# Metadata Might Make Language Models Better

Kaspar Beelen  The Alan Turing Institute  Daniel van Strien  British Library

(Working Paper, November 2022)

###### Abstract

This paper discusses the benefits of including metadata when training language
models on historical collections. Using 19th-century newspapers as a case
study, we extend the time-masking approach proposed by Rosin et al. (2022) and
compare different strategies for inserting temporal, political and
geographical information into a Masked Language Model. After fine-tuning
several DistilBERT on enhanced input data, we provide a systematic evaluation
of these models on a set of evaluation tasks: pseudo-perplexity, metadata
mask-filling and supervised classification. We find that showing relevant
metadata to a language model has a beneficial impact and may even produce more
robust and fairer models.

##  1 Introduction

Predictive language modelling has caused a paradigm shift within machine
learning and natural language processing. Especially with the advent of
transformer-based architectures, language models increased in size and became
more powerful. Representations derived from such models are often referred to
as “pixie dust”, which magically improve the performance on almost any NLP
application.111The term “pixie dust” was taken from the Stanford “Natural
Language Processing with Deep Learning” course by C. Manning,
https://web.stanford.edu/class/cs224n/

These revolutions in the field of Deep Learning have trickled down to Digital
History. Slowly but consistently, historians have been interrogating models
such as BERT, evaluating their use and assessing their limitations for
analysing the textual traces of the past. Since most off-the-shelf models are
trained on contemporary data—often opaque pools of text with little or no
information on their provenance—their application to history remains rather
problematic. A steady stream of recent papers started therefore to investigate
the merits of “historicizing” language models, which entailed training or
fine-tuning them on historical corpora. These attempts focused on learning
representations solely from text, modelling the probabilities of target words
only from textual context (either from the surrounding or previous tokens).
However, extra-linguistic information, often encoded as metadata, can be
critical for understanding the meaning of a document. This metadata has often
been created as the result of significant work by the custodians of these
collections and this type of curated metadata is usually not a given for the
large web-scraped datasets traditionally used for training language models.
This paper, therefore, discusses whether models that learn a relationship
between textual content and the historical context will yield better
representations of language. Attention to context outside of a text is
especially relevant for scholars in the computational humanities and digital
history, who strive to situate language use within specific historical,
ideological and social configurations.

To study language in relation to certain metadata variables, scholars
traditionally slice and dice their data before training models. To understand
diachronic semantic change, Bamler and Mandt (2017) compared embeddings
extracted from models trained on different epochs. Azarbonyad et al. (2017)
trained models on parliamentary speeches from left- and right-wing MPs to
detect ideologically contested concepts. Montariol et al. (2021) proposed
incremental fine-tuning as a strategy for detecting language change. Instead
of splitting the data, the method we present is more holistic: we avoid
training separate models for each community or epoch by inserting such
relevant contextual information directly into the learning process. This paper
investigates how including such metadata can improve language models, and
describes scenarios where this approach can make a difference. Our approach
builds on the recent “time masking” technique developed by Rosin et al.
(2022), which proved a simple but effective method for making models such as
BERT more sensitive to temporal contexts by pretending year information to the
input data. Our main contributions are threefold:

  * •

We extend the approach of Rosin et al. (2022) to other extra-linguistic
aspects such as political leaning. We refer to our methods as MDMA (MetaData
Masking Approaches).

  * •

We systematically compare different strategies for metadata masking.

  * •

We evaluate the benefits of MDMA for different NLP tasks and historical
applications.

##  2 Related Literature

Whilst a complete overview of developments of language models is out-of-scope
for this paper—and would require a much greater word limit—, we provide a
summary of key results, particularly as they relate to using language models
with historical text.

The use of language models pre-trained on large datasets has extended to a
wide range of NLP tasks. The word2vec Mikolov et al. (2013) and ELMo Peters et
al. (2018) papers outlined methods that allowed models to be trained on large
amounts of language data without supervision through the use of pre-training
tasks that only require examples of text data. Taking inspiration from methods
developed in the computer vision domain, Howard and Ruder Howard and Ruder
(2018) introduced ULMFIT as a method for enabling effective transfer learning
for NLP tasks. Compared to previous approaches such as word2vec, an advantage
of this approach is that you would no longer be required to train a model from
scratch to target a new domain. This means, for example, a model could more
readily be adapted for the historical domain by fine-tuning a ‘generic’
language model trained on Wikipedia. The final major development to note is
the mass adoption of transformers-based approaches to language models
popularised by ‘Attention is All you Need’Vaswani et al. (2017). Partially as
a result of the increased computational efficiency of transformer-based
language models, the size of these models has grown rapidly with models now
passing 500 billion parameters Smith et al. (2022).

###  2.1 Language models for historical text

Following a broader interest in using language models across various
downstream tasks, there has been a growing curiosity and exploration of the
potential use of language models for working with historical texts. There are
several potential challenges to adopting existing pre-trained language models
for working with historical texts. Two major challenges are:

  * •

OCR quality: historical text is often made machine-readable through the use of
Optical Character Recognition software. The errors introduced by these models
may negatively impact the quality of language models trained on this noisy
data van Strien et al. (2020); Jiang et al. (2021),

  * •

Semantic change: change in the use of language over time may reduce the
performance of models trained on contemporary data when applied to historic
texts Qiu and Xu (2022)

Various papers have evaluated the suitability of existing language models on
historical text data. Since many companies, with resources far exceeding most
humanities researchers or libraries, have already trained and released
language models, an ideal scenario would be to find that these models can
readily be used without the additional expense of training.

An evaluation of “zero shot” approaches, in which contemporary language models
are used without additional fine-tuning Toni et al. (2022), showed a
disappointing performance. An assessment of the effectiveness of using
transfer learning for named-entity-recognition found more positive results
from levering existing models Todorov and Colavizza (2020).

Several papers have either fine-tuned an existing language model on historic
data Beelen et al. (2021); Hosseini et al. (2021) or trained language models
from scratch on historic data Schweter et al. (2022); Bamman and Burns (2020);
Aprosio et al. (2022). An evaluation comparing these approaches Manjavacas and
Fonteyn (2022) found that training from scratch led to improved performance of
these language models. Overall, most of the literature explores the
application of language models to historic material changes only the
collections used for training. The training process itself largely remains
untouched.

##  3 Metadata Masking

Only a few recent papers have systematically experimented with including
extra-textual information such as time Rosin and Radinsky (2022); Dhingra et
al. (2022) or geographic information Hofmann et al. (2022) in language models.
Methodologically, these approaches focus on either adapting the input data and
masking strategies Rosin et al. (2022); Dhingra et al. (2022) or the model
architecture Hofmann et al. (2022); Rosin and Radinsky (2022). In this paper,
we focus on the former set of approaches, which is transferable to other
collections and easily implementable in existing frameworks such as the
Hugging Face Transformers Python library Wolf et al. (2020).

In this paper, we aim to demonstrate that MDMA anchor documents in their
historical context by learning a relationship between metadata and content.
The technique is rooted in masked language modelling, which predicts the
probability of a randomly masked token based on its surrounding textual
context. In a sequence of length $n$, the probability of a masked word at
position $i$ is $p(w_{i})=p(w_{i}|w_{1},...w_{i-1},w_{i+1},...,w_{n})$. By
adding metadata tokens to the input, for example, a label $l$ indexing the
political leaning of the newspaper, the metadata becomes part of the
computation of $p(w_{i})$:

$p(w_{i})=p(w_{i}|l,w_{1},...w_{i-1},w_{i+1},...,w_{n})$ (1)

In cases where we mask the metadata, the task almost resembles supervised
classification, in which we produce a label given a sequence of tokens:

$p(w_{i})=p(l|w_{1},...,w_{i-1},w,w_{i+1},...,w_{n})$ (2)

Label $l$ can be any (combination of) metadata field(s). The time masking
approach Rosin et al. (2022), which adds information such as the year of
publication to the input sequence, has proven to work well, especially for
newspapersDhingra et al. (2022), a genre of historical data that closely
registers the pulse of time by reporting on events that occurred in the recent
past. With time-maksing we encode the year by prepending it to each text
fragment when training the language model. The preprocessed text then looks as
follows:

> “1856 [DATE] These proposals, emanating as they do from the Western powers,
> either are, or they are not, such as Russia ought to accept.”

After training the model on such examples, we can gauge how predictions for
masked tokens change depending on the indicated year. Taking an example from
the Cheltenham Chronicle222This snippet was not observed in the training set.
from 1866 “This effort was due her Majesty’s being now in constant
communication with Lord Derby”, we mask the personal pronoun and calculate how
predictions differ based on the marked year of publication. In our case, where
we used 19th-century newspapers for creating a language model (see below), the
predictions will reflect whether a King or Queen was sitting on the British
throne, i.e. “1860 [DATE] This effort was due [MASK] Majesty’s…” returns ‘her’
as the most likely filler. When we change the time-stamp to 1810333i.e. “1810
[DATE] This effort…”, the model predicts ‘his’ as the correct pronoun.

Vice versa, models trained with MDMA can guess the date of a textual fragment
simply by masking the prepended metadata token. Revisiting the above quotation
about the Crimean war: if we mask the year token at the start of the sequence
(replace “1856” with “[MASK]”) the model predicts “1854”, “1855” and “1856” as
the model probable fillers. This allows us to automatically date historical
articles. In this paper, we evaluate both capabilities of the metadata-
enhanced language models.

##  4 Data Description

To evaluate different strategies for metadata-masking, we concentrated on a
small but openly accessible newspaper corpus which was created as part of a
Heritage Made Digital (HMD) newspaper digitisation
project.444https://blogs.bl.uk/thenewsroom/2019/01/heritage-made-digital-the-
newspapers.html. The HMD newspapers comprise around 2 billion words in total,
with the bulk of the articles originating from the (then) liberal paper The
Sun. Geographically, most papers are metropolitan (i.e. based in London), but
with the inclusion of The Northern Daily Times and Liverpool Standard, the
historical county of Lancashire is also represented. Even though HMD is an
unbalanced dataset—it massively over-represents a London, liberal
perspective—it provides nonetheless diverse enough materials for the purposes
of our experiments. Table 1 contains a more detailed overview of the corpus.

NLP | Title | Politics | Location | Tokens
---|---|---|---|---
2083 | The Northern Daily Times | NEUTRAL | LIVERPOOL | 14.094.212
2084 | The Northern Daily Times | NEUTRAL | LIVERPOOL | 34.450.366
2085 | The Northern Daily Times | NEUTRAL | LIVERPOOL | 16.166.627
2088 | The Liverpool Standard | CONSERVATIVE | LIVERPOOL | 149.204.800
2090 | The Liverpool Standard | CONSERVATIVE | LIVERPOOL | 6.417.320
2194 | The Sun | LIBERAL | LONDON | 1.155.791.480
2244 | Colored News | NONE | LONDON | 53.634
2642 | The Express | LIBERAL | LONDON | 236.240.555
2644 | National Register | CONSERVATIVE | LONDON | 23.409.733
2645 | The Press | CONSERVATIVE | LONDON | 15.702.276
2646 | The Star | NONE | LONDON | 163.072.742
2647 | The Statesman | RADICAL | LONDON | 61.225.215
Table 1:  Overview of the Heritage Made Digital newspaper collection.

##  5 Data Processing and Model Training

While faster than the glacially slow RNNs, training transformer-based language
models requires quite some computational resources. Running a grid search to
estimate the impact of every preprocessing strategy and hyperparameter is
simply not feasible (unless you have access to a battalion of software
engineers and myriad GPUs). We want this research to be useful, not wasteful,
and, therefore, we were guided by a sense of pragmatism and minimalism in the
design of our experiments. Many of our decisions to limit the range of our
experiments have to be understood in this spirit. At the same time, we are
confident that these limitations don’t hurt the quality of our findings.
Drawing on the assumption that “scaling laws“Kaplan et al. (2020) will mean
that relative improvements demonstrated using smaller models and datasets will
transfer to larger models, we do not train all the models on the full dataset.
Moreover, our goal is to evaluate the impact of different preprocessing
methods on the performance of language models. Even though the models
themselves remain open to improvement, their relative performance will help us
to develop best practices.

In all language modelling experiments, we used the HMD dataset uploaded on the
Hugging Face Hub.555https://huggingface.co/datasets/davanstrien/hmd_newspapers
Training a BERT model on the whole corpus would have taken us more than a week
(using a reasonably powerful Azure virtual machine with one GPU). To reduce
training time, we made the following decisions:

  * •

We downsampled the corpus to half a billion tokens by randomly selecting only
25% of the articles.666Data processing was done using the datasets Python
library Lhoest et al. (2021)

  * •

We opted for DistilBERT Sanh et al. (2019), a BERT descendant that contains
substantially fewer parameters whilst retaining most of BERT’s performance. As
a result of the reduced number of parameters, DistilBERT represents a more
appropriate choice when training a Masked Language Model with only half a
billion tokens.

  * •

As we lacked the data to train from scratch, we started with pre-trained
DistilBERT models, which we fine-tuned for one epoch (i.e. 81.789 optimization
steps).777 Each model evaluated in this paper was fine-tuned on the pre-
trained DistilBERT available via the Hugging Face model hub.

The combination of these choices shrank training time from more than a week to
around 24 hours per model.

With the OCR quality being overall rather questionable, we avoided sentence
splitting but chunked the text into units of hundred tokens. We furthermore
lower-cased all text but refrained from any additional processing to make
downstream comparison easier later on.

The main purpose of our experiments is to estimate how different strategies
for appending and masking metadata affect the quality of the language model.
By analysing in which scenarios MDMA improve the performance of the fine-tuned
DistilBERT models, we hope to provide an answer to the following questions:

  * •

RQ1: Should metadata be added as special tokens?: As metadata describes
content, we assess if the year of publication (or other metadata fields)
should be prepended as special or normal tokens to the input data. In the
former scenario, we added distinct year tokens such “[1850]” to the tokenizer
and each text fragment. The latter strategy adds that year as normal tokens
(i.e. “1850”). Special tokens set the metadata apart from the actual text.
Does this make it more difficult for the model to learn a relationship between
text and context?

  * •

RQ2: Should we change the masking probability?: Does the model perform better
if we mask the metadata at higher rates? In the standard scenario, we masked
all tokens (including the metadata fields) with the default probability of
0.15. For other models, we use different masking probabilities for text and
metadata: we increase the masking probability for the metadata tokens to 0.25
(or 0.75) but continue masking words in the actual text with the default value
of 0.15.

  * •

RQ3: Should we combine the metadata fields?: When describing a text document,
should we add all metadata attributes simultaneously? Or should we just add
one feature at a time? Which approach makes the model most sensitive to extra-
textual context?

To address these questions we trained the following series of models:

  * •

ERWT series: The Encoder Representations with Time or ERWT models.888Erwt is
Dutch for pea. uses the issue’s year of publication, e.g. 1810, 1811, as
metadata.

  * •

PEA series: The Political Enhanced representAtions or PEA models999Pea is
English for erwt. are trained using the newspaper’s declared political
orientation—such as liberal or conservative—as recorded in Victorian resources
or scholarly summaries Beelen et al. (2022).

  * •

PEATM: This model was trained using a combination of metadata fields (year,
political leaning and place of publication).

Table 2 gives an overview of all the DistilBERT models we fine-tuned and
evaluated in this paper. The metadata column records which fields were added
to each text segment. Special token indicates whether we added the values for
these fields as special tokens. With respect to masking, we initially used the
standard probability of 0.15 over all the tokens in a segment (including the
added metadata tokens). This strategy is referred to as NORMAL in table 2. For
some models, we experimented with increased metadata masking. For
ERWT_mask_25, we’ve hidden the year of publication 25% of the time and masked
the other tokens at the standard rate. The last column gives an example of the
formatted input data. To more accurately assess the impact of adding metadata
to the training routine we also created an HMDistilBERT model on the same set
of text fragments but without adding the year of publication or other
information to the input data. This allows us to distinguish between the
effects of fine-tuning on historical data and the contributions of MDMA.

name | metadata | special token | masking | example input
---|---|---|---|---
HMDistilBERT | NO | NO | NORMAL | Text Fragment…
ERWT | YEAR | NO | NORMAL | 1861 [DATE] …
ERWT_st | YEAR | YES | NORMAL | [1861] [SEP] …
ERWT_mask_25 | YEAR | NO | 0.25 | 1861 [DATE] …
ERWT_mask_75 | YEAR | NO | 0.75 | 1861 [DATE] …
PEA | POL | NO | NORMAL | liberal [POL] …
PEA_st | POL | YES | NORMAL | [lib] [POL] …
PEATM | YEAR, POL, LOC | NO | 0.25 | 1861 [Y] lib [P] lon [L] …
Table 2: Overview of the DistilBERT models the models that were fine-tuned for
the experiments in this paper.

Probably undertrained, the ERWT and PEA series are far from perfect. However,
we do assume that the relative differences in performance are a reliable
measure to assess the effects of different preprocessing and masking
approaches.101010We trained these models mainly for evaluation purposes but
released them also on the HuggingFace Hub:
https://huggingface.co/Livingwithmachines.

##  6 Evaluation

In this section, we investigate in greater detail how different strategies for
incorporating metadata play out in downstream applications and tasks: when and
how does adding extra-textual knowledge make a difference?

###  6.1 Pseudo-Perplexity

Perplexity is a standard metric to measure the performance of language
models.111111 As a reminder: lower perplexity scores indicate better quality.
Usually, the underlying likelihood is computed by iterating over a sequence
and, at each step, calculating the probability of the subsequent token.
However, for MLMs like DistilBERT, we computed the pseudo-perplexity, based on
pseudo-log-likelihood (PLL) as described by Salazar et al. (2019). To obtain
the PLL score, we mask each token one by one and use the whole surrounding
context to compute the probability. For a document $W$ and model with
parameters $\theta$ the PPL is equal to:

$PPL(W):=\sum_{t=1}^{|W|}logP_{MLM}(w_{t}|W_{\\\ t};\theta)$ (3)

We computed Pseudo Perplexity (PP) for a sequence $W$ of length $N$ as:

$PP(W):=-\frac{1}{N}\left(e^{PPL}\right)$ (4)

For our experiments, we made two changes to the PP calculations.

  * •

While perplexity is usually calculated over the whole corpus, we wanted to
scrutinize in which contexts MDMA enhances the quality of the ERWT models and
therefore computed a PP score for each chunk of $N$ tokens separately. We
report the average score across all fragments in table 3.121212We kept $N$
constant to either 128 or 64 for each experiment.

  * •

We prepended each segment of $N$ tokens with a year and separator token as
required by the model. For the original DistilBERT model we inserted the year
and ‘[SEP]‘ before the actual text. We then obtain the PP as described in
equation 4 except that we do not mask metadata tokens. For example: to obtain
the perplexity for the sequence “patronage of her majesty the queen” published
in 1841 for the ERWT model, we would, at step one, attach “1841 [DATE]” and
only start masking at “patronage”, (i.e. “1841 [DATE] [MASK] of her Majesty
the Queen”). Then move on to the second iteration by hiding the token “of”
(“1841 [DATE] patronage [MASK] her Majesty the Queen”), etc.

length | 64 |  | 128 |
---|---|---|---|---
model | mean | sd | mean | sd
DistilBERT | 354.40 | 376.32 | 229.19 | 294.70
HMDistilBERT | 32.94 | 64.78 | 25.72 | 45.99
ERWT | 31.49 | 61.85 | 24.97 | 44.58
ERWT_st | 31.69 | 62.42 | 25.03 | 44.74
ERWT_masked_25 | 30.97 | 61.50 | 24.59 | 44.36
ERWT_masked_75 | 31.02 | 61.41 | 24.63 | 44.40
PEA | 31.63 | 62.09 | 25.58 | 44.99
PEA_st | 31.65 | 62.19 | 25.59 | 44.99
Table 3: Pseudo Perplexity (PP) scores for each DistilBERT model. We compute
perplexity 2500 fragments of 64 tokens (left) and 1000 snippets of 128 tokens
(right). Lower scores indicate better performance.

As calculating perplexity was compute-intensive131313Even though not as
expensive as normal perplexity, see Salazar et al. (2019), we produced
perplexity scores for 1000 fragments of 128 tokens and 2500 chunks of length
64. We evaluated the models on exactly the same masked text sequences. Table 3
lists the resulting perplexity scores for each model. The impact of LM fine-
tuning on perplexity scores immediately catches the eye. The original
DistilBERT model performs very badly on this task, with PP scores almost a
magnitude of ten higher than HMDistilBERT. This begs the question whether
fine-tuning really “historicizes” language models or adapts them to the
peculiarities of textual data with omnipresent OCR errors. Notwithstanding,
the effect on the perplexity scores is substantial. Interestingly enough
HMDistilBERT performs systematically worse than any of the ERWT models, which
suggests that inserting metadata in the training routine produces a positive
effect, even though it is not comparable to the benefits introduced by fine-
tuning on historical data. The differences are smaller but they are
nonetheless significant.141414We compared the scores of HMDistilBERT with the
results of the ERWT models using a paired t-test with alternative hypotheses
set to “greater”.

Looking more closely at how the different time-masking strategies played out,
we can observe a slight preference for increased masking and using the
standard vocabulary instead of special year tokens. ERWT_st has a slightly
higher perplexity than the standard ERWT, which is however outperformed, to a
larger degree, by ERWT_masked_25. Similarly to Rosin et al. (2022), the
benefits of time-masking don’t seem to increase linearly, at least not in this
scenario, as ERWT_masked_75 scores slightly worse again.

Because we computed the PP for each segment of 64 tokens individually, we can
inspect how the score varies as a function of relevant independent variables.
Using a simple OLS regression model, we modelled the PP as a function of OCR
quality151515Please note the OCR quality scores were computed at the article
level and thus serve as a rough proxy for the chunk. These scores report the
confidence of the OCR software, rather than providing an ‘objective‘ extrinsic
measure of OCR quality., the decade in which the article was published and the
political leaning of newspapers. We created dummy variables for each political
label and treated the decade as a categorical variable.161616We used the
Python package statsmodels for model fitting. The R-style equation looked
like: $PP~{}OCR+C(decade)+con+rad+lib$. The results are reported in the
Appendix, see table 8. Obviously, OCR has the largest effect, reminding us of
the challenge historical data poses to language models. For HMDistilBERT we
expect the PP score to decrease with 265 when moving the OCR quality from 0
(very bad) to 1 (perfect). More importantly, MDMA seem to dampen the impact of
OCR. The coefficients for the OCR variable shrink when using ERWT models,
which suggests that training models with metadata may be more robust to OCR
noise, i.e. the quality of the predictions varies less as a function of OCR
errors. In addition, increased masking of metadata seems to produce even more
robust models. However, we have to warn that at this point the gains remain
small, and whether they increase when training longer on more data remains to
be seen. 171717 We also ran a regression model on the differences between
HMDistilBERT and each ERWT model, which suggested that these changes in the
parameter values are statistically significant. Of special interest to us was
understanding how perplexity varies in relation to “representativeness” of
(political) categories in our data: are articles from underrepresented groups,
such as the radical press, treated worse by the language model? We observe
that for each political leaning, the coefficients for the ERWT models shrink
compared to HMDistilBERT. Especially for the radical press, who constituted
only a small minority in our dataset, the relative difference is the largest:
from 60.49 (for HMDistilBERT) to 55.89 (for ERWT_masked_25), a percentage
change of around 7.5%. This may indicate that, in general, the perplexity for
the radical fragments is higher (i.e. it poses more challenges to the language
model compared to liberal papers), but inserting metadata tends to reduce the
damage somewhat. However, more detailed research is needed to understand the
actual implications of these findings for historical research.
181818Unfortunately, we did not observe similar consistent effects for the
decade variable, this will require further research.

###  6.2 Predicting Metadata with Mask-filling

ERWT and PEA models are versatile: they listen to the metadata cues, but can
also predict aspects of extra-linguistic context. In this experiment, we
evaluate the latter ability. We firstly zoom in on the ERWT series and assess
how well these models correctly guess the year of publication given a text
fragment. We do this by masking the metadata token of each fragment and taking
the filler with the highest probability as the predicted year of publication.
Initially, we evaluated our models on a random sample of 10,000 text chunks
with a length of 100 tokens. All the texts used for evaluation were not used
for training the language model.

We created a random and majority baseline to benchmark ERWT and PEA’s mask-
filling abilities. For the majority baseline, we chose the year (or the
political label) that appeared most frequently in the training data. Given how
unbalanced the data are, temporally as well as politically, the most common
label is not a trivial baseline to beat. As a random baseline, we sampled
randomly over the distribution of years (or political labels) in the training
set. Of course, a model specifically trained for predicting years or political
leaning may perform better than masking tokens with ERWT. However, the point
of our experiment is to investigate which preprocessing strategy delivers the
strongest model for this task. By adding the baseline, we only want to
demonstrate that the model is learning a non-trivial relation between text and
metadata during pre-training. Later, in section 6.3, we scrutinize whether
MDMA improves language models for supervised classification.

model | random | no year | bad ocr
---|---|---|---
random | 19.458 |  |  6.678*/32.343
majority | 13.929 |  |  4.488*/33.452
ERWT | 8.235 | 8.625 | 7.487
ERWT_st | 9.420 | 9.744 | 7.523
ERWT_masked_25 | 7.353 | 7.756 | 7.311
ERWT_masked_75 | 6.066 | 6.317 | 7.182
Table 4: Mean absolute error between predicted and actual year using the most
probable filler for the masked year token as the predicted year. Results were
computed on i.) a random sample (left), ii.) the same sample but with year
mentions removed (middle) iii.) articles from before 1830 with low OCR quality
(less than 0.5) (right). Scores obtained with a biased baseline are marked
with an asterisk.

Table 4 reports the results for masking the time metadata token. We calculated
the mean absolute error between the predicted and true year of
publication.191919We did look into the top $n$ predictions for each masked
metadata token, but this didn’t change our findings and interpretation, and
we, therefore, decided to remove these numbers from the paper to avoid the
tables becoming too unwieldy. All ERWT models yield substantially better
predictions than the baselines. When given a snippet of 100 words, the mean
distance between the predicted and actual timestamp varies between six or
eight years, which is vastly better than the baselines, which meander between
14 and 20 years. Put differently, comparing the worst ERWT model with the best
baseline, the error term increases by roughly 70% (from ca. 8.2 to 13.9
years). This clearly shows that by systematically appending temporal metadata
ERWT learns to situate documents in time. With respect to tokenization
strategy, the results indicate that adding distinctive year tokens to the
vocabulary slightly hurts the performance. The simpler preprocessing approach
is approximately one year closer to the actual year, which amounts to a small
but significant difference ( using a two-sided t-test). More importantly,
table 4 suggests that for the task of mask-filling, the more heavily masked
models deliver better results. The average error drops to just above 6 years
for ERWT_masked_75, which is well above its nearest competitor ERWT_masked_25.
Of course, a sceptical reader may rightfully doubt that ERWT really learns to
relate texts to time. Maybe the model relies on explicit markers, such as the
year mentioned in a text, to predict the date of publication. To evaluate
this, we reran the above experiments but stripped all references to years from
the text.202020We used the following regular expression
$r^{\prime}[789][0-9]{2}^{\prime}$ and substituted all matched patterns with
white space. As expected, ERWT’s performance drops, but only minimally so.
Across all cases, the model trained without special year tokens obtains the
best scores. The sample is referred to as “no year” in table 4.

To challenge our undertrained model even more, we evaluated its performance on
articles from before 1830 with poor OCR quality, also removing year references
(the “bad ocr” sample). The scores obtained from this sample are not
comparable with the random set. We gave the baselines access to information
the language model does not have: the biased majority baseline takes the most
frequent year appearing in the training set before 1830\. The biased random
approach similarly samples year labels from examples before 1830. In other
words: these baselines know that all snippets date from before 1830,
information ERWT does not have access to, as it was trained on the whole
1800-1880 period. Scores for biased baselines are marked with an asterisk. We
also report the results for the non-biased baselines which sample over the
complete year distribution. Luckily, the quality does not deteriorate
radically. Even in this situation, the ERWT models perform reasonably well:
the scores are a bit worse than those of the biased baselines, but well below
the unbiased ones. In any case, these results show that avoiding special year
tokens and using the standard vocabulary yields the best results.212121Relying
on the tokenizer’s standard vocabulary has one slight drawback: the model is
more likely to return tokens that are not years.

By training the PEA models, we also experimented with inserting political
information into the language model. We followed the same training strategy as
above, prepending the political orientation of a newspaper either as a
standard or a special token. Based on the same sample of 10,000 snippets, we
masked the token indexing the political leaning and used the filler with the
highest probability as the prediction. Again, we used majority and random as
baselines. Given that HMD is a very unbalanced dataset (the vast majority of
the articles are from liberal papers), we used macro f1, precision and recall
as metrics for evaluation.

sample | random |  |  | political |  |
---|---|---|---|---|---|---
model | macro f1 | precision | recall | macro f1 | precision | recall
random | 0.198 | 0.199 | 0.197 | 0.204 | 0.203 | 0.210
majority | 0.171 | 0.150 | 0.200 | 0.172 | 0.151 | 0.200
PEA | 0.474 | 0.665 | 0.413 | 0.469 | 0.668 | 0.412
PEA_st | 0.461 | 0.655 | 0.397 | 0.467 | 0.709 | 0.397
PEATM | 0.475 | 0.664 | 0.415 | 0.470 | 0.713 | 0.402
Table 5: Prediction of political leaning based on the most probable filler for
the masked metadata token. Results were computed on i.) a random sample
(left), ii. a sample that contained politically charged keywords (right).

In terms of their f1 scores, the PEA models did not perform very well, but
they were still substantially better than the baselines. Similarly to the ERWT
series, the masking strategy that relied on special tokens to index metadata
performed somewhat worse. We anticipated that the PEATM would outperform its
more simplistic peers, but to this was hardly the case. By observing multiple
metadata fields during pre-training, we expected the model to better handle
the confounding of the politics, place and time. Put differently, we thought
combined metadata might help PEA_TM to establish which aspects of language use
are symptomatic of the Metropolitan press and which reflect a paper’s liberal
orientation. However, the gains from PEATM were only minimal, unfortunately.

The absence of partisan signals in the data may also explain the low scores.
To evaluate this, we created a sample that contained at least one
ideologically charged “political” keyword.222222We used snippets that
contained the following terms: liberal(s), conservative(s) and tory/ies. From
previous experiments we knew that they are strongly related to the political
leaning of the newspaper. Also these results—again somewhat to our
surprise—did not change to a great extent. PEA still beats both the random and
majority baselines but it does not obtain higher f1 scores compared to the
random sample. A consistent finding across all these experiments was that
adding metadata as special tokens tends to hurt performance. Using the
tokenizer’s standard vocabulary is the better option.

###  6.3 Supervised Experiments: Classification and Regression

In this section, we evaluate how ERWT and PEA models perform on supervised
classification tasks with historical data. More precisely we gauge the extent
to which our models can predict political leaning, date, and animacy. For each
experiment, we load the pre-trained language models and change the
classification head to perform either multi-class classification (political
leaning), binary classification (animacy) or regression (time).

To detect partisanship in newspaper articles, we used excerpts of hundred
tokens that were not part of the language modelling task and trained a
classifier to predict the political leaning of the newspaper in which the
snippet appeared. Again, we care more about relative than absolute performance
on this task. We firstly trained models on a random sample, but because (as
observed above) many fragments lacked partisan signals, we also evaluated the
classifiers on fragments that contained political keywords, such as “liberal”
and “conservative”. Both the random and the political sample contain 15.000
fragments. 10200 were used for training, 1800 for validation and the remaining
3000 for testing. Each text snippet was appended with year metadata and
special tokens as required by the model. We used the same number of epochs
(10) and learning rate (0.0002) for each pre-trained model, to ensure the
comparison is fair.

sample | random |  | political |
---|---|---|---|---
model | f1 macro | accuracy | f1 macro | accuracy
DistilBERT | 0.631 | 0.800 | 0.639 | 0.822
HMDistilBERT | 0.663 | 0.828 | 0.694 | 0.843
ERWT | 0.667 | 0.830 | 0.692 | 0.846
ERWT_st | 0.670 | 0.820 | 0.691 | 0.846
ERWT_masked_25 | 0.671 | 0.831 | 0.690 | 0.847
ERWT_masked_75 | 0.672 | 0.831 | 0.688 | 0.846
PEA | 0.662 | 0.825 | 0.683 | 0.845
PEA_st | 0.668 | 0.827 | 0.682 | 0.844
PEATM | 0.672 | 0.833 | 0.691 | 0.850
Table 6: Performance of classifier trained on ca. 10,000 text snippets of
hundred tokens and political labels. We trained the classifier both on a
random sample (left) and a sample in which each snippet contained a political
keyword (right).

As can be surmised from table 6, we remain uncertain about the merits of MDMA
for downstream classification tasks. Pre-training on historical data does seem
to improve accuracy in general, however, the differences between HMDistilBERT
and the other models appear to be small and not consistently in the favour of
the ERWT/PEA series (especially for the random sample). From these preliminary
results, we gather that models which apply intenser time-masking may perform
better, with ERWT_maked_25 obtaining the highest f1 and accuracy scores for
this type of model. The models trained by only inserting political labels
perform worse than those that used time-masking——again somewhat to our
surprise. At the moment we don’t have a clear explanation. An issue here could
be that the hyperparameters, such as the learning rate, destroy valuable
knowledge present in the fine-tuned models. The strongest performer, albeit by
a small margin, is PEA_TM. Contrary to our earlier findings, combined metadata
may make language models better for sequence classification.

To assess how models trained with MDMA, perform on other genres—outside of the
newspaper universe—we evaluated them on atypical animacy dataset Tolfo et al.
(2020); Ardanuy et al. (2020). This corpus contains annotations on the animacy
of machines, i.e. we asked annotators if the machine in a given sentence was
portrayed as “alive” or not? The data is derived from a collection of 19th
Century books.232323https://www.bl.uk/collection-guides/digitised-printed-
books Given that this dataset is considerably smaller, we reduced the number
of epochs (to three) as well as the learning rate (to 0.0001).

model | f1 binary | accuracy
---|---|---
DistilBERT | 0.752 | 0.826
HMDistilBERT | 0.741 | 0.831
ERWT | 0.769 | 0.848
ERWT_st | 0.762 | 0.831
ERWT _masked_25 | 0.777 | 0.848
ERWT _masked_75 | 0.756 | 0.837
PEA | 0.752 | 0.837
PEA_st | 0.763 | 0.843
PEA_TM | 0.777 | 0.848
Table 7: Results for training a sequence classifier on the Atypical Animacy
dataset.

Table 7 largely paints a similar picture as we observed in the previous
classification experiment. There seems something to gain from pre-training on
historical texts with added metadata, but not that much. ERWT_masked_25 and
PEA_TM appear as the strongest competitors among the fine-tuned models, but
not miles ahead even compared to the standard DistilBERT model. In general,
the improvement in accuracy scores meanders around 2% between the off-the-
shelf model and the best ERWT/PEA variant.

Lastly, we experimented with year regression (see results in the Appendix) but
our findings tied in with the previous two experiments. Masking the year
field, yielded actually better scores than training a supervised classifier on
top of the ERWT pre-trained model. 242424As a disclaimer, we need to add that
in contrast to the previous evaluation tasks, supervised results are harder to
reproduce and can vary just by changing the random seed. The supervised models
are moreover very sensitive to hyperparameters and require further study to
become more robust.

##  7 Conclusions and Future Work

So, is MDMA all you need for training better language models? The findings in
this paper demonstrated that adding metadata can make language models more
sensitive to aspects of extra-textual context. and they appeared to be more
robust to bias in data and OCR noise (as suggested by the analysis of pseudo-
perplexity). Also, MDMA adds new functionality to MLMs. ERWT, for example,
could accurately situate fragments of newspaper texts in time (even though PEA
models struggled with detecting partisanship). Experiments with supervised
machine learning suggested that pre-training with metadata likely leads to
better accuracy scores, but the evidence remains mixed and somewhat fragile.

In general, this paper allows us to propose the following guidelines for
metadata masking:

  * •

RQ1: adding metadata as special tokens tends to diminish performance on our
evaluation tasks.

  * •

RQ2: masking metadata at higher rates (i.e. 25% of the time) generally
delivered the best results.

  * •

RQ3: combining metadata may provide more robust models for sequence
classification.

Since adding metadata to the training process is “cheap” in comparison to
other modifications, even modest improvements to the performance of the model
may warrant the use of this modified approach.

In future work, we will train models on a larger and more diverse newspaper
dataset (in terms of their political and geographical composition,) but also
include other collections such as books. We started to experiment with
approaches that require changes to the objective function (not just the input
data) as well. Lastly, we aim to extend our evaluation to different tasks that
use historical (newspaper) data, such as Named Entity Recognition or emotion
classification.

##  8 Appendix

parameter | model | coefficient | p-value
---|---|---|---
ocr | HMDist | -264.930 | 0.000
| ERWT | -253.228 | 0.000
| ERWT_ts | -255.035 | 0.000
| ERWT_masked_25 | -250.516 | 0.000
| ERWT_masked_75 | -250.432 | 0.000
pol_rad | HMDist | 60.481 | 0.000
| ERWT | 56.519 | 0.000
| ERWT_ts | 57.072 | 0.000
| ERWT_masked_25 | 55.877 | 0.000
| ERWT_masked_75 | 55.967 | 0.000
pol_con | HMDist | 37.560 | 0.000
| ERWT | 36.074 | 0.000
| ERWT_ts | 36.397 | 0.000
| ERWT_masked_25 | 35.526 | 0.000
| ERWT_masked_75 | 35.573 | 0.000
pol_lib | HMDist | 36.718 | 0.000
| ERWT | 35.115 | 0.000
| ERWT_ts | 35.461 | 0.000
| ERWT_masked_25 | 34.641 | 0.000
| ERWT_masked_75 | 34.694 | 0.000
dec_1820 | HMDist | -8.950 | 0.125
| ERWT | -6.989 | 0.210
| ERWT_ts | -7.462 | 0.186
| ERWT_masked_25 | -6.462 | 0.245
| ERWT_masked_75 | -6.602 | 0.234
dec_1860 | HMDist | 10.780 | 0.062
| ERWT | 12.088 | 0.029
| ERWT_ts | 11.750 | 0.035
| ERWT_masked_25 | 12.291 | 0.026
| ERWT_masked_75 | 11.990 | 0.029
Table 8: shows the coefficients and p-values obtained by running an OLS regression model with pseudo-perplexity as the dependent variable and decade, political leaning and OCR quality as predictors. | loss | mean absolute error | mean squared error | accuracy
---|---|---|---|---
DisilBERT | 354.606 | 15.148 | 354.606 | 0.022
HMDistilBERT | 354.235 | 15.323 | 354.235 | 0.022
ERWT | 313.965 | 14.556 | 313.965 | 0.018
ERWT_masked_25 | 323.114 | 14.469 | 323.114 | 0.020
ERWT_masked_75 | 338.570 | 15.287 | 338.570 | 0.014
Table 9: shows the error and accuracy for predicting the year of publication
given a text fragment

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We recall from [32, Section 7], the definition of the preprojective algebra
$\Pi({\mathbb{L}}_{n})$ of type ${\mathbb{L}}_{n}$. The algebra
$\Pi({\mathbb{L}}_{n})$ is given by the following quiver

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{1}}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{2}}$$\scriptstyle{\beta_{1}^{*}}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{3}}$$\scriptstyle{\beta_{2}^{*}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{n-2}}$$\scriptstyle{\beta_{3}^{*}}$$\textstyle{n-1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{n-1}}$$\scriptstyle{\beta_{n-2}^{*}}$$\textstyle{n\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{n-1}^{*}}$$\scriptstyle{\gamma}$

with the relation
$\gamma^{2}+\sum_{i=1}^{n}(\beta_{i}\beta^{*}_{i}-\beta^{*}_{i-1}\beta_{i-1})$,
where we set $\beta_{0}=\beta_{0}^{*}=\beta_{n}=\beta_{n}^{*}:=0$.
Representation theory of $\Pi({\mathbb{L}}_{n})$ and its deformations have
been studied by several researchers (see e.g., [12, 36]). The generalized
Dynkin diagrams and their preprojective algebras also appear in the study of
the quantum groups and tensor categories related to conformal field theory
(see e.g., [28, 40, 45]). We note that in this area the diagram
${\mathbb{L}}_{n}$ is usually named as $T_{n}$.

The main theorem of this section tells that the posets
$\textup{s}\tau\textup{-}\operatorname{tilt}\Pi({\mathbb{L}}_{n}),\
\operatorname{silt}\Pi({\mathbb{L}}_{n})$ are isomorphic to the Weyl group $W$
and the braid group $B$ of type ${\mathbb{B}}_{n}$ and ${\mathbb{C}}_{n}$
(equipped with the right weak order), which now we recall.

Recall that the underlying valued graphs
$|\Delta_{{\mathbb{B}}_{n}}|,|\Delta_{{\mathbb{C}}_{n}}|$ of the Dynkin
diagrams $\Delta_{{\mathbb{B}}_{n}},\Delta_{{\mathbb{C}}_{n}}$ of types
${\mathbb{B}}_{n},{\mathbb{C}}_{n}$ coincide to each other. It follows that
the Weyl groups and the braid groups of types ${\mathbb{B}}_{n}$ and
${\mathbb{C}}_{n}$ coincide to each other. For simplicity we set
$|\Delta|:=|\Delta_{{\mathbb{B}}_{n}}|=|\Delta_{{\mathbb{C}}_{n}}|$.

$|\Delta|:\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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3.0pt}}}}}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
64.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
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188.28003pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{n.}$}}}}}}}\ignorespaces}}}}\ignorespaces$

Recall that the _Weyl group_ $W:=W_{|\Delta|}$ is defined by the generators
$s_{i}$ and relations $(s_{i}s_{j})^{m(i,j)}=1$, where

$m(i,j):=\left\\{\begin{array}[]{ll}1&\mbox{if $i=j$,}\\\ 2&\mbox{if no edge
between $i$ and $j$ in $\Delta$,}\\\ 3&\mbox{if there is an edge
$i\stackrel{{\scriptstyle}}{{\mbox{---}}}j$ in $\Delta$,}\\\ 4&\mbox{if there
is an edge $i\stackrel{{\scriptstyle 4}}{{\mbox{---}}}j$ in $\Delta$.}\\\
\end{array}\right.$

(We refer to [13, 38, 39] for the background of Weyl groups and braid groups.)
We regard $W$ as a poset defined by the right weak order.

The braid group $B=B_{|\Delta|}$ is defined by generators $a_{i}$ and
relations $(a_{i}a_{j})^{m(i,j)}=1$ for $i\neq j$ (i.e. the difference with
$W_{\Delta}$ is that we do not require the relations $a_{i}^{2}=1$ for any
$i$). We regard $B$ as a poset defined by right-divisibility order.

Now we can state the main result of this section which gives a classification
of (2-term) tilting complexes for preprojective algebra of type
${\mathbb{L}}_{n}$.

######  Theorem 5.1.

Let $\Pi({\mathbb{L}}_{n})$ be the preprojective algebra of type
${\mathbb{L}}_{n}$.

  * (1)

We have the following anti-isomorphism of posets

$W\to\textup{s}\tau\textup{-}\operatorname{tilt}\Pi({\mathbb{L}}_{n}).$

  * (2)

We have the following anti-isomorphism of posets

$B\to\operatorname{silt}\Pi({\mathbb{L}}_{n}).$

####  5.1.2. Proof of Theorem 5.1

Recall that the preprojective algebra $\Pi({\mathbb{A}}_{n})$ of type
${\mathbb{A}}_{n}$ is given by the following quiver

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{1}}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{2}}$$\scriptstyle{\alpha_{1}^{*}}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{3}}$$\scriptstyle{\alpha_{2}^{*}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{n-2}}$$\scriptstyle{\alpha_{3}^{*}}$$\textstyle{n-1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{n-1}}$$\scriptstyle{\alpha_{n-2}^{*}}$$\textstyle{n\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{n-1}^{*}}$

with the mesh relation
$\sum_{i=1}^{n}(\alpha_{i}\alpha^{*}_{i}-\alpha^{*}_{i-1}\alpha_{i-1})$, where
we set $\alpha_{0}=\alpha_{0}^{*}=\alpha_{n}=\alpha_{n}^{*}:=0$. By [16,
Definition 4.6, Theorem 4.8], the algebra automorphism $\nu$ of
$\Pi({\mathbb{A}}_{n})$ given below is a Nakayama automorphism of
$\Pi({\mathbb{A}}_{n})$.

$\nu:e_{i}\mapsto e_{n-i+1},\ \alpha_{i}\mapsto\alpha^{*}_{n-i},\
\alpha_{i}^{*}\mapsto\alpha_{n-i}.$

Observe that the map $\nu$ has order $2$. Therefore we can take the Nakayama
twisted algebra $\Pi({\mathbb{A}}_{n})*\langle\nu\rangle$.

To prove Theorem 5.1, we need the following observation, which seems to be
well-known for experts and proved by a straightforward computation. But for
the convenience of the readers, we give a sketch of the proof.

######  Lemma 5.2 (cf. [32, Section 7]).

The following assertions hold.

  1. (1)

The skew group algebra $\Pi({\mathbb{A}}_{2n})*\langle\nu\rangle$ is Morita
equivalent to $\Pi({\mathbb{L}}_{n})$.

  2. (2)

The algebra $\Pi({\mathbb{L}}_{n})$ is symmetric.

###### Proof.

(1) For simplicity we set
$\Lambda:=\Pi({\mathbb{A}}_{2n}),G:=\langle\nu\rangle,A:=\Lambda*G$. The
subset $\\{e_{1},\dots,e_{n}\\}$ of vertexes is a complete set of the
representative of the quotient set
$\\{e_{1},\dots,e_{2n}\\}/\langle\nu\rangle$. It follows from Lemma 3.10 that
if we set $\epsilon:=\sum_{i=1}^{n}e_{i}$ then the algebra
$A^{\prime}:=\epsilon A\epsilon$ is a basic algebra which is Morita equivalent
to $A$. Thus we only have to prove $A^{\prime}\cong\Pi({\mathbb{L}}_{n})$.

For $i=1,2,\dots,n-1$ we set $\beta_{i}:=\alpha_{i},\
\beta_{i}^{*}:=\alpha^{*}_{i}$ and $\gamma:=\alpha_{n}(e_{n+1}*\nu)$ where we
regard these as elements of $A^{\prime}$. It follows from the mesh relation of
$\Lambda=\Pi({\mathbb{A}}_{2n})$ that for $i=1,2,\dots,n-1$ we have
$\beta_{i}\beta_{i}^{*}-\beta^{*}_{i-1}\beta_{i-1}=0$. It also follows from
the mesh relation of $\Lambda=\Pi({\mathbb{A}}_{2n})$ that
$\gamma^{2}=\alpha_{n}\alpha_{n}^{*}=\alpha^{*}_{n-1}\alpha_{n-1}=\beta^{*}_{n-1}\beta_{n-1}$.
Thus we have $\gamma^{2}-\beta^{*}_{n-1}\beta_{n-1}=0$.

Now we obtain a canonical algebra homomorphism $f:\Pi({\mathbb{L}}_{n})\to
A^{\prime}$. It is straightforward to check that elements
$\gamma,\beta_{i},\beta^{*}_{i}\ (i=1,2,\dots,n-1)$ are generators of the
algebra $A^{\prime}$ and hence that the map $f$ is surjective. Finally we can
compute the dimensions of both algebras and check that they are of the same
dimension. Thus we conclude that the map $f$ is an isomorphism.

(2) follows from (1) and Corollary 4.10. ∎

We recall the following result.

######  Theorem 5.3.

  * (1)

We have the following anti-isomorphism of posets

$W\to\operatorname{2-tilt}\Pi({\mathbb{A}}_{2n}).$

  * (2)

We have the following anti-isomorphism of posets

$B\to\operatorname{tilt}\Pi({\mathbb{A}}_{2n}).$

###### Proof.

They follows from [5, 47]. ∎

Now we see that our main theorem is a simple consequence of a combination of
the above preparations and our general result developed in the previous
sections.

###### Proof of Theorem 5.1.

By Lemma 5.2, $\Pi({\mathbb{A}}_{2n})*\langle\nu\rangle$ is Morita equivalent
to $\Pi({\mathbb{L}}_{n})$. Observe that the action
$\langle\nu\rangle\curvearrowright\Pi({\mathbb{A}}_{2n})$ preserves canonical
idempotent elements and trivial stabilizers. Thus, by applying Theorem 4.12,
we have a poset isomorphism

$\operatorname{2-tilt}\Pi({\mathbb{A}}_{2n})\cong\operatorname{2-tilt}\Pi({\mathbb{L}}_{n})\cong\textup{s}\tau\textup{-}\operatorname{tilt}\Pi({\mathbb{L}}_{n})\
\textnormal{and}\
\operatorname{tilt}\Pi({\mathbb{A}}_{2n})\cong\operatorname{silt}\Pi({\mathbb{L}}_{n}).$

Then the conclusion follows from Theorem 5.3. ∎

###  5.2. The generalized preprojective algebra $\Pi({\mathbb{C}}_{n})$ of
type ${\mathbb{C}}_{n}$ in the sense of [33]
($\operatorname{char}{\mathbf{k}}=2$ case).

In the previous section, we deal with the Nakayama twisted algebra
$\Pi({\mathbb{A}}_{2n})*\langle\nu\rangle$. A natural question is that how
about $\Pi({\mathbb{A}}_{2n+1})*\langle\nu\rangle$. The point here is that the
stabilizer group $\operatorname{stab}(n)$ of the middle vertex $n$ is a group
of order $2$ and its group algebra ${\mathbf{k}}\operatorname{stab}(n)$ is
local if and only if $\operatorname{char}{\mathbf{k}}=2$.

In this section, we deal with the Nakayama twisted algebra
$\Pi({\mathbb{A}}_{2n+1})*\langle\nu\rangle$ in the case
$\operatorname{char}{\mathbf{k}}=2$. A key observation given in the following
lemma tells that $\Pi({\mathbb{A}}_{2n+1})*\langle\nu\rangle$ is Morita
equivalent to the preprojective algebra $\Pi({\mathbb{C}}_{n})$ of type
${\mathbb{C}}_{n}$ in the sense of Geiss-Leclerc-Schröer in [33]. Thus the
same method with the previous section determines the posets of support
$\tau$-tilting modules and silting complexes over $\Pi({\mathbb{C}}_{n})$.

######  Lemma 5.4.

Assume that $\operatorname{char}{\mathbf{k}}=2$. Then the following assertions
hold.

  1. (1)

The action $\langle\nu\rangle\curvearrowright\Pi({\mathbb{A}}_{2n+1})$
preserves the canonical idempotents and has local stabilizers.

  2. (2)

The skew group algebra $A=\Pi({\mathbb{A}}_{2n+1})*\langle\nu\rangle$ is
Morita equivalent to the generalized preprojective algebra
$\Pi({\mathbb{C}}_{n})$ of type ${\mathbb{C}}_{n}$ in the sense of [33].

###### Proof.

(1) follows from Lemma 1.3.

(2) can be proved a direct computation similar to that in the proof of Lemma
5.2. ∎

######  Remark 5.5.

After finishing the first draft of this paper, R. Wang notified to us that she
and X-W Chen [21, Theorem B] independently obtained a general result which
gives a relationship between skew group algebras of preprojective algebras
(which is not necessarily the action on Nakayama automorphism) and the
generalized preprojective algebras in the sense of Geiss-Leclerc-Schröer. We
leave it to a future work to obtain a generalization of Theorem 5.6 below by
applying our results to the results of [21].

As a consequence, we obtain the following result by the same argument of
Theorem 5.1.

######  Theorem 5.6.

Assume that $\operatorname{char}{\mathbf{k}}=2$. Let $\Pi({\mathbb{C}}_{n})$
be the preprojective algebra of type ${\mathbb{C}}_{n}$.

  * (1)

We have the following anti-isomorphism of posets

$W\to\textup{s}\tau\textup{-}\operatorname{tilt}\Pi({\mathbb{C}}_{n}).$

  * (2)

We have the following anti-isomorphism of posets

$B\to\operatorname{silt}\Pi({\mathbb{C}}_{n}).$

######  Remark 5.7.

Fu-Geng [34] studied $\tau$-tilting theory of the generalized preprojective
algebras $\Pi(C,D)$ in the sense of [33] over a field ${\mathbf{k}}$ of
arbitrary characteristic and established a bijective correspondence between
the support $\tau$-tilting modules over $\Pi(C,D)$ and the corresponding Weyl
group $W(C)$.

The poset isomorphism of (1) in the above theorem can be looked as an
refinement of the bijection by Geng-Fu for $\Pi({\mathbb{C}}_{n})$ in the case
$\operatorname{char}{\mathbf{k}}=2$. We can expect that there are the similar
poset isomorphisms for any $\Pi(C,D)$ over a base field of arbitrary
characteristic.

We note that if $\operatorname{char}{\mathbf{k}}\neq 2$, then the Nakayama
twisted algebra $\Pi({\mathbb{A}}_{2n+1})*\langle\nu\rangle$ is Morita
equivalent to the preprojective algebra of type ${\mathbb{C}}_{n}$ given in
[32, Section 7].

###  5.3. The folded mesh algebra $K^{(m)}({\mathbb{A}}_{n})$ of type
${\mathbb{A}}_{n}$

Adachi-Kase [2] constructed two examples of algebras which are tilting
discrete but not silting discrete. One of them is given by the $m$-fold mesh
algebra $K^{(m)}({\mathbb{A}}_{n})$ of type ${\mathbb{A}}_{n}$, which is
called the stable Auslander algebra in [2]. Indeed, Adachi-Kase [2, Theorem
4.1] proved that $K^{(m)}({\mathbb{A}}_{n})$ is tilting discrete, but not
silting discrete, provided that $n,m\geq 5$, $n$ is odd and $m$ is coprime to
$n-1$.

It is natural to ask how is the case that $n$ is even. The same method with
Adachi-Kase works and we can show that $K^{(m)}({\mathbb{A}}_{n})$ is not
silting discrete even in the case that $n$ is even. As to tilting
discreteness, Adachi-Kase proved it by detailed and long analysis of tilting
modules over $K^{(m)}({\mathbb{A}}_{n})$. The aim of this section is that
using our methods developed in the paper, we show that in the case that $n$ is
even, the tilting poset of $K^{(m)}({\mathbb{A}}_{n})$ is isomorphic to that
of $\Pi({\mathbb{A}}_{n})$ and $K^{(m)}({\mathbb{A}}_{n})$ is tilting
discrete.

######  Theorem 5.8.

Let $K:=K^{(m)}({\mathbb{A}}_{n})$ of the $m$-fold mesh algebra of type
${\mathbb{A}}_{n}$. Assume that $n$ is even and that $m$ is coprime to $n-1$.
Then, there exists a poset isomorphism
$\operatorname{tilt}K\xrightarrow{\cong}\operatorname{tilt}\Pi({\mathbb{A}}_{n})$
that sends $K[l]$ to $\Pi({\mathbb{A}}_{n})[l]$ for any $l\in{\mathbb{Z}}$.
Consequently $K$ is tilting discrete.

Before giving a proof, we recall basic facts about the folded mesh algebras.
We basically follows the notation and conventions of [7, Section 3, 4].

Let $K({\mathbb{Z}}{\mathbb{A}}_{n})$ be the mesh algebra of the stable
translation quiver ${\mathbb{Z}}{\mathbb{A}}_{n}$ and $\tau$ the canonical
translation of ${\mathbb{Z}}{\mathbb{A}}_{n}$. Then recall that for a non-
negative integer $m$, the $m$-folded mesh algebra $K^{(m)}({\mathbb{A}}_{n})$
is defined to be the orbit category
$K^{(m)}({\mathbb{A}}_{n}):=K({\mathbb{Z}}{\mathbb{A}}_{n})/\langle\tau^{m}\rangle$
(regarded as an algebra). We also recall that the $1$-folded mesh algebra
$K^{(1)}({\mathbb{A}}_{n})$ is nothing but the preprojective algebra
$\Pi({\mathbb{A}}_{n})$.

We need the following lemma.

######  Lemma 5.9.

The Nakayama twisted algebra $K^{(m)}({\mathbb{A}}_{n})*\langle\nu\rangle$ is
Morita equivalent to the orbit category
$K({\mathbb{Z}}{\mathbb{A}}_{n})/\langle\tau^{m},\nu\rangle$.

###### Proof.

By general relationship between orbit categories and skew group algebras,
given in [10, Definition 3.6], we see that the Nakayama twisted algebra
$K^{(m)}({\mathbb{A}}_{n})*\langle\nu\rangle$ is Morita equivalent to the
orbit category
$(K({\mathbb{Z}}{\mathbb{A}}_{n})/\langle\tau^{m}\rangle)/\langle\nu\rangle$.
On the other hand, by [7, Theorem 4.2], the Nakayama automorphism $\nu$ of
$K^{(m)}({\mathbb{A}}_{n})$ is induced from that of
$K({\mathbb{Z}}{\mathbb{A}})$. It follows that
$(K({\mathbb{Z}}{\mathbb{A}}_{n})/\langle\tau^{m}\rangle)/\langle\nu\rangle\cong
K({\mathbb{Z}}{\mathbb{A}}_{n})/\langle\tau^{m},\nu\rangle$. ∎

Now we proceed the proof of the main theorem of this section.

###### Proof of Theorem 5.8.

First, we claim that the Nakayama twisted algebra
$K^{\prime}:=K*\langle\nu\rangle$ is Morita equivalent to
$\Pi({\mathbb{A}}_{n})*\langle\nu\rangle$ and hence is Morita equivalent to
$\Pi({\mathbb{L}}_{n/2})$. Indeed, from the explicit description of the
Nakayama functor $\nu$ of $K({\mathbb{Z}}{\mathbb{A}}_{n})$ given in [7,
Theorem 4.2], we observe that $\nu^{2}=\tau^{-n+1}$. Since $m$ and $n-1$ are
coprime to each other, we have $\tau\in\langle\tau^{m},\nu\rangle$. It follows
that $\langle\tau^{m},\nu\rangle=\langle\tau,\nu\rangle$ and that
$K({\mathbb{Z}}{\mathbb{A}}_{n})/\langle\tau^{m},\nu\rangle=K({\mathbb{Z}}{\mathbb{A}}_{n})/\langle\tau,\nu\rangle$.
Thanks to Lemma 5.9 and the fact
$K^{(1)}({\mathbb{A}}_{n})\cong\Pi({\mathbb{A}}_{n})$, we can conclude that
the desired statement holds.

It follows from the claim and Theorem 5.1 that
$\operatorname{silt}K^{\prime}\cong\operatorname{silt}\Pi({\mathbb{L}}_{n/2})\cong\operatorname{tilt}\Pi({\mathbb{A}}_{n})$.
We note that under this isomorphism, $K^{\prime}$ corresponds to
$\Pi({\mathbb{A}}_{n})$ Therefore, the interval $[K^{\prime}[l],K^{\prime}]$
in
$(\operatorname{silt}K^{\prime})^{\langle\nu\rangle\operatorname{mod}}\subset\operatorname{silt}K^{\prime}$
is finite for any $l\geq 0$. Since
$\operatorname{tilt}K=(\operatorname{silt}K)^{\nu}\cong(\operatorname{silt}K^{\prime})^{\langle\nu\rangle\operatorname{mod}}$,
we see that $K$ is tilting discrete. By [7, Theorem 4.2], the action
$\langle\nu\rangle\curvearrowright K$ preserves the canonical idempotent
elements and has trivial stabilizers. Finally applying Theorem 4.7, we
conclude that all silting complexes over $K^{\prime}$ is
$\langle\nu\rangle\operatorname{mod}$-stable and $\operatorname{tilt}K$ is
isomorphic to $\operatorname{tilt}\Pi({\mathbb{A}}_{n})$. ∎

In a subsequent work, we study $m$-fold mesh algebras of other types.

##  Appendix A Reinterpretation to $G$-graded algebras

In this section, we briefly explain that our results given in the main body of
the paper reinterpreted to results for a finite dimensional $G$-graded algebra
$A=\bigoplus_{g\in G}A_{g}$ via Cohen-Montgomery duality [23].

We also explain that in the special case that $A$ is strongly $G$-graded, we
can obtain results for the canonical inclusion $A_{e}\hookrightarrow A$ where
$A_{e}$ denotes the degree $e$-part of $A$ (where $e$ is the unit element of
$G$). These are generalizations of the results given in the paper [17] in
which the authors dealt with a finite dimensional strongly $G$-graded algebra
$A$ and often assumed that $A_{e}$ is selfinjective and that
$\operatorname{char}{\mathbf{k}}\nmid|G|$. Given a group $H$ and a normal
subgroup $N$ of $H$, the group algebra ${\mathbf{k}}H$ acquires a canonical
$G:=H/N$-grading such that $({\mathbf{k}}H)_{e}={\mathbf{k}}N$. Thus, our
results can be regarded as generalizations of results given in [43] for a
group extension $N\triangleleft H$.

###  A.1. $G$-graded algebras

Let $G$ be a finite group and $A:=\bigoplus_{g\in G}A_{g}$ a finite
dimensional $G$-graded algebra and $\operatorname{grmod}A$ the category of
finite dimensional $G$-modules.

We denote the dual Hopf algebra of the group Hopf algebra ${\mathbf{k}}G$ by
${\mathbf{k}}G^{*}:=\operatorname{Hom}_{{\mathbf{k}}}({\mathbf{k}}G,{\mathbf{k}})$.
By [23, Proposition 1.3], a finite dimensional $G$-graded algebra
$A=\bigoplus_{g\in A}A_{g}$ may be regarded as ${\mathbf{k}}G^{*}$-module
algebra. Let $\Lambda:=A\\#{\mathbf{k}}G^{*}$ be the smash product [23, p.
241]. Then by [23, Theorem 2.2] there is an equivalence
$\operatorname{mod}\Lambda\simeq\operatorname{grmod}A$.

By [23, Lemma 3.3], the group $G$ acts on $\Lambda$. One of the main result of
the paper [23, Theorem 3.5(Duality for coactions)] tells that the skew group
algebra $\tilde{A}:=\Lambda*G=(A\\#{\mathbf{k}}G^{*})*G$ is isomorphic to the
matrix algebra $M_{|G|}(A)$ of the order $|G|$. In particular the algebra
$\tilde{A}$ is Morita equivalent to $A$ and we have an equivalence
$\operatorname{mod}A\simeq\tilde{A}\operatorname{mod}$.

Then it is tedious but straightforward to check that the induction functor
$\operatorname{Ind}_{\Lambda}^{\tilde{A}}:\operatorname{mod}\Lambda\to\tilde{A}\operatorname{mod}$
corresponds to the functor $U:\operatorname{grmod}A\to\operatorname{mod}A$
which forgets the grading of $G$-graded $A$-modules via the equivalences given
above. Namely the left diagram is commutative up to natural isomorphism:

$\textstyle{\operatorname{mod}\widetilde{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\operatorname{mod}A}$$\textstyle{\operatorname{mod}\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Ind}_{\Lambda}^{\widetilde{A}}}$$\scriptstyle{\sim}$$\textstyle{\operatorname{grmod}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{U}$$\textstyle{\operatorname{mod}\widetilde{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\operatorname{Res}_{\Lambda}^{\tilde{A}}}$$\textstyle{\operatorname{mod}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V}$$\textstyle{\operatorname{mod}\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\operatorname{grmod}A.}$

It is also straightforward to check that the functor
$V:\operatorname{mod}A\to\operatorname{grmod}A$ defined below corresponds to
the restriction functor
$\operatorname{Res}_{\Lambda}^{\tilde{A}}:\tilde{A}\operatorname{mod}\to\operatorname{mod}\Lambda$
via the above equivalences: Let $N$ be an (ungraded) $A$-module, we define a
graded $A$-module $V(N)$ in the following way. The underlying
${\mathbf{k}}$-vector space of $V(N)$ is defined to be ${\mathbf{k}}G\otimes
N$. For $g\in G$, the degree $g$-component $V(N)_{g}$ is defined to be
$V(N)_{g}:=g\otimes N$. Finally the action of a homogeneous element $a_{h}\in
A_{h}$ on $g\otimes n$ for $n\in N$ is given by

$a_{h}\cdot_{V(N)}(g\otimes n):=hg\otimes a_{h}n.$

The assignment $N\mapsto V(N)$ gives a functor
$V:\operatorname{mod}A\to\operatorname{grmod}A$.

######  Definition A.1.

Let $N$ be an (ungraded) $A$-module and $X$ a $G$-module. We define an
$A$-module $X\otimes_{{\mathbf{k}}}^{G}N$ in the following way: The underlying
${\mathbf{k}}$-vector space is defined to be $X\otimes_{{\mathbf{k}}}N$. The
action of a homogeneous element $a_{g}\in A_{g}$ is given by

$a_{g}\cdot(x\otimes n):=gx\otimes a_{g}n.$

We note that $UV(N)={\mathbf{k}}G\otimes_{{\mathbf{k}}}^{G}N$.

######  Definition A.2.

An $A$-module $N$ is called ($\operatorname{mod}G$)-_stable_ if
$X\otimes_{{\mathbf{k}}}^{G}N\in\operatorname{add}N$ for any $G$-modules $X$.

We explain $G$-stability of $G$-graded $A$-modules $M$. Let $g\in G$. We
define the degree shift $M(g)$ of a graded $A$-module $M=\bigoplus_{g\in
G}M_{g}$ in the following way: The underlying $A$-module of $M(g)$ is defined
to be that of $M$. The grading is defined by $(M(g))_{h}:=M_{hg^{-1}}$. It is
straightforward to check that under the identification of graded $A$-modules
and $\Lambda=A\\#{\mathbf{k}}G^{*}$-modules the degree shift $M(g)$
corresponds to the twist ${}_{g}M$ given in Section 2.1.1.

$\operatorname{grmod}A\leftrightarrow\operatorname{mod}\Lambda,\
M(g)\leftrightarrow{}_{g}M.$

Thus we introduce the following terminology.

######  Definition A.3.

A graded $A$-module $M$ is called $G$-_stable_ if $M\cong M(g)$ for any $g\in
G$.

Using these preparations, we can reinterpret the results in the main body of
the paper for $G$-graded and ungraded $A$-modules. Among other things, we
point out the following reinterpretation of Theorem 3.3. We denote by
$(\textup{grs}\tau\operatorname{tilt}A)^{G}$ the poset of $G$-stable basic
graded support $\tau$-tilting $\Lambda$-module, by
$(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{\operatorname{mod}G}$ the
poset of ($\operatorname{mod}G$)-stable basic support $\tau$-tilting
$A$-modules. Then

######  Theorem A.4.

The adjoint pair $(U,V)$ induces an isomorphism of posets

$U:(\textup{grs}\tau\operatorname{tilt}A)^{G}\longleftrightarrow(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{\operatorname{mod}G}:V.$

###  A.2. Strongly graded case

Now we discuss the spacial case that a graded algebra $A=\bigoplus_{g\in
G}A_{g}$ is strongly graded [23, p.245]. In this section, we denote the degree
$e$-subalgebra by $\Lambda:=A_{e}$ (where $e$ is the unit element of $G$).
Then by [23, Theorem 2.2, Theorem 2.12], the category $\operatorname{grmod}A$
is equivalent to $\operatorname{mod}\Lambda$ via the functor
$(-)_{e}:\operatorname{grmod}A\to\operatorname{mod}\Lambda$ that takes the
degree $e$-part of graded $A$-modules. Its quasi-inverse is the functor
$A\otimes_{\Lambda}-:\operatorname{mod}\Lambda\to\operatorname{grmod}A$.

$(-)_{e}:\operatorname{grmod}A\rightleftarrows\operatorname{mod}\Lambda:A\otimes_{\Lambda}-$
(1-8)

Let $g\in G$. Under the equivalence, the degree shift functor $(g)$
corresponds to the functor $A_{g^{-1}}\otimes_{\Lambda}-$.

######  Definition A.5.

A $\Lambda$-module $M$ is called $G$-_stable_ if
$A_{g^{-1}}\otimes_{\Lambda}M\cong M$ for any $g\in G$.

The composition
$U\circ(A\otimes_{\Lambda}-):\operatorname{mod}\Lambda\to\operatorname{mod}A$
is nothing but the induction functor $\operatorname{Ind}_{\Lambda}^{A}$ and
that the composition $(-)_{e}\circ
V:\operatorname{mod}A\to\operatorname{mod}\Lambda$ is the restriction functor
$\operatorname{Res}_{\Lambda}^{A}$.

$\operatorname{Ind}_{\Lambda}^{A}=U\circ(A\otimes_{\Lambda}-):\operatorname{mod}\Lambda\rightleftarrows\operatorname{mod}A:(-)_{e}\circ
V=\operatorname{Res}_{\Lambda}^{A}.$

Using these preparations, we can specialize the results for a $G$-graded
algebra $A$ (that are obtained as reinterpretations of results in the main
body) to the induction and the reduction functors associated to the canonical
injection $\Lambda\hookrightarrow A$. As a consequence, we obtain
generalizations of the results given in the paper [17] in which the authors
dealt with a finite dimensional strongly $G$-graded algebra $A$ and often
assumed that $\Lambda=A_{e}$ is selfinjective and that
$\operatorname{char}{\mathbf{k}}\nmid|G|$.

Let $H$ be a group and $N$ a normal subgroup of $H$. We set $G:=H/N$. Then the
group algebra $A:={\mathbf{k}}H$ becomes a strongly $G$-graded algebra such
that $A_{e}={\mathbf{k}}N$ by setting $A_{g}:={\mathbf{k}}N\underline{g}$ for
$g\in G$ where $\underline{g}$ denotes a representative of $g$. Thus, our
results can be regarded as generalizations of results given in [43] for a
group extension $N\triangleleft H$.

Among other things, we point out the following theorem obtained as a
specialization of Theorem A.4.

######  Theorem A.6.

The adjoint pair
$(\operatorname{Ind}_{\Lambda}^{A},\operatorname{Res}_{\Lambda}^{A})$ induces
an isomorphism of posets

$\operatorname{Ind}_{\Lambda}^{A}:(\textup{s}\tau\textup{-}\operatorname{tilt}\Lambda)^{G}\longleftrightarrow(\textup{s}\tau\textup{-}\operatorname{tilt}A)^{\operatorname{mod}G}:\operatorname{Res}_{\Lambda}^{A}.$

######  Remark A.7.

Let $\Lambda$ be an algebra with an action of $G$ and $A:=\Lambda*G$ as in the
main body of the paper. Then if we define $A_{g}:=\Lambda*g$ for $g\in G$,
then $A:=\Lambda*G$ becomes a strongly $G$-graded algebra. Hence, applying
Theorem A.6 to this case, we can recover Theorem 3.3.

In a similar way, we can obtain results of the main body of the paper by first
establishing results for a $G$-graded algebra and functors $U,V$.

##  Appendix B A formalism by theory of Hopf algebras and tensor categories

In this section, we discuss a formal aspect of the paper including
($\operatorname{mod}G$)-action on $A=\Lambda*G$-modules given in Definition
2.2, by using theory of Hopf algebras and tensor categories. For unexplained
terminology, we refer [29].

Since the group algebra ${\mathbf{k}}G$ has a canonical structure of a Hopf
algebra, the module category
$\operatorname{mod}G=\operatorname{mod}{\mathbf{k}}G$ has a canonical
structure of a symmetric monoidal category
${\mathcal{C}}_{{\mathbf{k}}G}:=(\operatorname{mod}G,\otimes_{{\mathbf{k}}}^{G},{\mathbf{k}})$.
(We remark that the tensor product of the monoidal category
${\mathcal{C}}_{{\mathbf{k}}G}$ is usually denoted by
$\otimes_{{\mathbf{k}}}$.)

A pair $\tilde{\Lambda}=(\Lambda,\rho)$ of an algebra $\Lambda$ and an action
$\rho:G\curvearrowright\Lambda$ may be identified with an algebra object
$\tilde{\Lambda}$ of ${\mathcal{C}}_{G}$. It is well-known that the category
$\tilde{\Lambda}\operatorname{mod}_{{\mathcal{C}}_{{\mathbf{k}}G}}$ of
$\tilde{\Lambda}$-module objects in ${\mathcal{C}}_{{\mathbf{k}}G}$ is
equivalent to the category $\operatorname{mod}A$ of (ordinary) modules over
the skew group algebra $A:=\Lambda*G$. On the other hand, the tensor product
$\otimes_{{\mathbf{k}}}^{G}$ of ${\mathcal{C}}_{{\mathbf{k}}G}$ induces a
functor
$\otimes_{{\mathbf{k}}}^{G}:{\mathcal{C}}_{{\mathbf{k}}G}\times\operatorname{mod}_{{\mathcal{C}}_{{\mathbf{k}}G}}\tilde{\Lambda}\to\operatorname{mod}_{{\mathcal{C}}_{{\mathbf{k}}G}}\tilde{\Lambda}$,
which gives
$\operatorname{mod}_{{\mathcal{C}}_{{\mathbf{k}}G}}\tilde{\Lambda}$ a
structure of a ${\mathcal{C}}_{{\mathbf{k}}G}$-module category [29,
Proposition 7.8.10]. It is straightforward to check that under the equivalence
$\operatorname{mod}_{{\mathcal{C}}_{{\mathbf{k}}G}}\tilde{\Lambda}\simeq\operatorname{mod}A$,
this functor gives the tensor product $(X,N)\mapsto
X\otimes_{{\mathbf{k}}}^{G}N$ of Definition 2.2.

Let ${\mathbf{k}}G^{*}:=\operatorname{Hom}_{{\mathbf{k}}}(G,{\mathbf{k}})$ be
the dual Hopf algebra of ${\mathbf{k}}G$. An action
$\rho:G\curvearrowright\Lambda$ may be regarded as a map
$\rho:{\mathbf{k}}G\otimes\Lambda\to\Lambda$ that gives a
${\mathbf{k}}G$-module algebra structure on the algebra $\Lambda$. The induced
map $\rho^{*}:\Lambda\to{\mathbf{k}}G^{*}\otimes\Lambda$ gives $\Lambda$ a
structure of ${\mathbf{k}}G^{*}$-comodule algebra. For a $\Lambda$-module $M$
and a ${\mathbf{k}}G^{*}$-module $Y$, we can construct a $\Lambda$-module
$Y\otimes_{{\mathbf{k}}}^{{\mathbf{k}}G^{*}}M$ in the following way: the
underlying ${\mathbf{k}}$-vector space is defined to be the usual tensor
product $Y\otimes M$ of the underlying spaces. The multiplication
$\Lambda\otimes(Y\otimes_{{\mathbf{k}}}^{{\mathbf{k}}G^{*}}M)\to
Y\otimes_{{\mathbf{k}}}^{{\mathbf{k}}G^{*}}M$ is defined by the following
composition

$\Lambda\otimes(Y\otimes
M)\xrightarrow{\rho^{*}\otimes\operatorname{id}}{\mathbf{k}}G^{*}\otimes\Lambda\otimes(Y\otimes
M)\cong({\mathbf{k}}G^{*}\otimes Y)\otimes(\Lambda\otimes M)\to Y\otimes M$

where the third map is the tensor product of the action maps of $Y$ and $M$.
Let
${\mathcal{C}}_{{\mathbf{k}}G^{*}}=(\operatorname{mod}{\mathbf{k}}G^{*},\otimes_{{\mathbf{k}}}^{{\mathbf{k}}G^{*}},{\mathbf{k}})$
be the symmetric monoidal category of ${\mathbf{k}}G^{*}$-modules. Then the
induced functor
$\otimes_{{\mathbf{k}}}^{{\mathbf{k}}G^{*}}:{\mathcal{C}}_{{\mathbf{k}}G^{*}}\times\operatorname{mod}\Lambda\to\operatorname{mod}\Lambda$
gives $\operatorname{mod}\Lambda$ a structure of
${\mathcal{C}}_{{\mathbf{k}}G^{*}}$-module category.

The dual basis $\\{g^{*}\\}_{g\in G}\subset{\mathbf{k}}G^{*}$ of the canonical
basis $\\{g\\}_{g\in G}\subset{\mathbf{k}}G$ is a complete set of primitive
orthogonal idempotent elements of ${\mathbf{k}}G^{*}$ and hence
${\mathbf{k}}G^{*}\cong\prod_{g\in
G}{\mathbf{k}}g^{*}\cong{\mathbf{k}}^{\times|G|}$ is semi-simple as an
algebra. Thus a ${\mathbf{k}}G^{*}$-module $Y$ is a direct sum of simple
${\mathbf{k}}G^{*}$-modules ${\mathbf{k}}g^{*}$ for $g\in G$ and the tensor
product $Y\otimes_{{\mathbf{k}}}^{{\mathbf{k}}G^{*}}-$ is completely
determined by ${\mathbf{k}}g^{*}\otimes_{{\mathbf{k}}}^{{\mathbf{k}}G^{*}}-$
for $g\in G$. Let $g\in G$ and $M$ a $\Lambda$-module. Then the
$\Lambda$-module
${\mathbf{k}}g^{*}\otimes_{{\mathbf{k}}}^{{\mathbf{k}}G^{*}}M$ is isomorphic
to ${}_{g}M$. Hence, we see that $G$-stability for basic modules coincides
with ($\operatorname{mod}{\mathbf{k}}G^{*}$)-stability (defined in the same
way of Definition 2.2). We understand that the difference between the
characterization of $G$-stability (Corollary 2.14) and that of
($\operatorname{mod}G$)-stability (Proposition 2.15) is caused by the fact
that ${\mathbf{k}}G^{*}$ is semi-simple as an algebra but ${\mathbf{k}}G$ is
not in general. Finally, our main results Theorem 3.11 and Theorem 4.3 can be
looked as bijections between ($\operatorname{mod}{\mathbf{k}}G^{*}$)-stable
objects and ($\operatorname{mod}{\mathbf{k}}G$)-stable objects.

It is worth pointing a possible generalization obtained by replacing the group
Hopf algebra ${\mathbf{k}}G$ with a (finite dimensional) Hopf algebra $H$. The
above constructions works for an $H$-modules algebra $\Lambda$ and the smash
product algebra $A:=\Lambda\\#H$. Thus we can expect that we have bijections
between ($\operatorname{mod}H$)-stable objects and
($\operatorname{mod}H^{*}$)-stable objects. (Furthermore, we can expect more
general statements by using tensor categories as in [25].) Some of the result
of the paper may be easily generalized for a general Hopf algebra $H$ and its
module algebra $\Lambda$. However, we remark that one of the key step Lemma
2.23(2) does not hold true for a general Hopf algebra $H$ (see, e.g., [23,
Theorem 3.2], [48, Corollary 9.3.4]). Thus, we decided to leave
generalizations for Hopf algebras to our future research.

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