# A Simple and Plug-and-play Method for Unsupervised Sentence Representation
Enhancement

Lingfeng Shen
Johns Hopkins Iniversity &Haiyun Jiang
Tencent AI Lab &Lemao Liu
Tencent AI Lab &Shuming Shi
Tencent AI Lab

###### Abstract

Generating proper embedding of sentences through an unsupervised way is
beneficial to semantic matching and retrieval problems in real-world
scenarios. This paper presents Representation ALchemy (RepAL), an extremely
simple post-processing method that enhances sentence representations. The
basic idea in RepAL is to de-emphasize redundant information of sentence
embedding generated by pre-trained models. Through comprehensive experiments,
we show that RepAL is free of training and is a plug-and-play method that can
be combined with most existing unsupervised sentence learning models. We also
conducted in-depth analysis to understand RepAL.

##  1 Introduction

Learning high-quality sentence embeddings is a fundamental task in Natural
Language Processing (NLP) field Socher et al. (2011); Le and Mikolov (2014);
Kiros et al. (2015); Reimers and Gurevych (2019); Gao et al. (2021). In real-
world scenarios, especially when a large amount of supervised data is
unavailable, an approach that provides high-quality sentence embeddings in an
unsupervised paradigm is of great value.

Generally, the unsupervised sentence encoder (USE) can be categorized into two
paradigms. The first is pre-trained language model (PTM) Devlin et al. (2019);
Liu et al. (2019) based paradigm, which are naturally good unsupervised
sentence representation learning models. For example, BERT Devlin et al.
(2019) and BERT-like Liu et al. (2019); He et al. (2020); Raffel et al. (2020)
models, commit to design stronger pre-trained language models by self-training
with mask or next sentence prediction. While designing stronger PTMs for
better sentence representation is extremely expensive, time-consuming, and
labor-intensive. Based on PTMs, secondary trained, e.g., contrastive-based
methods Reimers and Gurevych (2019); Logeswaran and Lee (2018); Gao et al.
(2021), proved to be effective to further improve the representation quality
of sentences. For example, SimCSE Gao et al. (2021) minimizes the distance
between positive pairs of sentences and pulls away from the negative pairs of
sentences in the embedding space, which achieves promising performance.

This paper focuses on enhancing sentence embeddings generated by above two
paradigms in an supervised way. Our basic idea is to _refine sentence
representations by removing the redundant information from the levels of
sentences and corpus_. Considering corpus-level, there may be shared
information among the corpus, which may lead to homogenous properties for all
sentence embeddings, which diminishes the distinctiveness between sentences.
As for sentence-level, there are several trivial words within the sentence.
Such words are proven to bring a negative impact on downstream NLP tasks, like
Natural Language Inference (NLI) Mahabadi et al. (2020); Zhou and Bansal
(2020) and text classification Choi et al. (2020); Qian et al. (2021).

Therefore, we propose a simple, straightforward, and effective method called
representation alchemy (RepAL), which improves sentence representations
without training and extra resources. RepAL accepts raw sentence
representations as inputs, which are generated from existing unsupervised
sentence models. Then RepAL outputs refined representations by extracting two
redundant representations from different perspectives. Intuitively, it is like
_an alchemy that improves sentence representation by refinement_. It’s worth
mentioning that our proposed RepAL can be applied to almost USEs and is a
plug-and-play method in sentence embedding enhancement without extra training
cost. To verify, we perform extensive experiments on both English and Chinese
benchmarks, and results demonstrate the effectiveness of the proposed RepAL.

##  2 Related Work

Methods for unsupervised sentence learning have been extensively explored.
Early works are mainly based on distributional hypothesis Socher et al.
(2011); Le and Mikolov (2014). Hill Hill et al. (2016) proposed to learn
sentence representations with its internal structure. Then Pagliardini
Pagliardini et al. (2018) proposed Sent2Vec, a simple unsupervised model
allowing to compose sentence embeddings using word vectors.

Then strong pre-trained language model Devlin et al. (2019) emerged from the
blue. Such pre-trained models own potential to improve the quality of sentence
representation. However, models like BERT own strong anisotropy in their
embedding space, which means the sentence embeddings produced by BERT have
extremely high cosine similarity, leading to unsatisfactory performances on
sentence embedding.

Recently, contrastive learning began to play an important role in unsupervised
sentence representation learning Zhang et al. (2020); Yan et al. (2021); Meng
et al. (2021); Gao et al. (2021); Wang et al. (2021). Such methods are based
on the assumption that high-quality embedding methods should bring similar
sentences closer while pushing away dissimilar ones.

Specifically, the most relevant work is BERT-whitening Su et al. (2021a, a), a
post-processing method, and a detailed comparison between it and our work is
illustrated in Appendix A.

##  3 Methodology

###  3.1 Problem Formulation

In unsupervised sentence representation learning, we take a collection of
unlabeled sentences $\\{x_{i}\\}_{i=1}^{n}$, also we choose a suitable
unsupervised sentence learning model (e.g., BERT) as the encoder
$f(\cdot;\theta)$, where $\theta$ represents the trainable parameters in $f$.
Specifically, we have a carefully designed training objective $\mathcal{L}$
for unsupervised training, and $\theta$ is then fixed as $\theta_{0}$ where
$\theta_{0}=\operatorname{argmin}\mathcal{L}$. Finally, we obtain the sentence
representation $v_{i}$ for $x_{i}$ by feeding it into the encoder, i.e.,
$v_{i}=f(x_{i};\theta_{0})$.

RepAL plays its role in refining $v_{i}$ to $v_{i}^{\prime}$ with
$v_{i}^{\prime}=g(v_{i})$, instead of directly selecting $v_{i}$ for sentence
representation. RepAL aims to extract and refine two types of redundancy,
namely _sentence-level redundancy_ and _corpus-level redundancy_ ,
respectively. Sentence-level redundancy denotes the useless word information
hidden in the target sentence, which may bias the representations that reflect
the core semantics of the sentence. Corpus-level redundancy denotes the shared
redundant information in all sentence representations within the dataset,
making all the representations tend to be homogenous and thus reducing the
distinction.

RepAL generates $x_{i}^{*}$ by an operation called _partial mask_ on $x_{i}$,
then feed $x_{i}^{*}$ into the encoder $f(.;\theta_{0})$ to obtain sentence-
level redundancy embeddings $v_{i}^{*}$. Besides, RepAL produces a global
vector $\hat{v}$ as corpus-level redundancy embedding. Finally, RepAL
generates the refined embedding $v_{i}^{\prime}$ for downstream tasks through
the _embedding refinement_ operation by combining $v_{i},v_{i}^{*}$ and
$\hat{v}$.

###  3.2 Redundant Embedding Generation

In RepAL, we firstly detect redundant information and generate their
embeddings from the target sentence, which is a groundbreaking step in our
method and determines the performance. Moreover, we also conduct deep analyses
towards two kinds of redundancy, and defer them to Appendix C.

####  3.2.1 Sentence-level Redundancy

We apply a partial mask to extract the sentence-level redundancy.
Specifically, given a sentence $x_{i}=\\{w_{1},w_{2},\dots,w_{N}\\}$, partial
mask generates a partially masked sentence $x_{i}^{*}$, a mask version of
$x_{i}$, where _informative_ words in $x_{i}$ are replaced with [MASK] to
distill the trivial words from the sentence. We judge the words as informative
according to their TF-IDF Luhn (1958); Jones (1972) values111In RepAL, we use
the default function (based on TF-IDF) in Jieba toolkit to extract the
keywords within a sentence. calculated on a general corpus. In the following,
we generate partially masked sentence $x_{i}^{*}$, where only the keywords in
the sentence $x_{i}$ are masked, and $f(x_{i}^{*})$ as the corresponding
redundant embedding. Since the model is forced to see only the non-masked
context words, $f(x_{i}^{*})$ actually encode the information from the trivial
words (non-keywords), and the sentence-level redundant embedding is defined as
follows:

$x_{i}^{*}=\text{PartialMask}(x_{i},\text{keyword});\quad
v_{i}^{*}=f(x_{i}^{*})$ (1)

where $v_{i}^{*}$ is the sentence-level redundant embedding of $x_{i}$.
Diminishing $v_{i}^{*}$ from $x_{i}$ representation aims to de-emphasizes the
influence of trivial words when producing sentence embedding.

####  3.2.2 Corpus-level Redundancy

Given a sentence set $\mathcal{X}=\\{x_{i}\\}_{i=1}^{n}$, we feed all the
sentences to the encoder $f$, and take the average embedding $\hat{v}$ as its
corpus-level redundant embedding, which can be formally defined as follows:

$\hat{v}=\frac{\Sigma_{i=1}^{n}f(x_{i})}{n}$ (2)

where $\hat{v}$ is the corpus-level redundant embedding. Diminishing $\hat{v}$
from each sentence’s representation makes them more distinguishable to each
other.

###  3.3 Embedding Refinement

After generating redundant embedding, the embedding refinement operation can
be formalized via the conceptually simple subtraction operation, which is
defined as follows:

$v_{i}^{\prime}=f(x_{i})-\lambda_{1}\cdot v_{i}^{*}-\lambda_{2}\cdot\hat{v}$
(3)

where $f(x)$ corresponds to the original embedding of $x_{i}$, and $v_{i}^{*}$
and $\hat{v}$ represent the redundant embedding at two levels, respectively.
Since the two redundant embeddings typically do not contribute completely
equal to the embedding $v$, we introduce two independent hyper-parameters
$\lambda_{1}$ and $\lambda_{2}$ to balance the two terms.

Baseline | ATEC | BQ | LCQMC | PAWSX | STS-B | Avg
---|---|---|---|---|---|---
BERT | 16.51$\rightarrow$19.58 | 29.35$\rightarrow$32.89 | 41.71$\rightarrow$44.53 | 9.84$\rightarrow$11.28 | 34.65$\rightarrow$47.00 | 26.41$\rightarrow$31.06(+4.65)
RoBERTa | 24.61$\rightarrow$27.00 | 40.54$\rightarrow$39.51 | 70.55$\rightarrow$70.98 | 16.23$\rightarrow$16.98 | 63.55$\rightarrow$64.01 | 43.10$\rightarrow$43.70(+0.60)
RoFormer | 24.29$\rightarrow$25.07 | 41.91$\rightarrow$42.56 | 64.87$\rightarrow$65.33 | 20.15$\rightarrow$20.13 | 56.65$\rightarrow$57.23 | 41.57$\rightarrow$42.06(+0.49)
NEZHA | 17.39$\rightarrow$18.98 | 29.63$\rightarrow$30.53 | 40.60$\rightarrow$41.85 | 14.90$\rightarrow$15.43 | 35.84$\rightarrow$36.68 | 27.67$\rightarrow$28.69(+1.02)
W-BERT | 20.61$\rightarrow$23.29 | 25.76$\rightarrow$29.83 | 48.91$\rightarrow$50.01 | 16.82$\rightarrow$16.96 | 61.19$\rightarrow$61.46 | 34.66$\rightarrow$36.31(+1.65)
W-RoBERTa | 29.59$\rightarrow$30.44 | 28.95$\rightarrow$43.12 | 70.82$\rightarrow$71.39 | 17.99$\rightarrow$18.48 | 69.19$\rightarrow$70.92 | 43.31$\rightarrow$46.87(+2.56)
W-RoFormer | 26.04$\rightarrow$27.68 | 28.13$\rightarrow$42.63 | 60.92$\rightarrow$61.55 | 23.08$\rightarrow$23.05 | 66.96$\rightarrow$67.13 | 41.03$\rightarrow$44.38(+3.35)
W-NEZHA | 18.83$\rightarrow$21.33 | 21.94$\rightarrow$23.02 | 50.52$\rightarrow$52.01 | 18.15$\rightarrow$19.00 | 60.84$\rightarrow$60.82 | 34.06$\rightarrow$35.24(+1.18)
C-BERT | 26.35$\rightarrow$28.69 | 46.68$\rightarrow$48.02 | 69.22$\rightarrow$69.98 | 10.89$\rightarrow$12.03 | 68.89$\rightarrow$69.66 | 44.41$\rightarrow$45.68(+1.27)
C-RoBERTa | 27.39$\rightarrow$28.43 | 47.20$\rightarrow$47.14 | 67.34$\rightarrow$67.98 | 09.36$\rightarrow$10.55 | 72.02$\rightarrow$71.80 | 44.66$\rightarrow$45.18(+0.52)
C-RoFormer | 26.24$\rightarrow$27.68 | 47.13$\rightarrow$47.63 | 66.92$\rightarrow$67.85 | 11.08$\rightarrow$11.65 | 69.84$\rightarrow$69.73 | 44.24$\rightarrow$44.91(+0.67)
C-NEZHA | 26.02$\rightarrow$26.73 | 47.44$\rightarrow$48.02 | 70.02$\rightarrow$70.63 | 11.46$\rightarrow$11.80 | 68.97$\rightarrow$69.53 | 44.78$\rightarrow$45.34(+0.56)
Sim-BERT | 33.14$\rightarrow$33.48 | 50.67$\rightarrow$51.14 | 69.99$\rightarrow$72.44 | 12.95$\rightarrow$13.58 | 69.04$\rightarrow$69.55 | 47.16$\rightarrow$48.04(+0.88)
Sim-RoBERTa | 32.23$\rightarrow$33.10 | 50.61$\rightarrow$51.53 | 74.22$\rightarrow$74.77 | 12.25$\rightarrow$13.28 | 71.13$\rightarrow$72.20 | 48.09$\rightarrow$48.98(+0.89)
Sim-RoFormer | 32.33$\rightarrow$32.59 | 49.13$\rightarrow$49.46 | 71.61$\rightarrow$72.13 | 15.25$\rightarrow$15.69 | 69.45$\rightarrow$70.01 | 47.55$\rightarrow$48.02(+0.47)
Sim-NEZHA | 32.14$\rightarrow$32.52 | 46.08$\rightarrow$47.42 | 60.38$\rightarrow$60.51 | 16.60$\rightarrow$16.58 | 68.50$\rightarrow$69.19 | 44.74$\rightarrow$45.26(+0.52)
Table 1: The experimental results of RepAL on Chinese semantic similarity
benchmarks. The numbers before $\rightarrow$ indicate the performance without
RepAL and the numbers after $\rightarrow$ mean the performance with RepAL.
Blue numbers indicate RepAL improves the baseline.

##  4 Experiments

This section shows that our method can be adaptive to various USE and improves
their performance.

###  4.1 Baselines and Benchmarks

To verify the effectiveness of our method, we evaluate RepAL on both Chinese
and English benchmarks. To investigate whether our method can be applied to
various unsupervised sentence encoder (USE), we choose two kinds of encoders:
vanilla USE and secondary trained USE. For vanilla USE, we select BERT Devlin
et al. (2019), RoBerTa Liu et al. (2019), RoFormer Su et al. (2021b) and NEZHA
Wei et al. (2019) for Chinese; for English, we select BERTbase, BERTlarge
Devlin et al. (2019) and RoBERTabase Reimers and Gurevych (2019).
Specifically, we name the secondary trained USE equipped with whitening Huang
et al. (2021), ConSERT Yan et al. (2021), and SimCSE Gao et al. (2021) as
W-USE (e.g., W-BERT), C-USE (e.g., C-BERT), and Sim-USE (e.g., Sim-BERT),
respectively. Results of Sim-USE and C-USE are from our implementation.
Details of training SimCSE on Chinese benchmarks are deferred to Appendix B.

  * •

Chinese: We select five Chinese benchmarks for evaluation: AETC, LCQMC, BQ,
PAWSX, and STS-B. The details about them are deferred to the Appendix B.

  * •

English: We select STS task benchmarks as our English datasets, including STS
2012-2016 tasks Agirre et al. (2012, 2013, 2014, 2015, 2016), the STS
benchmark Cer et al. (2017) and the SICK-Relatedness dataset Marelli et al.
(2014).

###  4.2 Experimental Settings

The vanilla USE in our experiments is the same as their original settings.
Besides, we keep the settings baselines the same as their original ones. As
for hyper-parameters, we follow previous unsupervised work Gao et al. (2021);
Yan et al. (2021), and tune $\lambda_{1}$ and $\lambda_{2}$ based on the STS-B
dev set. The results are evaluated through Spearman correlation.

Baseline | STS-12 | STS-13 | STS-14 | STS-15 | STS-16 | Avg
---|---|---|---|---|---|---
BERT | 57.86$\rightarrow$59.55 | 61.97$\rightarrow$66.20 | 62.49$\rightarrow$65.19 | 70.96$\rightarrow$73.50 | 69.76$\rightarrow$72.10 | 63.69$\rightarrow$66.70(+3.01)
BERTl | 57.74$\rightarrow$59.90 | 61.16$\rightarrow$66.20 | 61.18$\rightarrow$65.62 | 68.06$\rightarrow$73.01 | 70.30$\rightarrow$74.72 | 62.62$\rightarrow$67.47(+4.85)
RoBERTa | 58.52$\rightarrow$60.88 | 56.21$\rightarrow$62.20 | 60.12$\rightarrow$64.10 | 69.12$\rightarrow$71.41 | 63.69$\rightarrow$69.94 | 60.59$\rightarrow$65.41 (+4.82)
W-BERT | 63.62$\rightarrow$64.50 | 73.02$\rightarrow$73.69 | 69.23$\rightarrow$69.69 | 74.52$\rightarrow$74.69 | 72.15$\rightarrow$76.11 | 69.21$\rightarrow$70.39 (+1.18)
W-BERTl | 63.62$\rightarrow$63.90 | 73.02$\rightarrow$73.41 | 69.23$\rightarrow$70.01 | 74.52$\rightarrow$75.18 | 72.15$\rightarrow$75.89 | 69.21$\rightarrow$70.39 (+1.18)
W-RoBERTa | 68.18$\rightarrow$68.85 | 62.21$\rightarrow$63.03 | 67.13$\rightarrow$67.69 | 67.63$\rightarrow$68.23 | 74.78$\rightarrow$75.44 | 67.17$\rightarrow$68.43 (+1.26)
C-BERT | 64.09$\rightarrow$65.01 | 78.21$\rightarrow$78.54 | 68.68$\rightarrow$69.04 | 79.56$\rightarrow$79.90 | 75.41$\rightarrow$75.74 | 72.27$\rightarrow$72.69 (+0.42)
C-BERTl | 70.23$\rightarrow$70.70 | 82.13$\rightarrow$82.54 | 73.60$\rightarrow$74.12 | 81.72$\rightarrow$82.01 | 77.01$\rightarrow$77.58 | 76.03$\rightarrow$76.48 (+0.45)
Sim-BERT | 68.93$\rightarrow$69.33 | 78.68$\rightarrow$78.93 | 73.57$\rightarrow$73.95 | 79.68$\rightarrow$80.01 | 79.11$\rightarrow$79.29 | 75.11$\rightarrow$75.44 (+0.33)
Sim-BERTl | 69.25$\rightarrow$69.60 | 78.96$\rightarrow$79.30 | 73.64$\rightarrow$73.92 | 80.06$\rightarrow$80.31 | 79.08$\rightarrow$79.42 | 75.31$\rightarrow$75.61 (+0.30)
Table 2: The experimental results of RepAL on English semantic similarity benchmarks. ‘Avg’ indicates the average performance of all English benchmarks including STS-B and SICK-R in Table 3, and BERTl means BERTlarge during the experiments. Baseline | STS-B | SICK-R
---|---|---
BERT | 59.04$\rightarrow$ 66.35 | 63.75$\rightarrow$ 64.55
BERTl | 59.59$\rightarrow$ 68.21 | 60.34$\rightarrow$ 64.61
RoBERTa | 55.16$\rightarrow$ 65.75 | 61.33$\rightarrow$ 63.61
W-BERT | 71.34$\rightarrow$ 71.45 | 60.60$\rightarrow$ 62.61
W-BERTl | 71.34$\rightarrow$ 69.56 | 60.60$\rightarrow$ 65.00
W-RoBERTa | 71.43$\rightarrow$ 72.03 | 58.80$\rightarrow$ 63.95
C-BERT | 73.12$\rightarrow$ 73.45 | 66.79$\rightarrow$ 67.15
C-BERTl | 77.48$\rightarrow$ 77.91 | 70.02$\rightarrow$ 70.51
Sim-BERT | 75.71$\rightarrow$ 76.00 | 70.12$\rightarrow$ 70.51
Sim-BERTl | 75.84$\rightarrow$ 76.11 | 70.34$\rightarrow$ 70.61
Table 3: The results of RepAL on STS-B and SICK-R

###  4.3 Results

As shown in Table 1, RepAL improves the baselines’ performance in most cases.
For example, RepAL produces 4.65%, 1.65%, 1.27%, and 0.88% improvement to
BERT, W-BERT, C-BERT, and Sim-BERT, respectively. Generally, as USE becomes
stronger, the improvements brought by RepAL decrease. Still, for strong
baselines like C-BERT and Sim-BERT, RepAL still makes progress over them.
Specifically, RepAL achieves $1.27\%$ and $0.88\%$ performance increase for
C-BERT and Sim-BERT, indicating the effectiveness of RepAL on extremely strong
baselines. The results on English benchmarks are listed in Table 2, 3, where
RepAL also obtains improvements over various USEs. Overall, both results on
Chinese and English benchmarks demonstrate the effectiveness of RepAL and
illustrate that RepAL is a plug-and-play method for sentence representation
enhancement.

###  4.4 Ablation study

This section investigates the individual effective of embedding refinement of
two different levels. As shown in Table 4, each of operation is beneficial and
combining them lead to stronger performance.

| BERT | W-BERT | C-BERT | S-BERT
---|---|---|---|---
RepAL | 31.06 | 36.31 | 45.68 | 48.04
w/o Sen | 29.28 | 34.03 | 43.49 | 46.09
w/o Cor | 30.42 | 35.78 | 45.03 | 47.63
Table 4: Ablation studies of RepAL on Chinese benchmarks. ‘S-BERT’ refers to
‘Sim-BERT’.

##  5 Conclusion

In this paper, we propose RepAL, a universal method for unsupervised sentence
representation enhancement. Based on the idea that de-emphasize redundant
information, RepAL extracts then refines redundant information for the
sentence embedding at sentence-level and corpus-level. Through a simple
embedding refinement operation, RepAL successfully achieves improvements on
both Chinese and English benchmarks and is proved to be a simple and plug-and-
play method in modern techniques for unsupervised sentence representation.

##  6 Limitation

RepAL is a universal method for sentence representation, and sentence
representation is eventually used for downstream tasks. However, RepAL does
not consider task-specific information since different downstream tasks may
have different preferences. Therefore, exploring task-specific modification on
RepAL is a future direction.

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##  Appendix A Difference between BERT whitening

Some post-processing methods have been proposed to improve the quality of
contextual sentence embeddings to solve such a problem. The post-processing
paradigm aims to enhance sentence embeddings through simple and efficient
operations without extra training or data. The most promising method is
whitening Huang et al. (2021), dedicated to transforming sentence embedding
into Gaussian-like embedding, which proved to be effective in sentence
embedding improvement. Specifically, the most relevant work to ours is
whitening Huang et al. (2021) since the corpus-level refinement is similar to
the average embedding subtraction in whitening. However, there are three
principal differences between such two works. Firstly, the motivation is
different. Whitening aims at transforming the sentence embedding to Gaussian-
like embedding for distance measurement on an orthogonal basis. Our method
starts a perspective of redundancy refinement, which aims to diminish the
impact of trivial words within a sentence during similarity calculation.
Second, the methodology is different. Our method additionally employs a
partial mask to filter the redundancy and introduce weight factors to control
the impact during embedding refinement. Lastly, the in-depth analysis shows
that our method aims to diminish the upper bound of the largest eigenvalue of
the embedding matrix and the impact of trivial words, which is irrelevant to
whitening’s effects.

##  Appendix B Details of Chinese benchmarks and training Chinese SimCSE

(1) ATEC: A semantic similarity dataset related to customer service; (2)
LCQMC: A dataset consisting problem matching across multiple domains; (3) BQ:
a dataset consisting problem matching related to bank and finance; (4) PAWSX
Yang et al. (2019) : The dataset contains multilingual paraphrase and non-
paraphrase pairs, we select the Chinese part; (5) STS-B: A Chinese benchmark
labeled by semantic correlation between two sentences.

As for training SimCSE on benchmarks, we delete the labels in ATEC, LCQMC, BQ,
PAWSX, and STS-B, and merge them into an unsupervised corpus. Then SimCSE is
launched on the merged corpus, the best checkpoint is selected according to
the STS-B dev set, which keeps the same setting in previous works Gao et al.
(2021); Yan et al. (2021). Specifically, we find that the best dropout rate
$r$ on Chinese corpus is about 0.3.

##  Appendix C Detailed Analysis and Discussion

The proposed RepAL enhances sentence embedding by filtering redundant
information from two levels: sentence-level and corpus-level. Despite the
presentations of the overall experiment results and analysis, the intrinsic
properties of RepAL remain unclear. In this section, we illustrate the reasons
why RepAL is effective in enhancing sentence embedding.

In Sec C.1, we provide the evidence about the impact of trivial words in
sentence embedding and show the capacity of our sentence-level embedding
refinement. In Sec C.2, we show why the corpus-level embedding refinement
enhances sentence embedding and illustrate the relation between the largest
eigenvalue and performance.

###  C.1 Sentence-level Refinement

We conduct analyses for sentence-level refinement (SR) as follows: we
investigate the impact of trivial words w/o RepAL, which explains the
necessity of removing such redundancy information and validates the
effectiveness of SR.

We first define the importance $H$ of word $w\in x_{i}$ in semantic similarity
calculation, which can be defined as follows:

$H(x_{i},x^{-}_{i};w)=Sim(x_{i},x^{-}_{i})-Sim(x_{i}/w,x^{-}_{i})$ (4)

where $x_{i}$ and $x^{-}_{i}$ are a pair of sentences and $x_{i}/w_{i}$ means
deleting the word $w_{i}$ from $x_{i}$. Note that we do not consider the words
in $x^{-}_{i}$ since it is equivalent to evaluation on more sentences. Then we
define the set of trivial words within $x_{i}$ as $S(x_{i})$, which are
unmasked by jieba. Thus we can define the redundancy overlap ratio $r(p_{i})$
of a sentence pair $p_{i}=(x_{i},x^{-}_{i})$ as follows:

$r(p_{i})=\frac{|S(x_{i})\cap T(x_{i})|}{|T(x_{i})|}$ (5)

where $T(x_{i})$ represents the top-5 words with highest importance $H$ in
$x_{i}$. $r(p_{i})$ is a metric to reflect the impact of trivial words in
semantic similarity between the sentence pair $p_{i}$, since higher $r(p_{i})$
indicates more trivial words are important towards semantic similarity
calculation. We randomly sample 300 sentence pairs from STS-B Cer et al.
(2017) and select BERT as the USE, and we calculate the average redundancy
overlap ratio $\hat{r}=\frac{\sum_{i=1}^{N}r(p_{i})}{N}$ w/o SR. The results
show that $\hat{r}$ reaches $10.2\%$ without SR, after applying SR, $\hat{r}$
drops to $7.1\%$222$\hat{r}$ changes since the inputs during similarity
calculation have changed when SR activates. After SR. Eq 4 becomes
$H(x_{i},x^{-}_{i};w)=Sim(G(x_{i}),G(x^{-}_{i}))-Sim(G(x_{i}/w),G(x^{-}_{i}))$
where $G(\cdot)$ represents SR operation. The results demonstrate that SR
diminishes the impact of trivial words when measuring semantic similarity.

Moreover, we select some representative words and evaluate their importance
w/o RepAL. As shown in Table 5, the results show that our SR indeed diminishes
the impact of such trivial words when calculating semantic similarity.

Word | No Refinement | With Refinement | $\Delta$
---|---|---|---
the | 1.02 | 0.56 | -0.46
a | 0.98 | 0.43 | -0.55
to | 0.59 | 0.32 | -0.27
in | 0.68 | 0.21 | -0.47
some | 0.60 | 0.31 | -0.29
with | 0.72 | 0.24 | -0.48
and | 0.99 | 0.61 | -0.38
Table 5: The importance of trivial words w/o sentence-level refinement.
$\Delta$ means the importance change.

###  C.2 Corpus-level Refinement

To investigate whether corpus-level refinement diminishes the upper bound of
the eigenvalue of embedding $E^{*}$, we make numerical experiments to dive
into the relationship between the performance (Spearman correlation),
$\lambda$ and the upper bound of the largest eigenvalue of $E^{*}$.

Specifically, we launch the experiments on six English benchmarks with
BERTbase. As shown in Figure LABEL:figure:lamda, when performance rises at
peak, the upper bound of the largest eigenvalue of the embedding matrix
$E^{*}$ is around the minimum, showing a coincidence between the two. The
numerical results show that the corpus-level refinement enhances sentence
embedding since it diminishes the largest eigenvalue of $E^{*}$. Previous
methods Huang et al. (2021) is equivalent to subtracting the average vector
with $\lambda=1.0$, which fails to suppress the largest eigenvalue of
embedding matrix extremely. However, our method chooses to subtract a larger
$\lambda$ with adaptive weight, further suppressing the upper bound of the
largest eigenvalue of the embedding matrix. The results show that the average
embedding subtraction needs an adaptive weight. Moreover, this also
illustrates why our method can still significantly improve the performance on
W-BERT with substantial progress.


# Inv-SENnet: Invariant Self Expression Network for clustering under biased
data

###### Abstract

Subspace clustering algorithms are used for understanding the cluster
structure that explains the dataset well. These methods are extensively used
for data-exploration tasks in various areas of Natural Sciences. However, most
of these methods fail to handle unwanted biases in datasets. For datasets
where a data sample represents multiple attributes, naively applying any
clustering approach can result in undesired output. To this end, we propose a
novel framework for jointly removing unwanted attributes (biases) while
learning to cluster data points in individual subspaces. Assuming we have
information about the bias, we regularize the clustering method by
adversarially learning to minimize the mutual information between the data and
the unwanted attributes. Our experimental re- sult on synthetic and real-world
datasets demonstrate the effectiveness of our approach.

Index Terms—  clustering, bias mitigation, subspace

##  1 Introduction

Most real world datasets often carry information arising from several
attributes. Some of these attributes are of no importance and do not represent
the goal of the task. Ideally, these attributes should be uncorrelated to the
features of the dataset that are important to fullfil the task. But due to
various reasons in many real world datasets, these biases appear to be highly
correlated with these features of importance. This mostly appears due to the
nature of the data collection pipeline[1]. When such datasets are used for
training data driven models the results thus acquired also get affected by
these biases.

This problem of performing inference when the data contains biases is studied
under the broader umbrella of bias mitigation. Most of the past work in this
area is aimed at solving this problem with supervision[2][3][4] i.e. when
there are labels pertaining to the desired task present during the training
time. In many naturally appearing datasets, it is very hard to label the data
particularly based on the factors of importance. Therefore to learn the
patterns from such datasets unsupervised methods, e.g. clustering, are used.
But when biases are present in the data the result of such unsupervised
methods is often highly correlated with the biases. In this paper, we try to
address this problem.

Among the data-driven approaches for clustering, subspace clustering shows
great promise. In subspace clustering the assumption is that the high
dimensional data lie on a union of low dimensional subspaces. The objective
here is to find separate subspaces for separate clusters of data points. Among
the family of subspace clustering algorithms Self-expression based algorithms
form the state of the art[5]. Self expression imposes the constraint that
every data point in the dataset can be explained through a linear combination
of all the other data point in the dataset,

$x_{i}=\sum_{i\neq j}c_{i,j}x_{j}$ (1)

Here $c_{ij}$ represents the coefficient of the $j^{th}$ datapoint w.r.t. its
contribution to reconstruct the $i^{th}$ datapoint. One of the reason why self
expression based methods are popular is because of the subsapce preserving
property of the coefficient matrix achieved under certain regularisation
function [6][7][8][9][10]. This means that $c_{ij}$ is only non-zero when the
$i^{th}$ and $j^{th}$ datapoints are in the same subspace. Most recent
advances in subspace clustering literature have focused on scalability and out
of sample clustering using neural networks. But even these algorithms fail
under the presence of biases.

More often we have labels pertaining to these sources of biases. Under the
assumption that the labels of the biases are known during the training, we
propose an information-theoretic inspired way of jointly learning the
clustering while ignoring the bias confounds in the data. We evaluate our
proposed method over synthetic and naturally occurring image datasets and show
superior performance than the current state of the art.

The remaining paper is organized as follows. Section 2 presents the problem
statement, section 3 presents the proposed solution, section 4 presents the
experiments and section 5 presents the discussion.

##  2 Problem Statement

Most clustering algorithms are required to be trained on the whole dataset to
find clusters. As the size of the dataset increases, they become hard to
scale. Scalable Subspace clustering algorithms show great promise in solving
this problem [11][12][13]. The method proposed in [5], based on self-
expression, scales well for large datasets and shows great performance on out-
of-sample data points. But under the presence of bias, these methods fail on
out-of-sample data points resulting in learning wrong self-expression
coefficients. In this paper, we focus on this problem and look toward
improving the existing method. Below we formulate the problem of biases in
clustering.

###  2.1 Formulation

Let $\boldsymbol{x}\in\mathbb{R}^{d}$ be the input data and
$\boldsymbol{X}=[\boldsymbol{x}_{1},\ldots,\boldsymbol{x}_{n}]\in\mathbb{R}^{d\times
n}$ represent the input data matrix where the $x_{j}$ represents different
data points while the rows represent the dimensions. We make the assumption
that the data, _i.e._ , the columns in $\boldsymbol{X}$ lie on the union of
$k$ low-dimensional subspaces
$\boldsymbol{x}_{j}\in\cup_{i=1}^{k}\mathcal{X}_{i}$. Let
$\boldsymbol{b}=[b_{1},\ldots,b_{2}]$ represent the set of bias labels, where
$b_{j}\in\\{0,1\\}$ indicate the existence of a bias factor for every
$\boldsymbol{x}_{j}$. Based on the self-expressive models [6] we can define
each data point in $\boldsymbol{X}$ as a linear combination of all the other
data points as [6],

$\boldsymbol{x}_{j}=\sum_{i\neq j}c_{ij}\boldsymbol{x}_{i}$ (2)

Here $c_{ij}$ represents the elements of the matrix $\boldsymbol{C}$, self-
expressive coefficient matrix, satisfying the following,

$\boldsymbol{X}=\boldsymbol{C}\boldsymbol{X},$ (3)

where the diagonal element of $\boldsymbol{C}$ equals 0. Hence learning of the
self expressive coefficient matrix could be formulated as the following
optimisation problem,

$\min_{\boldsymbol{C}}\mathcal{L}(\boldsymbol{X},\boldsymbol{C}\boldsymbol{X})-\mathcal{R}(\boldsymbol{C})\,,$
(4)

where $\mathcal{L}$ and $\mathcal{R}$ represent cost and regularization terms,
respectively. Therefore the $\boldsymbol{C}$ matrix, subspace preserving, can
be further used to get the affinity matrix
$\boldsymbol{A}=|\boldsymbol{C}|+|\boldsymbol{C}^{\top}|$. We can obtain the
clustering labels $\hat{\boldsymbol{s}}$ over $\boldsymbol{X}$ using the
affinity matrix in the spectral clustering algorithm [14].

Now, let $\boldsymbol{s}$ be the set of true (desired) cluster labels, and
$\mathcal{I}(\boldsymbol{x},\boldsymbol{x}^{\prime})$ be the mutual
information between $\boldsymbol{x}$ and $\boldsymbol{x}^{\prime}$. Suppose
$\hat{\boldsymbol{s}}$ are the cluster labels based on the $\boldsymbol{C}$
matrix we get from solving (4). Then due to the presence of biases present in
the data we can very possibly observe,

$\mathcal{I}(\hat{\boldsymbol{s}},\boldsymbol{b})\gg\mathcal{I}(\boldsymbol{s},\boldsymbol{b})$
(5)

The effect of bias on classification task is already well discussed and
studied in literature [2]. Therefore in task where supervision is not present,
its very likely that presence of strong biases in the data would lead to
clustering algorithm output getting biased too as formulated in (5). This
would likely result in $\hat{\boldsymbol{s}}\neq\boldsymbol{s}$. In this
contribution we aim at solving the problem in (4) while enforcing bias
mitigation as expressed in Eq. (5). To solve this problem we propose an
information theoretic based approach in the next section.

##  3 Bias mitigation for Self-expressive networks

In this section we discuss the elements of our proposed solution and finally
discuss the architecture and optimisation for Inv-SENnet.

Fig. 1: Architecture for INV-SENnet. Here
$k(\boldsymbol{x};\boldsymbol{\theta}_{u})$ and
$q(\boldsymbol{x};\boldsymbol{\theta}_{v})$ be key and the query net
respectively. $g(\boldsymbol{u}_{j};\boldsymbol{\theta})$ and
$g^{\prime}(\boldsymbol{v}_{i};\boldsymbol{\theta}_{g^{\prime}})$ are the bias
classifiers.

###  3.1 Self Expressive Network

In literature there are many optimisation framework already proposed that try
to solve (4) to learn the self expression coefficient matrix $C$. Most of the
past works have tried to solve (4) optimising the following loss,

$\mathcal{L}_{\mathrm{SE}}=\min_{c_{ij},i\neq
j}\|\boldsymbol{x}_{j}-\sum_{i\neq
j}c_{ij}\boldsymbol{x}_{i}\|^{2}_{2}-\sum_{i\neq j}r(c_{ij})$ (6)

where $r(\cdot)$ represents the regularisation term based on elastic net
regularisation [15]. The role of $r(\cdot)$ is to provide denser subspace
preserving coefficient matrix. Let
$f(\cdot):\mathbb{R}^{D}\times\mathbb{R}^{D}\rightarrow\mathbb{R}$ be the
function, parameterised by $\boldsymbol{\Theta}$ taking arguments
$\boldsymbol{x}_{i}$ and $\boldsymbol{x}_{j}$, and giving self-expression
coefficients $c_{i,j}$ in matrix $\boldsymbol{C}$, i.e.
$f(\boldsymbol{x}_{i},\boldsymbol{x}_{j};\boldsymbol{\Theta})=c_{ij}$.
Therefore we can rewrite (6) as the following optimisation,

$\mathcal{L}_{\mathrm{SE}}=\min_{\boldsymbol{\Theta}}\frac{\gamma}{2}\|\boldsymbol{x}_{j}-\sum_{i\neq
j}f(\boldsymbol{x}_{i},\boldsymbol{x}_{j};\boldsymbol{\Theta})\boldsymbol{x}_{i}\|^{2}_{2}-r(f(\boldsymbol{x}_{i},\boldsymbol{x}_{j};\boldsymbol{\Theta}))$
(7)

In [5], the authors propose the SENnet, which models
$f(\boldsymbol{x}_{i},\boldsymbol{x}_{j};\boldsymbol{\Theta})$ as neural
network expressed as

$\displaystyle
f(\boldsymbol{x}_{i},\boldsymbol{x}_{j};\boldsymbol{\Theta})=\alpha\mathcal{T}_{\beta}(\boldsymbol{u}^{\top}\boldsymbol{v})$
$\displaystyle\boldsymbol{u}_{i}=k(\boldsymbol{x}_{i};\boldsymbol{\theta}_{u})\in\mathbb{R}^{p}$
$\displaystyle\boldsymbol{v}_{i}=q(\boldsymbol{x}_{j};\boldsymbol{\theta}_{v})\in\mathbb{R}^{p}$

where,

$\mathcal{T}_{\beta}(\cdot)=\text{sign}(t)\max(0,|t|-\beta).$ (8)

Here $k$ and $q$ represents the key and the query networks, while
$\mathcal{T}$ is a soft threshholding operator.
$\boldsymbol{\Theta}=\\{\boldsymbol{\theta}_{u},\boldsymbol{\theta}_{v},\beta\\}$
represents the learnable parameters of the model. We highlight that SENet is
not designed to cluster the data when bias is present in the data. We improve
upon their model by adding a bias mitigation mechanism. The resulting
methodology leads to a learning algorithm capable of performing clustering and
mitigation jointly as described next.

###  3.2 Bias Mitigation

As discussed in Section 2.1, the presence of biases in the training dataset
leads to biases-dependent solutions for the optimization problem, and,
therefore, leads to biased clustering. Below we present the proposed bias
mitigation strategy. For this, we assume that the training dataset contains
labels for the bias classes. Note, however, that biases labels are not
required for the test set. Therefore the goal here is to propose a way of
learning $\boldsymbol{\Theta}$ s.t. $c_{ij}$ is invariant to the presence of
bias.

Referring to the proposed architecture in Fig. 1 we define
$g(k(\boldsymbol{x};\boldsymbol{\theta}_{u});\boldsymbol{\theta}_{g}):\mathbb{R}^{p}\to\mathbb{R}$
and
$g^{\prime}(q(\boldsymbol{x};\boldsymbol{\theta}_{v});\boldsymbol{\theta}_{g}^{\prime}):\mathbb{R}^{p}\to\mathbb{R}$
as the bias classification functions over $\boldsymbol{u}$ and
$\boldsymbol{v}$ that gives the $\boldsymbol{b}$ labels for the corresponding
$\boldsymbol{x}$. Therefore the optimisation framework in (7) can be modified
as:

$\mathcal{L}=\mathcal{L}_{\mathrm{SE}}-\lambda(\mathcal{I}(\boldsymbol{u},\boldsymbol{b})+\mathcal{I}(\boldsymbol{v},\boldsymbol{b})).$
(9)

Here $\lambda\in\mathbb{R}_{+}$ is a scalar hyper-parameter controlling the
bias mitigation component in the loss. Estimating $\mathcal{I}(\cdot)$
requires the joint distributions $p(\boldsymbol{b},\boldsymbol{v})$ and
$p(\boldsymbol{b},\boldsymbol{u})$. This makes the problem hard to solve and
often cases intractable. Mutual information can also be written in terms of
the conditional entropy such that

$\displaystyle\mathcal{I}(\boldsymbol{b},\boldsymbol{v})=\mathcal{H}(\boldsymbol{b})-\mathcal{H}(\boldsymbol{b}|\boldsymbol{v}),\qquad\mathcal{I}(\boldsymbol{b},\boldsymbol{u})=\mathcal{H}(\boldsymbol{b})-\mathcal{H}(\boldsymbol{b}|\boldsymbol{u}).$

Here, $\mathcal{H}$ represents the entropy i.e.
$\mathcal{H}(\boldsymbol{b})=log(P(\boldsymbol{b}))$. Note that
$\mathcal{H}(\boldsymbol{b})$ can be ignored when optimizing (9) since the
bias $\boldsymbol{b}$ is independent of the model parameters
$\boldsymbol{\Theta}$. To compute conditional entropies conditional densities
such as $P(\boldsymbol{b}|\boldsymbol{u})$ and
$P(\boldsymbol{b}|\boldsymbol{v})$ are required, but can be difficult to
obtain. In [2] the authors approximated the conditional distribution using a
variational distribution $Q$. Following a similar reasoning we can define two
distributions $Q$ and $Q^{\prime}$ to approximate
$P(\boldsymbol{b}|\boldsymbol{u})$ and $P(\boldsymbol{b}|\boldsymbol{v})$,
respectively. Now, let $\boldsymbol{\theta}_{g}$ and
$\boldsymbol{\theta}_{g^{\prime}}$ be the parameters of $Q$ and $Q^{\prime}$.
Thus, each parameter can be learned by minimizing the Kullback–Leibler
($\mathcal{D}_{KL}$) divergence between the respective distributions. For
$\boldsymbol{\theta}_{g}$ we have:

$\min_{\boldsymbol{\theta}_{g}}\mathcal{D}_{KL}[Q(g(k(\boldsymbol{x};\boldsymbol{\theta}_{u});\boldsymbol{\theta}_{g})|k(\boldsymbol{x};\boldsymbol{\theta}_{u}))\|P(\boldsymbol{b}|k(\boldsymbol{x};\boldsymbol{\theta}_{u}))]$
(10)

while an equivalent problem can be solved for
$\boldsymbol{v}=q(\boldsymbol{x};\boldsymbol{\theta}_{v})$, omitted here for
sake of space.

Now, we can define the bias classification optimization problem by combining a
cross-entropy loss and the KL in (10) as a regularization term. The goal of
this optimization is to not only make the bias classifier able to classify the
bias class accurately, but also to make the
$k(\boldsymbol{x};\boldsymbol{\theta}_{u})$ invariant of the bias. To solve
the second objective we can impose a regularisation on
$\boldsymbol{\theta}_{u}$ aiming at maximizing the negative of (10), hence
making it hard for the $g(\cdot)$ to classify the bias. Therefore the
optimisation becomes a $\min\max$ problem,

$\displaystyle\mathcal{L}_{\mathrm{bias}}^{\mathrm{key}}$
$\displaystyle=\min_{\boldsymbol{\theta}_{u}}\max_{\boldsymbol{\theta}_{g}}\mathbb{E}_{\boldsymbol{x}\sim
P_{X}(\cdot)}[\mathbb{E}_{\tilde{\boldsymbol{b}}\sim
Q(\cdot|k(\boldsymbol{x};\boldsymbol{\theta}_{u}))}[logQ(\tilde{\boldsymbol{b}}|k(\boldsymbol{x};\boldsymbol{\theta}_{u}))]]$
$\displaystyle-\mu\mathbb{E}_{\boldsymbol{x}\sim
P_{X}(\cdot)}[\mathcal{L}_{c}(\boldsymbol{b},g(k(\boldsymbol{x};\boldsymbol{\theta}_{u});\boldsymbol{\theta}_{g})]$
(11)

where $\mathcal{L}_{c}(\cdot)$ represents the cross entropy loss. The second
term in (11) is the relaxation term for the KL divergence condition in (10).
Similar discussion is presented in [2]. Equivalently, we can define the
minimax problem $\mathcal{L}_{\mathrm{bias}}^{\mathrm{query}}$ for optimizing
$\boldsymbol{\theta}_{v}$ and $\boldsymbol{\theta}_{g^{\prime}}$, also omitted
in this manuscript.

Let
$\boldsymbol{\Theta}_{g}=\\{\boldsymbol{\theta}_{g},\boldsymbol{\theta}_{g^{\prime}}\\}$
represent the set of all parameters of the bias classifiers. Therefore,
combining (11), for both key and query networks, and (6) we can rewrite (9)
as:

$\displaystyle\mathcal{L}$
$\displaystyle=\mathcal{L}_{\textrm{SE}}-\lambda(\mathcal{L}_{\textrm{bias}}^{\textrm{key}}+\mathcal{L}_{\textrm{bias}}^{\textrm{query}})$
$\displaystyle=\min_{\boldsymbol{\Theta}}\max_{\boldsymbol{\Theta}_{g}}\frac{\gamma}{2n}\sum_{j}[\|\boldsymbol{x}_{j}-\sum_{i\neq
j}\mathcal{T}_{\beta}(k(\boldsymbol{x}_{j};\boldsymbol{\theta}_{u})^{T}q(\boldsymbol{x}_{i};\boldsymbol{\theta}_{v}))\boldsymbol{x}_{i}\|_{2}^{2}$
$\displaystyle\quad-r(\mathcal{T}_{\beta}(k(\boldsymbol{x}_{j};\boldsymbol{\theta}_{u})^{T}q(\boldsymbol{x}_{i};\boldsymbol{\theta}_{v})))]$
$\displaystyle\quad-\lambda\mathbb{E}_{\boldsymbol{x}\sim
P_{\boldsymbol{X}}(\cdot)}[\mathbb{E}_{\tilde{\boldsymbol{b}}\sim
Q(\cdot|k(\boldsymbol{x};\boldsymbol{\theta}_{u}))}[\log
Q(\tilde{\boldsymbol{b}}|k(\boldsymbol{x};\boldsymbol{\theta}_{u}))]]$ (12)
$\displaystyle\quad+\mu\mathbb{E}_{\boldsymbol{x}\sim
P_{\boldsymbol{X}(\cdot)}}[\mathcal{L}_{c}(\boldsymbol{b},g(k(\boldsymbol{x};\boldsymbol{\theta}_{u});\boldsymbol{\theta}_{g})]$
$\displaystyle\quad-\lambda\mathbb{E}_{\boldsymbol{x}\sim
P_{\boldsymbol{X}}(\cdot)}[\mathbb{E}_{\tilde{\boldsymbol{b}}\sim
Q^{\prime}(\cdot|q(\boldsymbol{x};\boldsymbol{\theta}_{v}))}[\log
Q^{\prime}(\tilde{\boldsymbol{b}}|q(\boldsymbol{x};\boldsymbol{\theta}_{v}))]]$
$\displaystyle\quad+\mu\mathbb{E}_{\boldsymbol{x}\sim
P_{\boldsymbol{X}(\cdot)}}[\mathcal{L}_{c}(\boldsymbol{b},g^{\prime}(q(\boldsymbol{x};\boldsymbol{\theta}_{v});\boldsymbol{\theta}_{g}^{\prime})].$

###  3.3 Training

In (12) we present the overall loss function for the architecture proposed in
Fig. 1. In practice, we use two optimizers; one to solve the inner
maximisation and the second one to solve the outer minimisation. First we
compute the forward pass of the data through
$k(\boldsymbol{x}_{j};\boldsymbol{\theta}_{u})$ and
$q(\boldsymbol{x}_{i};\boldsymbol{\theta}_{v})$ and compute
$\mathcal{L}_{\mathrm{SE}}$. At the same time we also pass both
$\boldsymbol{u}_{j}$ and $\boldsymbol{v}_{i}$ through their respective bias
classifiers $g(\cdot)$ and $g^{\mathrm{\prime}}(\cdot)$ to get the bias
classification. We then compute the
$\partial{\mathcal{L}_{\mathrm{bias}}^{\mathrm{key}}}/\partial{\boldsymbol{\theta}_{g}}$
and
$\partial{\mathcal{L}_{\mathrm{bias}}^{\mathrm{query}}}/\partial{\boldsymbol{\theta}_{g^{\prime}}}$
and backpropagate through the bias classifiers $g(\cdot)$ and
$g^{\prime}(\cdot)$ respectively. Using the gradient reversal technique [16]
together with the gradient of $\mathcal{L}_{\mathrm{SE}}$ we compute,

$\displaystyle\nabla\boldsymbol{\theta}_{u}=\frac{\partial{\mathcal{L}_{\mathrm{SE}}}}{\partial{\boldsymbol{\theta}_{u}}}-\lambda\frac{\partial{\mathcal{L}_{\mathrm{bias}}^{\mathrm{key}}}}{\partial{\boldsymbol{\theta}_{u}}},\qquad\nabla\boldsymbol{\theta}_{v}=\frac{\partial{\mathcal{L}_{\mathrm{SE}}}}{\partial{\boldsymbol{\theta}_{v}}}-\lambda\frac{\partial{\mathcal{L}_{\mathrm{bias}}^{\mathrm{query}}}}{\partial{\boldsymbol{\theta}_{v}}}.$

for key and query network respectively. At the beginning of the training, the
bias classifiers converges quickly. But as the training goes on the $C$ matrix
becomes invariant of the biases and the bias classifier starts performing
poorly. This happens because the $\mathcal{I}(\boldsymbol{u},\boldsymbol{b})$
and $\mathcal{I}(\boldsymbol{v},\boldsymbol{b})$ goes down i.e the key and the
query net becomes good at unlearning the biases.

Fig. 2: Examples of Datasets with bias [Top-Rows: Biased set; Bottom-Rows:
Unbiased set]. We modified the MNIST and FashionMNIST dataset by randomly
adding color ([Red ,Green] for MNIST and [Blue, Yellow] for FashionMNIST) to
each dataset. For Dogs and Cats dataset, we sub-sampled dark dog and bright
cat images from the original dataset.

##  4 Experiment

We conduct experiments on real-world benchmarks to evaluate our model on
clustering biased data. We evaluate our method first on image dataset and then
discuss a case study on an fMRI dataset.

###  4.1 Setup

Network Architecture: We formulate query, key and bias networks as an MLP. For
query and key network, we create a three-layer MLP with ReLU and tanh(·) as
the activation functions. The number of hidden units in each layer of these
MLP are $\\{1024,1024,1024\\}$, and the output dimension is 1024. For the Bias
networks, we use a three-layer MLP of dimensions $\\{1024,512,256\\}$ followed
by the classification layer. We use ReLU activation function and batch
normalization layer between each fully-connected layer. We apply softmax
function to the output of the classification layer to compute class
probability scores. To optimize our model, we use Adam [17] optimizer with a
constant learning rate of $1e−3$ for the query and key networks and $1e-4$ for
the bias networks.

Datasets: To evaluate our method, we consider two scenarios where our data
samples are biased. In the first setup we intentionally add bias information
to existing benchmarks (MNIST [18] and FashionMNIST [19]). In the second case,
we consider a natural setting (Dogs and Cats dataset) where the bias
information is inherently present in the data in the form of undesired
attribute. Figure2 shows some of the samples from our constructed datasets.
Setting 1: We select MNIST and FashionMNIST dataset for evaluation. Both
benchmarks consists of grayscale image. We color each image red or green for
MNIST and blue and yellow for FashionMNIST to introduce spurious correlations.
We color each image in a manner that strongly correlates with original class
labels [20]. This allows us to effectively measure the generalization
properties of a given method. We create our training set in the following
manner: We first assign a binary label $y$ to the image based on the
digit/class i.e. we set $y=0$ for digits $0-4$ (classes $0-4$ for
FashionMNIST) and $y=1$ for digits $5-9$ (classes $5-9$ for FashionMNIST ). We
then flip the label with $25\%$ probability. After that we color the image as
one of the color according to its (possibly flipped) label. We flip the color
with a probability $e$. For both benchmarks we set $e=40\%$. . We use
scattering convolution transform[21] to generate features from MNIST and
fashionMNIST similar to [5] for optimal comparison.
Setting 2: For this setting, We select the Dogs and Cats dataset from the
kaggle competition [22] for evaluation. We followed the similar setting from
[2], wherein, we subsample the original training set consisting of 25K images
to create two subsets: a biased subset with dark colored dog images and bright
colored cat images, and another set without those type of images. In total,
our biased and unbiased subset contains 6378 and 8125 images respectively. For
feature extraction we used [23] as it learns to represent features in a union
of subspaces.

Metrics: For quantitative evaluation, we consider clustering accuracy (ACC),
normalized mutual information (NMI) [24] and adjusted rand index (ARI) [25].
These metrics are commonly used in the literature to evaluate clustering
methods.

###  4.2 Results

We show our experimental results in tables 1 and 2. We evaluate vanilla SENet
and INV-SENnet model for two settings: (a). out-of-distribution clustering and
(b). Mix-domain clustering. In both the tables $N_{i}$ represents the number
of training samples. For the first experiment, we want to test the
generalizability of the learned self expression coefficient matrices. To this
end, we first train the model on the biased version of each dataset and then
test it on the unbiased version. If a method is optimized to directly learn
self expressive coefficients using the data without handling the bias
information, it should fail to generalize to out-of-distribution samples where
the spurious correlation might be absent. From the results in table 1, we can
observe that having spurious correlation in a dataset can negatively affect
clustering performance. This is evident from the results of standard SENet
model, which fails to accurately cluster MNIST and FashionMNIST data based on
ground-truth categories. The model performance is only marginally better than
random chance ($10\%$ clustering accuracy for MNIST and FashionMNIST; $50\%$
clustering accuracy for Dogs and Cats). In contrast, we see considerable
improvement of our proposed method over the standard model. In particular, we
see a significant increase in all the three metrics for MNIST, while for
FashionMNIST we see moderate improvements.For Dogs and Cats dataset, we again
see improvement in performance when incorporating invariance to the bias
information. In the second experiment, we train and test on the same set that
contains both biased and unbiased data in equal proportions. We perform this
experiment to evaluate clustering when a dataset both biased and unbiased
samples. We see a significant drop in all three of the metric for all of the
benchmarks when comparing to Table 1. A very possible reason for this could be
the mix of domains during the test times. Even then we see that INV-SENnet
preforms better than SENnet on all the metrics. We see a significant
improvement in the NMI score while comparing both the models. This further
signifies that the clustering represents more of the desired attribute and
less of the bias.

Table 1: Out-of-distribution clustering performance Dataset | Method | $N_{i}$ | ACC (%) | NMI. (%) | ARI (%)
---|---|---|---|---|---
MNIST | SENnet | 10000 | 41.79 | 33.34 | 21.05
60000 | 36.32 | 25.14 | 16.14
Inv-SENnet | 10000 | 68.44 | 60.55 | 45.23
60000 | 78.04 | 67.53 | 58.56
Fashion-MNIST | SENnet | 10000 | 38.47 | 30.15 | 19.51
60000 | 46.73 | 42.46 | 31.18
Inv-SENnet | 10000 | 50.73 | 42.15 | 30.14
60000 | 56.91 | 44.68 | 34.41
Dogs and Cats | SENnet | 6738 | 64.07 | 07.59 | 06.94
| Inv-SENnet | 6738 | 78.53 | 25.32 | 32.4
Table 2: Mix-domain clustering performance Dataset | Method | $N_{i}$ | ACC (%) | NMI. (%) | ARI (%)
---|---|---|---|---|---
MNIST | SENnet | 15000 | 12.75 | 02.66 | 00.13
Inv-SENnet | 15000 | 35.29 | 25.84 | 11.31
Fashion-MNIST | SENnet | 15000 | 20.37 | 13.50 | 02.39
Inv-SENnet | 15000 | 33.70 | 30.49 | 14.75
Dogs and Cats | SENnet | 14953 | 32.68 | 44.97 | 22.90
Inv-SENnet | 14953 | 40.43 | 52.99 | 26.27

##  5 Conclusion

In this work we formulated a bias mitigation strategy aiming at minimizing the
mutual-information between bias and features. We applied this strategy for
subspace clustering when the data is confounded by identifiable biases. The
proposed InvSENnet was effective in learning bias-invariant subspace
clustering, reducing the influence of biases in different simulated and real
datasets, clearly outperforming its version without bias mitigation
capability. Future work will focus on extending this work to datasets with
modalities having multiple confounds such as EEG[26][27]and fMRI[28][29].

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