# Long radial coherence of electron temperature fluctuations in non-local
transport in HL-2A plasmas

Zhongbing Shi<EMAIL_ADDRESS>Southwestern Institute of Physics, Post Office
Box 432, Chengdu 610041, China  Kairui Fang  Southwestern Institute of
Physics, Post Office Box 432, Chengdu 610041, China  Jingchun Li, Zhaoyang Lu
Department of Earth and Space Sciences, Southern University of Science and
Technology, 518055 Shenzhen, Guangdong, People’s Republic of China  Xiaolan
Zou  CEA, IRFM, F-13108 Saint-Paul-lez-Durance, France  Jie Wen, Zhanhui Wang,
Xuantong Ding, Wei Chen, Zengchen Yang, Min Jiang Xiaoquan Ji, Ruihai Tong,
Yonggao Li, Peiwang Shi, Wulyv Zhong, and Min Xu  Southwestern Institute of
Physics, Post Office Box 432, Chengdu 610041, China

###### Abstract

The dynamics of long-wavelength ($k_{\theta}<1.4\mathrm{\ cm^{-1}}$),
broadband (20-200 kHz) electron temperature fluctuations
($\tilde{T}_{e}/T_{e}$) of plasmas in gas-puff experiments were observed for
the first time in HL-2A tokamak. In a relative low density ($n_{e}(0)\simeq
0.91\sim 1.20\times 10^{19}/m^{3}$) scenario, after gas-puffing the core
temperature increases and the edge temperature drops. On the contrary,
temperature fluctuation drops at the core and increases at the edge. Analyses
show the non-local emergence is accompanied with a long radial coherent length
of turbulent fluctuations. While in a higher density ($n_{e}(0)\simeq 1.83\sim
2.02\times 10^{19}/m^{3}$) scenario, the phenomena were not observed.
Furthermore, compelling evidence indicates that $\textbf{E}\times\textbf{B}$
shear serves as a substantial contributor to this extensive radial
interaction. This finding offers a direct explanatory link to the intriguing
core-heating phenomenon witnessed within the realm of non-local transport.

###### pacs:

: 52.55.Tn, 52.25.Fi, 52.55.Fa, 52.70.Gw

††preprint: APS/123-QED

##  I Introduction

One of the perplexing phenomena observed and extensively studied worldwide is
non-local transport, which has been investigated in various tokamaks
(TFTRkissick1994transient ,TEXTgentle1995strong /J-TEXTShi_2018 ,
RTPgalli1999non ; mantica1999nonlocal , HL-2Ahong2007observation , Alcator
C-modrice2013non , KSTARshi2017intrinsic , EASTliu2019dynamics , Tore
Suprazou2000edge , JETgoriniperipheral , AUGryter2000propagation , DIII-
Drodriguez2019predict ) and stellerators(W7-ASstroth1996 ,
LHDinagaki2006abrupt ). These experiments have consistently revealed a
peculiar long-range transient temperature response triggered by pulsed
interventions. These observations suggest that disturbances at the plasma’s
edge can perturb the deeper plasma equilibrium, challenging the conventional
understanding of transport governed by local parameters and the gradient-flux
relationida2015towards ; Inagaki_2013 .

In recent years, turbulence theories focusing on multiple-scaled nonlinear and
statistical dynamics have gained traction in addressing this enigma. These
include turbulence spreading modelshahm2004turbulence ; gurcan2005dynamics ;
lin2004turbulence ; wang2011turb ; hariri2016cold and the concept of self-
organized criticality (SOC)diamond1995dynamics ; kubota1997avalanche ;
wang2011turb ; chen2016dynamics . Moreover, several studies propose that the
transition between trapped electron modes (TEMs) and ion temperature gradient
modes (ITGs) may play a role in the critical nature of non-local
effectsgao2014non . Evidence has emerged indicating the coexistence of
rotation reversal, shear, and heat pinch phenomenarice2013non ;
shi2017intrinsic ; ida2015towards ; mantica2000heat . Notably, in the past
year, a new form of non-locality in the ion temperature channel was discovered
in J-TEXTShi2020Observation .

Despite more than two decades since its discovery, the mechanisms and
theoretical models behind non-locality remain enduring enigmas in the field.
In recent years, a particular perspective has emerged, challenging the
traditional understanding of non-local transport. This perspective, supported
by simulations, posits that the injection of a cold pulse may not actually
induce genuine non-local transportrodriguez2018explaining ; Angioni_2019 .
Instead, it suggests that non-locality can be explained as an outcome of the
transient changes in local density gradients, with distinct mechanisms
governing such behavior in plasmas dominated by TEMs versus ITGsli2019 ;
li2022 ; li2023 .

Undoubtedly, local drift-wave turbulent transport mechanisms stand as widely
accepted explanations for anomalous transport phenomena in tokamaks.
Gyrokinetic and gyrofluid theories have been extensively validated and serve
as robust frameworks for understanding these processesli2023 . However,
intriguingly, observations from experiments like EAST have revealed instances
where density fluctuations transiently increase in the core before any local
alterations in density and temperature occurliu2019dynamics . Such findings
hint at the possibility of a turbulence spreading modelhariri2016cold at play.
The question that looms large is how these turbulent fluctuations can evolve
more rapidly than local profiles or gradients, a puzzle that demands further
elucidation.

Hence, non-locality, to a certain extent, remains an enigma within the domain
of heat transport. This perplexity arises from the challenges in deciphering
thermal fluctuations within the turbulence. These fluctuations are
characterized by their low amplitude within the core region, their propensity
for long-distance effects, and the formidable obstacles posed by localized
measurements. Notably, the turbulence and its electron thermal fluctuation
(referred to as $\tilde{T}_{e}/T_{e}$) present formidable detection and
measurement challenges. This is exacerbated by the fact that the inherent
thermal noise level is quite substantial, exceeding 4$\%$ of the temperature
level, in stark contrast to the 1$\%$ threshold for $\tilde{T}_{e}/T_{e}$.

Thankfully, a ray of hope has emerged with the successful adaptation of
correlation techniques initially developed for astronomical detection
instruments into the realm of plasma diagnostics. A notable development on
this front occurred last year with the deployment of a correlation electron
cyclotron emission system (CECE) in the HL-2A tokamak. This advancement has
played a pivotal role in shedding light on the profound physics underlying the
non-local phenomenon. In this letter, we present a detailed analysis of
electron temperature fluctuations, unveiling several groundbreaking findings
for the first time.

Figure 1: Raw data in a non-local experiment (shot 37075). Time evolutions of
(a)electron density from far infrared interferometer(FIR). (b)Plasma current.
Electron temperature from ECE at (c) normalized radii $\rho\sim 0.10$ and (d)
$\rho\sim 0.80$. (e) Power of plasma radiation from bolometer and (f)Raw
signal of $H_{\alpha}$ radiation diagnostic. The perpendicular red lines
represent the injection on and off, the same below.

##  II Experimental Setup

This study is based on data obtained from the HL-2A tokamak, a medium-sized
device with a major radius (R) of 1.65 meters and a minor radius (a) of 0.4
meters LIU2003147 . The tokamak is equipped with a toroidal magnetic field
ranging from 1.2 to 2.7 Tesla and is capable of generating plasma currents of
up to 480 kiloamperes. The versatility of HL-2A allows for the investigation
of non-local transport effects through various experimental techniques,
including modulated electron cyclotron heating (ECH), supersonic molecular
beam injection (SMBI), pellet injection, gas puffing, fast current ramps, and
laser blow-off sun2008features . In this particular study, we focus on non-
local phenomena induced by gas puffing (GPs) in ohmic-heating plasmas
characterized by relatively low electron densities (approximately
$0.91-1.20\times 10^{19}/m^{3}$). Key diagnostic tools employed include a
multi-channel Electron Cyclotron Emission (ECE) system, which shares a common
sightline and horn antenna with the CECE system, and a Doppler BackScattering
(DBS) reflectometer located at a toroidal angle of 22.5 degrees relative to
the ECE window zhongbing2018progress . The DBS instrument provides valuable
information about plasma rotation, density fluctuations, and radial electric
fields.

Figure 2:  (a) Electron temperature from plasma edge to the core. The gas is
fed at 0 ms. Profiles of (b) electron temperature and (c) normalized $T_{e}$
gradient scale length at 2 ms before, 3 ms after , 14 ms after ($T_{e}(0)$
peaking) and 28 ms after (recovery).

##  III Experimental results

###  III.1 General non-local phenomenon in GP experiment

Figure 1 illustrates various parameters observed during a typical discharge
involving multiple Gas Puffs (GPs). In this experimental shot, the plasma is
subjected to ohmic heating, featuring a toroidal magnetic field configuration
of 1.34 T and a plasma current of approximately 150-153 kA. Each gas puff
lasts for about 1-2 ms, which is similar in duration to Supersonic Molecular
Beam Injections (SMBIs) but with lower pulse intensity. The on/off timing of
the GPs is indicated by the front and rear red lines perpendicular to the time
axis. It’s crucial to note that the time intervals between these pulses are
sufficiently long to ensure the independence of each non-local transport
effect.

Additionally, the plasma density (Figure 1(a)) is measured using a far-
infrared interferometer (FIR) li2017new . It’s evident that the line-averaged
density increases immediately after the GPs and then gradually decreases.
Simultaneously, the inner core electron temperature exhibits a bulge, followed
by a recovery over a duration of approximately 30 ms, while the edge
amplitudes undergo opposite changes. The evolution of plasma current,
radiation power, and $H_{\alpha}$ signals are also displayed.

The foundational investigation commences with equilibrium parameters because
local collisionality and gradients in the background are likely to influence
the turbulent state, subsequently affecting the measurements. Figure 2(a)
presents the time traces of electron temperature ($T_{e}$) signals at
different radial positions. It’s important to emphasize that the puff time is
normalized to zero with the onset of gas injection, facilitating a clearer
time series for signal comparison. The heat diffusivity induced by sawtooth
oscillations ranges from 4-6 $m^{2}/s$ at $\rho\simeq 0.5$, with no
discernible variation before and after the occurrence of non-local phenomena.
The puffs in the experiment are deployed beyond $\rho>0.7$, resulting in
immediate changes in edge profiles. Figure 2(b) and 2(c) display profiles of
electron temperature ($T_{e}$) and the normalized gradient scale length
$a/L_{T_{e}}$ before, during, and after the non-local effect. Negative time
values represent the period before puffing when parameters serve as
references. In Figure 2(b), we observe a rapid change in $T_{e}$ occurring
first outside the inverse radius, peaking with a delay of approximately 2-3
ms. Subsequently, the core region of $T_{e}$ begins to rise, with a maximum
increment of about $18\%$ (approximately 200 eV) observed in the central core
at around 14 ms. Regarding the normalized gradient scale length $a/L_{T_{e}}$
presented in Figure 2(c), local $a/L_{T_{e}}$ variations are consistent with
local $T_{e}$ over time. In contrast, $a/L_{T_{e}}$ displays a global increase
following the gas puff.

Figure 3: (a-c)Cross coherencies caculated during the nonlocal phases (w
nonlocal) and the steady state phases (w/o nonlocal) at different radii
and(d-f)Time sequences with $\tilde{T}_{e}/T_{e}$ (20-200kHz).

###  III.2 Non-locality in electron temperature fluctuations

Figure 3(a-c) illustrates the coherencies, while figure 3(d-f) displays
normalized time-domain fluctuations ($\tilde{T}_{e}/T_{e}(t)$) integrated
within the 20-200 kHz frequency range. Evidently, following the gas puff,
there is a rapid and significant increase in the amplitude of relative
electron temperature fluctuations, primarily observed at the peripheral radius
around $\rho\simeq 0.80$. This observation is substantiated by both the
coherency graphs and the $\tilde{T}_{e}/T_{e}(t)$ plots. Moving closer to the
location of the $T_{e}$ inverse radius, there is a diminished distinction
between the two phases. Conversely, at the edge, during the non-local phase,
$\tilde{T}_{e}/T_{e}(t)$ decreases in the central core. Reflecting on the
previous $T_{e}$ response, the behavior of these fluctuations aligns with the
$T_{e}$ increment/decrement, as a greater/lower fluctuation amplitude
corresponds to stronger/weaker turbulence and more/less heat transport (heat
loss), assuming other conditions remain constant. Notably, the CECE-measured
fluctuations have been previously linked to $\nabla T_{e}$ driving in ohmic
heated L-mode plasma, a concept widely accepted in drift-wave turbulence
theoriesfang2019eight .

Figure 4: Evolvement of local ($\rho=0.09$) (a) electron temperature (b)
normalized gradient length of $a/L_{T_{e}}$ and (c) relative fluctuations of
$\tilde{T}_{e}/T_{e}$. (d) Relation between $a/L_{T_{e}}$ and
$\tilde{T}_{e}/T_{e}$. Several core data points imply fluctuations are not
mainly driven by the temperature gradients.

Figure 5: (a)Relative electron temperature fluctuation profile of
$\tilde{T}_{e}/T_{e}(\%)$ (20-200kHz) . (b) Radial coherent length with
reference position at $\rho=0.80$ and (c) spatiotemporal structure of coherent
length in the low density scenario.

Figure 6: (a)Profile of $\tilde{T}_{e}/T_{e}$ .(b) Radial coherent length with
reference of $\rho=0.86$ in the high density ($n_{e}(0)\simeq 1.83\sim
2.02\times 10^{19}/m^{3}$) scenario. There is no obvious difference between
phase before and after the gas puff.

This study also undertakes a similar comparison between $a/L_{T_{e}}$ and
$\tilde{T}_{e}/T_{e}$. One of the outcomes, at $\rho\sim 0.09$, is presented
in figure 4. Interestingly, unlike the typical gradient-flux relation,
fluctuations in this scenario exhibit a decrease in amplitude alongside the
local gradient of $a/L_{T_{e}}$, with the latter remaining relatively
unchanged (slightly increasing). This finding suggests that fluctuations in
the region of $\rho<0.30$ are not primarily determined by local $a/L_{T_{e}}$;
in other words, the steady-state gradient driving is not applicable in this
context. For electrostatic turbulence, another possibility considered is the
role of $\nabla n_{e}$ driving. However, validating this mechanism presents
challenges, primarily due to limited means of locally measuring core density.
While the interferometer with tomographic inversion can reconstruct the
density profile, it may introduce distortion in the signal evolution in the
core due to edge data interference. Conversely, in the peripheral region, the
local gradient and $\tilde{T}_{e}/T_{e}$ can be associated since the
relationship between figure 2(c) and 3(d) demonstrates a positive correlation.
Notably, no significant changes were observed in coherency and fluctuation
evolution around the $T_{e}$ inverse radius.

Using multiple radial positions, the profile of $\tilde{T}_{e}/T_{e}$ can be
constructed, as depicted in Figure 5(a). In light of the preceding analyses,
it becomes apparent that the $\tilde{T}_{e}/T_{e}$ profile also exhibits an
inverse radius when computed with and without the non-local phases.
Furthermore, additional radial cross coherency analyses were conducted,
focusing on specific channels in relation to others, as illustrated in Figure
5(b). Specifically, the channel at $\rho=0.80$ was selected as a reference
point. The radial coherence length ($L_{c}$) was theoretically defined as the
radial span over which the integrated coherencies or fluctuations diminish to
$1/e$ of their amplitude.

Following the gas puffing, the radial coherence length experiences a
significant increase during the non-local phases, as evident in Figure 5(b).
During the steady phase, the observed radial coherence extends approximately
2.5 cm, but this length escalates to about 8.8 cm during the non-local phase.
Given the selected channel at $\rho=0.80$, this result suggests that the long
radial influence penetrates to approximately $\rho\sim 0.55$, precisely within
the vicinity of the $T_{e}$ inverse region. Furthermore, the observed radial
coherence effect persists, manifesting as a non-Gaussian tail extending
approximately 22 cm, signifying profound and deeper transient interactions.
This phenomenon also implies that long radial coherent turbulences are
established immediately after the gas puffs [Figure 5(c)].

In conjunction with the response time illustrated in Figure 3(d-f), the
alterations in coherent length are consistent with fluctuations in amplitude.
These observations of long-distance coherence or correlation likely constitute
a mechanism for radial transient transport. Furthermore, $L_{c}$ is determined
by integrating fluctuations within the 20-200 kHz frequency range to mitigate
the influence of low-frequency MHD interference. This represents a significant
advancement compared to previously reported macroscopic structures in some
instances.

It is noteworthy that the measurement principle of ECE involves utilizing two
adjacent channels with very close frequencies for correlation analysis to
eliminate the influence of noise. As illustrated in Figure 1 (see ref. [35]),
the bandwidth of the adjacent channels is 150 MHz, corresponding to a spatial
width of approximately 1 cm. Due to radiation broadening, there is some
overlap between measurements from these two channels. When the two channels
are measured independently, turbulence exhibits spatial correlation, while
noise lacks correlation, thus achieving the objective of noise reduction.
Moreover, in the high-frequency range, the correlation is relatively weak.
Although the correlation is weak around 100-200 kHz at $\rho\sim 0.8$, it
remains evident when comparing cases with and without non-local phenomena.
Additionally, to eliminate the influence near zero frequency, signals in the
0-20 kHz range were not considered. Therefore, a range of 20-200 kHz was
employed in this study. In fact, the choice of frequency range has no impact
on the results presented in Figure 3.

###  III.3 A comparison between low- and high-density scenario

Before delving into comparative scenarios, it is imperative to juxtapose the
performance of $\tilde{T}_{e}/T_{e}$ with density fluctuations derived from
DBS data. Within the DBS system, the meso-scale portion (20-100 kHz) of the
spectrum demonstrates a degree of consistency with $\tilde{T}_{e}/T_{e}$.
However, it is worth noting that due to the distinct wavenumbers detected
(with $k_{\perp}$ falling in the range of 4-10 $\mathrm{cm^{-1}}$ by the DBS
and $k_{\theta}$ remaining below 1.4 $\mathrm{cm^{-1}}$) via the CECE system,
higher-frequency ($\leq$100 kHz) density fluctuations exhibit dissimilar
behavior. This observation aligns with previous reports from HL-2Ashi2018roles
; ida2015towards .

It’s important to highlight that cross-correlation analysis between
$\tilde{n}_{e}$ and $\tilde{T}_{e}$ was not conducted in this study.
Nonetheless, an alternative and valuable physical parameter can be derived -
the perpendicular velocity, as it is closely related to the Doppler shift
($v_{\perp}(\rho,t)=2\pi f_{d}/k_{\perp}$). In a broad sense, $v_{\perp}$ can
be considered as the sum of $v_{E\times B}$ and $v_{\phi}$ when $v_{\phi}$ is
negligibly small compared to $v_{E\times B}$ at the plasma edge. Subsequently,
$v_{E\times B}$ is associated with the $E\times B$ shearing rate:

$\gamma_{E\times B}=(r/q)d(E_{r}/B_{p}R)/dr\sim\frac{\partial v_{E\times
B}}{\partial r}$ (1)

where $q$ represents the safety factor, and $B_{p}$ signifies the poloidal
magnetic field. As is well-established, turbulence suppression takes place
when the $E\times B$ shearing rate surpasses the linear growth rate of the
underlying instability, and converselyschirmer2006radial . Given that the
measured velocities are primarily influenced by $E\times B$ shear, all other
factors being constant, one can estimate the evolution of $\gamma_{E\times
B}$.

A high-density scenario ($n_{e}(0)\simeq 1.83\sim 2.02\times
10^{19}/m^{3},I_{p}\simeq 160kA$) was investigated while maintaining
consistent puff intensity. Using data from the DBS reflectometer, electron
temperature fluctuations in the high-density scenario were investigated and
are presented in figure 6. In figure 6(a), the profile for this scenario is
displayed, while figure 6(b) shows the radial coherent length. The
$\tilde{T}_{e}/T_{e}$ was also integrated with 20-200 kHz coherencies to
obtain the normalized coherence (20-200 kHz). The legend ”w/o nonlocal, puff”
indicates the period of $T_{e}$ drop after puffing, while ”w/o puff”
corresponds to the steady-state phase. In contrast to figure 5, the
$\tilde{T_{e}}/T_{e}$ and coherent length exhibit smaller increases during the
non-steady state. Importantly, the coherence entirely dissipates at $\rho\sim
0.65$. This suggests that edge puffing has minimal influence on the core in
the context of relatively higher-density plasma. For the central
$\tilde{T_{e}}/T_{e}$, the fluctuation amplitudes remain below the noise level
and cannot be reliably extracted.

Furthermore, Figure 7 displays the $E\times B$ shearing rates
($\gamma_{E\times B}$) for this particular scenario (shot 37079) alongside
data from shot 37075. The polarity of $v_{\perp}$ signifies ion or electron
diamagnetic drift direction, respectively. It’s evident that the $E\times B$
flow experiences transient variations after the puffs in both scenarios. In
shot 37079, $v_{\perp}$ at all detected radii decelerates, causing shear rates
to immediately decline with an inward propagation damping effect. Conversely,
in the low-density scenario, the reduction in $\gamma_{E\times B}$ persists in
the peripheral region. However, the propagation dynamics shift within
$\rho=0.52$. Instead, $v_{\perp}$ and $\gamma_{E\times B}$ show rapid
increases after the puffs, a phenomenon distinct from the high-density
scenario. Notably, the point at which shear rates begin to change aligns
closely with the $T_{e}$ inverse radius. In both scenarios, the trends in
shear rates in the central region are consistent with their impact on
$\tilde{T}_{e}/T_{e}(t)$.

These findings align closely with the critical gradient model described in
previous worksWangzhanhui2011 . To elucidate further, considering the radial
dimension, it can be expressed as follows:

$\displaystyle\frac{\partial
I}{\partial\hat{t}}-\frac{\partial}{\partial\hat{\rho}}(\hat{\chi}_{turb}\frac{\partial
I}{\partial\hat{\rho}})=\hat{\gamma}I-\hat{\beta}I^{2},$ (2)
$\displaystyle\frac{\partial\langle\hat{T}\rangle}{\partial\hat{t}}-\frac{\partial}{\partial\hat{\rho}}[(\hat{\chi}_{turb}+\hat{\chi}_{neo})\frac{\partial\langle\hat{T}\rangle}{\partial\hat{\rho}}]=\hat{S},$
(3)

where the normalized quantities are denoted with a superscript $\hat{}$ and
the normalized process was described in Wangzhanhui2011 . Here, $I$ is the
turbulence intensity, $\beta$ is local nonlinear damping rate,
$\hat{\chi}_{turb}=\hat{\chi}_{0}I/(1+\hat{A}^{2}/\hat{\kappa}_{c}^{2})$ is
the normalized turbulent thermal diffusivity, $\hat{\chi}_{neo}$ is the
normalized neoclassical thermal diffusivity , $\hat{S}$ is the normalized
heating source and
$\hat{\gamma}=\hat{\gamma}_{0}\hat{R}(\hat{A}-\hat{A}_{crit})\Theta(\hat{A}-\hat{A}_{crit})/(1+\hat{A}^{2}/\hat{\kappa}_{c}^{2})$
is the local growth rate of turbulent intensity with a Heviside function
$\Theta$. $\hat{A}=|\partial\langle\hat{T}\rangle|/\partial\hat{\rho}$ is the
magnitude of the local mean temperature (ion or electron) gradient. The
electric field shear is related to local parameter with
$\hat{\kappa}_{c}^{2}$. Notably, turbulence spreading exhibits a velocity of
$\hat{v}f\sim\sqrt{\hat{\gamma}\hat{\chi}_{turb}}$, which is comparable to the
Fisher front speedFISHER ; PhysRevLett.118.185002 . It has been demonstrated
in simulations that with strong shearing (smaller $\hat{\kappa}_{c}$) and weak
shearing, a bifurcation of $T_{e}$ can be achievedWangzhanhui2011 .

Figure 7:  Spatiotemporal plots of (a-b)Perpendicular velocities $v_{\perp}$,
(c-d)$E\times B$ shear rates $\gamma_{E\times B}$ and (e-h) absolute values of
$\gamma_{E\times B}$ with core $T_{e}$ increase (nonlocal phenomenon) in the
low density ($n_{e}(0)\simeq 0.91\sim 1.20\times 10^{19}/m^{3}$) scenario and
without core $T_{e}$ increase in the high density ($n_{e}(0)\simeq 1.83\sim
2.02\times 10^{19}/m^{3}$) scenario. The plus or minor value of $v_{\perp}$
represents the ion or electron diamagnetic drift direction, respectively. The
shearing rates increase abruptly inside inverse region in the non-local
scenario.

##  IV Conclusion and Discussion

This study sheds light on the intricate dynamics of turbulence, which play a
pivotal role in non-local transport effects in tokamak plasmas li_cpb ;
li_cpb1 ; jiao . Leveraging advanced core detection and measurement
techniques, our analyses unveil the behavior of non-local transport in low-
density plasmas, intricately linked with the broadband (20-200 kHz) electron
temperature fluctuations. Both our findings in the frequency domain through
cross-coherence analysis and the time-domain $\tilde{T}_{e}/T_{e}(t)$
demonstrate that transient changes are intimately associated with turbulent
dynamics. During non-locality phases, the $\tilde{T}_{e}/T_{e}$ temporal
profiles exhibit marked differences between the central and edge regions.
Consequently, we challenge the conventional understanding of local
$a/L_{T_{e}}$ driving mechanisms. Furthermore, our observations reveal the
rapid formation of long radial coherent fluctuations, with coherence lengths
comparable to the plasma size (approximately 0.5$a$). This phenomenon arises
immediately after pulse injection, effectively breaking down local transport
relations. In essence, transient transport can be initiated by distant
effects.

Simultaneously investigating plasma rotation, we identify a partial alignment
between $E\times B$ flow shear and the transient fluctuation changes, without
accounting for instability growth rates. However, in a comparative high-
density scenario, our observations manifest stark differences, with no clear
indications of core-edge transient interactions. When the non-local conditions
are met, particularly in scenarios characterized by lower density and larger
plasma current, we observe a simultaneous decrease in flow shearing rates and
rapid turbulence growth in the same region. This observation suggests that
flow shearing rates could significantly contribute to the formation of long
radial coherent structures due to their interplay with turbulence. Our direct
observations show an increase in core shear rates within 1-2 ms, which
corresponds to transient changes in $\tilde{T}_{e}/T_{e}$. Notably, this
propagation speed of amplitude change is on the order of $10^{2}$ m/s, far
exceeding the device’s global confinement time of 30 ms. Considering that the
inward coherent length extends to approximately 8.8 cm with the reference at
$\rho=0.80$, it encompasses a significant portion of the plasma’s minor
radius.

In addition, these observations are contingent on diagnostic compatibility,
which can be further optimized. For instance, the measured
$\tilde{T}_{e}/T_{e}$ profile exhibits a steep decay, reaching noise levels at
relatively peripheral radii. This limitation could be mitigated by employing a
smaller beam size or extending the Rayleigh length of the diagnostic sight.
Additionally, the $v_{\perp}$ of the flow will be detected in the central core
using a higher radio frequency of the reflectometer. The observed fluctuation
bifurcation hints at the potential formation of a barrier-like structure, akin
to a transient internal transport barrier (ITB) xu , in the inverse region.
Evidence of interactions between shear profiles and structures can also be
gleaned from comparative experiments, where the location of the barrier
(sudden change) varies. Importantly, these results are corroborated by their
disappearance or significant weakening in scenarios without non-locality.

In general, local plasma parameters play a pivotal role in driving
instabilities, such as pressure gradients, ion and electron temperature
gradients. Even in cases of non-diffusive momentum transport, which falls
under the umbrella of locality since these non-diffusive terms are determined
by local parameters, non-local transport exhibits a clear departure from local
closure. Non-local transport encompasses the influence of parameters located
far from where the transport occurs, challenging the notion that electrostatic
turbulence alone dictates heat flux. Instead, it is primarily determined by
the coupling between turbulence and electric fluctuations. Nonetheless, the
presence of shear and long radial coherence in electron temperature
fluctuations provides valuable insights. This study does not delve into other
processes, such as nonlinear multi-scale turbulent interactions, turbulence
spreading, and density-temperature bicoherence. However, it’s worth noting
that in the theory of self-organized criticality, enhanced avalanches can
occur when a long radial correlation is present pan2015evidence . This could
potentially serve as a mechanism for transient flow variations. To conduct a
comprehensive investigation, future experiments will explore electric field,
density, and temperature channels.

Acknowledgement The authors thank to Prof. Inagaki for useful analytic
discussions in APTWG meeting. This work is partly supported by National Key
R&D Program of China (Grant Nos. 2017YFE0301203, 2017YFE0301106), Sichuan
Science and Technology Program (Grant No. 2018RZ0123, 2021YFSY0044), National
Natural Science Foundation of China (Grant Nos. 12175055), and Shenzhen
Municipal Collaborative Innovation Technology Program-International Science
and Technology Cooperation Project (GJHZ20220913142609017).

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# Maximum likelihood thresholds of generic linear concentration models

Daniel Irving Bernstein, Steven J. Gortler, and Louis Theran

###### Abstract

The maximum likelihood threshold of a statistical model is the minimum number
of datapoints required to fit the model via maximum likelihood estimation. In
this paper we determine the maximum likelihood thresholds of generic linear
concentration models. This turns out to be the number one would expect from a
naive dimension count, which is surprising and nontrivial to prove given that
the maximum likelihood threshold is a semi-algebraic concept. We also describe
geometrically how a linear concentration model can fail to exhibit this
generic behavior and briefly discuss connections to rigidity theory.

##  1 Introduction

Fitting a model with relatively small datasets is a challenging problem that
arises in many applications [17, Chapter 18], including modeling networks
related to gene regulation [13, 21, 23] and metabolic pathways [18]. In this
paper, we are concerned with a special case of the following.

######  Question 1.1.

Let $\mathcal{M}$ be an $n$-variate Gaussian model, i.e. a parameterized
family of $n$-variate Gaussian density functions. What is the minimum $d$ such
that given a dataset consisting of $d$ datapoints,111assumed to be
independently and identically distributed according to a probability
distribution that is mutually absolutely continuous with respect to Lebesgue
measure on $\mathbb{R}^{n}$ then, almost surely, there exists some
$f\in\mathcal{M}$ of maximum likelihood?

The $d$ in Question 1.1 for a particular $\mathcal{M}$ is called the _maximum
likelihood threshold_ of $\mathcal{M}$. In this paper, we consider the
particular case of Question 1.1 that takes $\mathcal{M}$ to be a _linear
concentration model_. Such models consist of all $n$-variate normal
distributions whose inverse covariance matrix (often called the _concentration
matrix_) lies in a given linear subspace $L$ of $n\times n$ symmetric
matrices; we denote the corresponding model by $\mathcal{M}_{L}$. When $L$ is
defined by setting certain off-diagonal entries to zero, $\mathcal{M}_{L}$ is
called a _Gaussian graphical model_. Maximum likelihood thresholds of Gaussian
graphical models, and related quantities, have been studied extensively in the
algebraic statistics literature [4, 5, 6, 7, 14, 22].

The main result of this paper is Theorem 3.23 which gives the maximum
likelihood threshold $\mathcal{M}_{L}$ where $L$ is a _generic_ linear
subspace of symmetric matrices. In short, it says that for a generic $L$ the
maximum likelihood threshold can be computed by a naive dimension count, using
no more information about $L$ than its dimension. This is the nicest possible
answer one could hope for, but since we are working over the real field, it
does not immediately follow from algebro-geometric considerations.

One appealing feature of our approach is that we are explicit about which
linear subspaces fail to attain this naive bound. Linear concentration models
that appear in practice often exhibit behaviour that is different from the
generic one. Notably, the MLT of a Gaussian graphical model depends on more
than the number of edges in the associated graph. The results in this paper
can, then, be interpreted as describing geometric conditions on $L$ that lead
to more complicated behaviour of the MLT.

We begin with Theorem 2.1, which establishes two conditions that are
equivalent to the existence of a maximum likelihood estimate in
$\mathcal{M}_{L}$ given a particular dataset. We then use Theorem 2.1 to
establish a lower bound for the maximum likelihood threshold of
$\mathcal{M}_{L}$ in Proposition 3.15, and then again to establish that this
lower bound is also an upper bound in Proposition 3.21. Section 4 studies an
easily computable upper bound on the maximum likelihood threshold called the
_generic completion rank_. This generalizes a notion of the same name for
Gaussian graphical models. Using this this language, Theorem 3.23 says that
the generic completion rank of a generic linear space is equal to its maximum
likelihood threshold. Section 5 explores connections to rigidity theory.

###  1.1 Notation

In what follows, $\mathcal{S}^{n}$ denotes the vector space of $n\times n$
real symmetric matrices, $\mathcal{S}^{n}_{+}$ denotes the closed subcone of
positive semidefinite (PSD) matrices, and $\mathcal{S}^{n}_{++}$ denotes the
open subcone of positive definite matrices. A _linear concentration model_ is
a set of the form $L\cap\mathcal{S}^{n}_{++}$ where
$L\subseteq\mathcal{S}^{n}$ is a linear space. The identity matrix of size
$k\times k$ is denoted $I_{k}$.

##  2 Motivation in detail

Given datapoints $x_{1},\dots,x_{d}\in\mathbb{R}^{n}$ with sample covariance
matrix $S$, the _maximum likelihood estimate in the linear covariance model
for $L$_ is the solution, if it exists, to the following optimization problem
(see e.g. [17, p. 632])

—l— KTr(SK) - logdetK K ∈S^n_++ and K ∈L.

The _maximum likelihood threshold_ of a linear space
$L\subseteq\mathcal{S}^{n}$, denoted $\operatorname{MLT}(L)$, is the minimum
$d$ such that (2) has a solution for all generic $S\in\mathcal{S}^{n}_{+}$ of
rank $d$. Note that if $L$ does not contain a positive definite matrix, then
$\operatorname{MLT}(L)$ is infinite so we restrict our focus to linear spaces
that contain a positive definite matrix. The set $\mathcal{S}^{n}_{++}\cap L$
is called the _feasible region_ its elements are called _feasible points_.
Given an inner product space $V$ and a linear subspace $L\subseteq V$,
$L^{\perp}$ denotes the orthogonal complement of $L$ and $\pi_{L}:V\rightarrow
L$ denotes orthogonal projection onto $L$ with kernel $L^{\perp}$. Here, we
are mainly interested in the $\binom{n+1}{2}$-dimensional space
$\mathcal{S}^{n}$ with the inner product of $\Omega,S\in\mathcal{S}^{n}$ given
by $\operatorname{Tr}(\Omega S)$. If $\Omega$ and $S$ are both PSD, then
$\operatorname{Tr}(\Omega S)=0$ if and only if $\Omega S=0$

######  Theorem 2.1.

Let $S\in\mathcal{S}^{n}_{+}\setminus\\{0\\}$ and let
$L\subseteq\mathcal{S}^{n}$ be a linear subspace such that
$L\cap\mathcal{S}^{n}_{++}\neq\emptyset$. Then the following are equivalent:

  1. (1)

there is a (unique) optimum solution to the problem (2),

  2. (2)

there exists a matrix $K\in\mathcal{S}^{n}_{++}$ such that $S-K^{-1}\in
L^{\perp}$, and

  3. (3)

there does not exist a non-zero matrix $\Omega\in\mathcal{S}^{n}_{+}\cap L$
such that $\operatorname{Tr}(\Omega S)=\Omega S=0$.

In this case, the optimum is the unique matrix $K$ satisfying $K\in
L\cap\mathcal{S}^{n}_{++}$ and $S-K^{-1}\in L^{\perp}$.

###### Proof.

$(1)\implies(2)$: Let $K$ be the assumed optimal point. Since the objective
function is strictly convex, $K$ is the unique optimizer. Since $K$ is a
feasible point, $K\in L\cap\mathcal{S}^{n}_{++}$. The gradient of the
objective function at $K$ is $S-K^{-1}$. As $K$ is in the relative interior of
the feasible region, the principle of Lagrange multipliers says that the
gradient must be orthogonal to the constraints at an optimal value. Hence,
$S-K^{-1}\in L^{\perp}$, as desired.

$(2)\implies(3)$: Let $K$ satisfy $(2)$ and let
$\Omega\in\mathcal{S}^{n}_{+}\cap L$ such that $\operatorname{Tr}(\Omega S)=0$
be given. From $(2)$, $\Omega$ and the gradient $S-K^{-1}$ are orthogonal. We
then compute

$0=\operatorname{Tr}(\Omega(S-K^{-1}))=\operatorname{Tr}(\Omega
S)-\operatorname{Tr}(\Omega K^{-1})=\operatorname{Tr}(\Omega K^{-1})$

using bilinearity of the inner product. Since $\Omega$ and $K^{-1}$ are PSD,
the above implies that the column span of $K^{-1}$ lies in the kernel of
$\Omega$. Since $K^{-1}$ is full rank, it follows that $\Omega$ has an
$n$-dimensional kernel, so $\Omega=0$.

$(3)\implies(1)$: Suppose (3) holds and let $K\in L\cap\mathcal{S}^{n}_{++}$
(recall that we assume that such a $K$ exists). Define $L_{0}$ to be the
intersection of $L$ and the unit sphere in $\mathcal{S}^{n}$, i.e.

$L_{0}:=\\{\Omega\in L:\operatorname{Tr}(\Omega^{2})=1\\}.$

As $K$ is positive definite, for each $\Omega\in L_{0}$ there exists
$\lambda>0$ such that $K+\lambda\Omega\in L\cap\mathcal{S}^{n}_{++}$; define
$\lambda_{\Omega}$ to be the supremum over all such $\lambda$. Define
$f(K):=\operatorname{Tr}(SK)-\log\det K$. We will show that for any $T\in
L_{0}$,

$f(K+\lambda\Omega)\rightarrow\infty\quad{\rm
as}\quad\lambda\rightarrow\lambda_{\Omega}.$ (A)

It will then follow that $f(K+\lambda\Omega)>f(K)$ outside of a compact
neighborhood of $K$. The restriction of $f$ to this neighborhood has a global
minimum (as a continuous function on a compact set), which, by construction,
is a global minimum of $f$. In other words, the problem (2) has an optimal
solution. The rest of the proof establishes Equation (A) by a case analysis.

If $\Omega\notin\mathcal{S}^{n}_{+}$ then $\lambda_{\Omega}$ is finite, since
for large enough $t$, the signature of $K+t\Omega$ will agree with that of
$\Omega$. Hence $K+t_{\lambda}\Omega$ lies on the boundary of the PSD cone
$\mathcal{S}^{n}_{+}$, making it singular. Hence, for
$0<\lambda<\lambda_{\Omega}$, the term $\operatorname{Tr}(S(K+\lambda\Omega))$
is bounded. Then (A) follows since

$\log\det(K+\lambda\Omega)\to-\infty\qquad\text{as
$\lambda\to\lambda_{\Omega}$}.$

If $\Omega\in\mathcal{S}^{n}_{+}$ then $\lambda_{\Omega}=\infty$, since for
any $\lambda>0$, $K+\lambda\Omega$ is the sum of a PD and a PSD matrix, which
is again PD. Since $\det(K+\lambda\Omega)$ is a polynomial in $\lambda$ of
degree $n$, it follows that

$f(K+\lambda\Omega)=\operatorname{Tr}(SK)+\lambda\overbrace{\operatorname{Tr}(S\Omega)}^{>0}-O(\log(\lambda)).$

This diverges to $\infty$ as $\lambda\rightarrow\infty$. ∎

##  3 Generic linear subspaces

In this section we derive the maximum likelihood threshold of a generic linear
subspace $L\subseteq\mathcal{S}^{n}$. A note on our use of geometric and
topological language is in order. Let $V$ be a finite-dimensional vector
space. Throughout this paper, $V$ will either be $\mathbb{R}^{n}$, equipped
with the standard inner product, or $\mathcal{S}^{n}$, equipped with the
trace-norm inner product, i.e. where the inner product of matrices $A$ and $B$
is defined to be $\operatorname{Tr}(AB)$.

The _Zariski topology_ on $V$ is the topology whose closed sets are varieties.
We use the convention that varieties are the vanishing loci of systems of
polynomials. In particular, they need not be irreducible. The _Euclidean
topology_ on $V$ is the more familiar topology derived from the metric
associated to the inner product on $V$. The topology induced on the projective
space $\mathbb{P}(V)$ by the Zariski/Euclidean and the usual quotient map will
be referred to by the same name.

The Zariski/Euclidean topology on any subset $X$ of $V$ or $\mathbb{P}(V)$ is
the subspace topology on $X$ induced by the Zariski/Euclidean topology on $V$
or $\mathbb{P}(V)$. Topological statements that do not explicitly specify a
topology should be interpreted in the Euclidean topology. We will always
explicitly say so when we are using the Zariski topology. For any subset $X$
of $V$ or $\mathbb{P}(V)$, the Zariski topology is coarser than the Euclidean
topology.

Recall that the _interior_ of a subset $X$ of any topological space is the
union of all open sets contained in $X$. Now let $X\subseteq V$. The _affine
hull_ of $X$ is the intersection of all affine subspaces (i.e. translations of
linear subspaces) containing $X$. The affine hull of $X$ comes equipped with
the subspace topology induced by the Euclidean topology on $V$. The _relative
interior_ of $X$ is the interior of $X$ in this subspace topology. Every
convex set has non-empty relative interior.

######  Definition 3.1.

Let $X$ be an irreducible semi-algebraic set. We say that a statement is
_generically_ true on $X$, or holds at _generic points_ of $X$ if it is true
on a nonempty Zariski-open subset of $X$.

Definition 3.1 requires $X$ to be irreducible so that the set of points where
a generically true statement fails is nowhere-dense in $X$. The following
lemma give a sufficient condition for a semi-algebraic subset of a semi-
algebraic set to be contained in a Zariski-closed subset.

######  Lemma 3.2.

Let $X$ be a semi-algebraic subset of some real vector space $V$ and let
$E\subseteq X$ be a semi-algebraic subset. If for all $y\in E$ and for every
Euclidean neighborhood $U$ of $y$ in $X$, there exists some $x\in X\setminus
E$, then $E$ is contained in a Zariski-closed proper subset of $X$.

###### Proof.

First assume $X$ is smooth. As $E$ is semialgebraic, $E$ can be stratified
into finitely many smooth semialgebraic sets $E_{1},\dots,E_{k}$ [3]. Our
assumptions imply that $E$ is nowhere dense in $X$; the same therefore holds
for each $E_{i}$. Since each $E_{i}$ is a nowhere-dense smooth submanifold of
$X$, each $E_{i}$ has strictly lower dimension than $X$. Thus $E$ is of lower
dimension than $X$. Meanwhile, by the results in [8, Section 2.8], the real
Zariski closure $Z$ of $E$ in $V$ has the same dimension as $E$. Hence
$X\nsubseteq Z$, and so $Z\cap X$ is the desired subset.

If $X$ is not smooth, then the non-smooth locus of $X$ is contained a semi-
algebraic subset $W^{\prime}$ of $X$ of lower dimension (this follows from
Proposition 5.53 in [3]). We will denote the Zariski closure of $W^{\prime}$
in $X$ by $W$, which has the same dimension as $W^{\prime}$. Then applying the
previous argument to intersection of the smooth locus of $X$ with $E$ gives us
a Zarkiski-closed subset $Z$ of $X$. Then $Z\cup W$ contains $E$ and is
Zariski-closed in $X$. ∎

In order to talk about generic linear spaces (in the sense of Definition 3.1),
we need a way to view each linear space as an element of some semi-algebraic
set. A _Grassmannian_ , defined and explained below, is that semi-algebraic
set. A _generic linear space_ is a generic point of some Grassmannian.

######  Definition 3.3.

Let $V$ be a vector space and let $d\leq\dim(V)$ be an integer. The
_Grassmannian of $d$-planes in $V$_, denoted $\operatorname{Gr}(d,V)$ is the
set of all $d$-dimensional linear subspaces of $V$.

Given a field $\mathbb{F}$ and an $\mathbb{F}$-vector space $V$, we briefly
explain one way to view the Grassmannian $\operatorname{Gr}(d,V)$ as a
projective variety. For more details, see, e.g., [16, Lecture 6]. Let $L$ be a
$d$-dimensional linear subspace of an $n$-dimensional vector space $V$ over a
field $\mathbb{F}$ with a distinguished basis $\\{e_{1},\ldots,e_{n}\\}$. If
$v_{1},\ldots,v_{d}$ is a basis of $L$, the vector of the $\binom{n}{d}$
minors of the $d\times n$ matrix $M$ that has as its rows the coordinates of
the $v_{i}$ in the basis $\\{e_{j}\\}$ are called the _Plücker coordinates_ of
$L$. Changing basis of $L$ amounts to replacing $M$ by $BM$ for some
invertible $d\times d$ matrix $B$, so changing basis of $L$ scales the Plücker
coordinates. Conversely, a basis of $L$ can be reconstructed from its Plücker
coordinates, and the map that sends $L$ to its Plücker coordinates is a closed
embedding
$\operatorname{Gr}(d,V)\to\mathbb{P}\left(\mathbb{F}^{\binom{n}{d}}\right)$.
In what follows, we are interested in the case where $\mathbb{F}=\mathbb{R}$,
which makes $\operatorname{Gr}(d,V)$, additionally, a semi-algebraic set.

In what follows, we will use the idea of Lemma 3.2 on subsets of a
Grassmannian. Since it is easier to describe perturbations of bases instead of
points on the Grassmannian, we provide a lemma that lets us switch back and
forth.

######  Lemma 3.4.

Let $E\subseteq\operatorname{Gr}(d,V)$ be a semi-algebraic subset of the
Grassmannian of $d$-dimensional subspaces of an $n$-dimensional real inner-
product space $V$. Suppose for each $L\in E$, there exists a basis
$v_{1},\ldots,v_{d}$ of $L$, such that every neighborhood of
$(v_{1},\ldots,v_{d})\in V^{d}$ contains a point $(w_{1},\ldots,w_{d})$ so
that the span of $\\{w_{1},\dots,w_{d}\\}$ is not in $E$. Then $E$ is
contained is a Zariski-closed proper subset of $\operatorname{Gr}(d,V)$.

###### Proof.

Let $L\in E$ and let $B$ be a Euclidean neighborhood of $L$ in
$\operatorname{Gr}(d,V)$. Let $p:V^{d}\rightarrow\operatorname{Gr}(d,V)$ be
the map that sends a point $(v_{1},\ldots,v_{d})\in V^{d}$ to the Plücker
coordinates of its span. Because $p$ is continuous, the set $U:=p^{-1}(B)$ is
an open subset of $V^{d}$ that contains every basis of $L$. By hypothesis
there is some basis $(v_{1},\ldots,v_{d})$ of $L$ so that any neighborhood
$U^{\prime}$ of it contains a point $(w_{1},\ldots,w_{d})\in V^{d}$ with the
property that $p(w_{1},\ldots,w_{d})$ is not in $E$. In particular,
$B=p(U)\nsubseteq E$.

As $L\in E$ and $B$ were arbitrary, $E$ has empty Euclidean interior in
$\operatorname{Gr}(d,V)$ and is therefore of lower dimension The result then
follows as in the proof of Lemma 3.2. ∎

###  3.1 Lower bound on the MLT

The punchline of this subsection is Proposition 3.15 which lower-bounds the
maximum likelihood threshold of $\mathcal{M}_{L}$ for generic $L$.

To begin, we need the description of the faces of the PSD cone.

######  Lemma 3.5.

Given a $d$-dimensional linear subspace $V$ of $\mathbb{R}^{n}$, define

$F_{V}:=\\{A\in\mathcal{S}^{n}_{+}:Ax=0\ {\rm for\ all}\ x\in V\\}.$

Then $F_{V}$ is a face of $\mathcal{S}^{n}_{+}$ and is linearly isomorphic to
$\mathcal{S}^{n-d}_{+}$ (so in particular of dimension $\binom{n-d+1}{2}$).
Moreover, every face of $\mathcal{S}^{n}_{+}$ is of this form.

###### Proof.

See [20, Theorem A.2]. ∎

The following key lemma is a slight variation of results that can be found in
[2] and [19].

######  Lemma 3.6.

Let $d\leq n-1$. Let $L\subseteq\mathcal{S}^{n}$ be a linear space such that
$L\cap\mathcal{S}^{n}_{++}\neq\emptyset$. Assume $L$ has dimension $m$ where

$m\geq dn-\binom{d}{2}-n+d+1.$

Then $L$ contains a nonzero PSD matrix of rank $n-d$ or less.

###### Proof.

By assumption, $L$ contains a non-zero PSD matrix. Let $s$ be the smallest,
non-zero, rank of a PSD matrix in $L$. Suppose $s\geq(n-d+1)$ (which is $\geq
2$). Let $\Omega$ be such a matrix of rank $s$. Let $F$ be the unique face of
$\mathcal{S}^{n}_{+}$ containing $\Omega$ in its relative interior. Lemma 3.5
implies that $\dim(F)=\binom{s+1}{2}\geq\binom{n-d+2}{2}$. Therefore
$\dim(F)+\dim(L)$ is at least

$dn-\binom{d}{2}-n+d+1+\binom{n-d+2}{2}=\binom{n+1}{2}+2.$

Since $L$ intersects the relative interior $F$, this implies $\dim(F\cap
L)\geq 2$. Thus $L$ contains an affine line $l$ that does not go through the
origin, and $l$ has a one dimensional intersection with $F$. Meanwhile, since
$F$ is convex, line-free and closed, this line must intersect the relative
boundary of $F$ (in at least one of its two directions) at a non-zero point.
But any such intersection point has rank at most $s-1$, contradicting the
minimality of $s$. ∎

######  Lemma 3.7.

Let $n$ be a positive integer, let $d\leq n-1$ and let

$m\geq dn-\binom{d}{2}-n+d+1.$

Let $E\subseteq\operatorname{Gr}(m,\mathcal{S}^{n})$ be the set of linear
spaces that contain a positive definite matrix but do not contain a PSD matrix
of rank $n-d$. Then $E$ lies inside a Zariksi-closed proper subset of
$\operatorname{Gr}(m,\mathcal{S}^{n})$.

###### Proof.

Let $L\in E$. Lemma 3.6 implies that $L$ contains a non-zero PSD matrix
$\Omega$ with rank at most $(n-d)$. Assume that $\Omega$ has rank strictly
less than $n-d$. Given any neighborhood $U$ in $(\mathcal{S}^{n})^{d}$ of an
ordered basis $(\Omega_{1},\dots,\Omega_{d})$ such that $\Omega_{n}=\Omega$,
there exists some PSD $\Omega^{\prime}$ of rank $n-d$ such that
$(\Omega_{1},\dots,\Omega_{d-1},\Omega^{\prime})$ lies in $U$. In this case,
the span of $\\{\Omega_{1},\dots,\Omega_{d-1},\Omega^{\prime}\\}$ does not lie
in $E$. The desired result now follows from Lemma 3.4. ∎

######  Example 3.8.

We give an example where the $E$ in Lemma 3.7 is not empty. Take $d=1$, let
$n$ be even and let $L^{\prime}$ be a linear subspace of $\mathcal{S}^{n/2}$
of dimension $(n+2)/2$. Define $L$ to be the following linear subspace of
$\mathcal{S}^{n}$

$L=\left\\{\begin{pmatrix}A&0\\\ 0&A\end{pmatrix}|A\in L^{\prime}\right\\}.$

Then $L$ has dimension $n+2$ which abides by the lower bound in Lemma 3.7. If
$L^{\prime}$ contains a positive definite matrix, then so does $L$. However,
$L$ does not contain _any_ matrices of rank $n-d=n-1$ since it only contains
matrices of even rank.

We note that the behavior observed in Example 3.8 cannot happen when $L$ is
defined by setting certain off-diagonal entries to zero, i.e. the linear
spaces defining Gaussian graphical models This is because such linear spaces
include all diagonal matrices.

######  Definition 3.9.

For integers $1\leq d\leq n$, let $V(n,d)\subseteq\mathcal{S}^{n}$ denote the
quasi-affine variety of $n\times n$ symmetric matrices of rank $d$. Define

$B(n,d):=\\{(\Omega,S):\Omega\in V(n,n-d),S\in V(n,d)\ {\rm such\ that}\
\Omega S=0\\}$

and let $\pi_{n,d}:B(n,d)\rightarrow V(n,n-d)$ and
$\tau_{n,d}:B(n,d)\rightarrow V(n,d)$ be the canonical projection maps, i.e.
$\pi_{n,d}(\Omega,S)=\Omega$ and $\tau_{n,d}(\Omega,S)=S$.

######  Lemma 3.10.

Let $d,n$ be integers such that $1\leq d\leq n$. Then $B(n,d)$ is a smooth
submanifold of $\mathcal{S}^{n}\times\mathcal{S}^{n}$.

###### Proof.

For each $1\leq i\leq n-d$ and $1\leq j\leq d$ let $X(i,j)$ denote the subset
of $B(n,d)$ consisting of all pairs $(\Omega,S)$ such that $\Omega$ and $S$
have $i$ and $j$ positive eigenvalues, respectively. Since $X(i,j)$ is semi-
algebraic, it has a smooth point. Since the $X(i,j)$s form a partition of
$B(n,d)$, it is now enough to show that the group of linear automorphisms of
$B(n,d)$, i.e. the subgroup of the general linear group on
$\mathcal{S}^{n}\times\mathcal{S}^{n}$ that send $B(n,d)$ to itself, acts
transitively on each $X(i,j)$.

We now describe some invertible linear automorphisms of $B(n,d)$. If $Q$ is
any orthogonal matrix and $(\Omega,S)\in B(n,d)$, then

$(Q^{T}\Omega Q)(Q^{T}SQ)=Q^{T}0Q=0.$

Since $Q^{T}\Omega Q$ and $Q^{T}SQ$ are both symmetric, the map

$(\Omega,S)\mapsto(Q^{T}\Omega Q,Q^{T}SQ)$

on $\SS^{n}$ restricts to a linear automorphism of $B(n,d)$. Since $\Omega$
and $S$ are symmetric, their column spans are orthogonal complements. Thus if
$D$ is any diagonal matrix, the map

$(\Omega,S)\mapsto(D\Omega D,DSD)$

on $\mathcal{S}^{n}$ also restricts to a linear automorphism of $B(n,d)$.

Now let $(\Omega,S)\in X(i,j)$. Since the column-span of $\Omega$ is the
kernel of $S$ and vice versa, the spectral theorem provides a linear
automorphism of $B(n,d)$ sending $(\Omega,S)$ to

$\left(\begin{pmatrix}\Omega^{\prime}&0\\\
0&0\end{pmatrix},\begin{pmatrix}0&0\\\ 0&S^{\prime}\end{pmatrix}\right)$

where $\Omega^{\prime}$ and $S^{\prime}$ are diagonal and non-singular. By
permuting rows and columns and then scaling symmetrically by multiplying on
both sides by an appropriate diagonal matrix, we find a morphism
$\varphi_{\Omega,S}$ of $B(n,d)$ that sends $(\Omega,S)$ the unique point in
$X(i,j)$ of the form

$\left(\begin{pmatrix}A&0\\\ 0&0\end{pmatrix},\begin{pmatrix}0&0\\\
0&B\end{pmatrix}\right)$

where $A$ and $B$ are diagonal with $\pm 1$ diagonal entries in descending
order. Given two points $(\Omega_{1},S_{1})$ and $(\Omega_{2},S_{2})$ in
$B(n,d)$, we have a linear automorphism
$\varphi_{\Omega_{2},S_{2}}^{-1}\circ\varphi_{\Omega_{1},S_{1}}$ sending
$(\Omega_{1},S_{1})$ to $(\Omega_{2},S_{2})$. Thus the linear automorphism
group of $B(n,d)$ acts transitively on each $X(i,j)$. ∎

Because $B(n,d)$ is smooth, we can analyze it using basic tools from
differential topology which we now introduce. For a more background, see e.g.
[15]. Given a smooth manifold $X$ and a point $p\in X$, the tangent space of
$X$ at $p$ will be denoted $T_{p}X$. Let $Y,Z$ be smooth submanifolds of a
smooth manifold $X$. Given $p\in Y\cap Z$, one says that $Y$ and $Z$
_intersect transversally at $p$ in $X$_ if $T_{p}Y+T_{p}Z=T_{p}X$. If this
holds for every $p\in Y\cap Z$ then one simply says that $Y$ and $Z$ intersect
transversally in $X$. When $X$ is a vector space, we will identify $T_{p}X$
with $X$ for all $p\in X$. In this case, if $Y$ and $Z$ are Euclidean-open
subsets of vector subspaces of $X$, then $Y$ and $Z$ intersect transversally
if and only if they intersect transversally at a single $p\in Y\cap Z$. For
our purposes, the key utility of transversality is the following lemma.

######  Lemma 3.11 ([15, p. 30, inter alia]).

Let $X$ be a smooth manifold with smooth submanifolds $Y$ and $Z$ If $Y$ and
$Z$ intersect transversally, then $Y\cap Z$ is a smooth submanifold of $X$.
Moreover, at any $p\in Y\cap Z$, we have $T_{p}(Y\cap Z)=T_{p}Y\cap T_{p}Z$.

We now define some specialized notation that we will need throughout the rest
of this section. Let $1\leq d\leq n$ be integers and fix a linear subspace
$L\subseteq\mathcal{S}^{n}$. Assume that $L$ contains a matrix $\Omega_{0}$ of
rank $n-d$ and let $S_{0}$ be a matrix of rank $d$ such that
$\Omega_{0}S_{0}=0$, i.e. such that $(\Omega_{0},S_{0})\in B(n,d)$. The set
$\pi_{n,d}(\tau_{n,d}^{-1}(S_{0}))$ consists of all
$\Omega\in\mathcal{S}^{n}(n-d)$ with the same kernel as $\Omega_{0}$. In
particular, it does not depend on our choice of $S_{0}$ so we denote it by
$Z(\Omega_{0})$. Moreover, the Zariski closure of $Z(\Omega_{0})$ is a linear
space which we denote by $\overline{Z}(\Omega_{0})$. Therefore the tangent
space of $Z(\Omega_{0})$ at any point is $\overline{Z}(\Omega_{0})$. Lemma
3.12 below is well-known but we prove it anyway for the sake of clarity. It
implies that the dimension of $Z(\Omega_{0})$ is

$(n-d)^{2}-\binom{n-d}{2}=\binom{n-d+1}{2}.$

######  Lemma 3.12.

Let $W$ be a linear subspace of $\mathbb{R}^{n}$ of dimension $k$. The set
$\mathcal{K}$ of $n\times n$ symmetric matrices whose columns span $W$ has
dimension $k^{2}-\binom{k}{2}$.

###### Proof.

The dimension of $\mathcal{K}$ is the same as that of
$\mathcal{K}\cap\mathcal{S}^{n}_{+}$, which by Lemma 3.5 implies the lemma. ∎

What follows is a key technical novelty in this paper: we explore whether
$Z(\Omega)$ and $L$ intersect transversally. In Lemma 3.13 we show that this
property allows us, starting at some point $(\Omega_{0},S_{0})\in B(n,d)$, to
perturb $S_{0}$ in any direction, and to counter this with an appropriate
change to $\Omega_{0}$ so that the new pair remains in $B(n,d)$. This is akin
to the situation in rigidity theory where, starting with an infinitesimally
rigid framework that has an equilibrium stress, one can perturb the framework
and then appropriately track this with an updated equilibrium stress. (See
[11, Lemma 5.17].) This connection is further developed in Remark 3.22. On the
other hand, once we have reduced the problem to one of transversality, it is
easy to show (Lemma 3.14) that this must happen generically once the dimension
of $L$ is sufficiently high.

The next lemma show how transversality leads to this ability to perturb-and-
track within a neighborhood of $B(n,d)$.

######  Lemma 3.13.

Let $d,n$ be integers such that $1\leq d\leq n$. Let
$L\in\operatorname{Gr}(m,\mathcal{S}^{n})$ satisfy

  1. 1.

$L$ contains a matrix $\Omega_{0}$ of rank $n-d$, and

  2. 2.

$L$ intersects $Z(\Omega_{0})$ transversally in $\mathcal{S}^{n}$.

Let $S_{0}\in V(n,d)$ satisfy $\Omega_{0}S_{0}=0$. Then every neighborhood $W$
of $S_{0}$ in $V(n,d)$ satisfies the property that $\tau_{n,d}(W)$ contains a
neighborhood of $S_{0}$ in $V(n,d)$.

###### Proof.

The assumed transversality implies that we can express $\mathcal{S}^{n}$ as
the following direct sum

$\mathcal{S}^{n}=L\oplus\overline{Z}(\Omega_{0}).$

Thus every point in $\mathcal{S}^{n}\times\mathcal{S}^{n}$ can be written as
$(x,0)+(z,0)+(0,y)$ for some $x\in L$, $z\in\overline{Z}(\Omega_{0})$ and
$y\in\mathcal{S}^{n}$. Note that

$(x,0)+(0,y)\in L\times\mathcal{S}^{n}\qquad{\rm
and}\qquad(z,0)\in\overline{Z}(\Omega_{0})\times\\{0\\}.$

Meanwhile, $\overline{Z}(\Omega_{0})\times\\{0\\}$ is the tangent space to
$\overline{Z}(\Omega_{0})\times\\{S_{0}\\}$ at $(\Omega_{0},S_{0})$. Together,
this implies that $L\times\mathcal{S}^{n}$ intersects
$Z(\Omega_{0})\times\\{S_{0}\\}$ transversally in
$\mathcal{S}^{n}\times\mathcal{S}^{n}$ at $(\Omega_{0},S_{0})$.

By assumption, we have $S_{0}\in V(n,d)$ and $\Omega_{0}S_{0}=0$. So
$(\Omega_{0},S_{0})\in B(n,d)$ and $Z(\Omega_{0})\times\\{S_{0}\\}$ is a
smooth submanifold of $B(n,d)$. The above implies that
$L\times\mathcal{S}^{n}$ intersects $B(n,d)$ transversally in
$\mathcal{S}^{n}\times\mathcal{S}^{n}$. Lemma 3.11 then implies that
$(L\times\mathcal{S}^{n})\cap B(n,d)$ is a smooth manifold.

Now let $p\in T_{(\Omega_{0},S_{0})}B(n,d)$. Viewing $p$ as an element of
$\mathcal{S}^{n}\times\mathcal{S}^{n}$, transversality of the intersection of
$L\times\mathcal{S}^{n}$ and $Z(\Omega_{0})\times\\{S_{0}\\}$ in
$\mathcal{S}^{n}\times\mathcal{S}^{n}$ implies that there exist $q\in
L\times\mathcal{S}^{n}$ and $r\in\overline{Z}(\Omega_{0})\times\\{0\\}$ such
that $p=q+r$. Since $Z(\Omega_{0})\times\\{S_{0}\\}$ is a smooth submanifold
of $B(n,d)$, $r$ lies in $T_{(\Omega_{0},S_{0})}B(n,d)$. Since $p$ does too by
assumption, $q$ must also lie in $T_{(\Omega_{0},S_{0})}B(n,d)$. Since $q\in
L\times\mathcal{S}^{n}$ and $q\in T_{(\Omega_{0},S_{0})}B(n,d)$, then from
Lemma 3.11, $q\in T_{(\Omega_{0},S_{0})}[(L\times\mathcal{S}^{n})\cap
B(n,d)]$. We have therefore just shown that $(L\times\mathcal{S}^{n})\cap
B(n,d)$ and $Z(\Omega_{0})\times\\{S_{0}\\}$ intersect transversally as
submanifolds of $B(n,d)$ at $(\Omega_{0},S_{0})$.

Next, consider the (surjective) differential of $\tau_{n,d}$ at the point
$(\Omega_{0},S_{0})\in B(n,d)$:

$d_{(\Omega_{0},S_{0})}\tau_{n,d}:T_{(\Omega_{0},S_{0})}B(n,d)\rightarrow
T_{S_{0}}V(n,d).$

The space $T_{(\Omega_{0},S_{0})}Z(\Omega_{0})\times\\{S_{0}\\}$ is the kernel
of $d_{(\Omega_{0},S_{0})}\tau_{n,d}$. Meanwhile we have shown that

$T_{(\Omega_{0},S_{0})}B(n,d)=T_{(\Omega_{0},S_{0})}((L\times\mathcal{S}^{n})\cap
B(n,d))\oplus T_{(\Omega_{0},S_{0})}(Z(\Omega_{0})\times\\{S_{0}\\})$

Thus the restriction of $\tau_{n,d}$ to $(L\times\mathcal{S}^{n})\cap B(n,d)$
is a submersion onto $V(n,d)$ at $(\Omega_{0},S_{0})$ (which is also a smooth
manifold). The image of an open set under a submersion is locally open [15,
Page 20]. Thus, if $W$ is any neighborhood of $(\Omega_{0},S_{0})$ in
$(L\times\mathcal{S}^{n})\cap B(n,d)$, then $\tau_{n,d}(W)$ contains a
neighborhood of $S_{0}$ in $V(n,d)$. ∎

The next lemma says that generically, we will see a transverse intersection.

######  Lemma 3.14.

Let $d,m,n$ be integers satisfying $1\leq d\leq n$ and $m\geq
dn-\binom{d}{2}+1$. Let $E\subseteq\operatorname{Gr}(m,\mathcal{S}^{n})$
consist of all linear spaces $L$ satisfying

  1. 1.

$L$ contains a PSD matrix of rank $n-d$, and

  2. 2.

$Z(\Omega)$ and $L$ intersect non-transversally for all PSD $\Omega\in L$ of
rank $n-d$.

Then $E$ lies inside a Zariski-closed proper subset of
$\operatorname{Gr}(m,\mathcal{S}^{n})$.

###### Proof.

Let $L\in E$ and $\Omega\in L\cap\mathcal{S}^{n}_{+}(n-d)$. Define
$k:=\dim(L\cap\overline{Z}(\Omega))$ and observe that $k\geq 1$ since

$\dim(L)+\dim(Z(\Omega))\geq
dn-\binom{d}{2}+1+\binom{n-d+1}{2}=\binom{n+1}{2}+1.$

Thus we have a basis $\Omega=\Omega_{1},\dots,\Omega_{k}$ of
$L\cap\overline{Z}(\Omega)$. Let $\Omega_{k+1},\dots,\Omega_{m}$ be such that
$\Omega_{1},\dots,\Omega_{m}$ is a basis of $L$. Define $l$ to be the expected
dimension of the intersection of $L$ and $\overline{Z}(\Omega)$, i.e.

$l:=\dim(L)+\dim(Z(\Omega))-\binom{n+1}{2}.$

By assumption, $L$ intersects $Z(\Omega)$ non-transversely. Then $k\geq l+1$
and for almost all $\Omega_{l+1}^{\prime},\dots,\Omega_{k}^{\prime}$ the
intersection of the span of

$\Omega_{1},\dots,\Omega_{l},\Omega_{l+1}^{\prime},\dots,\Omega_{k}^{\prime},\Omega_{k+1},\dots,\Omega_{m}$

with $Z(\Omega)$ is transverse and non-empty, and thus not in $E$. In
particular, choose $\Omega_{l+1}^{\prime},\dots,\Omega_{k}^{\prime}$ by
perturbing each of $\Omega_{l+1},\dots,\Omega_{k}$ an arbitrarily small amount
off of $Z(\Omega)$. Meanwhile, $E$ is semi-algebraic by [3, Theorem 1.22].
Thus Lemma 3.4 then implies that $E$ lies in a Zariski-closed proper subset of
$\operatorname{Gr}(m,\mathcal{S}^{n})$. ∎

######  Proposition 3.15.

Let $m,d,n$ be integers such that $1\leq d\leq n$ and $m\geq
dn-\binom{d}{2}+1$ and let $G\subseteq\operatorname{Gr}(m,\mathcal{S}^{n})$
consist of all $L$ such that

$L\cap\mathcal{S}^{n}_{++}\neq\emptyset.$

Then there exists a Zariski-closed proper subset $E\subset G$ such that if
$L\notin E$ the following holds: there exists an open subset
$U\subseteq\mathcal{S}^{n}_{+}(d)$ such that for each matrix $S\in U$, there
exists a rank $(n-d)$ PSD matrix $\Omega\in\mathcal{S}^{n}_{+}\cap L$ such
that $\Omega S=0$.

###### Proof.

Let $E_{1},E_{2}$ denote the Zariski-closed proper subsets of
$\operatorname{Gr}(m,\mathcal{S}^{n})$ indicated in Lemmas 3.7 and 3.14.
Define $E:=(E_{1}\cup E_{2})\cap G$ and assume $L\notin E$. Lemmas 3.6 and 3.7
give us a matrix $\Omega^{\prime}_{0}\in L$ that is PSD of rank $n-d$. Lemma
3.14 then gives us a matrix $\Omega_{0}\in L$ that is PSD of rank $n-d$ such
that $L$ and $Z(\Omega_{0})$ intersect transversely.

Now let $M$ be a matrix whose columns are a basis for the kernel of
$\Omega_{0}$ and define the PSD matrix $S_{0}:=MM^{T}$. Let $N$ be a
neighborhood of $\Omega_{0}$ in $L\cap V(n,n-d)$. By intersecting with the
open subset of $V(n,n-d)$ consisting of PSD matrices, we may assume without
loss of generality that $N$ only contains PSD matrices. Then
$W:=\pi_{n,d}^{-1}(N)$ is an open subset of $(L\times\mathcal{S}^{n})\cap
B(n,d)$ by continuity and contains $(\Omega_{0},S_{0})$. Lemma 3.13 then
implies that $\tau_{n,d}(W)$ contains a neighborhood $U$ of $S_{0}$ in
$V(n,d)$. We may assume as before, without loss of generality, that $U$
consists only of PSD matrices. Then $U$ is the desired neighborhood. ∎

######  Example 3.16.

Consider the case where $d=4$, $n=10$ and $L$ is the space of $10\times 10$
symmetric matrices whose $ij$ entry is $0$ whenever $i\in\\{1,5\\}$ and
$j\in\\{6,10\\}$. This is the linear space corresponding to the Gaussian
graphical model on the complete bipartite graph $K_{5,5}$. In this case,
$\dim(L)=35\geq nd-\binom{d}{2}+1$. Let us consider the set $T$ of
$S\in\mathcal{S}^{n}_{+}(d)$ such that there exists a rank $(n-d)$ PSD matrix
$\Omega\in\mathcal{S}^{n}_{+}\cap L$ such that $\Omega S=0$. It can be shown
using $(d+1)$-connectivity of $K_{5,5}$ [1] and coning [10] that $T$ is not
empty. Meanwhile all of the $S\in T$ must satisfy some non-trivial algebraic
condition [9]. Thus, the conclusion of Proposition 3.15 does not hold, and
this $L$ must be in the exceptional set of that proposition. In particular, we
see that for all PSD $\Omega\in L$ of rank $n-d$, it must be the case that
$Z(\Omega)$ and $L$ intersect non-transversally. In other words, this $L$ is
in the exceptional set described in Lemma 3.14.

###  3.2 Upper bound on the MLT

The punchline of this subsection is Proposition 3.21 which tells us that the
upper bound from Proposition 3.15 on the MLT of $\mathcal{M}_{L}$ for generic
$L$ is also a lower bound. This direction is easier since it only requires
algebraic considerations, as opposed to semi-algebraic.

######  Lemma 3.17.

Let $n,d,m$ be integers such that $1\leq d\leq n$ and $m\leq nd-\binom{d}{2}$.
Let $E\subseteq\operatorname{Gr}(m,\mathcal{S}^{n})$ be the set of $L$s such
that for all $K\in\operatorname{Gr}(d,\mathbb{R}^{n})$, there exists a non-
zero $\Omega\in L$ such that $\ker(\Omega)\supseteq K$. Then $E$ lies in a
Zariski-closed proper subset of $\operatorname{Gr}(m,\mathcal{S}^{n})$.

###### Proof.

Let $K\in\operatorname{Gr}(d,\mathbb{R}^{n})$ and define

$\Lambda(K):=\\{\Omega\in\mathcal{S}^{n}:\ker(\Omega)\supseteq K\\}.$

Then $L\in E$ if and only if $L\cap\Lambda(K)\neq\\{0\\}$ for all $K$.
Moreover, $\Lambda(K)$ a linear space of dimension $\binom{n-d+1}{2}$ so
$m+\dim{\Lambda(K)}<\binom{n+1}{2}$. Any $L\in E$ can therefore be perturbed
to obtain a nearby $L^{\prime}\in\operatorname{Gr}(m,\mathcal{S}^{n})\setminus
E$ so Lemma 3.4 implies that $E$ lies within a Zariski-closed proper subset of
$\operatorname{Gr}(m,\mathcal{S}^{n})$. ∎

The following lemma characterizes the normal space of $V(n,d)$ at $S$ in its
embedding in $\mathcal{S}^{n}$.

######  Lemma 3.18.

Let $S\in V(n,d)$ and let $\Omega\in\mathcal{S}^{n}$. Then $\Omega S=0$ if and
only if $\operatorname{Tr}(\Omega X)=0$ for all $X\in T_{S}V(n,d)$.

###### Proof.

Let $W$ denote the smooth submanifold of $\mathbb{R}^{n\times n}$ consisting
of _all_ $n\times n$ matrices of rank $d$. Given $M\in W$, the tangent space
$T_{M}W$ consists of all matrices that map the kernel of $M$ into its image
[16, Example 14.16]. Since $S$ is symmetric, its kernel is the orthogonal
complement of its span. Therefore if $A\in T_{S}W$ and $x,y\in\ker(S)$, the
following holds

$0=y^{T}Ax=tr(y^{T}Ax)=tr(Axy^{T}).$ (B)

The orthogonal complement of a subspace $L$ of $\mathcal{S}^{n}$ will be
denoted by $L^{\perp_{s}}$, and that of a subspace $L$ of
$\mathbb{R}^{n}\times\mathbb{R}^{n}$ will be denoted $L^{\perp_{n}}$. We now
claim that (B) implies the following

$(T_{S}W)^{\perp_{n}}=\left\\{\sum_{i=1}^{(n-d)^{2}}\alpha_{i}x_{i}y_{i}^{T}:x_{i},y_{i}\in\ker(S)\
{\rm for\ all\ }i\ {\rm and\ }\alpha_{i}\in\mathbb{R}\right\\}.$ (C)

Indeed, the inclusion $\supseteq$ is clear. To see the $\subseteq$ inclusion,
first note that $(T_{S}W)^{\perp_{n}}$ has dimension $(n-d)^{2}$. It now
suffices to exhibit $(n-d)^{2}$ linearly independent matrices of the form
$xy^{T}$ for $x,y\in\ker(S)$. Let $x_{1},\dots,x_{n-d}$ be a basis of
$\ker(S)$. The set of matrices of the form $x_{i}x_{j}$ for $1\leq i\leq j\leq
n$ is linearly independent which can be seen by changing our basis of
$\mathbb{R}^{n}$ so that $x_{i}$ is the $i^{\rm th}$ standard unit vector.

Viewing $\mathcal{S}^{n}$ as a subspace of $\mathbb{R}^{n\times n}$, we claim
the following:

$(T_{S}V(n,d))^{\perp_{s}}=(T_{S}W)^{\perp_{n}}\cap\mathcal{S}^{n}.$ (D)

The inclusion
$(T_{S}V(n,d))^{\perp_{s}}\supseteq(T_{S}W)^{\perp_{n}}\cap\mathcal{S}^{n}$ is
clear.

To get the reverse inclusion, it now suffices to show that

$\dim(T_{S}W^{\perp_{n}}\cap\mathcal{S}^{n})\geq\dim((T_{S}V(n,d))^{\perp_{s}})=\binom{n-d+1}{2}.$

Let $x_{1},\dots,x_{n-d}$ be a basis of $\ker(S)$ and consider the following
set $\mathcal{B}$ of $\binom{n-d+1}{2}$ matrices

$\mathcal{B}:=\\{x_{i}x_{j}^{T}+x_{j}x_{i}^{T}:1\leq i\leq j\leq n\\}.$

By change of basis we can without loss of generality assume that $x_{i}$ is
the $i^{\rm th}$ standard basis vector. If $i\neq j$ then
$x_{i}x_{j}^{T}+x_{j}x_{i}^{T}$ is the matrix where every entry is zero aside
from $(i,j)$ and $(j,i)$ which are both $1$. If $i=j$, then
$x_{i}x_{j}^{T}+x_{j}x_{i}^{T}$ is the matrix that is all zeros aside from a
$2$ at the $(i,i)$ entry. Thus $\mathcal{B}$ is linearly independent and (C)
implies that $\mathcal{B}\subset(T_{S}W)^{\perp_{n}}\cap\mathcal{S}^{n}$.

In light of (D), the lemma follows if we prove the the following:

$\\{\Omega\in\mathcal{S}^{n}:\Omega
S=0\\}=(T_{S}W)^{\perp_{n}}\cap\mathcal{S}^{n}$

Suppose $\Omega\in(T_{S}W)^{\perp_{n}}\cap\mathcal{S}^{n}$ and define
$k:=\operatorname{rank}(\Omega)$. Then $\Omega$ has a spectral decomposition
$\Omega=U\Lambda U^{T}$ where $U$ is an $n\times k$ matrix with orthonormal
columns and $\Lambda$ is a nonsingular diagonal $k\times k$ matrix. From (C)
it follows that $\Omega S=0$.

Now suppose $\Omega\in\mathcal{S}^{n}$ and $\Omega S=0$. As before, we have a
spectral decomposition $\Omega=U\Lambda U^{T}$ where $U$ has linearly
independent columns so $U^{T}S=0$. Thus the columns of $U$ are in the kernel
of $S$. From its spectral decomposition, we see that $\Omega$ can be written
as a sum of outer products of vectors in the kernel of $S$ as in (C). This
implies that $\Omega\in(T_{S}W)^{\perp}$. ∎

######  Lemma 3.19.

Let $S\in V(n,d)$ and let $L\subseteq\mathcal{S}^{n}$ be a linear space. Then
there exists a nonzero $\Omega\in L$ such that $\Omega S=0$ if and only if
$\pi_{L}:V(n,d)\rightarrow L$ is not a submersion at $S$.

###### Proof.

In light of Lemma 3.18, the existence of some $\Omega\in L\setminus\\{0\\}$
such that $\Omega S=0$ is equivalent to the following

$\dim(L)\cap(T_{S}V(n,d))^{\perp}\geq 1$

which is equivalent to $\dim(\pi_{L}(T_{S}V(n,d)))<\dim L$ which is what it
means for $\pi_{L}:V(n,d)\rightarrow L$ to not be a submersion at $S$. ∎

######  Lemma 3.20.

Let $L\subseteq\mathcal{S}^{n}$ be a linear space. Suppose that there exists
an $S\in V(n,d)$ such that there is no nonzero $\Omega\in L$ satisfying
$\Omega S=0$. Let $F_{L}\subseteq V(n,d)$ consist of all $S\in V(n,d)$ such
that there is a nonzero $\Omega\in L$ satisfying $\Omega S=0$. Then $F_{L}$ is
a Zariski-closed proper subset of $V(n,d)$.

###### Proof.

Let $m$ denote the dimension of $L$. Then we can write $\pi_{L}=\tau\circ\rho$
where $\rho:V(n,d)\rightarrow\mathbb{R}^{m}$ is a morphism of varieties and
$\tau:\mathbb{R}^{m}\rightarrow L$ is a linear isomorphism. Lemma 3.19 implies
that $\pi_{L}$ is a submersion at $S$ and that $F_{L}$ is the subset of
$V(n,d)$ where $\rho$ is not submersive. This implies that $F_{L}$ is defined
by the vanishing of the $m\times m$ minors of a matrix of polynomials and that
these minors are not all identically zero. The lemma now follows. ∎

######  Proposition 3.21.

Let $n,d$ be integers such that $1\leq d\leq n$ and let
$L\subset\mathcal{S}^{n}$ be a linear subspace. Let $m$ denote the dimension
of $L$ and assume

$m\leq nd-\binom{d}{2}.$

Then there exists a Zariski-closed proper subset
$E\subseteq\operatorname{Gr}(m,\mathcal{S}^{n})$ such that if $L\notin E$ then
there exists a Zariski-closed proper subset $F_{L}\subset V(n,d)$ such that if
$S\in V(n,d)\setminus F_{L}$ then $\Omega S\neq 0$ for all $\Omega\in
L\setminus\\{0\\}$.

###### Proof.

Take $E\subset\operatorname{Gr}(m,\mathcal{S}^{n})$ the set indicated in Lemma
3.17. Then for any $L\in\operatorname{Gr}(m,\mathcal{S}^{n})\setminus E$,
there exists $K\in\operatorname{Gr}(d,\mathbb{R}^{d})$ be such that
$\ker(\Omega)\neq K$ for all $\Omega\in L$. Thus if $S\in V(n,d)$ has span
$K$, then $\Omega S\neq 0$ for all $\Omega\in L$. We can then apply Lemma 3.20
to obtain a set $F_{L}\subset V(n,d)$ such that if $S\notin F_{L}$, then
$\Omega S\neq 0$ for all $\Omega\in L$. ∎

######  Remark 3.22.

The ideas in Lemmas 3.19 and 3.18 and their proofs sheds a some light on the
non-transversal intersections of Lemma 3.14. Suppose that $Z(\Omega)$ and $L$
intersect non-transversally at some $\Omega$ with rank $n-d$. Then, as we are
dealing with linear spaces, this intersection is of dimension greater than its
expected value, $\max(0,m-(nd-\binom{d}{2}))$.

Let $S$ be a rank $d$ symmetric matrix such that $\Omega S=0$. Let $k$ be the
dimension of the linear space of symmetric matrices $\Phi\in L$ (of any rank)
so that $\Phi S=0$. Then in this case, $k$ will have dimension greater than
its expected value $\max(0,m-(nd-\binom{d}{2}))$.

In light of the proof of Lemma 3.19, this means that the kernel of $d\pi_{L}$
at $S$ is also of dimension greater than its expected value
$\max(0,nd-\binom{d}{2}-m)$. When $m\geq dn-\binom{d}{2}$, the expected kernel
dimension is $0$, and having a greater value can be interpreted as a kind of
unexpected “infinitesimal flexibility” of $\pi_{L}$ at $S$.

###  3.3 Putting it all together

######  Theorem 3.23.

Let $n\in\mathbb{N}$ and let $1\leq m\leq\binom{n+1}{2}$. Then there exists a
Zariski-closed proper subset $E\subset\operatorname{Gr}(m,\mathcal{S}^{n})$
such that if $L\in\operatorname{Gr}(m,\mathcal{S}^{n})$ is not in $E$ and
$L\cap\mathcal{S}^{n}_{++}\neq\emptyset$, then maximum likelihood threshold of
$L$ is the minimum $d$ such that

$m\leq nd-\binom{d}{2}.$

###### Proof.

For any $1\leq d\leq n$, Theorem 2.1 implies that the existence of a solution
to the optimization problem (2) is equivalent to the non-existence of a
nonzero $\Omega\in\mathcal{S}^{n}_{+}\cap L$ such that $\Omega S=0$. First
suppose $m\geq nd-\binom{d}{2}+1$. By Proposition 3.15, there exists a
Zariski-closed proper subset
$E_{1}\subset\operatorname{Gr}(m,\mathcal{S}^{n})$ such that if $L\notin
E_{1}$, then there exists an open set $U\subseteq\mathcal{S}^{n}_{+}$ such
that for all $S\in U$, there exists a nonzero $\Omega\in
L\cap\mathcal{S}^{n}_{+}$ such that $\Omega S=0$.

Now suppose instead that $m\leq nd-\binom{d}{2}$. Proposition 3.21 implies
existence of a Zariski-closed proper subset
$E_{2}\subset\operatorname{Gr}(m,\mathcal{S}^{n})$ such that if $L\notin
E_{2}$, then there exists a Zariski-closed proper subset
$F_{L}\subset\mathcal{S}^{n}_{+}(d)$ such that for
$S\in\mathcal{S}^{n}_{+}(d)\setminus F_{L}$ (and thus for almost all
$S\in\mathcal{S}^{n}_{+}(d)$) we have $\Omega S\neq 0$ for all nonzero
$\Omega\in L$. Letting $E$ be the union of the $E_{1}$ and $E_{2}$ over the
finite set of applicable $d$, we obtain the theorem. ∎

######  Remark 3.24.

We can now close the circle of ideas that started with Definition 3.1, the
definition of genericity. The set of linear concentration models, i.e. the
subset of $\operatorname{Gr}(d,\mathcal{S}^{n})$ consisting of all subspaces
that contain a positive definite matrix, is an irreducible semi-algebraic set.
Thus Theorem 3.23 gives the maximum likelihood threshold of a generic linear
concentration model.

We now discuss the ways that some $L\in\operatorname{Gr}(m,\mathcal{S}^{n})$
containing a positive definite matrix can have maximum likelihood threshold
strictly less than the $d$ given in Theorem 3.23. If $d$ satisfies $m\geq
dn-\binom{d}{2}+1$, then Proposition 3.15 can fail to apply to $L$ for two
reasons: either $L$ does not contain a PSD matrix of rank $n-d$, or for every
PSD $\Omega\in L$, $Z(\Omega)$ and $L$ intersect non-transversally. As we saw
in Lemmas 3.7 and 3.14, both conditions can be expressed by the vanishing of
certain polynomials on the Plücker coordinates of $L$. It is interesting to
note that Gaussian graphical models that fail to have the expected maximum
likelihood threshold always do so for the second reason. In particular, if
$L\subseteq\mathcal{S}^{n}$ is defined by setting certain off-diagonal entries
to zero, then $L$ always contains a diagonal matrix with $n-d$ nonzero
diagonal entries, all of which are positive.

For generic $L$ we see that there is a striking gap in behaviour at our
threshold for $m$. When

$m\leq nd-\binom{d}{2},$

then outside of some Zariski-closed $F_{L}\subset\mathcal{S}^{n}_{+}(d)$,
there is no non-zero $\Omega\in L$ _of any signature_ so that $\Omega S=0$. On
the other hand when

$m\geq nd-\binom{d}{2}+1$

then there is an open neighborhood $U\subset\mathcal{S}^{n}_{+}(d)$, so that
for $S\in U$ there is an $\Omega\in L$ that is PSD _and has rank $n-d$_ and
such that $\Omega S=0$.

The maximal rank $(n-d)$ of $\Omega$ is not needed in order to apply Theorem
2.1; all that is needed is that $\Omega$ is PSD and nonzero. That said, for
generic $L$, once $m$ is large enough to obtain a non-zero $\Omega$ over such
a $U$, we also get an $\Omega$ of rank $n-d$. For non-generic $L$, such as the
ones obtained by setting certain off-diagonal entries to zero, we can get
other behaviour. In particular, for some such $L$, there exist open
neighborhoods $U$ of the set of rank-$d$ PSD matrices where for each $S\in U$,
there exist non-zero PSD $\Omega\in L$ with $\Omega S=0$, but if $S$ does not
lie in a particular Zariski-closed set, then the PSD $\Omega\in L$ satisfying
$\Omega S=0$ have rank strictly less than $n-d$ [5, Section 6.1].

##  4 Generic completion rank

Theorem 2.1 is a generalization of certain results about _Gaussian graphical
models_ , which are linear concentration models of a particular type. We will
now define such models and discuss how Theorem 2.1 fits into what is known
about them. Each graph $G$ on vertex set $\\{1,\dots,n\\}$ with $e$ edges
defines a linear subspace $L_{G}$ of $\mathcal{S}^{n}$ as follows

$L_{G}:=\\{A\in\mathcal{S}^{n}:A_{ij}=0\ \text{for all non-edges}\ ij\ {\rm
of}\ G\\}.$

The dimension $m$ of $L_{G}$ is $e+n$. The corresponding Gaussian model,
denoted $\mathcal{M}_{G}$, is called the _Gaussian graphical model_ associated
to $G$. To our knowledge, such models were first studied by Dempster in [12],
where they proved equivalence of the first two conditions of Theorem 2.1 for
Gaussian graphical models. Uhler found an easily computable upper bound on the
maximum likelihood threshold of a given Gaussian graphical model [22].
Blekherman and Sinn called this upper bound the _generic completion rank_ in
[7] and provided the first known examples where it failed to be sharp. We now
extend the notion of generic completion rank to arbitrary linear covariance
models and show that it still gives an upper bound on the maximum likelihood
threshold.

######  Definition 4.1 ([7, Definition 1.1]).

The _generic completion rank_ of a linear subspace
$L\subseteq\mathcal{S}^{n}$, denoted $\operatorname{GCR}(L)$, is the minimum
$d$ such that $\pi_{L}(V(n,d))$ is full-dimensional in $L$.

When $L=L_{G}$ for some graph $G$, Definition 4.1 agrees the definition of
generic completion rank given in [7]. Below, we prove Theorem 4.3, which says
that generic completion rank upper-bounds the maximum likelihood thresholds,
as it does for graphs. Before proving that, we need the following lemma which
can be seen as the relaxation of Theorem 2.1 to allow for non-PSD $\Omega$. It
is applicable to generic completion ranks, as opposed to maximum likelihood
thresholds.

######  Lemma 4.2.

Let $L\subseteq\mathcal{S}^{n}$ be a linear subspace. Then $\pi_{L}(V(n,d))$
is full-dimensional in $L$ if and only if for every regular point $S\in
V(n,d)$ of the restriction of $\pi_{L}$ to $V(n,d)$, if $\Omega\in L$ and
$\Omega S=0$ then $\Omega=0$.

###### Proof.

First assume that $\pi_{L}(V(n,d))$ is full dimensional in $L$ and that $S$ is
a regular point of the restriction of $\pi_{L}$ to $V(n,d)$. Then Sard’s
Theorem [15] implies that $\pi_{L}(T_{S}V(n,d))=L$. Let $\Omega\in L$ satisfy
$\Omega S=0$, let $Y\in L$ and let $X\in T_{S}V(n,d)$ satisfy $\pi_{L}(X)=Y$.
Lemma 3.18 implies that $\operatorname{Tr}(\Omega X)=0$. Since orthogonal
projection in an inner product space is equivariant with respect to the inner
product, this implies that $\operatorname{Tr}(\pi_{L}(\Omega)\pi_{L}(X))=0$.
Since $\Omega\in L$ and $\pi_{L}(X)=Y$, this gives $\operatorname{Tr}(\Omega
Y)=0$. We have thus proven that $\Omega\in L^{\perp}$. Since $\Omega\in L$ as
well, $\Omega=0$.

Now assume that for every regular point $S$ of the restriction of $\pi_{L}$ to
$V(n,d)$, if $\Omega\in L$ and $\Omega S=0$ then $\Omega=0$. Let $S$ be a
regular point of this restriction. Lemma 3.18 implies that
$L\cap(T_{S}V(n,d))^{\perp}=0$ and thus
$L^{\perp}+T_{S}V(n,d)=\mathcal{S}^{n}$. Therefore $\pi_{L}(T_{S}V(n,d))=L$.
Then the submersion theorem [15] implies that $\pi_{L}(V(n,d))$ is full-
dimensional in $L$. ∎

######  Theorem 4.3.

Let $L\subseteq\mathcal{S}^{n}$ be a linear subspace. Then
$\operatorname{GCR}(L)$ is the minimum $d$ such that for all generic PSD $S$
of rank $d$, there does not exist an $\Omega\in L$ such that $\Omega S=0$.
Moreover, $\operatorname{GCR}(L)\geq\operatorname{MLT}(L)$.

###### Proof.

The first statement follows from Lemma 4.2, taking the non-generic locus to be
the singular points of $\pi_{L}$. The second statement now follows from part 3
of Theorem 2.1. ∎

######  Corollary 4.4.

The maximum likelihood threshold of a generic linear subspace
$L\subseteq\mathcal{S}^{n}$ is equal to its generic completion rank.

###### Proof.

Given Theorem 4.3, Theorem 3.23 implies that the generic completion rank of a
generic $L\in\operatorname{Gr}(m,\mathcal{S}^{n})$ is at least the minimum $d$
such that $m\geq dn-\binom{d}{2}$ and Proposition 3.15 implies that the
generic completion rank is at most this $d$. ∎

##  5 Rigidity Theory

The maximum likelihood threshold of a graph is intimately tied to its
rigidity-theoretic properties. The first connections therein were established
by Gross and Sullivant in [14] and further developed by the authors of this
paper and others in [5, 6]. The ideas explored in this paper are inspired by
such connections and we discuss some of them here.

Let $P\in\mathbb{R}^{d\times(n-1)}$. Associated to $P$ is the pinned
configuration ${\bf p}=({\bf p}_{1},\dots,{\bf p}_{n})$ where ${\bf p}_{n}=0$
and ${\bf p}_{i}$ is the $i^{\rm th}$ column of $P$ for $i<n$. The Gram matrix
of ${\bf p}$ is obtained from $P^{T}P$ by adding a row and column of zeros.
Let $G$ be a graph with vertex set $\\{1,\dots,n\\}$ and edge set $E$ with
cardinality $e$. The pair $(G,{\bf p})$ is called a _pinned framework_.

Let $L^{r}_{G}\subseteq\mathcal{S}^{n-1}$ ($r$ for rigidity) be the linear
space consisting of symmetric matrices $\Omega$ whose $i^{\rm th}$ row sum
vanishes whenever $\\{i,n\\}\notin E$ and $\Omega_{ij}=0$ whenever
$\\{i,j\\}\notin E$ for $i,j<n$. We will call matrices in $L^{r}_{G}$ _pinned
equilibrium stress matrices_ 222In the rigidity theory literature, an
equilibrium stress matrix for a graph $G$ on $n$ vertices refers to an
$n\times n$ symmetric matrix $\Omega$ with vanishing row sums and
$\Omega_{ij}=0$ for every non-edge $\\{i,j\\}$ of $G$. There is an invertible
linear map from our $(n-1)\times(n-1)$ pinned equilibrium stress matrices to
these $n\times n$ equilibrium stress matrices. In particular, given a
$(n-1)\times(n-1)$ pinned equilibrium stress matrix $\Omega$, define
$\Omega^{\prime}\in\mathcal{S}^{n}$ by adding an extra row and column such
that all row-sums vanish and that $\Omega^{\prime}_{in}=0$ whenever $i$ and
$n$ are non-adjacent in $G$. Since $\Omega$ and $\Omega^{\prime}$ have the
same rank, the eigenvalue interlacing theorem implies that $\Omega$ is PSD if
and only if $\Omega^{\prime}$ is. for the graph $G$. This space has dimension
$|E|$, which we can see by constructing a basis as follows. For $1\leq i<j<n$
such that $\\{i,j\\}\in E$, define $B_{ij}$ to be the $(n-1)\times(n-1)$
symmetric matrix of all zeros, aside from $1$ at the $(i,i)$ and $(j,j)$
entries and $-1$ at the $(i,j)$ and $(j,i)$ entries. For $1\leq i<n$ such that
$\\{i,n\\}\in E$, define $B_{in}$ to be the $(n-1)\times(n-1)$ symmetric
matrix of all zeros, aside from a $1$ on the $i^{\rm th}$ diagonal. Then
$\\{B_{ij}:\\{i,j\\}\in E\\}$ is a basis of $L_{G}$.

If $\Omega\in L^{r}_{G}$ and $\Omega P=\Omega S=0$, we say that $\Omega$ is a
_pinned equilibrium stress matrix_ for the framework $(G,{\bf p})$. The map
$\pi_{L^{r}_{G}}$ onto $L^{r}_{G}$ is naturally linearly isomorphic to the map
that takes a pinned framework $(G,{\bf p})$ to the squared lengths along the
edges of $G$. This isomorphism follows from the fact that squared edge length
measurements are computed as $\operatorname{Tr}(B_{ij}P^{T}P)$ for
$\\{i,j\\}\in E$. In particular, let $P,Q\in\mathbb{R}^{d\times(n-1)}$ with
corresponding pinned configurations ${\bf p},{\bf q}$ of $n$ points in
$\mathbb{R}^{d}$, obtained by adding the origin. The pinned configurations
${\bf p}$ and ${\bf q}$ will have the same squared lengths along the edges of
$G$ if and only if $\pi_{L^{r}_{G}}(P^{T}P)=\pi_{L^{r}_{G}}(Q^{T}Q)$. In this
case we say that ${\bf p}$ and ${\bf q}$ are _$G$ -equivalent_, or simply
_equivalent_ if $G$ is clear from context. Thus the structure of the fiber
$\pi_{L^{r}_{G}}^{-1}(\pi_{L^{r}_{G}}(P^{T}P))$ reveals rigidity information
about ${\bf p}$. Notably, if $L^{r}_{G}$ contains a PSD matrix of rank $n-k-1$
that is a pinned equilibrium stress matrix for $(G,{\bf p})$, then any
framework $(G,{\bf q})$ with the same edge lengths as $(G,{\bf p})$ has an
affine span of dimension at most $k$.

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