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Choosing the Number of Topics in LDA Models – A Monte Carlo Comparison of
Selection Criteria111Financial support from the German Research Foundation
(DFG) (WI 2024/8-1) and the National Science Centre (NCN) (Beethoven Classic
3: UMO-2018/31/G/HS4/00869) for the project TEXTMOD is gratefully
acknowledged. The project also benefited from cooperation within HiTEC Cost
Action CA 21163.

Victor Bystrov
University of Lodz
Rewolucji 1905r. 41, 90-214 Lodz, Poland
email<EMAIL_ADDRESS>

Viktoriia Naboka-Krell
Justus Liebig University Giessen
Licher Strasse 64, 35394 Giessen, Germany
email<EMAIL_ADDRESS>

Anna Staszewska-Bystrova
University of Lodz
Rewolucji 1905r. 37/39, 90-214 Lodz, Poland
email<EMAIL_ADDRESS>

Peter Winker
Justus Liebig University Giessen
Licher Strasse 64, 35394 Giessen, Germany
email<EMAIL_ADDRESS>
Abstract

Selecting the number of topics in LDA models is considered to be a difficult
task, for which alternative approaches have been proposed. The performance of
the recently developed singular Bayesian information criterion (sBIC) is
evaluated and compared to the performance of alternative model selection
criteria. The sBIC is a generalization of the standard BIC that can be
implemented to singular statistical models. The comparison is based on Monte
Carlo simulations and carried out for several alternative settings, varying
with respect to the number of topics, the number of documents and the size of
documents in the corpora. Performance is measured using different criteria
which take into account the correct number of topics, but also whether the
relevant topics from the DGPs are identified. Practical recommendations for
LDA model selection in applications are derived.

Key Words: Topic models, text analysis, latent Dirichlet allocation, singular
Bayesian information criterion, Monte Carlo simulation, text generation
JEL classification: C49

##  1 Introduction

Text data have been increasingly used in different applications lately. One of
the main challenges in working with text data is to structure and to quantify
these data. To this end, probabilistic topic modelling approaches are often
applied, as they allow to uncover hidden structures behind text data. One of
the best-known and widely used topic modelling approaches is Latent Dirichlet
Allocation (LDA) introduced by Blei2003. For some recent applications making
use of this method, see, e.g., Luedering2016, Thorsrud2020, Ellingsen2022, and
Savin2022.

LDA is an unsupervised method that builds on two basic assumptions. First, it
is assumed that each document in a corpus represents a mixture of topics. The
second assumption is that each topic is given by a mixture of words from the
vocabulary. The number of these topics, or themes, is a parameter to be set by
the researcher. Often this decision is based on human/expert judgment and is,
therefore, rather subjective. In order to account for possible subjectivity
and to allow for a more standardised estimation procedure, different
evaluation metrics have been developed for identifying an optimal number of
topics in LDA models. Some of them aim to minimize the similarity of different
topics [Cao2009], maximize the topic coherence [Mimno2011] or maximize the
goodness-of-fit between the estimated and the actual document-word frequencies
[Grossetti2022]. These criteria, however, often result in (substantially)
different numbers when applied to the same corpus. Their performance might
also differ across corpora depending on the underlying data set (see examples
in Section 2). Bystrov2022 propose to use a new measure for selecting an
optimal number of topics, namely the singular Bayesian information criterion
(sBIC). This information criterion reflects the trade-off between goodness-of-
fit and model complexity and showed promising results in a first application.

There have been some attempts to compare selected criteria based on individual
real datasets.222A notable exception including also a small scale Monte Carlo
simulation is Grossetti2022. In this paper, a comprehensive Monte Carlo (MC)
simulation is proposed, which allows a systematic evaluation going beyond
individual case reports by using a large number of datasets coming from well
defined data generating processes (DGP) with known properties. Thereby, we
consider three different data generating processes to reflect different types
of text data commonly used in applications. In a first step, we generate
corpora with a known (true) number of topics, to which LDA models with
different numbers of topics are fitted. Then, we apply the metrics to select
the number of topics and evaluate their performance over many MC replications.
To the best of our knowledge, no such systematic and comprehensive comparison
analysis of the metrics used for choosing the number of topics in LDA models
has been performed yet.

The contribution of this paper is threefold. First, with the sBIC we implement
a new measure for identifying the true number of topics in LDA models. Second,
we perform a proper MC simulation study to evaluate the proposed criterion as
well as other evaluation criteria commonly used in applications.333This
version of the paper is preliminary regarding the limited number of
replications conducted for the Monte Carlo simulations. Due to constraints in
available computational resources, current work focuses on extending the
number of replications to 1 000 for all setups. Third, we evaluate the
considered metrics quantitatively and qualitatively, i.e., we consider whether
the actual number of topics is approximated well, and also the content and
structure of the estimated topics.

The remainder of this paper is structured as follows. The considered model
selection criteria are described in Section 2. Section 3 presents the design
and the implementation details of the MC simulations. The results of the MC
simulations for three different DGPs are presented in Section 4, which is
divided in two subsections to address the main trade-off between number of
topics and coherence/structure of the uncovered topics. The final section
summarises the findings and provides recommendations for applications.

##  2 Model Selection Criteria for LDA

The selection of the optimal number of topics for LDA models can be based
either on measures of topic quality (similarity or coherence) or on measures
of goodness-of-fit and model complexity.

Let us consider an LDA model under a standard “bag-of-words” assumption. For a
document corpus $\mathcal{D}$ that consists of $J$ documents, each document
$j$ ($j=1,2,\ldots,J$) is a set of $N_{j}$ words, where the ordering of words
is ignored. The total number of words in the corpus is equal to
$N=\sum_{j=1}^{J}N_{j}$. The document corpus $\mathcal{D}$ can be
characterized by a $J\times I$ document-word frequency matrix
$X=\\{x_{ji}\\}_{j,i=1}^{J,I}$, where $x_{ji}$ is the frequency of word $i$
encountered in document $j$ and $I$ is the number of different words in the
vocabulary.

Under the “bag-of-words” assumption, an LDA model can be summarized by a
$J\times K$ matrix $\theta$ of document-topic frequencies and a $K\times I$
matrix $\beta$ of topic-word frequencies with the dimensions of these matrices
depending on the number of topics $K$. The estimated document-term matrix is a
product of estimates $\hat{\theta}$ and $\hat{\beta}$:
$\hat{X}=\hat{\theta}\times\hat{\beta}$. A set of candidate LDA models is
determined by the numbers of topics in candidate models:
$K\in\\{K_{min},\ldots,K_{max}\\}$.

In the following, we describe two popular semantic measures of topic quality,
which are often used in applications, and two recently developed goodness-of-
fit measures.

###  2.1 Topic Similarity

Following Cao2009, the optimal number of topics is often selected by
minimizing the average cosine similarity across topics:

$Cao\\_Juan(K)=\frac{\sum_{k=1}^{K}\sum_{l=k+1}^{K}corr(k,l)}{K\times(K-1)/2},$

where

$corr(k,l)=\frac{\sum_{i=1}^{I}\beta_{ki}\beta_{li}}{\sqrt{\sum_{i=1}^{I}\beta_{ki}^{2}}\sqrt{\sum_{i=1}^{I}\beta_{li}^{2}}},$

and $\beta_{ki}$ is the frequency of word type $i$ in topic $k$.

The average cosine similarity is extensively used for selecting the number of
topics in different text-as-data applications, e.g. analyzing scientific
articles to examine the evolution of research over time and identify future
fields of research [Loureiro2021, Tiba2018], analyzing the speeches by
Executive Board members of the European Central Bank [Hartmann2018],
investigating news data in the context of economic reforms [Lin2022],
analyzing and categorizing innovation projects [Dahlke2021].

###  2.2 Topic Coherence

Mimno2011 proposed a model selection procedure that maximizes the average
semantic coherence of topics:

$Mimno(K)=\frac{1}{K}\sum_{k=1}^{K}coh(k,\mathbf{i}^{(k)}),$

where $coh(k,\mathbf{i}^{(k)})$ is the coherence metric for topic $k$,

$coh(k,\mathbf{i}^{(k)})=\frac{2}{v\times(v-1)}\sum_{m=2}^{v}\sum_{n=1}^{m-1}\log\frac{f(i_{m}^{(k)},i_{n}^{(k)})+\epsilon}{f(i_{n}^{(k)})},$

$\mathbf{i}^{(k)}=(i_{1}^{(k)},\ldots,i_{v}^{(k)})$ is the list of the $v$
most frequent word types in topic $k$, $f(i)$ is the document frequency of
word $i$ (i.e., the number of documents with at least one token of type $i$),
and $f(i,i^{\prime})$ is the co-document frequency of words $i$ and
$i^{\prime}$ (i.e., the number of documents containing one or more tokens of
type $i$ and at least one token of type $i^{\prime}$). The default number of
the most probable words used is equal to 20. The smoothing parameter
$\epsilon$ is included to avoid taking the logarithm of zero and its default
value is equal to $e^{-12}$.

The average semantic coherence is also often used for selecting the number of
topics in applied topic mining analyzing, e.g., monetary policy speeches
[Ferrara2022], news data to forecast aggregated stock returns [Adammer2020],
energy market tweets with regard to their impact on market movements
[Polizos2022], or survey responses on the consequences of Covid-19 pandemic
[Kleinberg2020].

###  2.3 OpTop Criterion

Grossetti2022 proposed to use a goodness-of-fit statistic based on the
comparison of actual and estimated document-word frequencies. The frequency of
word type $i$ in document $j$ estimated in an LDA model with $K$ topics is

$\hat{x}^{(K)}_{ji}=\sum_{k=1}^{K}\hat{\theta}^{(K)}_{jk}\hat{\beta}^{(K)}_{ki}.$

Because the matrix of document-word frequencies is usually sparse,
Grossetti2022 suggest collapsing relatively unimportant words in a single
frequency bin. For document $j$, they order word types from the smallest to
the largest estimated frequency,
$(i^{(j)}_{1},i^{(j)}_{2},\ldots,i^{(j)}_{I})$ such that
$\hat{x}^{(K)}_{ji_{1}}$ $\leq$ $\hat{x}^{(K)}_{ji_{2}}$ $\leq$ $\ldots$
$\leq$ $\hat{x}^{(K)}_{ji_{I}}$, and select a sub-vector of relatively
unimportant word types $(i^{(j)}_{1},i^{(j)}_{2},\ldots,i^{(j)}_{p})$. The
cumulative frequency of relatively unimportant word types in document $j$
estimated in an LDA model with $K$ topics is

$\hat{x}^{(K)}_{j,min}=\sum_{i\in(i^{(j)}_{1},\ldots,i^{(j)}_{p})}\hat{x}^{(K)}_{ji},$

where $\hat{x}^{(K)}_{ji_{p}}$ is the largest frequency such that
$\sum_{i=i^{(j)}_{1}}^{i^{(j)}_{p}}\hat{x}^{(K)}_{ji}<x_{cutoff}$, and
$x_{cutoff}$ is a cumulative frequency cut-off value. Following Lewis and
Grossetti (2022), we use $x_{cutoff}=0.05$ as a baseline cut-off value. (For a
robustness check, we also consider a cut-off value of $0.20$) The resulting
goodness-of-fit statistic is

$OpTop(K)=\sum_{j=1}^{J}\left[(P_{j}+1)\left(\sum_{i\in(i^{(j)}_{p+1},\ldots,i^{(j)}_{I})}\frac{(\hat{x}^{(K)}_{ji}-x_{ji})^{2}}{\hat{x}^{(K)}_{ji}}+\frac{(\hat{x}^{(K)}_{j,min}-x_{j,min})^{2}}{\hat{x}^{(K)}_{j,min}}\right)\right],$
(2.1)

where $(i^{(j)}_{p+1},\ldots,i^{(j)}_{I})$ is a sub-vector of relatively
important word types in the $j$th document and $P_{j}$ is the length of this
sub-vector. Grossetti2022 propose to select an optimal number of topics by
minimizing the OpTop statistic (2.1) over a range of numbers of topics. Unlike
criteria proposed by Cao2009 and Mimno2011, the OpTop statistic is not a
semantic measure of topic quality, but a goodness-of-fit measure that can be
easily computed.

###  2.4 Singular Bayesian Information Criterion

The last model selection criterion – singular Bayesian information criterion
is a version of the Bayesian information criterion (BIC) which can be applied
to singular statistical models [Hayashi2020], for which it is more suitable
for model selection than BIC. The method was successfully applied by
Bystrov2022 for selecting parsimonous LDA models with coherent topics, however
the properties of the criterion as applied to LDA modelling have not been
studied in a simulation setup.

Computation of sBIC is based on several results from the literature. The first
one, is the decomposition of log-marginal likelihood of a text corpus
$\mathcal{D}$ with $K$ topics, described by Watanabe2009. In the context of an
LDA model, this representation can be written as

$\log L(\mathcal{D}|K)=\log
P(\mathcal{D}|\hat{\theta},\hat{\beta},K)-\lambda(K)\log(N)+(m(K)-1)\log\log(N)+O_{p}(1),$

where $\hat{\theta}$ and $\hat{\beta}$ are consistent estimators of the
document-topic and topic-word probability matrices, respectively, $\lambda(K)$
is a learning coefficient, measuring stochastic complexity of a model with $K$
topics, $m(K)$ is its multiplicity and $N$ stands for the total number of
words in the corpus.

In the next step, to deal with the problem of unknown values of $\lambda(K)$
and $m(K)$ which depend on the true value of $K$, model averaging described in
Drton2017 is applied. In this approach, the singular Bayesian information
criterion for an LDA model with $K$ topics can be defined as an approximation
of the log-marginal likelihood obtained by averaging of models with smaller
number of topics (see Drton2017):

$sBIC(K)=\log L^{\prime}(\mathcal{D}|K),$ (2.2)

where $L^{\prime}(\mathcal{D}|K)$ denotes the approximation following from the
Drton2017 procedure. To compute the marginal likelihood for every sub-model
with the number of topics $k$ where $k\leq K$, the formulas for the learning
coefficient $\lambda(k)$ and its multiplicity $m(k)$ derived for LDA by
Hayashi2021 are used. Model selection is then based on maximizing the sBIC
value.

As described by Bystrov2022 evaluation of (2.2) for some datasets may be
associated with numerical problems resulting from very small values of the
likelihood function. In order to avoid these problems, in the experiments we
use high-precision computations.

##  3 Monte Carlo Simulation

Despite the broad usage of the evaluation metrics described in the previous
section, there is no consensus yet on which metric performs best, when it
comes to choosing the number of topics. Given that the ground truth, i.e., the
actual data generating process is unknown in real applications, the
performance of the metrics can only be assessed based on a subjective analysis
of the uncovered themes. To account for this problem, a Monte Carlo simulation
study is required, for which the data are generated by a well defined DGP with
a known number of different topics.444The idea of using Monte Carlo
simulations for obtaining well-defined text corpora has been applied recently
by Wang2021 in the context of model selection for text classification tasks.
The authors use the generated data to evaluate the classification performance
of different topic models. This allows not only to compare the performance of
alternative metrics with regard to the number of topics identified, but also
to evaluate whether certain characteristics of the corpora such as number or
length of documents might affect the relative performance. Furthermore, it
enables to evaluate not only the number of topics identified, but also whether
these topics correspond closely to the topics underlying the DGP, i.e.,
focusing also on the content in an objective approach.

This section provides the details of the Monte Carlo simulation setup used for
the comparison of the methods described in Section 2. First, in Subsection 3.1
we present the general framework that is applied for each of three different
DGPs. Second, in Subsection 3.2 we describe the DGPs, which are derived from
actual corpora with typical characteristics of textual data used in
applications. Finally, Subsection 3.3 provides some technical implementation
details.

###  3.1 Procedure

The three DGPs used in the Monte Carlo simulations are designed to replicate
the characteristics of a given real document corpus. Figure 1 presents the
generic procedure which is applied to each of these DGPs.

Figure 1: Generic procedure for Monte Carlo simulations with a given selection
criterion

As described in Section 2, LDA is based on the assumption that each document
in a corpus is a distribution over a given number $K_{true}$ of latent topics
and each topic is a distribution over a fixed corpus vocabulary [Blei2003].
Thus, an LDA model is described by two matrices, the first containing the
probabilities of occurrence of each word in each topic (topic-word
distribution), and the second providing the probabilities of each topic
occurring in a single document (document-topic distribution). The approach for
generating Monte Carlo text corpora is based on these two matrices.

Therefore, in the first step of the procedure, an LDA model is estimated using
a given real document corpus with a number of topics that was used in previous
analysis of the selected corpus. In order to make sure that only distinct
topics will be used for the generation of Monte Carlo text corpora in the
following step, topics exhibiting a cosine similarity measure with other
topics larger than a selected cut-off value (95% or 99% percentile) are
dropped (see Appendix A for more details). This correction allows to reduce a
potential bias that can be induced by high topic similarity observed in the
generated data and ensures a more stable performance of all model selection
criteria.

For the remaining $K_{true}$ topics, the document-topic matrix is re-scaled to
ensure that topic weights add up to one and passed on to the second step of
text generation. The text generation process based on LDA is presented in
Algorithm 1. For each document in the original corpus, a new Monte Carlo
document is created with the same number of words and document-topic
distribution. For each word in this document, first, a topic is randomly
selected based on the known document-topic distribution. Then, the word is
randomly selected from the known vocabulary using the known topic-word
distribution. It should be noted that the algorithm as presented does not
exactly reproduce the generative procedure described in Blei2003, where
document-topic and topic-word frequency matrices are obtained using hyper
parameters. In applications, these hyper parameters are not often estimated,
and using flat priors would result in distributions of document-topic and
topic-word frequencies different from the ones actually observed. Therefore,
we follow a data-driven approach and try to replicate closely the properties
of the observed datasets serving as the benchmarks for our Monte Carlo corpora
and, consequently, use the estimated document-topic and topic-word frequency
matrices.

Algorithm 1 Text generation

1:for $document=1,2,\ldots,J$ do

2: $document\\_length=original\\_document\\_length$

3: for $word=1,2,\ldots,document\\_length$ do

4: Randomly select a topic from the document-topic distribution of the current
document

5: Randomly select a word from the topic-word distribution

6: Append the selected word to the current document

7: end for

8: Append the generated document to the corpus.

9:end for

Algorithm 1 is applied to each DGP with 500 Monte Carlo replications, i.e.,
500 corpora containing the same number of documents of same length as the
original corpus.

In the third step, for each criterion that we use, we estimate the LDA model
with number of topics within the interval
$[\max\\{2;K_{true}-20\\},K_{true}+20]$, where $K_{true}$ is the number of
topics used when generating text corpora from the DGPs. The maximum length of
the range of admitted values for the number of topics is equal to 40 with the
true number of topics $K_{true}$ of the DGP in the center of the interval if
larger than 20. Otherwise, the lower bound is set to 2, the lowest sensible
number of topics. This limited range of admitted values for the number of
topics is due to the high computational costs of model estimation. The optimal
number of topics is determined for each of the selected criteria based on
these estimated models.

For the final step, the comparison of the outcomes of different model
selection criteria in Subsection 4.1, we use descriptive statistics such as
standard deviation, mean, median, and skewness. For the visualisation of the
distributions over the number of topics determined according to the considered
criteria, we use histograms. Furthermore, in Subsection 4.2, we will also
provide information about the extent to which the topics used for generating
the texts are found when applying LDA with the number of topics selected by
the different criteria.

###  3.2 Data Generation Processes

The three DPGs used for the Monte Carlo simulations are related to three real
world corpora:

  * •

DGP 1 replicates the characteristics of a corpus consisting of scientific
papers published in the Journal of Economics and Statistics (JES).

  * •

DGP 2 reproduces features of the corpus consisting of abstracts submitted to
European Research Consortium for Informatics and Mathematics (ERCIM) and
Computational and Financial Econometrics (CFE) conferences.

  * •

DGP 3 reproduces the properties of a corpus containing Newsticker items from
heise online.

The data from JES used for DGP 1 cover the period from 1984 to 2020 and
consists of 704 documents with an average text length of about 3,000 words.
The size of the vocabulary for this corpus is equal to 3,911 words. The
collection focuses on scientific publications in empirical economics and
applied statistics. The initial number of topics selected was equal to 60 as
in Bystrov2022. After removing topics which were too similar, the final number
of topics used in DGP 1 is equal to 38 ($K_{true}=38$).

The conference abstract data used for DGP 2 cover the period from 2007 to 2019
and consists of 11,387 documents with an average text length of about 80
words. For this corpus the dictionary is composed of 1,796 words. The focused
nature of conference abstracts allows to expect a limited number of topics.
The initial number of topics selected for these data was equal to 20. This
number is reduced to 12 ($K_{true}=12$) after removing the topics that were
too close to each other.

The heise data used for DGP 3 cover the period from 1996 to 2021 and include
181,402 documents with an average length of about 120 words. The number of
words in the vocabulary for this corpus is equal to 4,675. The news platform
discusses a significant number of topics concerning technological advances.
The initial number of topics selected was equal to 120. Keeping the most
distinct topics, the final number of topics used in DGP 3 was equal to 70
($K_{true}=70$). In the analysis we use only the most recent 50,000 documents
from this corpus because using the whole dataset would increase the
computational costs for the Monte Carlo simulation beyond the available
capacities.

###  3.3 Details of Implementation

All Monte Carlo simulations were implemented using Python. To generate random
sequences used in the text generation stage (Algorithm 1), the random number
generator from Pythons’ numpy package was used
(https://numpy.org/doc/stable/reference/random/generator.html). LDA models
were estimated using the Gibbs sampler as implemented in the Python package
”lda” (https://pypi.org/project/lda/). For each corpus generated based on DGP
1, models with topic numbers in the interval $[18;58]$ were estimated, for
corpora generated based on DGP 2 - in the interval $[2;32]$, and for corpora
generated based on DGP 3 - in the interval $[50;90]$. Most other parameters of
the package were used at the default values. The number of iterations was set
to a relatively small value of 1 000 due to computational constraints.

The Cao_Juan and Mimno criteria were computed using the Python package
”tmtoolkit” (https://pypi.org/project/tmtoolkit/). The Python implementations
of the sBIC and OpTop model selection criteria were written by the authors.
Computations were performed using the high-performance-computing-cluster at
Justus Liebig University Giessen (justHPC)
(https://www.hkhlr.de/de/cluster/justhpc-giessen).555Code details can be found
in the Github repository for this paper at https://github.com/VikaNa/sBIC .

##  4 Results

This section summarizes the results of the Monte Carlo simulations. It is
divided into two subsections. The first (4.1) presents and discusses the
results on finding the optimal (true) number of topics. The second one (4.2)
focuses on topics’ contents and structure, and proposes a procedure to
analyse/evaluate both.

###  4.1 Number of Topics

The first set of results concerns the estimation of the number of topics $K$.
Figures 2, 3 and 4 present histograms for the numbers of themes selected by
the evaluation metrics described in Section 2 for all three considered DGPs.
In each of the histograms, the red vertical line depicts the true number of
topics ($K_{true}$) used for generating the corpora. Table 1 provides selected
descriptive statistics computed for these estimates. The shape and location of
histograms shown in Figures 2-4 suggest that sBIC is clearly the best method
for selecting the number of topics for DGP 1 and DGP 2, while it performs
similarly to the method of Cao2009 for DGP 3.

Figure 2: Comparison of evaluation metrics for DGP1 Figure 3: Comparison of
evaluation metrics for DGP2 Figure 4: Comparison of evaluation metrics for
DGP3

Statistics from Table 1 further indicate that the mean and the median of the
estimated number of topics for sBIC is in fact the closest to the true value
for all DGPs. For DGP 2 the median of the estimates provided by sBIC is the
actual number of themes. The performance of the metric differs for DGP 1 and
DGP 3. In the first case, sBIC tends to select too many topics and in the
second it chooses too few topics on average. The differences between the true
and the estimated values are relatively small, however both types of
estimation errors have their consequences. Overestimation means that some
artificial topics will be generated, while underestimation implies that a
number of relevant themes will be omitted. These issues are further discussed
in Section 4.2 where the structure and content of the estimated topics is
evaluated.

The performance of the OpTop statistic is rather poor as the procedure has a
strong tendency to select too many topics for each DGP for both cut-off values
of low-frequency words (5% and 20%). In each case, the mean and median values
of the estimates are very close to the maximum of the range of candidates for
the optimal number of topics. Such large overestimation errors mean that a
substantial number of topics that do not belong would be estimated. Since, as
noted by Mimno2011, there is a trade-off between obtaining many refined topics
and meaningful themes, the quality of these additional topics found by the
OpTop method might be expected to be rather low.

The working of the average cosine similarity (Cao_Juan) depends on the DGP.
The mean/median number of topics selected for DGP 1 is too large as compared
with the true number of topics, while the mean/median number of topics
selected for DGPs 2 and 3 is too low as compared to the true number of topics.
This outcome might depend on particular features of the DGPs (e.g. DGP 1
including a relatively small number of longer documents) which could be
subject to further analyses. On the whole, the estimation errors are larger
than for sBIC and smaller than in case of the OpTop procedure.

The unsystematic behaviour in terms of the tendency to over- or underestimate
can be also seen for the average semantic coherence (Mimno). The mean/median
number of topics selected for DGP 1 and DGP 3 are too large as compared with
the true number of topics, while there is severe underestimation problem for
DGP 2. The performance of this procedure seems to be quite unstable as in the
case of DGPs 1 and 3 the estimates have the largest variance as compared to
the remaining methods.

|  |  DGP1 ($K_{true}=38$) |  DGP2 ($K_{true}=12$) |  DGP3 ($K_{true}=70$)
---|---|---|---|---
sBIC | std | 1.23 | 1.35 | 4.09
mean | 40.15 | 12.28 | 65.43
median | 40.00 | 12.00 | 66.00
skewness | -0.24 | 0.00 | -0.04
Cao_Juan | std | 4.54 | 1.68 | 5.36
mean | 53.40 | 9.43 | 63.53
median | 55.00 | 9.00 | 64.00
skewness | -1.16 | -0.28 | 0.12
Mimno | std | 9.23 | 0.20 | 7.55
mean | 47.88 | 2.04 | 79.50
median | 50.00 | 2.00 | 81.00
skewness | -1.89 | 5.55 | -0.61
OpTop 5% | std | 2.30 | 0.59 | 2.06
mean | 55.81 | 31.70 | 88.03
median | 57.00 | 32.00 | 89.00
skewness | -1.22 | -2.17 | -1.28
OpTop 20% | std | 2.39 | 0.67 | 2.61
mean | 55.67 | 31.63 | 87.38
median | 56.00 | 32.00 | 88.00
skewness | -1.37 | -1.78 | -1.30
Table 1: Evaluation of different criteria

###  4.2 Content and Structure of Topics

While the selected number of topics delivers first general insights on the
performance of different criteria, this indicator does not contain information
on the correspondence between the topics used to generate the text corpora and
the topics obtained using the selected number of topics in the estimation
procedure. Therefore, the structure and the content of topics should be also
considered.666In applications, sometimes the quality of topics is analyzed
based on human judgment. For example, Morstatter2018 present an approach based
on existing measures of topic coherence and extending them by a measure of
topic consensus by humans. Although this approach delivers some measure of
interpretability by humans, the authors point out the need for automated and
reproducible measures of topic quality. To this end, we propose to consider
the problem as a classification task. This allows to compare the results
obtained using all the different selection criteria quantitatively making use
of well established performance metrics. We use precision and recall as
commonly used performance metrics. In standard applications, these are defined
as follows:

  * •

Recall describes how many relevant items are retrieved.

  * •

Precision indicates how many retrieved items are relevant.

In standard classification tasks, the length of predicted and actual labels is
the same. In our case it might be different, as the number of topics selected
by each of the considered evaluation metrics can deviate from the true number
of topics as described in the previous subsection. Thus, we define the True
Positive (TP) class as those topics that were correctly identified, i.e., true
topics which find their match in the set of estimated topics for the number of
topics indicated by the given selection criterion. Using this definition,
precision and recall can be defined and calculated as follows:

$\text{Recall}=\frac{|\text{TP}|}{K_{true}},$ (4.1)

where $|\text{TP}|$ denotes the cardinality of the set TP and $K_{true}$ is
the true number of topics in a particular DGP.

$\text{Precision}=\frac{|\text{TP}|}{K_{metric}},$ (4.2)

where $K_{metric}$ is the proposed number of topics according to the selection
criterion considered.

As there might be a trade-off between recall and precision, the F1 measure is
often used as a combined measure. F1 is calculated as follows:

$\text{F1}=2*\frac{\text{Precision}*\text{Recall}}{\text{Precision}+\text{Recall}}.$
(4.3)

For computing these measures, estimated topics have to be matched with true
topics from the DGP. This matching can be done using topic matching technique
proposed by Bystrov2022, the so-called best matching. For each “true” topic, a
match in the set of estimated topics is identified using the cosine similarity
measure. If one of the “true” topics finds several matches, we only consider
the matches with the highest cosine similarities. Obviously, a “best match”
does not have to be a sensible match, i.e., close to the true topic.
Therefore, we apply a threshold for the cosine similarity which has to be
surpassed in order to consider a match as being a sensible match. This
threshold is the same as used for the topic number reduction step for each DGP
described in Subsection 3.2 (see Appendix A for further details).

Table 2 describes the distribution of precision and recall for each DGP and
each evaluation metric.777As a robustness check we also calculate the
described performance metrics using cosine similarities instead of the binary
indicator match/no match. The procedure is described in Appendix C. The
results do not differ qualitatively. As mentioned before, our application
differs from standard classification problems as the number of true and
estimated topics might differ. Hence, the interpretation of the results is
slightly different. Here, a precision value of 1 means that all of the
estimated topics are sensible matches to some of the true topics. However, it
does not imply that all of the true topics are uncovered. Consequently, this
measure might overestimate the performance of a metric if it tends to
underestimate the true number of topics. For DGP 2, for example, the Mimno
metric is described by an average precision value of 1, while the average
recall value is 0.17. In the previous subsection, it was shown that the Mimno
metric tends to underestimate the true number of topics for DGP 2. Thus, the
high precision value only indicates that these few estimated topics are
related to the true topics. sBIC, on the other hand, shows relatively high
values for both recall and precision, 0.93 and 0.92 respectively indicating
that mostly true topics and most of the true topics are found.

As for recall, a value of 1 means that all of the true topics are uncovered by
the estimated topics. However, it does not imply that $K_{metric}=K_{true}$.
Consequently, this measure might lead to overestimation of the performance of
a metric if it tends to select too many topics. For DGP 1, for example, the
Cao_Juan metric reveals an average recall value of 1, while the average
precision value of 0.72 is substantially lower. Also in this example, sBIC
performs well with average recall and precision values of 0.99 and 0.94,
respectively.

To take account of the trade-off described above, it seems appropriate to
consider both evaluation metrics simultaneously. This is done making use of
the F1 score defined in equation 4.3 a the harmonic mean of recall and
precision. The interpretation of F1 is straightforward: the higher the values
the better the joint score for both recall and precision. The results indicate
that sBIC outperforms the other evaluation metrics for DGP 1 and DGP 2. For
DGP 3, according to the F1 score sBIC is found to perform similarly to the
Cao_Juan criterion, while still exhibiting some advantages compared to the
other criteria.

|  | Recall | Precision | F1
---|---|---|---|---
data | metric | mean | std | mean | std | mean | std
DGP1 | Cao_Juan | 1.00 | 0.00 | 0.72 | 0.07 | 0.83 | 0.04
Mimno | 0.96 | 0.12 | 0.78 | 0.09 | 0.85 | 0.06
OpTop 20% | 1.00 | 0.00 | 0.68 | 0.03 | 0.81 | 0.02
OpTop 5% | 1.00 | 0.00 | 0.68 | 0.03 | 0.81 | 0.02
sBIC | 0.99 | 0.01 | 0.94 | 0.03 | 0.97 | 0.01
DGP2 | Cao_Juan | 0.78 | 0.13 | 1.00 | 0.02 | 0.87 | 0.09
Mimno | 0.17 | 0.02 | 1.00 | 0.00 | 0.29 | 0.02
OpTop 20% | 1.00 | 0.00 | 0.38 | 0.01 | 0.55 | 0.01
OpTop 5% | 1.00 | 0.00 | 0.38 | 0.01 | 0.55 | 0.01
sBIC | 0.93 | 0.06 | 0.92 | 0.07 | 0.92 | 0.04
DGP3 | Cao_Juan | 0.87 | 0.05 | 0.96 | 0.03 | 0.91 | 0.02
Mimno | 0.93 | 0.04 | 0.83 | 0.05 | 0.87 | 0.02
OpTop 20% | 0.98 | 0.01 | 0.79 | 0.03 | 0.87 | 0.02
OpTop 5% | 0.98 | 0.01 | 0.78 | 0.02 | 0.87 | 0.02
sBIC | 0.88 | 0.04 | 0.95 | 0.03 | 0.91 | 0.02
Table 2: Descriptive statistics of recall, precision, and F1 scores

##  5 Conclusions and Outlook

Estimating LDA models requires making a number of decisions regarding
parameter settings. This paper considered the problem of selecting the value
of one of those essential parameters, viz. the number of topics discussed in
the text corpus. The main aim was to analyze the properties of various model
selection criteria with special focus on the recently proposed singular
Bayesian information criterion. The performance of the methods was examined
via Monte Carlo experiments using synthetic data generating processes based on
empirical text corpora which differed with respect to the number and length of
documents and the number of topics. The performance of different model
selection procedures was evaluated by not only examining the accuracy of
estimating the actual number of topics but also by analyzing the structure and
contents of the estimated topics.

Simulation results showed that the singular Bayesian information criterion
performed relatively well for all data generating processes considered in the
experiments. It was the best method for estimating the number of topics as it
was associated with the smallest estimation errors as compared to the
competitors. In addition, it resulted in topics with good content and
structure and performed in a relatively stable fashion for all data generating
processes. Across the DGPs, the working of the method based on sBIC was worst
for DGP 3 corresponding to a text corpus with a large number of short
documents and a substantial number of topics. In this setting, sBIC exhibited
a certain downward bias in the selected number of topics which might be taken
into account in applied work. The reasons for this finding and possible
adjustments to the method might be subject to further analyses.

The performance of the methods proposed by Cao2009 and Mimno2011 depended on
the DGP. For each of these methods, the experiments revealed cases of
systematic under- or overestimation of the true number of topics. The
estimation errors were larger than those found for sBIC and had some negative
consequences for the structure and content of the estimated topics. Dependence
on the DGP implies that reliability and stability of these methods cannot be
guaranteed in applied work unless further analyses will explain the relation
between features of a DGP and the model selection results. Despite these
drawbacks, the method of Cao2009 was still overall the second best approach to
LDA model selection in the experiments reported in this paper. It was found
that the method could be particularly useful for modelling collections of many
short texts related to a large range of topics.

The final set of conclusions relates to the OpTop criterion. It was shown that
the method tends to select models with an excessively large number of topics.
The estimation errors were very substantial and led to small precision and F1
metric values used for examining the content and structure of estimated
topics. These results imply that using this criterion in applied work can
result in obtaining some spurious topics, which do not correspond to the data
generating process. It seems that poor estimation properties of the OpTop
procedure could be improved by the introduction of an appropriate penalty for
model complexity (which increases with the number of topics) into the test
statistic formula. This adjustment constitutes a further direction of future
research.

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##  Appendix A Topic Number Reduction

The goal of the topic number reduction step in preparing our DGPs for the
Monte Carlo simulation was to use well separated topics allowing for a robust
comparison of the topics estimated with the underlying DGPs. The process of
topic number reduction comprises the following three steps:

  1. 1.

Starting with the estimated LDA for a given corpus, for each topic the most
similar other topic is identified using the standard matching proposed by
Bystrov2022.

  2. 2.

For deciding whether a pair of topics is “too similar”, i.e., will be excluded
before generating synthetic data within the Monte Carlo simulation, a
threshold value has to be defined. This value is also obtained by a data
driven approach. We calculate all pairwise cosine similarity scores for each
DGP providing $\frac{K^{2}-K}{2}$ typical values. Sorting them in increasing
order provides the distributions shown in Figure 5. Following the approach of
the “elbow” criterion, we set percentile values defining the cut-off value for
each DGP. These values are shown in the figure by the red horizontal line and
correspond to the 95% percentile for DGP2 and to the 99% percentile for DGPs 1
and 3, respectively.

(a) DGP1

(b) DGP2

(c) DGP3

Figure 5: Distribution of the pairwise cosine similarity values.

  3. 3.

All topics belonging to matched topic pairs above the cut-off value are
considered as being too similar and, consequently, are removed from the model
before starting the data generation within the Monte Carlo simulation. Figures
6, 7, and 8 show examples of pairs including redundant topics in each DGP,
which are eliminated by this method.

Figure 6: Similar topics in DGP 1 Figure 7: Similar topics in DGP 2 Figure 8:
Similar topics in DGP 3

##  Appendix B Recall and Precision

Figures 9, 10, and 11 exhibit the scatter plots of recall and precision values
for each DGP separately. Thereby, each point corresponds to one of the
simulated corpora. Consequently, there is a total of 300 points in each plot.
However, the evaluation metrics considered may result in the same recall and
precision scores for multiple corpora. Thus, some points may overlap.

Figure 9: Precision and recall for DGP1 Figure 10: Precision and recall for
DGP2 Figure 11: Precision and recall for DGP3

##  Appendix C Weighted Recall & Precision

As a robustness analysis, we report results for alternative definitions of
recall and precision. We identified a true positive (TP) for the measures in
Section 4.2, when the similarity of matched topics was above a predefined
threshold. Here, we use the actual cosine similarity scores instead, which
would be close to 1 for good matches. Hence, recall and precision values are
calculated as follows:

$\text{Recall}=\frac{\sum_{i=1}^{n}\text{cosine\\_similarity\\_score}_{i}}{K_{true}},$
(C.1)

$\text{Precision}=\frac{\sum_{i=1}^{n}\text{cosine\\_similarity\\_score}_{i}}{K_{metric}},$
(C.2)

where $K_{true}$ is the true number of topics in a particular DGP.
$K_{metric}$ is the proposed number of topics for the evaluation metric
considered. The numerator contains the sum of cosine similarity values of all
the $n$ identified matches. Therefore, recall presents the average cosine
similarity value among the matches relative to the true number of topics.
Precision presents the average cosine similarity value between the matches
relative to the estimated number of topics.

Table 3 summarizes the recall, precision, and F1 score values for this
alternative definitions of recall and precision. As expected, the values are
smaller than the values shown in Table 2 for the original definitions, but the
qualitative findings about the relative performance of the different criteria
remain unchanged. According to the F1 scores, sBIC performs best for DGP 1 and
DGP 2, while the average F1 scores are quite similar for all the considered
metrics in DGP 3, still with a minor advantage for Cao_Juan and sBIC.

|  | Recall | Precision | F1
---|---|---|---|---
data | metric | mean | std | mean | std | mean | std
DGP1 | Cao_Juan | 0.99 | 0.00 | 0.71 | 0.07 | 0.82 | 0.04
Mimno | 0.95 | 0.13 | 0.76 | 0.08 | 0.83 | 0.07
OpTop 20% | 0.98 | 0.00 | 0.67 | 0.03 | 0.80 | 0.02
OpTop 5% | 0.98 | 0.00 | 0.67 | 0.03 | 0.80 | 0.02
sBIC | 0.99 | 0.02 | 0.93 | 0.03 | 0.96 | 0.02
DGP2 | Cao_Juan | 0.76 | 0.14 | 0.96 | 0.03 | 0.84 | 0.10
Mimno | 0.11 | 0.02 | 0.66 | 0.02 | 0.19 | 0.02
OpTop 20% | 1.00 | 0.00 | 0.38 | 0.01 | 0.55 | 0.01
OpTop 5% | 1.00 | 0.00 | 0.38 | 0.01 | 0.55 | 0.01
sBIC | 0.92 | 0.07 | 0.91 | 0.06 | 0.91 | 0.05
DGP3 | Cao_Juan | 0.85 | 0.06 | 0.94 | 0.02 | 0.89 | 0.03
Mimno | 0.92 | 0.04 | 0.81 | 0.05 | 0.86 | 0.02
OpTop 20% | 0.98 | 0.02 | 0.78 | 0.03 | 0.87 | 0.02
OpTop 5% | 0.98 | 0.02 | 0.78 | 0.02 | 0.87 | 0.02
sBIC | 0.86 | 0.05 | 0.92 | 0.03 | 0.89 | 0.03
Table 3: Descriptive statistics of recall, precision, and F1 scores based on
cosine similarity

While recall and precision values of our standard implementation are discrete
leading to clustering of points in the scatter plots shown in Appendix B, the
weighted recall and precision values reported in this section are continuous
and each point is actually unique due to the differences in the cosine values,
although these might be minor. Therefore, we do not use the type of plots from
Appendix B taking into account the clustering, but standard scatter plots in
Figures 12, 13, and 14.

Figure 12: Precision and recall based on cosine similarity for DGP1 Figure 13:
Precision and recall based on cosine similarity for DGP2 Figure 14: Precision
and recall based on cosine similarity for DGP3


$\displaystyle=\frac{s}{\delta(x)c}|sc-t\delta(x)|\leq\frac{2}{c}|s\delta(y)-t\delta(x)|\leq\frac{2\delta(y)}{c}|s-t|+\frac{2t}{c}|\delta(y)-\delta(x)|$
$\displaystyle\leq 2|s-t|+4|\delta(y)-\delta(x)|.$

Overall, we have:

$\textstyle
s|\lambda_{A}(x)-\lambda_{A}(y)|\leq\sup|q^{\prime}|\,(2|s-t|+4\sup_{\overline{U}}|\delta^{\prime}|\,|x-y|).$

Now, since $1/({\epsilon\circ\pi}_{A})$ is smooth and since $\rho_{A}^{1/2}$
is Lipschitz continuous as in the proof of Corollary 4.9, we have:

$|s-t|=|l_{A}(x)-l_{A}(y)|\leq C_{l_{A}}|x-y|$

for some constant $C_{l_{A}}$. Hence, for some constant $C_{\lambda_{A}}$,

$|\lambda_{A}(x)s-\lambda_{A}(y)t|\leq C_{\lambda_{A}}|x-y|.$

Next, define $\varphi_{t}:l_{A}^{-1}([0,1])\to l_{A}^{-1}([0,1]),\,0\leq t\leq
1$ by $\varphi_{t}(H_{A}(x_{0},s))=H_{A}(x_{0},ts)$. Fix a point $a_{0}\in A$.
Then Remark 4.10 gives $l_{A}^{-1}([0,1])\simeq A\times CL$ near $a_{0}$.
Choosing coordinates on $L$, we have
$L\subset\textrm{$l$-sphere}\subset\mathbb{R}^{l+1}$. Thus, again near
$a_{0}$, we have $l_{A}^{-1}([0,1])\subset A\times\mathbb{R}^{l+1}$ where the
inclusion preserves the strata and is smooth on each stratum. Then we see
$\varphi_{t}$ can be given as $\varphi_{t}(a,u)=(a,tu)$ near $a_{0}$; in
particular, $\varphi_{t}$ is Lipchitz continous near $a_{0}$ uniformly with
respect to $t$.

Having the above estimates, it is now clear that $\widetilde{\pi}_{A}$ is
locally Lipchitz continuous (note
$\widetilde{\pi}_{A}(x)=\varphi_{(1-\lambda_{A}(x))l_{A}(x)}(x)$.) ∎

By definition, the _height_ of a stratum $A$ is the supremum of the lengths
$k$ of the chains $A=A_{k}>A_{k-1}>\cdots>A_{0}$, which is finite since $A>B$
implies $B$ has strictly smaller dimension than $A$ ([Mat12, Proposition
2.7.]).

For each integer $i$, define $\widetilde{\pi}_{i}:M\to M$ by
$\widetilde{\pi}_{i}|_{A(2\delta)}=\widetilde{\pi}_{A}|_{A(2\delta)}$ for the
strata $A$ of height $i$ and $\widetilde{\pi}_{i}=$ the identity elsewhere
(this is possible since our tubular neighborhoods of incomparable strata are
disjoint).

If $i$ is the height of $A$, let
$\delta_{i}=\delta\circ\widetilde{\pi}_{i+1}\circ\cdots\circ\widetilde{\pi}_{d}$.
Then

$A(\delta_{i})=(\widetilde{\pi}_{d}^{-1}\circ\cdots\circ\widetilde{\pi}_{i+1}^{-1})(A(\delta))$

since $l_{A}\circ\widetilde{\pi}_{B}=l_{A}$ for $A<B$.

Let $W(\delta)$ be the union of $A(\delta_{\operatorname{ht}A})$ over all the
strata $A$ of $W$. Then let

$\widetilde{\pi}=\widetilde{\pi}_{0}\circ\cdots\circ\widetilde{\pi}_{d}$

and $\pi=\widetilde{\pi}|_{W(\delta)}$.

Now, we have $\pi(W(\delta))\subset W$. More explicitly, we have that $\pi$ is
given by $\pi_{A}$’s in the following way. For $A<B$, we have
$\pi_{A}\circ\widetilde{\pi}_{B}=\pi_{A}$. Hence, if $i$ is the height of $A$
and $x$ in $A(\delta_{i})$, then we have
$(\widetilde{\pi}_{i+1}\circ\cdots\circ\widetilde{\pi}_{d})(x)$ is in
$A(\delta)$ and so

$(\widetilde{\pi}_{i}\circ\cdots\circ\widetilde{\pi}_{d})(x)=(\pi_{A}\circ\widetilde{\pi}_{i+1}\circ\cdots\circ\widetilde{\pi}_{d})(x)=\cdots=\pi_{A}(x).$

Since $\widetilde{\pi}_{j}|_{W}$ is the identity for any $j$, we have that
$\widetilde{\pi}$ (thus $\pi$) coincides with $\pi_{A}$ on $A(\delta_{i})$.

######  4.14 Lemma.

In the definition of $\widetilde{\pi}$, the function $\delta$ can be chosen so
that the interior of $W(\delta)$ is a neighborhood of $W$.

###### Proof.

For a stratum $A$ of height $i$, let
$A^{\prime}(\delta_{i})=A(\delta_{i})-\bigcup_{B>A}B(\delta_{\operatorname{ht}(B)})$.
Then $W(\delta)$ is the disjoint union of
$A^{\prime}(\delta_{\operatorname{ht}(A)})$ over all $A$. If $x$ is in
$A^{\prime}(\delta_{i})-A$, then $\delta_{i}>0$ near $x$. Thus, if we assume
$\delta_{i}$ decreases slowly compared to $l_{A}$, then
$A^{\prime}(\delta_{i})$ contains a point near $A$ unless the point is already
in $B(\delta_{\operatorname{ht}(B)})$ for some $B>A$. That is, $W(\delta)$
contains a neighborhood of $A$. Since the stratification on $W$ is locally
finite, such an assumption on $\delta$ need to be made again only finitely
strata at a time. The assertion is therefore clear. ∎

######  4.15 Proposition.

For some neighborhood $U$ of $W$ (namely the interior of $W(\delta)$),
$\pi:U\to W$ is a strong deformation retract; i.e., there is a homotopy
$h_{t}:U\to U$ from $\pi$ to the identity such that $h_{t}|_{W}$ is thr
identity. Moreover, $h_{t}$ can be taken to be smooth on $U-W$.

###### Proof.

For a stratum $A$ of $W$, if $x=H_{A}(x_{0},s)$ and $\delta(x)>0$, then let

$\widetilde{\pi}_{A,t}(x)=H_{A}(x_{0},(1-(1-t)\lambda_{A}(x))s)$

and $\widetilde{\pi}_{A,t}(x)=x$ for any other $x$. Note
$\widetilde{\pi}_{A,t}(U)\subset U$. Then define $\widetilde{\pi}_{t}$ as we
defined $\widetilde{\pi}$ from $\widetilde{\pi}_{A}$. Take $h_{t}:U\to U$ to
be the restriction of $\widetilde{\pi}_{t}$. ∎

An important consequence that we will need in the next section is:

######  4.16 Corollary (relative Poincaré lemma).

Let $\sigma$ be a continuous $k$-form252525By a _continuous $k$-form_, we mean
a continuous section of the $k$-th exterior power of $T^{*}U$. If it is
smooth, then it is a differential $k$-form., $k>0$ on some neighborhood $U$ of
$W$ such that $\sigma$ is smooth and closed on $U-W$. If $\sigma|_{W}=0$,
then, after shrinking $U$, there exists a continuous $(k-1)$-form $\psi$ on
$U$ such that $\psi$ is smooth on $U-W$ with $d\psi=\sigma$ and $\psi|_{W}=0$.

###### Proof.

Assume $U$ is small enough that $h_{t}:U\to U$ for $h_{t}$ in the preceding
proposition. Let $p:U\times[0,1]\to U$ denote the projection and $p_{*}$
denote the integration along fibers; i.e., in the local coordinates
$x_{1},\dots,x_{n},t$,

$p_{*}(f\,dt\wedge dx_{i_{1}}\wedge\cdots\wedge
dx_{i_{k-1}})=\left(\int_{0}^{1}f(\cdot,t)\,dt\right)dx_{i_{1}}\wedge\cdots\wedge
dx_{i_{k-1}}$

and $p_{*}(\alpha)=0$ for a form $\alpha$ not involving $dt$.

Then the above local formula implies that we have: on $U-W$,

$h_{1}^{*}\sigma-h_{0}^{*}\sigma=Jd\sigma+dJ\sigma$

where $J=p_{*}\circ h^{*}$ for $h:U\times[0,1]\to U,\,(x,t)\mapsto h_{t}(x)$.
Let $\psi=J\sigma$, which is defined on $U$ since $J$ does not involve
differentiation. Here, $h_{1}^{*}\sigma=\sigma$ since $h_{1}$ is the identity
and $d\sigma=0$ by assumption. On the other hand, since $h_{0}|_{U}=\pi$ and
since $\sigma|_{W}=0$, we have $h_{0}^{*}\sigma=0$ on $U$. Also, we have
$\psi|_{W}=0$ since $h_{t}(W)\subset W$. ∎

###### Proof of Theorem 4.1.

There is a small $0\neq t\in\mathbb{C}$ such that $p^{-1}(t)\subset$ the
interior of $W(\delta)$. Indeed, if this is not the case, we can choose a
sequence $x_{i}\not\in$ the interior of $W(\delta)$ such that $p(x_{i})\to 0$.
We have $x_{i}$ is in the pre-image under $p$ of some closed ball, which is
compact. Then, passing to a subsequence, we can assume $x_{i}$ is convergent
with limit in $W=p^{-1}(0)$. But then $x_{i}\in$ the interior of $W(\delta)$
for some large $i$, a contradiction.

Choose Whitney stratifications on $\mathfrak{X}-W$ and $\mathbb{C}$ so that
the origin is the only zero-dimensional stratum on $\mathbb{C}$ and $p$ maps
each stratum submersively to a stratum on $\mathbb{C}$; this is possible by
[GM88, Part 1, Theorem 1.7].262626In fact, since $\mathfrak{X}-W$ is a complex
variety, we can assume that the stratifications on $\mathfrak{X}-W$ and
$\mathbb{C}$ consist of smooth complex subvarieties. But this fact is not
needed for the proof. Together with the stratification on $W$, we then have
that $\mathfrak{X}$ is Whitney-stratified in such a way that $p$ maps each
stratum of $\mathfrak{X}$ submersively to a stratum.

Fix a stratum $A$ of $W$ of height $i$. Note $A(\delta_{i})\cap\mathfrak{X}-A$
has a Whitney stratification induced from $\mathfrak{X}$, as in the proof of
Proposition 4.8. Let $p_{A}=p|_{A(\delta_{i})\cap\mathfrak{X}-A}$. Then
$p_{A}$ maps each stratum submersively to a stratum. Thus, by Lemma 4.7, there
is a neighborhood $U$ of $t$ such that:

---
$\textstyle{p_{A}^{-1}(t)\times
U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{A}\times\operatorname{id}}$$\scriptstyle{\sim}$$\textstyle{p_{A}^{-1}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\pi_{A},p_{A})}$$\textstyle{A\times
U.}$

This implies that $\pi_{A}$ (thus $\pi$) is a submersion on each stratum of
$p_{A}^{-1}(t)$ to $A$ (note, as in the proof of Lemma 4.7, $(\pi_{A},p_{A})$
is a submersion on each stratum). We have a system of tubular neighborhoods of
the strata of $\mathfrak{X}$ that contains such a system for $W$ that we used
above. Then this system restricts to $p^{-1}(t)$. Hence, by Thom’s first
isotopy lemma (to be precise, its abstract version [Mat12, Corollary 10.2]),
$\pi|_{p^{-1}(t)}$ is a locally trivial fibration over each stratum; thus is a
finite covering over each stratum (as the fibers are finite sets). The theorem
is then the consequence of the following comparison theorem in étale topology.
∎

######  4.17 Theorem (SGA 4, Exposé XI, Théorèm 4.3. (iii)).

Let $X$ be a variety over $\mathbb{C}$.272727The theorem is often stated for a
smooth complex variety but smoothness is actually not needed. Then the functor
$X\mapsto X(\mathbb{C})$ determines an equivalence between the category of
finite étale coverings of $X$ and the category of finite (topological)
coverings of $X(\mathbb{C})$ in the classical topology.

Moreover, scheme structures on coverings are induced naturally.282828i.e., in
a manner analogous to, for example, manifold structures are induced naturally
on coverings of manifolds. In particular, if a covering a priori had a
manifold structure, then an induced smooth variety structure would coincide
with it.

###### Proof.

The moreover part is contained in the proof. ∎

##  5\. Moment maps and integrable systems

The notion of a moment map appears in the context of Hamiltonian group actions
(Hamiltonian dynamics) in symplectic geometry. In particular, there is a
moment map for a toric variety embedded in some projective space over the
complex numbers and, by the convexity theorem, the image is a convex polytope,
the same polytope $\Delta(1)$ associated to the toric variety (Definition
1.1), up to sign.

In fact, we can and do define a moment map over an algebraically closed field
of characteristic zero and we will see that the convexity result remains
valid. Over such a base field, composing the moment map with a map to a toric
variety then gives a map from a variety to the polytope $\Delta(1)$. Over
$\mathbb{C}$ at least, the components of the moment map constitute an integral
system. As in [HK15], we then see that the pull-back of them to gives an
integral system on the original variety.

###  5.1. Moment map

Recall the setup preceding Theorem 1.3; i.e. $\mathbb{G}_{m}^{r}$ is acting on
$\mathbb{P}^{q}$ linearly so that the homogeneous coordinates have weights
$a_{0},\dots,a_{q}$ in $\mathbb{Z}^{r}$. First assume the base field
$k=\mathbb{C}$ is the field of complex numbers. Then the real $r$-torus
$(S^{1})^{r}$ also acts on $\mathbb{P}^{q}$ in a way it preserves the
Hermitian structure (i.e. it act as unitary operators).

Let $\mathfrak{t}_{r}=\operatorname{Lie}((S^{1})^{r})$ be the Lie algebra of
the real $r$-torus. Then there is the moment map [Woo10, Example 3.4.2.]:

$\mu:\mathbb{P}^{q}\to\mathfrak{t}_{r}^{*}$

given by

$\langle\mu(x),\xi\rangle=\frac{\operatorname{Im}(\widetilde{x},\xi\cdot\widetilde{x})}{|\widetilde{x}|^{2}}$

where $\widetilde{x}$ is a representative in $\mathbb{C}^{q+1}$ of the point
$x$ and $(\ ,\ )$ is the standard inner product. Explicitly,

$\mu([z_{0}:\cdots:z_{q}])=\frac{1}{|z|^{2}}\sum_{j=0}^{q}a_{j}|z_{j}|^{2}$

since $\mathbb{G}_{m}^{r}$ acts on $\mathbb{C}^{q+1}$ with weights
$-a_{0},\dots,-a_{q}$. The explicit formula can be used to define a moment map
for a base field other than $\mathbb{C}$ if it still has characteristic zero.
Thus, assume the base field $k$ is an algebraically closed field of
characteristic zero. Then we can find a real closed field $F$ such that
$k=F(i),i^{2}=-1$ (see for example [Nag93] §5.1. Exercise 7.) Thus, as a
$F$-vector space, $k$ is $F^{2}$ respect to the basis $1,i$. Then we can
define the Euclidean norm $|z|$ of $z\in k$ by $|z|=\sqrt{x^{2}+y^{2}}$ when
$z=x+iy$. We then define the moment map

$\mu:\mathbb{P}^{q}\to\mathfrak{t}_{r}^{*}$

by the above formula, where we used the notation $\mathfrak{t}_{r}=F^{r}$ and
viewed each $a_{j}$ as a linear functional on $\mathfrak{t}_{r}$. Since
$F^{r}$ is an inner product space, $\mathfrak{t}_{r}^{*}$ is canonically
identified with $\mathfrak{t}_{r}$.

For an $(S^{1})^{r}$-submanifold of a complex projective space, the next lemma
is a part of the Kempf–Ness theorem, the theorem stating that a GIT quotient
can be viewed as a symplectic quotient. The original proof uses differential
geometry.

We recall that a point $x$ in $\mathbb{P}^{q}$ is said to be semistable if
either of the following equivalent conditions is met:

  * •

For a nonzero vector $\widetilde{x}\in k^{q+1}$ representing $x$, the orbit
closure $\mathbb{G}_{m}^{r}\cdot\widetilde{x}$ does not contain the origin
$0$.

  * •

There exists a nonconstant $\mathbb{G}_{m}^{r}$-invariant homogeneous
polynomial in homogeneous coordinates on $\mathbb{P}^{q}$ that does not vanish
at $x$.

The moment map gives another characterization.

######  5.1 Lemma (cf. [MFK94] Theorem 8.3.).

A point $x$ in $\mathbb{P}^{q}$ is a semistable point if and only if
$\mu^{-1}(0)$ intersects the orbit closure $\overline{\mathbb{G}_{m}^{r}\cdot
x}$.

###### Proof.

First suppose $\mu^{-1}(0)$ intersects $\overline{\mathbb{G}_{m}^{r}\cdot x}$;
i.e., there is a point $y$ in $\overline{\mathbb{G}_{m}^{r}\cdot x}$ such that
$\mu(y)=0$. We shall show $y$ is semistable. Suppose otherwise. Let $v$ be a
nonzero vector in $k^{q+1}$ that is a representative of $y$. By the
Hilbert–Mumford criterion, we can find a homomorphism
$\rho:\mathbb{G}_{m}\to\mathbb{G}_{m}^{r}$ such that $0$ is in the
(Zariski-)closure of $\rho(\mathbb{G}_{m})v$. Replacing $\rho$ by $\rho^{-1}$
if needed, we can assume $\rho(t)v\to 0$ as $t\to\infty$, in the sense of
specialization. Since $\mathbb{G}_{m}^{r}$ acts diagonally, the action of
$\mathbb{G}_{m}$ through $\rho$ is also diagonal:

$\rho(t)v=\operatorname{diag}(t^{-b_{0}},\dots,t^{-b_{q}})v$

for some integers $b_{i}$. Then, for each $i$, $t^{-b_{i}}v_{i}\to 0$ as
$t\to\infty$. This means $v_{i}=0$ if $b_{i}\leq 0$. On the other hand, the
moment map corresponding to the $\mathbb{G}_{m}$-action through $\rho$ is
$\rho^{*}\circ\mu$ where $\rho^{*}$ is the transpose of $\rho$ and then
$(\rho^{*}\circ\mu)(y)=|v|^{-2}\sum b_{j}|v_{j}|^{2}=0$. This is possible only
if $v_{j}=0$ for all $j$ such that $b_{j}>0$. Thus, $v=0$, a contradiction.
Hence, $y$ is semistable, proving the claim. Now, if $x$ is not semistable,
then $\overline{\mathbb{G}_{m}^{r}\cdot x}$ lies in the unstable locus
$X^{us}$ which is the complement of the semistable locus and thus $y$ is also
in $X^{us}$, a contradiction. Hence, $x$ is semistable.

Next, suppose $x$ is semistable, $y$ a point in
$\overline{\mathbb{G}_{m}^{r}\cdot x}$ such that $\mathbb{G}_{m}^{r}\cdot y$
is closed ($y$ is usually called a polystable point). Let $v$ be a nonzero
vector in $k^{q+1}$ representing $y$ and define the function $\psi_{v}$ on
$(k^{*})^{r}$ by $\psi_{v}(t)=|t\cdot v|^{2}.$ Since
$0\not\in\mathbb{G}_{m}^{r}\cdot v$, we see that $\psi_{v}$ is bounded from
below by a nonzero number. Since $\mathbb{G}_{m}^{r}\cdot v$ is closed,
$\psi_{v}$ admits a minimum $t$.292929Although $F$ may not be Cauchy-complete,
a semialgebraic continuious function on a bounded closed semialgebraic set
(over $F$) admits a minimum and a maximum. Replacing $v$ and $y$ by $t\cdot v$
and $t\cdot y$, without loss of generality, we shall assume $t=1$, the
identity element. We shall show $\mu(y)=0$; for that, we shall show
$(\rho^{*}\circ\mu)(y)=0$ for every one-parameter subgroup
$\rho:\mathbb{G}_{m}\to\mathbb{G}_{m}^{r}$. Recall that $k=F(i)$ and then
define the function $f:F_{>0}\to F$ by
$f(t)=\psi_{v}(\rho(t))=\sum_{j=0}^{q}t^{-2b_{j}}|v_{j}|^{2}$ where $b_{i}$
are as above. Then we have:

$\frac{d}{dt}|_{t=1}f(t)=-2\sum_{j=0}^{q}b_{j}|v_{j}|^{2}=-2|v|^{2}(\rho^{*}\circ\mu)(y).$

Since $f$ has a minimum at $t=1$, the above has to vanish:303030It is known
and is easy to see that the mean value theorem holds over a real closed field.
$(\rho^{*}\circ\mu)(y)=0$, proving the claim. ∎

######  5.2 Proposition.

Let $W\subset\mathbb{P}^{q}$ be a possibly non-normal toric variety as in the
setup preceding Theorem 1.3. Then we have
$\mu(W)\cap\mathbb{Q}^{r}=\Delta_{\mathbb{Q}}(S)$, $S$ the semigroup of $W$.
In particular, $\mu(W)$ is convex and coincides with $\Delta(1)$.

###### Proof.

Let $R$ be the homogeneous coordinate ring of $W$; i.e., the semigroup algebra
of $S$. For each point $\lambda\in\mathbb{Q}^{r}$, we have:

$\lambda\in\Delta_{\mathbb{Q}}(S)\Leftrightarrow(n,n\lambda)\in S\text{ for
some $n\in\mathbb{N}_{>0}$}\Leftrightarrow
R^{\lambda}_{+}:=\bigoplus_{n=1}^{\infty}R_{n,n\lambda}\neq 0.$

Thus, we need to show $R^{\lambda}_{+}\neq 0\Leftrightarrow\mu^{-1}(-\lambda)$
intersects $W$. First suppose $\lambda=0$. If $R^{\lambda}_{+}\neq 0$, then
there exists a nonconstant $\mathbb{G}_{m}^{r}$-invariant homogeneous
polynomial $f$ that does not vanish identically on $W$. In particular, $W$
admits a semistable point $x$. Then, by Lemma 5.1, $\mu^{-1}(0)$ intersects
the orbit closure $\overline{\mathbb{G}_{m}^{r}\cdot x}$. Since
$\overline{\mathbb{G}_{m}^{r}\cdot x}\subset W$, that means $\mu^{-1}(0)$
intersects $W$. Conversely, if there is a point $x$ in $\mu^{-1}(0)\cap W$,
then $\mu^{-1}(0)$ intersects the orbit at $x$, a fortiori, the orbit closure
at $x$; so, again, by Lemma 5.1, $x$ is semistable. This proves the case when
$\lambda=0$.

Next, assume $\lambda$ is in $\mathbb{Z}^{r}$ and consider the
$\mathbb{G}_{m}$-action on $R$ given by $t\cdot_{\lambda}f=t^{-\lambda}t\cdot
f$ for degree-one elements $f$ in $R$. Then $R^{\lambda}_{+}$ is the
$\cdot_{\lambda}$-invariant subspace of $R_{+}$. With respect to
$\cdot_{\lambda}$, the moment map looks $\mu_{\lambda}=\mu+\lambda$. It
follows that $\mu_{\lambda}^{-1}(0)=\mu^{-1}(-\lambda)$ and so the claim
follows from the early case. Finally, since the equation we need to show is
invariant under scaling, we are done. ∎

######  5.3 Remark (on convexity).

For manifolds, the above Proposition 5.2 in a more general setup is due to
Atiyah and Guillemin–Sternberg in the abelian case and due to Kirwan in the
non-abelian case ([Woo10, Theorem 8.1.5]). That $\mu(W)$ is convex is a
theorem of Atiyah ([Woo10, Theorem 8.2.1]) whose method of the proof is more
differential-geometric.

######  5.4 Example.

Consider the elliptic curve example. In that case, the generators of the
semigroup $S$ are
$\widehat{\nu}(x)=(1,1),\,\widehat{\nu}(y)=(1,0),\,\widehat{\nu}(z)=(1,3)$ and
thus we have:

$\mathbb{C}^{*}\to\mathbb{P}^{2},\,t\mapsto[t:1:t^{3}].$

The closure of the image is
$W=\operatorname{Proj}(\mathbb{C}[S])=V(y^{2}z=x^{3})$. The weight of the
$S^{1}$-action is $(1,0,3)\in\mathbb{Z}^{3}$ and so

$\mu([z_{0}:z_{1}:z_{2}])=\frac{1}{|z|^{2}}(|z_{0}|^{2}+3|z_{2}|^{2}).$

Hence, the moment image of $W$ is $[0,3]$. Also, we have
$\varphi:[x:y:z]\mapsto[y^{2}z:y^{3}:z^{3}]$ and so

$(\mu\circ\varphi)([x:y:z])=\frac{1}{|y^{2}z|^{2}+|y^{3}|^{2}+|z^{3}|^{2}}(|y^{2}z|^{2}+3|z^{3}|^{2}).$

The image is again the interval $[0,3]$ as expected.

This ends a quick discussion of a moment map on a toric variety. Finally,
composing $\varphi:X-B\to W$ with $\mu$ we get:

$\varphi_{\mu}:X-B\to\mathfrak{t}_{r}^{*}$

whose image is contained in the polytope $\Delta(1)$ associated to
$W\subset\mathbb{P}^{q}$. Moreover, if $W$ is the limit of a toric
degeneration of $X$, then the image is dense in the polytope for volume
reason.

###  5.2. Construction of integrable systems

We now want to recover the construction of an integrable system after Harada
and Kaveh [HK15] in our setup. We shall assume the base field $k=\mathbb{C}$
is the complex field.

We first recall some sympletic-geometry notions. Let $(M,\omega)$ be a
symplectic manifold. To each smooth function $f$ on $M$, by non-degeneracy, we
can have a unique vector field $X_{f}$ such that $i(X_{f})\omega=df$, where
$i(X_{f})$ denotes the interior product; i.e.,
$i(X_{f})\omega=\omega(X_{f},\cdot)$. We can then form a pair ping of two
smooth functions:

$\\{f,g\\}=\omega(X_{f},X_{g})$

called the _Poisson bracket of $f,g$_ ([DS08, Definition 18.5.]). A
calculation in local coordinates shows that it is a Lie bracket that also
satisfies the Leibniz rule. We then say smooth functions
$f_{1},\dots,f_{r},r=\frac{1}{2}\dim M$ form a _(completely) integrable system
on $M$_ if (1) $df_{1},\dots,df_{r}$ are linearly independent at each point
and (2) they pairwise Poisson-commute; i.e., $\\{f_{i},f_{j}\\}=0$ for any
$i,j$ ([DS08, Definition 18.10.]).

We have an extension of this notion to a stratified space:

######  5.5 Definition (integrable system in the stratified sense; cf.
Definition 1.1. in [HK15]).

Let $M$ be a manifold and $X\subset M$ a closed subset stratified by
submanifolds of $M$. A sympletic form $\omega$ on $X$ in the stratified sense
is a continuous 2-form on $M$313131By a continuous $k$-form we mean a
continuous (as opposed to smooth) section of the $k$-th exterior power of the
cotangent bundle. such that its restriction to each stratum $A$ of $X$ is
symplectic (smooth, closed and nondegenerate).

Then we say real-valued continuous functions $f_{1},\dots,f_{r}$ on $X$ form
an integrable system in the stratified sense with respect to $\omega$ if (1)
they are smooth functions on each stratum and (2) on each stratum $A$, they
pairwise Poisson-commute with respect to $\omega|_{A}$ and the span of their
differentials at each point has half the dimension of $A$.

######  5.6 Example.

Let $\mu:W\to\mathfrak{t}_{r}^{*}$ be the moment map as above. Then the
components of $\mu$

$(f_{1},\dots,f_{r}):W\overset{\mu}{\to}\mathfrak{t}^{*}_{r}\simeq\mathbb{R}^{r}$

form an integrable system on $W$ in the stratified sense, when $W$ is given
the stratification by torus orbits.

In §1 and §2, we studied morphisms to toric varieties. They can be used to
obtain an integrable system in the stratified sense:

######  5.7 Proposition.

Let $X$ be a complex algebraic variety of dimension $r$. Also, let
$W\subset\mathbb{P}^{n}$ be a possibly non-normal projective complex toric
variety of dimension $r$ (in the sense in §1) and assume that we have some
continuous map $\varphi:X\to W$ that has dense image in the classical
topology. Let $\omega_{\mathbb{P}^{n}}$ be the standard symplectic (i.e.,
Kähler) form on $\mathbb{P}^{n}$.

Assume that, for each orbit $A$ on $W$, $\varphi^{-1}(A)$ has a natural
reducible-variety structure so that $\varphi:\varphi^{-1}(A)\to A$ is finite
étale.

Then there exists an integrable system $f_{1},\dots,f_{r}$ on $X$ in the
stratified sense with respect to $\varphi^{*}(\omega_{\mathbb{P}^{n}}|_{W})$;
namely, $f_{1},\dots,f_{r}$ are given as:

$(f_{1},\dots,f_{r}):X\overset{\varphi}{\to}W\overset{\mu}{\to}\mathfrak{t}^{*}_{r}\simeq\mathbb{R}^{r}.$

###### Proof.

We only need to check that the $\varphi^{*}f_{i}$’s Possion-commute with
respect to the restriction of $\omega^{\prime}=\varphi^{*}\omega$ to a stratum
of $X$. But for any smooth functions $f,g$ on an orbit $A$, we have:

$\varphi^{*}{\\{f,g\\}}=\\{\varphi^{*}f,\varphi^{*}g\\}$

i.e., $\varphi^{*}$ is a homomorphism of Poisson algebras. Hence, the
assertion is clear. ∎

If $X$ is given a sympletic form $\omega_{X}$ on some open dense subset
$X_{0}$ (e.g., if $X\subset\mathbb{P}^{n}$, then the standard Kähler form of
$\mathbb{P}^{n}$ restricts to the regular locus of $X$), then, _a priori_ ,
$\omega_{X}$ need not coincide with the pull-back of $\omega_{W}$ under some
$\varphi:X_{0}\to W_{0}$ for some sympletic form $\omega_{W}$ on some open
dense subset $W_{0}$ of $W$.

In some situations (e.g., toric degenerations), there is a simple way to fix
this issue using the following generalization of a theorem of Weinstein
([Vai87, Theorem 3.4.17.], [DS08, Theorem 7.4.]).

######  5.8 Theorem (Weinstein).

Let $M,W$ be as in the previous section §4. Let $\omega_{i},i=0,1$ be two
continuous323232By a _continuous $k$-form_, we mean a continuous section of
the $k$-th exterior power of $T^{*}U$. If it is smooth, then it is a
differential $k$-form. $2$-forms on some neighborhood $U$ of $W$ such that
$\omega_{0}|_{W}=\omega_{1}|_{W}$. Assume $\omega_{0}$ is nondegenerate (so
$\omega_{0}$ is a sympletic form if smooth).

If $\omega_{i},i=0,1$ are locally Lipschitz continuous (not just continuous;
see Lemma 4.13 for the definition of local Lipschitz continuity) and they are
smooth closed forms on $U-W$, then, after we shrink $U$, there exists a
homeomorphism $f:U\to U$ such that $f^{*}{\omega_{1}}=\omega_{0}$ on $U$,
$f:U-W\to U-W$ is a diffeomorphism and $f:W\to W$ is the identity.

###### Proof.

The original proof (that uses Moser’s trick) goes through without a change,
because the relative Poincaré lemma (Corollary 4.16) holds for $W\subset M$
and the local Lipschitz continuity ensures we can solve a differential
equation. But, to be sure, we shall repeat the proof (specifically the proof
of [Vai87, Theorem 3.4.17.]).

Let $\sigma=\omega_{1}-\omega_{0}$ and $\omega_{t}=\omega_{0}+t\sigma$. For
some neighborhood $U$ of $W$, by Corollary 4.16 (relative Poincaré lemma), we
can find a continuous one-form $\psi$ on $U$ such that $\psi$ is smooth with
$d\psi=\sigma$ on $U-W$ and $\psi|_{W}=0$. Also, by the local formula in the
proof of Corollary 4.16, we see that $\psi$ is locally Lipschitz since
$\sigma$ is. Since $\omega_{t}|_{W}=\omega_{0}|_{W}$ is non-degenerate,
shrinking $U$, we can have that $\omega_{t},0\leq t<1$ are all non-degenerate.
The non-degeneracy means we can find unique continuous vector fields $\xi_{t}$
on $U$ such that $i(\xi_{t})\omega_{t}=-\psi$, where $i(v)$ denotes the
interior product; i.e., $i(v)\omega_{t}=\omega_{t}(v,\cdot)$. Note
$\xi_{t}|_{W}=0$ since $\psi|_{W}=0$. Also, $\xi_{t}$ is locally Lipschitz
uniformly with respect to $t$ since $\omega_{t}$ and $\psi$ are locally
Lipschitz.

The existence and uniqueness theorem for ordinary differential equations
applies to a vector field that is locally Lipschitz uniformly with respect to
time ([Tes12, Theorem 2.2.]). Thus, we get the unique solution $f_{t}:U\to U$
to the ODE with the initial condition:

$\frac{d}{dt}f_{t}(x)=\xi_{t}(f_{t}(x)),\,f_{0}(x)=x.$

We know that $f_{t}$ is defined for $t$ in some interval around $0$ and is a
homeomorphism (as $f_{-t}$ is the inverse). Let $\Omega\subset\mathbb{R}\times
U$ be the subset consisting of $(t,x)$ such that $x\in U$ and $t$ in the
maximal interval for which the solution $f_{t}(x)$ exists with the initial
condition $f_{0}(x)=x$. If $\xi_{t}$ is independent of $t$, then we know
$\Omega$ is open (e.g., [Tes12, Theorem 6.1.]). But an argument similar to the
one showing the openness in that case still carries over to show that $\Omega$
here is also open.

Since $\xi_{t}|_{W}=0$, we have that $\mathbb{R}\times W\subset\Omega$. Thus,
shrinking $U$, we can assume $[-1,1]\times U\subset\Omega$; that is,
$f_{t}:U\to U$ is defined for all $-1\leq t\leq 1$.

For any smooth form $\alpha$ on $U-W$, we have:

$\frac{d}{dt}f_{t}^{*}\alpha=f_{t}^{*}((i(\xi_{t})d+di(\xi_{t}))\alpha).$

(it comes from a Lie derivative; see [DS08, Exercise after Definition 6.3.]).
Then we have:

$\displaystyle\frac{d}{dt}f_{t}^{*}\omega_{t}$
$\displaystyle=\left(\frac{d}{ds}f_{s}^{*}\omega_{t}\right)|_{s=t}+\left(\frac{d}{ds}f_{t}^{*}\omega_{s}\right)|_{s=t}=f_{t}^{*}(di(\xi_{t})\omega_{t})+f_{t}^{*}\sigma$
$\displaystyle=f_{t}^{*}(-d\psi+\sigma)=0.$

That is, on $U-W$, we have $f_{t}^{*}\omega_{t}$ is independent of $t$ and so
$f_{1}^{*}\omega_{1}=\omega_{0}$. Take $f=f_{1}$. ∎

We now obtain the following stronger form of the main result of [HK15].

######  5.9 Theorem (improved Harada–Kaveh).

Let $M$ be a smooth complex variety (e.g.,
$M=\mathbb{P}^{n}\times\mathbb{A}^{1}$) and $\mathfrak{X}\subset M$ a closed
subvariety. Assume we have a surjective morphism
$p:\mathfrak{X}\to\mathbb{A}^{1}$ that is the restriction of a proper morphism
$M\to\mathbb{A}^{1}$. Let $W=p^{-1}(0)$ and equip it with a complex-algebraic
Whitney stratification.

Assume there is a sympletic form $\omega$ on $M$ such that $\omega|_{W}$ is a
sympletic form in the stratified sense (Definition 5.5).

Then there exist some $t\neq 0$ and a continuous surjective map

$\varphi:X:=p^{-1}(t)\to W$

such that (1) $\varphi^{*}(\omega|_{W})=\omega|_{X}$ and (2) for each stratum
$A$ on $W$, $\varphi^{-1}(A)$ has a natural reducible-variety structure so
that $\varphi:\varphi^{-1}(A)\to A$ is finite étale.

Moreover, if $W$ comes with a moment map $\mu:W\to\mathfrak{t}^{*}_{r}$ (e.g.,
$\mathfrak{X}$ is a toric degeneration), then

$(f_{1},\dots,f_{r}):X\overset{\varphi}{\to}W\overset{\mu}{\to}\mathfrak{t}^{*}_{r}\simeq\mathbb{R}^{r}$

is an integrable system in the stratified sense (Definition 5.5).

###### Proof.

Let $\pi:U\to W$ be the collapsing map constructed in the previous section,
$U$ the interior of $W(\delta)$. By Theorem 5.8 applied to
$\omega_{1}=\pi^{*}\omega$ and $\omega_{0}=\omega$ ($\omega_{1}$ is locally
Lipschitz by Lemma 4.13), after we shrink $U$, we find a diffeomorphism
$f:U-W\to U-W$ such that $f^{*}\pi^{*}\omega=\omega$ on $U-W$. As in the proof
of Theorem 4.1, for some small $t\neq 0$, we have $p^{-1}(t)\subset
U-W=f^{-1}(U-W)$. Then the composition

$\varphi:p^{-1}(t)\overset{f}{\to}U-W\overset{\pi}{\to}W$

has the required properties. ∎

######  5.10 Remark (Kähler metric).

The above theorem does not involve a Kähler metric, in contrast to [HK15].

######  5.11 Remark (Sympletic stratified space).

The notion of a sympletic stratified space; i.e., a stratified space with a
symplectic structure was introduced by R. Sjamaar and E. Lerman [SL91], who
showed that a symplectic reduction with respect to a singular value of the
moment map still has a structure of a sympletic stratified space. Among the
others, they also showed, in the setup of a sympletic stratified space, that
there is a natural Poisson bracket and that Hamiltonian dynamic still works.

Therefore, one possibility is to formulate Theorem 5.9 in terms of a sympletic
stratified space in the above sense or in some weaker sense. That requires to
have a stratified version of Theorem 5.8. Alternatively, it is possible that
the fixing using Proposition 5.8 is not actually needed; i.e., a careful
examination of the construction in the previous section might show the
construction preserves the sympletic structure. The latter possibility is not
unreasonable since, for example, a tubular neighborhood of a submanifold
admits a sympletic form that is an extension of that on the submanifold (see
[Vai87, Theorem 3.4.19.]).

We have stated (and proved) the above theorem with the weakest possible
assumptions on a toric degeneration. That is in part because of the following
intriguing problem:

######  5.12 Problem (proper toric degeneration).

Given a projective variety $X\subset\mathbb{P}^{n}$, construct an embedding
$\mathbb{P}^{n}\hookrightarrow M$ into a smooth toric variety and a closed
subvariety $\mathfrak{X}$ of $M$ and a proper surjective function $p$ on $M$
such that $W=p^{-1}(0)\cap\mathfrak{X}$ is the closure of a torus-orbit and a
fiber of $p|_{\mathfrak{X}}$ close to $W$ is isomorphic to $X$.

In particular when $M$ comes with a sympletic form that restricts to the
standard sympletic form on $\mathbb{P}^{n}$, Theorem 5.9 constructs an
integrable system on $X$ with respect to the standard sympletic structure.
Roughly the idea of the above is to replace a typical ambient space
$\mathbb{P}^{n}\times\mathbb{A}^{1}$ with a smooth toric variety, which should
give us some flexibility and also would avoid known counter-examples. It is
hoped that the flexibility permits the construction in [KMM22] (equivalently
Murata’s Ph.D. thesis) to yield a toric degeneration in the above sense.

We also note that, in view of Cox’s theorem (Example 1.8) $\mathbb{A}^{N}-B\to
M$, a closed subvariety of $M$ amounts to a closed subvariety of the affine
space $\mathbb{A}^{N}$ that is invariant under
$G=\operatorname{Hom}(\operatorname{Pic}(M),\mathbb{G}_{m})$. Hence, in the
above setup, we have a generalization of a Gröbner degeneration in the sense
that we consider $G$-invariant polynomials instead of homogeneous polynomials.
(We are not aware of discussion of such a generalization in literature, except
Knutson’s offhanded remark.)

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