL}(d)\times(\mathbb{C}^{\times})^{n})\cdot Z$, and
so $F(Z_{0},t)$ resides in one such orbit for all $t\geq 0$. For every
$g\in\operatorname{GL}(d)$, it holds that
$g/\operatorname{det}(g)^{1/d}\in\operatorname{SL}(d)$, and so normalizing
gives

$F(Z_{0},t)\in(\operatorname{SL}(d)\times(\mathbb{C}^{\times})^{n})\cdot
Z_{0}\qquad\forall t\geq 0.$

Next, consider the action of $\operatorname{SL}(d)\times\mathbb{C}^{\times}$
on $(\mathbb{C}^{d})^{\otimes n}$ defined by
$(g,a)\cdot(z_{1}\otimes\cdots\otimes z_{n}):=a(gz_{1}\otimes\cdots\otimes
gz_{n})$ and extending linearly. Then

$\tau(F(Z_{0},t))\in(\operatorname{SL}(d)\times\mathbb{C}^{\times})\cdot\tau(Z_{0})\qquad\forall
t\geq 0.$

Suppose $Z_{0}$ is full spark. Then by Lemma 9(a), $\tau(Z_{0})$ is semi-
stable under the action of $\operatorname{SL}(d)$. Proposition 6 then implies
that every point in
$\operatorname{cl}((\operatorname{SL}(d)\times\mathbb{C}^{\times})\cdot\tau(Z_{0}))$
is also semi-stable under the action of $\operatorname{SL}(d)$. Importantly,
this includes $\tau(Z_{\infty})$, where
$Z_{\infty}:=\lim_{t\to\infty}F(Z_{0},t)$. As the limit point of gradient
flow, $Z_{\infty}$ is a critical point of the frame potential, and by Lemma
9(b), $Z_{\infty}$ is a minimizer. It follows that $Z_{\infty}$ is a unit norm
tight frame. ∎

It remains to prove Lemma 9, which will require some technology from geometric
invariant theory. For the moment, let’s return to the general setting of a
reductive group $G$ acting linearly on a finite-dimensional complex vector
space $V$. To discern whether a point is unstable under the action of $G$, it
is convenient to work with small subgroups of $G$. A one-parameter subgroup of
$G$ is a homomorphism of algebraic groups $\lambda\colon\mathbb{C}^{\times}\to
G$, meaning there exists a decomposition $V=\bigoplus_{i\in I}V_{i}$ and
integer weights $w\colon I\to\mathbb{Z}$ such that for every $i\in I$, $v\in
V_{i}$, and $t\in\mathbb{C}^{\times}$, it holds that

$\lambda(t)\cdot v=t^{w(i)}v.$

Of course, we may conclude that $v\in V\setminus\\{0\\}$ is unstable under the
action of $G$ if there exists a one-parameter subgroup $\lambda$ of $G$ such
that

$\lim_{t\to 0}\lambda(t)\cdot v=0.$

Amazingly, the converse also holds:

######  Proposition 7 (Hilbert–Mumford criterion Hilbert:93 ; MumfordFK:94 )

A nonzero vector $v\in V$ is unstable under the action of $G$ if and only if
there exists a one-parameter subgroup $\lambda$ of $G$ such that

$\lim_{t\to 0}\lambda(t)\cdot v=0.$

######  Proof (of Lemma 9)

(a) Select any one-parameter subgroup
$\lambda\colon\mathbb{C}^{\times}\to\operatorname{SL}(d)$. By the determinant,
the integer weights $w\colon[d]\to\mathbb{Z}$ of $\lambda$ satisfy
$\sum_{i\in[d]}w(i)=0$. Reindex as necessary so that $w(1)\geq\cdots\geq
w(d)$. If $w(i)=0$ for every $i\in[d]$, then $\lambda(t)$ equals the identity
matrix for all $t\in\mathbb{C}^{\times}$, and so $\lim_{t\to
0}\lambda(t)\cdot\tau(Z)=\tau(Z)\neq 0$. Otherwise, $w(1)>0>w(d)$. Put
$D(t):=\operatorname{diag}\\{t^{w(i)}\\}_{i\in[d]}$ and take
$S\in\operatorname{GL}(d)$ such that $\lambda(t)=SD(t)S^{-1}$. Given a full
spark $Z\in\operatorname{S}(d,n)$, put
$\tilde{Z}:=[\tilde{z}_{1}\cdots\tilde{z}_{n}]:=S^{-1}Z$. Then

$\displaystyle S^{-1}\lambda(t)\cdot\tau(Z)$
$\displaystyle=D(t)S^{-1}\cdot(z_{1}\otimes\cdots\otimes z_{n})$
$\displaystyle=D(t)\tilde{z}_{1}\otimes\cdots\otimes D(t)\tilde{z}_{n}$
$\displaystyle=\bigg{(}\sum_{i_{1}\in[d]}t^{w(i_{1})}(\tilde{z}_{1})_{i_{1}}e_{i_{1}}\bigg{)}\otimes\cdots\otimes\bigg{(}\sum_{i_{n}\in[d]}t^{w(i_{n})}(\tilde{z}_{n})_{i_{n}}e_{i_{n}}\bigg{)}$
$\displaystyle=\sum_{i_{1},\ldots,i_{n}\in[d]}\bigg{(}t^{\sum_{j\in[n]}w(i_{j})}\prod_{j\in[n]}(\tilde{z}_{j})_{i_{j}}\bigg{)}e_{i_{1}}\otimes\cdots\otimes
e_{i_{n}}.$ (5)

We will use the fact that $Z$ (and thus $\tilde{Z}$) is full spark to identify
a component of (5) that does not vanish as $t\to 0$. By the Hilbert–Mumford
criterion, this suffices to prove the result.

Let $k$ denote the number of entries in the $d$th row of $\tilde{Z}$ that
equal zero, let $j_{1},\ldots,j_{k}\in[n]$ denote the corresponding column
indices, and index the remaining indices as
$\\{j_{k+1},\ldots,j_{n}\\}:=[n]\setminus\\{j_{1},\ldots,j_{k}\\}$. (In the
degenerate case where $k=0$, we may take $j_{\ell}=\ell$ for each
$\ell\in[n]$.) Writing $\tilde{Z}=[\tilde{z}_{1}\cdots\tilde{z}_{n}]$, then
full spark implies

$0\neq\det([\tilde{z}_{j_{1}}\cdots\tilde{z}_{j_{d}}])=\sum_{\sigma\in
S_{d}}\operatorname{sign}(\sigma)\prod_{t\in[d]}(\tilde{z}_{j_{t}})_{\sigma(t)}.$

It follows that there exists a permutation $\sigma\in S_{d}$ such that

$\prod_{t\in[d]}(\tilde{z}_{j_{t}})_{\sigma(t)}\neq 0.$

In addition, $(\tilde{z}_{j_{t}})_{d}\neq 0$ for every $t>k$ by construction,
and full spark implies $k<d$. As such, we consider the component of (5) with
indices

$i_{j}:=\left\\{\begin{array}[]{cl}\sigma(t)&\text{if $j=j_{t}$, $t\in[d]$}\\\
d&\text{otherwise}\end{array}\right.$

since

$\prod_{j\in[n]}(\tilde{z}_{j})_{i_{j}}=\bigg{(}\prod_{t\in[d]}(\tilde{z}_{j_{t}})_{\sigma(t)}\bigg{)}\bigg{(}\prod_{t\in\\{d+1,\ldots,n\\}}(\tilde{z}_{j_{t}})_{d}\bigg{)}\neq
0$

and

$\sum_{j\in[n]}w(i_{j})=\sum_{i\in[d]}w(i)+(n-d)w(d)=(n-d)w(d)<0,$

where the last inequality uses the assumption that $d<n$ and the fact that
$w(d)<0$. It follows that the $(i_{1},\ldots,i_{n})$ component of (5) diverges
as $t\to 0$.

(b) Suppose $Z\in\operatorname{S}(d,n)$ is a non-minimizing critical point of
the frame potential. By Corollary 1, there exist
$d^{\prime},n^{\prime}\in\mathbb{N}$ satisfying

$\frac{n^{\prime}}{d^{\prime}}>\frac{n}{d}$ (6)

such that $n^{\prime}$ of the columns of $Z$ form a tight frame for a
$d^{\prime}$-dimensional subspace of $\mathbb{C}^{d}$ that is orthogonal to
the other $n-n^{\prime}$ columns. Explicitly, there is a unitary matrix
$U\in\mathbb{C}^{d\times d}$ and a permutation matrix $P\in\mathbb{R}^{n\times
n}$ such that

$UZP^{*}=\left[\begin{array}[]{cc}X&0\\\ 0&Y\end{array}\right],$

where $X\in\mathbb{C}^{d^{\prime}\times n^{\prime}}$ satisfies
$XX^{*}=\frac{n^{\prime}}{d^{\prime}}I_{d^{\prime}}$. Put

$D(t):=\left[\begin{array}[]{cc}t^{d-d^{\prime}}I_{d^{\prime}}&0\\\
0&t^{-d^{\prime}}I_{d-d^{\prime}}\end{array}\right],$

and take the one-parameter subgroup
$\lambda\colon\mathbb{C}^{\times}\to\operatorname{SL}(d)$ defined by
$\lambda(t):=U^{-1}D(t)U$. Denoting $Z=[z_{1}\cdots z_{n}]$ and
$UZ=[\tilde{z}_{1}\cdots\tilde{z}_{n}]$, then

$\displaystyle U\lambda(t)\cdot\tau(Z)$
$\displaystyle=D(t)U\cdot(z_{1}\otimes\cdots\otimes z_{n})$
$\displaystyle=D(t)\tilde{z}_{1}\otimes\cdots\otimes
D(t)\tilde{z}_{n}=(t^{d-d^{\prime}})^{n^{\prime}}(t^{-d^{\prime}})^{n-n^{\prime}}\tilde{z}_{1}\otimes\cdots\otimes\tilde{z}_{n}.$

The exponent on $t$ simplifies to $n^{\prime}d-d^{\prime}n$, which is positive
by (6). This implies that $\lim_{t\to 0}\lambda(t)\cdot\tau(Z)=0$, as claimed.
∎

To conclude, we note that in Section 16.3 of the monograph Kirwan:online ,
Kirwan reports that $\tau(Z)$ is semi-stable under the action of
$\operatorname{SL}(d)$ precisely when

$\frac{|\\{j:z_{j}\in S\\}|}{\operatorname{dim}(S)}\leq\frac{n}{d}$

for every nontrivial subspace $S$ of $\mathbb{C}^{d}$. Interestingly, the
identical condition appears in Barthe’s theorem HamiltonM:21  to characterize
the real $Z\in\operatorname{S}(d,n)$ for which there exists
$A\in\operatorname{GL}(d)$ such that normalizing the columns of $AZ$ gives a
unit norm tight frame. While the original proof of Barthe’s theorem seems
difficult to parse Barthe:98 , this coincidence suggests a geometric proof of
the complex case using the fact that gradient flow sends semi-stable points to
unit norm tight frames.

###### Acknowledgements.

Some of the ideas in this project were conceived at the 2018 MFO Mini-Workshop
on Algebraic, Geometric, and Combinatorial Methods in Frame Theory. DGM was
partially supported by AFOSR FA9550-18-1-0107 and NSF DMS 1829955. TN was
partially supported by NSF DMS 2107808. SV was partially supported by NSF DMS
2044349, the NSF–Simons Research Collaboration on the Mathematical and
Scientific Foundations of Deep Learning (MoDL) (NSF DMS 2031985), and the
TRIPODS Institute for the Foundations of Graph and Deep Learning at Johns
Hopkins University. CS was partially supported by NSF DMS 2107700.

## References

  * (1) P.-A. Absil, K. Kurdyka, On the stable equilibrium points of gradient systems, Syst. Control. Lett. 55 (2006) 573–577.
  * (2) B. Alexeev, J. Cahill, D. G. Mixon, Full spark frames, J. Fourier Anal. Appl. 18 (2012) 1167–1194.
  * (3) F. Barthe, On a reverse form of the Brascamp–Lieb inequality, Invent. Math. 134 (1998) 335–361.
  * (4) J. J. Benedetto, M. Fickus, Finite normalized tight frames, Adv. Comput. Math. 18 (2003) 357–385.
  * (5) D. P. Bertsekas, Nonlinear Programming, 2nd ed., Athena Scientific, 1999.
  * (6) N. Boumal, V. Voroninski, A. Bandeira, The non-convex Burer–Monteiro approach works on smooth semidefinite programs, Adv. Neural Inf. Process. Syst. 29 (2016) 2757–2765.
  * (7) J. Cahill, M. Fickus, D. G. Mixon, M. J. Poteet, N. Strawn, Constructing finite frames of a given spectrum and set of lengths, Appl. Comput. Harmon. Anal. 35 (2013) 52–73.
  * (8) J. Cahill, D. G. Mixon, N. Strawn, Connectivity and irreducibility of algebraic varieties of finite unit norm tight frames, SIAM J. Appl. Algebra Geom. 1 (2017) 38–72.
  * (9) E. J. Candès, X. Li, M. Soltanolkotabi, Phase retrieval via Wirtinger flow: Theory and algorithms, IEEE Trans. Inform. Theory 61 (2015) 1985–2007.
  * (10) P. G. Casazza, J. Kovačević, Equal-norm tight frames with erasures, Adv. Comput. Math. 18 (2003) 387–430.
  * (11) I. Daubechies, A. Grossmann, Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986) 1271–1283.
  * (12) J. A. De Loera, T. B. McAllister, Vertices of Gelfand–Tsetlin polytopes, Discrete Comput. Geom. 32 (2004) 459–470.
  * (13) R. J. Duffin, A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952) 341–366.
  * (14) M. Fickus, D. G. Mixon, Tables of the existence of equiangular tight frames, arXiv:1504.00253.
  * (15) R. Ge, F. Huang, C. Jin, Y. Yuan, Escaping from saddle points—online stochastic gradient for tensor decomposition, Proc. Mach. Learn. Res. 40 (2015) 797–842.
  * (16) R. Ge, J. D. Lee, T. Ma, Matrix completion has no spurious local minimum, Adv. Neural Inf. Process. Syst. 29 (2016) 2981–2989.
  * (17) V. K. Goyal, J. Kovačević, J. A. Kelner, Quantized frame expansions with erasures, Appl. Comput. Harmon. Anal. 10 (2001) 203–233.
  * (18) L. Hamilton, A. Moitra, The Paulsen problem made simple, Isr. J. Math., to appear.
  * (19) D. Hilbert, Über die vollen Invariantensysteme, Math. Annalen, 42 (1893) 313–373.
  * (20) R. B. Holmes, V. I. Paulsen, Optimal frames for erasures, Linear Algebra Appl. 377 (2004) 31–51.
  * (21) F. C. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Mathematical Notes, vol. 31, Princeton U. Press, Princeton, NJ, 1984.
  * (22) K. Kreutz-Delgado, The Complex Gradient Operator and the $\mathbb{CR}$-Calculus, arXiv:0906.4835.
  * (23) T. Laurent, J. Brecht, Deep linear networks with arbitrary loss: All local minima are global, Proc. Mach. Learn. Res. 80 (2018) 2902–2907.
  * (24) D. Mumford, J. Fogarty, F. Kirwan, Geometric Invariant Theory, Springer Science & Business Media, 1994.
  * (25) T. Needham, C. Shonkwiler, Admissibility and frame homotopy for quaternionic frames, arXiv:2108.02275.
  * (26) T. Needham, C. Shonkwiler, Symplectic geometry and connectivity of spaces of frames, Adv. Comput. Math. 47 (2021) 5.
  * (27) T. Needham, C. Shonkwiler, Toric symplectic geometry and full spark frames, arXiv:2110.11295.
  * (28) M. Nouiehed, M. Razaviyayn, Learning deep models: Critical points and local openness, ICLR 2018, arXiv:1803.02968.
  * (29) S. Smale, Mathematical problems for the next century, Math. Intell. 20 (1998) 7–15.
  * (30) T. Strohmer, R. W. Heath, Grassmannian frames with applications to coding and communication, Appl. Comput. Harmon. Anal. 14 (2003) 257–275.
  * (31) J. Sun, Q. Qu, J. Wright, Complete dictionary recovery over the sphere I: Overview and the geometric picture, IEEE Trans. Inform. Theory 63 (2016) 853–884.
  * (32) J. Sun, Q. Qu, J. Wright, A geometric analysis of phase retrieval, Found. Comput. Math. 18 (2018) 1131–1198.
  * (33) P. M. L. Tammes, On the origin of number and arrangement of the places of exit on the surface of pollen-grains, Recueil Trav. Bot. Néerl. 27 (1930) 1–84.
  * (34) R. P. Thomas, Notes on GIT and symplectic reduction for bundles and varieties, in: Surveys in Differential Geometry, Vol. X: Essays in Geometry in Memory of S. S. Chern, International Press, 2006, pp. 221–273.
  * (35) J. J. Thomson, On the structure of the atom, Phil. Mag. 7 (1904) 237–265.
  * (36) L. Welch, Lower bounds on the maximum cross correlation of signals, IEEE Trans. Inform. Theory 20 (1974) 397–399.


# Investigation of the burst phenomenon in SiPMs at liquid nitrogen
temperature

M. Guarise11footnotetext: Corresponding author. M. Andreotti  A. Balboni  R.
Calabrese  D. Casazza  A. Cotta Ramusino  A. Corallo  S. Chiozzi  R. D’Amico
M. Fiorini  T. Giammaria  E. Luppi  L. Pierini  L. Tomassetti

###### Abstract

Burst effect of Silicon Photomultiplier (SiPM) at cryogenic temperatures have
been discovered few years ago looking at the dark count rate of SiPMs at
liquid nitrogen temperatures. Bursts are trains of single signals that happen
randomly and are clearly distinguishable from the primary DCR and correlated
noise because of their particular time distribution. In this article we
describe a detailed study related to both the external causes that triggers
bursts and to the phenomenon, internal to the sensor, that produces this dark
signals. We related the burst occurrence to the luminescence produced by some
trapping centers in the SiPMs when they are excited by ionizing radiation that
impinges on the sensor.

##  1 Introduction

Silicon Photomultipliers (SiPM) are highly sensitive solid state
photodetectors, which consist of a 2D array of small-size Avalanche
Photodiodes (APDs) working in Geiger mode, connected in parallel and joined
together on a common silicon substrate. They are frontier sensors with
challenging features of high sensitivity down to the single-photon level, fast
timing and high dynamic range while maintaining low-voltage operation,
mechanical robustness and insensitivity to magnetic fields [1, 2]. In
addition, cryogenic operation of SiPMs allows to keep the dark count rate
(DCR) at very low levels of mHz/mm2, with respect to hundreds of kHz/mm2 which
are typical at room temperature [3]. For these reasons, SiPMs have become
widely used in many fields of everyday life, such as in Lidar and PET scan,
and also in laboratory researches. In particular, in high energy physics,
SiPMs are used in the readout of scintillation light of tracking detectors [4]
and calorimeters[5], in cryogenic TPC like in the DUNE experiment [6], in
Cherenkov detectors[7], and in other applications[8].

A newly discovered phenomenon occurring in a few types of SiPM models of
Hamamatsu Photonics K.K. (HPK) and Fondazione Bruno Kessler (FBK) when
operated at liquid nitrogen (LN2) temperature, has been recently discovered
[9]. This phenomenon, called bursts effect, consists in trains of consecutive
avalanche events, characterized by a rate that is about $100$ times that of
the single-event uncorrelated dark counts, and results in an overall increase
of the DCR. Burst events start typically with a high-amplitude event (> 4p.e.)
and last for tents of millisecond. The number of events in the burst is
typically $\sim 100$ and the amplitude of events contained in the burst are
distributed around the single photoelectron (p.e.).

Since the origin and production mechanisms of this phenomenon is still
unknown, in this work we initially try to understand the external causes that
trigger the bursts and later we will focus our attention to the internal
phenomenon that produces the single events of the burst. We present the
experimental results concerning the tests performed with SiPMs at liquid
nitrogen temperature, to investigate the cause of the burst events and, in
particular, verify if they are related to the interaction of ionizing
radiations with the SiPM. The paper is organised as follows: in section 2 we
present the experimental set-up, in section 3 we describe the measurements
performed and their results and eventually in section 4 we discuss the
possible implications and further investigation that could be performed on
this topic.

##  2 Experimental setup

In order to study the behaviour of burst events and investigate their origin,
the HPK SiPM model S13360-9934 was used [9]. This sensor is characterized by a
sensitive area of $36\text{\,}{\mathrm{mm}}^{2}$ divided in 14331 cells with a
pitch of $75\text{\,}\mathrm{\SIUnitSymbolMicro m}$, its package is a surface
mount type with an hole wire bonding connection and a 150$\mu$m silicone resin
protection window. In particular, two samples identified by the serial numbers
$200032$ and $200034$ were involved in the tests. Their breakdown voltage is
respectively $51.56\text{\,}\mathrm{V}$ and $51.94\text{\,}\mathrm{V}$ at room
temperature, and $41.70\text{\,}\mathrm{V}$ and $42.14\text{\,}\mathrm{V}$ at
cryogenic one. The SiPM bias voltage is provided by the TTi PLH120-P power
supply, $5\text{\,}\mathrm{V}$ over the breakdown voltage.

The measurements are performed at cryogenic temperature, by using a dewar with
height of $60\text{\,}\mathrm{cm}$, internal diameter of
$20\text{\,}\mathrm{cm}$ and capacity of $14\text{\,}\mathrm{l}$, filled with
liquid nitrogen at a temperature of $77\text{\,}\mathrm{K}$. The dewar is
placed into a custom dark box covered by polyurethane-coated black fabric to
shield the sensors from external light.

The SiPMs are placed in a metal box and connected on custom AC amplifiers,
which allow both to provide the bias voltage to the photodetector and to
amplify the charge signals. SiPMs have been covered with teflon and black tape
in order to prevent optical crosstalk and photons from the scintillation of
Argon. The cryogenic AC amplifiers were designed by INFN Bologna and are
supplied by the TTi EX345T power supply at $3.3\text{\,}\mathrm{V}$. The
amplified signal is connected to the Tektronix MSO64B oscilloscope and the
following parameters are set: sampling rate of
$625\text{\,}\mathrm{M}\mathrm{S}\mathrm{/}\mathrm{s}$, bandwidth of
$20\text{\,}\mathrm{MHz}$ and trigger on the SiPM signal amplitude at 0.5p.e..
The oscilloscope is set to the fast frame acquisition mode, in which for every
trigger, a recorded waveform is stored in a temporary buffer before being
saved to disk. In this mode, the oscilloscope acquires a frame only when it
triggers an event, providing also the absolute time of the occurrence.
Typically, we record $5000$$\div$ $10\,000$ frames with a time window of
10$\mu$s.

In our experimental setup, the SiPM can be oriented differently in space. In
particular, we focused on configurations in which the SiPM is arranged
horizontally, i.e. with the active surface parallel to the ground, or
vertically, i.e. with the active area orthogonal to the floor. One or two
SiPMs can be involved in different tests; in the second case, the active areas
are placed one in front of each other at a distance of $6.25\pm
0.25\text{\,}\mathrm{mm}$, and the oscilloscope trigger is set on one SiPM
signal.

A scintillator coupled with a photomultiplier tube (PMT), biased by the High-
Voltage CAEN N470 module at $-1800\text{\,}\mathrm{V}$, can be place below the
dewar in order to identify the passage of cosmic ray. In addition, to study
the possible correlation between burst events and a source of ionizing
radiation, a thoriated tungsten electrode for TIG welding was used [10]. For
this last configuration, a pixel detector has been used to estimate the rate
of particles hitting the SiPM.

Several tests were performed by using the instruments previously reported and
arranging SiPMs in the different configurations described. An example of the
experimental setup used to performed a specific test in shown in figure 1. In
this case two SiPMs are placed vertically, one in front of the other, and the
radioactive source in the middle, to study the impact of a radiation source on
burst events.

Figure 1: Example of the experimental setup used to study burst events related
to a radioactive source. Two SiPMs are place vertically, one in front of the
other, and the thoriated tungsten electrodes in between.

##  3 Results

The idea was to verify if burst events are related to the interaction of
ionizing radiation with the SiPM. Since cosmic rays are a quite abundant
source of ionizing particles that can be easily exploited in laboratory, the
first test was performed to verify if burst events are related to the
interaction of cosmic particles with the SiPM volume. A SiPM is placed
horizontally, so that the active area is crossed by cosmic rays particles, and
a scintillator coupled to a PMT is placed $27\text{\,}\mathrm{cm}$ below the
dewar. If the hypothesis is true, coincidences between the SiPM signal,
related to the first event of a burst, and the PMT signal should be observed.
The test was successful, but few coincidences were observed due to the large
distance between the SiPM and the scintillator. An example of the coincidence
between the two signals is shown in figure 2. This is a clear evidence that
bursts can be triggered by cosmic rays. By repeating the measurement with SiPM
vertically oriented, no coincidences were observed as expected.

Figure 2: Coincidence between the SiPM signal related to the first event of a
burst and the PMT signal. This burst event is triggered by a cosmic ray.

To further validate our hypothesis and verify if a relevant number of burst
events are triggered by cosmic rays, other tests were performed by using two
SiPMs placed one in front of each other at a small distance. The idea is to
search for coincidences between the first event of a burst generated in the
first SiPM and that in the second SiPM. A coincidence between the two signals
represents a clear evidence of burst events generated by the same charged
particle, which crossed both the sensors. When SiPMs are vertical, no
coincidences were observe since there are no cosmic rays crossing both SiPMs.
Instead, in the horizontal configuration, $9$ coincidences were observed over
a time interval of $39\text{\,}\mathrm{min}$, resulting in an occurrence rate
of (3.8 $\pm$ 1.3) mHz. Thus it is reasonable to state that in general some
burst events are triggered by the interaction of cosmic ray particles with the
SiPM and these coincidence events, in particular, are due to those cosmic rays
which cross both SiPMs. The absolute cosmic rays integral intensity at sea
level is $\approx
70\text{\,}{\mathrm{m}}^{-2}\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{sr}}^{-1}$
[11]. However, the coincidence event are related to the passage of a cosmic
ray particle through both SiPMs, therefore by taking into account the cosmic
ray angular distribution and the SiPMs configuration, we estimated the
percentage of particles hitting the first SiPM which cross the second one too.
The Monte-Carlo simulation result is $45\pm 3\text{\,}\mathrm{\char
37\relax}$, where the error was estimated by repeating the simulation with
SiPMs staggered by $1\text{\,}\mathrm{mm}$ at a distance of $6$ or
$6.5\text{\,}\mathrm{mm}$. The expected cosmic ray coincidence rate is so
$3.0\pm 0.2\text{\,}\mathrm{mHz}$, in agreement with the measured rate.

Let’s focus on the burst rate of a single SiPM. By assuming that burst events
are due to the interaction of ionizing particles with the SiPM, their rate
($R_{burst}$) can be parameterized as the sum of the interaction rate of each
available radiation source. The SiPMs in our laboratory are normally subjected
to a ionizing particle flux from cosmic rays and environmental radioactivity.
We define $R_{cos}$ as the rate of burst events triggered by cosmic ray
particles, and $R_{env}$ as that due to environmental radioactivity,
therefore:

$R_{burst}=R_{cos}+R_{env}$ (3.1)

$R_{cos}$ depends on the azimuth angle ($\theta$), which describes the
inclination of the SiPM active area; it is maximum when the SiPM is
horizontally oriented and zero when it is vertically oriented. Instead, we
assumed that $R_{env}$ is constant and doesn’t depend on $\theta$. We
estimated $R_{cos}$ and $R_{env}$ by comparing the burst rate when the SiPM is
vertically and horizontally oriented, whose results are listed in table 1. The
burst rates were obtained by analyzing the timestamp files and refer to the
SiPM, whose signal was connected to the oscilloscope trigger.

If the SiPM is placed horizontally, its active area is crossed both by
particles from cosmic rays and environmental radioactivity, while in the
second case there is no cosmic rays contribution, but only the environmental
one. Therefore, $R_{env}$ is equal to the burst rate evaluated in the second
measurement (i.e. vertical configuration), while $R_{cos}$ is given by the
difference between the total burst rate obtained in the first measurement
(i.e. horizontal configuration) and $R_{env}$. The results are the following:

$R_{env}=$16\pm 1\text{\,}\mathrm{mHz}$\qquad\qquad R_{cos}=$5\pm
2\text{\,}\mathrm{mHz}$$ (3.2)

The measured $R_{cos}$ is in agreement, within the experimental error, with
the expected cosmic ray rate on the SiPM active area which is equal to
$6.50\text{\,}\mathrm{mHz}$. This result confirms that a part of the burst
events are due to the interaction of cosmic ray particles with the SiPM.

Configuration | Horizontal | Vertical
---|---|---
Acquired waveforms | $5000$ | $9909$
Acquisition time | $39\text{\,}\mathrm{min}$ | $78\text{\,}\mathrm{min}$
Coincidences | $9$ | $0$
Burst rate | $21\pm 2\text{\,}\mathrm{mHz}$ | $16\pm 1\text{\,}\mathrm{mHz}$
Table 1: Results obtained by placing two SiPMs one in front of the other in
horizontal and vertical configuration. The burst rate is referred to the SiPM
whose signal is connected to the oscilloscope trigger. Coincidences are
related to the first SiPM signals of the burst event.

If burst events are related to the interaction of ionizing particles with the
SiPM, the number of burst events should increase with the rate of ionizing
particles interacting with it, such as when it is placed near a radioactive
source. To verify this hypothesis, two SiPMs were placed one in front of the
other in vertical configuration and a thoriated tungsten electrode for TIG
welding was placed in between. As expected no coincidences between the burst
events triggered in the two SiPMs were observed, since no particles crossing
both devices are expected. In this case, the bust rate can be parameterized as
the sum of the contribution from environmental radioactivity and that from
thoriated tungsten electrode, called $R_{Th}$, while the cosmic ray
contribution is zero since the vertical configuration:

$R_{burst}=R_{env}+R_{Th}$ (3.3)

$R_{Th}$ was evaluated as the difference between the total burst rate measured
in this test, equal to $93\pm 7\text{\,}\mathrm{mHz}$, and $R_{env}$, which is
constant and was previously evaluated 3.2. The result is:

$R_{Th}=$77\pm 7\text{\,}\mathrm{mHz}$$ (3.4)

This burst rate contribution is in agreement, within the experimental error,
with the expected number of particles hitting the SiPM produced by the
thoriated tungsten electrodes, equal to $81\pm 1\text{\,}\mathrm{mHz}$. This
result further validates our hypothesis that bursts are triggered by ionizing
radiation that releases their energy in the SiPM.

###  3.1 Temporal distribution of events in a burst

Figure 3: Temporal distribution of single events inside the burst.
Superimposed an exponential decay curve fit. In this example, the dacey time
$\tau$ is (118$\pm$2)ms

We further investigate the temporal distribution of the single events in the
burst. To do that, we acquired many hours of data and we identified burst
events with the technique described in [9]. For each burst of events (in total
we analysed 549 bursts) we studied the rate of events inside the burst.
Looking at the arrival time of the single events in the burst, we can
reconstruct the time distribution of events as shown in figure 3. Here, we
divided the total temporal length of the burst in several slices and we
counted the number of events that happened in each bin ($Events(t)$). Given
the shape of the obtained trend, as shown in figure 3, we supposed a simple
exponential decay in the form:

$Events(t)=A(0)e^{-t/\tau}$ (3.5)

where A(0) is the amount of events in the first bin and $\tau$ is the decay
constant. We thus fit the temporal distribution with the equation 3.5
obtaining the decay constant and the initial amplitude for each of the burst
we found.

Finally we compute the global histogram of the decay time constant $\tau$ as
shown in figure 4. As clearly visible, the trend is Gaussian with a mean value
of (118.0$\pm$0.9) ms, and a sigma of (29.1$\pm$0.9) ms.

Figure 4: Histogram of the decay constant $\tau$. Superimposed a Gaussian fit.

The temporal behaviour of the events in the burst shown in figure 3 can be
interpreted as a luminescence of some trapping sites embedded into the sensor.
Given the long lifetime of this light signal and the single photon sensitivity
of SiPMs, we are able to count single photons produced in the luminescence
process. In this view, the differences in the initial number A(0) can be
related to the fact that the photon emitted in the fluorescence and the ones
seen by the active area of the detector are not equal for all the bursts.

##  4 Discussion and conclusions

In this article we present a further investigation on the burst effect of
SiPMs at LN2 temperature with respect to previous works [9, 12]. In particular
we present a detailed study of the possible causes of this phenomenon. We
investigated the relation of burst effect to the energy deposition of ionizing
radiation in the sensor through cosmic muons and a Th source. Results obtained
for different configurations, seem to confirm our hypothesis that bursts of
events are related to the charged particle interaction in the SiPM. A detailed
analysis about the typical time delay between events in the burst have also
been performed. We found that the events in a burst have a typical exponential
decay time distribution with a decay constant of (118.0$\pm$0.9)ms. This
results arises from a global analysis performed on almost 550 bursts observed
in the SiPM 13360 (75$\mu$m pitch) produced by Hamamatsu Photonics K.K.. The
results obtained about the internal mechanism of generation of the burst
events, can thus be related to the light emission from some trap centers in
the sensors that are activated when ionizing radiation interacts with the SiPM
volume. In this scenario, following processes of energy transfer, the energy
released by the incident particle can excite the trap centers whose long
lifetime emission at cold is then detected by the SiPM itself. The emission
centers involved in the process can be related to the particular material and
composition of the sensor but also on the kind of production processes used in
the fabrication of the SiPM. As already largely discussed in our previous
paper [9], we confirm that the type of protection resin cannot be the only
responsible for this luminescence as there are some sensors with the same
protection resin as the one tested for this article that do not show burst
effect. A possible explanation can be related to the manner the sensor and the
resin are manufactured and worked out during the entire production process.
For example mechanical or laser cutting can produce some defects that in some
cases can act as luminescence sites [13, 14]. Given the poor information about
the material composition and the fabrication process, we cannot give any claim
on the nature of trapping centers and a further investigations in this view
are necessary in close synergy with vendors.

## Acknowledgments

The present research has been supported and partially funded by the Italian
Research Center on High Performance Computing, Big Data and Quantum Computing
(ICSC) funded by MUR Missione 4 - Next Generation EU (NGEU), by the Italian
Ministero dell’Università e della Ricerca for the digital and environmental
transitions -M4C1 I.3.4 (CUP: F73C23000720006), by the department of Physics
and Earth Science University of Ferrara FIRD 2023. We would also like to
especially thank M. Cavallina and S. Squerzanti for the precious mechanical
work. The authors participating in the DUNE single phase photon detection
(SPPD) consortium acknowledge its support in making available the custom
sensors. We would like to thank also HPK Italy for the helpful discussions.

## References

  * [1] Stefan Gundacker and Arjan Heering.  The silicon photomultiplier: fundamentals and applications of a modern solid-state photon detector.  Physics in Medicine & Biology, 65:17TR01, 2020.
  * [2] Nicola Serra, Alessandro Ferri, A Gola, Tiziana Pro, Alessandro Tarolli, Nicola Zorzi, and Claudio Piemonte.  Characterization of new fbk sipm technology for visible light detection.  Journal of Instrumentation, 8(03):P03019, 2013.
  * [3] K Ozaki, S Kazama, M Yamashita, Y Itow, and S Moriyama.  Characterization of new silicon photomultipliers with low dark noise at low temperature.  Journal of Instrumentation, 16(03):P03014, 2021.
  * [4] Roel Aaij, Ahmed Sameh Wagih Abdelmotteleb, Carlos Abellan Beteta, Fernando Jesus Abudinén, Cyrille Achard, Thomas Ackernley, Bernardo Adeva, Marco Adinolfi, Patrik Harri Adlarson, Hossein Afsharnia, et al.  The lhcb upgrade i.  arXiv preprint arXiv:2305.10515, 2023.
  * [5] CMS collaboration et al.  Development of the cms detector for the cern lhc run 3.  arXiv preprint arXiv:2309.05466, 2023.
  * [6] M Andreotti, S Bertolucci, A Branca, C Brizzolari, G Brunetti, R Calabrese, P Carniti, D Casazza, C Cattadori, E Calvo, et al.  Cryogenic characterization of hamamatsu hwb mppcs for the dune photon detection system.  Journal of Instrumentation, 19(01):T01007, 2024.
  * [7] Thomas Frach.  Optimization of the digital silicon photomultiplier for cherenkov light detection.  Journal of instrumentation, 7(01):C01112, 2012.
  * [8] Frank Simon.  Silicon photomultipliers in particle and nuclear physics.  Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 926:85–100, 2019.
  * [9] M. Guarise.  A newly observed phenomenon in the characterisation of sipm at cryogenic temperature.  Journal of Instrumentation, 16:T10006, 2021.
  * [10] Hiroyuki SAITO.  Thorium-232 exposure during tungsten inert gas arc welding and electrode sharpening.  Industrial health, 41:273–278, 2003.
  * [11] R. L. Workman and Others.  Review of Particle Physics.  PTEP, 2022:083C01, 2022.
  * [12] T. Tsang.  Studies of event burst phenomenon with sipms in liquid nitrogen.  Journal of Instrumentation, 18:C01050, 2023.
  * [13] A Cremades and J Piqueras.  Luminescence study of structural changes induced by laser cutting in diamond films.  Journal of applied physics, 78(5):3353–3356, 1995.
  * [14] Gisele A dos Reis Benatto, Claire Mantel, Adrian A Santamaria Lancia, Peter B Poulsen, Søren Forchhammer, and Sergiu V Spataru.  Laser induced luminescence characterization of mechanically stressed pv cells.  In 2021 IEEE 48th Photovoltaic Specialists Conference (PVSC), pages 1949–1953. IEEE, 2021.