11institutetext:  1 The University of British Columbia, Vancouver, BC V6Z 1Z4,
Canada
2 Vector Institute, Toronto, ON M5G 1M1, Canada
11email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Universal Debiased Editing on Foundation Models for Fair Medical Image
Classification

Ruinan Jin 1122 Wenlong Deng 1122 Minghui Chen 1122 Xiaoxiao Li 1122

###### Abstract

In the era of Foundation Models’ (FMs) rising prominence in AI, our study
addresses the challenge of biases in medical images while using FM API,
particularly spurious correlations between pixels and sensitive attributes.
Traditional methods for bias mitigation face limitations due to the restricted
access to web-hosted FMs and difficulties in addressing the underlying bias
encoded within the FM API. We propose an U(niversal) D(ebiased) E(diting)
strategy, termed UDE, which generates UDE noise to mask such spurious
correlation. UDE is capable of mitigating bias both within the FM API
embedding and the images themselves. Furthermore, UDE is suitable for both
white-box and black-box FM APIs, where we introduced G(reedy) (Z)eroth-O(rder)
(GeZO) optimization for it when the gradient is inaccessible in black-box
APIs. Our whole pipeline enables fairness-aware image editing that can be
applied across various medical contexts without requiring direct model
manipulation or significant computational resources. Our empirical results
demonstrate the method’s effectiveness in maintaining fairness and utility
across different patient groups and diseases. In the era of AI-driven
medicine, this work contributes to making healthcare diagnostics more
equitable, showcasing a practical solution for bias mitigation in pre-trained
image FMs.

††footnotetext: This work is supported in part by the Natural Sciences and
Engineering Research Council of Canada (NSERC), Compute Canada and Vector
Institute.
Corresponding author: Xiaoxiao Li

##  1 Introduction

Using pre-trained models or encoders to convert complex input data into vector
representations is widely used in computer vision and natural language
processing (NLP) [26, 7]. This process transforms the original data into a
representative low-dimensional hidden space, preserving information for
downstream tasks. With the advancement of Foundation Models (FM), services
like Google MedLM [18], Voyage.ai, and ChatGPT provide data embedding
services, eliminating the need for specialized hardware or extensive training.
Their effectiveness is especially notable in medical fields, addressing data
scarcity and hardware constraints. Clinics can enhance these models with
minimal effort by fine-tuning custom classifiers on compact embedding output
from FM application programming interface (APIs), which provides machine
learning (ML) service as a function without users to train their own models
from scratch. However, the inherent biases in the APIs’ training data and
their model potentially harm marginalized groups by perpetuating gender, race,
and other biases, a problem that has been exposed in the NLP field [17]. The
biased embedding can compromise models’ performance on minority groups [4].
Therefore, it is essential to develop new solutions to attain fairness in
medical image embedding from pre-trained FM API.

Studies have been done to address the fairness issues in classification
problems using machine learning (ML) models. These methods can be categorized
into three approaches: 1)Model-based strategies update or remove bias-related
model parameters to mitigate bias, for example, using adversarial methods [1,
22, 10]; or prune parameters that are significant for sensitive attributes
(SAs) [23]; or update model to minimize the mutual information between target
and SA representations using disentanglement learning [3]. However, the
pretrained FM API services offer users very limited control over the model
parameters [8, 9]. Thus model-based approaches are infeasible under the
constraints. 2)Prediction calibration-based methods analyze the prediction
probability distribution of a classifier and apply different thresholds to
each subgroup to reduce the discrepancy among subgroups [5, 14, 16]. These
methods need custom thresholds for various tasks, classes, and groups, which
limits their generalizability. Moreover, a significant amount of validation
data is needed to determine the thresholds, rendering them less efficient in
medical imaging applications where data scarcity is an unavoidable problem.
3)Data-based strategies alleviate bias at the pre-training stage. Re-
distribution and re-weighting methods [15, 16] address unfairness by adjusting
the balance of subgroups. The effectiveness of redistributed training data is
limited because it can only affect the classification head, not the pre-
trained FM API embedding encoder. Recent data editing methods show strong
ability to reduce disparity among subgroups by removing sensitive information
from the input images [10, 21]. However, can leave the model unchanged, they
fail to address the underlying bias encoded within the FM API. Moreover, these
approaches either depend on disease labels and require significant
computational costs. Yao _et al._ introduce an image editing method
independent on targeted downstream tasks via sketching [25], but this method
suffers from subpar performance as not learnable.

In this work, we take two unique properties of pretrained FM API into
consideration. Firstly, we aim to emphasize the need to maintain the
flexibility of using FM without limiting it to a specific task while
maintaining good utility, thus focusing on the learnable data editing-based
method that is independent of the downstream task. We choose to edit on the
image space rather than on the FM API embeddings since the former provides
better control on medical image fidelity and enables interpretability on the
applied editing (as shown in Sec. 3.1). Secondly, given the constraint against
modifying parameters in the pre-trained API, we commit to not changing the FM
API model parameters, even with black-box access to the API. To the best of
our knowledge, no effective and universal111We refer ‘universal’ as debiasing
API embeddings for various classification tasks. method has been available to
address this important, timely, but challenging research question: How to
remove bias on medical images when using their embeddings from a pre-trained
FM API for various classification tasks?

To answer this question, we propose the Universal Debiased Editing, UDE, where
the resulted edit can be shared across all subjects to eliminate SA-related
information for various disease classification tasks. Specifically, we start
by using a pre-trained SA classifier, trained with the same FM API embeddings.
Then, we optimize a set of learnable parameters (referred UDE noise) to be
added to the images by confusing the SA classifier. Furthermore, we introduce
a greedy zeroth-order optimization strategy, GeZO, when APIs restrict gradient
propagation in the black-box setting. Lastly, we apply this UDE noise to input
images to generate fair embeddings and achieve unbiased disease classification
across various disease tasks. Extensive evaluations on disease classification
tasks show our method’s effectiveness in promoting fairness while preserving
utility.

##  2 Method

###  2.1 Problem Setting

Figure 1: Overview of universal debias editing pipeline. (a) We eliminate the
spurious correlation by breaking the connection between SA $\mathcal{A}$ and
image $\mathcal{X}$, ensuring the model relies solely on disease-related
information $\mathcal{Y}$. (b) Training UDE noise deceives the pre-trained SA
classifier in $\mathcal{A}$ using a frozen FM API and SA classifier, suitable
for both white-box and black-box API scenarios based on gradient
accessibility. (c) We demonstrate the use of UDE noise: users augment their
images with this noise, extract embeddings via the FM API, and proceed to
train fair disease classifiers. (d) The (G)reedy (Z)eroth-(O)rder (GeZO)
black-box editing method, selects the optimal perturbation via the Greedy
Gradient process when gradients is inaccessible. It tracks the best
perturbation using velocity in each local iteration to update the UDE noise
$\epsilon$.

This section outlines the problem of fairness in a binary medical image
classification task. Firstly, we define key variables: input images
$x\in\mathcal{X}$, binary disease labels $y\in\mathcal{Y}=\\{0,1\\}$, and
sensitive attribute $a\in\mathcal{A}=\\{1,...,|\mathcal{A}|\\}$ (i.e.
$|\mathcal{A}|=2$ for gender with male and female). As patients may not have
sensitive and disease labels simultaneously, we denote images with sensitive
attribute labels as $\\{x_{i},a_{i}\\}^{N}=[X_{A},A]$ and images with disease
label as $\\{x_{j},y_{j}\\}^{M}=[X_{T},Y]$, where $N$ and $M$ are the number
of samples. Then we denote the FM API as
$\phi:\mathcal{X}\rightarrow\mathcal{Z}$, to obtain the image embedding
$z=\phi(x)$; the disease classifier as $f:\mathcal{Z}\rightarrow\mathcal{Y}$
and SA classifier as $g:\mathcal{Z}\rightarrow\mathcal{A}$.

Fairness Issue: Fig. 1 (a) shows there exists an association between sensitive
attributes $\mathcal{A}$ and targets $\mathcal{Y}$ (dashed line). In the
Empirical Risk Minimization (ERM) [20], the model $f(\phi(\cdot))$
superficially consider $\mathcal{A}$ as a proxy of $\mathcal{Y}$, causing the
issue of fairness. This phenomenon persists even when there is an equal number
of data points for each class in both $\mathcal{Y}$ and $\mathcal{A}$, as
demonstrated in Sec. 3.1. Our approach aims to address these disparities using
universal debiased edit.

###  2.2 Universal Debias Editing on Image for Fair Disease Classification

In this section, we present our image editing approach aimed at reducing bias
via learning a universal noise tensor with the same dimensionality as the
image, which is called UDE noise. Then, we explain how to use this UDE noise
for fair disease classification.

Universal Debiased Editing. Given the spurious correlation between SA
$\mathcal{A}$ and disease $\mathcal{Y}$, we aim to remove this link through
intervention on $\mathcal{X}$ by adding minimal universal editing $\epsilon$,
known as UDE noise, concealing the spurious correlations while preserving
image utility. As shown in Fig. 1 (b), we first obtain a pre-trained SA
classifier for medical images by either leveraging an existing one (if
available) or using the collected $[X_{A},A]$ to train the classifier $g$ with
cross-entropy loss. Then, we freeze the classifier and leverage gradient
ascent to learn the minimal UDE noise $\epsilon$, which is constrained by
certain threshold to preserve image fidelity. $\epsilon$ is trained to deceive
the SA classifier $g$ without compromising useful information by solving the
following optimization problem:

$\displaystyle\epsilon=\arg\max_{\epsilon}\left[\frac{1}{N}\sum_{i=1}^{N}\mathcal{L}_{\texttt{CE}}(a_{i},g(\phi(x_{i}+\epsilon)))\right]\quad$
$\displaystyle\text{s.t.}~{}||\epsilon||_{2}<th,$ (1)

where $\mathcal{L}_{\texttt{CE}}$ is the cross-entropy loss, and $th$ is a
noise radius for the editing. To avoid manually determining $th$ and simplify
the optimization, we convert the constrain into a regularization term and
obtained the following objective:

$\displaystyle\mathcal{L}_{\epsilon}=-\left[\frac{1}{N}\sum_{i=1}^{N}\mathcal{L}_{\texttt{CE}}(a_{i},g(\phi(x_{i}+\epsilon)))\right]+\lambda||\epsilon||_{2}$
(2)

where $\lambda$ regularizes the magnitude of $\epsilon$. Sec. 3.4 explores its
effect in detail. Importantly, we also offer a zero-order strategy for black-
box APIs, as detailed in Sec. 2.3.

Fair Disease Classification Having obtained the UDE noise $\epsilon$ to
conceal the SA $\mathcal{A}$, we then freeze and add it to disease
classification images $X_{T}$. As shown in Fig. 1 (c), this action guarantees
that the FM generates a fair embedding devoid of sensitive information,
denoted as $\hat{z}_{i}=\phi(x_{i}+\epsilon)$. Finally, these fair embeddings
are leveraged by the subsequent disease classifier $f$ for prediction. To
train the classifier $f$, we employ a cross-entropy loss:

$\mathcal{L}_{T}=\frac{1}{M}\sum_{i=1}^{M}\mathcal{L}_{\texttt{CE}}(y_{i},f(\phi(x_{i}+\epsilon)),$
(3)

where classifier $f$ is the only trainable parameter and $M$ is the size of
$X_{T}$.

###  2.3 Greedy Zero-order Optimization for Black-box FM API

Zero-order (ZO) optimizations [2, 12], which avoid the need for gradient
computations, offer distinct advantages in optimizing black-box models.
However, these methods often exhibit slower convergence rates, particularly in
training large FM models. In contrast, our proposed UDE, concentrating on the
input space, offers a mitigation of this training efficiency issue of ZO
optimization.

Inspired by ZO-SGD [19] and recent MeZO [13] that employ in-place perturbation
updates, we propose a G(reedy) Z(ero-)O(rder) (GeZO) optimization specifically
for efficient UDE. GeZO employs in-place perturbations [19, 13] and greedily
updates with the gradient sign that achieves global optimal loss, thereby
accelerating the optimization process. As shown in Fig. 1 (d), the core
procedure of GeZO is the Greedy Gradient in the right box, where the gradient
of UDE’s objective (Eq. 2) is estimated for each local iteration
$r\in\\{1,...,|R|\\}$ in GeZO. Greedy Gradient takes the UDE noise from the
current local iteration ($\epsilon^{r}$) and estimates its gradient by
continuously adding minor perturbations ($\delta$) in different directions
(_e.g._ , $d_{1}$ and $d_{2}$) to it, _i.e._ , $d_{1}\cdot\delta$ and
$d_{2}\cdot\delta$. It then greedily selects the best direction perturbation
($d_{\texttt{best}}\cdot\delta$) that results in the smallest objective in
this process. Like MeZO [13], we integrate stochastic sampling for $C$ times.
Each time we calculate the objective using a subset of training data to avoid
sucking to the local optimum. Once the best direction and perturbation are
selected for each local iteration $r$, we add $d_{\texttt{best}}\cdot\delta$
to the current velocity, $v^{r}$. This velocity keeps track of the accumulated
updating direction and magnitude for $\epsilon$ for each local iteration $r$,
where the initial value for velocity $v^{1}$ is 0. As shown in the right part
of Fig. 1 (d), $v^{r+1}$ is updated by adding the best perturbation returned
by the Greedy Gradient. Finally, the UDE noise $\epsilon^{r+1}$ is updated
through adding the $v^{r+1}$. The local iterations $|R|$ is a hyper-parameter,
where a smaller iteration increases the efficiency at the cost of the more
biased gradient estimation. We investigate the effect of different $|R|$ in
Sec. 3.4. In actual implementation, we also introduced a momentum to update
the velocity to accelerate the convergence. While the key idea is presented in
this section, we provide a detailed step-by-step algorithm box for GeZO in
Alg. 1.

Algorithm 1 Greedy Zero-order (GeZO) Optimization

Setup: Local Iterations $R$, step size $s$, step size decay $k$, Edit from
last global epoch $\epsilon_{\texttt{epoch}}$, sample times $C$, best loss
$\mathcal{L}_{\texttt{best}}$, best direction $d_{\texttt{best}}$, momentum:
$\mu$, batch size $B$.

1:procedure UpdateEditionPerEpoch($\epsilon_{\texttt{epoch}}$):

2: $\epsilon^{1}\leftarrow\epsilon_{\texttt{epoch}}$, $s^{1}\leftarrow 0.01$,
$v^{1}\leftarrow 0$, $\mu\leftarrow 0.9$,
$\mathcal{L}_{\texttt{best}}\leftarrow\infty$, $k\leftarrow 0.95$

3: for $r=1\to R$ do

4:
$d_{\texttt{best}},\mathcal{L}_{\texttt{best}}\leftarrow$GreedyGradient($B,\epsilon^{r},s^{r}$)
$\triangleright$ find the $d_{\texttt{best}}$ achieve
$\mathcal{L}_{\texttt{best}}$

5: if $d_{\texttt{best}}\neq\text{None}$ then

6: $v^{r+1}\leftarrow\mu\cdot v^{r}+d_{\texttt{best}}$

7: $\epsilon^{r+1}\leftarrow\epsilon^{r}+v^{r+1}$, $s^{r+1}\leftarrow s^{r}$
$\triangleright$ Update $\epsilon$ with momentum

8: else

9: $s^{r+1}\leftarrow k\cdot s^{r}$ $\triangleright$ Reduce step size if no
improvement

10: return $\epsilon^{R}$ $\triangleright$ Return the updated
$\epsilon_{\texttt{epoch + 1}}$

11:procedure GreedyGradient($B,\epsilon^{r},s^{r}$)

12: $d_{\texttt{best}}\leftarrow\text{None}$

13: for $c=1\to C$ do $\triangleright$ Sample $C$ times

14: $\delta\leftarrow\text{RandomPerturbation}()\cdot s_{r}$ $\triangleright$
Generate scaled perturbation

15: for $d\in\\{-1,1\\}$ do

16: $\epsilon^{\prime}\leftarrow\epsilon^{r}+d\cdot\delta$ $\triangleright$
Apply perturbation in both directions

17:
$\mathcal{L}_{\epsilon}=-\left[\frac{1}{B}\sum_{i=1}^{B}\mathcal{L}_{\texttt{CE}}(a_{i},h(\phi(x_{i}+\epsilon^{\prime})))\right]+\lambda||\epsilon^{\prime}||_{2}$
$\triangleright$ One batch

18: if $\mathcal{L}_{\epsilon}<\mathcal{L}_{\texttt{best}}$ then

19: $\mathcal{L}_{\texttt{best}}\leftarrow\mathcal{L}_{\epsilon}$,
$d_{\texttt{best}}\leftarrow d\cdot\delta$ $\triangleright$ Update best loss
and direction

20: return $d_{\texttt{best}},\mathcal{L}_{\texttt{best}}$.

##  3 Experiement

###  3.1 Settings

Table 1: Training and Testing Set Distribution for Pneumonia, Edema, and Pleural Effusion. Diseases | Pneumonia | Edema | Pleural Effusion
---|---|---|---
Negative | Positive | Negative | Positive | Negative | Positive
M | F | M | F | M | F | M | F | M | F | M | F
$\\#$ Train Sample | 1500 | 150 | 150 | 1500 | 5000 | 500 | 500 | 5000 | 5000 | 500 | 500 | 5000
$\\#$ Test Sample | 100 | 100 | 100 | 100 | 200 | 200 | 200 | 200 | 200 | 200 | 200 | 200

Dataset. To demonstrate the generalizability of UDE. we adopt the CheXpert
dataset [6], a chest X-ray dataset with multiple disease labels, to predict
the binary label for Pleural Effusion, Pneumonia and Edema individually in
chest radiographs. All the ablation studies are performed on Pleural Effusion
classification task for space limit. We take gender bias (male and female) as
an example due to its broad impact on society and medical imaging analysis. To
demonstrate the effectiveness of bias mitigation methods, we follow [3] to
amplify the training data bias for each disease by (1) firstly dividing the
data into different groups according to the SA; (2) secondly calculating the
positive rate of each subgroup; (3) sampling subsets from the original
training dataset and increase each subgroup’s bias gap (more positive sample
in a subgroup). Then, we sample testing data with the same procedure, but
achieve an equal subgroup bias gap.

Evaluation metrics. We use the classification accuracy to evaluate the utility
of classifiers on the test set. To measure fairness, we employ equal
opportunity (EO) [5] and disparate impact (DI) [11] metrics. EO aims to ensure
equitable prediction probabilities across different groups, defined by the
sensitive attribute $a^{j}$, for a given class $y^{j}$. It quantifies the
disparity in true positive rates between groups:
$\text{EO}_{Y=y_{1}}=P(\hat{Y}=y^{1}|Y=y^{1},A=a^{1})-P(\hat{Y}=y^{1}|Y=y^{1},A=a^{2})$,
where a smaller gap signifies greater equality of opportunity. DI evaluates
the presence of indirect discrimination by measuring the ratio of positive
predictions across different groups:
$\text{DI}=\frac{P(\hat{Y}=1|A=a^{1})}{P(\hat{Y}=1|A=a^{2})}$. DI closing to 1
indicates the minimal disparity in positive prediction rates between the
groups. To align with EO metric, we employ the $|1-\textsc{DI}|$ to quantify
the fairness in percentage, with smaller values denoting greater fairness.

###  3.2 Implementation Details

Architecture. In our implementation, all methods use the same architecture. To
simulate the FM API, we utilize a publicly available self-supervised pre-
trained ViT-base model 222https://github.com/lambert-x/Medical_MAE, optimized
on X-ray images, given its superior performance metrics among all other
architectures [24]. Within our study, we fix the encoder of the ViT model and
only fine-tune the classification head, simulating the configuration detailed
in Sec. 2.1.

Univeresal Debiased Editing. For UDE, we first fine-tune the SA classifier,
$g$, using the feature encoded by the FM ($\phi(x_{i})$) with the training set
of Pleural Effusion in Table 1. We update it using Adam with a learning rate
(lr) of $0.0001$ for $50$ epochs. Second, we update the UDE noise through
implementing Eq. 2 by initializing $\epsilon$ as a trainable PyTorch parameter
with all entries initially set to zero. The parameter matches the input image
dimensions of $224\times 224$. If not otherwise mentioned, we choose
$\lambda=0.01$ in Eq. 2. It is optimized using the Adam optimizer with lr of
$0.01$ for $50$ epochs. Finally, we fine-tune the disease classifier, $f$,
with $\epsilon$ added on the input data as delineated in Eq. 3. We update it
using AdamW with lr of $1.25\times 10^{-4}$ for $50$ epochs.

GeZO. Here, we keep all parts for UDE the same as above, except using GeZO to
update the edit rather than the Adam, given that the gradient is not
accessible. The only key parameter that we are interested in is the local
iteration $T$, where we set $T=10$ if not otherwise mentioned. Furthermore,
the detailed implementation of the algorithm and hyperparameter setting for
GeZO can be found in Alg. 1.

###  3.3 Comparison with Baselines

Table 2: Comparision of binary prediction of Pleural Effusion, Pneumonia, and
Edema. We label the best performance in bold and the second-best performance
with underline. All the values are in percentage.

Diseases | Pleural Effusion | Pneumonia | Edema
---|---|---|---
$EO_{n}\downarrow$ | $EO_{p}\downarrow$ | DI $\downarrow$ | Acc $\uparrow$ | $EO_{n}\downarrow$ | $EO_{p}\downarrow$ | DI $\downarrow$ | Acc $\uparrow$ | $EO_{n}\downarrow$ | $EO_{p}\downarrow$ | DI $\downarrow$ | Acc $\uparrow$
ERM | 40.5 | 57.0 | 58.5 | 72.9 | 70.0 | 70.0 | 74.5 | 59.6 | 42.0 | 39.0 | 42.0 | 74.5
Sketch [25] | 44.0 | 52.0 | 57.5 | 66.3 | 64.0 | 61.0 | 72.6 | 56.3 | 44.0 | 15.0 | 15.5 | 67.0
Group DRO [16] | 41.0 | 58.0 | 59.5 | 72.3 | 60.0 | 56.0 | 64.4 | 61.0 | 40.5 | 40.5 | 43.3 | 74.8
Batch Samp. [15] | 41.5 | 50.5 | 51.8 | 74.3 | 64.0 | 72.0 | 76.6 | 60.5 | 39.0 | 43.0 | 44.8 | 76.0
BiasAdv [10] | 39.0 | 54.5 | 58.2 | 71.1 | 36.0 | 54.0 | 77.1 | 61.0 | 39.0 | 33.5 | 36.6 | 74.9
UDE | 28.5 | 23.0 | 25.6 | 75.1 | 38.0 | 25.0 | 34.7 | 61.3 | 27.5 | 17.0 | 19.7 | 75.6
UDE-GeZO | 37.0 | 25.0 | 28.5 | 72.8 | 35.0 | 27.0 | 33.7 | 61.5 | 34.0 | 15.5 | 17.2 | 75.1

####  3.3.1 Quantitative Analysis.

Table 2 summarizes the performances among different diseases and baselines.
The baselines include model-based strategy, like biasAdv 333In BiasAdv’s
implementation, we treat FM API as a white box as it requires access to FM’s
gradient and there is no zeroth-order optimization for it yet. [10]; data-
based strategies, like batch sampling [15] with data re-distribution and
sketch [25] with data generation; and prediction calibration-based strategy,
like Group DRO [16]. The details of these baselines are introduced in Sec. 1.
The results indicate that the UDE effectively balances fairness and utility.
Taking Pleural Effusion as the example to analyze, the white-box optimized UDE
outperforms all baselines. The $EO_{p}$ and DI decrease over 25% compared to
all the baselines, an indication of less disparity between the two genders.
Additionally, the utility not only maintains but also surpasses all other
groups, e.g., UDE’s accuracy (Acc) is higher than 2.2% compared to ERM. Given
we are using a balanced testing set, this increase is also a sign of better
generalization. Similar for Pneumonia and Edema, where UDE and UDE-GeZO takes
the best two performance for almost all entries. Although learned through
black-box optimization, our UDE-GeZO achieves comparable performance to the
standard optimization used in UDE. This demonstrates not only the validity of
GeZO but also the efficiency afforded by the small number of edition
parameters [13]. In Pneumonia, UDE-GeZO even slightly surpasses the
performance of UDE across most metrics.

These findings affirm the efficacy of UDE, where it not only facilitates fair
medical image embedding and training but also introduces better
generalizability in downstream classification tasks. Furthermore, the
universal applicability of the UDE is validated in different diseases, where
it consistently outperforms the baselines across three diseases in both
fairness and utility.

###  3.4 Ablation Studies

Effect of Regularization Coefficient. To investigate the effect of different
regularization coefficients’ ($\lambda$) effect, we vary $\lambda$ from 0 to
1. Fig. 2 (a) depicts the EO and the Acc for different $\lambda$s. The metrics
for ERM are labeled as horizontal dashed lines for convenience. As shown, both
the fairness and utility metrics remain relatively stable for $\lambda<1$,
consistently surpassing the ERM baseline. However, as $\lambda$ increases to
1, we observe a marked decline in accuracy below the ERM benchmark, along with
a notable increase in the EO metric for both classes. This performance
deterioration can be explained by the excessive constraint imposed by the
larger $\lambda$, which severely limits the magnitude of editing, thus
affecting the UDE’s functionality to learn useful disease features to improve
fairness and accuracy.

(a)

(b)

Figure 2: Ablation study of our methods: (a) Effect of different $\lambda$;
(b) Effect of different local epochs using GeZO.

Effect of Local Iterations in GeZO. In black-box API, total local iterations
$R$ in GeZO affect the optimization performance, wherein larger $R$ leads to
more accurate optimization at the cost of efficiency. Here, we examine the
effect of changing $R$, as shown in Fig. 2 (b). For fairness metrics,
increasing $R$ from 2 to 20 significantly reduces the EO score for both
classes, demonstrating a considerable debiasing impact with more local epochs.
This effect is attributed to increased perturbation sampling that expands the
search space with more local epochs, as introduced in Sec. 2.3. Meanwhile,
accuracy remains relatively stable, with minor fluctuations between 72% and
73%. Based on these findings, we choose $R=10$ for an optimal balance between
fairness and performance.

###  3.5 Visualization on UDE Noise

(a) (a)

(b) (b)

(c) (c)

Figure 3: Visualization of images with UDE noise with gaussian smoothing for
(a) Pleural Effusion; (b) Pneumonia; (c) Edema. For each row, images are
sequenced from left to right as follows: negative and positive images
optimized via white-box when the gradient is accessible, followed by negative
and positive images optimized via GeZO when gradient access is constrained. We
note that adding UDE noise onto the original image still provides visually
high fidelity. Figure 4: Visualization of chest X-rays and UDE noise patterns
(with gaussian smoothing) to interpret gender-discriminative image regions.
(a) The normalized UDE noise map, with larger noise highlighted by brighter
color, reveals gender-discriminative features. The large noise circled in
orange corresponds to the breast. The large noise circled in yellow reflects
artifacts on X-ray, such as text notations. (b) A female chest X-ray. (c) A
male chest X-ray.

This section presents the visualization of our added noise and images. Fig. 3
illustrates the preservation of high fidelity in images after adding noise,
while Fig. 4 interprets this noise. To enhance visual clarity, we apply a
Gaussian kernel to smooth out the noise, addressing the patch-like patterns
that arise from using a ViT as the encoder.

Fig. 3 demonstrates that images modified with noise maintain good fidelity for
various diseases, under both white-box and black-box optimization methods.
This suggests that the modified images continue to convey critical visual
disease information for human evaluation.

Fig. 4 (a) displays the top 20% of the normalized noise map, highlighting
larger noises in brighter colors and marking areas with significant noise
magnitude for easy identification. Fig. 4 (b) and (c) examine chest X-rays of
a female and a male, respectively. Notably, large noises, marked in orange,
align with areas potentially containing gender-specific information, such as
the breast region. Yellow circles highlight areas of artifacts, like machine
annotations on the X-ray. These noise maps enhance the interpretability of SA
discriminative features by clearly identifying significant noise locations.

##  4 Conclusion

In this study, we address a crucial, yet under-explored aspect of health
equity—the inherent bias in FM API’s usage for classification—through the
introduction of universal debiased editing. UDE effectively masks bias-
inducing pixels, enhancing fairness in API-generated embeddings. Furthermore,
GeZO tackles the challenge of the inaccessibility of the gradient in black-box
APIs by estimating gradients via perturbation. Future research will extend
UDE’s application across various FM APIs and settings, aiming to solidify its
role in promoting fairer machine learning practices.

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# Demonstration of electron focusing using electronic lenses in low-
dimensional system

Chengyu Yan<EMAIL_ADDRESS>London Centre for Nanotechnology, 17-19 Gordon
Street, London WC1H 0AH, United Kingdom
Department of Electronic and Electrical Engineering, University College
London, Torrington Place, London WC1E 7JE, United Kingdom  Micronova, Aalto
University, Tietotie 3, Otaniemi, Espoo, 02150,Finland
Michael Pepper  London Centre for Nanotechnology, 17-19 Gordon Street, London
WC1H 0AH, United Kingdom
Department of Electronic and Electrical Engineering, University College
London, Torrington Place, London WC1E 7JE, United Kingdom  Patrick See
National Physical Laboratory, Hampton Road, Teddington, Middlesex TW11 0LW,
United Kingdom
Ian Farrer  Department of Electronic and Electrical Engineering, University of
Sheffield, Sheffield, S1 3JD, United Kingdom
David Ritchie  Cavendish Laboratory, J.J. Thomson Avenue, Cambridge CB3 OHE,
United Kingdom
Jonathan Griffiths  Cavendish Laboratory, J.J. Thomson Avenue, Cambridge CB3
OHE, United Kingdom

###### Abstract

## Abstract

We report an all-electric integrable electron focusing lens in n-type GaAs. It
is shown that a pronounced focusing peak takes place when the focal point
aligns with an on-chip detector. The intensity and full width half maximum
(FWHM) of the focusing peak are associated with the collimation of injected
electrons. To demonstrate the reported focusing lens can be a useful tool, we
investigate characteristic of an asymmetrically gate biased quantum point
contact with the assistance of focusing lens. A correlation between the
occurrence of conductance anomaly in low conductance regime and increase in
FWHM of focusing peak is observed. The correlation is likely due to the
electron-electron interaction. The reported electron focusing lens is
essential for a more advanced electron optics device.

††preprint: APS/123-QED

## Introduction

During the past several years, electronic and optical technologies have seen
many encouraging developments. On the electronic end, emergent devices such as
spintronicsCFX05 ; WPI10 ; BDS12  and valleytronicsRTB07 ; SYC16  have been
proposed and realized; similarly, devices such as coherent optical memoryHDD04
and optical qubitPAL04  have laid the foundation of optical quantum
computation. Integrating optical and electronic properties into a single
platform or electron optics will provide a unique system for investigating
many phenomena emerging from their fusion. The wave nature of electrons in low
dimensions, especially ballistic electrons in clean semiconductors, could be
exploited for geometrical optical phenomena. In order to realize the potential
of electron optics, it is necessary to establish a mapping between the
fundamental optical components and their electronic counterparts. In this
regard, it is well known that quantum point contacts (QPCs) or other low-
dimensional electron sources are equivalent to a coherent optical sourceTLS00
; electronic spin polariser is inspired by an optical polariserDD90 ;
electronic cavity/mirror shares functional similarities with the optical
onesDTW01 . Electron focusing with the electrostatic lens has been
demonstrated with a double-concave lensSHU90 ; SSB90 . However, the primary
parameter that determines the focusing profile has not been addressed.

In the present work, we demonstrate an all-electric electron focusing lens
with a more intuitive design and identify the main factor that shapes the
focusing profile. A focusing peak occurs whenever the focal point, which is
gate-voltage tunable, spatially aligns with the on-chip detector. It is found
that the focusing profile is closely associated with the collimation of
injected electrons. To give an example of the potential usage of the
electronic focusing lens, we utilized this technique to investigate the
characteristic of an asymmetrically gate biased QPC. A correlation between the
occurrence of conductance anomaly in low conductance regime and increase in
FWHM of focusing peak is observed. The correlation is likely due to electron-
electron interaction.

Figure 1: Schematic of experiment setup and representative results of
electronic focusing. (a) Typical functionality of an optical double-convex
lens. If the incident light is parallel with the primary axis of the lens,
then it would be guided to the focal point $f_{0}$ (upper panel); on the other
hand, light with a non-zero incident angle with respect to the primary axis
would be guided to $f^{\ast}$ instead of $f_{0}$. (b) SEM image of the
experiment setup. The dimension of the split gates and lens gate of the imaged
device are the same as those used in the experiment; the difference relies on
that the gap between the split gates and lens gate are 300 nm for the imaged
device, and 100 nm (dev A) and 150 nm (dev B) for the measured devices. QPC1
and 2 are injectors; QPC3 is used as detector. The top-gate with a hollow area
is referred as lens gate, and it is patterned over a PMMA layer. Squares 1-5
at the edges of the mesa represent Ohmic contacts. An enlarged image, Fig. S1,
can be found in the supplementary information. (c) and (d) show representative
results before and after the sample was illuminated with a red LED. QPC 1-3
were set to G0. $I_{3}$ represents the signal through detector where a
noticeable enhancement, marked by the bold black arrow, was observed;
$I_{4}$+$I_{5}$ is the signal drained to Ohmic contact 4 and 5. It is
important to point out that, $I_{3}$ and $I_{4}$+$I_{5}$ were measured
simultaneously. It should be commented that the focusing peak is observable
either use a single injector or both.  Figure 2: Temperature dependence of
focusing signal. (a) QPC 1-3 were set to G0, the lattice temperature was
incremented from 20 mK to 1.6 K. Data have been offset vertically for clarity.
Inset shows a zoom-in of focusing peak at 20 mK and 1.6 K after removing a
linear background. (b) Normalized peak intensity $\gamma(T)/\gamma(1.6K)$ as a
function of temperature. $\gamma(T)$ was determined by subtracting a linear
background within the vicinity of focusing peak. The solid blue and magenta
lines show the calculated upper and lower bound of peak intensity using Eq.2
and 3 without adjustable parameters. Details about the calculation can be
found in the main text.  Figure 3: Focusing peak as a function of asymmetric
gate bias after illumination. (a) Conductance characteristic of QPC1 with
different asymmetric gate bias $\Delta V$; gate voltage $V_{sg}$ was applied
to the bottom arm of the split gate [refers to Fig.1 (b)], $V_{sg}+\Delta V$
was applied to the upper arm. Series resistance has not been removed. The red
arrow highlights the occurrence of a sub-0.7-anomaly. (b) Focusing peak with
different $\Delta V$ applied to QPC1. QPC1 was set to G0, the gate voltage
applied to the two arms of QPC1 were calibrated according to (a), QPC2 was not
used; QPC3 was under symmetric gate bias and fixed at G0. Data in (b) have
been offset vertically for clarity. (c) Peak position (blue) and FWHM (red) as
a function of $\Delta V$. It is necessary to mention a background has been
removed in determining the FWHM. The background can be determined from two
ways: 1. make a polynomial fitting in the vicinity of the focusing peak, as
shown in supplemental Fig. 1; 2. scales the data with both Ohmic 4 and 5
floating which does not include the correction due to focusing process, so
that at zero lens gate voltage (the scaled) $I_{3}$ is the same when Ohmic 4
and 5 are floating and grounded. The two methods lead to a similar conclusion.

## Experiment

To realize an electronic analogue of an optical focusing scheme, it requires
both collimated electron source and electron focusing lens [Fig. 1(a)]. The
highly collimated ballistic electronsMSB90 ; STO92  are injected by two QPCs,
i.e., QPC1 and 2 as shown in Fig. 1(b), whereas QPC3 functions as a detector.
Electron injection angle concentrates at 0∘ when the QPC is confined to low
conductance regime, but always has a finite angular spreadMSB90 ; STO92 .
Before being collected at QPC3, collimated electrons pass through an
electronic focusing lens defined via a top gate encapsulating a double-concave
shaped hollow regime [[Fig. 1(b)]; see supplementary information note 1 and 2
for comments on lens designSI19 ]. Electron refraction follows the Snell’s
lawSHU90 ; SSB90 . The relative refractive index
$N_{r}=\sqrt{\frac{n_{1}}{n_{2}}}$, which determines the location of focal
point, is adjustable via reducing $n_{2}$ by applying negative gate voltage
($n_{2}$ is the electron density under the top-gated regime; $n_{1}$ is the
density within the hollow area or raw 2D density).

The functionality of the electronic lens centered between the QPCs can be
verified by noticing an enhancement in the detected signal when the focal
point aligns with the saddle point of the detector (QPC3). To detect the real
focusing signal, the residual signals were simultaneously drained to Ohmic
contacts 4 and 5 in Fig. 1(b); otherwise, all the injected electrons would be
drained via the detector whether or not the focusing condition was matched.
Figure 1(c) shows a representative result with QPC 1-3 set to G0 (
G0=$\frac{2e^{2}}{h}$). The residual current $I_{4}$+$I_{5}$ measured at Ohmic
contacts 4 and 5 resembled a typical pinched-off behaviour; on the other hand,
$I_{3}$ measured at Ohmic contact 3 yielded a series of peaks. Peaks in
$I_{3}$ near pinched-off regime may arise from charging effect or scattering
at low electron density limit as suggested by their insensitivity against
transverse magnetic field [Fig.S5 of supplementary information]; the small
fluctuation can be a result of universal conductance fluctuation; peak
occurred at lens gate voltage $V_{lens}$ $\approx$ -0.90 V [marked by the bold
black arrow in Fig. 1(c)] was an indication of electron focusing. The focal
length $l_{f}$ at given $V_{lens}$ can be calculated from lensmaker’s
equation:

$\frac{1}{l_{f}}=(N_{r}-1)\times[\frac{1}{R_{1}}-\frac{1}{R_{2}}+\frac{(N_{r}-1)D}{N_{r}R_{1}R_{2}}]$
$None$

where $R_{1}$ = 2 $\mu$m and $R_{2}$ = -2 $\mu$m are the radius of right and
left surface of the lens, D = 2 $\mu$m is the thickness of lens, $N_{r}$ can
be extracted from capacitance modelYSU91 ; YTS18  as
$N_{r}=\sqrt{\frac{V_{pin}}{V_{pin}-V_{lens}}}$, where $V_{pin}$ indicates the
pinched-off voltage of the lens gate. Inserting $V_{pin}$ = -1.60 V and
$V_{lens}$ = -0.90 V, Eq.1 yielded that $N_{r}$ = 1.521 and $l_{f}$ = 2.17
$\mu$m which agrees very well with the lithographically defined distance (2.25
$\mu$m) between geometric centre of the lens and QPC3. The difference in the
value may arise from the fact that the effective dimension of the
(electrostatic) lens differs from the lithographically defined one; besides,
an offset of the lens along primary axis is also possible. An enhancement in
$I_{3}$ happened when the focal point of the electronic lens aligned with the
saddle point of QPC3. In addition, the width of the focusing peak also
revealed important insights. The focusing peak started forming once the focal
point was driven to the vicinity of the entrance of QPC3 and attenuating when
the focal point passed the exit; therefore a change in $l_{f}$ within this
range of $V_{lens}$ should match channel length of QPC3. The full-width half
maxima (FWHM) of the focusing peak suggested a change in $l_{f}$ of 313 nm
which is consistent with the lithographically defined channel length of 300
nm. The difference in values could probably arise from the finite angular
spread of injected electrons.

To further validate the existence of the focusing process, we presented data
after the sample was illuminated by a red LED in Fig. 1(d) with all QPCs set
to G0. According to Eq.(1), $N_{r}$ must remain the same when focusing peak
occurred before and after illumination. It was found that the only prominent
peak after illumination, the focusing peak, in $I_{3}$ happened at $V_{len}$ =
-1.08 V whereas $V_{pin}$ became -1.90 V as a result of increased 2D electron
density, these values suggest $N_{r}$ = 1.523 just as before illumination.

It might be concerned that the focusing peak may arise from trivial
electrostatic effects such as the cross-coupling between the split gates and
lens gate, or coherent effects such as universal conductance fluctuation.
These interpretations can be excluded by the data obtained with different
injector conductance, different combination of grounding scheme (for instance,
both Ohmic 4 and 5 are floating or grounded), temperature and transverse
magnetic field dependence [see note 4 of supplementary information for
detailed discussion].

### Temperature dependence of the focusing peak

The evolution of focusing peak with lattice temperature elevated from 20 mK to
1.6 K is presented in Fig. 2(a). The focusing peak intensity $\gamma$ weakened
with increasing temperature and followed the trend based on electron
collimation as shown in Fig. 2(b). To avoid the constant pre-factor which
could not be calculated directly, $\gamma(T)$ was normalized against
$\gamma(1.6K)$. $\gamma$ was closely associated with electron collimation, it
was suggested thatMMH92

$\gamma(T)/\gamma(1.6K)=exp\\{-l[l_{m}(T)^{-1}-l_{m}(1.6K)^{-1}]\\}$ $None$

where $l$ is electron propagation length, $l_{m}(T)$ is the mean free path for
electron-electron scattering at given temperature $T$, which is defined as
$l_{m}(T)=v_{F}\times\tau_{ee}(T)$ where $v_{F}$ is the Fermi velocity and
$\tau_{ee}(T)$ is electron-electron scattering timeMMH92

$\frac{1}{\tau_{ee}(T)}=\frac{E_{F}}{h}(\frac{k_{B}T}{E_{F}})^{2}[ln(\frac{E_{F}}{k_{B}T})+ln(\frac{2q_{TF}}{k_{F}})+1]$
$None$

where $E_{F}$ is the Fermi energy, $k_{B}$ is the Boltzmann constant, $q_{TF}$
is the Thomas-Fermi screening wave vector, $k_{F}$ is Fermi wave vector. $l$
can be obtained by assuming electrons follow the classical trajectory
connecting QPC1 (QPC2) and QPC3 with refraction at the lens taken into
account, whereas $l_{m}(T)$ is fully determined by $T$ and electron density in
the area between injectors and detector. The non-uniform electron density
underneath the lens (including the top-gated regime and hollow area) was a
complex situation. To simplify the calculation, we assume a uniform top-gate,
without hollow regime, is patterned between the injector and detectors. The
upper bound of the calculation [blue solid line in Fig.2(b)] corresponds to
$V_{lens}$ = -0.90 V applied to the virtual uniform top-gate, whereas lower
bound is given by zero-gate voltage [red solid line in Fig.2(b)]. It turned
out that the experimental data lied between the lower and upper bound but much
closer to the upper bound.

The agreement between $l$ and classical trajectory length (through a double-
convex lens) further supported that peak at $V_{len}$ = -0.90 V arose from
electron focusing. The temperature dependence data also indicate that the
focusing peak is not due to trivial coherent effect [supplementary information
note 4].

## Focusing peak with asymmetrically gate biased injector

Figure 4: Simulated confinement potential and electric field. (a) and (b) show
simulated confinement potential and absolute value of electric field with
$\Delta V$ = -1 V. The blue trace considers electrostatic contribution only
whereas the red trace is with correction due to electron-electron interaction.
Inset highlights results around the potential minimum.

After confirming the lens was capable of focusing electrons, we utilized the
technique to investigate the property of an asymmetrically gate biased QPC
where lateral spin-orbit coupling (LSOC) was suggested to play a vital
roleDRW09 ; BKF12 ; CHS15 ; VSO16 ; DJC17 , even in GaAsPPZ18 .

The behaviour of injector QPCs with asymmetric gate bias was characterized
using standard conductance measurement. Taking QPC1 as an example, if gate
voltage $V_{sg}$ was applied to the bottom arm of the split gate [refers to
Fig.1 (b)], $V_{sg}+\Delta V$ would be applied to the upper arm. $\Delta V$
was kept negative so that the 1D channel shifted towards the primary axis [Fig
1(a)] instead of the tips of the lens, to avoid diffraction at the tipsHHH00 .
A short plateau-like feature, as marked by the red arrow in Fig.3(a), below
0.7-anomaly started forming with increasing $\Delta V$ similar to previous
observationsDRW09 ; BKF12 ; CHS15 ; VSO16 ; DJC17 ; PPZ18 . We denoted this
feature as sub-0.7-anomaly hereafter. However, the sub-0.7-anomaly was
unlikely to arise from previously proposed LSOCDRW09 ; BKF12 ; CHS15 ; VSO16 ;
DJC17 ; PPZ18  . In GaAs electron gas, the ground-state electrons propagate
along the potential minimum of the 1D channel. For the studied device, the
lateral electric field at the potential minimum is about 100 V/m with $\Delta
V$ = -1 V according to simulation in Fig. 4 [supplementary information note
8], which is insufficient to generate noticeable LSOC. Ideally speaking the
electric field at the potential minimum should be zero, the finite value
obtained here is likely due to the finite grid spacing used in the simulation.

Additional insight into the observed sub-0.7-anomaly can be extracted from the
focusing experiment. In this experiment QPC1 (QPC2 was not used) was set to G0
under different $\Delta V$; on the other hand, QPC3 was under symmetric bias
and fixed at G0. The results are summarized in Fig. 3(b) and (c). It was clear
that the focusing peak centred at -1.08 V regardless of $\Delta V$. The
robustness of focusing peak position indicates that electron injection angle
from a QPC tends to concentrate at 0∘ even with asymmetric confinement.
Assuming that the injection angle concentrates at a non-zero angle $\alpha$,
then the injected electrons should be guided to $f^{\ast}$ instead of the
focal point [see upper panel of Fig. 1(a)]; to make $f^{\ast}$ align with the
saddle point of detector, an adjustment on $V_{lens}$ is necessary, for
instance the focusing peak should occur at $V_{lens}$ = -1.17 V assuming
$\alpha$ = 6∘. Although the central position of focusing peak showed no
explicit dependence on $\Delta V$, FWHM of focusing peak almost increased
monotonically against $\Delta V$, as shown in Fig. 3(c). The broadening of
FWHM suggests a larger angular spread of injected electrons; in other words, a
reduction in collimation [supplementary information note 7]. FWHM started
increasing rapidly at $\Delta V$ = -0.4 V which was roughly the same $\Delta
V$ to trigger the sub-0.7-anomaly.

We suggest that the occurrence of sub-0.7-anomaly and increase in FWHM of
focusing peak are possibly due to electron-electron interaction. Applying the
asymmetric gate-bias $\Delta V$ results in a change in the effective length of
the 1D channel, whereas the electron-electron interaction (e-e interaction)
especially the exchange part is sensitive to the channel length, it has been
shown conductance anomaly can occur between 0.8$\times$ $\frac{2e^{2}}{h}$ to
0.4$\times$ $\frac{2e^{2}}{h}$ depends on the channel lengthJYB06 . On the
other hand, for a symmetrically gate-biased QPC, the angle $\alpha$ within
which the electrons are highly collimated is given byBH89 ; RCC00 ,

$\alpha=\pm
arcsin(\sqrt{\frac{E_{F}-E_{b}}{E_{F}}}\times\frac{W_{min}}{W_{max}})$ $None$

where $E_{b}$ is the potential at the saddle point where minimum 1D channel
width $W_{min}$ occurs, $W_{max}$ is the critical channel width where electron
transport still remains non-adiabatic. It is found that $\alpha$ changes from
5.24∘ ($\Delta$V = 0) to 5.17∘ ($\Delta$V = -1 V) without e-e interaction; on
the other hand, $\alpha$ increases from 5.77∘ ($\Delta$V = 0) to 6.77∘
($\Delta$V = -1 V) after taking e-e interaction into account. It seems that
e-e interaction is an essential ingredient for an observable change in FWHM
[see note 9 of supplementary information for a more detailed discussion].

It is also important to check the role of disorder, which can result in
multiple irregular features on conductance characteristic. If we try to
understand the conductance measurement in Fig.3 (a) based on disorder, it is
natural to think the magenta trace (the third one from the right side), where
several irregular features are observed, corresponds to the case where the
effect of disorder is most substantial. The smoothing of the two most right
traces indicates the channel is moving away from the disorder. It is then
expected that the FWHM of focusing peak should follow a non-monotonic trend if
disorder plays a primary role, however, the experimental result shows a
monotonic trend.

## Extending the focusing scheme to material with strong spin-orbit coupling

It is helpful to apply the focusing scheme to material with strong intrinsic
spin-orbit coupling (SOC). The strong intrinsic SOC can causes lateral motion,
so that the two spin-branches tend to move along the opposite edges of the 1D
channel where the lateral electric field is noticeableDRW09 . LSOC is induced
as a result of the lateral electric field.

It has been shown that a small out-plane magnetic field can make angular
distribution centered at a finite angle rather than 0∘UTK94  . The effective
out-plane magnetic field induced by intrinsic SOC can have similar influence.
Therefore, it is expected that the angular spread has two peaks at $\pm\theta$
in the presence of intrinsic SOC ($\pm$ sign depends on spin orientation)
instead of a single peak at 0∘ when intrinsic SOC is absent. The spin branches
will thus lead to two focusing peaks. Information on both the intrinsic SOC
and LSOC can be extracted from the focusing profile.

## Conclusion

We have developed a double-convex electron focusing lens which is an essential
component for a complete tool kit of electron optics. A focusing peak occurred
when the focal point of the lens aligned with the detector, the intensity of
focusing was closely associated with the degree of electron collimation. Using
the focusing lens, we found that the injection angle of 1D electrons tends to
concentrate at 0∘ even with considerable asymmetric gate bias $\Delta V$.
However, the angular spread broadened with increasing $\Delta V$. The
increment of FWHM was correlated with the occurrence of a sub-0.7-anomaly,
possibly due to electron-electron interaction. The focusing scheme is ready to
be extended to material with strong intrinsic spin-orbit interaction, where it
allows to selectively polarized electron spin in the injector/detector and
study the interplay between local spin states by monitoring the evolution in
corresponding focusing peak.

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The authors gratefully acknowledge assistance on low-temperature measurement
setup from Sanjeev Kumar. This work was supported by the Engineering and
Physical Sciences Research Council (EPSRC), U.K. Author contributions
CY designed and performed the experiments; CY and MP analyzed the data; PS
fabricated the devices; JG did e-beam lithography; IF and DR grew the wafer;
CY wrote the paper with inputs from MP, and other co-authors. All authors read
and approved the final manuscript. Additional Information
Competing Interests: The authors declare no competing interests.