11institutetext: The University of Hong Kong, Hong Kong SAR, China
11email: {yhchen<EMAIL_ADDRESS>11email: {gyin,
<EMAIL_ADDRESS>Wake Forest University School of Medicine
22email<EMAIL_ADDRESS>
# cDP-MIL: Robust Multiple Instance Learning via Cascaded Dirichlet Process

Yihang Chen Equal contribution11 Tsai Hor Chan††footnotemark:  11 Guosheng Yin
11 Yuming Jiang 22 Lequan Yu Corresponding Author11

###### Abstract

Multiple instance learning (MIL) has been extensively applied to whole slide
histopathology image (WSI) analysis. The existing aggregation strategy in MIL,
which primarily relies on the first-order distance (e.g., mean difference)
between instances, fails to accurately approximate the true feature
distribution of each instance, leading to biased slide-level representations.
Moreover, the scarcity of WSI observations easily leads to model overfitting,
resulting in unstable testing performance and limited generalizability. To
tackle these challenges, we propose a new Bayesian nonparametric framework for
multiple instance learning, which adopts a cascade of Dirichlet processes
(cDP) to incorporate the instance-to-bag characteristic of the WSIs. We
perform feature aggregation based on the latent clusters formed by the
Dirichlet process, which incorporates the covariances of the patch features
and forms more representative clusters. We then perform bag-level prediction
with another Dirichlet process model on the bags, which imposes a natural
regularization on learning to prevent overfitting and enhance
generalizability. Moreover, as a Bayesian nonparametric method, the cDP model
can accurately generate posterior uncertainty, which allows for the detection
of outlier samples and tumor localization. Extensive experiments on five WSI
benchmarks validate the superior performance of our method, as well as its
generalizability and ability to estimate uncertainties. Codes are available at
https://github.com/HKU-MedAI/cDPMIL.

###### Keywords:

Multiple Instance Learning Whole Slide Images Bayesian Nonparametric Method
Uncertainty Estimation

##  1 Introduction

Whole slide histopathology images (WSIs) provide enriched information for
modern diagnosis of diseases. However, manually traversing through the WSIs
for clinical diagnosis would be challenging for pathologists as such images
are in giga-pixels (e.g., with a typical size of 80,000 $\times$ 80,000).
Recently, the multiple instance learning (MIL) paradigm has demonstrated
success in modeling WSIs with modern deep learning techniques. Multiple
instance learning (MIL) assumes that the data can be divided into sets of
instances, referred to as bags, and aims to learn the label of each bag, which
consists of multiple instances, with supervision solely provided by the bag
label [1]. MIL is often known as a weakly supervised learning problem because
pixel-wise annotations are limited due to the enormous size of WSIs.

Existing MIL-based methods on WSI analysis mainly focus on designing effective
algorithms to learn the inter-relations of patches, where attention-based [30,
38, 55, 48, 13] and graph-based [61, 28, 7, 12, 16, 9] methods demonstrate
significant success. Effective aggregation algorithms are then developed to
pool the patch-level features to bag-level for predictions on slides. However,
several challenges remain: First, the aggregation methods adopted by existing
paradigms based on first-order distance (e.g., the mean difference of
centroids) fail to provide accurate estimation of the bag-level feature
distribution, which hinders the downstream task performance based on these
pooled features. Second, the bag-level prediction model would mostly suffer
from overfitting due to the limited observations of bags, leading to poor
testing performance and generalizability. Third, most existing methods are
based on a deterministic (i.e., frequentist) framework, which fails to capture
the epistemic uncertainty (i.e., predictive confidence of the fitted MIL
model).

Figure 1:  Illustration of a cascaded Dirichlet process model on whole slide
images. We first pool the WSI patches with latent Gaussian distributions, and
then categorize the pooled WSIs into $K$ classes.

To tackle the limitations of the existing works, we propose a novel framework
for MIL — Cascaded Dirichlet Process Multiple Instance Learning (cDP-MIL),
which is a multi-stage Bayesian nonparametric framework learning the instance-
level and bag-level representations. Fig. 1 illustrates our cDP design. We
tackle instance-level feature pooling with a deep neural network parameterized
Dirichlet process, which effectively clusters the local features. The
posterior log-likelihood estimated by the patch-level DP can also interpret
the contribution of each patch to the slide-level prediction, which enables
the localization of patches of interest (e.g., tumor patches). We then design
a DP predictive module to perform bag-level predictions, which can also
quantify predictive uncertainties and generalize well to unseen data.
Moreover, we parametrize the mean and covariance of each Gaussian component
with a deep neural network and adopt a stochastic variational inference schema
to optimize the parameters with gradient descent. This facilitates fast
convergence and accurate approximation of the distributions in each component,
leading to improved predictive performance. Experiments on five histopathology
image datasets with cancer classification, subtyping, and localization tasks
validate the superior performance of our proposed framework. Thorough ablation
analysis validates the contributions of the proposed modules and robustness to
variations of key components and hyperparameters. We also validate the
capability of cDP-MIL to capture predictive uncertainty through out-of-
distribution (OOD) detection and generalization tasks.

##  2 Related Work

Bayesian Nonparametric Methods. Bayesian nonparametric methods aim to provide
optimal solutions from an infinite dimensional parameter space. They select
the best subset of parameters over an infinite space to best explain finite
sample observations. There has been extensive development on Bayesian
nonparametric methods in statistics, including Gaussian process (GP) [44, 39,
25, 33, 49], Dirichlet process (DP) [19, 34, 31, 5], and Pólya tree [43].
Echraibi et al. [19] combine DP with the Gaussian mixture model (GMM) and
update the prior parameters and variational posterior in closed forms owing to
the reparameterization trick [37]. DP also shows outstanding performance in
clustering due to its flexible determination of the number of clusters and
robust cluster allocation mechanism (from the richer-gets-richer
characteristic of DP, i.e., a sample has a higher probability of being
allocated to the clusters that have more observations) [46, 5, 19]. Existing
works widely adopt Markov chain Monte Carlo (MCMC) methods, e.g., Gibbs
sampling, to fit the DP model, which is not scalable to modern deep learning
frameworks as their time complexity is high. Recent advances in variational
inference [5] make deep Bayesian nonparametric methods possible by introducing
a surrogate variational objective from the probabilistic assumptions.

Multiple Instance Learning.  Multiple Instance Learning (MIL) serves as a
general framework for handling a series of objects, such as text instances,
natural images, and medical images (e.g., WSIs) [40, 28, 61, 13, 12, 23, 51].
Existing MIL paradigms divide WSIs into patches and perform slide-level
prediction through pooling on instance-level features. Attention-based MIL
(ABMIL) [30] employs the attention mechanism to aggregate these features,
enabling accurate predictions as well as providing interpretability. In
addition, there are approaches that model slides using graphs and obtain bag-
level predictions through the utilization of graph convolutional methods [7,
12, 28, 60, 61]. However, these methods are mostly deterministic and lack the
ability to estimate the inherent uncertainty within the fitted model.
Furthermore, the design of regularization poses significant challenges, as
these methods are often prone to overfitting.

Feature Aggregation in MIL. Due to the weak supervision nature of MIL, it is
difficult to accurately estimate the bag-level distribution from many
unlabelled instances, thus making the design of an effective aggregation
mechanism necessary. Many existing works attempt to design a pooling module
that effectively aggregates instance-level features into the bag level [28,
30, 55, 7]. HEAT [7] designs a pooling method based on the predicted patch
type to mitigate imbalanced observations of normal and tumor patches. However,
most of these methods operate on the first-order difference of the instances
in a bag, such as the mean difference, and neglect the covariance structures
of the feature embeddings. The predicted clusters may suffer from inherited
observational bias and decrease the quality of aggregated representations.

Bayesian Methods in MIL.  Recently, Bayesian methods [34, 58, 25] have emerged
as a promising solution to addressing the aforementioned challenges. These
methods offer natural regularization in the learning process while also
providing the ability to quantify uncertainty by sampling from the posterior
predictive distribution. BayesMIL [58] designs a Bayesian attention framework
for WSI classification by placing a mixture of log-normal distribution on
instances to allow for uncertainty estimation and better interpretability.
However, most probabilistic assumptions (e.g., Gaussian distribution) in
conventional Bayesian methods are strong and do not fit satisfactorily to
large WSI analysis, prompting the need to design deep nonparametric methods.
Nonparametric Bayesian methods also excel in MIL due to their capability of
feature representation learning [5, 59, 19, 34, 25, 36, 8]. Some methods
incorporate GP or DP into MIL learning, which, however, mainly focus on text
analysis, so that their adaptivity to large-scale WSI analysis is unknown.
Furthermore, they are also limited to MCMC methods or traditional variational
inference methods for posterior computation, which makes the computational
complexity very high. Moreover, these methods do not evaluate predictive
uncertainty inherited from the models.

##  3 Methodology

In order to capture features of WSIs from different levels more effectively,
our method consists of two major designs: an aggregation module based on DP to
form latent clusters, and a DP classifier to extract the most representative
features from latent clusters for bag-level predictions. The combination of
these two modules formulates a cascaded DP design which allows for more
effective allocations of weights to features. Fig. 2 shows an overview of our
proposed framework and Algorithm 1 presents the detailed workflow.

Figure 2: The overall framework of our proposed cascaded DP model. We first
use Otsu’s segmentation to outline tissue regions and select foreground
instances for MIL learning. We then design an aggregation module based on the
Dirichlet process to cluster instance-level features into latent clusters.
Finally, a prediction module based on DP would perform predictions based on
the distribution learned for each cluster.

1:Input:

2:Number of bags $B$, number of instances per bag $\\{N_{1},\ldots,N_{B}\\}$

3:Extracted features $\\{\mathbf{X}_{b}\\}_{b=1}^{B}$, bag-level labels
${\mathbf{Y}}=\\{Y_{1},\ldots,Y_{B}\\}$,

4:Hyperparameters
$\\{\mathbf{V}_{0},\mathbf{m}_{0},\kappa,T,K,\eta_{1},\eta_{2},p\\}$,

5:Parameter set of network $\mathbf{\Psi}=\\{\psi_{t}\\}_{t=1}^{T}$ and
$\mathbf{\Upsilon}=\\{\mathbf{\upsilon}_{k}\\}_{k=1}^{K}$ to encode mean and
covariance.

6:Output:

7:Trained MIL model $\mathcal{M}$

8:Initialize $\varphi\in\mathbb{R}^{B\times K}$

9:Initialize $\mathbf{\Phi}=[\mathbf{\phi}_{1},\ldots,\mathbf{\phi}_{B}],$
with $\mathbf{\phi}_{b}\in\mathbb{R}^{N_{b}\times T}$

10:for each training epoch do

11: for each bag $b$ do

12: $\mathbf{\phi}_{b}\leftarrow$ Fit-DP
(${\mathbf{X}}_{b},\mathbf{\phi}_{b},\mathbf{\Psi},T,\eta_{1}$)
$\triangleright$ Pooling by Algorithm 2

13: Cluster assignments
$\mathbf{Z}_{b}=\\{z_{bj}=\mathop{\arg\max}\limits_{t}\phi_{bjt},\ j\leq
N_{b},t\leq T\\}$

14: Centroids
$\mathbf{C}_{b}=\text{concat}(\text{mean}\\{\mathbf{x}_{bj},z_{bj}=t\\},t\leq
T)\in\mathbb{R}^{T\times p}$

15: end for

16: $\mathbf{C}\leftarrow\text{Concatenate}(\\{\mathbf{C}_{b}\\})$

17: $\varphi\leftarrow$ Fit-DP
$(\mathbf{C},\varphi,\mathbf{\Upsilon},K,\eta_{2},\mathbf{Y})$
$\triangleright$ Prediction by Algorithm 2

18:end for

19:return trained $\mathcal{M}$.

Algorithm 1 Our proposed cDP-MIL framework

###  3.1 Preliminaries

Multiple Instance Learning (MIL). Given the $b$-th bag of instances (e.g.,
WSIs) $\mathbf{X}_{b}=(\mathbf{x}_{b1},\ldots,\mathbf{x}_{bN_{b}})$, where
$\mathbf{x}_{bk}\in\mathbb{R}^{d}$ is the $d$-dimensional embedding of the
$k$-th instance, $N_{b}$ is the number of instances of the $b$-th bag, and the
$b$-th bag has label $Y_{b}$. The dataset
$\mathcal{D}=\\{\mathbf{X}_{b},Y_{b}\\}_{b=1}^{B}$ is composed of $B$ such
bag-level pairs. The objective of MIL is to learn an optimal model
$\mathcal{M}$ for predicting the bag-level label with the bag of instances as
input.

###  3.2 Cascaded Dirichlet Process for WSI Analysis

We design a cascaded Dirichlet process model for learning the instance-level
and bag-level representations. Instance-level aggregation is performed using
the first DP module by assuming that similar patches have the same latent
cluster label. We then aggregate cluster features into bag-level features,
where new latent cluster labels correspond to our task directly (e.g., each
cluster represents tumor/normal for binary cancer classification). Fig. 3
presents our proposed model in plate notation.

Dirichlet Process.  A Dirichlet process, denoted as $\text{DP}(\alpha,H)$, is
a random probability measure on the sample space $\mathcal{X}$, such that for
any measurable finite partition $S=\\{B_{i}\\}_{i=1}^{K}$ of the sample space,

$\displaystyle(X(B_{1}),X(B_{2}),\ldots,X(B_{K}))\sim\text{Dir}(\alpha
H(B_{1}),\alpha H(B_{2})\ldots,\alpha H(B_{K}))$

where $\alpha$ is a positive concentration rate, $H$ is the base probability
measure and $\text{Dir}(\cdot)$ stands for Dirichlet distribution.

DP Aggregation Module on Patch Level. With the features extracted for each
instance, we design an aggregation mechanism based on DP to aggregate
instance-level features to the bag level. We approximate DP using the stick-
breaking formulation, i.e., for the $b$-th WSI,

$\displaystyle\beta_{t}\sim\text{Beta}(1,\eta_{1}),\quad\pi_{t}=\beta_{t}\prod_{l=1}^{t-1}(1-\beta_{l}),$
(1)

where $\pi_{t}$ is the probability assigned to each cluster of similar
patches. We have $z|\mathbf{\pi}\sim\text{Cat}(\cdot|\mathbf{\pi})$,
characterizing the cluster assignment from a multinomial distribution based on
the cluster probability $\mathbf{\pi}$. Let
$\mathbf{\theta}_{t}=(\mathbf{\mu}_{t},\mathbf{\Sigma}_{t})$ be the mean and
variance parameters of the normal density of the $t$-th cluster, and we place
a normal-inverse-Wishart (NIW) prior on $\mathbf{\theta}_{t}$. We assume a
Wishart prior on the precision matrix $\mathbf{\Lambda}_{t}$ (i.e.,
$\mathbf{\Lambda}_{t}={\mathbf{\Sigma}_{t}}^{-1}$),

$\displaystyle
f_{\mathbf{\Lambda}_{t}}(\mathbf{\Lambda}_{t};\kappa,{\mathbf{V}_{t}})=\dfrac{1}{2^{\kappa
p/2}|\mathbf{V}_{t}|^{\kappa/2}\Gamma_{p}\left({\kappa}/{2}\right)}|\mathbf{\Lambda}_{t}|^{(\kappa-p-1)/2}e^{-\text{tr}({\mathbf{V}_{t}}^{-1}\mathbf{\Lambda}_{t})/2},$

where $\kappa>0$, $\mathbf{\Lambda}_{t}$ is a positive semi-definite matrix
and $\mathbf{V}_{t}\in\mathbb{R}^{p\times p}$ is a fixed symmetric positive
definite matrix, and $\Gamma_{p}$ is the multivariate gamma function. Given
the covariance matrix, $\mathbf{\theta}_{t}$ is distributed as NIW, with the
joint density function

$\displaystyle
p(\mathbf{\mu}_{t},\mathbf{\Sigma}_{t};\kappa,\mathbf{m}_{t},\mathbf{V}_{t})=\mathcal{N}\left(\mathbf{\mu}_{t};\mathbf{m}_{t},\mathbf{\Sigma}_{t}/\kappa\right)f_{\mathbf{\Lambda}_{t}}(\mathbf{\Lambda}_{t};\kappa,\mathbf{V}_{t}).$
(2)

Figure 3: The probabilistic model of the cascaded DP in plate notation on the
$b$-th bag.

Because it is computationally infeasible to model an infinite number of
clusters posited by the DP, we truncate the number of DP clusters in each bag
by $T$. The statistical properties of the mixed distribution are not affected
by the truncation [5] as corroborated by Theorem 1 in the supplementary
materials.

DP Prediction Module on Slide Level. Patches in a bag are fitted to the
respective latent clusters of that slide after the DP aggregation module. To
estimate predictive uncertainty, we design a DP predictive module to classify
the centroids of the clusters into $K$ classes (the maximum number of clusters
is truncated at $K$). We define a stick-breaking process for the bag-level DP
as

$\displaystyle\nu_{k}\sim\text{Beta}(1,\eta_{2}),\quad\pi_{k}=\nu_{k}\prod_{l=1}^{k-1}(1-\nu_{l}).$
(3)

The bag-level DP learns $K$ mixtures of Gaussian distributions, where each
mixture component has an NIW prior distribution as defined in Eq. (2). The
classification can be performed by predicting the cluster assignment of each
centroid with the largest log responsibility (i.e., the log-likelihood
weighted by the DP mixture weights, see Eq. (1) in the supplementary
materials) and then taking the average on different centroids.

###  3.3 Posterior Computation via Variational Inference

Directly inferring the cDP model through MCMC (e.g., Gibbs sampling) would be
infeasible since the time complexity is high. Therefore, we adopt a
variational inference procedure to compute the posterior distribution of the
cDP.

1:Input:

2:Features $\mathbf{X}\in\mathbb{R}^{n\times p}$, prior $\mathbf{\phi}$,
encoder network $\mathbf{\Psi}$,

3:Number of components $T$, concentration parameter $\eta$, optional label $Y$

4:Output:  Updated log-responsibility $\mathbf{\phi}$

5:Initialize $\mathbf{\gamma}_{1},\mathbf{\gamma}_{2}$

6:repeat

7: Encode mean $\mu_{\mathbf{\Psi}}$ and covariance
$\mathbf{\Sigma}_{\mathbf{\Psi}}$ with $\mathbf{\Psi}$

8: Update $\gamma_{1,t}\leftarrow
1+\sum_{j=1}^{n}\phi_{jt},\quad\gamma_{2,t}\leftarrow\eta+\sum_{j=1}^{n}\sum_{r=1}^{t}\phi_{jr}$

9: $\mathbb{E}_{\beta\sim
q}[\log\pi_{t}]\leftarrow\text{digamma}(\gamma_{1,t})-\text{digamma}(\gamma_{1,t}+\gamma_{2,t})$

10: $\phi_{jt}\leftarrow\mathbb{E}_{\beta\sim q}[\log\pi_{t}]+\mathbb{E}[\log
p(\mathbf{x}_{j})]+\mathbb{H}[q_{\psi_{t}}(\cdot|z_{j}=t,\mathbf{x}_{j})]$

11: Compute ELBO by Eq. (4)

12: Backpropagate ELBO to $\mathbf{\Psi}$

13:until convergence

14:return $\mathbf{\phi}$

Algorithm 2 Fit Dirichlet Process

Parameterization.  Unlike the conventional deep DP methods that heavily rely
on the parameterization tricks to learn a diagonal covariance matrix [19, 34],
we learn the full covariance matrix for better approximation of the
distribution of each component. We parameterize the mean and covariance matrix
with deep neural networks — a network $\psi_{t}$ that encodes both the mean
$\mu_{\psi_{t}}(x_{i})$ and covariance matrix
${\mathbf{\Sigma}}_{\psi_{t}}(x_{i})$ for each component $t$. We then obtain
$\mathbf{\Psi}=\\{\psi_{t}\\}_{t=1}^{T}$ as the set of trainable parameters.
This enables the bag label information to be backpropagated to the mean and
covariance of the DP for clustering through variational inference.
Furthermore, the neural network generates distinct mean and covariance for
each input sample. This allows for inductive distribution learning that the
model can generate proper means and covariance matrices for unseen samples.

Variational Objective.  Since the posterior of the proposed DP is intractable
(i.e., no closed-form solutions), we adopt a variational posterior $q$ to
approximate the true posterior distribution with the variational posterior. We
proceed with a mean-field variational inference by factorizing the joint
distribution into marginal distributions. The variational family of the DP
model that we optimize during variational inference is given by

$\displaystyle
q(\mathbf{\beta},\mathbf{\nu},\mathbf{\theta},\mathbf{\Lambda},z)=\prod_{t=1}^{T-1}q(\beta_{t})\prod_{k=1}^{K}q(\nu_{k})\prod_{t=1}^{T}q(\mathbf{\theta}_{t})\prod_{t=1}^{T}q(\mathbf{\Lambda}_{t})\prod_{b=1}^{B}q(z_{b}).$

The surrogate loss of backpropagation can be obtained by negating the evidence
lower bound (ELBO), which is computed by the log responsibility and the KL
divergence of the Wishart prior,

$\begin{split}\mathcal{L}(\mathbf{\Psi})=&\text{KL}(q(\mathbf{\beta})\|p(\mathbf{\beta}))+\text{KL}(q(\nu)\|p(\nu))+\text{KL}(q(\mathbf{\theta})\|p(\mathbf{\theta}))\\\
&+\text{KL}(q(\mathbf{\Lambda})\|p(\mathbf{\Lambda}))+\sum_{b}\text{KL}(q(z_{b})\|p(z_{b})),\end{split}$
(4)

where $\text{KL}(q(z_{b})\|p(z_{b}))$ can be interpreted as the supervised
loss (e.g., cross-entropy loss for classification). If no supervision label is
provided for the $b$-th sample, $\text{KL}(q(z_{b})\|p(z_{b}))$ can be set as
zero. Then the problem is reduced to learning the likelihood of inputs to the
assumed DP model. The detailed parameter update procedure of the variational
posterior and the closed-form KL divergences of the assumed distributions can
be found in the supplementary materials.

Prediction.  We derive the predictive distribution of the Dirichlet process
model [19] at patch-level as

$\begin{split}p(z=t|\mathbf{x})&\propto\int
p(\mathbf{x}|\mathbf{\theta}_{t})q(\mathbf{\theta}_{t}|z=t)d\mathbf{\theta}_{t}\\\
&\propto\mathbb{E}_{\beta\sim
q(\beta)}[\pi_{t}(\beta)]\times\mathbb{E}_{q_{\psi_{t}}}[p(\mathbf{x}|z=t)],\end{split}$
(5)

where $\mathbf{x}$ is the feature embedding of the testing sample, $p(\cdot)$
refers to Gaussian density function and $z$ refers to the cluster assignment.
We obtain the logits from this posterior distribution and combine them with
the logits from the labeled data. We can design a DP predictive module from
Eq. (5) and use the ELBO to train $\mathbf{\Psi}$ for patch-level prediction.
Similarly, slide-level prediction can be performed by Eq. (5) with respective
parameters of the slide-level DP.

###  3.4 Patch Score and Uncertainty Estimation

Patch scores and model uncertainty can be defined using patch-level DP and
slide-level DP, respectively. For tumor/normal binary classification, the log-
likelihood output by patch-level DP could be used as the score for each patch,

$\displaystyle
p(\mathbf{x}_{j})=\sum_{t=1}^{T}\phi_{t}\mathbb{E}_{q_{\psi_{t}}}[p(\mathbf{x}_{j}|z_{j}=t)].$
(6)

indicating how likely the patch is connected with tumors, where
$\mathbf{x}_{j}$ is the feature embedding and $z_{j}$ is the latent cluster
assignment of the $j$-th patch. Patch-level scores have great interpretability
because they can help to localize the tumor region. Adopting a deep Bayesian
nonparametric framework allows for uncertainty estimation through posterior
predictive distributions. We adopt the log-likelihood from DP as the measure
of uncertainty. The log-likelihood of the slide-level DP can be defined by the
posterior mixture density function of the DP,
$p(\mathbf{x}_{b})=\sum_{k=1}^{K}\phi_{k}\mathbb{E}_{q_{\psi_{k}}}[p_{X}(\mathbf{X}_{b}|z_{b}=k)],$
where $\mathbf{X}_{b}$ is the feature embedding and $z_{b}$ is the latent
cluster assignment of the $b$-th slide.

Table 1: Summary of datasets under two different tasks. Classification | Tumor | Normal
---|---|---
TCGA–COAD | 863 | 109
TCGA–BRCA | 1569 | 394
Camelyon 16 | 160 | 239
Subtyping | Type I | Type II
TCGA–ESCA | 131 | 107
TCGA–BRCA | 283 | 1420
TCGA–NSCLC | 540 | 512

##  4 Experiments

###  4.1 Experiments Setup

Datasets and Tasks. We evaluate the performance of our method on five WSI
analysis benchmarks — TCGA–COAD, TCGA–BRCA, TCGA–ESCA, and TCGA–NSCLC from the
TCGA project [53] and Camelyon 16 [3]. On average, each WSI in the TCGA
datasets consists of 2231 patches at magnification level 20 (24401 patches for
each WSI in Camelyon 16). We evaluate our method on cancer subtyping and
cancer classification tasks. For the cancer subtyping task, all WSIs in the
TCGA–ESCA dataset are assigned the label “Type I: adenocarcinoma” or the label
“Type II: squamous cell carcinoma” while the WSIs in TCGA-BRCA are assigned
“Type I: Ductal Carcinoma” or “Type II: Lobular Carcinoma”. As for TCGA–NSCLC,
there are two subtypes of cancer, namely, “Type I: Lung Adenocarcinoma (LUAD)”
and “Type II: Lung Squamous Cell Carcinoma (LUSC)”. For the cancer
classification task, all the cases are divided into “Normal” and “Tumor”
classes. We use 10-fold cross-validation to validate our model on these
datasets. Table 1 presents a summary of the datasets. Additional training
details and data split can be found in the supplementary materials.

In general, models potentially capture the epistemic and aleatoric
uncertainties well, if they can satisfactorily detect the OOD samples and
generalize performances on these samples. We validate the capability of
uncertainty estimation of our method with OOD detection tasks, and the
generalizability of our method on OOD generalization tasks. For the OOD
detection task, we compare with common measures of uncertainty such as maximum
confidence, entropy, and differential entropy. Specifically, we treat COAD as
the in-distribution dataset and BRCA as the OOD dataset, and adopt the area
under the receiver operating curve (AUROC) and the area under the precision-
recall curve (AUPR) as the evaluation metrics. For OOD generalization, we
train models on the COAD dataset and validate on the BRCA dataset on the
cancer binary classification task.

Preprocessing and Petraining. We adopt the commonly used Otsu’s thresholding
algorithm [12] to outline the tissue regions and use sliding windows to crop
each WSI into non-trivial patches (trivial patches are mostly background). We
then use a pretrained encoder to extract the features of each patch.
Specifically, we adopt a SimCLR pretrained KimiaNet [61] on 7,126 whole slide
image datasets to generate the patch features
$\\{\mathbf{X}_{b}\\}_{b=1}^{B}$.

Evaluation Metrics.  We use accuracy, F1-score, and AUROC to evaluate the
classification performance. Percentage values are used for the reported
metrics with standard deviations in brackets. For all metrics, a higher value
in percentage or a lower value in standard deviation indicates a more accurate
or stable method.

Table 2:  LUSC/LUAD subtypes classification results [%] of our cDP-MIL and other methods on the TCGA-NSCLC dataset. Model | Accuracy | AUROC
---|---|---
ABMIL [30] | 77.2 | 86.6
DSMIL [38] | 80.6 | 89.3
CLAM-SB [41] | 81.8 | 88.2
CLAM-MB [41] | 84.2 | 93.8
TransMIL [48] | 88.4 | 96.0
cDP-MIL | 93.5 | 97.4

Competitive Methods. We make comparisons with a variety of competitors,
including ABMIL [30], DSMIL [38], PatchGCN [12], GTNMIL [61], H2-MIL [28],
BayesMIL [58], Remix [55], CLAM [41], TransMIL [48], and HEAT [7], with
detailed descriptions of each method given in the supplementary materials.

###  4.2 Quantitative Results on Predictive Performance

We first evaluate cDP-MIL by slide-level classification. Tables 2, 3 and 4
present the performance of our model on cancer subtyping and cancer
classification. Compared with other state-of-the-art (SOTA) WSI or MIL
frameworks [30, 38, 61, 28, 7], our proposed method shows outstanding
performance on both tasks, indicating the capability of cDP-MIL to regularize
the learning process and reduce the train-to-test generalization error.

Furthermore, comparing cDP-MIL with another Bayesian MIL framework, Bayes-MIL
[58], our method shows large improvements in these tasks, which is evidence of
the possible strength that the model excels in leveraging inter-cluster
information. Hence, these experiments demonstrate the superior ability of our
method in capturing significant features (i.e., the richer-gets-richer
property) of WSI and infer the possible latent distributions of features.

Table 3: Cancer subtyping results [%] of our method and other methods on the
TCGA–ESCA and TCGA–BRCA datasets.

| TCGA–ESCA | TCGA–BRCA
---|---|---
Model | AUROC | Accuracy | Macro-F1 | AUROC | Accuracy | Macro-F1
ABMIL [30] | 92.5 (2.0) | 89.9 (2.8) | 89.8 (2.8) | 70.5 (23.9) | 69.0 (28.2) | 62.6 (23.4)
DSMIL [38] | 92.5 (1.7) | 87.3 (2.0) | 86.3 (2.0) | 81.5 (10.8) | 79.9 (18.5) | 72.7 (17.1)
ReMix [55] | 92.5 (7.2) | 90.0 (8.1) | 90.3 (7.7) | 75.0 (8.3) | 71.0 (14.3) | 67.9 (5.4)
PatchGCN [12] | 88.6 (3.5) | 92.1 (2.3) | 92.3 (2.4) | — | — | —
GTNMIL [61] | 89.7 (4.7) | 81.2 (4.8) | 89.2 (4.9) | 61.8 (13.9) | 82.0 (2.8) | 62.3 (18.3)
H2-MIL [28] | 92.1 (3.9) | 88.2 (5.8) | 88.0 (5.8) | 78.3 (11.1) | 87.0 (2.7) | 46.5 (0.8)
HEAT [7] | 92.8 (2.5) | 90.1 (4.5) | 90.3 (1.9) | 68.4 (6.5) | 86.1 (3.6) | 85.6 (13.3)
BayesMIL [58] | 82.4 (9.5) | 78.3 (10.0) | 80.3 (9.7) | 82.8 (5.2) | 85.1 (2.8) | 91.3 (1.6)
cDP-MIL | 94.0 (1.7) | 92.7 (1.1) | 93.3 (0.7) | 83.6 (0.2) | 87.2 (0.2) | 92.3 (0.1)

Table 4: Cancer classification performance of our method and other methods on
the TCGA–COAD, TCGA–BRCA, and Camelyon 16 datasets, with standard deviations
in brackets.

| Camelyon 16 | TCGA–COAD | TCGA–BRCA
---|---|---|---
Model | AUROC | Accuracy | F1 | AUROC | Accuracy | F1 | AUROC | Accuracy | F1
ABMIL [30] | 95.9 (3.6) | 93.0 (6.7) | 91.1 (4.9) | 95.9 (2.3) | 95.9 (0.9) | 97.7 (2.2) | 97.7 (1.7) | 95.7 (1.1) | 97.4 (1.6)
DSMIL [38] | 95.5 (4.3) | 94.0 (4.5) | 92.3 (4.9) | 93.4 (0.2) | 95.4 (0.5) | 97.5 (0.9) | 95.8 (0.5) | 96.3 (1.4) | 97.8 (2.0)
Remix [55] | 91.0 (6.0) | 92.5 (4.6) | 92.5 (4.7) | 94.3 (3.4) | 96.0 (4.6) | 92.8 (5.9) | 96.1 (0.7) | 95.8 (2.6) | 93.0 (3.4)
PatchGCN [12] | 69.9 (2.1) | 81.3 (3.2) | 88.9 (1.5) | 91.1 (5.3) | 97.1 (2.0) | 98.8 (1.0) | 96.2 (1.7) | 95.8 (2.6) | 93.0 (3.4)
GTNMIL [61] | 57.9 (5.8) | 54.3 (3.6) | 51.8 (9.9) | 97.3 (2.6) | 98.1 (1.3) | 95.9 (2.4) | 94.7 (1.0) | 94.5 (0.2) | 93.7 (1.7)
H2MIL [28] | 83.1 (11.1) | 81.9 (10.7) | 79.0 (13.4) | 99.7 (0.4) | 99.2 (0.5) | 97.4 (1.7) | 97.9 (2.7) | 98.0 (1.5) | 97.6 (2.2)
HEAT [7] | 78.3 (0.4) | 84.4 (0.5) | 80.0 (0.3) | 99.7 (0.2) | 99.4 (0.3) | 98.7 (0.4) | 98.8 (0.7) | 98.3 (0.5) | 98.7 (0.7)
BayesMIL [58] | 94.6 (4.0) | 87.5 (4.1) | 82.2 (7.3) | 98.7 (1.4) | 96.9 (2.3) | 98.2 (1.3) | 99.5 (0.03) | 98.1 (0.4) | 98.3 (0.2)
cDP-MIL | 98.2 (0.01) | 94.8 (0.6) | 93.5 (0.6) | 99.7 (0.2) | 99.4 (0.1) | 98.9 (0.1) | 99.8 (0.02) | 98.5 (0.4) | 99.2 (0.2)

Table 5: Performance [%] of cDP-MIL on out-of-distribution (OOD) detection
(compared to baseline uncertainty measures) and OOD generalization, with
TCGA–COAD as the in-distribution dataset and TCGA–BRCA as the OOD dataset.

| OOD Detection |  | OOD Generalization
---|---|---|---
Measure | AUROC | AUPR | Model | AUROC | Accuracy | F1
Max Confidence | 76.55 | 82.92 | ABMIL [30] | 59.12 | 83.78 |  91.12
Entropy | 75.25 | 80.67 | DSMIL [38] | 72.91 | 81.38 |  88.82
Differential Entropy | 68.39 | 86.54 | Remix [55] | 50.65 | 85.37 |  91.78
Log Resp. | 83.85 | 91.19 | cDP-MIL | 85.31 | 89.89 |  94.24

###  4.3 Out-of-Distribution Detection and Generalizability

Moreover, we validate the ability of our model in uncertainty estimation (by
detecting OOD samples) and generalizability (by generalization to OOD
datasets), as shown in Table 5. We observe that using the log-likelihood from
the posterior responsibility gives the best estimate of the uncertainty, and
our method has satisfactory generalizability to other datasets compared with
SOTA methods [30, 38, 55].

###  4.4 Qualitative Results

We also study the interpretability performance with the localization task on
the Camelyon 16 dataset. In Fig. 4, we provide a visualization of the patches
in WSIs along with their corresponding scores. Our method can accurately
locate tumor regions, as indicated by the ground truth annotations. It also
achieves better localization performance compared to HEAT [7], which
highlights its effectiveness in predicting instance-level labels.

Figure 4: Tumor region localization. Left: Ground truth; middle: HEAT; right: cDP-MIL. Ground truth regions are outlined with red boundaries. Lighter yellow indicates more important regions.  Figure 5: Performance of cDP-MIL with different concentration parameters $\eta_{1}$ and $\eta_{2}$. Left: COAD; middle: BRCA; right: Camelyon 16. Table 6: Performance [%] of different pooling methods using the same DP prediction module on the Camelyon 16 dataset for cancer binary classification. Method | AUROC | Accuracy | F1
---|---|---|---
Mean Pool | 77.6 (2.9) | 72.3 (1.4) | 66.1 (4.2)
Max Pool | 96.3 (0.7) | 90.8 (1.4) | 87.7 (1.6)
PL Pool | 78.3 (2.3) | 84.3 (4.3) | 80.0 (6.2)
$k$-means | 91.6 (1.5) | 87.8 (1.0) | 83.8 (1.9)
cDP-MIL | 98.2 (0.02) | 94.8 (0.6) | 93.5 (0.6)

###  4.5 Ablation Analysis

Compared with Different Aggregation Algorithms. We validate the effectiveness
of DP aggregation of cDP-MIL by comparing it with different aggregation
methods including $k$-means, max pooling, mean pooling, and PL pool [7]. We
change only the aggregation methods while keeping other components intact. As
shown in Table 6, our proposed DP pooling method obtains superior performance
over the baseline pooling methods, which validates its contribution to the
framework.

Table 7: Performance [%] of MLP and DP Classifier using the same DP
aggregation module on the Camelyon 16, TCGA–COAD, and TCGA–BRCA datasets for
cancer binary classification.

| Camelyon 16 | TCGA–COAD | TCGA–BRCA
---|---|---|---
Classifier | AUROC | Accuracy | F1 | AUROC | Accuracy | F1 | AUROC | Accuracy | F1
MLP | 96.3 | 92.5 | 90.6 | 96.6 | 96.9 | 98.3 | 96.9 | 94.4 | 96.7
DP Classifier | 98.2 | 94.8 | 93.5 | 99.7 | 99.4 | 98.8 | 99.8 | 98.5 | 99.2

Effect of DP Classifier. We also study the contribution of our proposed DP
classifier when predicting the bag-level labels. We compare our DP classifier
with MLP, which is commonly used in MIL models. Table 7 shows that our
proposed DP classifier (parameterized with the neural network) obtains a
satisfactory performance, which validates the contribution of this proposed
module. Note that our DP classifier can also accurately estimate the
predictive uncertainty.

Concentration parameters $\eta_{1}$ and $\eta_{2}$. The concentration
parameters $\eta_{1}$ and $\eta_{2}$ are those of beta distributions in the
stick-breaking process, which determines how actively we initiate new clusters
when performing pooling and predictions. The smaller the $\eta$, the higher
the probability that most points concentrate in only a few clusters. Fig. 5
presents the change in performance as we take different values of $\eta$. We
observe that our method is in general robust to changes in $\eta$. For
extremely small $\eta$, the model concentrates on the few top clusters quickly
without actively exploring other potential clusters, leading to suboptimal
performance. When $\eta$ becomes larger, the instance embeddings would
eventually converge to the top clusters (i.e., clusters with the highest
likelihood), since the patches in WSIs mostly have distinctive characteristics
(e.g., tumor vs normal patches). Therefore, $\eta_{1}$ and $\eta_{2}$ are
fine-grained hyperparameters which do not affect the learning performance
much.

Effects of Maximum Number of Mixture Components $T$.  We study the effects of
the maximum number of components of the DP aggregation module.

Figure 6: Performance of cDP-MIL on the ESCA dataset with different values of
$T$.

Fig. 6 presents the results of our method on TCGA–ESCA with different numbers
of maximum mixture components $T$. We observe the performance varies less when
the maximum number of components is set high. Unlike conventional clustering
methods (e.g., $k$-means) that require a predefined number of components,
clustering with DP can automatically select the number of clusters. This
reduces the reliance on the number of components and improves the flexibility
in feature pooling, which leads to more accurate bag-level prediction.

##  5 Conclusion

To solve the problem of multiple instance learning, we propose a novel
framework based on a cascade of Dirichlet processes. The DP aggregation module
on the patch level can capture distributional characteristics of features more
accurately by utilizing high-order information (e.g., covariance) while
scaling up to large datasets through stochastic variational inference. In
addition, our slide-level DP enables uncertainty estimation while maintaining
satisfactory performance. The empirical results show that our method achieves
superior performance to the state-of-the-art WSI analysis methods and
accurately estimates uncertainty to detect OOD samples.

## Acknowledgement

This work was partially supported by the Research Grants Council of Hong Kong
(27206123 and T45-401/22-N), the Hong Kong Innovation and Technology Fund
(ITS/274/22), and the National Natural Science Foundation of China (No.
62201483).

## References

  * [1] Andrews, S., Tsochantaridis, I., Hofmann, T.: Support vector machines for multiple-instance learning. Advances in Neural Information Processing Systems 15 (2002)
  * [2] Antoniak, C.E.: Mixtures of dirichlet processes with applications to bayesian nonparametric problems. The Annals of Statistics pp. 1152–1174 (1974)
  * [3] Bejnordi, B.E., Veta, M., Van Diest, P.J., Van Ginneken, B., Karssemeijer, N., Litjens, G., Van Der Laak, J.A., Hermsen, M., Manson, Q.F., Balkenhol, M., et al.: Diagnostic assessment of deep learning algorithms for detection of lymph node metastases in women with breast cancer. JAMA 318(22), 2199–2210 (2017)
  * [4] Bianchi, F.M., Grattarola, D., Alippi, C.: Mincut pooling in graph neural networks (2019)
  * [5] Blei, D.M., Jordan, M.I.: Variational inference for dirichlet process mixtures. Bayesian Analysis 1(1), 121–144 (2006)
  * [6] Brancati, N., Anniciello, A.M., Pati, P., Riccio, D., Scognamiglio, G., Jaume, G., De Pietro, G., Di Bonito, M., Foncubierta, A., Botti, G., et al.: Bracs: A dataset for breast carcinoma subtyping in h&e histology images. Database 2022, baac093 (2022)
  * [7] Chan, T.H., Cendra, F.J., Ma, L., Yin, G., Yu, L.: Histopathology whole slide image analysis with heterogeneous graph representation learning. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 15661–15670 (2023)
  * [8] Chan, T.H., Lau, K.W., Shen, J., Yin, G., Yu, L.: Adaptive uncertainty estimation via high-dimensional testing on latent representations. Advances in Neural Information Processing Systems 36 (2024)
  * [9] Chan, T.H., Wong, C.H., Shen, J., Yin, G.: Source-aware embedding training on heterogeneous information networks. Data Intelligence 5(3), 611–635 (2023)
  * [10] Chen, R.J., Chen, C., Li, Y., Chen, T.Y., Trister, A.D., Krishnan, R.G., Mahmood, F.: Scaling vision transformers to gigapixel images via hierarchical self-supervised learning. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 16144–16155 (2022)
  * [11] Chen, R.J., Lu, M.Y., Shaban, M., Chen, C., Chen, T.Y., Williamson, D.F., Mahmood, F.: Whole slide images are 2d point clouds: Context-aware survival prediction using patch-based graph convolutional networks. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part VIII 24. pp. 339–349. Springer (2021)
  * [12] Chen, R.J., Lu, M.Y., Shaban, M., Chen, C., Chen, T.Y., Williamson, D.F., Mahmood, F.: Whole slide images are 2d point clouds: Context-aware survival prediction using patch-based graph convolutional networks. In: International Conference on Medical Image Computing and Computer-Assisted Intervention. pp. 339–349. Springer (2021)
  * [13] Chen, R.J., Lu, M.Y., Weng, W.H., Chen, T.Y., Williamson, D.F., Manz, T., Shady, M., Mahmood, F.: Multimodal co-attention transformer for survival prediction in gigapixel whole slide images. In: Proceedings of the IEEE/CVF International Conference on Computer Vision. pp. 4015–4025 (2021)
  * [14] Chen, T., Kornblith, S., Norouzi, M., Hinton, G.: A simple framework for contrastive learning of visual representations. In: International Conference on Machine Learning. pp. 1597–1607. PMLR (2020)
  * [15] Chen, Y., Zhao, W., Yu, L.: Transformer-based multimodal fusion for survival prediction by integrating whole slide images, clinical, and genomic data. In: 2023 IEEE 20th International Symposium on Biomedical Imaging (ISBI). pp. 1–5. IEEE (2023)
  * [16] Chen, Y., Zhao, W., Yu, L.: Transformer-based multimodal fusion for survival prediction by integrating whole slide images, clinical, and genomic data. In: 2023 IEEE 20th International Symposium on Biomedical Imaging (ISBI). pp. 1–5. IEEE (2023)
  * [17] Contributors, M.: MMSelfSup: Openmmlab self-supervised learning toolbox and benchmark. https://github.com/open-mmlab/mmselfsup (2021)
  * [18] Dosovitskiy, A., Beyer, L., Kolesnikov, A., Weissenborn, D., Zhai, X., Unterthiner, T., Dehghani, M., Minderer, M., Heigold, G., Gelly, S., Uszkoreit, J., Houlsby, N.: An image is worth 16x16 words: Transformers for image recognition at scale (2021)
  * [19] Echraibi, A., Flocon-Cholet, J., Gosselin, S., Vaton, S.: On the variational posterior of dirichlet process deep latent gaussian mixture models. In: ICML Workshop on Invertible Neural Networks, Normalizing Flows, and Explicit Likelihood Models (2020)
  * [20] Ferguson, T.S.: A bayesian analysis of some nonparametric problems. The Annals of Statistics pp. 209–230 (1973)
  * [21] Gal, Y., Ghahramani, Z.: Dropout as a bayesian approximation: Representing model uncertainty in deep learning. In: International Conference on Machine Learning. pp. 1050–1059. PMLR (2016)
  * [22] Gao, H., Ji, S.: Graph u-nets. In: International Conference on Machine Learning. pp. 2083–2092. PMLR (2019)
  * [23] Graham, S., Vu, Q.D., Raza, S.E.A., Azam, A., Tsang, Y.W., Kwak, J.T., Rajpoot, N.: Hover-net: Simultaneous segmentation and classification of nuclei in multi-tissue histology images. Medical Image Analysis 58, 101563 (2019)
  * [24] Gustafsson, F.K., Danelljan, M., Schon, T.B.: Evaluating scalable bayesian deep learning methods for robust computer vision. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) Workshops (June 2020)
  * [25] Haußmann, M., Hamprecht, F.A., Kandemir, M.: Variational bayesian multiple instance learning with gaussian processes. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. pp. 6570–6579 (2017)
  * [26] He, K., Fan, H., Wu, Y., Xie, S., Girshick, R.: Momentum contrast for unsupervised visual representation learning. In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. pp. 9729–9738 (2020)
  * [27] Hoffman, M.D., Blei, D.M., Wang, C., Paisley, J.: Stochastic variational inference. Journal of Machine Learning Research (2013)
  * [28] Hou, W., Yu, L., Lin, C., Huang, H., Yu, R., Qin, J., Wang, L.: H2-mil: Exploring hierarchical representation with heterogeneous multiple instance learning for whole slide image analysis. In: Thirty-sixth AAAI conference on artificial intelligence (2022)
  * [29] Hughes, M.C., Sudderth, E.: Memoized online variational inference for dirichlet process mixture models. Advances in Neural Information Processing Systems 26 (2013)
  * [30] Ilse, M., Tomczak, J., Welling, M.: Attention-based deep multiple instance learning. In: International Conference on Machine Learning. pp. 2127–2136. PMLR (2018)
  * [31] Ishwaran, H., Zarepour, M.: Exact and approximate sum representations for the dirichlet process. Canadian Journal of Statistics 30(2), 269–283 (2002)
  * [32] Jiang, Z., Zheng, Y., Tan, H., Tang, B., Zhou, H.: Variational deep embedding: A generative approach to clustering. Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence (2017)
  * [33] Kandasamy, K., Neiswanger, W., Schneider, J., Poczos, B., Xing, E.P.: Neural architecture search with bayesian optimisation and optimal transport. Advances in Neural Information Processing Systems 31 (2018)
  * [34] Kandemir, M., Hamprecht, F.A., et al.: Instance label prediction by dirichlet process multiple instance learning. In: Proceddings of The Conference on Uncertainty in Artificial Intelligence. pp. 380–389 (2014)
  * [35] Kendall, A., Gal, Y.: What uncertainties do we need in bayesian deep learning for computer vision? Advances in Neural Information Processing Systems 30 (2017)
  * [36] Kim, M., De la Torre, F.: Gaussian processes multiple instance learning. In: International Conference on Machine Learning. pp. 535–542. Citeseer (2010)
  * [37] Kingma, D.P., Welling, M.: Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114 (2013)
  * [38] Li, B., Li, Y., Eliceiri, K.W.: Dual-stream multiple instance learning network for whole slide image classification with self-supervised contrastive learning. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 14318–14328 (2021)
  * [39] Li, Z., Xi, T., Deng, J., Zhang, G., Wen, S., He, R.: Gp-nas: Gaussian process based neural architecture search. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 11933–11942 (2020)
  * [40] Liu, P., Fu, B., Ye, F., Yang, R., Ji, L.: Dsca: A dual-stream network with cross-attention on whole-slide image pyramids for cancer prognosis. Expert Systems with Applications 227, 120280 (2023)
  * [41] Lu, M.Y., Williamson, D.F., Chen, T.Y., Chen, R.J., Barbieri, M., Mahmood, F.: Data-efficient and weakly supervised computational pathology on whole-slide images. Nature Biomedical Engineering 5(6), 555–570 (2021)
  * [42] Malinin, A., Gales, M.: Predictive uncertainty estimation via prior networks. Advances in Neural Information Processing Systems 31 (2018)
  * [43] Mauldin, R.D., Sudderth, W.D., Williams, S.C.: Polya trees and random distributions. The Annals of Statistics pp. 1203–1221 (1992)
  * [44] Rasmussen, C.E., Williams, C.K., et al.: Gaussian processes for machine learning, vol. 1. Springer (2006)
  * [45] Rezende, D.J., Mohamed, S., Wierstra, D.: Stochastic backpropagation and approximate inference in deep generative models. In: International Conference on Machine Learning. pp. 1278–1286. PMLR (2014)
  * [46] Ronen, M., Finder, S.E., Freifeld, O.: Deepdpm: Deep clustering with an unknown number of clusters. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 9861–9870 (2022)
  * [47] Shah, S.A., Koltun, V.: Deep continuous clustering. arXiv preprint arXiv:1803.01449 (2018)
  * [48] Shao, Z., Bian, H., Chen, Y., Wang, Y., Zhang, J., Ji, X., et al.: Transmil: Transformer based correlated multiple instance learning for whole slide image classification. Advances in Neural Information Processing Systems 34, 2136–2147 (2021)
  * [49] Snoek, J., Larochelle, H., Adams, R.P.: Practical bayesian optimization of machine learning algorithms. Advances in Neural Information Processing Systems 25 (2012)
  * [50] Wang, J., Lukasiewicz, T.: Rethinking bayesian deep learning methods for semi-supervised volumetric medical image segmentation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 182–190 (June 2022)
  * [51] Wang, S., Zhu, Y., Yu, L., Chen, H., Lin, H., Wan, X., Fan, X., Heng, P.A.: Rmdl: Recalibrated multi-instance deep learning for whole slide gastric image classification. Medical Image Analysis 58, 101549 (2019)
  * [52] Wang, Z., Ni, Y., Jing, B., Wang, D., Zhang, H., Xing, E.: Dnb: A joint learning framework for deep bayesian nonparametric clustering. IEEE Transactions on Neural Networks and Learning Systems 33(12), 7610–7620 (2021)
  * [53] Weinstein, J.N., Collisson, E.A., Mills, G.B., Shaw, K.R., Ozenberger, B.A., Ellrott, K., Shmulevich, I., Sander, C., Stuart, J.M.: The cancer genome atlas pan-cancer analysis project. Nature genetics 45(10), 1113–1120 (2013)
  * [54] Yang, B., Fu, X., Sidiropoulos, N.D., Hong, M.: Towards k-means-friendly spaces: Simultaneous deep learning and clustering. In: International Conference on Machine Learning. pp. 3861–3870. PMLR (2017)
  * [55] Yang, J., Chen, H., Zhao, Y., Yang, F., Zhang, Y., He, L., Yao, J.: Remix: A general and efficient framework for multiple instance learning based whole slide image classification. Medical Image Computing and Computer Assisted Intervention–MICCAI 2022: 25th International Conference, Singapore, September 18–22, 2022, Proceedings, Part II (2022)
  * [56] Yang, L., Cheung, N.M., Li, J., Fang, J.: Deep clustering by gaussian mixture variational autoencoders with graph embedding. In: Proceedings of the IEEE/CVF international conference on computer vision. pp. 6440–6449 (2019)
  * [57] Yee Whye Teh, Michael I Jordan, M.J.B., Blei, D.M.: Hierarchical dirichlet processes. Journal of the American Statistical Association 101(476), 1566–1581 (2006). https://doi.org/10.1198/016214506000000302, https://doi.org/10.1198/016214506000000302
  * [58] Yufei, C., Liu, Z., Liu, X., Liu, X., Wang, C., Kuo, T.W., Xue, C.J., Chan, A.B.: Bayes-mil: A new probabilistic perspective on attention-based multiple instance learning for whole slide images. In: The Eleventh International Conference on Learning Representations (2022)
  * [59] Zhang, C., Yin, G.: Bayesian nonparametric analysis of restricted mean survival time. Biometrics 79(2), 1383–1396 (2023)
  * [60] Zhao, W., Wang, S., Yeung, M., Niu, T., Yu, L.: Mulgt: Multi-task graph-transformer with task-aware knowledge injection and domain knowledge-driven pooling for whole slide image analysis. In: Thirty-seventh AAAI conference on artificial intelligence (2023)
  * [61] Zheng, Y., Gindra, R.H., Green, E.J., Burks, E.J., Betke, M., Beane, J.E., Kolachalama, V.B.: A graph-transformer for whole slide image classification. IEEE Transactions on Medical Imaging 41(11), 3003–3015 (2022)


#  Flying electron spin control gates

Paul L. J. Helgers  Paul-Drude-Institut für Festkörperelektronik, Leibniz-
Institut im Forschungsverbund Berlin e.V., Hausvogteiplatz 5-7, 10117 Berlin,
Germany  NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-
Wakamiya, Atsugi, Kanagawa 243-0198, Japan  James A. H. Stotz
<EMAIL_ADDRESS>Paul-Drude-Institut für Festkörperelektronik, Leibniz-
Institut im Forschungsverbund Berlin e.V., Hausvogteiplatz 5-7, 10117 Berlin,
Germany  Department of Physics, Engineering Physics & Astronomy, Queen’s
University, Kingston, ON, K7L3N6 Canada  Haruki Sanada  Yoji Kunihashi  NTT
Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi,
Kanagawa 243-0198, Japan  Klaus Biermann  Paulo V. Santos  santos@pdi-
berlin.de Paul-Drude-Institut für Festkörperelektronik, Leibniz-Institut im
Forschungsverbund Berlin e.V., Hausvogteiplatz 5-7, 10117 Berlin, Germany

###### Abstract

The control of "flying" (or moving) spin qubits is an important functionality
for the manipulation and exchange of quantum information between remote
locations on a chip. Typically, gates based on electric or magnetic fields
provide the necessary perturbation for their control either globally or at
well-defined locations. Here, we demonstrate the dynamic control of moving
electron spins via contactless gates that move together with the spin. The
concept is realized using electron spins trapped and transported by moving
potential dots defined by a surface acoustic wave (SAW). The SAW strain at the
electron trapping site, which is set by the SAW amplitude, acts as a
contactless, tunable gate that controls the precession frequency of the flying
spins via the spin-orbit interaction. We show that the degree of precession
control in moving dots exceeds previously reported results for unconstrained
transport by an order of magnitude and is well accounted for by a theoretical
model for the strain contribution to the spin-orbit interaction. This flying
spin gate permits the realization of an acoustically driven optical
polarization modulator based on electron spin transport, a key element for on-
chip spin information processing with a photonic interface.

##  I Introduction

The spin field-effect transistor proposed by Datta and Das [1] relies on the
precession of moving (or flying) electron spins around the effective magnetic
field $\vec{B}_{\mathrm{SO}}({\bf k})$ associated with the spin-orbit (SO)
interaction, which depends on electron momentum ${\hbar\bf k}$.
$\vec{B}_{\mathrm{SO}}$ can be electrically controlled by an electrostatic
gate via the Bychkov-Rashba effect,[2] thus opening the way for dynamic spin
manipulation by electric fields. Spin transistors based on the electrical spin
control have so far been demonstrated only for ballistic spin transport along
short ($<2~{}\mu$m) channels [3, 4].

Figure 1:  Flying control gate for electron spins. A surface acoustic wave
(SAW) excited by an interdigital acoustic transducer (IDT) is applied along a
quasi-one dimensional (1D) semiconductor channel. The piezoelectric potential
of the SAW creates a moving quasi-zero dimensional (0D) potential dot, which
captures spin-polarized, photoexcited electrons and holes and transports them
along the channel with the SAW velocity. Simultaneously, the SAW strain field
induces an effective spin-orbit magnetic field $B_{SO}$ at the carrier
transport sites with amplitude proportional to the SAW field. The latter acts
as a flying spin gate to control the spin precession rate.

SO-based spin control over long transport distances, which normally takes
place in the diffusive regime, faces two main challenges. The first is
D’yakonov-Perel’ (DP) spin dephasing[5] associated with the momentum-
dependence of $\vec{B}_{\mathrm{SO}}$. Approaches to reduce DP spin dephasing
and enable long-range spin transport lengths ($\ell_{s}$) include the
engineering the SO interaction [6, 7, 8, 9, 10, 7, 11] as well as exploitation
of motional narrowing effects [5]. The latter takes advantage of the inverse
dependence of the DP dephasing rate on the carrier scattering time, which can
be achieved via increased momentum scattering by impurities [12] or at the
boundaries of narrow transport channels (i.e., channel widths less than the
precession period, $L_{\mathrm{SO}}$ under $\vec{B}_{\mathrm{SO}}$). This
latter approach has been realized using quantum wire channels [13, 14] as well
as by enclosing the spins within moving potential dots [15, 16].

The second major challenge for spin control is to devise field configurations
to simultaneously drive spin motion and generate a tunable
$\vec{B}_{\mathrm{SO}}$ for controlled spin precession. An elegant solution is
offered by the carrier transport by moving potential dots produced by a
surface acoustic wave (SAW) along a one-dimensional (1D) channel. Figure 1
depicts an example based on a quantum wire (QWR) transport channel. Here, the
moving piezoelectric potential modulation produced by the SAW stores photo-
excited electrons and holes at different SAW subcycles and transports them
with the acoustic velocity. The spatial separation of electrons and holes
prevents recombination [17] and, simultaneously, also suppresses spin
relaxation due to the electron-hole exchange interaction [18]. The ambipolar
SAW transport can thus transfer electron spins over long distances (up to
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$) while enabling optical spin
readout by detecting the polarization of photons emitted by the recombination
of the transported carriers [19, 20, 21, 22, 16].

Figure 2:  Optically detected transport of spins in planar quantum wires. (a)
Sidewall quantum wires (QWRs) formed by the epitaxial overgrowth of a quantum
well (QW) on a GaAs (001) substrate structured with shallow ridges [30].
Photoexcited carriers are transported along the QWR by surface acoustic waves
(SAWs) generated by interdigital acoustic transducers (IDTs). (b)
Photoluminescence (PL) spectra recorded outside (red) and on (black) a ridge
sidewall showing the emission lines of the QW ($1.548\text{\,}\mathrm{eV}$)
and QWR ($1.521\text{\,}\mathrm{eV}$), respectively. (c)-(e) Profiles of the
right (black) and left (red) circularly polarized PL along the QWR axis
($x_{\mathrm{QWR}}$ coordinate) recorded under the configuration illustrated
in the upper panel (c) in the absence and (d)-(e) under a SAW. The PL was
excited by a right-circularly polarized laser spot focused at
$x_{\mathrm{QWR}}=0$ for (c) and (d), and at $x_{\mathrm{QWR}}=4.5~{}\mu$m for
(e).

Concomitantly with the transport, the SAW strain [23, 24, 25, 26, 27] and
piezoelectric fields at the carrier location induce a $\vec{B}_{\mathrm{SO}}$
contribution, which moves congruently with the carriers [cf. Fig. 1]. The SAW
amplitude acts, therefore, as a contactless, flying spin gate, which
dynamically controls the rate of spin precession during transport. In this
work, we demonstrate that long-range acoustic transport can be combined with a
high degree of dynamical control of the spin precession rate (by over 250%) if
the flying spins are transported while confined within micron-sized moving
potential dots. This degree of precession control exceeds by over an order of
magnitude previous results for acoustic transport in 2D quantum well (QW)
channels [28, 21]. We also show that the precession rate is mainly mediated by
the acoustic strain imparted by the SAW in the storage phase of the electron
spins during transport. The latter essentially enables the carrier wave to
also act as a flying spin gate controlled by the SAW amplitude.

The study addresses two types of acoustically defined, moving potential dots.
In one system, the dots are formed by propagating a SAW along a quasi-planar
GaAs quantum wire (QWR) [as illustrated in Fig. 1]. We have also investigated
spin transport and manipulation in moving potential dots created by the
interfering piezoelectric fields of orthogonal SAW beams (denoted as dynamic
quantum dots, DQDs[29, 15]). For both types of moving dots, we experimentally
demonstrate the flying spin gates using optically excited spins that can be
acoustically transported over large distances (tens of microns) with the spin
precession rate controlled over a wide dynamic range dictated by the SAW
amplitude (Sec. II) and, notably, without external electric or magnetic
fields. The measured precession rates are well accounted for by an analytical
model for the SO fields generated by the SAW strain and piezoelectric fields,
from which the strain-related SO parameters can be experimentally determined
(Sec. II). As a further check of consistency, we show that spin precession
rates and their dependence on the acoustic fields are also in good agreement
with microscopic calculations of the spin splittings under the SAW field using
a tight-binding approach. Interestingly, while the moving dot geometry enables
precession control, a substantial enhancement of the spin lifetime due to
motional narrowing is only observed for the DQDs. The limited spin lifetimes
in the QWRs, in contrast, is attributed to the fact that the positive impact
of lateral confinement on the spin lifetime is offset by spin scattering at
the lateral, compositional interfaces. Even so, the dramatic ability of the
flying spin gates to control the precession frequency by the SAW amplitude
demonstrates a processor for optically encoded polarization information based
on the dynamic control of electron spins during acoustic transport.

Figure 3:  Acoustic control of the precession rate of moving spins. Spin
polarization, $\rho_{s}$, during transport along (a) a quantum well (QW,
excitation wavelength $\lambda_{L}=776$ nm, $P_{exc}$ =
$15\text{\,}\mathrm{\SIUnitSymbolMicro W}$, 20 K), (b) a quantum wire (QWR,
$\lambda_{L}=776$ nm, $P_{exc}$ = $30\text{\,}\mathrm{\SIUnitSymbolMicro W}$
($3\text{\,}\mathrm{W}\text{\,}{\mathrm{m}}^{-1}$) and $P_{exc}$ =
$150\text{\,}\mathrm{\SIUnitSymbolMicro W}$
($80\text{\,}\mathrm{W}\text{\,}{\mathrm{m}}^{-1}$), 20 K), and (c) dynamic
dots (DQDs, $\lambda_{L}=776$ nm, 12 K) . The upper panels illustrate the
experimental setup: spin-polarized electrons (brown) are optically injected by
a circularly polarized laser ($L$, red) an transported by a SAW along the
$[110]$ direction (QW and QWR) or along the $[010]$ direction (dynamic dots).
The curves correspond to different SAW powers $P_{\mathrm{SAW}}$.

##  II Results

### Acoustic spin transport

The structure of the QWR samples is illustrated in Fig. 2(a). As described in
the Methods section, the GaAs QWRs are fabricated by combining steps of
surface patterning and overgrowth by molecular beam epitaxy (MBE). In this
process, the growth of an (Al,Ga)As/GaAs/(Al,Ga)As QW stack over a patterned
ridge leads to the formation of a thicker GaAs region (the QWR) at the ridge
sidewalls, which is electrically connected to the QW[30]. The
photoluminescence features of the sample are summarized in Fig. 2(b). Here,
the red and black curves compare PL spectra recorded under confocal excitation
and detection on the QW region and on the ridge sidewall (corresponding to the
QWR position), respectively. The former shows a single PL line with a
linewidth (full-width-at-half-maximum, FWHM) of $4\text{\,}\mathrm{meV}$
associated with the electron-heavy hole QW exciton at
$1.546\text{\,}\mathrm{eV}$. The spectrum recorded on the ridge sidewall shows
the excitonic emission from the QWR at $1.521\text{\,}\mathrm{eV}$ with a FWHM
of $4.6\text{\,}\mathrm{meV}$ together with a second line at
$1.548\text{\,}\mathrm{eV}$ (FWHM of 5.4 meV). The latter stems from the QW
regions near the QWR, which are slightly thinner than those further from the
ridge sidewalls [30]. The energy difference between these two lines yields a
lateral confinement energy for electrons (holes) in the QWR of approximately
$22\text{\,}\mathrm{meV}$ ($4\text{\,}\mathrm{meV}$).

Figure 4:  Spin dynamics under acoustic fields. (a) Spin transport length,
$\ell_{s}$, and (b) angular precession frequency, ${\Omega_{\mathrm{SO}}}$, as
a function of the strain amplitude $u_{xx,0}$ for transport along the QW
(black solid squares), QWR (red solid dots), and DQDs on a 30 nm QW (green
solid triangles). The orange triangles are for DQDs in a 20 nm-thick QW [28].
The open symbols display data for transport by a single SAW (open squares) and
by DQDs (open triangles) in a 20 nm thick QW[21]. Note that the
${\Omega_{\mathrm{SO}}}$ is scaled for the DQD data to account for the higher
velocity. The open and solid ${\Large\star}$’s yield the
${\Omega_{\mathrm{SO}}}$ obtained from tight-binding (TB) calculations for the
studied QWs and QWRs, respectively. The dashed lines in (a) and (b) are a
guide to the eye. The solid red and green lines in (b) are fits to Eqs. (2)
and (3) of the QWR and DQD data, respectively.

The optical detection of acoustically driven spin transport along the QWR is
illustrated in Figs. 2(c)-(e). The experiments were carried out using the
geometry depicted in the upper right panel by exciting spins using a right-
hand circularly polarized laser beam with energy below the QW resonance. The
black and red profiles display the spatial distribution of the integrated PL
from the QWR [detection window from $1.517\text{\,}\mathrm{eV}$ to
$1.526\text{\,}\mathrm{eV}$, cf. Fig. 2(a)] with right- and left-hand circular
polarizations, respectively. In the absence of a SAW [Fig. 2(c)], the emission
is restricted to the regions around the excitation spot and has a net right-
hand circular polarization. The PL profiles can be fitted with a Gaussian
characterized by a FWHM of $4\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The
increased spatial PL spread compared to the size of the laser spot is
attributed to the diffusion of the spin polarized carriers along the QWR axis.

Figures 2(d) and 2(e) display the corresponding profiles acquired in the
presence of a SAW that captures the spin polarized carriers and transports
them to trap centers at $x_{\mathrm{QWR}}=10~{}\mu$m, where the electrons and
holes recombine. The trapping and recombination centers are defect regions
along the QWR[30]. The trapping mechanism, which leads to emission at the same
energy as the QWR, is attributed to carrier capture at centers at the
interface between the QW (or QWR) layer and the (Al,Ga)As barrier layers
assisted by the transverse component of the SAW piezoelectric field, $F_{z}$
[30, 31]. These centers can capture carriers of one polarity during one half-
cycle of the SAW and release them during the passage of carriers of the
opposite polarity in the subsequent SAW half-cycle when $F_{z}$ reverses its
sign. The centers provide efficient recombination centers to stop the
transport and monitor the PL polarization. Note that the PL polarization
depends on the transport distance $\Delta x$ changing from right- [Fig. 2(e)]
to left-hand circular polarization [Fig. 2(d)] as $\Delta x$ increases from 5
to 10 $\mu$m. This is due to the larger precession angle under the SO field
accumulated while travelling a longer distance.

### Spin precession control

The procedure depicted in Figs. 2(c)-(e) was applied to determine the spatial
dependence of the spin polarization $\rho_{s}$ on the SAW amplitude. The
panels of Fig. 3 summarize $\rho_{s}$ profiles for acoustic transport in three
geometries, which are illustrated in the corresponding upper panels: (a) along
the QW, (b) along the QWR and (c) using DQDs. For the QW and QWR, the
transport distance $x$ is along the $[110]$-direction, but it is along the
$[010]$-direction for the DQDs. In all cases, $\rho_{s}$ oscillates with a
period that reduces with increasing SAW amplitudes, which demonstrates the
operation principle of the acoustic spin gate. In particular, a distinct
reversal of the spin polarization can be observed for both the QWR
($x=12.5\mu$m) and DQD ($x=25\mu$m) geometries, with the latter occurring well
within the spin coherence length of transport.

The solid lines are fits of the experimental data to an exponentially decaying
cosine function of the form

$\rho_{s}(x)=\rho_{s}(0)\cos\left(\frac{\Omega_{SO}}{v_{SAW}}\Delta
x\right)e^{-\Delta x/\ell_{s}}.$ (1)

The fit parameters are the total SO angular precession frequency
$\Omega_{\mathrm{SO}}$ as well as the characteristic spin transport length
$\ell_{s}$. $\Omega_{\mathrm{SO}}$ can be expressed in terms of the
oscillation period $L_{SO}=2\pi v_{\mathrm{SAW}}/\Omega_{SO}$. The fit
parameters $\ell_{s}$ and $\Omega_{\mathrm{SO}}$ are displayed as filled
symbols in Fig. 4 as a function of the SAW amplitude (stated in terms of the
amplitude of the uniaxial strain component
$u_{xx,0}\propto\sqrt{P_{\mathrm{SAW}}}$, see definition below). For the QWR
(solid red dots), $\ell_{s}\sim 7~{}\mu$m is independent of the SAW amplitude,
and this value is equal to the one measured in the absence of a SAW111see
Supplemental Material, Sec. SM2 at URL for details about the determination of
the spin transport length. In contrast, $\ell_{s}$ for the QW (solid black
squares) increases with SAW power [cf. Fig. 4(a)]–a behaviour which will be
further addressed below. For the DQDs (solid green triangles), $\ell$ is
considerably larger and comparable to the maximum measured transport distance.

A further remarkable result is the strong dependence of
${\Omega_{\mathrm{SO}}}$ on the SAW amplitude [cf. Fig. 4(b)] for transport
both along the QWRs and by DQDs. These structures thus act as acoustically
controlled spin logic and polarization modulators, where the gate is
determined by the acoustic amplitude. Their behaviour contrasts with that of
the QW data as well as with previous acoustic transport studies of QWs and DQD
on thinner QWs (open symbols in Fig. 4 [29, 21]), where the SAW-induced
changes in $\Omega_{\mathrm{SO}}$ are typically less than 10%.

### SAW-related spin-orbit fields

In order to quantify the SO fields induced by the SAW, we first note that the
strain field of a Rayleigh SAW along ${\bf\vec{x}}||[110]$ consists of two
uniaxial strain components $u_{xx}=u_{xx,0}\cos{(\phi_{\mathrm{SAW}})}$ and
$u_{zz}=u_{zz,0}\cos{(\phi_{\mathrm{SAW}})}$ as well as a phase-shifted shear
component $u_{xz}=u_{xz,0}\sin{(\phi_{\mathrm{SAW}})}$[33]. Here,
${\bf\vec{z}}||[001]$ is the QW growth axis,
$\phi_{\mathrm{SAW}}=(k_{\mathrm{SAW}}x-\omega_{\mathrm{SAW}}t)$ is the SAW
phase, and $k_{\mathrm{SAW}}=2\pi/{\lambda_{\mathrm{SAW}}}$ and
$\omega_{\mathrm{SAW}}$ are the SAW wave vector and angular frequency,
respectively. During transport, the SAW piezoelectric potential
$\Phi_{\mathrm{SAW}}=\Phi_{SAW,0}\cos(\phi_{\mathrm{SAW}})$ captures electrons
around the SAW phase $\phi_{\mathrm{SAW}}=0$ [cf. Fig. 5(a)] with a strain
field ($u_{xx}+u_{zz}$), where they are also subjected to a transverse
piezoelectric field $F_{z}$ proportional to $u_{xx}$222see Supplemental
Material, Sec. SM5 at URL for details about the contributions of the SAW
strain fields to the spin-orbit interaction.

Figure 5:  Tight-binding calculations of the spin precession rate. SAW phase
($\phi_{\mathrm{SAW}}$) dependence of the (a) electronic piezoelectric energy,
$-e\Phi_{\mathrm{SAW}}$, and of the (b) electron-spin precession frequency
determined by the tight-binding approach for a QW with the same thickness as
the QWR under $P_{\mathrm{SAW}}=100$ W/m. The dashed ($\Omega_{\mathrm{D}}$)
and solid lines ($\Omega_{\mathrm{R}}$) are the corresponding Dresselhaus and
piezoelectrically induced Rashba contributions, respectively, obtained from
Eq. (2).

The lateral dimensions of the QWRs and DQDs are much larger than the QWR (or
QW) thickness. For the determination of the ${\Omega_{\mathrm{SO}}}$ in QWRs
and DQDs, we will neglect the role of the lateral confinement (and of the
lateral interfaces) and assume that ${\Omega_{\mathrm{SO}}}$ is essentially
equal to the one in a QW of the same thickness. We now address the different
SO mechanisms acting on spins acoustically transported along the $x$-direction
of a QW with the SAW velocity $v_{\mathrm{SAW}}$. The effective electron wave
vector $k_{x}=m^{*}v_{\mathrm{SAW}}/\hbar$ is determined by the effective
electron mass $m^{*}$ ($\hbar$ is the reduced Planck’s constant)333see
Supplemental Material, Sec. SM3 at URL for details about the contributions of
the SAW strain fields to the spin-orbit interaction. The bulk inversion
asymmetry (BIA) of the III-V lattice induces an intrinsic SO field (the
Dresselhaus term[36]), which leads to the precession of the moving spins with
an angular frequency $\hbar{\Omega_{\mathrm{D}}}\sim\gamma
k_{x}\left(\pi/w_{z}\right)^{2}$. Here, $w_{z}$ denotes the extension of the
electron wave function along $z$. The structural inversion asymmetry (SIA)
induced by the SAW gives rise to precession components related to the strain
field, $\hbar{\Omega_{\mathrm{S}}}=\frac{1}{2}C_{3}u_{xx}k_{x}$ [20], as well
as to piezoelectric field $F_{z}$,
$\hbar{\Omega_{\mathrm{R}}}=2r_{41}F_{z}k_{x}$. The terms $\gamma$, $r_{41}$,
and $C_{3}$ are material parameters quantifying the strength of the different
SO contributions (cf. Sec. SM3). The previous expressions can be combined such
that the spin precession frequency for transport by a single SAW beam
($\Omega^{(QW)}_{\mathrm{SO}}$) and by DQDs ($\Omega^{(DQD)}_{\mathrm{SO}}$)
can be stated as: [20, 21]

$\displaystyle\Omega^{(QW)}_{\mathrm{SO}}$ $\displaystyle=$
$\displaystyle\Omega_{\mathrm{D}}{\bf\hat{y}}-(\Omega_{\mathrm{R}}+\Omega_{\mathrm{S}}){\bf\hat{y}}$
(2) $\displaystyle\Omega^{(DQD)}_{\mathrm{SO}}$ $\displaystyle=$
$\displaystyle\sqrt{2}\Omega_{\mathrm{D}}{\bf\hat{x}^{\prime}}-2(\Omega_{\mathrm{R}}+\Omega_{\mathrm{S}}){\bf\hat{y}^{\prime}}$
(3)

In Eq. (3), ${\bf\hat{x}^{\prime}=x-y}$ and ${\bf\hat{y}^{\prime}=x+y}$ are
the unit vectors parallel and perpendicular to the DQD propagation while
$\Omega_{\mathrm{R}}$ and $\Omega_{\mathrm{S}}$ refer to precession
frequencies for a single SAW beam along ${\bf\hat{x}}$. The $\sqrt{2}$ term in
this equation arises from the fact that the propagation velocity of the DQDs
equals to the vector sum of the velocities of the individual SAW beams.

Since $F_{z}\propto u_{xx}$ (hence
$\Omega_{\mathrm{R}}\propto\Omega_{\mathrm{S}}$), Eq. (2) predicts a linear
dependence of $\Omega^{(QW)}_{\mathrm{SO}}$ on the SAW amplitude. For a
quantitative comparison, we first note that while the $\gamma=$ $(17\pm
2)\text{\times}{10}^{-30}\text{\,}\mathrm{eV}\text{\,}{\mathrm{m}}^{3}$ [29]
and $r_{41}$ = -$(59\pm 7)\text{\times}{10}^{-21}\text{\,}\mathrm{\text{ $e$
}}\text{\,}{\mathrm{m}}^{2}$ [20] have well-established values, the reported
values of the strain-related parameter $C_{3}$ span a wide range (e.g., from
$0.81\text{\,}\mathrm{eV}\text{\,}\mathrm{nm}$ in n-type doped bulk GaAs [24,
26] to $0.31\text{\,}\mathrm{eV}\text{\,}\mathrm{nm}$ in a GaAs(001) QW [27]).

The red solid line in Fig. 4(b) is a fit of the QWR data to Eq. (2) and is
then used to determine $C_{3}=$$-2.6\pm
0.1\text{\,}\mathrm{eV}\text{\,}\mathrm{nm}$ (details in Sec. SM5). While
larger than previously reported values, this $C_{3}$ value reproduces very
well the measured data for DQDs (after considering the scaling of
${\Omega_{\mathrm{SO}}}$ due to the larger $v_{\textrm{DQD}}$) and shows more
reasonable agreement with the dashed lines in Fig. 4(b) from tight-binding
calculations. The relative signs of the terms of Eq. (2) is also consistent
with the distinctive slope of the red solid line in Fig. 4(b). These terms
assumed that the electron spins were experiencing the maximum
${\Omega_{\mathrm{SO}}}$ as calculated in Fig. 5(b), which supports the
assumption that the electrons have a phase purity near
$\phi_{\mathrm{SAW}}=0$.

### Tight-binding calculations of the spin splittings

In order to verify the predictions of the analytical model, we have also
calculated the SO field ${\Omega_{\mathrm{SO}}}$ in QWs using the tight-
binding (TB) method [37, 38]. The dots in Fig. 5(b) display the
${\Omega_{\mathrm{SO}}}$ dependence on $\phi_{\mathrm{SAW}}$ calculated for
the QWR under $P_{\mathrm{SAW}}=100$ W/m. In the TB calculations, the QWRs
were modelled as a thicker QW, thus neglecting the effects of the weak lateral
confinement. The total SO component ${\Omega_{\mathrm{SO}}}$ oscillates around
an average value $\Omega_{D}$ following the harmonic dependence of the SAW
fields. To compare with the experiments, we further assume that the electrons
are stored around the phases $\phi_{\mathrm{SAW}}=0$ of minimum electronic
piezoelectric energy $-e\Phi_{\mathrm{SAW}}$, as illustrated in Fig. 5(a). The
SAW amplitude dependence of ${\Omega_{\mathrm{SO}}}$ at these phases
calculated by the TB method is displayed by the $\star$’s in Fig. 4(b) for the
QWR and QW widths. The calculated linear dependence of
${\Omega_{\mathrm{SO}}}$ on strain amplitude, which corresponds to the
parameter $C_{3}$ in Eq.(2), does not depend on the thickness of the
structures and is, thus, similar for the QW and QWR. The sign of the $C_{3}$
parameter determined by the TB method agrees with the experiments, thus
reproducing the increase of the precession rate with SAW amplitude. Its
magnitude $|C_{3}|=1.65$ eVnm is slightly (33%) smaller than the
experimentally determined one, but still well above previously published
experimental values.

Figure 6:  Dimensionality effects on spin dephasing. (a) Spin transport
length, $\ell_{s}$, and (b) precession frequency ratio
$r_{{\Omega_{\mathrm{SO}}}}={\Omega_{\mathrm{SO}}}(w_{y})/{\Omega_{\mathrm{SO}}}(w_{y}\rightarrow\infty)$
as a function of the channel width, $w_{y}$, as determined from Monte-Carlo
simulations of the acoustic spin transport under SO fields with BIA and SIA
symmetries. The simulations were carried out assuming a mobility
$\mu=4m^{2}/(Vs)$ and temperature $T=10$ K.

### Dimensionality effects on the spin dynamics

One interesting observation in connection with Figs. 4 and 5(b) is that the
spin transport lengths in the QWRs are much shorter than the ones in the DQDs,
despite the much smaller lateral confinement dimensions in the former
structures. To understand the impact of the lateral confinement and motional
narrowing effects on the spin dynamics, we have carried out Monte-Carlo
simulations of the acoustic spin transport for channels with different widths
$w_{y}$. The results are summarized in Fig. 6. As expected from motional
narrowing in the regime of 1D spin transport
($w_{y}<L_{\mathrm{SO}}^{\mathrm{(2D)}}$), $\ell_{s}$ increases with
decreasing channel width as $\ell_{s}\propto w^{-n_{w}}$ with $n_{w}=2$ [13].
In addition, we observe similar dependencies of $\ell_{s}$ on the channel
width for BIA and SIA SO fields.

By using $L_{\mathrm{SO}}^{\mathrm{(2D)}}=\ell_{s}\sim 12~{}\mu$m (as for the
QW in Fig. 4), the enclosure of electrons in DQDs defined by
${\lambda_{\mathrm{SAW}}}=5.6~{}\mu$m should increase $\ell_{s}$ by a factor
of $(2L_{\mathrm{SO}}^{\mathrm{(2D)}}/{\lambda_{\mathrm{SAW}}})^{2}\sim 18$.
Such a mechanism qualitatively explains the long spin transport lengths by
DQDs. It also predicts huge spin lifetimes and transport lengths for the QWRs,
which are, however, not observed in the experiments. A possible explanation
lies on the nature of the lateral confinement potential: while purely
electrostatic for DQDs, it is imposed by structural interfaces in the QWRs.
Spin dephasing due to frequent scattering events at these interfaces (known as
Elliot-Yafet processes[39]), as previously postulated for QWRs,[14, 40] as
well as interface-related SO fields may thus limit $\ell_{s}$ in structural
channels.

A further remarkable experimental result is the weak SAW amplitude dependence
of the spin precession rate ${\Omega_{\mathrm{SO}}}$ for transport in QW
structures as compared to QWRs and DQDs [cf. Figs. 4(a) and 5(a)]. This
behaviour in QWs also contrasts with the TB predictions as well as with the
model leading to Eq. (2). Furthermore, it is at odds with the Monte-Carlo
simulations illustrated in Fig. 6(b), which yield the dependence of the spin
precession length on channel width. Here,
$r_{{\Omega_{\mathrm{SO}}}}={\Omega_{\mathrm{SO}}}(w_{y})/{\Omega_{\mathrm{SO}}}(w_{y}\rightarrow\infty)$
is the ratio between the precession frequencies in a channel with finite width
$w_{y}$ to the one in an unconstrained channel. One finds that
$r_{{\Omega_{\mathrm{SO}}}}$ is essentially independent of the channel width.
This result, which applies for SO fields of both BIA and SIA symmetries, can
be qualitatively understood by taking into account that the spin precession
rate ${\Omega_{\mathrm{SO}}}(w_{y})$ along the SAW propagation direction ($x$)
is determined by the carrier velocity component along $x$, which remains equal
to the SAW velocity as the carriers diffuse in the lateral direction 444see
Supplemental Material, Sec. SM6 at URL for details about the dependence of the
precession rate determined by the Monte-Carlo simulations on the channel
width.

We now briefly address a mechanism that can account for the weak dependence of
${\Omega_{\mathrm{SO}}}$ on SAW amplitude observed for the QW. This mechanism
is based on fluctuations in the storage phase $\phi_{\mathrm{SAW}}$ of the
carriers in the SAW potential during the transport. The driving force for this
transport is the longitudinal component of the SAW piezoelectric field given
by $F_{x}=-e\partial\Phi_{\mathrm{SAW}}/\partial x$. In the previous sections,
we have assumed that the electron spins remain at the SAW phase
$\bar{\phi}_{\mathrm{SAW}}=0$ corresponding to the minimum of the
piezoelectric energy $-e\Phi_{\mathrm{SAW}}$, as illustrated in Fig. 5(a). At
these phases, however, $F_{x}$ vanishes. To sustain a steady-state motion at
the SAW velocity, both electrons (superscript $e$) and holes ($h$) must
concentrate around a phase $\bar{\phi}^{(i)}_{\mathrm{SAW}}$ ($i=e,h$) of the
SAW potential satisfying
$\sin(\bar{\phi}^{(i)}_{\mathrm{SAW}})=r^{(i)}_{d}=v_{\mathrm{SAW}}/(\mu^{(i)}F_{z})$
[rather than at $\phi_{\mathrm{SAW}}=0$, cf. Fig. 5(a)]. At these phases, the
SAW-induced SO fields are reduced by a factor
$\cos(\bar{\phi}^{(i)}_{\mathrm{SAW}})=(1-(r^{(i)}_{d})^{2})$ with respect to
their maximum at $\phi_{\mathrm{SAW}}=0$. As a result, the effective spin
precession frequency reduces and becomes dependent on the effective carrier
mobility. Due to the narrower thickness $w_{z}$, the transport mobility
$\mu^{(i)}$ in the QWs is expected to be lower than in the QWR and DQD cases
(the ambipolar mobility scales with $w_{z}^{-6}$ in undoped QWs [42]), thus
reducing the SAW-induced SO fields relative to the Dresselhaus contribution.
As mentioned in connection with Fig. 4(a), $\ell_{s}$ increases with SAW power
for the spins in the QW, thus indirectly indicating a connection between the
mobility and the spin dynamics. According to this mechanism, the lower
efficiency of the spin gates in the QW arises from the lower ambipolar
mobility relative to the QWR and the DQDs due to its smaller thickness.
Further studies are, however, required to quantify the impact of this effect.

##  III Discussion

In this work, we have demonstrated the ability to acoustically transport and,
simultaneously, control the polarization of electrons spins stored within
moving potential dots defined by SAWs in GaAs QWs and QWRs. Within the flying
control gates, the spins precess around a SAW-induced SO field, and the
precession rate can be dynamically changed by a factor of over 2.5 by varying
the SAW amplitude, which enabled the controlled flip of the spin polarization.
The experimental results for the precession rates markedly exceed previous
results and agree well with the predictions of an analytical model for the
SAW-induced spin-orbit fields as well as with microscopic tight-binding
calculations. The latter of which enable a precise determination of the strain
related spin-orbit parameters. We have also addressed the mechanisms governing
spin relaxation while considering the role of the carrier mobility and of the
lateral interfaces on the spin dynamics. Here, the shorter spin transport
lengths measured in structurally defined QWRs as compared to electrostatically
defined DQDs are attributed to spin dephasing via frequent scattering at the
lateral (Al,Ga)As interfaces. Details of the scattering mechanisms are,
however, presently unknown and calls for additional structural studies of the
QWR interfaces.

The high degree of dynamical spin control in QWRs and DQDs demonstrated here
enables dynamic control by simply changing the acoustic amplitude of the
carrier wave. An obvious and important advantage is that both spin transport
and manipulation can be performed through this flying spin gate in a single
structure (e.g., a QWR) without requiring extra components (e.g.,
electrostatic gates for Rashba spin control). The experiments in Fig. 3(b)
directly illustrate an electro-optical polarization modulators based on this
approach. Here, the circular polarization of the PL can be set to an arbitrary
value by simply varying the SAW amplitude. This functionality can be naturally
extended to the transport of single spins[22] as well as for the generation of
polarized single photons[15], thus enabling the realization of a spin-based
quantum information processor with a photonic interface.

##  IV Methods

### QWR Sample Fabrication

The planar QWRs used in this work were fabricated using MBE by overgrowing a
10 nm-thick QW on a GaAs(001) substrate pre-patterned with shallow
(approximately 30 nm high) rectangular ridges [cf. Fig. 2(a)] [30]. The
anisotropic nature of the MBE growth induces a local thickening of the QW at
the ridge sidewalls oriented along the ${\bf\vec{x}}||[110]$-direction along
the surface, thus forming a QWR parallel to this sidewall. The thickness and
width of the QWR are $25\pm 5\text{\,}\mathrm{nm}$ and $200\pm
5\text{\,}\mathrm{nm}$, respectively, as determined by scanning transmission
electron microscopy [30]. The moving potential dots for carrier and spin
transport are created by propagating a SAW with a wavelength
${\lambda_{\mathrm{SAW}}}=4~{}\mu$m (frequency of $726$ MHz at
$15\text{\,}\mathrm{K}$ and velocity $v_{\mathrm{SAW}}=2904$ m/s) along the
wire axis. The SAW is generated by a split-finger interdigital transducer
deposited on the sample surface. Its amplitude will be quantified in terms of
the linear acoustic power density $P_{\mathrm{SAW}}$, defined as the ratio
between the coupled acoustic power and the width of the SAW beam. During the
acoustic transport, the carriers are confined within dots with dimensions
equal to the QWR width and less than $\sim{\lambda_{\mathrm{SAW}}}/2$ along
the directions perpendicular and parallel to the SAW propagation,
respectively. Acoustic spin transport was also investigated in the QW
embedding the QWR. In this case, carrier motion is unconstrained in the
direction perpendicular to the transport.

### DQD Sample Fabrication

The DQDs were created via the interference of two orthogonal SAW beams
propagating along the ${\bf\vec{x}}||[110]$ and ${\bf\vec{y}}||[\bar{1}10]$
directions of a 30 nm thick QW on GaAs(001) [15]. The interference of the
piezoelectric fields of the SAWs creates an array of DQDs propagating along
the ${\bf\vec{x^{\prime}}}||[010]$ surface direction [15]. DQDs yield the
longest, so far, reported acoustic spin transport distances [15, 43]. SAWs
with a wavelength of ${\lambda_{\mathrm{SAW}}}=5.6~{}\mu$m were used and yield
DQDs with dimensions of approximately
${\lambda_{\mathrm{SAW}}}/2\times{\lambda_{\mathrm{SAW}}}/2$ propagating with
a velocity $v_{\textrm{DQD}}=\sqrt{2}v_{\mathrm{SAW}}=$ 4115 m/s.

### Photoluminescence measurements

The spectroscopic photoluminescence (PL) studies of the spin transport were
performed at low temperatures (10-20 K) in a microscopic PL setup with radio-
frequency (rf) wiring for SAW excitation. The spins were optically excited
using a circularly polarized laser beam with tunable wavelength ($\lambda_{L}$
between 760 and 808 nm) focused onto a $\sim 2~{}\mu$m wide spot on the sample
surface. The PL emitted along the SAW path with left- ($I_{L}$) and right-hand
($I_{R}$) circular polarization was collected with spatial resolution and used
to determine the spin polarization [44]
$\rho_{s}=({I_{R}-I_{L}})/({I_{R}+I_{L}})$ 555see Supplemental Material, Sec.
SM1 at URL for further details about the experimental determination of the
spin polarization. Since hole-spin relaxation in QWs and QWRs is typically
much faster than the one for electrons [46], $\rho_{s}$ essentially reflects
the electron spin dynamics.

### Tight-binding Calculations

The analytical model for the SO fields applies for a QW with infinite
potential barriers and, thus, neglects the effects of the QW interfaces. In
addition, the model neglects the impact of the SAW field on the electronic
states, which can be significant for the valence band states in wide QWs [47].
The the SO field ${\Omega_{\mathrm{SO}}}$ contributions were calculated in QWs
using the tight-binding (TB) method [37, 38]. The effects of the SAW were
taken into account by using the strain and piezoelectric fields to determine
the atomic positions and on-site potentials, respectively, within the TB
supercell 666see Supplemental Material, Sec. SM4 at URL for further details
about the tight-binding calculation procedure.

### Monte Carlo spin dynamics

In the calculations, the spins are assumed to move along the $x$-direction
with the SAW velocity while having a random motion along the $y$-direction
with mean-free-path $\ell_{p}$ and thermal velocity $v_{p}$ defined by the
electron mobility $\mu$ and temperature 777see Supplemental Material, Sec. SM6
at URL for details about the Monte-Carlo simulations. Figure 6(a) and (b)
display the simulated dependence of $\ell_{s}$ and ${\Omega_{\mathrm{SO}}}$
for transport in channels with different widths $w_{y}$ under BIA and SIA SO
fields. The SO fields were assumed to have an amplitude
${\Omega_{\mathrm{SO}}}^{\mathrm{(2D)}}=2\pi
v_{\mathrm{SAW}}/L_{\mathrm{SO}}^{\mathrm{(2D)}}$ dictated by a spin
precession period $L_{\mathrm{SO}}^{(2D)}=12~{}\mu$m similar to the ones
measured for QWs in Fig. 3. We further assumed $\mu=4m^{2}/(Vs)$ and $T=4$ K,
which yield $\ell_{p}=0.1~{}\mu$m.

##  V Author Contributions

P.L.J.H. performed the experiments and analysis on the QW and QWR samples, and
J.A.H.S. performed the experiments and analysis using the DQDs. K.B. performed
the MBE growth of the samples. Initial discussions and experimental design and
development between H.S., Y.K., and P.L.J.H. were critical inputs to the
presented results. P.V.S. led the theory calculations and supervised the
project. P.L.J.H., P.V.S., and J.A.H.S. wrote the manuscript with input from
the other authors.

###### Acknowledgements.

We thank S. Ludwig for discussions and for a critical reading of the
manuscript, A.C. Poveda for the help with the graphics, as well as S.
Rauwerdink for the help in the fabrication of the samples. J. Stotz would also
like to acknowledge the Natural Science and Engineering Research Council of
Canada and the Alexander von Humboldt Foundation. This project has received
funding from the European Union’s Horizon 2020 research and innovation program
under grant agreement No 642688.

## References

  * Datta and Das [1990] S. Datta and B. Das, Electronic analog of the electro-optic modulator, Appl. Phys. Lett. 56, 665 (1990).
  * Bychkov and Rashba [1984] Y. A. Bychkov and E. I. Rashba, Oscillatory effects and the magnetic susceptibility of carriers in inversion layers, J. Phys. C 17, 6039 (1984).
  * Koo  _et al._ [2009] H. C. Koo, J. H. Kwon, J. Eom, J. Chang, S. H. Han, and M. Johnson, Control of spin precession in a spin-injected field effect transistor, Science 325, 1515 (2009).
  * Chuang  _et al._ [2015] P. Chuang, S.-C. Ho, L. W. Smith, F. Sfigakis, M. Pepper, C.-H. Chen, J.-C. Fan, J. P. Griffiths, I. Farrer, H. E. Beere, G. A. C. Jones, D. A. Ritchie, and T.-M. Chen, All-electric all-semiconductor spin field-effect transistors, Nat. Nanotechnol. 10, 35 (2015).
  * D’yakonov and Perel’ [1971] M. I. D’yakonov and V. I. Perel’, Spin orientation of electrons associated with the interband absorption of light in semiconductors, SPJETP 33, 1053 (1971).
  * Ohno  _et al._ [1999] Y. Ohno, R. Terauchi, T. Adachi, F. Matsukura, and H. Ohno, Spin relaxation in GaAs(110) quantum wells, Phys. Rev. Lett. 83, 4196 (1999).
  * Hernández-Mínguez  _et al._ [2012] A. Hernández-Mínguez, K. Biermann, R. Hey, and P. V. Santos, Electrical suppression of spin relaxation in GaAs(111)B quantum wells, Phys. Rev. Lett. 109, 266602 (2012).
  * Bernevig  _et al._ [2006] B. A. Bernevig, J. Orenstein, and S.-C. Zhang, Exact SU(2) symmetry and persistent spin helix in a spin-orbit coupled system, Phys. Rev. Lett. 97, 236601 (2006).
  * Koralek  _et al._ [2009] J. D. Koralek, C. P. Weber, J. Orenstein, B. A. Bernevig, S.-C. Zhang, S. Mack, and D. D. Awschalom, Emergence of the persistent spin helix in semiconductor quantum wells, Nature 458, 610 (2009).
  * Walser  _et al._ [2012] M. P. Walser, C. Reichl, W. Wegscheider, and G. Salis, Direct mapping of the formation of a persistent spin helix, Nat. Phys. 8, 757 (2012).
  * Balocchi  _et al._ [2011] A. Balocchi, Q. H. Duong, P. Renucci, B. L. Liu, C. Fontaine, T. Amand, D. Lagarde, and X. Marie, Full electrical control of the electron spin relaxation in GaAs quantum wells, Phys. Rev. Lett. 107, 136604 (2011).
  * Kikkawa and Awschalom [1999] J. M. Kikkawa and D. D. Awschalom, Lateral drag of spin coherence in gallium arsenide, Nature 397, 139 (1999).
  * Kiselev and Kim [2000] A. A. Kiselev and K. W. Kim, Progressive suppression of spin relaxation in two-dimensional channels of finite width, Phys. Rev. B 61, 13115 (2000).
  * Holleitner  _et al._ [2006] A. W. Holleitner, V. Sih, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Suppression of spin relaxation in submicron InGaAs wires, Phys. Rev. Lett. 97, 036805 (2006).
  * Stotz  _et al._ [2005] J. A. H. Stotz, R. Hey, P. V. Santos, and K. H. Ploog, Coherent spin transport via dynamic quantum dots, Nat. Mater. 4, 585 (2005).
  * Sanada  _et al._ [2013] H. Sanada, Y. Kunihashi, H. Gotoh, K. Onomitsu, M. Kohda, J. Nitta, P. V. Santos, and T. Sogawa, Manipulation of mobile spin coherence using magnetic-field-free electron spin resonance, Nat. Phys. 9, 280 (2013).
  * Rocke  _et al._ [1998] C. Rocke, A. O. Govorov, A. Wixforth, G. Böhm, and G. Weimann, Exciton ionization in a quantum well studied by surface acoustic waves, Phys. Rev. B 57, R6850 (1998).
  * Maialle  _et al._ [1993] M. Z. Maialle, E. A. de Andrada e Silva, and L. J. Sham, Exciton spin dynamics in quantum wells, Phys. Rev. B 47, 15776 (1993).
  * Sogawa  _et al._ [2001] T. Sogawa, P. V. Santos, S. K. Zhang, S. Eshlaghi, A. D. Wieck, and K. H. Ploog, Transport and lifetime enhancement of photoexcited spins in GaAs by surface acoustic waves, Phys. Rev. Lett. 87, 276601 (2001).
  * Hernández-Mínguez  _et al._ [2014] A. Hernández-Mínguez, K. Biermann, R. Hey, and P. V. Santos, Spin transport and spin manipulation in (110) and (111) GaAs quantum wells, Phys. Status Solidi B 251, 1736 (2014).
  * Sanada  _et al._ [2011] H. Sanada, T. Sogawa, H. Gotoh, K. Onomitsu, M. Kohda, J. Nitta, and P. V. Santos, Acoustically induced spin-orbit interactions revealed by two-dimensional imaging of spin transport in GaAs, Phys. Rev. Lett. 106, 216602 (2011).
  * Bertrand  _et al._ [2016] B. Bertrand, S. Hermelin, S. Takada, M. Yamamoto, S. Tarucha, A. Ludwig, A. D. Wieck, C. Bäuerle, and T. Meunier, Fast spin information transfer between distant quantum dots using individual electrons, Nat. Nanotechnol. 11, 672 (2016).
  * D’yakonov  _et al._ [1979] M. I. D’yakonov, I. A. Merkulov, and V. I. Perel’, Optical-orientation anisotropy produced in semiconductors by quadrupole splitting of the spin levels of the lattice nuclei, SPJETP 49, 160 (1979).
  * Pikus  _et al._ [1988] G. E. Pikus, V. A. Marushchak, and A. N. Titkov, Spin splitting of energy bands and spin relaxation of carriers in cubic III-V crystals (review), Sov. Phys. Semicond. 22, 115 (1988).
  * Kato  _et al._ [2004] Y. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Coherent spin manipulation without magnetic fields in strained semiconductors, Nature 427, 50 (2004).
  * Beck  _et al._ [2006] M. Beck, C. Metzner, S. Malzer, and G. H. Doehler, Spin lifetimes and strain-controlled spin precession of drifting electrons in zinc blende type semiconductors, Europhys. Lett. 75, 597 (2006).
  * English  _et al._ [2011] D. J. English, P. G. Lagoudakis, R. T. Harley, P. S. Eldridge, J. Hübner, and M. Oestreich, Strain-induced spin relaxation anisotropy in symmetric (001)-oriented GaAs quantum wells, Phys. Rev. B 84, 155323 (2011).
  * Biermann  _et al._ [2010] K. Biermann, O. D. D. Couto, Jr., E. Cerda, H. B. de Carvalho, R. Hey, and P. V. Santos, Spin transport in (110) GaAs-based cavity structures, J. Supercond. Novel Magn. 23, 27 (2010).
  * Stotz  _et al._ [2006] J. A. H. Stotz, R. Hey, P. V. Santos, and K. H. Ploog, Spin transport and manipulation by mobile potential dots in GaAs quantum wells, Physica E 32, 446 (2006).
  * J.Helgers  _et al._ [2019] P. L. J.Helgers, H. Sanada, Y. Kunihashi, A. Rubino, C. Christopher J. B. Ford, K. Biermann, and P. V. Santos, Sidewall quantum wires on GaAs (001) substrates, Phys. Rev. Applied 11, 064017 (2019).
  * Lazić  _et al._ [2012] S. Lazić, R. Hey, and P. Santos, Mechanisms for non-classical light emission from acoustically populated (311)A GaAs quantum wires, New J. of Phys. 14, 013005 (2012).
  * Note [1] See Supplemental Material, Sec. SM2 at URL for details about the determination of the spin transport length.
  * de Lima, Jr. and Santos [2005] M. M. de Lima, Jr. and P. V. Santos, Modulation of photonic structures by surface acoustic waves, Rep. Prog. Phys. 68, 1639 (2005).
  * Note [2] See Supplemental Material, Sec. SM5 at URL for details about the contributions of the SAW strain fields to the spin-orbit interaction.
  * Note [3] See Supplemental Material, Sec. SM3 at URL for details about the contributions of the SAW strain fields to the spin-orbit interaction.
  * Dresselhaus [1955] G. Dresselhaus, Spin-orbit coupling effects in zinc blende structures, Phys. Rev. 100, 580 (1955).
  * Santos and Cardona [1994] P. V. Santos and M. Cardona, Comment on: Observation of spin precession in GaAs inversion layers using antilocalization, Phys. Rev. Lett. 72, 432 (1994).
  * Santos  _et al._ [1995] P. V. Santos, M. Willatzen, M. Cardona, and A. Cantarero, Tight-binding calculations of spin splittings in semiconductor superlattices, Phys. Rev. B 51, 5121 (1995).
  * Elliott [1954] R. J. Elliott, Theory of the effect of spin-orbit coupling on magnetic resonance in some semiconductors, Phys. Rev. 96, 266 (1954).
  * Alsina  _et al._ [2008] F. Alsina, J. A. H. Stotz, R. Hey, U. Jahn, and P. V. Santos, Acoustic charge and spin transport in GaAs quantum wires, Phys. Status Solidi C 9, 2907 (2008).
  * Note [4] See Supplemental Material, Sec. SM6 at URL for details about the dependence of the precession rate determined by the Monte-Carlo simulations on the channel width.
  * Vörös  _et al._ [2005] Z. Vörös, R. Balili, D. W. Snoke, L. Pfeiffer, and K. West, Long-distance diffusion of excitons in double quantum well structures, Phys. Rev. Lett. 94, 226401 (2005).
  * Stotz  _et al._ [2008] J. A. H. Stotz, P. V. Santos, R. Hey, and K. H. Ploog, Mesoscopic spin confinement during acoustically induced transport, New J. Phys. 10, 093013 (2008).
  * Dyakonov [2008] M. I. Dyakonov,  _Spin physics in semiconductors_ (Springer, 2008).
  * Note [5] See Supplemental Material, Sec. SM1 at URL for further details about the experimental determination of the spin polarization.
  * Baylac  _et al._ [1995] B. Baylac, X. Marie, T. Amand, M. Brousseau, J. Barrau, and Y. Shekun, Hole spin relaxation in intrinsic quantum wells, Surf Sci 326, 161 (1995).
  * Santos  _et al._ [2004] P. V. Santos, F. Alsina, J. A. H. Stotz, R. Hey, S. Eshlaghi, and A. D. Wieck, Band mixing and ambipolar transport by surface acoustic waves in GaAs quantum wells, Phys. Rev. B 69, 155318 (2004).
  * Note [6] See Supplemental Material, Sec. SM4 at URL for further details about the tight-binding calculation procedure.
  * Note [7] See Supplemental Material, Sec. SM6 at URL for details about the Monte-Carlo simulations.