# Enabling Edge Artificial Intelligence via
Goal-oriented Deep Neural Network Splitting ††thanks: This work was supported
by MIUR under the PRIN Liquid Edge Contract, by the ANR under the France 2030
program, grant ”NF-NAI: ANR-22-PEFT-0003”, by the European Union under the
Italian National Recovery and Resilience Plan (NRRP) of NextGenerationEU,
partnership on “Telecommunications of the Future” (PE00000001 - program
“RESTART”).

Francesco Binucci1,2, Mattia Merluzzi3, Paolo Banelli1, Emilio Calvanese
Strinati3, Paolo Di Lorenzo2,4
1Department of Engineering, University of Perugia, Via G. Duranti 93, 06128,
Perugia, Italy
2Consorzio Nazionale Interuniversitario per le Telecomunicazioni, Viale G.P.
Usberti, 181/A, 43124, Parma, Italy
3CEA-Leti, Université Grenoble Alpes, F-38000 Grenoble, France
4DIET department, Sapienza University of Rome, via Eudossiana 18, 00184, Rome,
Italy

###### Abstract

Deep Neural Network (DNN) splitting is one of the key enablers of edge
Artificial Intelligence (AI), as it allows end users to pre-process data and
offload part of the computational burden to nearby Edge Cloud Servers (ECSs).
This opens new opportunities and degrees of freedom in balancing energy
consumption, delay, accuracy, privacy, and other trustworthiness metrics. In
this work, we explore the opportunity of DNN splitting at the edge of 6G
wireless networks to enable low energy cooperative inference with target delay
and accuracy with a goal-oriented perspective. Going beyond the current
literature, we explore new trade-offs that take into account the accuracy
degradation as a function of the Splitting Point (SP) selection and wireless
channel conditions. Then, we propose an algorithm that dynamically controls SP
selection, local computing resources, uplink transmit power and bandwidth
allocation, in a goal-oriented fashion, to meet a target goal-effectiveness.
To the best of our knowledge, this is the first work proposing adaptive SP
selection on the basis of all learning performance (i.e., energy, delay,
accuracy), with the aim of guaranteeing the accomplishment of a goal (e.g.,
minimize the energy consumption under latency and accuracy constraints).
Numerical results show the advantages of the proposed SP selection and
resource allocation, to enable energy frugal and effective edge AI.

###### Index Terms:

Goal-oriented communications, resource allocation, stochastic optimization,
DNN splitting.

##  I Introduction

6G will offer the fusion of cyber physical spaces, mobile connect-compute
networks, Artificial Intelligence (AI) and Machine Learning (ML). These new
services will bond communication and computing for a continuous data
distillation, processing and exchange among heterogeneous intelligent agents
and network infrastructure resources [1],[2] that will be flexibly accessed on
demand. Also, they must be enabled with stringent constraints in terms of
energy consumption, latency, reliability and trustworthiness among others. To
this end, Multi-access Edge Computing (MEC) moves computational and storage
resources at the edge of wireless networks, thus keeping computation as close
as possible to the data source. Among several services, we focus on Artificial
Intelligence (AI) at the edge, whose goal is to enable AI algorithms (learning
and inference) in a cooperative and distributed way, involving end devices,
MEC resources, and central clouds [3]. In this paper we focus on the inference
phase, i.e., edge inference (EI). The latter opens new challenges to the
orchestration of the network resources, which include an optimal choice for i)
splitting computations between local devices and remote ones,  ii) data
compression, and iii) communication design (e.g., bandwidth, transmit power,
interference management, etc.). For instance, an interesting goal is to reduce
the computational burden at the local edge devices (EDs), by optimally
providing enough MEC resources to perform a task, with a prescribed level of
reliability, and within predefined latency limits. In this context, goal-
oriented communications (GOCs) represent an emerging topic [1, 4, 5, 6, 7, 8]
that will help mitigating the unsustainable increase of data traffic, by
transmitting the minimum amount of information necessary to perform a specific
task with target effectiveness [9] (i.e., to achieve a specific goal). The
enablers of GOCs span from physical layer aspects up to the application ones,
with a cross-layer perspective where bits, information distortion, and
application performance are jointly optimized, in order to strike the best
trade-offs among energy consumption, latency, and learning performance. This
paper proposes a goal-oriented computation split between EDs and Edge Cloud
Servers (ECSs), building on the concept of Deep Neural Network (DNN) splitting
[10]. Specifically, splitting a DNN means to separate it into a head and tail
structure, to perform part of the computation at an ED (e.g., the head at the
device collecting data), and the remaining portion at another node (e.g., the
tail at an ECS). This can be seen as a cooperation problem, where the costs
(i.e., latency and energy expenditure due to the computation/transmission) are
broken down between the head and the tail focusing, for instance, on the
energy/latency trade-offs, on the quality of the wireless channels [11], or on
the target goal-effectiveness.

Related works. The benefits of DNN splitting strategies in MEC environments
have been already investigated in some recent works. The authors of [12] and
[13] consider DNN partitioning schemes to minimize the latency and the user
equipment energy consumption, respectively.

Figure 1: Illustration of the considered DNN splitting scenario.

Some works, like [5], witness how goal-oriented compression schemes can be
seen as a specific case of DNN splitting. In these scenarios, the EDs run an
encoder network, while the ECS employs a decoder network. The encoders
compress data taking into account the informativeness with respect to the
output of a specific inference task, retrieved by the decoder. Along this
line, in [14, 15], we propose an inference architecture where a convolutional
neural network is split between a Convolutional Encoder (CE), used to perform
a goal-oriented compression at the device-side, and a Convolutional Classifier
(CC) to complete the inference task at the ECS side. Although these works
consider the dynamic optimization of the transmission and computational
resources, they do not address the problem of optimally and dynamically
selecting the splitting point (SP) of the employed architectures. Differently,
the work in [11], proposes a dynamic SP selection strategy that reduces the
energy consumption of a user equipment to perform an edge-assisted image
classification task with target delay constraints. However, [11] does not
consider the performance degradation (e.g., the effects on the accuracy)
introduced by the noisy wireless transmission, and its dependency on the SP
selection.

Our contribution. In this work, we focus on goal-oriented DNN splitting to
enable low energy and timely cooperative edge inference with target delay and
accuracy performance, dynamically and jointly optimizing the SP selection and
the network resources. Specifically, focusing on an image classification task,
we first evaluate the impact of noise injection at different SPs during the
inference phase. Then, taking a step forward from our previous work [11], we
incorporate this knowledge in an optimal resource allocation strategy, which
dynamically selects also the best SP, with the goal of minimizing the ED
energy consumption under latency and accuracy constraints. To the best of our
knowledge, this is the first work that takes into account the accuracy
degradation due to noisy transmission of intermediate extracted features,
depending on the SP selection and wireless channel conditions. The proposed
algorithmic framework hinges on Lyapunov stochastic optimization, which
enables efficient adaptive solutions, without needing any prior knowledge of
the statistics of the wireless channels and the task arrival process. Finally,
numerical results assess the performance of our method, illustrating its
benefits in terms of energy-accuracy trade-off on realistic image
classification tasks.

##  II System model

Let us consider the system model illustrated in Fig. 1, composed by a single
ED (e.g., a camera) equipped with computing and communication resources, and
an Access Point (AP) connected to an ECS through an ideal backhaul. Both the
ED and the ECS are embarked with the same Deep Neural Network (DNN) model,
which is assumed to be trained and ready to treat inference requests on the
data collected by the ED. In a slotted fashion, the ED generates a batch
$\mathcal{B}$ of $card(\mathcal{B})$ new patterns to be inferred (e.g., a set
of images). The inference can partly take place at the ED and at the ECS,
thanks to DNN splitting. In particular, for each request, the ED selects an SP
based on the current status of communication and computation resources
(availability, quality of wireless channels, etc.). The choice of the SP
affects the size (dimensionality) of the data to be transmitted and the
portion of computations to be performed locally, thus impacting the energy
consumption of the ED to process and transmit, as well as the delay.
Therefore, once an SP $k$ is selected, the ED processes data up to SP $k$, and
transmits the extracted features to the AP. Then, the AP transfers the
features to the ECS, which resumes the computations up to the last layer of
the DNN, to output the inference result (e.g., a set of labels). The set of
SPs is denoted by $\mathcal{J}=\\{0,1,\ldots,k,\ldots,J\\}$, where $0$ denotes
the full offloading case (raw data are transmitted), and $J$ the full local
computation. Each SP $k$ can be represented by a tuple
$\mathbf{s}_{k}=[L_{k},F_{k}]$, with $L_{k}$ the number of its output
features, and $F_{k}$ the number of FLOPS needed to process data across layers
between SP $k-1$ and SP $k$

###  II-A Local computation model: Delay and energy consumption

We consider a slotted system indexed by $t=1,2,\ldots,$. At the beginning of
each slot, a new request is issued and a control decision is taken on the
system resources (e.g., communication and computing). At the ED side, denoting
by $f_{l}(t)$ the CPU clock frequency at time instant $t$, and by $\eta_{l}$
the number of FLOPS per CPU cycle (hardware dependent), and assuming that the
whole batch $\mathcal{B}(t)$ is serially handled through the selected SP $k$,
the local computing delay to extract features can be written as follows:

$D^{\text{comp}}_{l}(t)=\begin{cases}\displaystyle\frac{card(\mathcal{B}(t))\sum_{j=0}^{k}F_{j}}{\eta_{l}f_{l}(t)},\;\text{if}\;k(t)>0\\\
0,\;\text{otherwise}\end{cases}$ (1)

Similarly, using the typical cubic model for the CPU dynamic power consumption
[16], the local computing energy consumption is modeled as

$E_{l}^{\text{comp}}=\begin{cases}\displaystyle\frac{card(\mathcal{B}(t))\sum_{j=0}^{k}F_{j}\kappa
f_{l}^{2}(t)}{\eta_{l}},\;\text{if}\;k(t)>0\\\
0,\;\text{otherwise},\end{cases}$ (2)

with $\kappa$ the effective switched capacitance of the processor [16].

###  II-B Transmission model: Delay and energy consumption

In this work, we assume an analog goal-oriented transmission [15]. The impact
of DNN splitting with digital modulations is left for future work, although
analog modulations are becoming popular in the context of edge learning
problem with the concept of over-the-air computation for, e.g., federated
learning [17]. Therefore, we assume a base-band PAM signal expressed as
follows [18]:

$u(\tau)=\sum\nolimits_{k=-\infty}^{+\infty}a_{k}p_{g}(\tau-kT_{s}),$ (3)

where $a_{k}\in\mathbb{C}$ is the $k$-th transmitted symbol, $p_{g}(\tau)$ is
the pulse shaper, and $S_{r}={1}/{T_{s}}$ is the symbol rate. We consider a
Squared Root Raised Cosine (SRRC) shaping pulse $p_{g}(\tau)$, with two-sided
baseband noise-equivalent bandwidth $W=S_{r}$, roll-off factor $\beta=0.25$
[18], and unit energy. This way, we can assume an Inter-Symbol Interference
(ISI)-free system. We consider analog modulations, where the outputs $z_{k}$
of a selected SP $k$ are multiplexed on the I-Q components, and _directly_
mapped into the (analog) QAM symbols by $a_{k}\\!=\\!z_{2k-1}\\!+\\!jz_{2k}$.
The transmitted signal $u(\tau)$ is received through a Single-Output (SISO)
noisy flat-fading wireless channel $h(t)\in\mathbb{C}$ and, after matched-
filtering and one-tap equalization, it is passed to the ECS for
classification. Then, the instantaneous SNR at time instant $t$ is

$\gamma(t)=\frac{p_{\text{tx}}(t)|h(t)|^{2}}{W(t)N_{0}},$ (4)

where $N_{0}$ is the noise power spectral density (PSD), $W(t)$ is the
bandwidth, and $p_{\text{tx}}(t)$ is the transmit power. In each slot $t$, we
optimize the transmit power, the bandwidth, and the SP. For a generic SP $k$,
the overall communication delay can be written as follows

$D^{\text{tx}}(t)=\begin{cases}\displaystyle\frac{(1+\beta)L_{k}card(\mathcal{B}(t))}{2W(t)},\;\text{if}\;k(t)<J\\\
0,\;\text{otherwise}.\end{cases}$ (5)

Then, the transmit energy consumption is

$E^{\text{tx}}(t)=p_{\text{tx}}(t)D^{\text{tx}}(t).$ (6)

###  II-C Remote computation delay

Once the data related to SP $k$ are received by the AP and transferred to the
ECS, the computation is resumed from $k$ up to the output layer of the DNN.
Then, denoting by $f_{r}(t)$ the CPU cycle frequency of the ECS at time $t$,
and by $\eta_{r}$ the number of FLOPS per CPU cycle, the remote computation
delay can be written as:

$D^{\text{comp}}_{r}=\begin{cases}\displaystyle\frac{\left(\sum_{j=0}^{J}F_{j}-\sum_{j=0}^{k}F_{j}\right)card(\mathcal{B}(t))}{\eta_{r}f_{r}(t)},\;\text{if}\;k(t)<J\\\
0,\;\text{otherwise},\end{cases}$

In this paper, as in [11], we assume that $f_{r}(t)$ cannot be optimized, but
it is an exogenous variable that depends on the time-varying availability of
the ECS. Finally, the end-to-end delay can be written as the sum of all
components

$D^{\text{tot}}(t)=D_{l}^{\text{comp}}(t)+D^{\text{tx}}(t)+D_{r}^{\text{comp}}(t),$
(7)

while the overall ED energy consumption reads as

$E^{\text{tot}}(t)=E_{l}^{\text{comp}}(t)+E^{\text{tx}}(t)$ (8)

###  II-D Inference accuracy: The effect of noise and SP selection

In this section, we evaluate the accuracy of a DNN as a function of the
splitting point at different SNRs during wireless communication. As in [11],
we employ a pre-trained version of the MobileNetV2 network[19], which is
composed of 20 possible SPs, including the full offloading and the full local
computation case. We perform an image classification task based on the Intel-
Scene classification data-set[20], containing 17000 RGB images of landscapes,
belonging to 6 different classes. We include $14000$ samples in the training-
set and $3000$ samples in the test-set. The images have been resized to
$224\times 224$ pixels, in order to match the input-size of the classification
network, and the last layer has been resized to meet the number of possible
classes contained in the data-set. The model was fine-tuned by training it for
30 epochs, considering Adam as optimizer with a learning-rate
$lr=$1\text{\times}{10}^{-3}$$. As usual in classification tasks, we employ
the well-known cross-entropy loss as objective function. Fig. 2 shows the
accuracy in the inference phase adding white Gaussian Noise at different SPs,
to mimic different SNR conditions. Overall, we can notice that the outputs of
the top-layers (i.e., the ones deeper in the architecture) are less affected
by the noise with respect to the output of the bottom-layers. Indeed,
according to [21], the discrimination ability of DNNs tends to grow as we go
deeper in the network architecture. Thus, the effect of the noise is mitigated
in the final layers of the network. However, we can also notice that some SPs
(e.g., 3, 7) may exhibit an irregular behaviour. Indeed, as reported in [21],
for some network architectures (e.g., residual networks) the data-separation
law could present irregular behaviours, although representing a valid
approximation. Fig. 2 suggests that, under noisy wireless communication, the
SP selection should take into account energy, delay and accuracy, and not only
energy and delay as in [11]. Therefore, in the next section, we focus on
devising an algorithm that jointly selects SP, as well as wireless and
computing resource, to balance energy, delay, and accuracy.

Figure 2: Accuracy vs Splitting Point for different average SNRs
$\overline{\gamma}$.

##  III Problem formulation and solution

Due to the energy resource scarcity typical of EDs, the aim of this work is to
minimize the ED energy consumption in (8) under average delay constraints as
in [11], and a target accuracy constraint. A long-term problem is formulated:
—s—[2] Φ(t)lim_T→∞ 1T ∑_t=1^T E{E^tot(t)} (a) lim_T→∞ 1T ∑_t=1^T
E{D^tot(t)}≤D^avg(b) lim_T→∞ 1T ∑_t=1^T E{G(k(t), γ(t)}≥G^avg (c) p_tx(t) :=
γ(t) N0W(t)—h(t)—2 ≤p_tx^max, ∀t (d) 0 ≤W(t) ≤W_max⋅1(k(t)¡J), ∀t (e) k(t) ∈J,
γ(t) ∈R_0^+, ∀t (f) f_l^min⋅1(k(t)¿0) ≤ f_l(t)≤ f_l^max⋅1(k(t)¿0), ∀t
where $\Phi(t)=[W(t),k(t),\gamma(t),f_{l}(t)]$ is the set of the optimization
variables. $(a)$ is the long-term latency constraints, while $(b)$ is the
accuracy constraint; $(c)$ and $(d)$ encompass the constraints on the
transmission power and bandwidth, respectively, while $(e)$ imposes the SP
$k(t)$ to be chosen from the set $\mathcal{J}$ of all possible SPs and the SNR
constraint; finally, $(f)$ imposes the CPU cycle frequency of the ED to be
chosen between a minimum and a maximum value, while being $0$ in case of full
offloading decision.

To solve problem, we build on Lyapunov stochastic optimization. First, we
define two virtual queues to handle the long-term constraints $(a)$-$(b)$. The
virtual queues evolve as:

$\begin{split}(a)\;Z(t+1)&=\max(0,Z(t)+\mu(D^{\text{tot}}(t)-D^{\text{avg}}))\\\
(b)\;Y(t+1)&=\max(0,Y(t)+\lambda(G^{\text{avg}}-G(t))),\end{split}$ (8)

where $\mu$, and $\lambda$ are positive step-sizes. Then we build the Lyapunov
function and the associated Lyapunov Drift plus penalty (DPP) function

$\begin{split}L(\Theta(t))&=Z^{2}(t)+Y^{2}(t)\\\
\Delta_{p}(\Theta(t))&=\Delta(\Theta(t))+V\mathbb{E}\\{E^{\text{tot}}(t)|\Theta(t)\\},\end{split}$
(9)

where $\Theta(t)=\\{Z(t),Y(t)\\}$, and
$\Delta(\Theta(t))=\mathbb{E}\\{L(t+1)-L(t)|\Theta(t)\\}$ is the conditional
Lyapunov Drift [22], while the parameter $V$ controls the trade-off between
objective function minimization and virtual queues stability. Removing
expectations and exploiting upper-an upper bound of the DPP [22] that is
omitted due the lack of space, the optimization decouples among different
slots, with virtual queues representing dynamic penalties that drive the
system towards constraint guarantees. In particular, the mean rate stability
of each virtual queue111For a virtual queue $G$ is defined as
$\lim_{T\to\infty}\frac{\mathbb{E}\\{G(T)\\}}{T}=0$. guarantees the associated
constraint [22]. At each slot, based on the observation of virtual queues and
random context parameters, we solve the following problem (cf. (7), (8)):

$\displaystyle\underset{\Phi(t)}{\min}\;\;\mu Z(t)D^{\text{tot}}(t)-\lambda
Y(t)G(k(t),\gamma(t))+VE^{\text{tot}}(t)$ (10) s.t. $(c)$-$(f)$ of (III)

Problem (10) is a mixed integer convex program. In particular, variables
$\gamma(t)$ and $k(t)$ are integer, while the problem is convex in $W(t)$ and
$f_{l}(t)$. Also, once the SP is fixed, the problem can be split into a
communication and a computation sub-problem.

###  III-A Communication sub-problem solution

Since the main goal of this work is to show the importance of a goal-oriented
SP selection, we perform an exhaustive search over $\gamma(t)$ and $k(t)$,
while we reserve less complex optimization strategies to future work,
fundamental in multiple users scenarios. Once $\gamma(t)$ and $k(t)$ are
fixed, the problem is convex in $W(t)$, which can be easily computed in closed
form:

$W^{*}(\gamma,k)=\min\left(\frac{p_{\text{tx}}^{\max}|h(t)|^{2}}{\gamma(t)N_{0}},W_{\max}\right)\cdot\mathbbm{1}(k(t)<J),$
(11)

i.e., we always transmit with the maximum available bandwidth, since the
objective function is monotonically decreasing with respect to $W$.

###  III-B Computation sub-problem solution

The optimal ED clock frequency can be easily computed by nulling the
derivative of the objective function and imposing the KKT conditions [23],
thus ending up the following closed form expression

$f_{l}^{*}(t,k(t))=\left[\sqrt[3]{\frac{\mu Z(t)}{2\kappa
V}}\right]_{f_{l}^{\min}\cdot\mathbbm{1}(k(t)>0)}^{f_{l}^{\max}\cdot\mathbbm{1}(k(t)>0)}$
(12)

Finally, given the optimal bandwidth and clock frequency for each pair
$(k(t),\gamma(t))$, the optimal solution can be found as

$[W^{*}(t),\gamma^{*}(t),k^{*}(t)]=\underset{W^{*},f_{l}^{*}(\gamma,k),\gamma,k}{\text{argmin}}\;\mathcal{O}(t),$
(13)

with $\mathcal{O}(t)$ denoting the objective function in (10).

##  IV Simulation Results

In this section, we assess the performance of the developed dynamic resource
allocation strategy. We run the optimization strategies for $N=10^{4}$ slots,
averaging results by removing the transient interval. Similarly as in our
previous works [15], we model the accuracy function in (III)
$G(k(t),\gamma(t))$ using a Look-UP table (LUT) obtained through some
simulations on the test-set of the Intel-Scene classification data-set[20].

Computing Model: the maximum CPU clock frequency of the ED is
$f_{l}^{\max}=1.4$ GHz, and a conversion factor $\eta_{l}=50$ FLOPS per CPU
cycle (cf. (1)) is assumed. We consider an effective switched capacitance
$\kappa=$1.097\text{\times}{10}^{-27}$\left[\frac{s}{\text{cycles}}\right]^{3}$.
Instead, the ECS is equipped with a maximum clock frequency $f_{r}^{\max}=4.5$
GHz and $\eta_{r}=2000$ FLOPS per CPU cycle. The ECS availability varies
across the slots according to $f_{r}(t)=\alpha_{r}(t)f_{r}^{\max}$, where
$\alpha_{r}(t)\sim\mathcal{U}(0,1)$. The computational costs associated to the
inference network have been computed considering the model in [19] and stored
in specific LUTs used to drive the optimization strategy. The features
dimensionality and the number of FLOPS needed to compute across SPs can be
also found in [11, Fig. $1$]

Wireless communication model: we consider an ED with a maximum transmit power
$p_{\text{tx}}^{\max}=300$ mW, and a bandwidth $W_{\max}=10$ MHz; the noise
PSD is set to $N_{0}=-174$ dbm/Hz, with a receiver noise figure $F=5$ dB. The
channel is characterized by Rayleigh fading with unit variance, while the path
loss is set to different values visible in the illustrated results. The number
$card(\mathcal{B}(t))$ of images generated at each instant is modelled as a
Poisson process with $\lambda=5$. In all simulations we set
$\gamma(t)\in\\{-5,-4,-3,-2,0,5,10,20\\}$ dB.

In the first set of results we highlight the benefits of the dynamic selection
of both the Splitting Point $k$ and the SNR $\gamma$, compared to strategies
that fix one or the other. We set a latency constraint $D^{\text{avg}}=50$ ms
and we select the parameter $V$ (cf. (9)) that minimizes the energy under such
constraints, i.e., all comparisons are performed with the same end-to-end
delay. We test the allocation strategy considering different accuracy
constraints, namely $\mathcal{G}=\\{70,75,80,85\\}$ %. We assume different
propagation conditions, corresponding to $115,120,125$ dB of path loss.

Figure 3: Energy saving of our method with respect to benchmark strategies

In Fig. 3, we show the energy saving achieved with our method with respect to
three benchmarks: i) full local computation (FLC - solid lines): , i.e. all
computations are performed locally at the device ii) Best fixed SP (BFSP -
dotted lines): the SP is fixed and the SNR is optimized vi the adaptive
algorithm; to make a fair comparison, we illustrate the results obtained with
the SP achieving the lowest energy consumption; iii) Best fixed SNR (BFSNR -
dashed lines): the SNR is fixed and the SP is optimized; again, the SNR
achieving the lower energy consumption is considered for fair comparison.

Comparison with FLC:  First, we can notice how our method achieves
considerable energy saving in terms of energy consumption with respect to
performing all computations locally. The gain obtained by the proposed
strategy varies in the range $[88,95]\%$, depending on the path loss and the
accuracy constraints. As excepted, the higher the path loss, the lower is the
energy saving due to the need of increasing the transmission energy
consumption, but also the fact that more computations need to be performed
locally to achieve the target accuracy (cf. Fig. 2).

Comparison with BFSP:  with respect to the strategy with BFSP, the energy
saving is still remarkable, ranging from $20\%$ and $60\%$. However,
differently from the previous benchmark, it decreases as a function of the
path loss. This is due to the reduced degrees of freedom in pushing
computations locally or towards the edge. In fact, with the dynamic SP
selection strategy (i.e., our method), the transmission cost can be mitigated
balancing between local and remote processing in a dynamic and adaptive
fashion.

Figure 4: Average splitting point vs Accuracy for different channel
conditions.

Comparisons with BFSNR: we note that our method achieves an appreciable
reduction in terms of energy consumption, in the order of $10$-$20\%$. The
impact of the path loss on this gain is negligible. Also, the impact of the
dynamic selection of the SNR is lower with respect to the dynamic optimization
of the Splitting Point (cf dashed Fig. 3). We point out that, to better
evaluate the impact of the dynamic SNR selection, $\gamma(t)$ should be
considered as a continuous optimization variable while in this preliminary
work we are treating it as discrete. We leave this analysis for future
investigations.

On the impact of the target accuracy on the SP selection: as an additional
informative result, in Fig. 4 we plot the average SP selection (i.e., a
measure of the depth of local computations) obtained by the proposed strategy
as a function of the accuracy, for different path loss conditions. As
expected, increasing the accuracy constraint forces the system to autonomously
choose a deeper splitting point (cf. Fig. 2). Also, the depth of the average
splitting point increases as function of the path loss, as already remarked in
[11]. Indeed, as the channel conditions worsen, the transmission cost
increases, thus encouraging the ED to perform more local processing, of course
at the cost on increased energy.

Comparison with accuracy unaware strategies:  finally, to better highlight the
benefits of the goal-oriented SP selection, we test the dynamic resource
allocation following the same philosophy of [11], removing the accuracy
constraint (cf. $(b)$ in (III)) and focusing only on energy and latency.

TABLE I: Effective accuracy of the accuracy unaware strategy. $PL[dB]$ | 115 | 120 | 125
---|---|---|---
$G_{eff}[\%]$ | 17.302 | 22.17 | 33.40

As witnessed by Table I, the accuracy-unaware strategy leads to a dramatic
reduction in terms of learning performance. With a lower path loss, the system
chooses the first SPs of the network, which are also the most affected by the
noise. Conversely a higher path loss naturally pushes the strategy to select
deeper SPs, with a consequent accuracy increase.

##  V Conclusion and future work

We proposed a dynamic optimization strategy for goal-oriented DNN splitting
with latency and accuracy guarantees. First, we investigated the effect of SP
selection and wireless communication noise on the accuracy of a cooperative
inference task. Building on the acquired knowledge of this behavior, we
tailored a dynamic algorithm that jointly optimizes DNN splitting,
communication and computation resources. Numerical results show the
effectiveness of the proposed approach to meet the prescribed constraints also
in noisy environments, and encourages to further investigate this research
line. Future research directions include the evaluation of the proposed
strategy including outage probability constraints, the joint optimization of
the ED and the ECS resources, and the extension to multi-user scenarios should
be investigated. Also, early exiting as a way of reducing energy and/or
latency is worth being investigated [24].

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# Automorphic cohomology of Mumford-Tate domains

Mohammad Reza Rahmati  Centro de Investigacion en Matematicas, A.C.
<EMAIL_ADDRESS>
###### Abstract.

The article discusses some general motivations from Hodge theory toward
automorphic representations. We explain the semistability of Hodge bundles and
the parabolicity of these bundles. We end up with interesting ideas between
Mumford-Tate groups of Hodge structures and non abelian group cohomology.

###### Key words and phrases:

Automorphic forms and representations, Mumford-Tate groups and domains,
Unitary representation, Langlands Conjecture, Hodge theory, Polarization of
mixed Hodge structure, Peterson inner product

###### 1991 Mathematics Subject Classification:

14G35

## Introduction

Consider a variation of Hodge structure

$\Phi:S\to\Gamma_{\mathbb{Z}}\backslash D$

where $S$ is a smooth base manifold and $\Gamma$ is a discrete group. $\Phi$
may also be regarded as the period map associated to a family of polarized
Hodge structure on a vector space $V$ defined over $\mathbb{Q}$. $D$ is called
a period domain and it is simply known that it is a hermitian symmetric
complex manifold. There are naturally defined Hodge bundles $F^{p}$ of the
Hodge structure on $V$, and also the endomorphism bundle $\mathfrak{g}=End(V)$
on $D$. The corresponding local systems are
$\mathcal{V}:=\Gamma\backslash(D\times\mathcal{V})$ and
$\mathcal{G}:=\Gamma\backslash(D\times\mathfrak{g})$, respectively. Lets,
denote the Hermitian metric by $h$ to be any of the induced metric from that
of the upper half plane $H$. One way to explain the complex structure on $D$
is to embed it in its compact dual $\check{D}$, which is the set of all Hodge
filtrations on $V$ with the same Hodge numbers satisfying the first Riemann-
Hodge bilinear relation. $\check{G}$ is a homogeneous complex manifold. There
are $G_{\mathbb{C}}$-homogeneous vector bundles $F^{p}\to\check{D}$, called
Hodge bundles whose fiber at a given point $F^{\bullet}$ is $F^{p}$. Over
$D\subset\check{D}$ we have $V^{p,q}=F^{p}/F^{p+1}$, which are homogeneous
vector bundles for the action of $G_{\mathbb{R}}$. They are Hemitian vector
bundles with $G_{\mathbb{R}}$-invariant Hermitian metric given in each fiber
by the polarization form. The space of functions on $D$ can be identified with
the $\Gamma_{\mathbb{Z}}$-automorphic functions on $D$. A basic example of
this is a variation $V=V^{1,0}\oplus V^{0,1}$ obtained from the middle
cohomology of an elliptic fibration of curves. In this case $D=H$ is the upper
half plane, $\Gamma_{\mathbb{Z}}=Sl_{2}(\mathbb{Z})$ and the functions on the
Hodge domain $Sl_{2}(\mathbb{Z})\backslash H$ are the usual modular forms,
that are holomorphic functions on $H$ satisfying;

(1) $f(\dfrac{a\tau+b}{c\tau+d})=(c\tau+d)^{n}f(\tau)$

Around a cusp one sets $q=e^{2\pi i\tau}$ and expands in Laurent series
$f(q)=\sum_{n}a_{n}q^{n}$. The finiteness condition at the cusps are then
$a_{n}=0$ for $n<0$. If we consider the Deligne extension $V_{e}^{n,0}$ the
the modular forms are sections of $V_{e}^{n,0}\to\Gamma\backslash H$ that
extend to $H_{e}^{n,0}\to\overline{\Gamma\backslash H}$. A modular form is a
cusp form if $a_{0}=0$, i.e it vanishes at the cusps. This condition is
equivalent to $\int_{\Gamma\backslash H}\parallel\psi\parallel d\mu<\infty$,
[2].

Let $f,\ g$ be two cusp forms of weight $n$ on $H$ the upper half plane. By
putting

(2) $\langle f,g\rangle=\int_{\Gamma\backslash
H}f(z)\overline{g(z)}y^{n-2}dxdy,\qquad x=Re(z),\ y=Im(z)$

one obtains a Hermitian scalar product on the space of modular forms which is
positive and non-degenerate, invariant by $\Gamma=PSl_{2}(\mathbb{Z})$. One
can check that $\langle T(k)f,g\rangle=\langle f,T(k)g\rangle$, where $T(k)$
is the k-th Hecke operator defined by

(3) $T(k).f=k^{n-1}\displaystyle{\sum_{a\geq 0,\ 0\leq
b<d}^{ad=n}d^{-n}f(\dfrac{az+b}{d})}$

A vector valued $\mathcal{A}^{p,q}$-automorphic form
$f\in\mathcal{A}^{p,q}(D,\mathfrak{g})$ satisfies the following condition;

(4)
$f(\gamma.z)\gamma^{\prime}(z)^{p}\overline{\gamma^{\prime}(z)^{q}}=Ad(\rho)f(z),\qquad\gamma\in\Gamma$

where $\rho:\pi_{1}\to Gl(V)$ is the representation of the monodromy on the
parameter base space. $\mathcal{A}^{p,q}(D,\mathfrak{g})$ is vector space of
smooth $C^{\infty}$-forms on $D$ with values in $\mathfrak{g}$. The Peterson
inner product between two modular forms on the upper half plane can be
naturally stated for the vector valued automorphic forms. The hyperbolic
metric descended from $H$ defines the Hodge $*$-operator on $\mathcal{V}$ and
$\mathcal{G}$. Then the Laplace operator and the harmonic forms can be defined
in the same fashion as in Hodge theory. The hermitian metrics
$h_{\mathcal{V}}$ and $h_{\mathcal{G}}$ define the Hodge star operators on the
corresponding space of smooth $(p,q)$-forms. The Laplace operator is defined
by $\Delta=\bar{\partial}^{*}\bar{\partial}$ where
$\bar{\partial}^{*}:=-*\bar{\partial}*$. It is a self adjoint operator on the
space of $L^{2}$-sections of the corresponding bundles. The identification
$\mathcal{V}:=\Gamma\backslash(H\times\mathcal{V})$ and
$\mathcal{G}:=\Gamma\backslash(H\times\mathfrak{g})$ identify these Laplace
operators with the Laplace operator on $H$ acting on the space of
$\Gamma$-automorphic functions. The Hodge inner product is

(5) $\langle f,g\rangle=\int_{H}f(z)\wedge{*g(z)}dxdy=\int_{\Gamma\backslash
H}tr(f(z)\wedge g(z)^{*})y^{p+q-2}dxdy$

where $f^{*}=\bar{f}^{t}$. In the case of usual automorphic functions the
right hand side is Peterson inner product (2). This may be regarded as a
hermitian form induced from the hyperbolic metric on the upper half plane,
[5].

The aforementioned identification of $\mathcal{V}$ and $\mathcal{G}$
identifies this inner product with the one induced by the Hodge inner product,
defined by the Hodge star operator. If we add this fact from the mixed Hodge
metric theorem that the polarization form on the Hodge structures are unique,
it follows that the polarization form and the one induced from Peterson inner
product must be the same up to a constant factor.

##  1\. Moduli of Hodge structures and Automorphic cohomology

Let $h^{p,q}$ be the Hodge numbers of a Hodge structure

$\phi:\mathbb{U}(\mathbb{R})\to Aut(V_{\mathbb{R}})$

of weight $n$ and period domain $D$ with compact dual $\check{D}$. The group
$G_{\mathbb{R}}=Aut(V_{\mathbb{R}},Q)$ is a real simple Lie group that acts
transitively on $D$. The isotropy group $H$ of a reference polarized Hodge
structure $(V,Q,\phi)$ would be a compact subgroup of $G_{\mathbb{R}}$ that
contains a compact maximal torus $T$. One has

(6) $D=\\{\phi:S^{1}\to G_{\mathbb{R}}\ ;\ \phi=g^{-1}\phi_{0}g\\}$

It follows that $H=Z_{\phi_{0}}(G_{\mathbb{R}})$, the centralizer of
$\phi_{0}(S^{1})$. An easy exercise in linear algebra shows

(7) $H\cong\begin{cases}U(h^{2m+1})\times...\times
U(h^{m+1,m})\qquad\qquad\qquad n=2m+1\\\ U(h^{2m})\times...\times
U(h^{m+1,m-1})\times\mathcal{O}(h^{m,m})\qquad n=2m\end{cases}$

The group $G_{\mathbb{C}}$ is a complex simple Lie group that acts
transitively on $\check{D}$. The subgroup $P$ that stabilizes a
$F_{0}^{\bullet}$ is a parabolic subgroup with $H=G_{\mathbb{R}}\cap P$. The
case $n=1$ is classical and one knows that $D=H_{g}$ the Siegel generalized
upper half space $=\\{Z\in M_{g\times g}:Z=^{t}Z,\ Im(Z)>0\\}$.

The Lie algebra $\mathfrak{g}$ of the simple Lie group $G_{\mathbb{C}}$ is a
$\mathbb{Q}$-linear subspace of $End(V)$, and the form $Q$ induces on
$\mathfrak{g}$ a non-degenerate symmetric bilinear form

(8) $B:\mathfrak{g}\times\mathfrak{g}\to\mathbb{C}$

which upto scale is just the Cartan-Killing form
$\text{tr}(\text{ad}(x)\text{ad}(y))$. For each point $\phi\in D$

(9)
$\text{Ad}\phi:\mathbb{U}(\mathbb{R})\to\text{Aut}(\mathfrak{g}_{\mathbb{R}},B)$

is a Hodge structure of weight $0$ on $\mathfrak{g}$. This Hodge structure is
polarized by $B$.

Associated to each nilpotent transformation $N\in\mathfrak{g}$ one defines a
limit mixed Hodge structure. The local system $\mathfrak{g}\to\Delta^{*}$ is
then equipped with the monodromy $T=e^{\text{ad}N}$ and Hodge filtration
defined with respect to the multi-valued basis of $\mathfrak{g}$ by
$e^{\log(t)\frac{N}{2\pi i}}F^{\bullet}$. It gives a limit MHS
$(\mathfrak{g},F^{\bullet},W(N)_{\bullet})$. The polarizing form gives perfect
pairings

(10) $B_{k}:Gr_{k}^{W(N)}\mathfrak{g}\times
Gr_{-k}^{W(N)}\mathfrak{g}\to\mathbb{Q},\qquad B_{k}(u,v)=B(v,N^{k}v)$

defined via the hard Lefschetz isomorphism
$N^{k}:Gr_{-k}^{W(N)}\mathfrak{g}\cong Gr_{k}^{W(N)}\mathfrak{g}$, [2], [3].

######  Proposition 1.1.

The Hodge polarization of the Hodge bundles $\mathcal{V}$ and $\mathcal{G}$
are given by the Peterson Inner product as discussed in Sec (1). The Peterson
inner product induces the Cartan-Killing form
$B:\mathfrak{g}\times\mathfrak{g}\to\mathbb{C}$ on the fibers of
$\mathcal{G}$, and also the graded forms
$\langle.,.\rangle_{k}:PGr_{k}^{W(N)}\mathfrak{g}\times
PGr_{-k}^{W(N)}\mathfrak{g}\to\mathbb{Q},\
\langle.,.\rangle_{k}:=\langle\mathfrak{l}^{k}u,v\rangle=\langle
v,\mathfrak{l}^{k}v\rangle$ when the fibration is involved with the monodromy
$T=e^{\text{ad}(N)}$, where $\mathfrak{l}$ is induced from $ad(N)$. Moreover
the action of the Hecke operators induce self adjoint linear transformations
on $\mathfrak{g}$ which can be identified as $T(k)=\mathfrak{l}_{k}$. The
vector space $\mathcal{S}$ of complex automorphic forms (representations) with
coefficients in a Hodge bundle on a period (Mumford-Tate) domain has a mixed
Hodge structure which is polarized by the above form. (Riemann-Hodge bilinear
relations for automorphic forms) Let the primitive subspaces of $\mathcal{S}$
be defined by

$P_{l}:\ker(N^{l+1}):Gr_{l}^{W}\mathcal{S}\to Gr_{-l-2}^{W}\mathcal{S}$

Have a pure Hodge structure

$P_{l}=\displaystyle{\bigoplus_{p+q=k+l}S^{p,q}}$

of weight $k+l$ (for some $k>0$) polarized by the bilinear form
$\langle.,.\rangle_{l}:P_{l}\times P_{l}\to\mathbb{C}$ induced by Peterson
inner product and satisfies

  * •

$\langle\psi,\eta\rangle_{l}=0,\ \psi\in S^{p,q},\eta\in S^{r,s}$ unless
$r=p,s=q$.

  * •

$*\times\langle\psi,C_{l}.\psi\rangle_{l}>0$ for any $\psi\neq 0$ in
$S^{p,q}$, where $C$ is the corresponding Weil operator and $*$ is a complex
constant.

###### Proof.

The proof follows from the discussion in section (1) and (2) and the
uniqueness of the polarization form for mixed Hodge structures as well of the
operators $\mathfrak{l}_{k}$ known as mixed Hodge metric. ∎

We try to apply the above situation to the cohomology of period domains, where
the cohomology classes are automorphic forms. Let
$\mathfrak{g}_{\mathbb{C}}=\mathfrak{g}_{\mathbb{R}}\otimes\mathbb{C}=Lie(G_{\mathbb{R}}\otimes\mathbb{C})$
be a complex semi-simple Lie algebra,
$\mathfrak{h}=\mathfrak{t}\otimes\mathbb{C}$ a Cartan subalgebra,
$\mathfrak{t}=Lie(T)$, and $K_{\mathbb{C}}$ a complex Lie group corresponding
to the unique maximal compact subgroup $K\subset G_{\mathbb{R}}$. We denote
$\mathcal{U}(\mathfrak{g}_{\mathbb{C}})$ to be the universal enveloping
algebra of $\mathfrak{g}_{\mathbb{C}}$. We assume the action of
$K_{\mathbb{C}}$ will be locally finite, and its differential agrees with the
corresponding subspace of $\mathcal{U}(\mathfrak{g}_{\mathbb{C}})$. One may
match these data with the case $D=G_{\mathbb{R}}/H$ is a general Mumford-Tate
domain siting in the diagram

(11) $\begin{CD}G_{\mathbb{R}}/T@>{}>{}>G_{\mathbb{C}}/B\\\
@V{}V{}V@V{}V{}V\\\
D=G_{\mathbb{R}}/H@>{}>{}>G_{\mathbb{C}}/P=\check{D}\end{CD}$

with $T$ a maximal torus, $B$ a Borel subgroup, and horizontal arrows to be
inclusions, [2].

######  Definition 1.2.

A Harish-Chandra (HC)-module $M$ is a
$(\mathfrak{g}_{\mathbb{C}},K_{\mathbb{C}})$-module that is finite as
$\mathcal{U}(\mathfrak{g}_{\mathbb{C}})$-module with an admissible
$K_{\mathbb{C}}$-action.

We propose to show when the automorphic cohomology $H^{q}(D,L_{\mu})$ of a
period domain gives a HC-module. Let
$\mathcal{H}=\mathcal{U}(\mathfrak{h}_{\mathbb{C}})$. The Weyl group $W$ of
$(\mathfrak{g}_{\mathbb{C}},\mathfrak{h}_{\mathbb{C}})$ acts on $\mathcal{H}$
and gives an isomorphism $Z(\mathfrak{g}_{\mathbb{C}})\cong\mathcal{H}^{W}$,
where the upper-index means the elements fixed by $W$. For each
$\mu\in\mathfrak{h}_{\mathbb{C}}^{*}$ the homomorphism
$\chi_{\mu}:Z(\mathfrak{g}_{\mathbb{C}})\to\mathbb{C},\ z\mapsto z(\mu)$ is
called an infinitesimal character. A result of Harish-Chandra says that any
character of $Z(\mathfrak{g}_{\mathbb{C}})$ is an infinitesimal character, and
$\chi_{\mu}=\chi_{\mu^{\prime}}$ iff $\mu=w(\mu^{\prime})$ for some $w\in W$.
Lets fix a set of positive roots $\Phi^{+}$ of
$(\mathfrak{g}_{\mathbb{C}},\mathfrak{h}_{\mathbb{C}})$. Note that we have

(12)
$\mathfrak{g}_{\mathbb{C}}=\mathfrak{t}_{\mathbb{C}}\oplus\mathfrak{n}^{+}\oplus\mathfrak{n}^{-},\qquad\mathfrak{n}^{+}=\oplus_{\alpha\in\Phi^{+}}\mathfrak{g}^{\alpha}$

Any root $\mu\in\Phi^{+}$ defines an integrable almost complex structure on
$G_{\mathbb{R}}/T$ as well as a Borel subalgebra
$\mathfrak{b}=\mathfrak{h}\oplus\mathfrak{n}^{-}$ and a line bundle
$L_{\mu}=G_{\mathbb{C}}\times_{B}\mathbb{C}$. The sections of a vector bundle
$\mathcal{E}=G_{\mathbb{R}}\times_{T}E\to D$ already defined by the
representation $r:T\to Aut(E)$ are $E$-valued functions $f:G_{\mathbb{R}}\to
E$ such that the right action of $T$ is given by $f(g.t)=r(t^{-1})f(g)$ where
. Now take

(13) $\mathcal{E}_{\mu}=\Lambda^{q}T^{0,1^{*}}D\otimes L_{\mu}$

Identifying $T_{s}^{0,1}=\mathfrak{n}$ obtain
$A^{0,q}(D,L_{\mu})=(C^{\infty}(G_{\mathbb{R}})\otimes\Lambda^{q}\mathfrak{n}^{*}\otimes
L_{\mu})^{T}$. We abbreviate this by
$(C^{\infty}(G_{\mathbb{R}})\otimes\Lambda^{q}\mathfrak{n}^{*})_{-\mu}$.
Summarizing the identifications of the complexes of $G_{\mathbb{R}}$-modules
we get

(14)
$A^{0,q}((D,L_{\mu}),\bar{\partial})=(C^{\infty}(G_{\mathbb{R}})\otimes\Lambda^{q}\mathfrak{n}^{*}),\delta),\qquad
H^{q}(D,L_{\mu})=H^{q}(\mathfrak{n},C^{\infty}(G_{\mathbb{R}}))_{-\mu}$

See [2] for details.

######  Theorem 1.3.

[GS] Let $\rho=\sum_{\alpha\in\Phi^{+}}\alpha/2$. If $\mu+\rho$ is singular,
i.e. $\exists\alpha\in\Phi,\ (\mu+\rho,\alpha)=0$, then

$H^{k}(D,\mathcal{E})=0,\qquad\forall k$

If $\mu+\rho$ is nonsingular, then let $\omega$ be the element of the Weyl
group that carries $\mu+\rho$ into the highest Weyl chamber. Let
$l=\sharp\\{\alpha\in\Delta_{+}\ |\ \omega(\alpha)\text{ is negative }\\}$.
Then

$H^{k}(D,\mathcal{E})=\begin{cases}0\qquad\qquad\qquad k\neq l\\\
\mathcal{E}_{\omega(\mu+\rho)-\rho},\qquad\ k=l\end{cases}$

If $X=\Gamma\backslash D$, and $\mathcal{F}$ be the induced automorphic vector
bundle on $X$ from $\mathcal{E}$, then the corresponding automorphic
cohomology satisfies

$H^{k}(Y,\mathcal{F})=\begin{cases}0,\qquad k\neq\alpha(\pi)\\\
\mathbb{C}^{r},\ \ \ k=\alpha(\pi)\end{cases}$

where $r=c(D)v(X)\dim(E_{\omega(\mu+\rho)-\rho})$, the constant $c(D)$ only
depends to $D$, $v(D)$ is volume of the fundamental domain of $\Gamma$, and

$\alpha(\pi)=\sharp\\{\alpha\in\Delta_{+}-\Phi\
|(\lambda,\alpha>0\\}+\sharp\\{\alpha\in\Delta_{+}-\Phi\
|(\lambda,\alpha<0\\}$

It is well-known that the cohomology ring $H^{*}(D,\mathcal{E})$ is generated
by the chern classes of Hodge bundles on $D$. Such a chern class can be
written as a constant multiple of

$\Theta_{D}=B\wedge^{t}\bar{B}-C\wedge^{t}\bar{C}$

$B=\displaystyle{\sum_{\rho,\alpha\in\Delta_{\mathfrak{v}}-\Phi,(\lambda,\alpha)>0}\nu_{\rho}\otimes
e_{\alpha}\omega_{\rho}^{*}\otimes\omega^{\alpha}}$

$C=\displaystyle{\sum_{\sigma,\alpha\in\Delta_{\mathfrak{v}}-\Phi,(\lambda,\alpha)<0}\nu_{\sigma}\otimes
e_{\alpha}\omega_{\sigma}^{*}\otimes\omega^{\alpha}}$

where $B,C,B^{\prime},C^{\prime}$ are matrices of $(1,0)$-forms. $e_{\alpha},\
\alpha\in\Delta$ are root vectors of $G$, $\omega^{\alpha}$ are dual roots,
$\lambda$ is the highest weight. $\omega_{\rho}$ is a basis for $\mathfrak{g}$
which is an extension of a basis on $\mathfrak{k}$, $\Phi$ is defined via the
root decomposition of the subalgebra
$\mathfrak{k}=\mathfrak{h}\oplus_{\beta\in\Phi}\mathfrak{g}_{\beta}$. Finally
$\omega_{\rho}^{*}$ is the dual basis and $e_{\alpha}\omega_{\rho}^{*}$ is the
contragredient representation, [GS].

##  2\. Hodge bundles vs Stable Parabolic bundles

Let $X$ be a compact connected Riemann surface of genus $g$ and $\emptyset\neq
S\subset X$ a divisor. A parabolic bundle $E$ with parabolic divisor $S$
consists of the data of a filtration (quasi-parabolic condition)
$E_{s}=E_{s,1}\supset...\supset E_{s,l_{s}}\supset 0,\ \forall s\in S$, and
rational numbers $0\leq\alpha(s)<...<\alpha_{l_{s}}<1$ called parabolic
weights. Fix a positive integer $r$. The parabolic degree of $E$ is defined by

(15) $\text{Par-deg}(E)=\deg(E)+\sum_{s\in
S}\sum_{i}\alpha_{i(s)}(s)\dim(E_{s,i}/E_{s,i+1})$

Then every sub-bundle of $E$ is parabolic in a natural way. We call a
parabolic bundle semi-stable if the factor $(\text{par-deg}/\text{rank})$ is
non-increasing when passing to sub-bundles. In case this number is decreasing
for every proper sub-bundles we call the bundle stable. A Hermitian structure
on a parabolic bundle $E$ is a Hermitian structure on $E_{|_{X\diagdown S}}$
with the extra condition that around each $s\in S$ for any section $\sigma$ of
$E$ defined around $s$, if $\sigma(s)$ is non-zero in $E_{s,i}$, then

(16) $\|s\|=f(z)|z|^{\alpha_{i}(s)}$

with positive real valued $f$. The definition of a parabolic bundles can be
generalized for the higher dimensional base. In this case the set $S$ is a
divisor. One may apply this definition to the Hodge filtration on period
domains, see the Proposition 2.2 of this section.

######  Theorem 2.1.

(Mehta-Seshadri) A parabolic bundle $E$ of rank $k$ and parabolic degree $0$
is stable if and only if it is isomorphic to a bundle $E^{\rho}$, where
$\rho:\Gamma\to U(k)$ is an irreducible unitary representation of the group
$\Gamma$ admissible with respect to the weights and multiplicities of the
parabolic structure of $E$. Moreover, parabolic bundles $E^{\rho_{1}}$ and
$E^{\rho_{2}}$ are isomorphic if and only if the representations $\rho_{1}$
and $\rho_{2}$ are equivalent.

A unitary representation $\rho:\Gamma\to U(k)$ is called admissible with
respect to a given set of weights and multiplicities at $S$ if for each $i$,
we have $\rho(S_{i})=U_{i}D_{i}U_{i}^{-1}$, where $S_{i}$ is a generator for
the local monodromy at $P_{i}\in S$, with unitary $U_{i}\in U(k)$ and
$D_{i}=\exp(2\pi i.diag[\alpha_{1}^{i},...,\alpha_{r_{i}}^{i}])$, where each
$\alpha_{l}^{i}=\alpha_{l}(P_{i})$ is repeated $k_{l}^{i}=k_{l}(P_{i})$ times.

######  Proposition 2.2.

[5] The bundles $\mathcal{V}_{D}:=\mathcal{V}\otimes\mathcal{O}_{D},\
\mathcal{G}_{D}:=\mathcal{G}\otimes\mathcal{O}_{D}$ as well as the Hodge
bundles $\mathcal{F}^{p}$ are parabolic bundles over $D$.

###### Proof.

(sketch) Admissible matrices are parametrized by the flag varieties
$F_{i}=U(k)/U(k_{1})\times U(k_{2})\times...\times U(k_{r_{i}})$. The group
$\Gamma$ acts on the trivial bundle $H\times\mathbb{C}^{k}$ by the rule
$(z,\gamma)\mapsto(\gamma z,\rho(\gamma)v)$. Take the sheaf of its bounded
$(\Gamma,\rho)$-sections around the cusps. The direct image of this sheaf
under $H\to X$ is a locally free sheaf of rank $k$. The parabolic structure at
the image of cusps is defined by the matrices $\rho(S_{i})$. This gives a
parabolic vector bundle on the Riemann-Surface $X$. Loosely speaking this
bundle is the extension of the bundle
$E^{\rho}=\Gamma\backslash(H\times\mathbb{C}^{k})\to\Gamma\backslash H$. This
proves the parabolicity of $\mathcal{V}$. The proof for $\mathcal{G}$ is
similar. The result for the Hodge bundles follows from the trivial fact that
the holomorphic subbundles of a parabolic bundle are also parabolic. ∎

The standard Hermitian metric in $\mathbb{C}^{k}$ defines a $\Gamma$-invariant
metric on the trivial vector bundle $H\times\mathbb{C}^{k}\to H$. It extends
as a (pseudo)-metric to the bundle $E=E^{\rho}$. Explicitly, we choose
$\sigma_{i}\in Sl_{2}(\mathbb{R})$ such that $\sigma_{i}(\infty)=x_{i}$ and
$\sigma_{i}^{-1}S_{i}\sigma_{i}=\left(\begin{array}[]{cc}1&\pm 1\\\
0&1\end{array}\right)$, and make the change of coordinates $\zeta=e^{2\pi
i\sigma_{i}^{-1}z}$ at $P_{i}\in\Gamma\backslash H$. Then the metric $h$ is
given by a diagonal matrix
$(\zeta^{2\alpha_{1}^{i}},...,\zeta^{2\alpha_{r_{i}}^{i}})$, [5].

Another question we are interested in is to study the variation of the
(almost) complex structure on the homogeneous space $D$. For this we make the
following general set up. Let $G$ be a reductive Lie group defined over
$\mathbb{R}$ and
$\mathfrak{g}:=Lie(G)=\mathfrak{h}\oplus\bigoplus_{\alpha}\mathfrak{g}_{\alpha}$
its root decomposition, with $\alpha\in R$ and $T$ a maximal torus in
$G_{\mathbb{C}}$. Also consider $V\subset G$ a closed Lie subgroup with

(17)
$\mathfrak{v}=Lie(V_{\mathbb{C}})=\mathfrak{h}\oplus\bigoplus_{\beta\in\Delta}\mathfrak{g}_{\beta}$

Any almost complex structure on $G/V$ gives

$J:\mathfrak{g}_{\mathbb{R}}/\mathfrak{v}_{\mathbb{R}}\to\mathfrak{g}_{\mathbb{R}}/\mathfrak{v}_{\mathbb{R}},\qquad
J^{2}=-id$

$J(\theta(v).X)=\theta(v).J(X),\qquad v\in V,\
X\in\mathfrak{g}_{\mathbb{R}}/\mathfrak{v}_{\mathbb{R}}$,

where $\theta$ is the isotropy representation on $V$. The restriction of the
isotropy representation $\theta$ to $T$ is the adjoint representation giving
(17). It follows that $J$ leaves each isotropycal $\mathfrak{g}_{\beta}$
invariant, and reduces to multiplication by $\pm i$ on $\mathfrak{g}_{\beta},\
\beta\in R-\Delta$. Set

(18) $C:=\\{\alpha\in R-\Delta|J|_{\mathfrak{g_{\alpha}}}=i.\ id\\}$

A choice of $G$-invariant almost complex structure on $G/V$ gives rise to such
a $C$ such that $\alpha\in R-\Delta$ implies only one of $\alpha$ or $-\alpha$
is in $C$. It is possible to choose a system of positive roots $R_{+}$ such
that $C\subset R_{+}$. In this case $R_{+}$ is called the system of positive
roots adopted to invariant almost complex structure. This criteria generalizes
to arbitrary homogeneous vector bundles on $G/V$ associated to a
representation $\rho:V\to Gl(H)$ on a vector space $H$. One can apply the
above criteria to the system of Hodge bundles on $D$ or $\check{D}$ and obtain
a sequence of subsets $C_{p}$ of the $R$ adopted to the whole filtration,
$C_{p}\subset C_{p-1}$. The theorem states that this procedure is also
reversible. That is any choice of this adopted series provides complex
structure on each $F^{p}$.

######  Proposition 2.3.

[4] Let $\rho:V\to Gl(H)$ be an $n$-dimensional complex representation and
$E:=G\times_{V}E$ the associated vector bundle on $G/V$. Then there exists a
1-1 correspondence between $G$-invariant almost complex structures on $E$ and

$Hom_{V}^{J}(\ \bigoplus_{\alpha\in
C}\mathfrak{g}_{\alpha},\mathfrak{g}^{\prime}):=\\{f\ |\ f\circ J=-J\circ
J\\}$

where $\mathfrak{g}^{\prime}=\mathfrak{gl}_{n}(\mathbb{C})$, with a choice of
$C$ satisfying (18). Moreover such almost complex structure, will be always a
complex (holomorphic) structure.

The proof of this theorem follows from a careful diagram chasing in the
natural split exact sequence of $V$-modules

(19)
$0\to\mathfrak{g}^{\prime}\to\mathfrak{p}=(\mathfrak{g}\times\mathfrak{g}^{\prime})/\mathfrak{h}_{\mathbb{R}}\to(\mathfrak{g}/\mathfrak{v})_{\mathbb{R}}\to
0$

The $G$-invariant holomorphic structures then correspond to all possible
extensions of $ad\rho:\mathfrak{v}\to\mathfrak{gl}(H)$ to
$ad\rho:\mathfrak{v}\bigoplus_{\alpha\in
C}\mathfrak{g}_{-\alpha}\to\mathfrak{gl}(H)$. In the case of Hodge bundles the
complex structures of the system of Hodge bundles correspond the nummber
successive extensions when you add the roots in $C_{p-1}\setminus C_{p}$ at
each step of the Hodge filtration.

######  Theorem 2.4.

[4] Let $G$ be a reductive connected Lie group defined over $\mathbb{R}$, and
$V$ a closed subgroup such that $G/V$ is an irreducible Hermitian symmetric
space. Let $\rho$ be any irreducible representation of $V$. Then the
associated homogeneous vector bundle is stable.

Theorem 2.4 states that for any holomorphic subbundle $L$ of $F^{p}$ we have

$\dim(L).c_{1}(F^{p})-\dim(F^{p}).c_{1}(L)>0$

Applying this theorem to the situation of period domains we get,

######  Corollary 2.5.

The bundles $\mathcal{V}_{D},\mathcal{G}_{D}$ as well as the Hodge bundles
$\mathcal{F}^{p}$ are stable vector bundles over $D$.

##  3\. Non-abelian cohomology and MT-groups of Hodge structures

(1) Let $G$ be a group and $A$ (not necessarily abelian) another group on
which $G$ acts on the left. Write $A$ multiplicatively. $H^{0}(G,A)$ is by
definition the group $A^{G}$ of elements of $A$ fixed by $G$. A 1-cocycle
would be a map $s\mapsto a_{s}$ from $G\to A$ such that
$a_{st}=a_{s}.s(a_{t})$. Two cocycles $a_{s}$ and $b_{s}$ are equivalent if
there exists $a\in A$ such that $b_{s}=a^{-1}.a_{s}.s(a)$ for all $s\in G$.
This defines an equivalence relation and the quotient set is denoted by
$H^{1}(G,A)$. It is a pointed set with a distinguished element of the unit
cocycle $a_{s}=1$. Here $s(-)$ means the action of $s\in G$, and $a_{s}$ is
the value of the cocycle $a$ at $s\in G$. These two definitions agree with the
usual definitions of cohomology of $G$ when $A$ is abelian. These
constructions are also functorial in $A$ and $G$. If $0\to A\to B\to C\to 0$
is an exact sequence of non-abelian $G$-modules, then we have the following
exact sequence of pointed sets

$H^{0}(G,A)\to H^{0}(G,B)\to H^{0}(G,C)\to H^{1}(G,A)\to H^{1}(G,B)\to
H^{1}(G,C)$

when the group $G$ is Galois group of a (not necessarily finite) field
extension and $A$ a topological $G$-module such that

$A=\bigcup A^{H}$

when $H$ runs through the open normal subgroups of $G$, we define

$H^{1}(G,A):=\lim_{\rightarrow}H^{1}(G/H,A^{H})$

There is a simple geometric interpretation for $H^{1}(G,A_{K})$: It is the set
of classes of principal homogeneous spaces for $A$, defined over $k$ which
have a rational point over $K$, [6] page 123. As an example, take
$G=\mathbb{Z}/2=Gal(\mathbb{C}/\mathbb{R})$ acting naturally on a set of
tensors $A_{\mathbb{R}}$. Then any element in
$H^{1}(\mathbb{Z}/2,A_{\mathbb{C}})$ is determined by an involution map
$-1:a_{-1}\mapsto*\in A_{\mathbb{R}}$. This shows
$H^{1}(\mathbb{Z}/2,A_{\mathbb{C}})=A_{\mathbb{R}}$, [6].

If $V$ be a vector space over $k$, provided with a fixed tensor of $x$ of type
$(p,q)$, i.e. $x\in\bigotimes^{p}V\otimes\bigotimes^{q}V^{*}$ where $V^{*}$ is
the dual of $V$. two pairs $(V,x),(V,x^{\prime})$ are called $k$-isomorphic if
there is a $k$-linear isomorphism $f:V\to V^{\prime}$ such that
$f(x)=x^{\prime}$. Denote by $A_{k}$ the group of these automorphisms. Let
$K/k$ be a Galois extension with Galois group $G$. Write $E_{V,x}(K,k)$ for
the set of $k$-isomorphism classes that are $K$-isomorphic to $(V,x)$. The
group $G$ acts on $V_{K}$ by $s.(x\otimes\lambda)=x\otimes s.\lambda$. It also
acts on $f:V\to V^{\prime}$ by $s.f=s\circ f\circ s^{-1}$. If we put

$p_{s}=f^{-1}\circ s\circ f\circ s^{-1},\qquad s\in G$

the map $s\mapsto p_{s}(f)$ is a 1-cocycle in $H^{1}(G,A_{K})$, [6].

######  Theorem 3.1.

([6] page 153) The map $\theta:E_{V,x}(K/k)\to H^{1}(G,A_{K})$ defined by

$f\mapsto p_{s}(f)$

is a bijection.

######  Theorem 3.2.

([6] page 153) The set $H^{1}(G,Aut(Q,K))$ is in bijective correspondence with
the classes of quadratic $k$-forms that are $K$-isomorphic to $Q$.

The above definition can be generalized in this way that instead of
considering a single tensor
$T^{p,q}=\bigotimes^{p}V\otimes\bigotimes^{q}V^{*}$ one may consider a a sum
of such tensors that is a subset $T$ as

$T\subset T^{\bullet,\bullet}=\oplus_{p,q}T^{p,q}$

The proofs will proceed exactly the same and we obtain

######  Theorem 3.3.

Assume $T$ is a set of tensors for a vector space $V$. The map
$\theta:E_{V,T}(K/k)\to H^{1}(G,A_{T,K})$ defined by

$f_{T}\mapsto p_{s}(f_{T})$

is a bijection, where $A_{T,K}$ is the group of $K$-automorphisms of all the
tensors in $T$.

(2) To begin with let $V$ be finite dimensional $\mathbb{Q}$-vector space, and
$Q$ a non-degenerate bilinear map $Q:V\otimes V\to\mathbb{Q}$ which is
$(-1)^{n}$-symmetric for some fixed $n$. A Hodge structure is given by a
representation $\phi:\mathbb{U}(\mathbb{R})\to Aut(V,Q)_{\mathbb{R}}$. The
Mumford-Tate group of the Hodge structure $\phi$ denoted $M_{\phi}(R)$ is the
smallest $\mathbb{Q}$-algebraic subgroup of $G=Aut(V,Q)$ with the property
$\phi(\mathbb{U}(R))\subset M_{\phi}(R)$. $M_{\phi}$ is a simple, connected,
reductive $\mathbb{Q}$-algebraic group. If $F^{\bullet}\in\check{D}$ the
Mumford-Tate group $M_{F^{\bullet}}$ is the subgroup of $G_{\mathbb{R}}$ that
fixes the Hodge tensors.

We are going to investigate the relation between nonabelian cohomology
discussed above, and classifying spaces for Hodge structures. By definition
$H^{0}(G_{\mathbb{R}},Hg_{F^{\bullet}}^{\bullet,\bullet})=M_{F^{\bullet}}$.
The relation $H^{1}(\mathbb{Z}/2,G_{\mathbb{C}})=G_{\mathbb{R}}$ is trivial.
By applying Theorem 4.3 to the above construction we get the following
results.

######  Theorem 3.4.

(1) $H^{1}(\mathbb{Z}/2,M_{F^{\bullet}})=M_{\phi}$.

(2) $H^{1}(Gal(\mathbb{C}/\mathbb{Q}),Aut(Q,\mathbb{C}))=\check{D}$

(3) $H^{1}(Gal(\mathbb{R}/\mathbb{Q}),Aut(Q,\mathbb{R}))=D$

###### Proof.

(1) follows from the example at (1), and the fact that $M_{\phi}$ is the
subgroup of $G$ that fixes pointwise the algebra of Hodge tensors, cf. [MGK].

(2) Follows from Theorem 3.3, noting that
$\check{D}=G_{\mathbb{C}}/Stab_{G_{\mathbb{C}}}(F_{0})$.

(3) This also follows from Theorem 3.3, and similar identity
${D}=G_{\mathbb{R}}/Stab_{G_{\mathbb{R}}}(F_{0})$. . ∎

We have the short exact sequence

$0\to\mathbb{Z}/2\to Gal(\mathbb{C}/\mathbb{Q})\to
Gal(\mathbb{R}/\mathbb{Q})\to 0$

where the second non-zero map is the restriction

$0\to H^{1}(\mathbb{Z}/2,Aut(Q,\mathbb{C})^{Gal(\mathbb{R}/\mathbb{Q})})\to
H^{1}(Gal(\mathbb{C}/\mathbb{Q}),Aut(Q,\mathbb{C}))$

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\to
H^{1}(Gal(\mathbb{R}/\mathbb{Q}),Aut(Q,\mathbb{C}))^{\mathbb{Z}/2}$

where the second map is the restriction. The first map is called inflation
map. The first item is $Aut(Q,\mathbb{R})$, and the second item is $\check{D}$
by Theorem 3.2. Thus we have

$0\to Aut(Q,\mathbb{R})\to\check{D}\to
H^{1}(Gal(\mathbb{R}/\mathbb{Q}),Aut(Q,\mathbb{C}))^{\mathbb{Z}/2}$

as exact sequence of sets with distinguished unit elements.

######  Theorem 3.5.

(1) $H^{1}(\mathbb{U}_{\mathbb{R}},G_{\mathbb{R}})=D$, where $G_{\mathbb{R}}$
acts on $G_{\mathbb{R}}=Aut(Q,\mathbb{R})$ by $g:T\mapsto g^{t}Tg$.

(2) $H^{1}(\mathbb{U}(\mathbb{R}),Aut(\mathfrak{g}_{\mathbb{R}},B))=D$, where
$\mathbb{U}_{\mathbb{R}}$ acts by $g.X=g^{t}Xg,\
X\in\mathfrak{g}_{\mathbb{R}}$, and $B$ is the killing form.

###### Proof.

(1) The cocycle condition is equivalent to $\mathbb{U}_{\mathbb{R}}\to
G_{\mathbb{R}}$ being a homomorphism and the boundary condition is when two
such homomorphism are conjugate by an element of $G_{\mathbb{R}}$. Then, the
theorem is consequence of that, $D$ is isomorphic to the set of conjugacy
classes of the isotropy group of a fixed Hodge structure.

(2) Again the cocycle condition is the cocycles are
$Hom(\mathbb{U}_{\mathbb{R}},Aut(B))$, and the coboundray condition becomes
when two such homomorphisms are conjugate by an automorphism of $B$. Regarding
$B$ as a tensor then the theorem follows from the known fact that $M_{\phi}$
is the subgroup of $G$ with the property that $M_{\phi}$-stable subspaces
$W\subset T_{\phi}^{a,b}$ are exactly the sub-Hodge structures of these tensor
space. ∎

For each point $\phi\in D$, the adjoint representation

$Ad\phi:\mathbb{U}(\mathbb{R})\to Aut(\mathfrak{g}_{\mathbb{R}},B)$

induces a Hodge structure of weight $0$ on $\mathfrak{g}$. This Hodge
structure is polarized by the killing form
$B:\mathfrak{g}\times\mathfrak{g}\to\mathbb{C}$. and it is a sub-Hodge
structure of $\check{V}\times V$. Therefore (2) is analogue of (1) applied to
this vector space.

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