# Persona-aware Generative Model for Code-mixed Language

1Ayan Sengupta, 2Md Shad Akhtar, and 1Tanmoy Chakraborty
1Dept. of Electrical Engg., IIT Delhi, India; 2Dept. of CSE, IIIT-Delhi,
India;
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

Code-mixing and script-mixing are prevalent across online social networks and
multilingual societies. However, a user’s preference toward code-mixing
depends on the socioeconomic status, demographics of the user, and the local
context, which existing generative models mostly ignore while generating code-
mixed texts. In this work, we make a pioneering attempt to develop a persona-
aware generative model to generate texts resembling real-life code-mixed texts
of individuals. We propose Persona-aware Generative Model for Code-mixed
Generation, PARADOX, a novel Transformer-based encoder-decoder model that
encodes an utterance conditioned on a user’s persona and generates code-mixed
texts without monolingual reference data. We propose an alignment module that
re-calibrates the generated sequence to resemble real-life code-mixed texts.
PARADOX generates code-mixed texts that are semantically more meaningful and
linguistically more valid. To evaluate the personification capabilities of
PARADOX, we propose four new metrics – CM BLEU, CM Rouge-1, CM Rouge-L and CM
KS. On average, PARADOX achieves $1.6$ points better CM BLEU, $47\%$ better
perplexity and $32\%$ better semantic coherence than the non-persona-based
counterparts.

###### Index Terms:

Code-mixed generation, Personification of language, Transformers.

##  I Introduction

Code-mixing (aka code-switching) appears when two or more languages are used
interchangeably in a single utterance. It is common in multilingual societies
like India, where more than $24\%$ of the population speaks in more than one
language [1]. Code-mixing is even more prevalent on social media. Informal
usage of code-mixed languages on social media platforms like Twitter,
Facebook, YouTube, and other online social networks, gives rise to script-
mixing, in which a user can use a single script (e.g., Roman) or multiple
scripts (e.g., Devanagari for Hindi and Roman for English) within the same
text [2]. Recent literature has made significant efforts to understand
syntactic structure and semantics from code-mixed texts [3, 4, 5]. Similar
attempts have been made for pragmatic tasks – humour, sarcasm and hate
detection in the code-mixed regime [6, 7].

Text generation models must understand the syntax and semantics of texts, and
preserve semantic coherence during generation. Previous studies utilized
recurrent neural networks with generative models [8], as well as self-
attention-based pre-trained language models [9] for generating monolingual
texts. However, such an effort is limited in the code-mixed regime.
Previously, linguistic theories [10, 11], transfer learning [11], and
autoencoding [12] based approaches have been used to generate code-mixed texts
from parallel corpora or reference data. However, none of these methods
incorporates user information while generating code-mixed texts. Unlike
traditional languages, code-mixing is a derived language whose adoption
depends on different socioeconomic, demographic, and linguistic factors [13,
14]. Figure 1 demonstrates the code-mixing behaviour among Indian users on
Twitter and YouTube in terms of adoption and patterns in code-mixing. We
visualize the mean and standard deviation of the Code-Mixing Index (CMI) [15]
and the length of tweets/comments posted by different users. The distributions
show how different users conceive and prefer code-mixing.

Figure 1: User-specific distribution of CMI and text lengths across different
platforms. CMI is calculated as the fraction of minority language words in a
text. For instance, CMI of the text “I don’t want your nautanki” (“I don’t
want your gimmick”) is $\frac{1}{5}=0.2$, the fraction of Hindi (minority
language in this example) words in the text. Texts skewed toward
monolingualism, i.e., having an unequal proportion of words between different
languages, tend to have lower CMI than multilingual texts.

User persona plays a vital role in generation models, particularly in
personalized generation settings, such as conversational agents and
recommendation engines. Several studies have contributed towards persona-based
dialogue generation [16, 17], personalized story generation [18], and other
sub-tasks in text generation. Being a conversational language, personification
of code-mixing could deem appropriate for conversational systems such as
recommender engines, mental health counselling bots, and event booking
applications.

This motivates us to develop PARADOX111PARADOX: Persona-aware Generative Model
for Code-mixed Generation., a novel persona-attributed generative model for
code-mixed text generation. It aims to generate personalized code-mixed texts
by leveraging users’ historical utterances. It uses a Transformer-based
encoder-decoder architecture to learn the semantics of code-mixed generation.
The model utilizes a novel persona encoder to encode a user persona from their
behavioural preferences. Instead of projecting the user’s persona onto a
static space, PARADOX projects it onto a probabilistic latent space and
captures the contextual persona based on their historical persona.
Additionally, PARADOX uses an alignment module to re-align decoder outputs to
generate coherent texts.

We evaluate PARADOX against the vanilla Transformer in terms of both the
quality and coherence of generated texts. To quantify the extent of
personification in the code-mixed generation, we propose four metrics – CM
BLEU, CM Rouge-1, CM Rouge-L, and CM KS (Here CM stands for code-mixing). On
average, PARADOX achieves $1.6$ points better CM BLEU than the non-persona
counterpart. We also conduct detailed human evaluation, concluding that
PARADOX generates code-mixed texts $32\%$ more semantically coherent than the
vanilla Transformer model. PARADOX can imitate a user’s linguistic preference
$4\%$ better than the non-persona-based Transformer model.

Contributions: The major contributions of this paper are summarized below:

  * •

We make a pioneering effort in utilizing user persona in code-mixed text
generation. Compared to existing approaches, PARADOX does not require parallel
corpora or reference data for code-mixed text generation.

  * •

We propose a probabilistic persona encoder module that learns the latent
persona of a user from historical contexts. PARADOX captures the user persona
implicitly and does not require explicit persona features such as user
demographic information to encode user behaviours.

  * •

We design an alignment module to automatically induce alignments between
different subwords. Empirical results show that it improves the coherence of
generated texts.

  * •

We propose three metrics influenced by supervised machine translation – CM
BLEU, CM Rouge-1, and CM Rouge-L for evaluating the personification of code-
mixed generation models. We also propose CM KS as a distance measure to
evaluate code-mixing generation models.

  * •

Finally, we collect a large-scale longitudinal dataset from Twitter and
YouTube, primarily monolingual Hindi and Hindi-English code-mixed texts. This
dataset will be open-sourced.

Reproducibility: We have made the source code, datasets and steps to reproduce
the results public at https://github.com/victor7246/PARADOX.

Figure 2: PARADOX: Transformer encoder-decoder architecture with persona
encoder multi-headed (M.H.) fused attention (FAME), the contextual persona
encoder, and the alignment modules.

##  II Related Works

Code-mixed text generation has garnered much interest in recent times. Pratapa
et al. [10] explored equivalence constraint (EC) theory to generate Spanish-
English code-mixed texts from monolingual corpora. Their linguistic theory-
based approach showed superiority over recurrent neural networks in complex
code-mixed text generation. Rizvi et al. [19] recently developed GCM, a
toolkit that utilizes different linguistic theories to generate code-mixed
texts. Motivated by embedding matrix theory, Srivastava et al. [20] proposed
rule-based methods to generate Hindi-English code-mixed texts. Alternative
approaches use generative models – generative adversarial networks [21], or
variational autoencoder (VAE) [22]. Towards this, Garg et al. [23] explored
recurrent neural networks with SeqGAN pre-training for generating Mandarin-
English code-mixed data. Samanta et al. [12] developed a VAE-based method to
generate realistic and coherent Hindi-English code-mixed texts. Recently
Mondal et al. [24] proposed a Transformer-based encoder-decoder model for a
controllable code-mixed generation. Although multi-lingual language models
[25, 26, 27] demonstrated abilities in understanding low-resource languages,
their effectiveness on low-resource code-mixed generation remains unexplored.
Gupta et al. [11] devised a semi-supervised approach to transfer knowledge
from XLM to generate synthetic Hindi-English code-mixed texts.

Major Limitations of Existing Studies. Despite their popularity in several
generative applications, personalization remains neglected in the code-mixed
generation. The existing code-mixed generation models utilize parallel corpora
to understand the switching patterns and generate code-mixed texts
synthetically. These limitations motivate us to develop a code-mixed language
generation model that can automatically learn the language’s semantics and
capture the linguistic preferences of users while generating texts. Our
proposed method, PARADOX improves the quality of generated texts and preserves
the real-life phenomenon of code-mixing among users.

##  III Proposed Methodology

Here we explain PARADOX, and the code-mixing (hereby denoted as CM) generation
process utilizing user persona. PARADOX utilizes a persona encoder to encode a
user’s persona implicitly based on her historical utterances. It further
projects the persona onto a probabilistic latent space to capture the user’s
contextual persona. PARADOX employs an alignment module to re-calibrate the
output sequences that help our model understand the language of code-mixing
and enable the model to generate coherent texts.

###  III-A PARADOX Architecture

PARADOX, as shown in Figure 2(a), consists of four components – (a) an
encoder, (b) a contextual persona encoder, (c) a decoder, and (d) an alignment
module.

#### The Encoder with FAME

The encoder is used to jointly learn the semantics of a code-mixed text and a
user’s global persona based on the previous comments/tweets. A comment/tweet
${\mathbf{X}}_{u}$ by user $u$ is first tokenized using byte-pair encodings
[28] into $\langle x_{1},x_{2},...,x_{n}\rangle$. The initial contextual
embedding of a token $x_{i}$ is conditioned with user persona as

$\widetilde{Emb_{(x_{i},u)}}=Emb_{x_{i}}+PE_{i}+Emb_{u}$

, Where $Emb_{x_{i}}$ is the initial token embedding, $PE_{i}$ is the
positional encoding at position $i$, and $Emb_{u}$ is the user’s global
persona embedding captured through a static embedding layer. We use a stacked
encoder, in which each encoder consists of multi-headed fused attention
(FAME), followed by a residual, batch-normalization, and pointwise feed-
forward layers. Sengupta et al. [1] introduced FAME by combining scaled dot-
product attention [29] and outer-product attention [30] and showed to be
effective in capturing both semantics and morphology of code-mixed texts.

#### Contextual Persona Encoder

We project the static persona embedding onto a probabilistic latent space for
generating the contextual persona embedding for each user in a given context.
Formally, we generate a contextual persona embedding

$\widetilde{Emb_{u}}\sim
q_{\phi}(z|Emb_{u})=\mathcal{N}(\mu_{u},\,\sigma_{u}^{2}).$

Towards this, we define two linear projection matrices to learn the
distribution location and scale parameters as

$\mu_{u}=Emb_{u}.W_{\mu};$

$\sigma_{u}=Emb_{u}.W_{\sigma}.$

Following the reparameterization trick [22], we define the final generated
persona encoding as:

$\widetilde{Emb_{u}}=\mu_{u}+\epsilon_{u}\odot\sigma_{u}$

, Where $\epsilon_{u}$ is the random noise, independently drawn from
$\mathcal{N}(0,1)$.

We obtain the hidden representation of token $x_{i}$ conditioned on the
contextual persona encoding as

$\widetilde{h_{(x_{i},u)}}=h_{(x_{i},u)}+\widetilde{Emb_{u}}$

, Where $h_{(x_{i},u)}$ is the final hidden representation obtained from the
final layer of the encoder.

#### The Decoder

We adopt the Transformer decoder conditioned on the contextual user persona.
Similar to the original Transformer decoder, we use a stacked decoder,
initialized with the encoded output sequence. Drawing the motivation from
autoregressive generative language models like GPT2 [31], the decoder’s
objective is to predict the next token, conditioned on all the previous
tokens. The input to the decoder is the encoded input sequence concatenated
with positional encoding. Each decoder block consists of masked multi-headed
FAME, a residual connection, and a normalization layer. We also deploy multi-
headed FAME to attend to each decoder token with the encoded input tokens
$\widetilde{h_{(x_{i},u)}}$. For each decoder input position $j$, we generate
a hidden representation $h_{(j,u)}^{(dec)}\in{\mathbb{R}}^{|V|}$, representing
the output token at $(j+1)^{th}$ position; $|V|$ is the vocabulary size of the
decoder.

#### The Alignment Module

The final layer of PARADOX is an alignment module that learns the latent
alignment matrix and re-aligns the outputs generated by the decoder. The
primary objective behind using alignments in generative models is to learn the
global semantic similarity between different tokens explicitly. We use two
projection matrices $W^{Q}$ and $W^{K}$ to project the decoder token embedding
matrix $Emb^{(dec)}$ into two different subspaces. The alignment matrix is
defined as,

$\small\mathcal{A}=softmax\left(\frac{Q\cdot K^{T}}{\sqrt{d}}\right)$ (1)

where $Q=Emb^{(dec)}\cdot W^{Q}$, $K=Emb^{(dec)}\cdot W^{K}$, and $d$ is the
hidden size of the decoder. This operation resembles the scaled dot-product
attention mechanism [29]. However, as opposed to attention, we compute the
global context by considering the original embedding space of all tokens.
Finally, the re-aligned hidden representation is derived as,

$\widetilde{h_{(j,u)}}^{(dec)}=h_{(j,u)}^{(dec)}\cdot\mathcal{A}+h_{(j,u)}^{(dec)}$
(2)

This hidden representation is finally fed to a softmax layer to convert the
outputs into probabilities.

For the sake of simplicity, we denote the combination of text encoder and
persona encoder as ‘encoder’ and the combination of Transformer decoder and
the alignment module as ‘decoder’, throughout this paper. The generative model
is trained w.r.t. the decoder reconstruction cross-entropy loss. The
contextual persona encoder gives rise to a variational KL-divergence loss. We
use a variational hyperparameter $\lambda$ to assign its weightage in the
final computed loss.

Algorithm 1 Code-Mixed Text Generation with PARADOX

1:Trained model $\mathcal{M}=(enc,dec)$, user id $u$, historical utterance
$x_{u}$, prompt word $\\{w_{1}\\}$, decoder vocabulary $V$

2:$max\\_length\in\mathbb{N}$

3:

4:$L\leftarrow\\{w_{1}\\}$

5:$\tilde{w}\leftarrow\emptyset$

6:$i=m$

7:while $\tilde{w}\neq[SEP]$ and $i<max\\_length$ do

8: $h_{(x_{u},u)}=enc(x_{u},u)$

9: $P_{i+1}=dec(L,h_{(x_{u},u)})$

10: $\tilde{w}\leftarrow\operatorname*{arg\,max}_{V}P_{i+1}$

11: $L\leftarrow L\cup\\{\tilde{w}\\}$

12:Return L

###  III-B Code-Mixed Generation

We adopt an auto-regressive generation technique to generate new code-mixed
texts for different users. To encode the user’s historical persona, we use the
user’s last utterance (comment/tweet) in the encoder. Based on the decoder
input, we devise a cold-start generation strategy. In the cold-start
generation, we pass a seed word as input to the decoder and generate the rest
of the text. We formally report the text generation process in Algorithm 1.

##  IV Experimental Setup

This section elaborates on the experimental setup we adopt to evaluate our
model and baselines on personalized code-mixed generation.

###  IV-A Datasets

To the best of our knowledge, no existing longitudinal dataset is available
for Hindi-English code-mixed. A longitudinal dataset is required to study the
temporal evolution of a language. Although some datasets in the literature
consist of Hindi-English code-mixed texts collected from various online social
networks, most of them do not contain any user-specific information, making
them unsuitable for our study. To overcome this limitation, we have collected
code-mixed texts from the two most popular mediums where Indians are engaged –
Twitter and YouTube. From Twitter, we have collected over $0.8$ million in
tweets starting from the year $2011$, till date, from which we filter only
tweets originating from Mumbai and Delhi metropolitan regions, two cities with
the largest Hindi population. We have used Twitter API for academic research
with full archival access222https://api.twitter.com/2/tweets/search/all.
Further, for relevance, we have restricted ourselves to tweets related to
‘Cricket’, ‘Bollywood’, ‘Politics’, and ‘Government’. Starting $2014$, Twitter
automatically tags the language of a tweet. We have selected tweets with only
non-empty language tags. This gives us a total of $226480$ tweets from $19782$
users.

From YouTube, we have chosen two channels –
NishaMadhulika333https://www.youtube.com/c/nishamadhulika (a popular chef
based out of India with more than $12.7$ million followers), and
T-Series444https://www.youtube.com/aashiqui2 (a popular Hindi music record
channel started in $1983$ having more than $200$ million followers). We have
selected $42$ videos from the NishaMadhulika channel and $69$ from the
T-Series that were first posted in $2011$. We have scraped all comments
corresponding to these videos, accounting for $144822$ comments from $99998$
users.

For both datasets, we use a pre-trained language model open-sourced with
Huggingface555https://huggingface.co/sagorsarker/codeswitch-hineng-lid-lince,
that was fine-tuned on Hindi-English parts-of-speech (PoS) and language
identification (LID) tasks. Using this model, we label each token in each text
with the corresponding language (Hindi or English) and their associated PoS.
We tag a text as code-mixed only when the text contains at least one Hindi
verb written in either Devanagari or Roman script. We select users who have at
least three utterances in their entire timeline. Finally, we are left with
$18126$ tweets (from $2241$ users) and $8957$ YouTube comments (from $1349$
users).

We remove all the HTML tags (‘¡.*?¿’ patterns), URLs, emoticons, user mentions
(starting with ‘@’), and hashtags (staring with ‘#’). For simplicity, we
remove all numeric values from texts, as well. Finally, we convert all texts
to lower case. We highlight the key statistics of the datasets in Table I. We
use a $75$-$25$ split for training and validation.

Dataset | #Texts | #Users | Mean text length | Mean CMI
---|---|---|---|---
Twitter | 18126 | 2241 | 21.77 | 0.41
YouTube | 8957 | 1349 | 28.89 | 0.36

TABLE I: Dataset statistics, with mean text CMI and the average text lengths,
demonstrating the extent of code-mixing.

###  IV-B Evaluation Metrics

We adopt intrinsic and extrinsic evaluation metrics to evaluate our model in
terms of semantic understanding of code-mixing language and the ability to
personify code-mixing for different users.

For the intrinsic evaluation, we use perplexity, a metric that measures the
predictive power of a language model, compared against ground truth. We
calculate perplexity as $e^{loss}$, with $loss$ being the cross-entropy
reconstruction loss on the validation data. A lower perplexity score indicates
better reconstructibility and ability to learn the semantics of a generative
model.

Unlike Gupta et al. [11], we do not have any labelled gold data for evaluating
our generative model. Therefore, traditional supervised evaluation metrics –
BLEU [32], Rouge [33] can not be used directly to evaluate the personification
aspects of code-mixed generation models. Similarly, other extrinsic evaluation
measures such as Multilingual index (M Index) [34], Burstiness and Span
Entropy [35] can not be used, as these metrics are predominantly used to
evaluate the ability to capture corpus-level switching patterns of generative
models. To overcome the limitations of the existing evaluation metrics, we
propose four metrics for benchmarking generated code-mixed texts against the
historical utterances by different users. We devise CM BLEU by calculating the
BLEU score between the candidate and reference language sequences. For example
- consider a candidate code-mixed text “mujhe park janaa hai” (“I want to go
to the park”) and a reference text “mujhe rice aur curry khana hai” (“I want
to eat rice and curry”). Using the LID model, we can extract the corresponding
language sequences {Hi, Hi, Hi, Hi} and {Hi, En, Hi, En, Hi,, Hi, Hi, Hi, Hi,
Hi, Hi } from the candidate and reference texts, respectively. Therefore,
considering only the unigram and bigram overlaps between the candidate and the
reference, we calculate the CM BLEU score666Can be calculated and validated
using https://www.nltk.org/api/nltk.translate.bleu_score.html of $0.606$. If
we use a different reference text “I don’t want your nautanki” (Translation -
“I don’t want your gimmick”) with the corresponding language sequence {En, En,
En, En, Hi}, the CM BLEU reduces to $0.218$. The proposed metric could
calculate the similarity between the switching patterns demonstrated in the
candidate text and the historical references by calculating the overlap
between the language sequences. Similarly, we compute CM Rouge-1 and CM
Rouge-L by computing Rouge-1 and Rouge-L scores between the candidate and
reference language sequences.

Figure 3: Pearson correlation between different evaluation measures on the
validation dataset.

Additionally, we leverage the user-level historical CMI to evaluate the
linguistic patterns of the generated texts. If a user historically prefers
monolingualism over multilingualism, we want the generative model to learn the
pattern and generate texts with a lower CMI value for the user. Towards this,
we propose CM KS, a metric that computes the Kolmogorov-Smirnov distance
between generated and original CMI distribution of users. We highlight the
relationships between these metrics by calculating the Pearson correlation
between these measures, reported in Figure 3. Strong negative correlations
between perplexity and CM BLEU, CM Rouge-1, and CM Rouge-L indicate that
understanding semantics is essential to personify and replicate the switching
patterns. Therefore, by learning semantics well, the generative models can
learn the code-mixing patterns for different users and generate texts that
imitate users’ linguistic patterns. On the other hand, the correlations
between CM KS and other metrics are weak, indicating that the linguistic
preferences of users have no apparent linear relationships with switching
patterns.

###  IV-C Training Details

For all the models across all the experiments, we use a maximum text length of
$40$. PARADOX consists of $6$ encoder and decoder layers, with hidden sizes of
$768$ in all the layers. For multi-headed FAME and masked multi-headed FAME
blocks, we use total $8$ heads with $Dropout$ probabilities set as $0.1$. The
total number of parameters is $296$M. We use $6$ encoder and $6$ decoder
layers in the Transformer model, with $8$ heads in each multi-headed attention
block. For training PARADOX, We set the persona encoding variational weight
$\lambda=0.5$. All the models are trained for $50$ epochs with an early
stopping condition on validation loss with the patience of $10$. We set
$batch\\_size=4$ in all experiments during training and validation. We use
Adam optimizer with a learning rate of $4e-4$ and
$\beta_{1}=0.9,\beta_{2}=0.98$ for both PARADOX and Transformer. We use one
Tesla P100 and one Tesla V100 GPU for running all our experiments. For
PARADOX, each training and validation iteration takes $\sim 0.18$ and $\sim
0.12$ seconds, respectively. Strubell et al. [36] proposed estimation of power
usage and carbon emission behind running deep learning experiments. Following
that guidelines, we estimate a total power usage of $23.56$ kWh and an
equivalent CO2 emission of $22.46$ pounds.

##  V Comparative Analysis

In this section, we report the performances of PARADOX and the non-persona-
based Transformer model in terms of the intrinsic and extrinsic evaluation
measures. We report the average scores obtained in three experiments
initialized with different seed values.

We report the intrinsic evaluation results in Table II. PARADOX achieves
$43\%$ better perplexity on the Twitter dataset than the vanilla Transformer.
On the YouTube dataset, the margin is even higher ($45\%$). A lower validation
perplexity shows PARADOX’s strong ability to understand code-mixing semantics
and generate texts of different linguistic variations.

Model |  Perplexity $\downarrow$
---|---
| Twitter | YouTube
Transformer | 680.07 | 473.84
PARADOX | 297.43 | 292.44
(-) Contextual Persona | 320.79 | 295.26
(-) Alignment | 337.07 | 363.77
(-) FAME | 839.52 | 382.00

TABLE II: Intrinsic evaluation of the models based on perplexity
($\downarrow$: lower value indicates better performance).

Model |  CM BLEU $\uparrow$ |  CM Rouge-1 $\uparrow$ |  CM Rouge-L $\uparrow$ |  CM KS $\downarrow$
---|---|---|---|---
| Twitter | YouTube | Twitter | YouTube | Twitter | YouTube | Twitter | YouTube
Transformer | 22.21 | 29.36 | 58.69 | 61.02 | 51.10 | 57.27 | 0.42 | 0.37
PARADOX | 23.64 | 29.02 | 59.71 | 61.62 | 52.24 | 57.83 | 0.36 | 0.34
(-) Contextual Persona | 24.06 | 30.67 | 60.03 | 62.44 | 52.52 | 58.71 | 0.35 | 0.34
(-) Alignment | 23.56 | 30.67 | 59.84 | 62.12 | 52.22 | 58.43 | 0.32 | 0.37
(-) FAME | 20.37 | 29.36 | 55.78 | 60.80 | 49.16 | 57.37 | 0.36 | 0.37

TABLE III: Extrinsic evaluation of Transformer and PARADOX in terms of
preserving user-level switching patterns ($\uparrow$ (resp. $\downarrow$):
higher (resp. lower) value indicates better performance).

Table III highlights the extrinsic measures across all the generative models.
On the Twitter dataset, PARADOX achieves $1.43$ points better CM BLEU than the
Transformer model. On the YouTube dataset, however, PARADOX without contextual
persona, performs the best and outperforms the Transformer model with a margin
of $1.31$. In terms of the Rouge measures, PARADOX performs consistently
better than the non-persona counterpart with an average margin of $0.8$
points. In terms of distance-based measures, PARADOX performs significantly
better than the Transformer model on both datasets. Overall, PARADOX achieves
$4\%$ lower CM KS distance than Transformer. Lower KS distance indicates the
importance of utilizing user persona in generating user-specific code-mixed
texts.

Our ablation study shows the effectiveness of fused attention, contextual
persona module, and the alignment module in PARADOX. Adding FAME improves
validation perplexity by $57\%$. Similarly, the contextual user persona and
alignment modules improve validation perplexity by $29\%$ and $40\%$,
respectively, justifying their contributions to modelling low-resource
language. Therefore, having only the persona module is not sufficient for
understanding the language of personalized code-mixing. However, we observe
that having a contextual persona could hurt the ability of the model to
preserve the switching patterns for different users. This could be attributed
to users demonstrating minimal variability in their linguistic pattern, the
trend we highlighted in Figure 1. Therefore, a random exploration with the
contextual persona module could change the switching patterns observed for
different users. In this case, reusing the static persona for generating texts
could deem semantically and linguistically more meaningful.

#### Human Evaluation.

We perform a human evaluation study to evaluate the code-mixed texts generated
by PARADOX and the vanilla Transformer. We randomly sample $24$ examples from
each of these models and ask $30$ human evaluators777Evaluators are proficient
with Hindi-English code-mixed language and their age ranges in $21-35$. to
rate these examples based on Semantic coherence and Linguistic quality.
Semantic coherence measures the meaningfulness of the code-mixed texts,
whereas, with linguistic quality, we measure their structural validity. Both
the scores ranged between $1$-$5$, $1$ being the lowest, and $5$ being the
highest. Table IV presents the average semantic coherence and linguistic
quality scores, along with Fleiss’s Kappa [37] scores among the annotators. We
observe that PARADOX displays a better semantic coherence ($32\%$ better), as
well as better linguistic quality ($29\%$ better) than the Transformer model.
We observe fair agreement (Kappa $0.13$ for semantic coherence and Kappa
$0.14$ for linguistic quality) among the annotators for both models.

Model | Semantic Coherence $\uparrow$ | Linguistic Quality $\uparrow$
---|---|---
Transformer | 2.34 | 2.32
PARADOX | 3.08 | 3.00

TABLE IV: Human evaluation of the models.

##  VI Analysis of Code-Mixed Generation

Figure 4: Distribution of top Hindi verbs and nouns for Tweets. w.o. alignment
is the ablation of PARADOX without the alignment module.

Model |  Generated Text | Semantic Coherence | Linguistic Quality
---|---|---|---
PARADOX |  CM: are bhai apne! | 3.70 | 3.60
Eng: Hey my brother! |  |
2-4 |  CM: bhai bahut thik ho gaya | 4.27 | 4.27
|  Eng: brother is very well |  |
Transformer |  CM: rockstar walo se samjha media…….. | 3.07 | 3.57
Eng: Media understood by rockstars |  |
2-4 |  CM: boy likha hai thanks bhai | 3.57 | 3.53
|  Eng: Boy has written thanks brother |  |

TABLE V: Examples of code-mixed texts generated by PARADOX and Transformer
with human annotated average Semantic coherence and linguistic quality scores.

User ID |  Generated Text | Historical Avg. CMI | Generated CMI
---|---|---|---
3 |  CM: salman ji please aapse milna hai | 0.28 | 0.33
Eng: Salman, I want to meet you, please
264 |  CM: salman ka fan ho na. | 0.50 | 0.40
Eng: You are Salman’s fan, right?
762 |  CM: ye politics kr raha kya kr rahi h or truth show | 0.15 | 0.30
Eng: Is this politics, or truth show
2226 |  CM: ye fb ho jaye bhaijaan | 0.04 | 0.00
Eng: This has became fb, brother

TABLE VI: Example of different prompted (prompts highlighted with blue)
generation for different users. Different CMI indicates the difference in
prompted generation based on the user persona.

We further study the quality of code-mixed texts generated by different
models. We evaluate the linguistic traits of the texts generated by different
generative models. Towards this, we inspect the distribution of frequent nouns
and verbs for each model. Figure 4 shows the distribution of top Hindi verbs
and nouns from the Twitter dataset. Interestingly, PARADOX with the alignment
module, can replicate the word distribution observed in the real dataset. On
the other hand, without the alignment module, the generative model could
hallucinate and unrealistically use common phrases in incorrect contexts. This
highlights the effectiveness of the alignment module in recalibrating output
tokens and generating semantically meaningful texts. Although the dimension of
the alignment matrix is $|V|\times|V|$, with $|V|$ being the decoder
vocabulary size, the learnable parameters are of order $O(d^{2})$, where $d$
is the hidden dimension. The total number of additional parameters introduced
by the alignment module is $0.025\%$ of the entire network, which is
insignificant compared to other modules.

We further analyze a few sample texts generated by the generative models. We
report examples of texts generated by PARADOX and the Transformer, along with
the average semantic coherence and linguistic quality scores annotated by the
annotators in Table V. Transformer usually picks top nouns in the corpus and
generates texts around them without considering the syntax of the code-mixed
language resulting to incoherent texts in many cases. On the other hand,
PARADOX preserves the grammar of code-mixed texts with a more human-alike
switching pattern. It shows that PARADOX maintains the grammar of the base
language (Hindi in this case), attributing to a more coherent and reliable
generation. The examples highlight the key differences between the texts
generated by persona-based PARADOX and non-persona-based Transformer models
regarding text quality. Table VI further shows the personalized generation by
our model. With the same prompt (e.g., ‘salman’ in the first example), PARADOX
can understand different personas of users and can generate texts suited for
different users. The high similarity between the historical average of the
user CMIs and the generated CMIs indicates the model’s ability to understand
the linguistic preferences of users. This highlights the effectiveness of
personification in code-mixed text generation.

##  VII Conclusion

This paper described a personalized code-mixed generation model for generating
human-alike code-mixed texts. We highlighted the need for a personalized
generation under the pretext of code-mixing. Towards this, we devised a novel
persona-aware encoder-decoder model coupled with a novel alignment module for
generating more realistic and coherent Hindi-English code-mixed texts, the
first attempt toward personalized code-mixed generation. Empirical analyses
would benefit the research community in developing robust and reliable
language models for low-resource languages. Although our empirical study has
shown the effectiveness of persona-attributed text generation, currently
PARADOX captures only contextual persona, ignoring other explicit factors. Not
only does this restrict our model in cold-start generation (text generation
for new users without any history), but also it fails to consider the co-
association among users. In conversational settings, particularly, this can
deem essential. Another limitation of PARADOX is the inability to determine
the temporal evolution of a user’s persona driven by external factors. PARADOX
captures the user persona and its evolution solely from contextual
information. At the same time, a user’s linguistic preferences can also be
driven by other external socio-demographic and economic factors varying over
time, which our model currently undermines.

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# The Dirichlet problem for degenerate fully nonlinear elliptic equations on
Riemannian manifolds

Ri-Rong Yuan School of Mathematics, South China University of Technology,
Guangzhou 510641, China
Email address<EMAIL_ADDRESS>
###### Abstract

We derive the existence of $C^{1,1}$-solutions to the Dirichlet problem for
degenerate fully nonlinear elliptic equations on Riemannian manifolds under
appropriate assumptions.

##  1 Introduction

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 2$ with
smooth boundary $\partial M$, $\bar{M}:=M\cup\partial M$, with the Levi-Civita
connection $\nabla$. Let $\chi$ be a smooth symmetric $(0,2)$-tensor. Let $f$
be a symmetric function defined in an open symmetric convex cone
$\Gamma\subset\mathbb{R}^{n}$ containing positive cone, i.e.

$\Gamma_{n}:=\\{\lambda\in\mathbb{R}^{n}:\mbox{ each component
}\lambda_{i}>0\\}\subseteq\Gamma$

and with vertex at the origin and with boundary $\partial\Gamma\neq\emptyset$.

In this paper, we consider the Dirichlet problem

$\displaystyle f(\lambda(\nabla^{2}u+\chi))=\psi\mbox{ in }M,\,$
$\displaystyle u=\varphi\mbox{ on }\partial M$ (1.1)

with degenerate right-hand side

$\displaystyle\inf_{M}\psi=\sup_{\partial\Gamma}f,$ (1.2)

where $\lambda(\nabla^{2}u+\chi)$ denote the eigenvalues of $\nabla^{2}u+\chi$
with respect to $g$, and

$\sup_{\partial\Gamma}f=\sup_{\lambda_{0}\in\partial\Gamma}\limsup_{\lambda\rightarrow\lambda_{0}}f(\lambda).$

In addition, some standard assumptions are imposed as follows:

$\displaystyle f\in C^{\infty}(\Gamma)\cap C(\overline{\Gamma}),\mbox{ where
}\overline{\Gamma}=\Gamma\cup\partial\Gamma,$ (1.3) $\displaystyle
f_{i}(\lambda):=\frac{\partial f}{\partial\lambda_{i}}(\lambda)>0\mbox{ in
}\Gamma,\,$ $\displaystyle\forall 1\leq i\leq n,$ (1.4) $\displaystyle f\mbox{
is concave in }\Gamma.$ (1.5)

For $M=\Omega\subset\mathbb{R}^{n}$ and $\chi=0$, the Dirichlet problem for
degenerate Monge-Ampère equation has been studied by Guan [7] (with
homogeneous boundary data) and Guan-Trudinger-Wang [9], which was later
extended by Ivochkina-Trudinger-Wang [14] to more general degenerate Hessian
equations. In a series of papers [10, 11, 13], Harvey-Lawson proposed the
concepts of strict $F$ and $\widetilde{F}$ convexity assumptions on the
boundary then derived with Perron method the existence and uniqueness of
continuous solutions to the Dirichlet problem which have broader forms than
that of (1.1). See also the Survey [12]. For more progress and open problems
on degenerate nonlinear elliptic equations, we also refer the readers to [8].

This paper is a complement to these works. We derive $C^{1,1}$ weak solutions
to the Dirichlet problem for degenerate equations on Riemannian manifolds
subject to

$\displaystyle(-\kappa_{1},\cdots,-\kappa_{n-1})\in\overline{\Gamma}_{\infty}\mbox{
in }\partial M.$ (1.6)

Here $\kappa_{1},\cdots,\kappa_{n-1}$ denote the principal curvatures of the
boundary, and $\overline{\Gamma}_{\infty}$ is the closure of

$\Gamma_{\infty}=\left\\{\lambda^{\prime}\in\mathbb{R}^{n-1}:(\lambda^{\prime},\lambda_{n})\in\Gamma\right\\}.$

First we present some notion. We say $u\in C^{2}(\bar{M})$ is admissible if

$\displaystyle\lambda(\nabla^{2}u+\chi)\in\Gamma\mbox{ in }\bar{M}.$

We say $\underline{u}$ is a subsolution of the Dirichlet problem (1.1) if

$\displaystyle f(\lambda(\nabla^{2}\underline{u}+\chi))\geq\psi\mbox{ in
}\bar{M},\quad\underline{u}=\varphi\mbox{ on }\partial M.$ (1.7)

Furthermore, it is called a strictly subsolution if

$\displaystyle f(\lambda(\nabla^{2}\underline{u}+\chi))>\psi\mbox{ in
}\bar{M},\quad\underline{u}=\varphi\mbox{ on }\partial M.$ (1.8)

######  Theorem 1.1.

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 2$ with
smooth boundary satisfying (1.6). Suppose (1.3)-(1.5) hold. For the
$\varphi\in C^{2,1}(\partial M)$, $\psi\in C^{1,1}(\bar{M})$ satisfying (1.2),
we assume that there exists a $C^{2,1}$ admissible strictly subsolution to the
Dirichlet problem (1.1). Then the Dirichlet problem admits a weak solution
$u\in C^{1,1}(\bar{M})$ with

$\lambda(\nabla^{2}u+\chi)\in\overline{\Gamma}\mbox{ in }\bar{M},\quad\Delta
u\in L^{\infty}(\bar{M}).$

Condition (1.4) ensures (1.1) to be elliptic at any admissible solutions,
while (1.5) implies that the operator $F(A)=f(\lambda(A))$ is concave with
respect to $A$ when $\lambda(A)\in\Gamma$. Consequently, according to Evans-
Krylov theorem [3, 15] and classical Schauder theory, higher order estimates
for admissible solutions follow from

$|u|_{C^{2}(\bar{M})}\leq C.$ (1.9)

When $\chi=0$ and $M=\Omega$ is a smooth bounded domain in $\mathbb{R}^{n}$,
the estimate (1.9) was established by Caffarelli-Nirenberg-Spruck [1] for the
Dirichlet problem under assumptions that the principal curvatures
$\kappa_{1},\cdots,\kappa_{n-1}$ of $\partial\Omega$ satisfy

$\displaystyle(\kappa_{1},\cdots,\kappa_{n-1})\in\Gamma_{\infty}$

and $f$ satisfies proper assumptions including a unbounded condition. The
bounded case has been further studied by Trudinger [18] on
$\Omega\subset\mathbb{R}^{n}$. On general closed Riemannian manifolds, the
estimate (1.9) for equation (1.1) with $\chi=g$ has been obtained by Li [16]
when the underlying manifold admits nonnegative sectional curvature, by Urbas
[19] when replacing nonnegative sectional curvature assumption by certain
extra assumptions on $f$ including

$\displaystyle f_{j}(\lambda)\geq\delta\sum_{i=1}^{n}f_{i}(\lambda)\mbox{ if
}\lambda_{j}\leq 0.$ (1.10)

The Dirichlet problem (1.1) has been studied by Guan [4], where he imposed
either one of such two assumptions, beyond the subsolution assumption. We
shall remark that in [16, 19, 4], condition (1.10) and the assumption of
nonnegative sectional curvature are only used to prove gradient estimate. The
author [21]***The paper [21] is essentially extracted from [arXiv:2011.08580]
and [arXiv:2101.04947]. proposed partial uniform ellipticity then confirm the
key condition (1.10) when

$\displaystyle\lim_{t\rightarrow+\infty}f(t\lambda)>f(\mu)\mbox{ for any
}\lambda,\mbox{ }\mu\in\Gamma,$ (1.11)

thereby extending the a priori estimates to a large amount of Hessian type
equations on Riemannian manifolds. (See [22] for more extension). However, in
the literature cited above, the right-hand sides of equations are assumed to
satisfy a nondegenerate condition

$\inf_{M}\psi>\sup_{\partial\Gamma}f.$ (1.12)

Consequently, the estimates established there don’t apply to degenerate
equations.

In order to prove Theorem 1.1, besides the gradient estimate mentioned above,
the other key issue is to bound the second order derivatives at the boundary
in terms of a constant depending not on $(\delta_{\psi,f})^{-1}$, where

$\delta_{\psi,f}=\inf_{M}\psi-\sup_{\partial\Gamma}f.$

Such a boundary estimate is fairly delicate. To this end, following some idea
from [20]†††The paper [20] is essentially extracted from [arXiv:2203.03439]
and the first parts of [arXiv:2001.09238; arXiv:2106.14837]., we achieve such
two ingredients simultaneity through deriving a quantitative boundary
estimate. It is very different from previous works [7, 9, 14].

######  Theorem 1.2.

Assume that $(f,\Gamma)$ satisfies (1.4), (1.5) and (1.11). Let $(M,g)$ be a
compact Riemannian manifold with smooth boundary subject to (1.6). Let
$\varphi\in C^{\infty}(\partial M)$ and $\psi\in C^{\infty}(\bar{M})$ satisfy
(1.12). Suppose in addition that the Dirichlet problem (1.1) admits a
$C^{2}$-admissible subsolution $\underline{u}\in C^{2}(\bar{M})$ satisfying
(1.7). Then for any admissible solution $u\in C^{3}(M)\cap C^{2}(\bar{M})$ to
(1.1), we have

$\displaystyle\sup_{\partial M}\Delta u\leq C(1+\sup_{M}|\nabla u|^{2})$
(1.13)

where $C$ is a constant depending not on $(\delta_{\psi,f})^{-1}$.

The paper is organized as follows. In Section 2 we summarize some lemmas. They
are key ingredients in proof of Theorem 1.2. In Sections 3 and 4 we derive the
quantitative boundary estimate. Finally, we complete the proof of Theorem 1.1
in Section 5.

### Acknowledgements

The author was supported by the National Natural Science of Foundation of
China, Grant No. 11801587.

##  2 Preliminaries

The lemma below plays an important role in the proof of Proposition 3.1.

######  Lemma 2.1 ([5, Lemma 2.2]).

Suppose (1.4) and (1.5) hold. Let $K$ be a compact subset of $\Gamma$ and
$\beta_{0}>0$. There is a constant $\varepsilon>0$ such that, for $\mu\in K$
and $\lambda\in\Gamma$, when $|\nu_{\mu}-\nu_{\lambda}|\geq\beta_{0}$,

$\displaystyle\sum_{i=1}^{n}f_{i}(\lambda)(\mu_{i}-\lambda_{i})\geq
f(\mu)-f(\lambda)+\varepsilon(1+\sum_{i=1}^{n}f_{i}(\lambda)).$ (2.1)

Here $\nu_{\lambda}=Df(\lambda)/|Df(\lambda)|$ denotes the unit normal vector
to the level set,
$\\{\lambda^{\prime}\in\Gamma:f(\lambda^{\prime})=f(\lambda)\\}$, passing
through $\lambda$, where $Df(\lambda)=(f_{1}(\lambda),\cdots,f_{n}(\lambda))$.

The following lemmas are key ingredients for quantitative boundary estimate
for pure normal derivatives.

######  Lemma 2.2 ([20]).

‡‡‡The lemma was proposed in [arXiv:2203.03439] that was reorganized as a part
of [20].

Let $A$ be an $n\times n$ Hermitian matrix

$\left(\begin{matrix}d_{1}&&&&a_{1}\\\ &d_{2}&&&a_{2}\\\ &&\ddots&&\vdots\\\
&&&d_{n-1}&a_{n-1}\\\
\bar{a}_{1}&\bar{a}_{2}&\cdots&\bar{a}_{n-1}&\mathrm{{\bf
a}}\end{matrix}\right)$ (2.2)

with $d_{1},\cdots,d_{n-1},a_{1},\cdots,a_{n-1}$ fixed, and with $\mathrm{{\bf
a}}$ variable. Denote the eigenvalues of $A$ by
$\lambda=(\lambda_{1},\cdots,\lambda_{n})$. Let $\epsilon>0$ be a fixed
constant. Suppose that the parameter $\mathrm{{\bf a}}$ in $A$ satisfies the
quadratic growth condition

$\displaystyle\mathrm{{\bf
a}}\geq\frac{2n-3}{\epsilon}\sum_{i=1}^{n-1}|a_{i}|^{2}+(n-1)\sum_{i=1}^{n-1}|d_{i}|+\frac{(n-2)\epsilon}{2n-3}.$
(2.3)

Then the eigenvalues (possibly with a proper permutation) behave like

$\displaystyle|d_{\alpha}-\lambda_{\alpha}|<\epsilon,\mbox{ }\forall
1\leq\alpha\leq n-1;\quad 0\leq\lambda_{n}-\mathrm{{\bf a}}<(n-1)\epsilon.$

The concavity of $f$ yields that

$\displaystyle\mbox{For any }\lambda,\mbox{ }\mu\in\Gamma,\mbox{
}\sum_{i=1}^{n}f_{i}(\lambda)\mu_{i}\geq\limsup_{t\rightarrow+\infty}f(t\mu)/t.$

Inspired by this observation, the author [20] introduced the following
conditions:

$\displaystyle\mbox{For any }\lambda\in\Gamma,\mbox{
}\lim_{t\rightarrow+\infty}f(t\lambda)>-\infty,$ (2.4) $\displaystyle\mbox{For
any }\lambda\in\Gamma,\mbox{ }\limsup_{t\rightarrow+\infty}f(t\lambda)/t\geq
0.$ (2.5)

Obviously, it leads to

######  Lemma 2.3 ([20]).

Suppose $f$ satisfies (1.5) and (2.5). Then

$\displaystyle\sum_{i=1}^{n}f_{i}(\lambda)\mu_{i}\geq 0\mbox{ for any
}\lambda,\mbox{ }\mu\in\Gamma.$ (2.6)

In particular

$\displaystyle\sum_{i=1}^{n}f_{i}(\lambda)\lambda_{i}\geq
0,\quad\forall\lambda\in\Gamma.$ (2.7)

If, in addition, $\sum_{i=1}^{n}f_{i}(\lambda)>0$ then

$\displaystyle\sum_{i=1}^{n}f_{i}(\lambda)\mu_{i}>0\mbox{ for any
}\lambda,\mbox{ }\mu\in\Gamma.$ (2.8)

We now give characterizations of concave functions satisfying (1.11).

######  Lemma 2.4 ([20]).

In the presence of (1.4) and (1.5), the following statements are equivalent to
each other.

  1. (1)

$f$ satisfies (1.11).

  2. (2)

$f$ satisfies (2.4).

  3. (3)

$f$ satisfies (2.5).

  4. (4)

$f$ satisfies (2.6).

  5. (5)

$f$ satisfies (2.7).

  6. (6)

$f$ satisfies (2.8).

##  3 Quantitative boundary estimate for mixed derivatives

For a point $x_{0}\in\partial M$, we shall choose local coordinates

$\displaystyle x=(x_{1},\cdots,x_{n})$ (3.1)

with origin at $x_{0}$ such that, when restricted to $\partial M$,
$\frac{\partial}{\partial x_{n}}$ is normal to $\partial M$; moreover, we
assume $g_{ij}(x_{0})=\delta_{ij}$.

We will carry out the computations in such local coordinates, and set

$\nabla_{i}=\nabla_{\frac{\partial}{\partial
x_{i}}},\quad\nabla_{ij}=\frac{\partial^{2}}{\partial x_{i}\partial
x_{j}}-\Gamma_{ij}^{k}\frac{\partial}{\partial x_{k}},$

with a similar convention for higher derivatives, where $\Gamma^{k}_{ij}$ are
the Christoffel symbols

$\nabla_{\frac{\partial}{\partial x_{i}}}\frac{\partial}{\partial
x_{j}}=\Gamma_{ij}^{k}\frac{\partial}{\partial x_{k}}.$

######  Proposition 3.1.

Suppose, in addition to (1.4), (1.5) and (1.11), that the data $\varphi\in
C^{3}(\partial M)$, $\psi\in C^{1}(\bar{M})$ satisfies (1.12), and that there
is an admissible subsolution $\underline{u}\in C^{2}(\bar{M})$. Then for any
$x_{0}\in\partial M$, under local coordinate system (3.1), the admissible
solution $u$ of Dirichlet problem (1.1) must satisfy

$\displaystyle|\nabla_{\alpha n}u(x_{0})|\leq C(1+\sup_{M}|\nabla
u|),\quad\forall 1\leq\alpha\leq n-1$ (3.2)

where $C$ is a uniform positive constant depending not on
$(\delta_{\psi,f})^{-1}$.

###  3.1 Useful formulas and notation

Throughout this paper we use the notation

$\mathfrak{g}=\nabla^{2}u+\chi,\quad\mathfrak{\underline{g}}=\nabla^{2}\underline{u}+\chi.$

By direct computation one has

$\displaystyle\nabla_{ij}\nabla_{k}u=\frac{\partial^{3}u}{\partial
x_{i}\partial x_{j}\partial
x_{k}}-\Gamma_{ij}^{l}\frac{\partial^{2}u}{\partial x_{k}\partial x_{l}},$
(3.3) $\displaystyle\nabla_{ijk}u=$
$\displaystyle\frac{\partial^{3}u}{\partial x_{i}\partial x_{j}\partial
x_{k}}-\frac{\partial\Gamma_{ij}^{l}}{\partial x_{k}}\frac{\partial
u}{\partial x_{l}}-\Gamma_{ij}^{l}\Gamma_{kl}^{p}\frac{\partial u}{\partial
x_{p}}-\Gamma_{ij}^{l}\nabla_{kl}u-\Gamma_{ki}^{l}\nabla_{lj}u-\Gamma_{kj}^{l}\nabla_{il}u.$
(3.4)

Let $\sigma(x)$ be the distance function from $x\in M$ to $\partial M$,
$\rho(x)$ be the distance from $x$ to $x_{0}$. We denote

$\Omega_{\delta}=\\{x\in M:\rho(x)<\delta\\}.$

The boundary value condition, $u=\underline{u}=\varphi$ on $\partial M$,
implies that

$\displaystyle\nabla_{\alpha}(u-\varphi)=\frac{\nabla_{\alpha}\sigma}{\nabla_{n}\sigma}\nabla_{n}(u-\varphi)\mbox{
on }\partial M\cap\overline{\Omega}_{\delta},$ (3.5)
$\displaystyle\nabla_{\alpha\beta}u=\nabla_{\alpha\beta}\underline{u}+\frac{\nabla_{\alpha\beta}\sigma}{\nabla_{n}\sigma}\nabla_{n}(u-\underline{u})\mbox{
at }x_{0}.$ (3.6)

Let $h$ be the solution to

$\displaystyle\Delta h+\mathrm{tr}_{g}\chi=0\mbox{ in }M,\quad h=\varphi\mbox{
on }\partial M.$ (3.7)

The maximum principle also yields

$\displaystyle\underline{u}\leq u\leq h\mbox{ in }\partial M,$ (3.8)

then at $x_{0}$

$\displaystyle\nabla_{n}\underline{u}\leq\nabla_{n}u\leq\nabla_{n}h.$ (3.9)

In particular, we have $C^{0}$-bound and boundary gradient estimate

$\sup_{M}|u|+\sup_{\partial M}|\nabla u|\leq C.$ (3.10)

Under local coordinates (3.1), the tangential operator on boundary is given by

$\mathcal{T}=\pm\left({\frac{\partial}{\partial
x_{\alpha}}}-\eta\frac{\partial}{\partial x_{n}}\right),\mbox{ for
}1\leq\alpha\leq n-1,$ (3.11)

where $\eta={\nabla_{\alpha}\sigma}/{\nabla_{n}\sigma}$. The (3.5) means that

$\mathcal{T}(u-\varphi)=0\mbox{ on }\partial M\cap\overline{\Omega}_{\delta}.$
(3.12)

###  3.2 Completion of the proof of Proposition 3.1

Let $\mathcal{L}$ be the linearized operator of equation (1.1) at $u$.
Locally, it is given by

$\displaystyle\mathcal{L}v=F^{ij}\nabla_{ij}v\quad\mbox{ for }v\in C^{2}(M),$

where $F^{ij}=\frac{\partial F}{\partial a_{ij}}(\mathfrak{g})$. First, we
have the following lemma.

######  Lemma 3.2.

Let $u\in C^{3}(M)\cap C^{1}(\bar{M})$ be an admissible solution to equation
(1.1). For some small $\delta>0$, we have

$\displaystyle\left|\mathcal{L}(\mathcal{T}(u-\varphi))\right|\leq
C\left(1+(1+\sup_{M}|\nabla
u|)\sum_{i=1}^{n}f_{i}+\sum_{i=1}^{n}f_{i}|\lambda_{i}|\right),\mbox{ in
}\Omega_{\delta}$ (3.13)

where $C$ depends on $|\varphi|_{C^{3}(M)}$, $|\chi|_{C^{1}(M)}$,
$\psi_{C^{1}(M)}$ and other known data (but not on $(\delta_{\psi,f})^{-1}$).

###### Proof.

Differentiating the equation one has

$\displaystyle F^{ij}\nabla_{ij\alpha}u=\nabla_{\alpha}\psi.$

Combining with (3.3)-(3.4) one derives (3.13). ∎

Proposition 3.1 can be proved by constructing barrier functions similar to
that used in [20] in complex variables. The construction of this type of
barriers follows [6]. Let’s take

$\displaystyle\widetilde{\Psi}=$ $\displaystyle
A_{1}\sqrt{b_{1}}(\underline{u}-u)-A_{2}\sqrt{b_{1}}\rho^{2}+A_{3}\sqrt{b_{1}}(N\sigma^{2}-t\sigma)$
$\displaystyle+\frac{1}{\sqrt{b_{1}}}\sum_{\tau<n}|\nabla_{\tau}(u-\varphi)|^{2}+\mathcal{T}(u-\varphi),$

where $b_{1}=1+\sup_{M}|\nabla u|^{2}$.

Let $\delta>0$ and $t>0$ be sufficiently small such that $N\delta-t\leq 0$
(where $N$ is a positive constant sufficiently large to be determined),
$\sigma$ is $C^{2}$ in $\Omega_{\delta}$ and

$\displaystyle\frac{1}{2}\leq|\nabla\sigma|\leq 2,\mbox{
}|\mathcal{L}\sigma|\leq C_{2}\sum_{i=1}^{n}f_{i},\mbox{
}|\mathcal{L}\rho^{2}|\leq C_{2}\sum_{i=1}^{n}f_{i}\mbox{ in }\Omega_{\delta}$
(3.14)

and

$\displaystyle\mathcal{L}(N\sigma^{2}-t\sigma)=(2N\sigma-t)\mathcal{L}\sigma+2NF^{ij}\nabla_{i}\sigma\nabla_{j}\sigma\mbox{
in }\Omega_{\delta}.$ (3.15)

Furthermore, we can choose $\delta$ and $t$ small enough such that
$|2N\delta-t|$ is small.

By straightforward calculation and $|a-b|^{2}\geq\frac{1}{2}|a|^{2}-|b|^{2}$,
one derives

$\displaystyle\mathcal{L}(\sum_{\tau<n}|\nabla_{\tau}(u-\varphi)|^{2})\geq\sum_{\tau<n}F^{ij}\mathfrak{g}_{\tau
i}\mathfrak{g}_{\tau
j}-C_{1}^{\prime}\sqrt{b_{1}}\sum_{i=1}^{n}f_{i}|\lambda_{i}|-C_{1}^{\prime}b_{1}\sum_{i=1}^{n}f_{i}-C_{1}^{\prime}\sqrt{b_{1}}.$
(3.16)

By [4, Proposition 2.19], there is an index $r$ such that

$\displaystyle\sum_{\tau<n}F^{ij}\mathfrak{g}_{\tau i}\mathfrak{g}_{\tau
j}\geq\frac{1}{2}\sum_{i\neq r}f_{i}\lambda_{i}^{2}.$

Using Lemma 2.4 and
$\sum_{i=1}^{n}f_{i}(\underline{\lambda}_{i}-\lambda_{i})\geq 0$ respectively,
we obtain

$\sum_{i=1}^{n}f_{i}|\lambda_{i}|=2\sum_{\lambda_{i}\geq
0}f_{i}\lambda_{i}-\sum_{i=1}^{n}f_{i}\lambda_{i}<2\sum_{\lambda_{i}\geq
0}f_{i}\lambda_{i},$
$\sum_{i=1}^{n}f_{i}|\lambda_{i}|=\sum_{i=1}^{n}f_{i}\lambda_{i}-2\sum_{\lambda_{i}<0}f_{i}\lambda_{i}<\sum_{i=1}^{n}f_{i}\underline{\lambda}_{i}-2\sum_{\lambda_{i}<0}f_{i}\lambda_{i}.$

In conclusion, we have

$\displaystyle\sum_{i=1}^{n}f_{i}|\lambda_{i}|\leq\frac{\epsilon}{4\sqrt{b_{1}}}\sum_{i\neq
r}f_{i}\lambda_{i}^{2}+\frac{C\sqrt{b_{1}}}{\epsilon}\sum_{i=1}^{n}f_{i}.$
(3.17)

By (3.14), (3.15), (3.16), Lemma 3.2 and (3.17), we obtain

$\displaystyle\mathcal{L}(\widetilde{\Psi})\geq$ $\displaystyle
A_{1}\sqrt{b_{1}}\mathcal{L}(\underline{u}-u)+\frac{1}{2\sqrt{b_{1}}}\sum_{i\neq
r}f_{i}\lambda_{i}^{2}+A_{3}\sqrt{b_{1}}\mathcal{L}(N\sigma^{2}-t\sigma)$
(3.18) $\displaystyle-
C_{1}-C_{1}\sum_{i=1}^{n}f_{i}|\lambda_{i}|-\left(A_{2}C_{2}+C_{1}\right)\sqrt{b_{1}}\sum_{i=1}^{n}f_{i}$
$\displaystyle\geq$ $\displaystyle
A_{1}\sqrt{b_{1}}\mathcal{L}(\underline{u}-u)+2NA_{3}\sqrt{b_{1}}F^{ij}\nabla_{i}\sigma\nabla_{j}\sigma$
$\displaystyle+A_{3}(2N\sigma-t)\sqrt{b_{1}}\mathcal{L}\sigma-C\sqrt{b_{1}}\sum_{i=1}^{n}f_{i}-C.$

Next we will prove

$\displaystyle\mathcal{L}\widetilde{\Psi}\geq 0,\mbox{ in }\Omega_{\delta}$

for $0<\delta\ll 1$, if we appropriately choose $A_{1}\gg A_{2}\gg A_{3}>1$,
$N\gg 1$ and $0<t\ll 1$.

Case I: If $|\nu_{\lambda}-\nu_{\underline{\lambda}}|\geq\beta_{0}$, then by
Lemma 2.1 we have

$\displaystyle\sum_{i=1}^{n}f_{i}(\underline{\lambda}_{i}-\lambda_{i})\geq\varepsilon\left(1+\sum_{i=1}^{n}f_{i}\right),$
(3.19)

where we take
$\beta_{0}=\frac{1}{2}\min_{\bar{M}}dist(\nu_{\underline{\lambda}},\partial\Gamma_{n})$
as above, $\varepsilon$ is the positive constant in Lemma 2.1. Taking
$A_{1}\gg A_{3}\gg 1$ we obtain

$\mathcal{L}\widetilde{\Psi}\geq 0\mbox{ on }\Omega_{\delta}.$

Case II: Suppose that $|\nu_{\lambda}-\nu_{\underline{\lambda}}|<\beta_{0}$.
Then $\nu_{\lambda}-\beta_{0}\vec{\bf 1}\in\Gamma_{n}$ and

$\displaystyle f_{i}\geq\frac{\beta_{0}}{\sqrt{n}}\sum_{j=1}^{n}f_{j}.$ (3.20)

By (3.14), we have $|\nabla\sigma|\geq\frac{1}{2}$ in $\Omega_{\delta}$, then

$\displaystyle
2A_{1}N\sqrt{b_{1}}F^{i\bar{j}}\sigma_{i}\sigma_{\bar{j}}\geq\frac{A_{1}N\beta_{0}\sqrt{b_{1}}}{2\sqrt{n}}\sum_{i=1}^{n}f_{i}\mbox{
on }\Omega_{\delta}.$ (3.21)

On the other hand, $\mathcal{L}(\underline{u}-u)\geq 0$. Thus

$\mathcal{L}(\widetilde{\Psi})\geq 0\mbox{ on }\Omega_{\delta},\mbox{ if
}A_{1}N\gg 1.$

The boundary value condition $u-\varphi=0$ on $\partial M$ implies
$\mathcal{T}(u-\varphi)=0$ and $|\nabla_{\tau}(u-\varphi)|\leq C\rho$ on
$\partial M\cap\overline{\Omega}_{\delta}$. Thus

$\displaystyle\widetilde{\Psi}=$ $\displaystyle
A_{1}\sqrt{b_{1}}(\underline{u}-u)-A_{2}\sqrt{b_{1}}\rho^{2}+A_{3}\sqrt{b_{1}}(N\sigma^{2}-t\sigma)$
$\displaystyle+\frac{1}{\sqrt{b_{1}}}\sum_{\tau<n}|\nabla_{\tau}(u-\varphi)|^{2}+\mathcal{T}(u-\varphi)\leq
0,\mbox{ on }\partial M\cap\overline{\Omega}_{\delta}.$

Note that $\rho=\delta$ and $\underline{u}-u\leq 0$ on
$M\cap\partial\Omega_{\delta}$. Hence, if $A_{2}\gg 1$ then
$\widetilde{\Psi}\leq 0$ on $M\cap\partial\Omega_{\delta}$, where we use
$N\delta-t\leq 0$. Therefore $\widetilde{\Psi}\leq 0$ in $\Omega_{\delta}$ by
applying maximum principle. Together with $\widetilde{\Psi}(0)=0$, one has
$\nabla_{\nu}\widetilde{\Psi}(0)\leq 0$. Thus

$\displaystyle\nabla_{\nu}\mathcal{T}(u-\varphi)(0)\leq
C^{\prime}(1+\sup_{M}|\nabla u|).$

Here we use (3.10). Therefore

$\displaystyle\pm\nabla_{\alpha n}u\leq C(1+\sup_{\bar{M}}|\nabla u|),\mbox{
at }x_{0},$ (3.22)

where $C$ depends only on $|\varphi|_{C^{3}(M)}$, $|\underline{u}|_{C^{2}(M)}$
$|\psi|_{C^{1}(M)}$ and other known data (but not on $\sup_{M}|\nabla u|$).

Moreover, the constant $C$ in (3.22) does not depend on
$(\delta_{\psi,f})^{-1}$.

##  4 Quantitative boundary estimate for pure normal derivatives

Given $x_{0}\in\partial M$. As in (3.1), we choose local coordinate
$x=(x_{1},\cdots,x_{n})$ with origin at $x_{0}$ such that, when restricted to
$\partial M$, $\frac{\partial}{\partial x_{n}}$ is a inner normal vector to
$\partial M$, and $g_{ij}(x_{0})=\delta_{ij}$. Furthermore, we assume
$\\{\mathfrak{\underline{g}}_{\alpha\beta}\\}$ is diagonal at $x_{0}$.

When imposing assumption (1.6) on boundary we can derive

######  Proposition 4.1.

Let $(M,g)$ be a compact Riemannian manifold with smooth boundary satisfying
(1.6). Suppose, in addition to (1.4), (1.5), (1.11) and (1.12), that there
exists an admissible subsolution $\underline{u}\in C^{2}(\bar{M})$. Then any
admissible solution $u\in C^{2}(\bar{M})$ of the Dirichlet problem (1.1)
satisfies

$\displaystyle\nabla_{nn}u(x_{0})\leq
C\left(1+\sum_{\alpha=1}^{n-1}|\mathfrak{g}_{\alpha
n}|^{2}\right),\quad\forall x_{0}\in\partial M,$

where $C$ does not depend on $(\delta_{\psi,f})^{-1}$.

###### Proof.

In what follows the discussion is done at $x_{0}$, and the Greek letters
$\alpha,\beta$ range from $1$ to $n-1$. By (3.6)

$\displaystyle\mathfrak{g}_{\alpha\beta}=\underline{\mathfrak{g}}_{\alpha\beta}+\nabla_{n}(u-\underline{u})\nabla_{\alpha\beta}\sigma$
(4.1)

The proof consists of two steps.

Step 1. As in [20] there exist two uniform positive constants
$\varepsilon_{0}$, $R_{0}$ depending on $\mathfrak{\underline{g}}$ and $f$,
such that

$\displaystyle
f(\mathfrak{\underline{g}}_{11}-\varepsilon_{0},\cdots,\underline{\mathfrak{g}}_{(n-1){(n-1)}}-\varepsilon_{0},R_{0})\geq\psi,\quad$
(4.2)
$\displaystyle(\mathfrak{\underline{g}}_{11}-\varepsilon_{0},\cdots,\underline{\mathfrak{g}}_{(n-1){(n-1)}}-\varepsilon_{0},R_{0})\in\Gamma.$

Step 2. Next, we apply Lemmas ‡ ‣ 2.2 and 2.4 derive the quantitative boundary
estimates for double normal derivative. Let’s denote

$I(r)=\left(\begin{matrix}1&&&0\\\ &\ddots&&\vdots\\\ &&1&0\\\
0&\cdots&0&r\end{matrix}\right)$
$A(R)=\left(\begin{matrix}\mathfrak{g}_{11}&\mathfrak{g}_{12}&\cdots&\mathfrak{g}_{1(n-1)}&\mathfrak{g}_{1n}\\\
\mathfrak{g}_{21}&\mathfrak{g}_{22}&\cdots&\mathfrak{g}_{2(n-1)}&\mathfrak{g}_{2n}\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
\mathfrak{g}_{(n-1)1}&\mathfrak{g}_{(n-1)2}&\cdots&\mathfrak{g}_{{(n-1)}{(n-1)}}&\mathfrak{g}_{(n-1)n}\\\
\mathfrak{g}_{n1}&\mathfrak{g}_{n2}&\cdots&\mathfrak{g}_{n{(n-1)}}&R\end{matrix}\right)$
$\underline{A}(R)=\left(\begin{matrix}\mathfrak{\underline{g}}_{11}&&&&\mathfrak{g}_{1n}\\\
&\mathfrak{\underline{g}}_{22}&&&\mathfrak{g}_{2n}\\\ &&\ddots&&\vdots\\\
&&&\mathfrak{\underline{g}}_{{(n-1)}{(n-1)}}&\mathfrak{g}_{(n-1)n}\\\
\mathfrak{g}_{n1}&\mathfrak{g}_{n2}&\cdots&\mathfrak{g}_{n{(n-1)}}&R\end{matrix}\right)$
$B=\left(\begin{matrix}\nabla_{11}\sigma&\nabla_{12}\sigma&\cdots&\nabla_{1(n-1)}\sigma&0\\\
\nabla_{21}\sigma&\nabla_{22}\sigma&\cdots&\nabla_{2(n-1)}\sigma&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
\nabla_{(n-1)1}\sigma&\nabla_{(n-1)2}\sigma&\cdots&\nabla_{{(n-1)}{(n-1)}\sigma}&0\\\
0&0&\cdots&0&0\end{matrix}\right)$

Let $0<\varepsilon_{1}\ll 1$ satisfy

$\varepsilon_{1}\nabla_{n}(u-\underline{u})<\frac{\varepsilon_{0}}{8}.$

The assumption (1.6) on boundary implies that, for the fixed
$\varepsilon_{1}\in\mathbb{R}^{+}$ given above, there exists a uniform
positive constant $r_{0}$ such that

$B+\varepsilon_{1}I(r_{0}/\varepsilon_{1})\in\Gamma.$ (4.3)

We fix $\varepsilon_{0}$, $\varepsilon_{1}$, and $r_{0}$ that we have chosen.
From (4.2) we see that

$\displaystyle\lambda(\underline{A}(R)-\varepsilon_{1}\nabla_{n}(u-\underline{u})I(r_{0}/\varepsilon_{1}))\in\Gamma,\quad
R\geq R_{0}+r_{0}\nabla_{n}(u-\underline{u}).$ (4.4)

On the other hand, the identity (4.1) means that

$\displaystyle
A(R)=\left[\underline{A}(R)-\varepsilon_{1}\nabla_{n}(u-\underline{u})I(r_{0}/\varepsilon_{1})\right]+\nabla_{n}(u-\underline{u})[B+\varepsilon_{1}I(r_{0}/\varepsilon_{1})].$

For simplicity, we denote

$\underline{H}(R)=\underline{A}(R)-\varepsilon_{1}\nabla_{n}(u-\underline{u})I(r_{0}/\varepsilon_{1}).$

Notice that $\nabla_{n}(u-\underline{u})\geq 0$. According to Lemma 2.4 we
obtain

$\displaystyle f(A(R))\geq f(\underline{H}(R)).$ (4.5)

Let’s pick the parameter $\epsilon=\frac{\varepsilon_{0}}{8}$ in Lemma ‡ ‣
2.2, and we assume

$\displaystyle
R_{c}=\frac{8(2n-3)}{\varepsilon_{0}}\sum_{\alpha=1}^{n-1}|\mathfrak{g}_{\alpha
n}|^{2}+(n-1)\sum_{\alpha=1}^{n-1}\left(|\mathfrak{\underline{g}}_{\alpha\alpha}|+\frac{\varepsilon_{0}}{8}\right)+\frac{(n-2)\varepsilon_{0}}{8(2n-3)}+R_{0}+r_{0}\nabla_{n}(u-\underline{u}),$

where $\varepsilon_{0}$ and $R_{0}$ are the constants from (4.2).

Lemma ‡ ‣ 2.2 applies to $\underline{H}(R_{c})$ and the eigenvalues of
$\underline{H}(R_{c})$ (possibly with an order) shall behave like

$\displaystyle\lambda(\underline{H}(R_{c}))\in\left(\mathfrak{\underline{g}}_{1\bar{1}}-\frac{\varepsilon_{0}}{4},\cdots,\mathfrak{\underline{g}}_{(n-1)\overline{(n-1)}}-\frac{\varepsilon_{0}}{4},R_{c}\right)+\overline{\Gamma}_{n}\subset\Gamma.$
(4.6)

Applying (1.4), (4.2), (4.5) and (4.6), we hence derive

$\displaystyle F(A(R_{c}))\geq F(\underline{H}(R_{c}))\geq
f(\mathfrak{\underline{g}}_{11}-\frac{\varepsilon_{0}}{4},\cdots,\mathfrak{\underline{g}}_{(n-1){(n-1)}}-\frac{\varepsilon_{0}}{4},R_{c})>\psi$

which then gives $\mathfrak{g}_{nn}(x_{0})\leq R_{c}.$

∎

##  5 Proof of existence results

The condition (1.11) is necessary for the blow-up argument in [17].

######  Lemma 5.1 ([20]).

Suppose $f$ satisfies (1.3)-(1.5). Then it obeys (1.11).

###### Proof.

Since $f$ satisfies (1.3), we have

$\displaystyle\lim_{t\rightarrow 0^{+}}f(t\vec{\bf 1})=f(\vec{\bf
0})>-\infty.$ (5.1)

For $\lambda\in\Gamma$ there is $t_{\lambda}>0$ such that
$f(t_{\lambda}\vec{\bf 1})=f(\lambda)$. Thus

$f(\lambda)\geq f(\vec{\bf 0}).$

Consequently, (1.11) holds according to Lemma 2.4.

∎

As shown in (3.10), we have $C^{0}$-estimate and boundary gradient estimate

$\sup_{M}|u|+\sup_{\partial M}|\nabla u|\leq C.$

From Propositions 3.1 and 4.1 we deduce Theorem 1.2. Namely,

$\displaystyle\sup_{\partial M}\Delta u\leq C(1+\sup_{M}|\nabla u|^{2}).$

By $\Gamma\subset\Gamma_{1}$, $\Delta u>-\mathrm{tr}_{g}\chi.$ Together with
the second estimate,

$\displaystyle\sup_{M}\Delta u\leq C(1+\sup_{M}|\nabla u|^{2}+\sup_{\partial
M}|\Delta u|)$

announced in [17, Section 8], we obtain

$\displaystyle\sup_{M}\Delta u\leq C(1+\sup_{M}|\nabla u|^{2}).$ (5.2)

With such a second order estimate at hand, we can derive the gradient estimate
via a blow-up argument (see [2, 17]), thereby establishing (1.9). Notice
furthermore that the obtained estimates are independent of
$(\delta_{\psi,f})^{-1}$, we can prove Theorem 1.1 by standard approximate
method.

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