AWGN short = AWGN, long = additive white Gaussian noise, list = Additive White
Gaussian Noise, tag = abbrev ADMM short = ADMM, long = alternating direction
method of multipliers, list = Alternating Direction Method of Multipliers, tag
= abbrev MGMC short = MGMC, long = multi-group multi-casting, list = multi-
group multi-casting, tag = abbrev SGMC short = SGMC, long = single-group
multi-casting, list = single-group multi-casting, tag = abbrev AoA short =
AoA, long = angle-of-arrival, list = Angle-of-Arrival, tag = abbrev AoD short
= AoD, long = angle-of-departure, list = Angle-of-Departure, tag = abbrev KKT
short = KKT, long = Karush-Kuhn-Tucker, list = Karush-Kuhn-Tucker, tag =
abbrev MMF short = MMF, long = max-min-fairness, list = max-min-fairness, tag
= abbrev WMMF short = WMMF, long = weighted max-min-fairness, list = max-min-
fairness, tag = abbrev BB short = BB, long = base band, list = Base Band, tag
= abbrev BC short = BC, long = broadcast channel, list = Broadcast Channel,
tag = abbrev BS short = BS, long = base station, list = Base Station, tag =
abbrev BR short = BR, long = best response, list = Best Response, tag = abbrev
CB short = CB, long = coordinated beamforming, list = Coordinated Beamforming,
tag = abbrev CC short = CC, long = coded caching, list = Coded Caching, tag =
abbrev CE short = CE, long = channel estimation, list = Channel Estimation,
tag = abbrev CoMP short = CoMP, long = coordinated multi-point transmission,
list = Coordinated Multi-Point Transmission, tag = abbrev CRAN short = C-RAN,
long = cloud radio access network, list = Cloud Radio Access Network, tag =
abbrev CSE short = CSE, long = channel specific estimation, list = Channel
Specific Estimation, tag = abbrev CSI short = CSI, long = channel state
information, list = Channel State Information, tag = abbrev CSIT short = CSIT,
long = channel state information at the transmitter, list = Channel State
Information at the Transmitter, tag = abbrev CU short = CU, long = central
unit, list = Central Unit, tag = abbrev D2D short = D2D, long = device-to-
device, list = Device-to-Device, tag = abbrev DE-ADMM short = DE-ADMM, long =
direct estimation with alternating direction method of multipliers, list =
Direct Estimation with Alternating Direction Method of Multipliers, tag =
abbrev DE-BR short = DE-BR, long = direct estimation with best response, list
= Direct Estimation with Best Response, tag = abbrev DE-SG short = DE-SG, long
= direct estimation with stochastic gradient, list = Direct Estimation with
Stochastic Gradient, tag = abbrev DFT short = DFT, long = discrete fourier
transform, list = Discrete Fourier Transform, tag = abbrev DoF short = DoF,
long = degrees of freedom, list = Degrees of Freedom, tag = abbrev DL short =
DL, long = downlink, list = Downlink, tag = abbrev GD short = GD, long =
gradient descent, list = Gradeitn Descent, tag = abbrev IBC short = IBC, long
= interfering broadcast channel, list = Interfering Broadcast Channel, tag =
abbrev i.i.d. short = i.i.d., long = independent and identically distributed,
list = Independent and Identically Distributed, tag = abbrev JP short = JP,
long = joint processing, list = Joint Processing, tag = abbrev LOS short =
LOS, long = line-of-sight, list = Line-of-Sight, tag = abbrev LS short = LS,
long = least squares, list = Least Squares, tag = abbrev LTE short = LTE, long
= Long Term Evolution, tag = abbrev LTE-A short = LTE-A, long = Long Term
Evolution Advanced, tag = abbrev MIMO short = MIMO, long = multiple-input
multiple-output, list = Multiple-Input Multiple-Output, tag = abbrev MISO
short = MISO, long = multiple-input single-output, list = Multiple-Input
Single-Output, tag = abbrev MAC short = MAC, long = multiple access channel,
list = Multiple Access Channel, tag = abbrev MSE short = MSE, long = mean-
squared error, list = Mean-Squared Error, tag = abbrev MMSE short = MMSE, long
= minimum mean-squared error, list = Minimum Mean-Squared Error, tag = abbrev
mmWave short = mmWave, long = millimeter wave, list = Millimeter Wave, tag =
abbrev MU-MIMO short = MU-MIMO, long = multi-user MIMO, list = Multi-User
MIMO, tag = abbrev OTA short = OTA, long = over-the-air, list = Over-the-Air,
tag = abbrev PSD short = PSD, long = positive semidefinite, list = Positive
Semidefinite, tag = abbrev QoS short = QoS, long = quality of service, list =
Quality of Service, tag = abbrev RCP short = RCP, long = remote central
processor, list = Remote Central Processor, tag = abbrev RRH short = RRH, long
= remote radio head, list = Remote Radio Head, tag = abbrev RSSI short = RSSI,
long = received signal strength indicator, list = Received Signal Strength
Indicator, tag = abbrev RX short = RX, long = receiver, list = Receiver, tag =
abbrev SCA short = SCA, long = successive-convex-approximation, list =
Successive-Convex-Approximation, tag = abbrev SG short = SG, long = stochastic
gradient, list = Stochastic Gradient, tag = abbrev SIC short = SIC, long =
successive interference cancellation, list = Successive Interference
Cancellation, tag = abbrev SNR short = SNR, long = signal-to-noise-ratio, list
= Signal-to-Noise Ratio, tag = abbrev SDR short = SDR, long = semi-definite-
relaxation, list = semi-definite-relaxation, tag = abbrev SINR short = SINR,
long = signal-to-interference-plus-noise ratio, list = Signal-to-Interference-
plus-Noise Ratio, tag = abbrev SOCP short = SOCP, long = second order cone
program, list = Second Order Cone Program, tag = abbrev SSE short = SSE, long
= stream specific estimation, list = Stream Specific Estimation, tag = abbrev
SVD short = SVD, long = singular value decomposition, list = Singular Value
Decomposition, tag = abbrev TDD short = TDD, long = time division duplex, list
= Time Division Duplex, tag = abbrev TX short = TX, long = transmitter, list =
Transmitter, tag = abbrev UE short = UE, long = user equipment, list = User
Equipment, tag = abbrev UL short = UL, long = uplink, list = Uplink, tag =
abbrev ULA short = ULA, long = uniform linear array, list = Uniform Linear
Array, tag = abbrev UPA short = UPA, long = uniform planar array, list =
Uniform Planar Array, tag = abbrev WMMSE short = WMMSE, long = weighted
minimum mean-squared error, list = Weighted Minimum Mean-Squared Error, tag =
abbrev WMSEMin short = WMSEMin, long = weighted sum MSE minimization, list =
Weighted sum MSE Minimization, tag = abbrev WBAN short = WBAN, long = wireless
body area network, list = Wireless Body Area Network, tag = abbrev WSRMax
short = WSRMax, long = weighted sum rate maximization, list = Weighted Sum
Rate Maximization, tag = abbrev

# Low-Subpacketization Multi-Antenna
Coded Caching for Dynamic Networks

MohammadJavad Salehi1, Emanuele Parinello2, Hamidreza Bakhshzad Mahmoodi1, and
Antti Tölli1
This work is supported by the Academy of Finland under grant no. 318927
(6Genesis Flagship), and by Vaikuttavuussäätiö under the project Directional
Data Delivery for Wireless Immersive Digital Environments (3D-WIDE).  1Centre
for Wireless Communications, University of Oulu, 90570 Oulu, Finland
2Communication Systems Department, Eurecom, Sophia Antipolis, 06410 Biot,
France
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Multi-antenna coded caching combines a global caching gain, proportional to
the total cache size in the network, with an additional spatial multiplexing
gain that stems from multiple transmitting antennas. However, classic
centralized coded caching schemes are not suitable for dynamic networks as
they require prior knowledge of the number of users to indicate what data
should be cached at each user during the placement phase. On the other hand,
fully decentralized schemes provide comparable gains to their centralized
counterparts only when the number of users is very large. In this paper, we
propose a novel multi-antenna coded caching scheme for dynamic networks, where
instead of defining individual cache contents, we associate users with a
limited set of predefined caching profiles. Then, during the delivery phase,
we aim at achieving a combined caching and spatial multiplexing gain,
comparable to a large extent with the ideal case of fully centralized schemes.
The resulting scheme imposes small subpacketization and beamforming overheads,
is robust under dynamic network conditions, and incurs small finite-SNR
performance loss compared with centralized schemes.

###### Index Terms:

coded caching, MIMO communications, low-subpacketization, dynamic networks

##  I Introduction

Wireless networks are under continuous pressure to support increasing volumes
of multimedia content [1] and pave the way for the emergence of new
applications such as wireless immersive viewing [2]. For the efficient
delivery of such multimedia content, Maddah-Ali and Niesen proposed the idea
of _coded caching_ for increasing the data rates by exploiting cache content
across the network [3]. In a single-stream downlink network, coded caching
enables boosting the achievable rate by a multiplicative factor proportional
to the cumulative cache size in the entire network. This speedup is achieved
through multicasting of carefully created codewords to different groups of
users, such that each user can use its cache content to remove unwanted parts
from the received signal. Motivated by the growing importance of multi-antenna
communications [4], cache-aided MISO setting was later explored in [5, 6],
where it was revealed that the same coded caching gain could be achieved
together with the spatial multiplexing gain. This cumulative gain was then
shown in [7] to be optimal with the underlying assumptions of uncoded cache
placement and one-shot linear data delivery.

Following the introduction of MISO coded caching, many later works in the
literature addressed its important scaling and performance issues. Notably, in
[8], the authors showed that using optimized beamformers instead of zero-
forcing, one could achieve a better rate for communications at the finite-SNR
regime. Similarly, the exponentially growing subpacketization issue (i.e., the
number of smaller parts each file should be split into) was addressed
thoroughly in the literature. Interestingly, while reducing subpacketization
is challenging in single-antenna setups [9], MISO setups allow reducing the
subpacketization substantially without affecting the maximum achievable DoF
[10]. Of course, this reduced subpacketization modestly decreases the
achievable rate, especially at the finite-SNR regime [11]. Nevertheless, it
does not harm the achievable DoF and yet enables a much simpler optimized
beamformer design [12].

There exists another critical obstacle preventing the practical implementation
of coded caching schemes, especially in dynamic network setups. All the
schemes mentioned so far are centralized, i.e., the cache contents of the
users are dictated by a central server. However, the server needs prior
knowledge of the number of users to indicate what data should be cached at
each user during the placement phase. This makes it impossible to implement
caching techniques in dynamic networks where the users are allowed to
join/leave the network at any moment. One possible solution to this issue is
to use fully decentralized schemes, such as [13], where the cache contents of
each user are assumed to be fully random and codewords are built to achieve
the caching gain to the maximum possible extent. Unfortunately, the caching
gain of such decentralized solutions is comparable with centralized schemes
only asymptotically, i.e., when the number of users (or the size of each file)
is very large. Hence, the problem remains largely unresolved for practical
networks with a moderate number of users.

In this paper, we take a different approach to this problem by introducing a
hybrid centralized/decentralized coded caching scheme. In this scheme, the
cache contents of users are dictated by a central server, but the users can
join/leave the network at any time. Each user is assigned with a _caching
profile_ that indicates which contents should be cached at its memory.
Multiple users may be assigned with the same profile, making the resulting
setup similar to shared-cache models where multiple users have access to the
same cache storage (cf. [14]). However, due to the dynamic nature of the
network, the _length_ of each profile, defined as the number of users
associated with that profile, can vary during the time and be independent from
the length of other profiles. Then, we introduce a new delivery algorithm,
inspired by the low-subpacketization scheme in [12], for enabling a cumulative
caching and spatial multiplexing gain, comparable to a large extent with the
ideal case of fully centralized schemes. Of course, this paper is not the
first one introducing such a hybrid scheme; a similar approach is introduced
in [15] for single-antenna setups. However, here, we extended the results to
multi-antenna setups and do so with minor subpacketization and beamforming
overheads.

We emphasize that, this paper does _not_ aim at proposing an information-
theoretic DoF-optimal scheme for dynamic networks. Instead, the goal is to
provide a practical scheme with an appropriate performance at the finite-SNR
regime. In this regard, we leave most of the theoretical analyses of the
proposed scheme to the extended version of this paper, and use numerical
simulations here to investigate how our new scheme performs in comparison with
centralized schemes applied to static network setups. Nevertheless, we provide
helpful insights about the underlying reasons for any observation of an
improved/deteriorated performance.

In this paper, we use boldface lower- and upper-case letters to denote vectors
and matrices, respectively. Sets are shown with calligraphic letters. Similar
to vectors, it is assumed that the order of members in a set is important and
preserved. ${\mathbf{A}}[i,j]$ is the element at row $i$ and column $j$ of
matrix ${\mathbf{A}}$, and ${\mathbf{a}}[i]$ and ${\mathcal{A}}[i]$ are the
$i$-th elements of vector ${\mathbf{a}}$ and set ${\mathcal{A}}$,
respectively. $[K]$ represents the set of integers $\\{1,2,...,K\\}$, and
${\mathcal{A}}\backslash{\mathcal{B}}$ denotes the elements of set
${\mathcal{A}}$ which are not in set ${\mathcal{B}}$. Other notations are
defined as they are used in the text.

##  II System Model

We consider a MISO dynamic setup where a group of single-antenna users request
data from a multi-antenna server. The server has spatial DoF of
$\alpha$.111The real number of antennas may be larger than the spatial DoF.
However, we assume the spatial DoF at the users and the server is _set_ to one
and $\alpha$, respectively. In [8], it is thoroughly studied how spatial DoF
value affects the performance of coded caching schemes at finite- and high-SNR
regimes. The users request data from a library ${\mathcal{F}}$ of size $N$
files, where the size of each file is $f$ bits. Every user has a cache memory
large enough to store a portion $0<\gamma<1$ of the entire library. We define
the parameter $P$ to be the smallest integer such that $P\gamma$ is also an
integer. For example, if $\gamma=\frac{1}{4}$, then, $P=4$.

The users are allowed to join and leave the network at any time. Upon joining
the network, every user $k$ is associated (e.g., randomly) with a profile
index $p(k)\in[P]$. The parameter $p(k)$ indicates which subset of data should
be cached at the cache memory of user $k$ (more detailed explanation is
provided shortly after). After being associated with the profile index $p(k)$,
user $k$ starts filling up its cache memory by downloading data from the
server. We assume this cache filling process does not affect simultaneously
ongoing data delivery processes.222This is the case, for example, if the cache
is filled when excess transmission capacity becomes available at the server
(e.g., when the network traffic load is low), or if the users join the network
from specific locations with large transmission capacities (e.g., when
transmit antennas are located near the entrance door of the application area
in an extended reality use case). After all the required data is downloaded in
the cache memory of user $k$, this user is identified by the server to be
eligible to participate in upcoming coded caching (CC) data delivery
procedures.

Data requests are revealed to the server at specific time intervals. At the
beginning of each time interval, a subset of users reveal their requested data
from the library ${\mathcal{F}}$.333The reason for assuming that a _subset_ of
users request data is that in some applications, the requests are made only
when the user conditions are changed. For example, consider an extended
reality application where the requests are made only when the users move to a
different location. We denote the file requested by user $k$ as $W(k)$. The
server then carries out data delivery in two subsequent phases:

  1. 1.

CC data delivery, where the server builds a set of transmission vectors using
multi-antenna coded caching techniques and transmits them to eligible users
(that have filled their cache memories with the intended data);

  2. 2.

Unicast data delivery, where the server transmits all the remaining requested
data, not delivered in the CC phase.

The first phase is designed to enable a combined global caching and spatial
multiplexing gain, comparable with the ideal case of fully centralized
schemes. However, in the unicast phase, only the local caching gain is
achieved together with the spatial DoF. In this paper, without loss of
generality, we consider only a specific request interval and drop the interval
index. We assume the whole process is repeated at every time interval after a
new subset of users reveal their requests.

The CC delivery phase is based on the RED scheme in [12]. This scheme allows
achieving the largest possible combined DoF of caching and spatial
multiplexing with a very small subpacketization requirement. It also enables
implementing high-performance optimized beamformers with very low complexity
(using uplink-downlink duality). The same technique is used in this paper to
design optimized beamformers for both delivery phases. As discussed earlier,
we analyze the finite-SNR performance here, and leave theoretical analysis for
the extended version of this paper. In this regards, we use the total required
delivery time (over both phases) as our metric. Symmetric rate is then defined
as the inverse of the total delivery time.

Cache Placement. The content cached at each user $k$ is defined by its
associated profile $p(k)$. Multiple users may be associated with the same
profile, and hence, cache the same content. In order to indicate which data
elements should be stored for each profile, we use a $P\times P$ binary
placement matrix ${\mathbf{V}}$. The first row of ${\mathbf{V}}$ has
${\bar{t}}=P\gamma$ consecutive ones (other elements are zero), and the row
$1<p\leq P$ is a circular shift of the row $p-1$ to the right by one unit.
Given ${\mathbf{V}}$, we split each file $W\in{\mathcal{F}}$ into $P$ equal-
sized _packets_ $W_{p}$. Then, if ${\mathbf{V}}[p,p(k)]=1$ for some $k$ and
$p$, user $k$ is instructed to cache packets $W_{p}$ of every file
$W\in{\mathcal{F}}$.

######  Example 1.

Consider a MISO setup where $\gamma=\frac{1}{4}$, and hence, $P=4$ and
${\bar{t}}=1$. Then, the placement matrix is

${\mathbf{V}}=\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{bmatrix}\;.$

So, each file is split into $P=4$ packets, and, for example, if $p(k)=1$ for
some user $k$, then it has to cache packet $W_{1}$ from every file
$W\in{\mathcal{F}}$.

##  III CC Delivery Phase

After the users reveal their requests, the server first performs a CC delivery
phase where it uses coded caching techniques to build a set of codewords and
transmit them to eligible users. Assume the number of users associated with
profile $p$ and making requests at the considered time instant (i.e., the
length of profile $p$) is $\eta_{p}$. The server then chooses a common
parameter $\hat{\eta}$ to indicate the number of users participating in the CC
delivery phase. The system level impact of choosing $\hat{\eta}$ is clarified
later in this section. Then, for every profile $p$:

  1. 1.

if $\eta_{p}>\hat{\eta}$, then $\hat{\eta}-\eta_{p}$ users associated with
profile $p$ are selected randomly and excluded from the CC phase;

  2. 2.

if $\eta_{p}\leq\hat{\eta}$, then $\hat{\eta}-\eta_{p}$ phantom users are
assigned with the profile $p$.

Phantom users are non-existing users that appear when codewords are built and
are excluded from the actual delivery [12]. Although phantom users incur some
DoF loss, they enable a larger beamforming gain that can actually improve the
achievable rate, compared with schemes with a larger DoF, at the finite-SNR
regime (cf. [12, 8]).

After the above-mentioned process, all profiles will have the same length of
$\hat{\eta}$. Then, we consider a _virtual_ network with $P$ users, coded
caching gain ${\bar{t}}=P\gamma$, and spatial DoF of
$\bar{\alpha}=\lceil\frac{\alpha}{\hat{\eta}}\rceil$, where the virtual user
${\bar{k}}\in[P]$ has the cache content corresponding to the profile
$p={\bar{k}}$ in the original network. For this virtual network, we build the
transmission vectors using the RED scheme in [12],444The RED scheme requires
the spatial DoF to be larger than or equal to the caching gain. So, here we
assume $\bar{\alpha}\geq{\bar{t}}$. Relaxing this condition is left for the
extended version of the paper. resulting in a total number of $P(P-{\bar{t}})$
transmission vectors $\bar{{\mathbf{x}}}_{j}^{r}$, where $j\in[P-{\bar{t}}]$
and $r\in[P]$. In fact, the virtual network would require $P$ transmission
_rounds_ , where at each round, $P-{\bar{t}}$ individual transmissions are
done. The transmission vector $\bar{{\mathbf{x}}}_{j}^{r}$ is explicitly built
as

$\bar{{\mathbf{x}}}_{j}^{r}=\sum_{{\bar{n}}\in[{\bar{t}}+\bar{\alpha}]}{\bar{W}}_{\bar{{\mathbf{p}}}_{j}^{r}[{\bar{n}}]}^{{\bar{q}}}(\bar{{\mathbf{k}}}_{j}^{r}[{\bar{n}}])\bar{{\mathbf{w}}}_{\bar{{\mathcal{R}}}_{j}^{r}({\bar{n}})}\;,$
(1)

where $\bar{{\mathbf{p}}}_{j}^{r}$ and $\bar{{\mathbf{k}}}_{j}^{r}$ are the
packet and user index vectors, and $\bar{{\mathcal{R}}}_{j}^{r}({\bar{n}})$ is
the interference indicator set used for the ${\bar{n}}$-th data term, in the
$j$-th transmission of round $r$. Also,
$\bar{{\mathbf{w}}}_{\bar{{\mathcal{R}}}_{j}^{r}({\bar{n}})}$ is the optimized
beamforming vector suppressing data at every user in
$\bar{{\mathcal{R}}}_{j}^{r}({\bar{n}})$, ${\bar{W}}({\bar{k}})$ denotes the
file requested by the virtual user ${\bar{k}}$, and ${\bar{q}}$ is the
subpacket index which is initialized to one and increased every time a packet
appears in a transmission vector. All these parameters are exactly defined in
[12]. Specifically, $\bar{{\mathbf{p}}}_{j}^{r}$ and
$\bar{{\mathbf{k}}}_{j}^{r}$ are built such that the graphical representation
of the transmission vectors follows two perpendicular circular shift
operations over a grid. The following example clarifies how the transmission
vectors are built for the virtual network.

######  Example 2.

Consider the network in Example 1, and assume the spatial DoF is $\alpha=4$.
Also, assume $\eta_{1}=\eta_{2}=2$ and $\eta_{3}=\eta_{4}=3$. If we choose
either $\hat{\eta}=2$ or $\hat{\eta}=3$, the virtual network will have $P=4$
users, coded caching gain ${\bar{t}}=1$ and spatial DoF $\bar{\alpha}=2$.
Then, the data delivery will require four rounds, with three transmissions at
each round. According to [12], user and packet index vectors for the first
round are

$\displaystyle\bar{{\mathbf{k}}}_{1}^{1}=[\bar{1},\bar{2},\bar{3}],\quad\bar{{\mathbf{k}}}_{2}^{1}=[\bar{1},\bar{3},\bar{4}],\quad\bar{{\mathbf{k}}}_{3}^{1}=[\bar{1},\bar{4},\bar{2}]\;,$
$\displaystyle\bar{{\mathbf{p}}}_{1}^{1}=[\bar{2},\bar{1},\bar{1}],\quad\bar{{\mathbf{p}}}_{2}^{1}=[\bar{3},\bar{1},\bar{1}],\quad\bar{{\mathbf{p}}}_{3}^{1}=[\bar{4},\bar{1},\bar{1}]\;,$

resulting in the transmission vectors

$\displaystyle\bar{{\mathbf{x}}}_{1}^{1}$
$\displaystyle={\bar{W}}_{\bar{2}}^{\bar{1}}(\bar{1})\bar{{\mathbf{w}}}_{\bar{3}}+{\bar{W}}_{\bar{1}}^{\bar{1}}(\bar{2})\bar{{\mathbf{w}}}_{\bar{3}}+{\bar{W}}_{\bar{1}}^{\bar{1}}(\bar{3})\bar{{\mathbf{w}}}_{\bar{2}}\;,$
$\displaystyle\bar{{\mathbf{x}}}_{2}^{1}$
$\displaystyle={\bar{W}}_{\bar{3}}^{\bar{1}}(\bar{1})\bar{{\mathbf{w}}}_{\bar{4}}+{\bar{W}}_{\bar{1}}^{\bar{2}}(\bar{3})\bar{{\mathbf{w}}}_{\bar{4}}+{\bar{W}}_{\bar{1}}^{\bar{1}}(\bar{4})\bar{{\mathbf{w}}}_{\bar{3}}\;,$
$\displaystyle\bar{{\mathbf{x}}}_{3}^{1}$
$\displaystyle={\bar{W}}_{\bar{4}}^{\bar{1}}(\bar{1})\bar{{\mathbf{w}}}_{\bar{2}}+{\bar{W}}_{\bar{1}}^{\bar{2}}(\bar{4})\bar{{\mathbf{w}}}_{\bar{2}}+{\bar{W}}_{\bar{1}}^{\bar{2}}(\bar{2})\bar{{\mathbf{w}}}_{\bar{4}}\;,$

where brackets for interference indicator sets are dropped for notational
simplicity. For clarification, we use a tabular view, borrowed from [12], to
graphically represent the transmission vectors in Figure 1. In this
representation, columns and rows denote user and packet indices, respectively.
A darkly shaded cell means the packet index is cached at the user, while a
lightly shaded cell indicates that (part of) the packet index is transmitted
to the user. As can be seen in Figure 1, the transmission vectors in a single
round are built using circular shift operations of the lightly shaded cells
over non-colored cells of the table, in two perpendicular directions.

${\bar{W}}_{1}$${\bar{W}}_{2}$${\bar{W}}_{3}$${\bar{W}}_{4}$$\underset{\rightarrow}{\textrm{{user}}}$$\bar{1}$$\bar{2}$$\bar{3}$$\bar{4}$
(a) $\bar{{\mathbf{x}}}_{1}^{1}$

${\bar{W}}_{1}$${\bar{W}}_{2}$${\bar{W}}_{3}$${\bar{W}}_{4}$$\underset{\rightarrow}{\textrm{{user}}}$$\bar{1}$$\bar{2}$$\bar{3}$$\bar{4}$
(b) $\bar{{\mathbf{x}}}_{2}^{1}$

${\bar{W}}_{1}$${\bar{W}}_{2}$${\bar{W}}_{3}$${\bar{W}}_{4}$$\underset{\rightarrow}{\textrm{{user}}}$$\bar{1}$$\bar{2}$$\bar{3}$$\bar{4}$
(c) $\bar{{\mathbf{x}}}_{3}^{1}$

Figure 1: Graphical representation of transmission vectors for the virtual
network

After the transmission vectors are built for the virtual network, we need to
_elevate_ them to be applicable to the original network. We first need to
define a few new parameters; $b$ is the remainder of the division of $\alpha$
by $\hat{\eta}$, i.e.,
$b=\alpha-\hat{\eta}\lfloor\frac{\alpha}{\hat{\eta}}\rfloor$, and the set
${\mathcal{D}}_{{\bar{k}}}$ consists of the set of user indices in the real
network that are associated with the profile $p={\bar{k}}$. Then, we have two
elevation mechanisms, depending on the value of $b$.

1) If $b=0$, i.e., if $\frac{\alpha}{\hat{\eta}}$ is an integer, every vector
$\bar{{\mathbf{x}}}_{j}^{r}$ is elevated into one transmission vector
${\mathbf{x}}_{j}^{r}$ for the original network. To do so, each term inside
the summation in (1) is replaced by

${\bar{W}}_{\bar{{\mathbf{p}}}_{j}^{r}[{\bar{n}}]}^{{\bar{q}}}(\bar{{\mathbf{k}}}_{j}^{r}[{\bar{n}}])\bar{{\mathbf{w}}}_{\bar{{\mathcal{R}}}_{j}^{r}({\bar{n}})}\rightarrow\sum_{m\in[\hat{\eta}]}W_{\bar{{\mathbf{p}}}_{j}^{r}[{\bar{n}}]}^{q}({\mathcal{D}}_{\bar{{\mathbf{k}}}_{j}^{r}[{\bar{n}}]}[m]){\mathbf{w}}_{{\mathcal{R}}_{j}^{r}({\bar{n}},m)}\;,$

where

${\mathcal{R}}_{j}^{r}({\bar{n}},m)=\bigcup_{{\bar{k}}\in\bar{{\mathcal{R}}}_{j}^{r}({\bar{n}})}\\!\\!{\mathcal{D}}_{{\bar{k}}}\;\;\cup\;\;{\mathcal{D}}_{\bar{{\mathbf{k}}}_{j}^{r}[{\bar{n}}]}\;\;\backslash\;\;\\{{\mathcal{D}}_{\bar{{\mathbf{k}}}_{j}^{r}[{\bar{n}}]}[m]\\}\;.$
(2)

In other words, every data term intended for some user ${\bar{k}}$ in the
virtual network is replaced by $\hat{\eta}$ data terms intended for all users
associated with profile $p={\bar{k}}$ in the real network, and the inter-
stream interference between these terms is suppressed by beamforming vectors.
Note that each interference indicator set
$\bar{{\mathcal{R}}}_{j}^{r}({\bar{n}})$ in the virtual network consists of
$\bar{\alpha}-1$ virtual users, and hence, using (2), every interference
indicator set in the real network will have
$(\bar{\alpha}-1)\hat{\eta}+(\hat{\eta}-1)=\bar{\alpha}\hat{\eta}-1=\alpha-1$
users. So, suppressing the interference with the beamformers is possible as
the spatial DoF for the original network is $\alpha$. Summarizing the
discussions, if $b=0$, the transmission vectors for the original network are
built as

${\mathbf{x}}_{j}^{r}=\sum_{{\bar{n}}\in[{\bar{t}}+\bar{\alpha}]}\sum_{m\in[\hat{\eta}]}W_{\bar{{\mathbf{p}}}_{j}^{r}[{\bar{n}}]}^{q}({\mathcal{D}}_{\bar{{\mathbf{k}}}_{j}^{r}[{\bar{n}}]}[m]){\mathbf{w}}_{{\mathcal{R}}_{j}^{r}({\bar{n}},m)}\;.$
(3)

2) If $b\neq 0$, i.e., when $\alpha$ is _not_ divisible by $\hat{\eta}$, every
transmission vector $\bar{{\mathbf{x}}}_{j}^{r}$ is elevated into $\hat{\eta}$
transmission vectors ${\mathbf{x}}_{j,s}^{r}$, $s\in[\hat{\eta}]$, for the
original network. For elevation, we first define the _base_ and _extended_
interference indicator sets as

${\mathcal{B}}_{j,s}^{r,b}({\bar{n}},m)=\bigcup_{{\bar{k}}\in\bar{{\mathcal{R}}}_{j}^{r}[{\bar{n}}]}\\!\\!{\mathcal{D}}_{{\bar{k}}}\;\;\bigcup_{i\in{\mathcal{E}}_{s}^{b}(\hat{\eta})}\\{{\mathcal{D}}_{\bar{{\mathbf{k}}}_{j}^{r}[{\bar{t}}+\bar{\alpha}]}[i]\\}$

and

${\mathcal{R}}_{j,s}^{r,b}({\bar{n}},m)\\!=\\!\begin{cases}&\\!\\!\\!\\!\\!\\!{\mathcal{B}}_{j,s}^{r,b}({\bar{n}},m)\cup{\mathcal{D}}_{\bar{{\mathbf{k}}}_{j}^{r}[{\bar{n}}]}\backslash\\{{\mathcal{D}}_{\bar{{\mathbf{k}}}_{j}^{r}[{\bar{n}}]}[m]\\}\quad\\!{\bar{n}}<{\bar{t}}+\bar{\alpha}\\\
&\\!\\!\\!\\!\\!\\!{\mathcal{B}}_{j,s}^{r,b}({\bar{n}},m)\backslash\\{{\mathcal{D}}_{\bar{{\mathbf{k}}}_{j}^{r}[{\bar{n}}]}[m]\\}\quad\quad\quad\quad\;\;{\bar{n}}={\bar{t}}+\bar{\alpha}\end{cases}$

respectively, where ${\mathcal{E}}_{s}^{b}(\hat{\eta})$ denotes the first $b$
elements of $[\hat{\eta}]$, after they are circularly shifted to the left by
$s$ units (recall that the order of members in a set is preserved). In other
words,

${\mathcal{E}}_{s}^{b}(\hat{\eta})=\big{(}([b]+s-2)\mod\hat{\eta}\big{)}+1\;.$
(4)

Then, the elevated transmission vectors are built as

${\mathbf{x}}_{j,s}^{r}=\sum_{{\bar{n}}\in[{\bar{t}}+\bar{\alpha}]}\sum_{m\in[\hat{\eta}]}W_{\bar{{\mathbf{p}}}_{j}^{r}[{\bar{n}}]}^{q}({\mathcal{D}}_{\bar{{\mathbf{k}}}_{j}^{r}[{\bar{n}}]}[m]){\mathbf{w}}_{{\mathcal{R}}_{j,s}^{r,b}({\bar{n}},m)}\;.$
(5)

As an explanation, from the ${\bar{t}}+\bar{\alpha}$ total data terms within
the transmission vector $\bar{{\mathbf{x}}}_{j}^{r}$, we substitute each of
the first ${\bar{t}}+\bar{\alpha}-1$ terms with $\hat{\eta}$ terms for the
real network (as we did for the case $b=0$). However, the last data term is
only replaced by $b$ terms, out of the $\hat{\eta}$ options, chosen through a
circular shift operation (implemented using
${\mathcal{E}}_{s}^{b}(\hat{\eta})$ in the equations). The following example
clarifies the elevation process for both cases of $b=0$ and $b\neq 0$, and
enlightens the effect of phantom users.

######  Example 3.

Consider the same network in Example 1, with the profile lengths mentioned in
Example 2. Let us assume ${\mathcal{D}}_{\bar{1}}=[1,2]$,
${\mathcal{D}}_{\bar{2}}=[3,4]$, ${\mathcal{D}}_{\bar{3}}=[5,6,7]$, and
${\mathcal{D}}_{\bar{4}}=[8,9,10]$. For this network, we find transmission
vectors resulting from the elevation of the vectors in (2), for both cases of
$\hat{\eta}=2,3$.

If $\hat{\eta}=2$, no phantom users are needed. However, we should exclude two
users, associated with profiles $p=3,4$, from the CC delivery phase. Assume
users 7 and 10 are excluded. As $b=0$, transmission vectors
$\bar{{\mathbf{x}}}_{1}^{1}$ and $\bar{{\mathbf{x}}}_{1}^{2}$ in (2) are
elevated as

$\displaystyle{\mathbf{x}}_{1}^{1}=$ $\displaystyle
W_{2}^{1}(1){\mathbf{w}}_{5,6,2}+W_{2}^{1}(2){\mathbf{w}}_{5,6,1}+W_{1}^{1}(3){\mathbf{w}}_{5,6,4}$
$\displaystyle+W_{1}^{1}(4){\mathbf{w}}_{5,6,3}+W_{1}^{1}(5){\mathbf{w}}_{3,4,6}+W_{1}^{1}(6){\mathbf{w}}_{3,4,5}\;,$
$\displaystyle{\mathbf{x}}_{2}^{1}=$ $\displaystyle
W_{3}^{1}(1){\mathbf{w}}_{8,9,2}+W_{3}^{1}(2){\mathbf{w}}_{8,9,1}+W_{1}^{2}(5){\mathbf{w}}_{8,9,6}$
$\displaystyle+W_{1}^{2}(6){\mathbf{w}}_{8,9,5}+W_{1}^{1}(8){\mathbf{w}}_{5,6,9}+W_{1}^{1}(9){\mathbf{w}}_{5,6,8}\;,$

and their graphical representation will be as shown in Figure 3. Comparing
Figures 1 and 3, we can see that the elevation mechanism stretches the
transmission vectors horizontally.

$W_{1}$$W_{2}$$W_{3}$$W_{4}$$\underset{\rightarrow}{\textrm{{user}}}$${1}$${2}$${3}$${4}$${5}$${6}$${8}$${9}$
${{\mathbf{x}}}_{1}^{1}$ \- CC Delivery

$W_{1}$$W_{2}$$W_{3}$$W_{4}$$\underset{\rightarrow}{\textrm{{user}}}$$7$$10$
Left for Unicast

$W_{1}$$W_{2}$$W_{3}$$W_{4}$$\underset{\rightarrow}{\textrm{{user}}}$${1}$${2}$${3}$${4}$${5}$${6}$${8}$${9}$
${{\mathbf{x}}}_{2}^{1}$ \- CC Delivery

$W_{1}$$W_{2}$$W_{3}$$W_{4}$$\underset{\rightarrow}{\textrm{{user}}}$$7$$10$
Left for Unicast

Figure 3: Elevated versions of $\bar{{\mathbf{x}}}_{1}^{1}$ (top) and
$\bar{{\mathbf{x}}}_{2}^{1}$ (bottom) - The case $b=0$

Now, let us assume $\hat{\eta}=3$. In this case, no user is excluded from the
CC delivery phase, but two phantom users $\check{1}$ and $\check{2}$ are
associated with profiles $p=1$ and $p=2$, respectively. As $b=1$, each
transmission vector for the virtual network is elevated into $\hat{\eta}=3$
vectors for the real network. Here we only mention the vectors resulting from
$\bar{{\mathbf{x}}}_{1}^{1}$, which are built as

$\displaystyle{\mathbf{x}}_{1,1}^{1}\\!=\\!W_{2}^{1}(1){\mathbf{w}}_{2,\check{1},5}\\!+\\!W_{2}^{1}(2){\mathbf{w}}_{1,\check{1},5}\\!+\\!W_{2}^{1}(\check{1}){\mathbf{w}}_{1,2,5}+$
(6)
$\displaystyle\;\;\;W_{1}^{1}(3){\mathbf{w}}_{4,\check{2},5}\\!+\\!W_{1}^{1}(4){\mathbf{w}}_{3,\check{2},5}\\!+\\!W_{1}^{1}(\check{2}){\mathbf{w}}_{3,4,5}\\!+\\!W_{1}^{1}(5){\mathbf{w}}_{3,4,\check{2}},$
$\displaystyle{\mathbf{x}}_{1,2}^{1}\\!=\\!W_{2}^{2}(1){\mathbf{w}}_{2,\check{1},6}\\!+\\!W_{2}^{2}(2){\mathbf{w}}_{1,\check{1},6}\\!+\\!W_{2}^{2}(\check{1}){\mathbf{w}}_{1,2,6}+$
$\displaystyle\;\;\;W_{1}^{2}(3){\mathbf{w}}_{4,\check{2},6}\\!+\\!W_{1}^{2}(4){\mathbf{w}}_{3,\check{2},6}\\!+\\!W_{1}^{2}(\check{2}){\mathbf{w}}_{3,4,6}\\!+\\!W_{1}^{1}(6){\mathbf{w}}_{3,4,\check{2}},$
$\displaystyle{\mathbf{x}}_{1,3}^{1}\\!=\\!W_{2}^{3}(1){\mathbf{w}}_{2,\check{1},7}\\!+\\!W_{2}^{3}(2){\mathbf{w}}_{1,\check{1},7}\\!+\\!W_{2}^{3}(\check{1}){\mathbf{w}}_{1,2,7}+$
$\displaystyle\;\;\;W_{1}^{3}(3){\mathbf{w}}_{4,\check{2},7}\\!+\\!W_{1}^{3}(4){\mathbf{w}}_{3,\check{2},7}\\!+\\!W_{1}^{3}(\check{2}){\mathbf{w}}_{3,4,7}\\!+\\!W_{1}^{1}(7){\mathbf{w}}_{3,4,\check{2}}.$

The graphical representation of this elevation mechanism is shown in Figure 4,
where the columns representing phantom users are hatched for better
clarification. Comparing with the base transmission vectors for the virtual
network in Figure 1, the elevation mechanism works through horizontal
stretching followed by an extra circular shift operation over a block of user
indices (users 5,6,7 in this example). Finally, we should note that the effect
of phantom users should be removed before the real transmission. To do so, we
simply remove phantom indices from the transmission vectors. So, for example,
instead of ${\mathbf{x}}_{1,1}^{1}$ in (6), we transmit

$\displaystyle{\mathbf{x}}_{1,1}^{1}=$ $\displaystyle
W_{2}^{1}(1){\mathbf{w}}_{2,5}+W_{2}^{1}(2){\mathbf{w}}_{1,5}+W_{1}^{1}(3){\mathbf{w}}_{4,5}$
$\displaystyle+W_{1}^{1}(4){\mathbf{w}}_{3,5}+W_{1}^{1}(5){\mathbf{w}}_{3,4}\;.$

$W_{1}$$W_{2}$$W_{3}$$W_{4}$$\underset{\rightarrow}{\textrm{{user}}}$${1}$${2}$$\check{1}$${3}$${4}$$\check{2}$${5}$${6}$${7}$${8}$${9}$${10}$

(a) ${{\mathbf{x}}}_{1,1}^{1}$ \- CC Delivery

$W_{1}$$W_{2}$$W_{3}$$W_{4}$$\underset{\rightarrow}{\textrm{{user}}}$${1}$${2}$$\check{1}$${3}$${4}$$\check{2}$${5}$${6}$${7}$${8}$${9}$${10}$

(b) ${{\mathbf{x}}}_{1,2}^{1}$ \- CC Delivery

$W_{1}$$W_{2}$$W_{3}$$W_{4}$$\underset{\rightarrow}{\textrm{{user}}}$${1}$${2}$$\check{1}$${3}$${4}$$\check{2}$${5}$${6}$${7}$${8}$${9}$${10}$

(c) ${{\mathbf{x}}}_{1,3}^{1}$ \- CC Delivery

Figure 4: Elevated versions of $\bar{{\mathbf{x}}}_{1}^{1}$ \- The case $b\neq
0$

As can be seen in Example 3, the choice of $\hat{\eta}$ can heavily affect
both performance and complexity metrics such as the required subpacketization
and number of transmissions. A smaller $\hat{\eta}$ reduces the chance of
requiring phantom users, but more real users are excluded from the CC delivery
phase, and hence, the achievable coded caching gain is reduced. On the other
hand, while with a larger $\hat{\eta}$ we can include more users in the CC
delivery phase, in specific transmissions, the achievable DoF can become too
small after the effect of phantom users is removed (in Example 3,
${\mathbf{x}}_{1}^{1}$ has a DoF of six, but for ${\mathbf{x}}_{1,s}^{1}$,
$s\in[3]$ the DoF is reduced to five). In fact, if the distribution of profile
lengths $\eta_{p}$ and the choice of $\hat{\eta}$ are such that we need many
phantom users, the achievable DoF in some transmissions may fall below the
spatial DoF $\alpha$ (i.e., we lose DoF by using coded caching techniques).
One solution to this issue is to check the achievable DoF at every
transmission vector after removing the effect of phantom users, and avoid the
transmission if the DoF falls below $\alpha$. The data terms intended to be
sent with such transmission vectors are then transmitted during the subsequent
unicast delivery phase.

##  IV Unicast Delivery Phase

During the unicast delivery phase, we fulfill the request of users excluded
from the CC delivery phase. Meantime, we also transmit any other data term not
sent during the CC delivery phase. This includes, for example, data terms in
transmission vectors with a DoF value smaller than $\alpha$ (due to the
presence of phantom users), or data terms re-transmitted because of channel
errors. We use a greedy mechanism (which is not necessarily optimal) to
achieve an appropriate spatial DoF for every transmission vector in this
phase. To do so, we first split the files requested by users excluded from the
CC delivery phase into the same number of subpackets we needed in the CC
delivery phase (this subpacketization value is discussed shortly after). Then,
for every user $k$, we define $u(k)$ to be the number of subpackets that
should be delivered to user $k$, and sort users by their $u(k)$ values in the
descending order. Now, we select $\max\\{\alpha,U\\}$ first users, where $U$
is the total number of users for which $u(k)>0$, and deliver one subpacket to
each selected user, with a single transmission vector. The inter-stream
interference in the transmission vector is suppressed by beamforming vectors
(which is possible as the spatial DoF is $\alpha$). Finally, we update $u(k)$
values and repeat the same procedure, until all the remaining data terms are
transmitted.

######  Example 4.

Consider the network in Example 3, and assume $\hat{\eta}=2$. In this case, we
don’t need any phantom users, but the users 7 and 10 are excluded from the CC
delivery phase (as also shown in Figure 3). As will be shown later, the total
subpacketization in this case is $P({\bar{t}}+\bar{\alpha})=12$. However, in
the placement phase, each file is already split into $P=4$ packets, and one
packet is cached at each user. So, during the delivery, we need to further
split each packet into $\frac{12}{4}=3$ subpackets and deliver nine subpackets
to each of these users. So, $u(k)=9$ for $k=7,10$ and zero otherwise, and
$U=2$. Then, we need nine transmission vectors, where each vector delivers one
subpacket to each of the users 7 and 10. For example, the first and second
vectors are built as
$W_{1}^{1}(7){\mathbf{w}}_{10}+W_{1}^{1}(10){\mathbf{w}}_{7}$ and
$W_{1}^{2}(7){\mathbf{w}}_{10}+W_{1}^{2}(10){\mathbf{w}}_{7}$, respectively.

##  V Performance Analysis

As stated earlier, the goal of this paper is _not_ to provide a theoretically
optimal scheme. Instead, we aim at proposing a practical coded caching
solution for dynamic networks where the users are allowed to join and leave
the network. In this regard, our scheme has a small subpacketization
requirement and enables a very simple optimized beamformer design using
uplink-downlink duality [12], and hence, can be applied to networks with a
large ($O(10^{2})$) number of users. Also, as we will verify through
simulations, it is able to provide a comparable performance with the non-
dynamic scenario.

The subpacketization requirement of our scheme depends on the value of $b$. If
$b=0$ (i.e., if $\alpha$ is divisible by $\hat{\eta}$), the total
subpacketization is $P({\bar{t}}+\bar{\alpha})$. This simply follows the fact
that in this case, every transmission vector for the virtual network is
elevated into exactly one vector for the original network, and no extra
splitting is required. However, if $b\neq 0$, the scheme needs a larger
subpacketization of $P(\hat{\eta}{\bar{t}}+\alpha)$. Analyzing the required
subpacketization in this case is lengthy due to the extra circular shift
operation in the scheme and removed here due to the lack of space. It can be
seen that in both cases, the required subpacketization is pretty small and
grows linearly with the total number of users in the CC delivery phase.

Simulation results for the proposed scheme are provided in Figures 5 and 6. We
have done simulations for a MISO network with 50 single-antenna users, where
the cache ratio at each user is $\gamma=0.1$ (i.e., $P=10$), and the spatial
DoF is set to $\alpha=10$. The number of transmitting antennas at the server
can be any number larger than $\alpha$; it is assumed to be 12 here. For the
distribution of $\eta_{p}$, we consider uniform distribution (i.e.,
$\eta_{p}=5$ for $p\in[10]$) as well as three other scenarios, where the non-
uniformity of $\eta_{p}$ values increase from scenario 1 to 3. The exact
values of $\eta_{p}$ for these scenarios are shown in Table I. In all
simulations, optimized beamformers are used and both CC and unicast delivery
phases are implemented. For a better comparison, in the figures we have also
included the rate curve for the case all the required data is delivered
through the unicast delivery phase (i.e., there is no coded caching gain).

$0$$5$$10$$15$$20$$25$$30$$20$$40$$60$SNR [dB]Symmetric Rate [nats/s]Uniform,
$\hat{\eta}=5$Scenario 1, $\hat{\eta}=3$Scenario 1, $\hat{\eta}=5$Scenario 1,
$\hat{\eta}=7$Scenario 3, $\hat{\eta}=5$Scenario 3, $\hat{\eta}=9$Scenario 3,
$\hat{\eta}=10$No Coded Caching

Figure 5: Performance comparison for various $\hat{\eta}$ values, scenarios 1 and 3 Scenario |  $\eta_{p}$ values | Standard deviation
---|---|---
1 | 5,4,5,5,4,3,6,6,5,7 | 1.15
2 | 9,3,1,4,5,7,2,6,5,8 | 2.58
3 | 8,3,8,0,4,10,7,4,0,6 | 3.40
TABLE I: Simulation scenarios with non-uniform profile lengths

$0$$5$$10$$15$$20$$25$$30$$20$$40$$60$SNR [dB]Symmetric Rate [nats/s]Uniform,
$\hat{\eta}=5$Scenario 1, $\hat{\eta}=7$Scenario 2, $\hat{\eta}=9$Scenario 3,
$\hat{\eta}=10$No Coded Caching

Figure 6: Performance comparison for $\hat{\eta}=\max\eta_{p}$, scenarios 1,
2, and 3

In Figure 5, the effect of the $\hat{\eta}$ parameter on the performance in
scenarios 1, 3 is shown. In general, choosing a larger $\hat{\eta}$ value (up
to $\max\eta_{p}$) improves the performance. This is because with increased
$\hat{\eta}$, a better coded caching gain, strong enough to cover for the DoF
loss of phantom users, is achieved. Of course, it should be noted that in
scenario 3, the case $\hat{\eta}=9$ provides slightly (about one percent)
better rate compared with $\hat{\eta}=10$. This is because as $\eta_{p}$
distribution becomes more non-uniform, by choosing $\hat{\eta}=\max\eta_{p}$
we need to add too many phantom users, increasing the chance of DoF loss.
Nevertheless, even for the very non-uniform case of scenario 3, this
performance loss is negligible, and one can safely consider
$\hat{\eta}=\max\eta_{p}$ in all cases. Of course, a more theoretical
analysis, due for the extended version of this paper, would better clarify the
probable performance loss of selecting $\hat{\eta}=\max\eta_{p}$.

Finally, in Figure 6, we compare the performance of the proposed scheme with
the case of uniform $\eta_{p}$ distribution (i.e., the maximum possible coded
caching gain), for scenarios 1, 2, and 3, when $\hat{\eta}=\max\eta_{p}$. As
can be seen, the proposed scheme performs very well, with loss of less than
ten percent for the considered network setup, compared with the uniform case.
This is because with $\hat{\eta}=\max\eta_{p}$, we try to maximize the
achievable global caching gain. Also, when DoF falls due to phantom users,
optimized beamformers enable a larger beamforming gain (as the size of the
interference indicator set becomes smaller than $\alpha-1$), thus compensating
for the minor performance loss. Interestingly, in the low-SNR regime and for
scenario 1, the proposed scheme even outperforms the uniform case by a small
margin. This is because the beamforming gain is more effective in this regime,
as discussed in [8].

##  VI Conclusion and Future Work

We introduced a new multi-antenna coded caching scheme for dynamic networks
where the users are allowed to join and leave the network freely. Instead of
dictating individual caching profiles for users, we associated them with a set
of predefined caching profiles. Data delivery is then performed in two
consecutive phases, with the goal of maximizing the achievable performance.
The proposed scheme requires a small subpacketization, enables easy
implementation of optimized beamformers, and its performance and robustness
under dynamic conditions is shown through simulations. The current paper
provides a proof-of-concept, with many theoretical analyses and optimizations
due for future work.

## References

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Weinberg Proceedings of the International Astronomical Union: LaTeX Guidelines
for authors

17 2021 10.1017/xxxxx

Proceedings IAU Symposium

# New Dimensions of Galactic Chemical Evolution

David H. Weinberg1 1Department of Astronomy and CCAPP, Ohio State University,
140 W. 18th Ave., Columbus, OH 43210, USA. email<EMAIL_ADDRESS>

###### Abstract

Dramatic recent progress in understanding galactic chemical evolution (GCE)
has been driven partly by direct observations of the distant past with HST and
JWST and partly by archeaological interpretation of stellar abundances from
giant high-resolution spectroscopic surveys (APOGEE, GALAH) and the
complementary power of Gaia astrometry and photometry. Focusing on
archaeology, I give a rapid-fire, and I hope synthesizing, review of work my
collaborators and I have done on theoretical modeling and observational
interpretation. I discuss (1) the interleaved but distinguishable roles of
stellar scale astrophysics and galactic scale astrophysics in governing GCE,
(2) the use of abundance ratio trends to empirically infer nucleosynthetic
yields, (3) the uncertainty in the overall scale of yields and its degeneracy
with the importance of galactic outflows, (4) the emergence of equilibrium in
GCE, (5) the dimensionality of the stellar distribution in chemical abundance
space, and (6) insights from chemical abundances on the early history of the
Milky Way, including measurements of the intrinsic scatter of abundance ratios
in metal-poor stars ($-2\leq[{\rm Fe}/{\rm H}]\leq-1$) suggesting that a
typical halo star at this metallicity is enriched by the products of $N\sim
50$ supernovae mixed over $\sim 10^{5}M_{\odot}$ of star-forming gas.

###### keywords:

Galaxy: abundances – Galaxy: evolution – galaxies: abundances – galaxies:
evolution

††editors: F. Tabatabaei, B. Barbuy & Y. Ting, eds.

##  1 Introduction

I am a relative latecomer to the field of galactic chemical evolution (GCE),
after working on large scale structure and other aspects of galaxy formation
through most of my career. This late arrival has advantages and disadvantages.
The principal advantage is fresh perspective informed by my experience in
other fields. The principal disadvantage is my sometimes limited knowledge of
what has come before, which in a voluminous field like GCE is a lot (e.g.,
Tinsley 1980, Prantzos & Aubert 1995, Pagel 1997, Kobayashi et al. 2006,
Matteucci 2012). Fortunately, I have had wonderful collaborators to help
mitigate my ignorance. Even more fortunately, dramatic advances in GCE data
sets have created radically new opportunities for modeling and interpretation.
For me personally, the APOGEE survey (Majewski et al. 2017) of SDSS-III
(Eisenstein et al. 2011), SDSS-IV (Blanton et al. 2017), and SDSS-V (Kollmeier
et al. 2017) has been most important, but similar advances are coming from
GALAH (Buder et al. 2021), Gaia (Gaia Collaboration 2023), SEGUE (Yanny et al.
2013, Rockosi et al. 2023), LAMOST (Luo et al. 2015), H3 (Conroy et al. 2019),
and numerous observing programs that target specific stellar populations in
the Milky Way and its neighbors.

It is a long time since I wrote a conference proceedings, but IAU Symposia
have always been the gold standard of these, and this is my first opportunity
to contribute to one! As in my talk, I will use this opportunity to run
lightly over a lot of ground, and to make editorial comments I might never get
past a stern journal referee.

##  2 The Structure of the GCE Problem

A GCE model relies on “stellar scale” astrophysics that determines the IMF-
averaged yields and delay time distribution (DTD) of different enrichment
channels, and on “galactic scale” astrophysics such as accretion and star
formation history, star formation laws, outflows, radial gas flows, and
redistribution of stars by radial migration, disk heating, and mergers. For
compactness, I will refer to these components as SSA and GSA, respectively.
Distinct GCE models may differ in their SSA assumptions, their GSA
assumptions, or both. Model predictions can be tested against a wide range of
observables in the Milky Way and other galaxies, including abundance ratio
trends (e.g., $[\alpha/{\rm Fe}]$ vs. $[{\rm Fe}/{\rm H}]$), distribution
functions (e.g., $p([{\rm Fe}/{\rm H}])$, $p([\alpha/{\rm Fe}])$), and the
dependence of these on a stellar population’s age, galactic position, and
kinematics.

Abbreviation | Meaning
---|---
AGB | asymptotic giant branch
CCSN | core collapse supernova(e)
DTD | delay time distribution
GCE | galactic chemical evolution
GSA | galactic scale astrophysics
IMF | initial mass function
ISM | interstellar medium
SFE | star formation efficiency
SFH | star formation history
SFR | star formation rate
SNIa | Type Ia supernova (e)
SSA | stellar scale astrophysics
Table 1: A partial list of abbreviations used

This structure is reminiscent (to this reminiscer, anyway) of galaxy formation
in the late ’80s and early ’90s. Models depended both on poorly known
cosmological inputs — such as $\Omega_{m}$, $\Omega_{b}$, the primordial power
spectrum, and the nature of dark matter — and on galactic scale modeling of
gas cooling, angular momentum, feedback, outflows, mergers, and so forth. They
could be tested against observations of galaxy clustering and galaxy
properties over a range of redshifts, and the interesting challenge was to
learn about both cosmology and galaxy scale physics despite the tradeoffs
between the two.

If nucleosythetic yields were a solved theoretical problem, then our GCE
models could take them as robust inputs and focus on constraining GSA — rather
like galaxy formation today, where the uncertainties in cosmological
parameters are mostly negligible compared to the uncertainties in
“gastrophysics.” However, the physics challenges of nucleosynthesis
calculations and the variance of predictions from study to study are large
enough that I usually take these predictions as “serving suggestions” rather
than recipes to be followed to the letter.

In GCE, one thing my collaborators and I have tried to do is to separate
constraints on SSA and GSA by drawing on complementary observables. Loosely
speaking (see, e.g., Andrews et al. 2017, Weinberg et al. 2017), models
predict that abundance ratio trends depend most strongly on yields (SSA),
while distribution functions depend on star formation history (SFH), stellar
radial migration (Schoenrich & Binney 2009, Hayden et al. 2015, Loebman et al.
2016, Johnson et al. 2021), and other GSA.

APOGEE abundances show that the median trends of [X/Mg] vs. [Mg/H] are nearly
universal throughout the Milky Way disk (Weinberg et al. 2019) and bulge
(Griffith et al. 2021a), provided one separately examines the high-$\alpha$
and low-$\alpha$ populations. This universality directly demonstrates the
dominant role of SSA in determining these trends, since GSA features like SFH
and gas flows are almost certainly different between the outer disk, inner
disk, and bulge. Mg is a better reference element than Fe for this purpose
because it has a single astrophysical source, massive stars exploding as core
collapse supernovae (CCSN), whereas Fe comes from both CCSN and Type Ia
supernovae (SNIa). The separation between high-$\alpha$ and low-$\alpha$
populations is really caused by the difference in Fe from SNIa, not by
differences in $\alpha$ elements themselves. For other elements, the
separation of the two [X/Mg] sequences depends on the relative contribution of
CCSN (or, more generally, prompt sources) vs. SNIa (or, more generally,
delayed sources) to element X. By formalizing this idea in a “2-process
model,” we have used observed trends in APOGEE and GALAH to derive empirical
constraints on the IMF-averaged yields of CCSN and SNIa for many elements
(Weinberg et al. 2019, 2022, Griffith et al. 2019, 2022). For elements in
which the delayed contribution comes from asymptotic giant branch (AGB) stars
rather than SNIa, this decomposition is only approximate. Isolating CCSN
yields from other contributions enables sharper tests of theoretical models of
CCSN and black hole formation (Griffith et al. 2021b).

In a related vein, Johnson et al. (2023) exploit the dependence of abundance
ratios on SSA to derive empirical constraints on the metallicity dependence of
nitrogen yields from the trend of [N/O] vs. [O/H] observed in the Milky Way
and external galaxies. Separating the contributions of massive stars and
intermediate mass stars to the IMF-averaged yield is trickier, but we show
that one can get useful empirical constraints on this decomposition from the
separation of [N/O] ratios between high-$\alpha$ and low-$\alpha$ stellar
populations in the Milky Way disk (Vincenzo et al. 2021; J. Roberts et al., in
prep.).

One key difference between present-day GCE and early ’90s galaxy formation is
that abundance “measurements” are themselves derived by fitting intricate
stellar atmosphere models to observed spectra, making them subject to complex
theoretical and observational systematics. Therefore, when deducing lessons
about chemical evolution from abundance data, one must also allow for the
possibility of observational systematics that are comparable in magnitude to
GCE model differences. One important example is the difference between [O/Fe]
or [O/Mg] trends found in optical vs. near-IR spectroscopic surveys (Griffith
et al. 2019, 2022), or even between analyses of the same near-IR spectra that
adopt different $T_{\rm eff}$ and log$g$ calibrations (Hayes et al. 2022).
IMF-averaged CCSN yield models predict that [O/Mg] is nearly independent of
metallicity (Andrews et al. 2017), as found in APOGEE. If the sloped [O/Mg]
trend found in optical surveys is correct, it has sharp implications for the
origin of O, Mg, or both.

##  3 Yields, Outflows, and Equilibrium

While abundance ratios provide a powerful tool for constraining yields, or
more generally SSA, with little sensitivity to galactic scale uncertainties,
there is a critical degeneracy between the overall scale of yields and the
mass-loading factor $\eta\equiv\dot{M}_{\rm out}/\dot{M}_{*}$ of galactic
outflows. This degeneracy can be easily understood in one-zone GCE models,
which assume a fully mixed gas reservoir so that the abundances of newly
forming stars depend on time but not on spatial position. These models have a
long history (e.g., Talbott & Arnett 1972, Larson 1972). Here I summarize some
of the lessons from the analytic solutions of Weinberg et al. (2017, hereafter
WAF).

For an element like O (or Mg) whose production is dominated by massive stars,
one can approximate enrichment as instantaneous with a net IMF-averaged yield
$y_{\rm O}^{\rm cc}$ that includes CCSN and massive star winds. The evolution
of oxygen mass in the gas reservoir follows

$\displaystyle\dot{M}_{\rm O}$ $\displaystyle=$ $\displaystyle y_{\rm O}^{\rm
cc}\dot{M}_{*}-Z_{\rm O}\dot{M}_{*}-\eta Z_{\rm O}\dot{M}_{*}+rZ_{\rm
O}\dot{M}_{*}$ (1) $\displaystyle=$ $\displaystyle y_{\rm O}^{\rm
cc}\dot{M}_{*}-(1+\eta-r)\dot{M}_{*}Z_{\rm O}~{}.$ (2)

Here $Z_{\rm O}=M_{\rm O}/M_{g}$ is the oxygen mass fraction of the ISM, and
the terms on the r.h.s. represent new O production, loss of O from the ISM to
star formation and outflow, and return of O to the ISM by recycling of stellar
envelopes. Although some recycling comes from lower mass stars with long
lifetimes, approximating it as instantaneous is usually accurate compared to
calculations that include full time dependence. For a Kroupa (2001) IMF, the
recycling factor $r\approx 0.4$, roughly half of it from stars with
$M>8M_{\odot}$.

As first argued by Larson (1972), if there is ongoing gas accretion that
sustains star formation, then the ISM abundance does not grow indefinitely but
instead reaches an equilibrium when the source and sink terms of Eq. 2
balance. For an exponential star formation history, $\dot{M}_{*}\propto
e^{-t/\tau_{\rm sfh}}$, and a time-independent star formation efficiency,
${\rm SFE}=\tau_{*}^{-1}=\dot{M}_{*}/M_{g}$, one can show that the equilibrium
abundance is

$Z_{\rm O,eq}={y_{\rm O}^{\rm cc}\over 1+\eta-r-\tau_{*}/\tau_{\rm sfh}}~{}.$
(3)

The full time-dependence is

$Z_{\rm O}(t)=Z_{\rm O,eq}\left(1-e^{-t/\bar{\tau}}\right)$ (4)

with

$\bar{\tau}={\tau_{*}\over 1+\eta-r-\tau_{*}/\tau_{\rm sfh}}~{}.$ (5)

In the molecular gas of present-day star forming galaxies a typical SFE
timescale is $\tau_{*}\approx 2\,{\rm Gyr}$ (Leroy et al. 2008, Sun et al.
2023), so the correction $\tau_{*}/\tau_{\rm sfh}$ is often small compared to
$1+\eta-r$.

Importantly, one can also find analytic solutions for Fe enrichment by SNIa if
one approximates the DTD as $R_{\rm Ia}(t)\propto e^{-(t-t_{d})/\tau_{\rm
Ia}}$ after a minimum delay time $t_{d}$ (equation 53 of WAF), which allows
one to calculate the evolution of $[\alpha/{\rm Fe}]$ and $[{\rm Fe}/{\rm H}]$
for realistic cases. The introduction of the new timescale $\tau_{\rm Ia}$
makes the solutions more complex than Eq. 4, but the behavior remains
intuitive. One can accurately approximate a $t^{-1.1}$ power-law DTD (Maoz &
Graur 2017) as a sum of two exponentials, and the presence of short-$\tau_{\rm
Ia}$ and long-$\tau_{\rm Ia}$ components leads to interesting behavior in some
scenarios. WAF also present analytic solutions for a linear exponential SFH,
$\dot{M}_{*}\propto te^{-t/\tau_{\rm sfh}}$, and for cases with sudden changes
of $\eta$ or $\tau_{*}$, and their equation (117) provides a general way to
model more complex star formation histories by combining the solutions for
exponential histories.

For understanding the yield-outflow degeneracy, equations 3-5 convey the key
points. The ISM abundance in equilibrium depends on $y_{\rm O}^{\rm cc}$ and
$\eta$ because the latter controls the rate at which metals are lost in
galactic winds. The timescale $\bar{\tau}$ for approaching equilibrium is
typically short compared to the age of the galaxy, though it can become long
if $\eta$ is small and star formation is inefficient.

Figure 1: (Left) IMF-averaged net yield of O (circles) or Mg (triangles)
computed from the CCSN models of Griffith et al. (2021b) for a Kroupa (2001)
IMF (solid) or a Kroupa, Tout, & Gilmore (1993) IMF (dotted), with four
scenarios for black hole formation: all stars born with $8M_{\odot}\leq M\leq
120M_{\odot}$ explode as CCSN; stars with $8M_{\odot}\leq M\leq 40M_{\odot}$
explode; or the explosion landscapes predicted by Sukhbold et al. (2016) for
two different neutrino-driven central engines. Yields are scaled to the
corresponding solar abundance. For the same black hole scenario the predicted
yields of the two IMFs differ by a factor of three. For a given IMF, the
difference between the scenarios with the most or least black hole formation
is also a factor of three. (Right) Evolution of the O abundance for the yields
marked by the blue and red circle in the left-hand panel, $y_{\rm O}^{\rm
cc}/Z_{{\rm O},\odot}=2.4$ and 0.85, respectively, with correspondingly chosen
outflow mass loading factors $\eta=1.8$ or $\eta=0$. Solid curves use an SFE
timescale $\tau_{*}=2\,{\rm Gyr}$ typical of molecular gas in local galaxies,
while dotted curves use a longer SFE timescale $\tau_{*}=6\,{\rm Gyr}$. All
curves use the analytic solution (Eq. 4) for an exponential SFH and assume
$\tau_{\rm sfh}\rightarrow\infty$ (constant SFR).

For a Kroupa IMF and the CCSN yields of Chieffi & Limongi (2004) and Limongi &
Chieffi (2006), the predicted yield is $y_{\rm O}^{\rm cc}\approx 0.015$
(i.e., $1.5M_{\odot}$ of oxygen produced per $100M_{\odot}$ of stars formed).
Achieving solar oxygen abundance $Z_{{\rm O},\odot}\approx 0.006$ (Asplund et
al. 2009) therefore requires strong outflows, with $\eta\approx 2$, a long-
standing result in theoretical models of the mass-metallicity relation (e.g.,
Finlator & Davé 2008, Peeples & Shankar 2011, Zahid et al. 2012) as well as my
group’s GCE models (e.g., Andrews et al. 2017, WAF, Johnson et al. 2021). I
was initially puzzled that the “Trieste group” (which by now spans many
nations) was able to produce successful GCE models with no outflows (e.g.,
Chiappini et al. 1997, Matteucci & Francois 1998, Minchev et al. 2013, Spitoni
et al. 2019), but I eventually learned it is because they adopt an IMF-
averaged O yield that is about $3\times$ lower than mine, with $y_{\rm O}^{\rm
cc}\approx Z_{{\rm O},\odot}$ (Minchev, private communication). In their case,
the principal reason for this much lower yield is the assumption of a steeper
IMF (Miller & Scalo 1979, Kroupa et al. 1993), with $\phi(M)\propto M^{-2.7}$
at high masses compared to $M^{-2.3}$ for a Kroupa (2001) IMF.

In addition to the IMF, a key source of uncertainty in CCSN yields is black
hole formation (Sukhbold et al. 2016; Griffith et al. 2021b). Production of O,
Mg, and other $\alpha$-elements grows rapidly with progenitor mass, but if the
most massive stars implode to black holes then they never release these
products to the ISM. The left panel of Fig. 1, adapted from Griffith et al.
(2021b), shows their IMF-averaged O and Mg yields for a Kroupa (2001) IMF or
the much steeper Kroupa et al. (1993) IMF, and for four different scenarios of
black hole formation: the complex “landscapes” predicted for two neutrino-
driven central engines (W18 and N20) by Sukhbold et al. (2016), a sharp
transition from explosion to implosion at $M=40M_{\odot}$, or no black hole
formation. For a given IMF, the yields predicted for these four scenarios span
a factor of three, and for a given black hole scenario the difference between
these two IMFs is also a factor of three.

The right panel of Fig. 1 shows $Z_{\rm O}(t)$ from Eq. 4 for two different
combinations of yield and $\eta$: $y_{\rm O}^{\rm cc}/Z_{{\rm O},\odot}=2.4$
and $\eta=1.8$ (blue) and $y_{\rm O}^{\rm cc}/Z_{{\rm O},\odot}=0.85$ and
$\eta=0$ (red). In each case we assume constant SFR ($\tau_{\rm
sfh}\rightarrow\infty$) and $\tau_{*}=2\,{\rm Gyr}$ (solid) or $6\,{\rm Gyr}$
(dotted). As expected, both the high-yield/high-$\eta$ and low-
yield/low-$\eta$ combinations can produce $Z_{\rm O}\approx Z_{{\rm O},\odot}$
at late times. These examples and Eq. 5 also explain why equilibrium is a
central concept in my group’s GCE models (and earlier work such as Finlator &
Davé 2008, Davé et al. 2012) but is not nearly so evident in the Trieste
models. If $\eta\approx 2$ then the timescale for reaching equilibrium is
fairly short, even with $\tau_{*}=6\,{\rm Gyr}$ (blue dotted line). If
$\eta=0$ then the timescale is longer, and $Z_{\rm O}$ may still be climbing
at $t=10-13\,{\rm Gyr}$. For the $\eta=0$, $\tau_{*}=6\,{\rm Gyr}$ model (red
dotted line), the abundance remains well below equilibrium even at $t=12\,{\rm
Gyr}$. The impact of $\eta$ and $\tau_{*}$ on time-evolution is amplified for
Fe because of the delayed SNIa contribution.

The scale of yields is therefore tightly connected to the importance of
outflows and the emergence of equilibrium. Unfortunately, black hole formation
and other theoretical uncertainties make it difficult to predict the absolute
scale of CCSN yields with confidence. Deuterium is one possible way forward
because its theoretical yield is perfectly known: stars destroy whatever D
they are born with, making $y_{D}=0$ (all deuterium is primordial).
Unfortunately, the large sightline-to-sightline variance of D/H (Linsky et al.
2006), plausibly interpreted as a sign of dust depletion onto PAH molecules,
makes determining the intrinsic ISM D/H ratio difficult. For the ISM D/H value
advocated by Linsky et al. (2006), the implied values of $\eta$ and $y_{\rm
O}^{\rm cc}$ are fairly high, similar to the blue models in Fig. 1 (Weinberg
2017). The mean Fe yield of CCSN (Rodriguez et al. 2022) offers another,
empirical route towards determining the overall yield scale (Weinberg, in
prep.). There are uncertainties in going from the measured quantity to
$\alpha$-element yields such as $y_{\rm O}^{\rm cc}$, but the Rodriguez et al.
measurement suggests a value intermediate between the two cases shown in Fig.
1, requiring mild outflows ($\eta\approx 1$) to reach solar abundance.

##  4 Dimensionality and Scatter

Despite the yield-outflow degeneracy, abundance ratio trends (e.g., [X/Mg] vs.
[Mg/H]) provide strong empirical constraints on yield ratios, and by
exploiting the separation of high-$\alpha$ and low-$\alpha$ populations one
can start to separate the contribution of distinct astrophysical processes
(e.g., CCSN vs. SNIa vs. AGB). The addition of stellar ages — from
asteroseismology, or parallax + photometry, or spectroscopic diagnostics
trained on these methods — helps further separate prompt and delayed
enrichment channels and provides leverage on SFE and SFH. Models with
different SFH but the same yields and SFE produce similar abundance ratio
tracks, but they produce different metallicity distribution functions because
the number of stars formed at a given metallicity is proportional to the SFR.
By fitting one-zone models to stellar abundances of dwarf spheroidals or
stellar streams one can get interesting SFH constraints from the metallicity
distribution alone (Sandford et al. 2022), and these become sharper and more
robust if the data include $[\alpha/{\rm Fe}]$ ratios and stellar ages
(Johnson et al. 2022).

With large, homogeneous data sets, one can measure the scatter around mean
abundance trends to gain additional diagnostics of yields, enrichment history,
mixing of stellar populations by mergers or migration, and mixing of metals
within the star-forming ISM. Measuring intrinsic scatter accurately is
challenging because one must carefully assess the observational contribution,
including scatter caused by differential abundance systematics across the
stellar sample.

Scatter in the high-dimensional space of stellar abundance ratios is a nuanced
concept, linked to the dimensionality of the stellar distribution in this
space. One of the earliest investigations of this dimensionality was led by
Yuan-Sen Ting, who showed that 6-9 principal components were required to
explain the abundance distribution of available high-resolution data sets
(Ting et al. 2012; for related approaches see Andrews et al. 2012, 2017 and
Patil et al. 2022). The dimensionality of the distribution is connected to the
number of distinct processes that contribute to the observed abundances, but
the connection is not a simple one. For example, in a one-zone model with a
fully mixed gas reservoir, all stellar abundances depend on a single parameter
(time) even if many enrichment channels are operating. To produce scatter one
needs multiple enrichment channels and either imperfect ISM mixing (Krumholz &
Ting 2018) or the mixing of stellar populations that experience different
relative contributions from these channels.

The success of the 2-process model in fitting individual abundance patterns of
Milky Way disk stars in APOGEE and GALAH shows that much of the star-to-star
variation is explained by two components, one describing overall enrichment
and a second describing the ratio of CCSN/SNIa enrichment (Weinberg et al.
2022; Griffith et al. 2022). Thus, to a surprisingly good approximation one
can predict all of a star’s APOGEE or GALAH abundances from its [Mg/H] and
[Mg/Fe] alone. In a similar fashion, one can predict these abundances to
surprising accuracy from [Fe/H] and age (Ness et al. 2019). It would be
interesting but ultimately disappointing if this 2-dimensional description
were perfect.

Fortunately, it isn’t. My collaborators and I have investigated this issue in
two complementary ways, by examining the conditional distribution
$p({\rm[X/Fe]}|{\rm[Fe/H],[Mg/Fe]})$ and its generalizations in APOGEE (Ting &
Weinberg 2022) and by examining the residuals $\Delta([{\rm X/H}])$ of stellar
abundances relative to a 2-process fit in APOGEE and GALAH (Weinberg et al.
2022, Griffith et al. 2022). Each of these studies covers 15-16 elemental
abundances. Key findings include:

  1. 1.

The observed total scatter (intrinsic + observational) around a 2-d fit for
high-S/N APOGEE stars ranges from $\approx 0.01-0.02$ dex (Mg, O, Si, Ca, Fe,
Ni) to $\approx 0.1$ dex (Na, K, V, Ce), with a median of 0.03 dex.

  2. 2.

While inferring intrinsic scatter from total scatter requires precise
knowledge of observational uncertainties, the correlation of abundance
residuals provides robust evidence for additional dimensions in abundance
space.

  3. 3.

One must condition on at least seven elements (e.g., Fe, Mg, O, Ni, Si, Ca,
Al) before the residual correlations of remaining APOGEE elements are
consistent with observational noise.

  4. 4.

Much of the $[\alpha/{\rm Fe}]$ scatter within the high-$\alpha$ and
low-$\alpha$ disk populations is caused by CCSN/SNIa variation within those
populations at fixed [Fe/H]. These variations in turn are correlated with age.

  5. 5.

APOGEE 2-process residuals show correlations among Ca, Na, Al, K, Cr, and Ce
and separately among Ni, V, Mn, and Co.

  6. 6.

GALAH 2-process residuals show correlations between Ba and Y, expected because
both have large AGB $s$-process contributions, and significant correlations of
these two with Zn.

  7. 7.

Ba and Y in GALAH and Ce in APOGEE show increasing positive residuals at young
stellar ages, implying a late-time enhancement of AGB enrichment relative to
SNIa enrichment. Intriguingly, Na shows a similar pattern in APOGEE even
though it is not expected to have a large AGB contribution.

  8. 8.

Residual abundances provide a powerful method for comparing stellar
populations, including dwarf satellites, streams, and star clusters, because
they control for star-by-star differences in overall metallicity and
$[\alpha/{\rm Fe}]$ that can otherwise mask subtler differences in individual
elements.

##  5 The Early History of the Milky Way

Much of the discussion above focuses on the regime of the Milky Way thin and
thick disks. The defining principle of IAU 377 was to connect studies of the
early Milky Way to those of high-redshift galaxies. I will conclude with three
recent GCE studies of the early Milky Way, each of which I mentioned in my
talk.

While searches for low metallicity stars have often focused on the nearby
Galactic halo, theoretical models predict that the oldest stars should be
concentrated near the center of the Galaxy (White & Springel 2000). Deriving
metallicities from Gaia BP/RP spectra with a machine learning algorithm
trained on APOGEE/Gaia overlap, Rix et al. (2022) show that stars with $[{\rm
M}/{\rm H}]<-1.5$ are indeed centrally concentrated, with a Gaussian radial
extent $\sigma_{R}$ of only 2.7 kpc. Over the range $-2\leq[{\rm M}/{\rm
H}]\leq-1$ the observed metallicity distribution follows $d\log n/d[{\rm
M}/{\rm H}]=1$. In this regime, far below equilibrium, one can ignore the loss
terms in Eq. 2 and derive a fairly general prediction

${d\log n\over d[{\rm M}/{\rm H}]}=\left[1-\left({M_{*}\over
M_{g}}\right)\left({\tau_{*}\dot{M}_{g}\over M_{g}}\right)\right]^{-1}$ (6)

The combination $(\tau_{*}\dot{M}_{g}/M_{g})$ corresponds to the fractional
change of the gas reservoir mass over one SFE timescale. The observed slope of
unity matches the value theoretically expected for a small stellar mass
fraction and/or a slowly changing gas reservoir.

APOGEE disk abundances show a plateau at $[{\rm Mg}/{\rm Fe}]\approx 0.35$ for
stars with $-1.2\leq[{\rm Fe}/{\rm H}]\leq-0.6$, which is conventionally
interpreted as representing the CCSN yield ratio. However, after using
kinematic data to remove accreted halo stars, Conroy et al. (2022) find a more
complex evolution in the H3 survey. For $-2.5\leq[{\rm Fe}/{\rm H}]\leq-1.5$
the typical $[\alpha/{\rm Fe}]$ declines from $\approx 0.6$ to $\approx 0.3$,
then rises slightly for $-1.5\leq[{\rm Fe}/{\rm H}]\leq-0.5$. With similar
kinematic cuts, the APOGEE data are consistent with this behavior. Conroy et
al. (2022) interpret this trend with a model in which the true CCSN plateau is
at $[\alpha/{\rm Fe}]\approx 0.6$, the initial decline in $[\alpha/{\rm Fe}]$
is caused by SNIa enrichment with a very low SFE ($\tau_{*}\approx 50\,{\rm
Gyr}$), and the rise in $[\alpha/{\rm Fe}]$ for $[{\rm Fe}/{\rm H}]>-1.5$ is
caused by rapidly accelerating SFE during which $\tau_{*}$ drops from
$50\,{\rm Gyr}$ to $2\,{\rm Gyr}$. This “simmer-to-boil” transition also
coincides with a rapid increase in the angular momentum of stellar populations
over the range $-1.5<[{\rm Fe}/{\rm H}]<-0.9$, also found in the analyses of
Belokurov & Kravtsov (2022) and Rix et al. (2023). While there is still much
to be decoded in these data, it appears that we are seeing both the kinematic
and chemical birth of the Milky Way disk in these archaeological studies.

Figure 2: Crosses show the RMS intrinsic scatter in [X/Fe] at fixed [Fe/H]
measured in a sample of 86 subgiant halo stars with $-2\leq[{\rm Fe}/{\rm
H}]\leq-1$. Circles show the RMS scatter predicted for random draws from the
high-mass IMF producing $N=10$ (blue), 50 (green), or 100 (yellow) CCSN, using
the mass-dependent CCSN yields of Sukhbold et al. (2016). For most elements,
the measured scatter corresponds to the predicted scatter for $N\approx 50$.
From Griffith et al. (2023).

The scatter of disk star abundances around 2-process predictions is small, but
we may reasonably expect scatter to be larger at low metallicities that probe
early phases of chemical evolution. Griffith et al. (2023) measure abundances
of 12 elements in 86 metal-poor subgiants ($-2\leq[{\rm Fe}/{\rm H}]\leq-1$),
with a survey strategy and analysis procedures designed specifically to allow
robust determination of the intrinsic scatter. We find RMS intrinsic scatter
in [X/Fe] at fixed [Fe/H] ranging from 0.04 dex (Cr) to 0.16 dex (Na), with a
median of 0.08 dex. One plausible interpretation of this scatter is from
stochastic sampling of the IMF, as any individual star is enriched not by the
IMF-averaged CCSN yield but by the particular set of supernovae that enriched
its birth environment.

The Griffith et al. measurements are well explained if the number of CCSN
contributing to the enrichment of a given star is $N\sim 50$ (Fig. 2). The
stochastic CCSN model roughly (though not perfectly) predicts the variation of
scatter from element to element, and it also roughly explains the correlations
of element fluctuations from star to star (fig. 15 of Griffith et al. 2023).
Producing the median metallicity of the sample with $N=50$ CCSN requires that
the products of these CCSN be diluted by about $10^{5}M_{\odot}$ of gas,
orders of magnitude smaller than the likely mass of the Galaxy at this epoch.
There are factor-of-three level uncertainties in this argument, and even the
basic interpretation of scatter as a stochastic IMF sampling effect may be
incorrect. Nonetheless, this analysis illustrates what we can hope to learn by
studying scatter around abundance trends as well as the trends themselves. If
our stochastic sampling interpretation is correct, then the early Galaxy was
divided into many ($10^{4}-10^{6}$) chemically disconnected regions, with each
newly formed star sampling only the nucleosynthetic products of its own
elemental horizon.

## Acknowledgements

I am grateful to the many collaborators from whom and with whom I have learned
the subject of GCE, especially (and in approximately chronological order)
Jennifer Johnson, Ralph Schoenrich, Brett Andrews, Jon Holtzman, Emily
Griffith, James Johnson, Jon Bird, Fiorenzo Vincenzo, Yuan-Sen Ting, Hans-
Walter Rix, Juna Kollmeier, Charlie Conroy, Nathan Sandford, Dan Weisz, and
many others in the SDSS collaboration. I thank the IAU organizers for the
opportunity to present some of this work to an amazing group of participants
from around the world. My GCE work has been supported by the U.S. National
Science Foundation, most recently by NSF AST-1909841. It has relied critically
on data from the Sloan Digital Sky Survey (www.sdss.org).

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