# Compressibility of Distributed
Document Representations

Blaž Škrlj  Jožef Stefan Institute, Ljubljana, Slovenia
and Jožef Stefan International Postgraduate School, Ljubljana, Slovenia
Email<EMAIL_ADDRESS>Matej Petković  Jožef Stefan Institute, Ljubljana,
Slovenia
Email<EMAIL_ADDRESS>
###### Abstract

Contemporary natural language processing (NLP) revolves around learning from
latent document representations, generated either implicitly by neural
language models or explicitly by methods such as doc2vec or similar. One of
the key properties of the obtained representations is their dimension. Whilst
the commonly adopted dimensions of 256 and 768 offer sufficient performance on
many tasks, it is many times unclear whether the default dimension is the most
suitable choice for the subsequent downstream learning tasks. Furthermore,
representation dimensions are seldom subject to hyperparameter tunning due to
computational constraints. The purpose of this paper is to demonstrate that a
surprisingly simple and efficient recursive compression procedure can be
sufficient to both significantly compress the initial representation, but also
potentially improve its performance when considering the task of text
classification. Having smaller and less noisy representations is the desired
property during deployment, as orders of magnitude smaller models can
significantly reduce the computational overload and with it the deployment
costs. We propose CoRe, a straightforward, compression-agnostic framework
suitable for representation compression. The CoRe’s performance is showcased
and studied on a collection of 17 real-life corpora from biomedical, news,
social media, and literary domains. We explored CoRe’s behavior when
considering contextual and non-contextual document representations, different
compression levels, and 9 different compression algorithms. Current results
based on more than 100,000 compression experiments indicate that recursive
Singular Value Decomposition offers a very good trade-off between the
compression efficiency and performance, making CoRe useful in many existing,
representation-dependent NLP pipelines.

##  I Introduction

Contemporary machine learning methods increasingly rely on the quality of
latent representations, produced during e.g., training of deep neural network
models or _dimensionality reduction techniques_. The common denominator to
many models used in practice throughout science and industry is a rather
arbitrary selection of the embedding dimension – it appears widely accepted
that a sufficiently high embedding dimension is preferred (e.g. 256 or 768).
However, in many practical scenarios such as the development of embedded
systems, mobile and online learning, model compactness is the desired property
[1, 2]. There has been research targeted at finding e.g., sufficient neural
network architectures for, e.g., mobile deployment [3], and, similarly, the
distillation of existing large neural language models [4]. Albeit succeeding
at their tasks, such research endeavors do not emphasize the actual properties
of the obtained representations, but rather the model itself. In recent years,
the learned representations have been actively studied, offering insights into
how existing representations can be compressed for many practical
applications, including image [5] and online text classification [6].

Finally, as the manual annotations are potentially expensive, the domain of
self-supervised learning explores to what extent can a neural network-based
system learn relevant representations without any supervision [7] The purpose
of this paper is to explore, in the domain of document representation
learning, to what extent can an existing representation be efficiently
compressed with as little performance loss as possible, in _self-supervised_
manner – without human annotations.

The contributions of this work are: i) We propose CoRe, an embedding-agnostic
methodology for automatic recursive compression of latent document
representations. CoRe can construct up to two orders of magnitude
($768\rightarrow 8$) smaller representations whilst maintaining the
performance within a few percentage points, offering drastic memory
consumption reduction for down-stream learning. ii) The proposed methodology
is evaluated on 17 real-life data sets, where we explore to what extent
_contextual_ and _non-contextual_ document representations can be _compressed_
with multiple linear and non-linear dimensionality reduction methods, offering
novel insights into the compressibility of different types of document
representations. iii) We demonstrate that a very efficient SVD-based recursive
compression can offer better-than-initial performance resulting from lower-
dimensional representations.

The remainder of this work is structured as follows. In Section II we discuss
the related work. In Section III, the proposed CoRe methodology is presented,
followed by the description of the considered compression algorithms (Section
IV), empirical evaluation setting (Section V) and the experimental results
(Section VI). We conclude with discussion and conclusions in Section VII. The
project’s repository is accesible here.

##  II Related Work

The proposed work builds on the ideas from the fields of representation
learning and model compression. The notion of representation learning has been
considered throughout different sub-domains of machine learning and has
received increasing interest in the last years. Neural network-based embedding
learning is becoming the prevailing method for obtaining representations
(embeddings) of graphs or their nodes [8], images [9] and documents [10, 11].
In recent years, two main types of document embeddings emerged, namely the
ones based on contextual neural language models [12, 13] and the non-
contextual ones based on earlier word representation learning techniques [14].
Commonly used _ad hoc_ dimensionalities of the representations of, e.g., 768,
256, and 128 yield satisfying results, however, are commonly not rigorously
inspected, possibly due to high computational costs of manual inspection of a
model’s behavior concerning this hyperparameter.

Exploration of how language models can be compressed has been an active effort
for more than a decade [15]. As a result, the development of methods that
automatically explore a given representation’s properties and attempt to
further reduce it is an ongoing research endeavor [6, 16]. With the advent of
neural language models, _distillation_ re-emerged as a form of model
compression [17]. Similarly, the pruning of existing models was also shown to
be a viable alternative [18, 19]. By selecting the representative subword
space, very memory-efficient models can be obtained [20]. Similarly, high
compressibility of neural language models can be obtained via low-rank
approximations [6, 21]. Recently, attempts to exploit external knowledge to
compress transformer-based models such as BERT [22] were also investigated
[23]. Finally, the idea of _self-supervised_ learning has recently been
explored in the context of language models, albeit originating in the image
domain [24]. For example, the Albert [25] exploits the inter-sentence
coherence to obtain better and more compact language models. Albeit many
compression-related ideas have been proposed and already offered model
improvements, to our knowledge no systematic evaluation of different
representation compression algorithms has yet been conducted. Furthermore,
another rationale for this paper is that it is not clear whether recursive
compression is a better option if compared to a direct projection to a lower
dimension, which we explored systematically in this work.

##  III Compression of Representations (CoRe)

Figure 1: Schematic overview of recursive autoencoding, as explored in this
work. The initial representation is recursively compressed
($d_{0},d_{1},d_{2}$) to a compact, however expressive representation. This
example shows two compression steps only (for readability purposes).

The proposed approach (schematically shown in Figure 1) investigates to what
extent a given latent representation can be compressed to maintain similar
functionality compared to the initially obtained representation. This work
builds on the idea of _lossy compression_ – the obtained, more compressed
representations can be of lower quality, as long as that quality is sufficient
for a given _practical application_. Albeit there exist many compression
benchmarks, normally the reconstruction error of the origin e.g., data set is
measured. However, as the purpose of _learned_ representations is to enable
association between input data and e.g., a collection of targets (e.g.,
genres), the quality of the compressed representations is estimated via
extrinsic evaluation with the logistic regression classifier.

###  III-A Recursive autoencoding

Next, we formally describe the incremental reduction of a given
representation’s dimension – the key idea of CoRe. Let
$\boldsymbol{E}\in\mathbb{R}^{|D|\times d_{0}}$ represent a
$d_{0}$-dimensional representation of a set of documents $D$ – a corpus. Let
$p(d)\in\mathbb{R}$ be the performance of the classifier learned from
$d$-dimensional embeddings of the documents. The purpose of this paper is to
explore how $p(d)$ changes with $d$ and to find $d_{i}$-dimensionional
representation, such that $p(d_{i})$ is at most $\epsilon$ worse (but can be
better) than the initial performance $p(d)_{0}$. We further define a function
$\textsc{EMB}:\mathbb{R}^{|D|\times d}\rightarrow\mathbb{R}^{|D|\times
d_{i}}$. For example, $\textsc{EMB}(\boldsymbol{E},512,256)$ projects the
space $\boldsymbol{E}$ from 512 to 256 dimensions. The key idea of this paper
explores the following recurrence relation:
$d_{i+1}=\max\\{d_{i}/\kappa,\kappa\\}$. The parameter $\kappa$ denotes the
dimension reduction factor; the larger the $\kappa$, the more the space is
reduced at each step of the optimization. After $i>0$ steps, CoRe next creates
the compressed representation
$\boldsymbol{E}_{i}=\textsc{EMB}(\boldsymbol{E}_{i-1},d_{i-1},d_{i})$, where
$\boldsymbol{E}_{0}=\boldsymbol{E}$. Here, on each call of the EMB function,
we learn how to reduce the dimension of the output of the previous call,
starting from the initial data frame $\boldsymbol{E}$ and proceeding towards
lower dimensions.

The key idea revolves around _recursive construction_ of incrementally smaller
representations. Notice that up to this point, we did not discuss the
properties of EMB – the proposed procedure is _agnostic_ to the embedding
algorithm.

###  III-B Time and Space Complexity

Let $d_{0}$ denote the initial representation’s dimension. Next, let
$k\sim\log_{\kappa}(d_{0})-1$ be the number of recursive steps. Then, the
computational complexity is
$\mathcal{O}\left(\gamma\cdot|D|\cdot\sum_{i=1}^{k}d_{i-1}\cdot d_{i}\right)$,
which can be simplified to
$\mathcal{O}\left(\gamma\cdot|D|\cdot\kappa^{2k+1}\right)$, provided that
$\kappa$ is lower-bounded by some constant, e.g., $\kappa\geq 2$. Here, the
$d_{i-1}$ and $d_{i}$ correspond to the dimensions of the input and hidden
layer in a single-layer architecture, $\gamma$ to the number of epochs and
$|D|$ to the number of documents. Potentially interesting is also the
complexity w.r.t. $\kappa$, i.e. the reduction term. By assuming $d_{0}$ is an
exponent of $\kappa$ (commonly the case, e.g., $2^{9}$), the complexity can
also be expressed as $\mathcal{O}(\kappa^{2k+1})$, indicating that recursive
reduction is not much more expensive than its first step, which already
requires $\kappa^{2k+1}$ steps. Indeed, the exact value of the above sum is
$\kappa^{3}(\kappa^{2k}-1)/(\kappa^{2}-1)$, thus the proportion of the
computations done in the first step of the iteration is approximately
$(\kappa^{2}-1)/\kappa^{2}$ which is always greater than $75\%$ if $\kappa\geq
2$. In terms of space complexity, the complexity is bounded as
$\mathcal{O}(|D|\cdot d_{0})$. The implementation, however, offers an option
to hold all intermediary compression steps (and produce them), which results
in additional space overhead which was in the considered experiments not
problematic.

##  IV Representation compression algorithms

In the following section we discuss the representation compression techniques
we considered in this work intending to systematically explore the space of
low-dimensional document representations. The considered methods are
summarised in Table I.

TABLE I: Overview of the considered compression algorithms. If no reference is
given, the algorithm was built for this work.

Compression approach | Description
---|---
UMAP [26] | Non-linear compression based on manifold theory
Sparse random projections [27] | Johnson Lindenstrauss lemma-informed projections
Truncated SVD [28] | Singular value decomposition
Cluster-aggregation (mean, median, max) | Clustering into $d$ dimensions followed by aggregation
Neural autoencoder - small/large | Neural autoencoders of various complexities
Random subspaces (inspired by [29, 30]) | Random subspace of dimension $d$

The widely adopted methods such as UMAP [26] were shown to perform
competitively to most learning-based approaches such as e.g., t-SNE [31],
hence this and similar approaches are not included in this work. As
contributions of this work, we implemented the following approaches, of which
performance we believe offers additional insights into the representations’
properties. The _Neural-small_ and _Neural-large_ are two differently sized
autoencoder architectures. A single layer example can be stated as

$\textsc{AE}(\textbf{D}_{r})=\textbf{W}_{\textrm{out}}^{T}\textrm{SoftSign}(\textrm{BN}(\textrm{Dropout}(\textbf{W}_{\textrm{emb}}^{T}\textbf{D}_{r}+b_{0}))).$
(1)

where $\boldsymbol{D}_{r}$ is a dense representation of $D$ documents. The
SoftSign activation is defined as $\text{SoftSign}(x)=\frac{x}{1+|x|}$, The BN
(BatchNorm) is defined as:
$\textrm{BN}(x)=\frac{\textbf{x}-\mathbb{E}[\textbf{x}]}{\sqrt{\mathrm{Var}[\textbf{x}]+\varepsilon}}.$
The $\varepsilon$ is a small constant required for numeric stability. The goal
of AE is to learn the association
$\textsc{AE}(\textbf{D}_{r})\approx\textbf{D}_{r}$,
$\textsc{AE}:\mathbb{R}^{|D|\times d}\rightarrow\mathbb{R}^{|D|\times d}$. To
obtain a low-dimensional representation, forward pass is considered only up to
the first hidden layer, i.e.,

$\textsc{EMB}(\textbf{D}_{r})=\textbf{W}_{\textrm{emb}}^{T}\textbf{D}_{r}+b_{0}\quad(\textsc{EMB}:\mathbb{R}^{|D|\times
d}\rightarrow\mathbb{R}^{|D|\times d_{i}}).$

Note how no activations are employed, ensuring non-activated representations.
The weight updates are considered as follows. The main hypothesis of this work
explores whether incremental reconstruction of latent spaces of lower
dimension indeed preserves the performance. As such, the autoencoder attempts
(at each step) to overfit the representation, and is hence optimized until the
loss is $\approx 0$. Note that being able to reconstruct a given input data
set with zero error can be related to _lossless compression_ ; thus, CoRe
effectively explores whether incremental steps of theoretically lossless
compression yield an useful, low-dimensional (lossy) representation.

Next, we implemented a variant of the random subspaces algorithm, which can be
summarised in the following two simple steps. First, randomly select $d_{i}$
dimensions from the initial representation. Create a subspace-based only on
these representations and perform $l_{2}$ normalization across samples. This
approach serves as a very simple/naïve baseline.

The third branch of approaches we implemented was aimed at exploring whether
the incremental aggregation of detected structures can serve as a simple-to-
use compression technique. Here, the two-step procedure operates as follows.
First, the KMeans++ [32] algorithm is used on the transposed initial matrix
$\textbf{D}_{0}^{T}$ to partition the dimensions into $d_{i}$ clusters. For
each detected cluster, an aggregation function is applied. We considered max,
mean, and median-based aggregations. Intuitively, this procedure should
maintain the key parts of the feature space that are of relevance to
maintaining the initial structure. The compression can thus be summarised as:

$\textbf{D}_{i}=\textrm{AGGREGATE-
CLUSTERS}(\textrm{KMeans++}(\textbf{D}_{0}^{T},d_{i})).$

This approach explores whether the macro structure of the embeddings offers
sufficient expressive power. Note that the recursive version of this algorithm
incrementally reduces the feature space (instead of using $\textbf{D}_{0}$ at
each step, it uses the prior representation $\textbf{D}_{i+1}$). Finally, we
included the sparse random projections algorithm as a baseline [27].

TABLE II: Data sets used: their domain and the numbers of documents $D$,
tokens $T$, maximal document length $M$, and unique labels $L$.

Data set | $D$ | $T$ | $M$ | $L$ | topic
---|---|---|---|---|---
bbc [33] | 1406 | 49418 | 343 | 4 | news topics
subjects | 1786 | 104972 | 227 | 4 | text topics
CNN-news | 2107 | 144077 | 605 | 7 | news topics
pan-2017 [34] | 3599 | 575670 | 1478 | 2 | gender detection
insults | 3946 | 29133 | 288 | 2 | insult detection
questions [35] | 5452 | 13278 | 37 | 6 | question categories
SMSSpam [36] | 5571 | 17744 | 88 | 2 | SMS messages spam detection
MedRelation | 8361 | 16796 | 99 | 2 | medical relations detection
AAAI2021 [37] | 8473 | 41387 | 149 | 2 | fake news detection
mbti | 8675 | 313038 | 419 | 16 | personality type detection (from text)
yelp | 10000 | 76482 | 592 | 5 | review classification
hatespeech [38] | 10557 | 29982 | 195 | 4 | white supremacists hate speech detection
semeval-2019 [39] | 13240 | 39714 | 155 | 2 | offensive speech prediction
articles | 19990 | 285144 | 10692 | 20 | news classification
sarcasm [40] | 28619 | 50241 | 166 | 2 | sarcasm detection
authors | 53678 | 29997 | 1008 | 45 | Victorian era author detection
drugs-condition [41] | 70406 | 96879 | 435 | 150 | drug side effects classification

##  V Empirical evaluation

The experiments were aimed to explore the compressibility of DistilBERT [42]
and doc2vec [14] representations concerning both the considered compression
level, as well as the compression (autoencoding) algorithm used.

For each compression algorithm (and each data set) we conducted stratified
three-fold cross-validation. All experiments were repeated three times, which
was possible due to the utilization of the SLING supercomputing
infrastructure. Similarly, we report the performance for each compression
step. Statistical significance is assessed via critical distance diagrams [43]
which compare average ranks achieved with a given method. We conducted the
compression experiments on 17 different real-life data sets, shown in Table
II.

Figure 2: Overall compression performance for doc2vec-based initial
representations and compression (autoencoding) algorithms. The x-axis denotes
the number of compression steps (dimension halvings as $\kappa=2$), and y-axis
the difference in (micro) F1 performance w.r.t. the initial representation.
The method names with “-dir” correspond to direct projection into a given
dimension (non-recursive compression).

###  V-A Evaluation of representation quality

We are interested in a given compressed representation’s relation to the
performance of the initial representation. Hence, we introduce $\epsilon F1$,
a measure which is computed as: $\epsilon
F1=F1_{\textrm{compressed}}-F1_{\textrm{initial}}$. If $\epsilon F1>0$, the
compressed representation is _better_ by $\epsilon F1$ when compared to the
original (initial) representation’s performance. If $\epsilon F1<0$, the
original representation is better. For multi-class problems, we considered the
micro $F1$ score. Note that the scores are reported in range $[0,1]$ and not
as percentages. The classifier used in this work is Logistic Regression with
regularization parameter C set to 1 [44]. The default dimension of
representations was set to 768, as commonly adopted in the literature. Each
computational job had at most 2 hours to finish on a 16GB (RAM) 8 core virtual
machine with no GPU.

##  VI Results

Due to space limitations, we comment on all representations, but only show the
graphs for non-contextual ones (doc2vec). In Figure 2, the overall impact of
compression on the performance can be observed. It can be observed that both
recursive, as well as non-recursive representations offer sufficient
performance when considering multiple compression steps (above the red line of
5% margin), however, recursive compression offers superior, in the case of
doc2vec also better-than-origin representation’s performance.

Figure 3: doc2vec results. The lines entail statistically (p = 0.05; Friedman-
Nemenyi) indistinguishable performance.

The results indicate that the first few compression steps can even be
beneficial for the final representation’s quality, however, once very low-
dimensional representations are considered, surprisingly, cluster-max-direct
yielded the highest-positioned curve, indicating good average performance.
This observation only holds for non-contextual representations. For the
contextual ones, the SVD-based approaches perform consistently well.

When comparing all considered approaches (their average performance across the
data sets), the first conclusion is that contextual representations (BERT)
outperform the non-contextual ones (doc2vec). The other main conclusions are
i) that recursive SVD’s performance is relatively consistent throughout
different levels of compression, ii) that random subspaces are not sufficient
for preserving the structure (consistent last places), iii) that the main
competitors to SVD are neural autoencoder-based approaches, which are more
computationally expensive, albeit being able to perform non-linear
decomposition, and iv) that recursive _linear_ representation performs well
across different compression levels ($\kappa$).

We next present the critical distance diagram [43] in Figure 3. The recursive
application of SVD consistently offers the best results, regardless of the
representation type. Interestingly, the clustering-based reductions were more
effective when considering contextual representations (cluster-mean-direct).
Overall, however, autoencoder-based representations similarly performed
consistently better than e.g., the UMAP-based ones. The results in tabular
form (for SVD-based compression) are given in Table III.

TABLE III: SVD-based compression results across the data sets and $\kappa$
levels. The results where the relative performance w.r.t. the origin
representation was $\geq 0$ are highlighted. Note how some of the data sets
are easier to compress than others (e.g., _articles_ vs. _hatespeech_).

| Dataset | AAAI2021 | CNN-news | MedRelation | SMSSpam | articles | authors | bbc | drugs | hatespeech | insults | mbti | pan-2017 | questions | sarcasm | semeval-2019 | subjects | yelp
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Compression-representation | Compr. steps |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
SVD (BERT) | 1 | -0.0 | -0.004 | -0.003 | 0.001 | -0.018 | -0.028 | 0.0 | -0.024 | 0.013 | 0.002 | 0.003 | 0.004 | -0.023 | -0.019 | 0.006 | 0.0 | 0.022
| 2 | -0.007 | -0.021 | -0.036 | 0.001 | -0.03 | -0.064 | 0.003 | -0.055 | 0.009 | 0.013 | 0.012 | -0.003 | -0.051 | -0.039 | 0.013 | 0.0 | 0.025
| 3 | -0.017 | -0.04 | -0.09 | -0.001 | -0.049 | -0.111 | 0.0 | -0.108 | 0.012 | 0.005 | 0.025 | 0.018 | -0.112 | -0.073 | 0.004 | 0.002 | 0.02
| 4 | -0.038 | -0.08 | -0.106 | -0.009 | -0.093 | -0.166 | -0.009 | -0.175 | 0.011 | -0.009 | 0.02 | 0.011 | -0.16 | -0.108 | -0.017 | -0.013 | 0.019
| 5 | -0.054 | -0.148 | -0.134 | -0.025 | -0.195 | -0.212 | -0.02 | -0.276 | 0.004 | -0.01 | 0.016 | -0.011 | -0.21 | -0.13 | -0.021 | -0.018 | 0.006
| 6 | -0.07 | -0.234 | -0.141 | -0.026 | -0.323 | -0.242 | -0.057 | -0.358 | -0.006 | -0.02 | 0.021 | 0.003 | -0.256 | -0.163 | -0.04 | -0.027 | -0.002
| 7 | -0.099 | -0.324 | -0.138 | -0.04 | -0.458 | -0.281 | -0.119 | -0.459 | -0.008 | -0.02 | 0.009 | -0.052 | -0.335 | -0.24 | -0.057 | -0.069 | -0.023
| 8 | -0.201 | -0.535 | -0.158 | -0.058 | -0.531 | -0.298 | -0.216 | -0.513 | -0.007 | -0.031 | 0.008 | -0.077 | -0.425 | -0.28 | -0.073 | -0.11 | -0.052
| 9 | -0.207 | -0.575 | -0.322 | -0.073 | -0.59 | -0.298 | -0.42 | -0.516 | -0.006 | -0.043 | 0.006 | -0.099 | -0.407 | -0.303 | -0.084 | -0.195 | -0.085
SVD (doc2vec) | 1 | 0.006 | 0.014 | 0.014 | 0.001 | -0.012 | -0.012 | 0.001 | -0.011 | 0.002 | 0.002 | 0.054 | 0.016 | -0.001 | -0.001 | 0.002 | -0.005 | -0.001
| 2 | 0.006 | 0.015 | 0.014 | 0.001 | -0.0 | -0.019 | 0.0 | -0.039 | 0.002 | 0.002 | 0.106 | 0.033 | -0.001 | -0.001 | 0.001 | -0.005 | -0.003
| 3 | 0.003 | 0.036 | 0.009 | 0.001 | -0.005 | -0.02 | -0.006 | -0.088 | 0.002 | 0.002 | 0.112 | 0.043 | -0.001 | -0.006 | -0.007 | -0.008 | 0.004
| 4 | -0.008 | 0.062 | -0.003 | 0.001 | -0.019 | -0.049 | -0.007 | -0.128 | 0.001 | 0.001 | 0.094 | 0.039 | -0.001 | -0.019 | -0.028 | -0.008 | 0.001
| 5 | -0.014 | 0.058 | -0.024 | 0.001 | -0.051 | -0.166 | -0.003 | -0.162 | 0.002 | -0.009 | 0.039 | 0.02 | -0.006 | -0.033 | -0.039 | -0.005 | -0.004
| 6 | -0.027 | 0.054 | -0.075 | -0.006 | -0.149 | -0.341 | -0.0 | -0.198 | -0.001 | -0.013 | -0.118 | 0.032 | -0.022 | -0.042 | -0.04 | -0.026 | -0.024
| 7 | -0.032 | 0.006 | -0.147 | -0.015 | -0.257 | -0.518 | -0.005 | -0.256 | 0.001 | -0.02 | -0.174 | -0.004 | -0.064 | -0.123 | -0.041 | -0.036 | -0.053
| 8 | -0.034 | -0.182 | -0.145 | -0.033 | -0.428 | -0.657 | -0.066 | -0.317 | 0.004 | -0.018 | -0.24 | -0.185 | -0.127 | -0.164 | -0.041 | -0.085 | -0.117
| 9 | -0.037 | -0.351 | -0.149 | -0.069 | -0.531 | -0.756 | -0.284 | -0.362 | 0.004 | -0.021 | -0.255 | -0.18 | -0.208 | -0.207 | -0.041 | -0.09 | -0.122
SVD-dir (BERT) | 1 | -0.0 | -0.004 | -0.003 | 0.001 | -0.018 | -0.028 | 0.0 | -0.023 | 0.013 | 0.002 | 0.003 | 0.004 | -0.022 | -0.019 | 0.006 | 0.0 | 0.022
| 2 | -0.008 | -0.019 | -0.049 | 0.001 | -0.027 | -0.064 | 0.0 | -0.056 | 0.009 | 0.007 | 0.013 | -0.002 | -0.042 | -0.04 | 0.007 | 0.0 | 0.022
| 3 | -0.019 | -0.042 | -0.081 | 0.0 | -0.048 | -0.11 | -0.003 | -0.106 | 0.009 | 0.004 | 0.031 | 0.018 | -0.108 | -0.074 | 0.002 | 0.002 | 0.016
| 4 | -0.039 | -0.078 | -0.105 | -0.009 | -0.095 | -0.167 | -0.014 | -0.175 | 0.01 | -0.007 | 0.019 | 0.011 | -0.16 | -0.109 | -0.016 | -0.013 | 0.018
| 5 | -0.056 | -0.146 | -0.134 | -0.027 | -0.192 | -0.212 | -0.02 | -0.276 | 0.004 | -0.009 | 0.017 | -0.011 | -0.213 | -0.131 | -0.021 | -0.018 | 0.003
| 6 | -0.065 | -0.233 | -0.141 | -0.026 | -0.323 | -0.242 | -0.058 | -0.357 | -0.005 | -0.02 | 0.02 | -0.001 | -0.255 | -0.162 | -0.039 | -0.027 | -0.004
| 7 | -0.099 | -0.323 | -0.138 | -0.04 | -0.458 | -0.281 | -0.119 | -0.459 | -0.008 | -0.02 | 0.009 | -0.054 | -0.333 | -0.24 | -0.057 | -0.069 | -0.024
| 8 | -0.201 | -0.535 | -0.158 | -0.058 | -0.531 | -0.299 | -0.216 | -0.513 | -0.007 | -0.031 | 0.008 | -0.078 | -0.424 | -0.279 | -0.073 | -0.11 | -0.053
| 9 | -0.207 | -0.575 | -0.322 | -0.073 | -0.59 | -0.298 | -0.42 | -0.516 | -0.006 | -0.043 | 0.006 | -0.099 | -0.407 | -0.303 | -0.084 | -0.195 | -0.086
SVD-dir (doc2vec) | 1 | 0.007 | -0.012 | -0.008 | 0.003 | -0.005 | -0.016 | 0.004 | -0.009 | 0.001 | 0.0 | 0.069 | 0.03 | 0.001 | -0.003 | -0.003 | 0.0 | 0.002
| 2 | 0.006 | -0.011 | -0.008 | 0.003 | 0.007 | -0.023 | 0.003 | -0.036 | 0.001 | 0.0 | 0.125 | 0.052 | 0.0 | -0.004 | -0.003 | 0.0 | 0.002
| 3 | 0.004 | 0.008 | -0.014 | 0.003 | 0.004 | -0.025 | -0.004 | -0.086 | 0.0 | 0.0 | 0.132 | 0.059 | 0.001 | -0.008 | -0.011 | -0.002 | 0.007
| 4 | -0.007 | 0.037 | -0.022 | 0.003 | -0.011 | -0.054 | -0.0 | -0.126 | -0.001 | -0.002 | 0.112 | 0.056 | 0.0 | -0.02 | -0.032 | -0.003 | 0.003
| 5 | -0.016 | 0.033 | -0.048 | 0.003 | -0.041 | -0.169 | 0.004 | -0.158 | 0.001 | -0.009 | 0.059 | 0.038 | -0.003 | -0.034 | -0.043 | 0.004 | -0.004
| 6 | -0.028 | 0.03 | -0.098 | -0.003 | -0.143 | -0.345 | -0.003 | -0.201 | -0.002 | -0.015 | -0.101 | 0.05 | -0.021 | -0.045 | -0.045 | -0.022 | -0.022
| 7 | -0.033 | -0.013 | -0.158 | -0.012 | -0.248 | -0.522 | -0.004 | -0.256 | 0.001 | -0.02 | -0.157 | 0.013 | -0.06 | -0.12 | -0.044 | -0.033 | -0.05
| 8 | -0.033 | -0.206 | -0.162 | -0.031 | -0.419 | -0.661 | -0.062 | -0.315 | 0.004 | -0.019 | -0.219 | -0.166 | -0.124 | -0.166 | -0.044 | -0.076 | -0.111
| 9 | -0.038 | -0.372 | -0.167 | -0.065 | -0.524 | -0.76 | -0.279 | -0.36 | 0.004 | -0.022 | -0.236 | -0.163 | -0.208 | -0.211 | -0.044 | -0.079 | -0.115

Tabular results are in alignment with Figure 3. We also observed both drastic
performance reductions relatively quickly (e.g., the _authors_ data set), but
also very gradual decline (_hatespeech_). Interestingly, the _drugs-condition_
data set, comprised of very long documents, was very hard to compress,
resulting always in $\epsilon F1<0$.

As an ablation study, we visualized the _bbc_ data set projected to 2D from
different higher dimensions (compression levels) in Figure 4.

Figure 4: Incremental reduction of dimension with recursive SVD and cluster
visualization in 2D (bbc). The 2D visualizations were for higher dimensions
obtained with default UMAP parameters.

The observation indicating that in some data sets very low dimensions preserve
the initial class structure if compressed in recursive manner with CoRe
indicate, that many real-life problems are potentially over-represented in
terms of the dimension commonly considered, and could be analysed via much
more compact (and efficient) representations.

##  VII Discussion and Conclusions

In Section VI, we demonstrated that both contextual and non-contextual
representations can be compressed by a factor of up to 100x (e.g., on the
_bbc_ data set), whilst maintaining the representation’s classification
performance. The observed compressibility varied mostly with respect to the
number of documents considered for learning. In this work, we also observed
the net positive $\epsilon F1$ (Table III). In terms of the compression
performance, we observed that linear recursive compression performed
surprisingly well (SVD). Overall, the non-contextual representations (doc2vec-
based) could be more easily compressed – in some cases (Figure 2), only tens
of dimensions were needed to retain a net positive $\epsilon F1$. On the
contrary, contextual representations are according to our experiments harder
to compress. Overall, however, tens of dimensions were identified by some of
the best-performing methods as enough to retain $5\%\;\epsilon F1$ margin,
which could already be of practical relevance. An example where lower-
dimensional representations could be practical and speed up the overall
learning process are AutoML systems.

As the purpose of this work was to explore whether a technique requiring
minimal or zero hyperparameter tunning can already improve the performance, we
believe that according to the current results, recursive SVD-based compression
is the most suitable one, even though UMAP and neural methods have the
potential to perform even better (at the cost of the additional hyperparameter
tuning).

Further work includes investigation of a larger collection of neural language
model-based representation to confirm the current results. Finally, we
identified as an open problem the _margin identification_ , i.e., how to
efficiently determine $\kappa$ without computing all intermediary
compressions. Identifying $\kappa$ upfront could offer automation of the
compression, and, when combined with recursive SVD which required zero
hyperparameter tunning, provide a powerful _post-hoc_ method for optimizing
representation-based learners.

## Acknowledgment

We thank Tomáš Mikolov for his invaluable comments. The work was supported by
the Slovenian Research Agency (young researcher grant (BŠ), grant P2-0103),
and by the European Commision (grants 825153, N2-0078, 952215, 825619).

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# Star Formation by Supernova Implosion

Leonard E. C. Romano Universitäts-Sternwarte, Fakultät für Physik, Ludwig-
Maximilians-Universität München, Scheinerstr. 1, D-81679 München, Germany
Max-Planck-Institut für extraterrestrische Physik, Giessenbachstr. 1, D-85741
Garching, Germany  Excellence Cluster ORIGINS, Boltzmannstr. 2, D-85748
Garching, Germany  Andreas Burkert  Universitäts-Sternwarte, Fakultät für
Physik, Ludwig-Maximilians-Universität München, Scheinerstr. 1, D-81679
München, Germany  Max-Planck-Institut für extraterrestrische Physik,
Giessenbachstr. 1, D-85741 Garching, Germany  Excellence Cluster ORIGINS,
Boltzmannstr. 2, D-85748 Garching, Germany  Manuel Behrendt Universitäts-
Sternwarte, Fakultät für Physik, Ludwig-Maximilians-Universität München,
Scheinerstr. 1, D-81679 München, Germany  Max-Planck-Institut für
extraterrestrische Physik, Giessenbachstr. 1, D-85741 Garching, Germany

###### Abstract

Recent hydrodynamical simulations of the late stages of supernova remnant
(SNR) evolution have revealed that as they merge with the ambient medium, SNRs
implode, leading to the formation of dense clouds in their center. While being
highly chemically enriched by their host SNR, these clouds appear to have
similar properties as giant molecular clouds, which are believed to be the
main site of star formation. Here, we develop a simple model, in order to
estimate the efficiency of the star formation that might be triggered by the
implosion of SNRs. We separately consider two cases, cyclic star formation,
maintained by the episodic driving of feedback from new generations of stars;
and a single burst of star formation, triggered by a single explosion. We find
that in the cyclic case, star formation is inefficient, with implosion-
triggered star-formation contributing a few percent of the observed star-
formation efficiency per free-fall timescale. In the single-burst case, higher
star-formation efficiencies can be obtained. However, while the implosion-
triggered process might not contribute much to the overall star-formation, due
to the high chemical enrichment of the birth clouds, it can explain the
formation of a significant fraction of metal-rich stars.

Dense interstellar clouds (371) – Star formation(1569) – Interstellar medium
(847) – Supernova remnants (1667)

††software: Julia v1.6.5 (Bezanson et al., 2014), Matplotlib v3.5.1 (Caswell
et al., 2021)

##  1 Introduction

Star formation is easy. Giant Molecular Clouds (GMCs) with masses $\sim
10^{4}\,-\,10^{7}\,\text{M}_{\odot}$ (Williams & McKee, 1997; Miville-
Deschênes et al., 2017) fragment, producing gravitationally unstable cores
that collapse (e.g. Jeans, 1902; McKee & Ostriker, 2007; Rosen et al., 2020)
and form stars. Yet, a closer look reveals that our understanding of the
processes linking these stages of star formation and even the formation of the
GMCs themselves is far from being predictive (see e.g. Lada et al., 2013;
Chevance et al., 2020). Indeed, the processes underlying star formation remain
to be one of the biggest puzzles in the current framework of galaxy formation
and evolution.

One of these processes, which could explain the formation of star-forming GMCs
is so-called triggered star-formation(Elmegreen, 1998, 2011). The underlying
idea being that shocks, e.g. due to Supernovae (SNe) or other feedback
processes, could compress the ambient gas and drive it towards gravitational
instability (GI).

Observational evidence suggests that stars form spatially and temporally
correlated (Gaczkowski et al., 2015, 2017; Krause et al., 2018; Zucker et al.,
2022; Ratzenböck et al., 2023; Verma et al., 2023), pointing towards triggered
star-formation scenarios. Meanwhile, many of the proposed triggers have been
confirmed in numerical simulations (Dobbs et al., 2022; Herrington et al.,
2023; Horie et al., 2024).

Recently, Romano et al. (2024) have found that besides the classical
triggering by shock-compression, SNe will eventually implode due to the
pressure of the interstellar medium (ISM) and form a dense, highly chemically
enriched cloud in their center, which might collapse and form stars. In this
letter, we leverage their results to give an estimate of the amount of star
formation that can be expected from this process.

##  2 Methods

Figure 1: A schematic overview of SN-implosion-triggered star-formation. In
this process, an explosion from a central source drives a shock (I), which
eventually implodes (II) and leads to the formation of a central cloud (III).
The central cloud grows and eventually becomes gravitationally unstable,
leading to star formation (IV). Depending on whether there are newly-formed,
massive stars, this cycle can repeat. The arrows connecting the different
stages are decorated with the corresponding transition timescales defined in
equations 5, 6, 9 and 11. The cloud formation time $t_{\text{cf}}$ is measured
from the time of the explosion so the time between implosion and cloud
formation is $t_{\text{cf}}-t_{\text{launch}}$.

A common way to quantify the star-formation efficiency is the so-called star-
formation efficiency per free-fall time $\epsilon_{\text{ff}}$. It is defined
as the ratio of the star-formation timescale and the free-fall timescale (see
e.g. Schinnerer & Leroy, 2024, for a review):

$\epsilon_{\text{ff}}=\frac{t_{\text{ff}}}{t_{\text{sf}}}\,.$ (1)

The free-fall timescale

$t_{\text{ff}}=\sqrt{\frac{3\pi}{32G\rho}}\sim
44.9\,n_{0}^{-0.5}\,\text{Myr}\,,$ (2)

where $n_{0}$ is in units of cm-3. $t_{\text{ff}}$ depends only on the density
$\rho=\mu\,m_{\text{H}}\,n_{0}$ of the ambient medium, where $\mu=1.4$ is the
mean atomic weight and $m_{\text{H}}$ is the mass of a hydrogen atom. On the
other hand, the star-formation timescale

$t_{\text{sf}}=\left(\frac{1}{\rho}\left.\frac{\text{d}\rho}{\text{d}t}\right|_{\text{sf}}\right)^{-1}\,,$
(3)

depends on the details of the processes that convert gas into stars.

Here we derive an expression for the timescale of SN-implosion-triggered star-
formation. We consider two cases. In both cases a central stellar population
drives a shock which will implode due to the pressure of the ISM, leading to
the formation of a central cloud (Romano et al., 2024). Once the cloud becomes
gravitationally unstable it collapses and forms new stars. In the first,
cyclic case, we consider the long term average of stellar populations that can
produce enough massive stars to keep maintaining a continuous cycle. In the
second, single-burst case we consider a stellar population or a single star
driving a single iteration of SN-implosion-triggered star-formation, which may
then either continue on indefinitely as in the cyclic case, or cease due to
the lack of newly formed massive stars. The main difference between the two
scenarios is that in the former, the number of SN explosions per cycle is
fixed by the expected stellar mass that is formed by each cycle, while in the
latter it is kept as a free parameter. A schematic overview of such a cycle is
shown in Figure 1.

We utilize the results of recent simulations by Romano et al. (2024). Their
results include timescales for various stages of SN Remnant (SNR) evolution
and cloud formation as well as information about various cloud properties as a
function of time after its formation (see e.g. their figures 7 and 8).

Besides these results, we need to make the following simplifying assumptions:

  1. 1.

Stellar populations follow the canonical Initial Mass Function (IMF)
$\xi_{\text{star}}\left(m\right)$ from Kroupa & Jerabkova (2021).

  2. 2.

Implosion-triggered star-formation can only be sustained if it leads to the
formation of at least one massive ($m_{\star}>8\,\text{M}_{\odot}$) star,
which can trigger another SN implosion.

  3. 3.

The ambient medium is initially at rest, with a uniform density of
$n_{\text{H}}=n_{0}$ and returns to this state by the end of each cycle.

  4. 4.

We neglect the role of early feedback (i.e. stellar winds and ionizing stellar
radiation) and assume that all stars in the stellar population explode
simultaneously in a common location, injecting a total energy of
$E_{\text{SN}}=10^{51}\,\text{erg}$ per SN into the ambient medium.

  5. 5.

We neglect all other sources of star formation and stellar feedback.

The star-formation-rate per unit volume for this process can be written as

$\left.\frac{\text{d}\rho}{\text{d}t}\right|_{\text{sf}}=\frac{M_{\star\text{,formed}}}{t_{\text{cycle}}V_{\text{exp}}}\,,$
(4)

where the stellar mass formed per cycle
$M_{\star\text{,formed}}=\epsilon_{\star}\,M_{\text{cloud}}$ is parameterized
as a fraction $\epsilon_{\star}$ of the mass of the central cloud prior to
collapse, $V_{\text{exp}}=4\pi/3\,R_{\text{exp}}^{3}$ is the volume traced by
the shock during its expansion, and

$t_{\text{cycle}}=t_{\text{delay}}+t_{\text{cf}}+t_{\text{GI}}+t_{\text{ff,cloud}}\,,$
(5)

is the time it takes to complete one cycle.

The fraction of cloud mass that is converted into stars $\epsilon_{\star}$ has
been constrained both from observations (Lee et al., 2016; Chevance et al.,
2020) and simulations (Grudić et al., 2022; Farias et al., 2024). It shows
large variation with a typical value of $\epsilon_{\star}\sim 10\,\%$. Due to
its large uncertainties, we treat this number as a model parameter, with a
fiducial value of $\epsilon_{\star}^{\text{fid}}=10\,\%$.

There is some delay before the massive stars in a newly formed population
explode. This time delay has been computed by Saitoh (2017) under the single
stellar population approximation and is typically in the range of
$t_{\text{delay}}\sim 2.2\,-\,4.6\,\text{Myr}$. We treat this timescale as a
model parameter with a fiducial value of $t_{\text{delay}}^{\text{fid}}\sim
3\,\text{Myr}$. However, we note that for stellar populations with exclusively
long-lived massive stars, the time delay can be over an order of magnitude
longer.

For $n_{0}\gtrsim 1$ the cloud-formation timescale (Romano et al., 2024)

$t_{\text{cf}}\sim 4\,n_{0}^{-0.15}\,N_{\text{SN}}^{0.3}\,\text{Myr}\,,$ (6)

where $N_{\text{SN}}$ is the number of SNe exploding simultaneously.

After its formation the central cloud evolves and its mass grows roughly like

$M_{\text{cloud}}\sim
20\,\left(\frac{t}{1\,\text{Myr}}\right)^{2.6}\,\text{M}_{\odot}\,,$ (7)

and its virial parameter drops

$\alpha_{\text{vir}}\sim
6000\,n_{0}^{-1}\,\left(\frac{t}{1\,\text{Myr}}\right)^{-1.75}\,\text{M}_{\odot}\,.$
(8)

The cloud collapses when it becomes gravitationally unstable, i.e.
$\alpha_{\text{vir}}\left(t_{\text{GI}}\right)\sim 1$, which leads to

$\displaystyle t_{\text{GI}}\sim 144\,n_{0}^{-0.57}\,\text{Myr}~{},$ (9)
$\displaystyle M_{\text{cloud}}\left(t_{\text{GI}}\right)\sim 8\times
10^{6}\,n_{0}^{-1.49}\,\text{M}_{\odot}\,~{}.$ (10)

After the onset of collapse, the cloud is expected to collapse on a free-fall
timescale, which depends on the density of the cloud at $t_{\text{GI}}$.

We find that the overdensity of the cloud relative to the ambient medium does
not follow a simple pattern, but its overall magnitude oscillates in time,
with typical values in the range of $\chi\sim 2-4$ and no clear correlation
with the simulation parameters. We thus treat the overdensity as a model
parameter with a fiducial value of $\chi^{\text{fid}}=3$, which is justified
by our results in section 3 that show that the value of $\chi$ only has a
marginal effect on the star-formation efficiency. The collapse timescale of
the cloud is thus

$t_{\text{ff,cloud}}=\chi^{-0.5}\,t_{\text{ff}}\sim
44.9\,\chi^{-0.5}\,n_{0}^{-0.5}\,\text{Myr}\,.$ (11)

In the meantime the blastwave expansion has affected a spherical region of
radius $R_{\text{exp}}$. Assuming that the blastwave expands as a momentum-
conserving snowplow, i.e. $R\propto t^{1/4}$, we estimate the size of the
affected region as

$R_{\text{exp}}\sim
R_{\text{launch}}\,\left(\frac{t_{\text{cf}}+t_{\text{GI}}+t_{\text{ff,cloud}}}{t_{\text{launch}}}\right)^{1/4}\,,$
(12)

where $R_{\text{launch}}\sim
51.3\,n_{0}^{-0.27}\,N_{\text{SN}}^{0.3}\,\text{pc}$ and
$t_{\text{launch}}\sim 0.98\,n_{0}^{-0.11}\,N_{\text{SN}}^{0.27}\,\text{Myr}$
(Romano et al., 2024).

The scaling of the cloud mass eq. 10 implies that above a certain density
$n_{\text{max}}$, the cloud mass becomes so low that it becomes unlikely to
form massive stars, which could trigger the next cycle of star formation.

By normalizing the IMF to the stellar mass formed from the collapsing cloud,
we find that the number of massive stars is

$N_{\text{SN}}=\int_{8\,\text{M}_{\odot}}^{\infty}\xi_{\text{star}}\left(m\right)\text{d}m\sim
1.08\,\frac{\epsilon_{\star}M_{\text{cloud}}}{100\,\text{M}_{\odot}}\,,$ (13)

which in combination with eq. 10 implies that the maximum density at which one
can expect the formation of at least one massive star is

$n_{\text{max}}\sim 1070\,\epsilon_{\star}^{0.67}\,\text{cm}^{-3}\,.$ (14)

##  3 Results

Figure 2: Star-formation efficiency per free-fall time as a function of
density for various models in the cyclic scenario. In panels (a), (b) and (c)
the star-formation efficiency $\epsilon_{\star}$, cloud overdensity $\chi$ and
stellar-population lifetime $t_{\text{delay}}$ are respectively varied, while
the remaining two parameters are kept at their respective fiducial values of
$\left(\epsilon_{\star}^{\text{fid}},\chi^{\text{fid}},t_{\text{delay}}^{\text{fid}}\right)=(0.1,3,3\,\text{Myr})$.
The blue line and shaded area correspond to the value of
$\epsilon_{\text{ff}}$ and its uncertainty derived from observations on scales
of $\sim 100\,\text{pc}$ (Schinnerer & Leroy, 2024).

In the previous section we have described a model to estimate the star-
formation efficiency per free-fall time $\epsilon_{\text{ff}}$ for SN-
implosion-triggered star-formation. Here we explore how our model compares to
observational estimates of $\epsilon_{\text{ff}}$ and its dependency on the
model parameters.

###  3.1 Cyclic Star-Formation

In the cyclic scenario, the number of SNe per cycle is determined by eqs. 13
and 10. In the limit of $t_{\text{GI}}/t_{\text{cycle}}\sim 1$, we thus find

$\epsilon_{\text{ff}}\propto\epsilon_{\star}^{0.3}\,n_{0}^{-0.22}\,.$ (15)

In Figure 2 we show how $\epsilon_{\text{ff}}$ varies with respect to the
model parameters $\epsilon_{\star}$, $\chi$ and $t_{\text{delay}}$ and compare
it to the most recent observational estimate of $\epsilon_{\text{ff}}$ as
reported by Schinnerer & Leroy (2024).

Panel (a) shows the variation of $\epsilon_{\text{ff}}$ with respect to
$\epsilon_{\star}$. The range of densities where cyclic star-formation can be
triggered becomes increasingly narrow as $\epsilon_{\star}$ is reduced, as
expected from eq. 14. Furthermore, it becomes clear that for fixed density,
$\epsilon_{\text{ff}}$ scales as expected from eq. 15, since $t_{\text{cf}}\ll
t_{\text{GI}}+t_{\text{ff,cloud}}$ for $n_{0}\geq 1$.

Panel (b) shows the variation of $\epsilon_{\text{ff}}$ with respect to
$\chi$. Higher $\chi$ generally implies higher $\epsilon_{\text{ff}}$, with a
diminishing effect as $\chi\gtrsim 4$. The effect diminishes because
$t_{\text{GI}}\sim 3\,t_{\text{ff}}$ and both timescales scale similarly with
density. Thus, reducing $t_{\text{ff,cloud}}\propto t_{\text{ff}}$ will reduce
the importance of $t_{\text{ff,cloud}}$ relative to the much longer
$t_{\text{GI}}$, but hardly affect $\epsilon_{\text{ff}}$.

Panel (c) shows the variation of $\epsilon_{\text{ff}}$ with respect to
$t_{\text{delay}}$. For $t_{\text{delay}}\ll 10\,\text{Myr}$ only a slight
reduction in $\epsilon_{\text{ff}}$ is noticeable at high ambient densities.
For longer delay times, comparable to $t_{\text{ff}}$, the effect is stronger.
Since the details of cloud formation are independent of $t_{\text{delay}}$ and
only $t_{\text{cycle}}$ depends on it affine-linearly,
$\epsilon_{\text{ff}}\propto t_{\text{delay}}^{-1}$ in the limit where
$t_{\text{delay}}\gg t_{\text{exp}}$, i.e. high densities or high
$t_{\text{delay}}$. However, we do not expect this limit to play a big role,
as extremely high values of $t_{\text{delay}}\gtrsim 40\,\text{Myr}$ are
unexpected and even at the maximum density eq. 14
$t_{\text{delay}}<t_{\text{exp}}$ indicating that the role of
$t_{\text{delay}}$ is almost negligible for SN-implosion-triggered star-
formation.

The range of values of $\epsilon_{\text{ff}}$ obtained with the fiducial set
of parameters is about two orders of magnitude below that derived from
observations $\epsilon_{\text{ff}}^{\text{obs}}=0.5^{+0.7}_{-0.3}\,\%$ with a
relatively weak scaling with density $\epsilon_{\text{ff}}\propto
n_{0}^{-0.22}$. The fact that we cannot explain $\epsilon_{\text{ff}}\sim
0.5\,\%$ is however not a problem and rather a feature as we expect several
other processes such as classical triggered star-formation (see e.g.
Elmegreen, 1998) or spontaneous gravitational collapse (Kennicutt, 1989; McKee
& Ostriker, 2007) to contribute to the star formation as well.

Given that the mass of the central cloud is significantly smaller than the
total swept up mass, it is not surprising that the contribution to the star
formation is minor. However, it should be noted, that the stars formed in the
implosion cloud are expected to be more chemically enriched than the bulk of
the stars formed by other processes (Romano et al., 2024). Moreover, we note
that the process is self-regulated, i.e. the cloud makes up a larger fraction
of the swept up mass for fewer SNe, which motivates us to study the single-
burst scenario, which we explore below.

###  3.2 Single-Burst Star-Formation

Figure 3: Same as Figure 2, but in the case of the single-burst scenario.
Panel (c) shows the dependency of $\epsilon_{\text{ff}}$ with respect to
$N_{\text{SN}}$, with a fiducial value of $N_{\text{SN}}^{\text{fid}}=1$.

In the single-burst scenario, we consider $\epsilon_{\text{ff}}$ for a single
iteration of SN-implosion-triggered star-formation. Such an iteration could be
triggered by the collective feedback of an entire stellar population or the
explosion of a single star, which includes both type-Ia and type-II SNe. It is
thus reasonable to measure $t_{\text{cycle}}$ from the time of the explosion,
as the delay time bears no meaning in this case.

As opposed to the cyclic case, in the single-burst scenario $N_{\text{SN}}$ is
not determined by eq. 13, but instead is treated as a free parameter. This
implies, that now in the limit of $t_{\text{GI}}\gg t_{\text{cf}}$,

$\epsilon_{\text{ff}}\propto\epsilon_{\star}\,N_{\text{SN}}^{-0.7}\,n_{0}^{-1.26}\,.$
(16)

These scalings are reflected by panels (a) and (c) in Figure 3. Panel (b)
shows, that as for the cyclic case, the effect of $\chi$ is small, since
$t_{\text{cycle}}$ is dominated by $t_{\text{GI}}$.

Due to the strong scaling with density, the range of values of
$\epsilon_{\text{ff}}$ obtained with the fiducial set of parameters spans four
orders of magnitude. At low densities, star formation is very efficient with
$\epsilon_{\text{ff}}\sim 10\,\%$ exceeding the observational mean by over an
order of magnitude, while at higher densities it becomes increasingly
inefficient.

The fact that $\epsilon_{\text{ff}}$ can significantly exceed the global
average derived from observations does, however, not cause any conceptual
problems, as the single-burst case should only occur occasionally. Thus, while
the single-burst scenario might contribute to the scatter, it is not expected
to dominate the global average.

##  4 Discussion

In the previous section we have shown the efficiency of SN-implosion-triggered
star-formation in two different scenarios. Here, we discuss some of the
limitations of our model, and the implications of our findings in the broader
context of galaxy evolution.

###  4.1 Limitations

The model considered in this letter is a detailed analysis of the implications
of the results of Romano et al. (2024). As such, the limitations applying to
their results also apply to this model.

Neglecting the turbulent nature of the ISM likely affects both the timescales
involved as well as the properties of the central cloud. Similarly, as
discussed by Romano et al. (2024) including more complete physics models for
processes, such as magnetic fields, cosmic rays, and heat conduction, could
qualitatively modify the cloud-formation process. Finally, the inclusion of
early stellar feedback and realistic time delays between SNe can further delay
the implosion and thus cloud formation (see e.g. Fichtner et al., 2024),
leading to a lower star-formation efficiency.

A model is only as good as its assumptions. It is thus worthwhile to consider
the role of our model assumptions.

For sufficiently massive clouds, assuming an IMF seems to be justified by the
observed statistics of stellar populations (Kroupa & Jerabkova, 2021).
However, for clouds with small masses, the IMF is not well sampled in the
high-mass end and the number of massive stars that are formed per cycle will
be dominated by stochasticity (see e.g. Grudic et al., 2023). Nonetheless,
since our cyclic model considers a long-term average, populations with
$N_{\text{SN}}>1$ should – over a larger number of iterations – sample the
high-mass end of the IMF well enough to justify our treatment. We have checked
eq. 13 for different shapes of the IMF commonly used in the literature (e.g.
the list of IMFs in Saitoh, 2017) and have found only minor differences; The
result that there is about 1 SN per $100\,M_{\odot}$ of newly formed stars is
on strong footing.

Clearly it is unreasonable to expect the ambient medium to return exactly to
its initial state of a uniform medium with constant density at rest.
Nonetheless, in the context of the simulations of Romano et al. (2024) it is
revealed that after cloud formation the density in the region between the
shock and the cloud indeed approaches the initial density of the ISM as the
cavity is refilled with backflowing material from the ever broadening front of
the blastwave, while the remaining kinetic energy is dissipated through small-
scale shocks, leaving a nearly static, and slightly turbulent medium.

Likewise, we expect the density in the cloud to be locally enhanced after its
collapse. However, as long as the SNR can break out of the cloud before shell
formation the dynamics are not expected to be strongly influenced.
Furthermore, in a more complete picture early feedback might counteract this
density enhancement and with more realistic delays between the SNe the central
mass distribution is expected to have only a small effect compared to the
potentially large time delay before the last SN of the population explodes.

In a more realistic galactic environment, however, these considerations are
confounded by several factors. The complex interplay of neighboring shocks,
gravitational torques, shear, and stratification effects is expected to lead
to significant inhomogeneities in the density and velocity distribution.

We expect the analysis of the single-burst case to be less affected, since the
ISM in the immediate vicinity of the SNR is not expected to change
dramatically during one cycle.

On the other hand, in the cyclic case, we consider the long-term average,
which means that necessarily all of these effects will have enough time to
affect the dynamics. As long as these effects do not prevent cloud formation
or disrupt the central clouds before they can collapse, we expect the effect
to be small. Nonetheless, in order to be certain a more thorough analysis is
required.

Finally, we have neglected the role of other sources of star formation and
feedback. While theoretical considerations suggest that individual SN shocks
are an unlikely candidate for triggered star-formation (Elmegreen, 2011), we
anyway consider the potential contribution, given the uncertainty of the
subject.

Classical, triggered star-formation has been estimated to be delayed by about
a free-fall timescale (Elmegreen, 1998). For low ambient densities, we thus do
not expect it to strongly affect the mass budget for cloud- and subsequent
star-formation by SN implosion, while at sufficiently high densities, the
onset of triggered star-formation might be before the onset of cloud formation
and thus reduce the mass budget for cloud formation.

On the other hand, we expect feedback from massive stars formed by classical,
triggered star-formation to interfere with the process in several ways.
Feedback that is sourced sufficiently far away from the explosion center might
heat the ambient medium and increase its pressure, leading to an earlier
implosion and thus accelerate the star-formation process. However, if the
source of the feedback is too close, it might disrupt the cloud before it can
form stars and delay or even prevent further star formation, leading to a
reduction of the free-fall efficiency.

###  4.2 Implications for Galaxy Evolution

We have shown that SN-implosion-triggered star-formation can contribute
$\lesssim 5\,\%$ of the globally averaged star formation. While this might not
seem like a large contribution, it should be noted that a bulk fraction
($\gtrsim 10\,\%$) of the highly enriched SN ejecta are expected to be locked
up in the central, star-forming cloud. Here we discuss the potential
implications for metal-rich star-formation.

The increment in metallicity per cycle can be expressed in terms of the
fraction of ejecta material locked up in the cloud
$M_{\text{ej,cl}}/M_{\text{cl}}$, the metallicity of the ejecta
$Z_{\text{ej}}$ and the metallicity of the ambient medium $Z_{0}$:

$\delta
Z=\frac{M_{\text{ej,cl}}}{M_{\text{cl}}}\left(Z_{\text{ej}}-Z_{0}\right)\,.$
(17)

The simulations of Romano et al. (2024) suggest that $\gtrsim 10-50\,\%$ of
the ejecta end up in the cloud before it collapses.

Shimizu et al. (2019) have computed the yields from various feedback processes
for a Chabrier IMF with mass range 0.1 to 120 M⊙ (Chabrier, 2003) using celib
(Saitoh, 2017). They find that for metallicities $Z\gtrsim 10^{-6}$, stellar
populations eject $\sim 10\,\%$ of their mass in type-II SNe, with
$Z_{\text{ej}}\sim 15-20\,\%$. In the more uncertain case of primordial
metallicity the stellar populations eject $\lesssim 90\,\%$ of their mass in
type-II SNe, with $Z_{\text{ej}}\sim 33\,\%$. While their IMF differs slightly
from the one we have adopted here, we expect the results to only change
slightly.

Combining these results and relating the mass of the stellar population to the
cloud mass through $\epsilon_{\star}$ we find that for sufficiently high
metallicity $\delta Z\gtrsim 10^{-3}\,\epsilon_{\star}$ and for primordial
metallicity $\delta Z\gtrsim 3\times 10^{-2}\,\epsilon_{\star}$.

A single cycle takes about $t_{\text{cycle}}\lesssim
10^{7}\,-10^{8}\,\text{yr}$, so over the lifetime of the universe there could
have been $\lesssim 100-1000$ cycles, during most of which the metallicity
would have been non-primordial. Neglecting dilution effects, we thus expect a
maximum enrichment of $\Delta Z_{\text{max}}=N_{\text{cycle}}\times\delta
Z\sim\left(0.1-1\right)\,\epsilon_{\star}$, slightly higher than the
metallicities of the most metal-rich stars that have been observed (see e.g.
Do et al., 2015; Sextl et al., 2024; Rix et al., 2024). As the chemical
enrichment in this scenario is primarily due to type-II SNe, we expect an
elevated level of $\alpha$-elements in the resulting stellar populations.

Turbulent mixing of ejecta and pristine gas at the bubble-shell interface of
the SNR could reduce the metal budget available to star formation. Moreover,
the increment in the metallicity of the ambient medium depends on the degree
of mixing between the left-over cloud gas and the pristine ambient medium. On
the other hand, we have neglected other enrichment mechanisms such as
enrichment due to type-Ia SNe and stellar winds from asymptotic giant branch
and massive stars (Kobulnicky, 1999; Saitoh, 2017).

##  5 Concluding Remarks

We have analyzed the efficiency of SN-implosion-triggered star-formation and
its contribution to the global star formation. Our analysis reveals that if
maintained, star-formation from SN implosion is quite inefficient,
contributing only about $\lesssim 5\,\%$ of the observed star-formation
efficiency per free-fall time $\epsilon_{\text{ff}}$. Nonetheless, because of
the projected high metal enrichment of the thus formed stars, we expect this
process to contribute significantly to the formation of metal-rich stars.

While our idealized model is useful for obtaining a rough estimate of the
contribution from SN-implosion-triggered star-formation, more detailed models
are required to explore the role of the galactic environment and external
sources of feedback.

We conclude that SN-implosion-triggered star-formation offers a compelling,
well-motivated pathway to the formation of metal-rich stars.

This research was funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy – EXC 2094 –
390783311.

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