# Robust Transductive Few-shot Learning via Joint Message Passing and
Prototype-based Soft-label Propagation

Jiahui Wang, Qin Xu111Corresponding authors., Bo Jiang111Corresponding
authors., and Bin Luo
Computer Science and Technology, Anhui University, China
wjh<EMAIL_ADDRESS>
###### Abstract

Few-shot learning (FSL) aims to develop a learning model with the ability to
generalize to new classes using a few support samples. For transductive FSL
tasks, prototype learning and label propagation methods are commonly employed.
Prototype methods generally first learn the representative prototypes from the
support set and then determine the labels of queries based on the metric
between query samples and prototypes. Label propagation methods try to
propagate the labels of support samples on the constructed graph encoding the
relationships between both support and query samples. This paper aims to
integrate these two principles together and develop an efficient and robust
transductive FSL approach, termed Prototype-based Soft-label Propagation
(PSLP). Specifically, we first estimate the soft-label presentation for each
query sample by leveraging prototypes. Then, we conduct soft-label propagation
on our learned query-support graph. Both steps are conducted progressively to
boost their respective performance. Moreover, to learn effective prototypes
for soft-label estimation as well as the desirable query-support graph for
soft-label propagation, we design a new joint message passing scheme to learn
sample presentation and relational graph jointly. Our PSLP method is
parameter-free and can be implemented very efficiently. On four popular
datasets, our method achieves competitive results on both balanced and
imbalanced settings compared to the state-of-the-art methods. The code will be
released upon acceptance.

##  1 Introduction

Few-shot learning (FSL), which aims to develop a learning model with the
generalization ability to new classes by using a few support samples, has
attracted considerable attention [9, 24, 25] in recent years. FSL tasks are
usually set in both inductive and transductive manners. In inductive FSL, each
query example is assumed to be classified independently of other examples at
inference time. In contrast, transductive FSL methods infer the label jointly
for all samples in the query set and usually demonstrate better performance
than inductive FSL. In this paper, we focus on the transductive FSL task.

In the past ten years, many methods have been developed for transductive FSL
tasks [32, 44, 13, 26, 43, 2, 14, 21, 17]. Overall, these methods can be
roughly categorized into two types, i.e., prototype-based and label
propagation-based methods. Prototype-based methods [2, 15, 23, 16, 26, 42]
generally first learn a representative prototype for each category and then
determine the label of each query sample by using the distance metrics between
it and prototypes. For example, Boudiaf _et al_. [2] propose to use class
prototypes as the initial parameters of the classifier and optimize them by
maximizing mutual information. Simon _et al_. [23] suggest to project features
into a subspace to obtain better prototype representations. Tian _et al_. [26]
propose to obtain a refined prototype feature by obtaining a probability
distribution matrix via conditional transport. The main benefit of prototype-
based methods is that they generally perform robustly w.r.t class-imbalanced
transductive FSL tasks. However, one main limitation of this type of methods
is that they usually fail to fully exploit the relationships of query samples
for their label prediction.

In contrast, label propagation-based transductive FSL methods try to propagate
the labels of support samples on the constructed graph encoding the
relationships between support and query samples, which thus can fully exploit
the relationships of both support and query samples. For instance, Rodríguez
_et al_. [21] propose to simultaneously use label propagation for feature
embedding and classification. Lazarou _et al_. [13] introduce iteratively
graph updating and label propagation. Fu _et al_. [14] suggest incorporating
learnable parameters into the graph structure and optimizing it with a loss
function after propagation. However, although label propagation-based methods
can obtain better performance, they usually perform insensitively w.r.t the
class-imbalanced transductive FSL problem.

Our aim in this paper is to integrate the above two principles together and
develop an efficient and robust transductive FSL approach termed Prototype-
based Soft-label Propagation (PSLP). To be specific, we first estimate a soft-
label presentation for each query sample by leveraging prototype
representation. Then, we conduct soft-label propagation on the query-support
graph (QSGraph). Both prototype representation and soft-label learning are
conducted progressively to boost their respective performance. Moreover, to
learn effective prototype representation for soft-label estimation as well as
QSGraph for soft-label propagation, we design a new joint message passing
(JMP) scheme to jointly learn sample presentation and QSGraph structure.
Overall, the main advantages of the proposed approach are three aspects. (1)
Effectiveness: The proposed PSLP method simultaneously integrates the merits
of label propagation and prototype-based metric principle with our JMP module
to refine prototype representation and graph construction. Thus, it addresses
transductive FSL tasks effectively. (2) Robustness: Our method conducts soft-
label propagation across all samples, allowing for the adaptive adjustment of
the propagation strength for each class to fit the class-imbalanced setting,
which thus performs robustly w.r.t class-imbalanced problems. (3) Efficiency:
The proposed method does not involve any learnable parameters, which can be
implemented very efficiently when compared with many other related works, as
validated in Experiments.

Note that recent work ProtoLP [43] is most related to our work. It also
combines prototype learning and label propagation. However, our PSLP is
significantly different from ProtoLP as follows. First, ProtoLP mainly
exploits prototypes for graph construction and feature representation and
needs a linear projection to obtain the final label prediction. In contrast,
our PSLP leverages prototypes for soft-label generation explicitly and thus is
much lighter than ProtoLP. Second, ProtoLP propagates the ‘known’ labels of
support samples while our PSLP aims to propagate soft labels for all samples.
Third, we further design a new JMP module to jointly learn sample
presentation, prototype representation, and QSGraph structure for our soft-
label propagation process. We will show in experiments that our PSLP performs
obviously better than ProtoLP on both learning accuracy and efficiency.

Overall, the main contributions of this paper are summarized as follows,

  * •

We propose a new general transductive FSL scheme by combining prototype
representation with label propagation and developing a novel soft-label
propagation approach.

  * •

We design a novel joint message propagation, which fully leverages the global
information between all samples, resulting in optimized sample feature
mappings and robust graph structures.

  * •

We demonstrate the effectiveness, robustness and efficiency of our method on
four commonly used benchmarks, achieving competitive results under different
settings compared to SOTA.

##  2 Related Works

Few-shot learning aims to train a model that can be generalized to new classes
by a small number of labeled samples. Existing works often employ meta-
learning strategies [34, 8] to train networks. Metric-based [24, 25, 29],
optimization-based [19, 6, 22], and data-based [1, 36, 38] methods are the
main research directions of current meta-learning methods. The vast majority
of these methods focus on the inductive setting, where the query example is
classified independently of the others. Our method focuses on transductive
few-shot learning [17, 21, 32, 44], which refers to simultaneously using
samples with both labeled and unlabeled labels for inference on the basis of
few-shot learning, requiring the network to effectively utilize the intrinsic
connections between all samples. Existing transductive few-shot learning
methods can be roughly classified into Prototype-based Methods and Label
propagation-based methods, respectively. Below, we introduce these two types
of methods.

Prototype-based Methods. In transductive few-shot learning, prototype-based
methods typically estimate class centers as prototypes and measure the
distance between samples and prototypes for classification [24]. To enhance
the feature representation of prototypes, Simon _et al_. [23] suggest mapping
features to a subspace. Lichtenstein _et al_. [15] reduce the distribution
difference between novel and base classes by applying ICA and PCA for
dimensional reduction to samples, leading to more accurate prototype
estimation. Liu _et al_. [16] correct prototypes using pseudo-labels assigned
to samples. Hu _et al_. [10] and Zhu _et al_. [42] propose a classifier to
update the prototype iteratively by sinkhorn distance [5] with the constrain
of class-balanced prior. Some methods optimize parameters during testing to
obtain improved prototype feature representations. For instance, Boudiaf _et
al_. [2] initialize prototype features as the initial parameters of the
classifier and optimize them by maximizing mutual information and minimizing
cross-entropy loss. Tian _et al_. [26] utilize a learnable conditional
transport metric to optimize label assignment matrices iteratively, obtaining
more accurate prototype representations. Prototype-based methods generally
perform robustly on class-imbalanced transductive FSL tasks but are weak in
fully exploiting the global relationships of all samples for label prediction.

Label Propagation-based Methods. Label propagation [41, 40] is widely used in
transductive few-shot learning [17, 21, 13, 14] that leverages the
correlations between sample features to construct graph structures and
propagates labels from labeled samples to unlabeled samples. For instance,
Rodríguez _et al_. [21] utilize label propagation for both feature mapping and
label prediction. Lazarou _et al_. [13] assume a uniform distribution of query
samples across classes and optimize label propagation results with optimal
transport [5]. Fu _et al_. [14] propose to optimize learnable class centers
and biases through backpropagation during label propagation to mitigate the
impact of imbalanced setting. However, few works integrate the design of
prototype-based methods and label propagation-based methods. Zhu _et al_. [43]
propose to build the graph by prototypes and optimizing the projection of
labels by label propagation. However, these works typically only use support
samples as the label propagation source, it defaults to a class-balanced prior
assumption, making it difficult to handle cases where the number of query
samples per class is imbalanced [28]. Our method propagates soft labels by all
samples, which can adapt well to imbalanced setting, and the simplified
prototype-based propagation steps also result in a more efficient inference
speed.

Figure 1: The process of the proposed method. First, we optimize the feature
representation and the query-support graph by joint message passing. Then, we
use the refined features and graph for prototype-based soft-label propagation
to alternately update the labels and prototypes, and output the prediction
results.

##  3 Methodology

Problem Formulation. Under the few-shot scenario, let us assume that we have a
large labeled base dataset
$\mathcal{D}_{base}={\\{(\boldsymbol{x}^{b}_{i},y^{b}_{i})\\}}_{i=1}^{n_{base}}$
with $N_{base}$ classes, where $n_{base}$ denotes the number of base class
samples, and $\boldsymbol{x}_{i}^{b}$ and $y_{i}^{b}$ represent the sample
features extracted by the backbone network $f_{\theta}$ and their
corresponding labels, respectively. During the training phase, we train
$f_{\theta}$ on $\mathcal{D}_{base}$ with the supervision of labels. Another
dataset of novel classes
$\mathcal{D}_{novel}={\\{(\boldsymbol{x}_{i},y_{i})\\}}_{i=1}^{n_{novel}}$ are
provided for evaluating classification performance. Here, the novel class
labels $y_{i}$ are categories that the network has never encountered before
and satisfy $N_{novel}\cap N_{base}=\varnothing$. During the inference phase,
for an $N$-way $K$-shot task, we randomly sample support set
$\mathcal{S}={\\{(\boldsymbol{x}^{s}_{i},y_{i})\\}}_{i=1}^{NK}$ and query set
$\mathcal{Q}={\\{\boldsymbol{x}^{q}_{i}\\}}_{i=1}^{M}$ from
$\mathcal{D}_{novel}$ to construct multiple tasks, indicating the support set
contains $N$ unseen categories, each with $K$ labeled samples, and the $M$
unlabeled samples in the query set all belong to these $N$ categories. We
focus on the transductive setting, where we employ $f_{\theta}$ to extract
features and simultaneously utilize all samples from $\mathcal{S}$ and
$\mathcal{Q}$, and predict labels $\tilde{y}$ of the samples in $\mathcal{Q}$.

Overview of Method. In this section, we propose our Prototype-based Soft-label
Propagation (PSLP) method for the transductive FSL problem. The proposed
method mainly contains 1) Joint Message Passing to learn effective prototype
representation and construct the query-support graph (QSGraph) and 2) Soft-
label Propagation on QSGraph to achieve label prediction for query samples.
The diagram of our proposed approach is shown in Fig. 1.

###  3.1 Joint Message Passing

In order to leverage structural information among all samples in both support
and query set, we propose Joint Message Passing (JMP) to refine sample
representation and graph structure simultaneously. Specifically, we first
concatenate support and query samples to obtain the initial sample features
$X^{(0)}$ as:

$X^{(0)}=\mathrm{Concat}([\boldsymbol{x}^{s}_{1},\cdots,\boldsymbol{x}^{s}_{NK}],[\boldsymbol{x}^{q}_{1},\cdots,\boldsymbol{x}^{q}_{M}])$
(1)

Using $X^{(0)}$, we then construct an initial query-sample graph (QSGraph) as:

$\displaystyle A^{(0)}_{ij}$
$\displaystyle=\mathrm{exp}\big{(}-\gamma\|\boldsymbol{x}^{(0)}_{i}-\boldsymbol{x}^{(0)}_{j}\|_{2}^{2}\big{)}$
(2)

where $\gamma$ represents the scaling factor. Then, we jointly update the
sample feature representation $X$ and QSGraph $A$ by progressively repeating
the following two steps.

Step1: Feature Representation Learning. Inspired by work [39], we employ a
message passing mechanism to learn each sample feature representation by
aggregating the information of its neighboring samples on QSGraph.
Specifically, the optimized features can be achieved by using the message
passing as follows:

$X^{(t)}\leftarrow(I+L^{(t-1)})^{k}X^{(t-1)}$ (3)

where $t=1,2\cdots T_{\mathrm{JMP}}$ is the iteration step,
$L^{(t-1)}=D^{-\frac{1}{2}}A^{(t-1)}D^{-\frac{1}{2}}$ and
$D=\mathrm{diag}(\sum_{j}^{NK+M}a_{ij})$ represents the degree matrix. $k\geq
1$ is a hyperparameter to control the size of the receptive field. Using Eq.3,
each sample learns its context-aware representation by aggregating the
information from its $k$-hop neighbors.

Step2: QSGraph Refining. Based on the above learned representation $X^{(t)}$,
we can refine the QSGraph structure $A^{(t)}$ to capture the relationships
between all query and sample samples. To avoid the over-smoothing issue in the
following label propagation process, we adopt nearest neighbor graph building
strategy, i.e., each sample node only connects to its $B$-nearest neighbors as
follows:

$A^{(t)}_{ij}\leftarrow\begin{cases}\mathrm{exp}(-\gamma\|\boldsymbol{x}^{(t)}_{i}-\boldsymbol{x}^{(t)}_{j}\|^{2}_{2})&\mathrm{if}\
\boldsymbol{x}^{(t)}_{j}\in\mathrm{NN}_{B}(\boldsymbol{x}^{(t)}_{i})\\\
0&\mathrm{otherwise}\end{cases}$ (4)

where $\mathrm{NN}_{B}(\boldsymbol{x})$ denotes the set of the $B$-nearest
neighbors of sample $\boldsymbol{x}$.

In practice, the above step 1 and step 2 are alternatively implemented to
boost their respective performance. After $T_{\mathrm{JMP}}$ iteration, the
refined sample feature representation $X$ and QSGraph structure $A$ are used
for the following prototype representation and soft-label propagation. We set
$T_{\mathrm{JMP}}$ to 1 for efficiency consideration. Note that the whole
process of the above JMP does not involve any learnable parameters, ensuring
efficient inference in practical applications.

###  3.2 Soft-label Propagation

To integrate prototype-based method with label propagation, we propose
Prototype-based Soft-label Propagation (PSLP). Overall, the proposed PSLP
module involves 1) generating soft-label representations for query samples, 2)
performing soft-labels on QSGraph to refine/predict the final labels for
queries. These two processes are progressively performed, leading to stable
propagation results. The soft-label propagation treats all samples as the
propagation source simultaneously and adaptively controls the strength of
label propagation for each class, which thus can adapt well to class-
imbalanced setting, as validated in Experiments. To be specific, we first
define the initial prototypes by using the sample features obtained from the
above JMP module as follows:

$\boldsymbol{c}^{(0)}_{n}=\frac{1}{K}\sum\nolimits_{\boldsymbol{x}^{\prime}_{s}\in\mathcal{S}_{n}}\boldsymbol{x}^{\prime}_{s}$
(5)

where
$X^{\prime}=\\{\boldsymbol{x}^{\prime}_{1},\boldsymbol{x}^{\prime}_{2}\cdots\boldsymbol{x}^{\prime}_{NK}\\}$
denotes the learned sample representations of the above JMP module and
$\mathcal{S}_{n}$ denotes the samples of class $n$ in the support set. Then,
we alternately update prototypes
$C^{(0)}=\\{\boldsymbol{c}^{(0)}_{1},\boldsymbol{c}^{(0)}_{2}\cdots,\boldsymbol{c}^{(0)}_{N}\\}$
and the label matrix $Z$ by using soft-label propagation.

Step1: Prototype-based Soft-label Generation. To involve all samples in label
propagation, we first generate soft-labels to the query samples based on the
prototypes. The soft-labels of query samples
$\tilde{Z}^{(t)}_{q}\in\mathbb{R}^{M\times N}$ can be obtained based on their
similarities to prototypes $C^{t}$ as:

$\tilde{Z}_{qn}^{(t)}=\frac{\mathrm{exp}(-\gamma\|\boldsymbol{x}_{q}-\boldsymbol{c}^{(t)}_{n}\|^{2}_{2})}{\sum_{n}^{N}\mathrm{exp}(-\gamma\|\boldsymbol{x}_{q}-\boldsymbol{c}^{(t)}_{n}\|^{2}_{2})}\quad$
(6)

Since $\sum_{n}\tilde{Z}_{qn}^{(t)}=1$ and $\tilde{Z}_{qn}^{(t)}\geq 0$, we
call them as soft-label representation. For support samples, since their
labels are known, we simply obtain $Z^{\prime(t)}_{s}\in\mathbb{R}^{NK\times
N}$ of them from the known labels $y_{s}$ as follows:

$Z^{\prime(t)}_{sn}=\mathrm{OneHot}(y_{s})=\begin{cases}1&\mathrm{if}\
y_{s}=n\\\ 0&\mathrm{otherwise}\end{cases}\quad$ (7)

The label matrix
$Z^{(t)}=[Z^{{}^{\prime}(t)},\tilde{Z}^{(t)}]\in\mathbb{R}^{(NK+M)\times N}$
is obtained by concatenating the label matrices of both support and query
samples.

Step2: Soft-label Propagation. After obtaining the label matrix
$Z^{(t)}$generated under the guidance of prototypes, we then conduct label
propagation on the above constructed QSGraph. Inspired by work [41], the
propagation result is obtained via a closed-form solution as:

$Z^{(t+1)}\leftarrow(I-\alpha L)^{-1}Z^{(t)}$ (8)

where $L=D^{-\frac{1}{2}}AD^{-\frac{1}{2}}$ denotes the laplacian matrix of
the refined QSGraph and parameter $\alpha\in[0,1)$ represents the propagation
strength. In contrast to vanilla label propagation relying solely on the
labels of support samples, our method utilizes all samples as propagation
sources to obtain more robust propagation results. Finally, we normalize the
obtained label matrix as

$Z_{qn}^{(t+1)}=\frac{Z_{qn}^{(t+1)}}{\sum_{n}^{N}Z_{qn}^{(t+1)}}$ (9)

Step3: Prototype Rectification. After we obtain the more accurate label
prediction via the above propagation process, we can update prototypes via the
learned soft-labels. Specifically, we update the prototype proportionally to
ensure the stability as:

$C^{(t+1)}=(1-\beta)C^{(t)}+\beta(Z^{{(t+1)}})^{\top}X$ (10)

where
$C^{(t)}=\\{\boldsymbol{c}^{(t)}_{1},\boldsymbol{c}^{(t)}_{2}\cdots,\boldsymbol{c}^{(t)}_{N}\\}$
and hyperparameter $\beta$ balances two terms.

By alternately conducting the above steps for $T_{\mathrm{PSLP}}$ times, both
prototypes and label matrices can be accurately estimated. We finally use the
output label matrix $Z^{T_{\mathrm{PSLP}}}_{q}$ to predict the labels
$\tilde{y}_{q}$ of the query samples as:

$\tilde{y}_{q}=\mathrm{arg}\max_{n}(Z_{qn}^{T_{\mathrm{PSLP}}})$ (11)

The overall process is outlined in Algorithm 1. It is worth noting that the
QSGraph is obtained through JMP and remains unchanged in PSLP, so the
propagation matrix $P=(I-\alpha L)^{-1}$ in Eq.8 can be obtained outside the
iterations to improve inference speed. Based on the above progressive
learning, both prototypes and label propagation can be iteratively optimized
to boost their performance. Also, for class-imbalanced setting, according to
Eq.8, each query sample propagates its soft-label, allowing the propagation
strength of each category to adaptively adjust based on the number of query
samples. Moreover, our PSLP method is parameter-free (only contains some
hyperparameters) and has simple update steps, typically converging in only 5
to 10 iterations and resulting in high efficiency.

1

Input: Features
$X=\\{\boldsymbol{x}_{1}^{s},..,\boldsymbol{x}_{NK}^{s},..{\boldsymbol{x}_{NK+M}^{q}}\\}$,
Support labels $Y=\\{y_{1},\cdots,y_{NK}\\}$, Hyperparamters
$k,B,\alpha,\beta$

Output: Query Labels $\tilde{y}_{q}$

2 for  _$t=1:T_{\mathrm{JMP}}$_ do

// Feature Representation Learning

$X^{(t)}\leftarrow(I+L^{(t-1)})^{k}X^{(t-1)}$;

// Eq.3

// Query-Support Graph Refining

$A^{(t)}\leftarrow\textsc{Update}(X^{(t)},B)$;

// Eq.4

3

4 end for

// Initialize Prototypes

$\boldsymbol{c}^{(0)}_{n}=\frac{1}{K}\sum_{\boldsymbol{x}^{\prime}_{s}\in\mathcal{S}_{n}}\boldsymbol{x}^{\prime}_{s}$
;

// Eq.5

// Compute Propagation Matrix

5 $P=(I-\alpha L)^{-1}$;

6 for  _$t=1:T_{\mathrm{PSLP}}$_ do

// Prototype-based Soft-label Generation

$Z^{(t)}\leftarrow\textsc{Generate}(X,C^{(t)},Y)$;

// Eq.6,7

// Soft-label Propagation

$Z^{(t)}\leftarrow\textsc{Normalize}(PZ^{(t)})$;

// Eq.7,8,9

// Prototype Rectification

$C^{(t+1)}\leftarrow(1-\beta)C^{(t)}+\beta(Z^{{(t)}})^{\top}X$;

// Eq.10

7

8 end for

return _$\tilde{y}_{q}=\mathrm{arg}\max_{n}(Z_{qn}^{T_{\mathrm{PSLP}}})$_ ;

// Prediction

9

Algorithm 1 PSLP algorithm

###  3.3 Comparison with Related Works

Label propagation methods have usually been employed for transductive FSL
problems. For example, iLPC [13] progressively adds the query samples with the
highest confidence to the support set for subsequent iterative propagation. AM
[14] introduces learnable parameters and learnable class centroids during the
label propagation and optimizes them through multiple loss functions based on
test time learning. ProtoLP [43] utilizes prototypes to construct a graph
structure and optimize the projection of the label through label propagation.
Compared with the related works above, the main differences are as follows.
First, our proposed PSLP propagates the label of all samples to achieve robust
results, which can also adapt well to the class-imbalanced setting. Second,
PSLP utilizes soft-label propagation to gradually rectify the prototypes and
assign the soft-labels to query samples, which leads to more stable and robust
learning. Third, the proposed JMP captures the global relations of all samples
to jointly learn better prototype and QSGraph representation.

##  4 Experiments

###  4.1 Datasets and Settings

To verify the effectiveness of the proposed method, the experiments are
conducted on four commonly used datasets: mini-ImageNet [30], tiered-ImageNet
[20], CUB-200-2011 [31], and CIFAR-FS [3, 12]. Following the prior works [27],
we adopt the same data partitioning, pre-processing procedures, and pre-
trained backbone networks [7, 37] for feature extraction, i.e., ResNet-12
[33], ResNet-18 [2], and WideResNet-28-10 [18]. The dimension of extracted
features is subsequently reduced to 40 by PCA. We implement the experiments on
10,000 tasks with 1-shot and 5-shot settings, where the number of query
samples $M$ is set to 75, and the average classification accuracy along with
95% confidence interval is reported for evaluation. In the class-balanced
setting, in line with [42, 43], the Sinkhorn-Knopp algorithm [5] is employed
in Eq.9, and each category has $\frac{M}{N}=15$ query samples. In the class-
imbalanced setting, following [28], query samples are generated following a
Dirichlet distribution with parameter $\alpha_{dir}=2$. The hyperparameters
are set as follows: for all tasks $B=8$, $\gamma=10$ , $T_{\mathrm{PSLP}}=10$,
for balanced setting $\alpha=0.7$, $\beta=0.6$, $k=4$, and for imbalanced
setting $\alpha=0.9$, $\beta=0.2$, $k=1$.

Table 1: Comparison results of transductive FSL methods in the class-balanced
setting on mini-ImageNet, tiered-ImageNet and CUB-200-2011 datasets. The best
and second best performance are marked in bold and underline respectively. *:
our implementation.

Method Publication Backbone mini-ImageNet tiered-ImageNet CUB-200-2011 1-shot
5-shot 1-shot 5-shot 1-shot 5-shot EPNet [21] ECCV’2020 ResNet-12
66.50$\pm$0.89 81.06$\pm$0.60 76.53$\pm$0.87 87.32$\pm$0.64 82.85$\pm$0.81
91.32$\pm$0.41 LR+ICI [33] CVPR’2020 66.85$\pm$0.92 79.26$\pm$0.55
80.79$\pm$0.84 87.92$\pm$0.50 88.06$\pm$0.79 92.53$\pm$0.35 iLPC [13]
ICCV’2021 69.79$\pm$0.99 79.82$\pm$0.55 83.49$\pm$0.84 89.48$\pm$0.47
89.00$\pm$0.70 92.47$\pm$0.35 EASE [42] CVPR’2022 70.47$\pm$0.30
80.73$\pm$0.16 84.54$\pm$0.27 89.63$\pm$0.15 80.11$\pm$0.21 93.13$\pm$0.11
ProtoLP* [43] CVPR’2023 70.22$\pm$0.27 80.93$\pm$0.16 83.87$\pm$0.25
89.73$\pm$0.15 89.99$\pm$0.20 93.22$\pm$0.10 PSLP(Ours) - 70.94$\pm$0.29
81.27$\pm$0.16 84.90$\pm$0.26 90.02$\pm$0.15 90.38$\pm$0.20 93.41$\pm$0.10 TIM
[2] NIPS’2020 ResNet-18 73.92$\pm$0.27 85.04$\pm$0.15 79.94$\pm$0.26
88.53$\pm$0.15 82.19$\pm$0.26 90.79$\pm$0.12 ReRep [4] ICML’2021
76.54$\pm$0.26 85.20$\pm$0.14 82.58$\pm$0.26 88.68$\pm$0.16 85.36$\pm$0.23
90.99$\pm$0.12 AM [14] Arxiv’2023 77.35 85.47 83.40 89.07 86.64 91.78 TCPR*
[35] NIPS’2022 75.99$\pm$0.26 84.39$\pm$0.15 82.65$\pm$0.26 88.26$\pm$0.16
84.90$\pm$0.23 90.10$\pm$0.11 ProtoLP* [43] CVPR’2023 77.26$\pm$0.27
85.07$\pm$0.14 82.99$\pm$0.26 88.70$\pm$0.15 85.86$\pm$0.24 90.52$\pm$0.11
noHub-S [27] CVPR’2023 76.68$\pm$0.28 84.67$\pm$0.15 83.09$\pm$0.27
88.43$\pm$0.16 85.81$\pm$0.24 90.52$\pm$0.12 PSLP(Ours) - 77.65$\pm$0.28
85.32$\pm$0.14 83.65$\pm$0.27 88.96$\pm$0.15 86.31$\pm$0.24 91.07$\pm$0.11
EPNet [21] ECCV’2020 WRN-28-10 70.74$\pm$0.85 84.34$\pm$0.53 78.50$\pm$0.91
88.36$\pm$0.57 87.75$\pm$0.70 94.03$\pm$0.33 LaplacianShot [44] ICML’2020
74.86$\pm$0.19 84.13$\pm$0.14 78.56$\pm$0.21 87.00$\pm$0.15 87.50$\pm$0.19
92.30$\pm$0.10 ReRep [4] ICML’2021 80.04$\pm$0.23 87.64$\pm$0.12
84.30$\pm$0.25 90.01$\pm$0.15 89.78$\pm$0.18 92.20$\pm$0.10 iLPC [13]
ICCV’2021 83.05$\pm$0.79 88.82$\pm$0.42 88.50$\pm$0.75 92.46$\pm$0.42
91.37$\pm$0.63 93.93$\pm$0.30 EASE [42] CVPR’2022 83.00$\pm$0.21
88.92$\pm$0.13 88.96$\pm$0.26 92.63$\pm$0.16 91.68$\pm$0.19 94.12$\pm$0.09
TCPR* [35] NIPS’2022 81.27$\pm$0.24 87.80$\pm$0.13 81.89$\pm$0.27
87.95$\pm$0.16 91.91$\pm$0.18 94.25$\pm$0.10 noHub-S [27] CVPR’2023
82.00$\pm$0.26 88.03$\pm$0.13 82.85$\pm$0.27 88.31$\pm$0.16 92.63$\pm$0.18
94.69$\pm$0.09 ProtoLP* [43] CVPR’2023 82.53$\pm$0.24 88.95$\pm$0.13
88.55$\pm$0.22 92.61$\pm$0.13 91.51$\pm$0.18 94.01$\pm$0.10 AM [14] Arxiv’2023
83.40$\pm$0.74 89.08$\pm$0.40 88.31$\pm$0.73 92.60$\pm$0.39 91.32$\pm$0.60
94.14$\pm$0.29 PUTM [26] ICCV’2023 81.40 88.10 86.80 92.20 91.20 93.70
PSLP(Ours) - 83.36$\pm$0.25 89.27$\pm$0.13 89.22$\pm$0.22 92.82$\pm$0.13
91.89$\pm$0.18 94.19$\pm$0.10

###  4.2 Comparison with State-of the Art

####  4.2.1 Class-Balanced transductive FSL

The classification comparison results by different backbones on mini-ImageNet,
tiered-ImageNet, CUB-200-2011 datasets are shown in Table 1, and the results
on CIFAR-FS are shown in Table 2. From the tables, the following observation
can be drawn: (1) The proposed PSLP outperforms the prototype-based methods,
including EASE [42], TIM [2], PUTM [26], which demonstrates the effective
combing label propagation with prototype learning. In addition, the proposed
joint message passing utilizes an efficient approach that can improve the
feature representation ability. (2) Our PSLP obtains a better accuracy
compared to the label propagation-based methods, including EPNet [21],
LaplacianShot [44], iLPC [13], ProtoLP [43]. This evidently demonstrates the
effectiveness of soft-label propagation for updating the prototypes and
labels. Compared to iLPC and ProtoLP, our approach simplifies the model
iteration steps and achieves consistent improvements of approximately 0.3% in
each setting. (3) In comparison with the methods that require excessive
learnable parameters to train through backpropagation, such as AM [14] and
noHub-S [27], the proposed PSLP achieves competitive results. Compared with
the nohub-S and AM, our method is more efficient and achieves higher inference
performance.

Table 2: Comparation result of transductive FSL methods in the class-balanced
setting on Cifar-FS dataset. *: our implementation.

Cifar-FS Method Backbone 1-shot 5-shot LR+ICI [33] ResNet-12 75.36$\pm$0.97
84.57$\pm$0.57 iLPC [13] 77.14$\pm$0.95 85.23$\pm$0.55 EASE [42]
78.41$\pm$0.29 85.67$\pm$0.11 ProtoLP* [43] 78.08$\pm$0.24 85.66$\pm$0.17
PSLP(Ours) 78.90$\pm$0.29 86.02$\pm$0.17 S2M2-R [18] WRN-28-10 74.81$\pm$0.19
87.47$\pm$0.13 LR+ICI* [33] 84.88$\pm$0.79 89.75$\pm$0.48 iLPC [13]
86.51$\pm$0.75 90.60$\pm$0.48 EASE [42] 87.60$\pm$0.23 90.60$\pm$0.16 ProtoLP*
[43] 87.51$\pm$0.23 90.68$\pm$0.15 AM [14] 86.91 90.80 PUTM [26] 86.40 90.40
PSLP(Ours) 87.87$\pm$0.23 90.88$\pm$0.15

####  4.2.2 Class-Imbalanced transductive FSL

Table 3: Comparison result of transductive FSL methods in class-imbalanced
setting on mini-ImageNet and tiered-ImageNet datasets. *: our implementation.

Method mini-ImageNet tiered-ImageNet 1-shot 5-shot 1-shot 5-shot PT-MAP [10]
63.2 69.3 67.8 73.2 LaplacianShot [44] 69.8 83.6 72.6 87.7 TIM [2] 70.9 83.2
77.8 88.3 BD-CSPN [16] 71.2 83.4 76.3 87.2 $\alpha$-TIM [28] 70.9 84.9 77.7
89.9 iLPC [13] 74.3 83.3 81.6 89.0 BAVAR [11] 74.2 85.6 80.6 89.3 AM [14] 72.0
85.6 78.9 88.7 ProtoLP* [43] 73.7 85.4 81.0 89.0 PUTM [26] 73.8 85.7 81.1 90.4
PSLP(Ours) 75.9 89.2 82.7 92.5

Table 4: Comparison result of transductive FSL methods in the class-imbalanced
setting on CUB-200-2011 dataset. *: our implementation.  CUB-200-2011
---
Method | ResNet-18 | WRN-28-10
1-shot | 5-shot | 1-shot | 5-shot
PT-MAP [10] | 63.2 | 69.3 | 67.8 | 73.2
LaplacianShot [44] | 73.7 | 87.7 | 84.2 | 91.4
TIM [2] | 74.8 | 86.9 | 82.8 | 90.5
BD-CSPN [16] | 74.5 | 87.1 | 85.1 | 91.5
$\alpha$-TIM [28] | 75.7 | 89.8 | 83.6 | 92.8
iLPC [13] | 78.7 | 86.9 | 86.4 | 90.8
BAVAR [11] | 82.0 | 90.7 | 85.7 | 90.8
AM [14] | 78.6 | 89.9 | - | -
ProtoLP* [43] | 76.8 | 86.4 | 82.3 | 90.6
PUTM [26] | 78.9 | 89.3 | 86.9 | 92.2
PSLP(Ours) | 78.9 | 90.2 | 86.2 | 93.3

In Table 3 and Table 4, we present the accuracy comparison between our PSLP
and SOTA methods in the class-imbalanced setting. From the results, we can
note that (1) Compared to the general transductive FSL methods, such as TIM
[2], ProtoLP [43], and BD-CSPN [16], our method achieves state-of-the-art
performance, demonstrating its effectiveness and robustness under class-
imbalanced setting. (2) In comparison with the work specifically addressing
the class imbalance in transductive FSL, such as $\alpha$-TIM [28], PUTM [26],
BAVAR [11], and AM [14], our method provides higher accuracy on mini-ImageNet
and tiered-ImageNet, further verifying the robustness of proposed PSLP. This
is because PSLP can adaptively adjust the strength of label propagation for
each class via soft labels of query samples. (3) Compared to the methods that
rely on prior distributions of samples for each class, our method yields
competitive results. For example, Compared to iLPC [13], which introduces
label propagation with a class-balanced prior, our PSLP achieves up to 5.9%
performance improvement. Compared with BAVER, which also relies on a class-
imbalance prior, our method obtains superior results on mini-ImageNet and
tiered-ImageNet and competitive results on CUB dataset, further proving the
advantage of freedom from prior assumptions of our method.

###  4.3 Ablation Experiments

####  4.3.1 Components Analysis

To verify the effectiveness of our proposed JMP and PSLP, we conduct the
ablation studies. The experimental results are shown in Table 5. We can
observe that (1) The proposed JMP achieves better results on different
datasets with different settings as it jointly learns improved prototype
representations and a more robust query-support graph structure. (2) Adopting
the proposed PSLP instead of the vanilla label-propagation yields a
significant improvement in accuracy. This is attributed to PSLP, which
alternately optimizes the prototypes and labels through soft-label propagation
and provides a more accurate estimation of the prototypes and labels for each
sample. (3) When the JMP and PSLP are involved, further accuracy improvement
is achieved.

Table 5: Component ablations of the proposed method on WRN-28-10 in balanced setting. JMP and PSLP represent joint message passing and prototype-based soft-label propagation respectively.  Component |  mini-ImageNet |  tiered-ImageNet
---|---|---
JMP | PSLP | 1-shot | 5-shot | 1-shot | 5-shot
|  | 75.29 | 87.65 | 82.12 | 91.44
✓ |  | 76.70 | 89.02 | 83.46 | 92.62
| ✓ | 83.24 | 89.25 | 89.19 | 92.79
✓ | ✓ | 83.36 | 89.27 | 89.22 | 92.82

####  4.3.2 Sensitivity to Hyperparameters

Figure 2: Mean Accuracy (%) of 5-way 1-shot tasks with different
hyperparameters on mini-ImageNet.

To further figure out the influence of hyperparameters, we experiment with
different values of hyperparameters on mini-ImageNet dataset. The
hyperparameters are $k$, $B$, $\alpha$, and $\beta$, which represent the size
of the receptive field, the number of neighbors for each sample in JMP, the
strength of label propagation, and the balance term of prototypes in PSLP,
respectively. The results are shown in Figure 2. From the figure, we can
observe that it leads to a decrease in accuracy when $\alpha$ and $\beta$ are
too low, which can be alleviated by increasing the number of iterations.
Furthermore, although $k$ and $B$ are low, the sufficiently aggregated
information from similar samples can be obtained by JMP, proving the
feasibility of sparse graph construction. Overall, our method is less affected
by hyperparameters choices, with accuracy fluctuating within 0.5% for most
choices, which demonstrates that our method is not sensitive to the choice of
these hyperparameters.

####  4.3.3 Robustness in Real-world Scenarios

Table 6: Mean accuracy (%) of 5-way 5-shot tasks with different query numbers
$M$ in the class-imbalanced setting on mini-ImageNet dataset. * : our
implemenetation.

Method Query Number 15 45 75 105 135 $\alpha$-TIM* [28] 83.68 85.02 85.67
86.10 86.18 ProtoLP* [43] 97.15 88.89 85.37 83.35 82.24 PUTM* [26] 84.77 85.04
85.14 85.03 85.04 PSLP(Ours) 98.32 91.53 89.23 87.77 86.96

A crucial aspect for practical applications is the robustness of method under
different class-imbalanced setting. To evaluate the robustness of the proposed
PSLP, we vary the number of query samples for each class or the class
distribution as it is not always known for real-world scenarios. From Table 6,
as $M$ decreases, label-class samples dominate, allowing methods based on
label propagation, such as ProtoLP and our PSLP, to achieve higher accuracy.
Compared to ProtoLP [43], our method significantly improves accuracy. The
$\alpha$-TIM [28] and PUTM [26] give lower accuracy, which is mainly due to
the insufficient number of samples available for test-time training. Although
the accuracy of our method declines slightly when the proportion of labeled
samples decreases as $M$ increases, it is still higher than the compared
methods. This verifies the robustness of our method in real-world scenarios.

####  4.3.4 Inference Speed

Table 7: Inference time (s) for 1000 tasks of 5-way 5-shot on mini-ImageNet by CPU. Tasks are build with the features extracted by backbones.  Backbone |  $\alpha$-TIM | PUTM | ProtoLP | Ours
---|---|---|---|---
ResNet12 | 90.58 | 454.93 | 7.16 | 1.54
WRN-28-10 | 119.27 | 477.29 | 9.02 | 1.86

To validate the efficiency of our method, we implement the $\alpha$-TIM [28],
ProtoLP [43], PUTM [26] on the same CPU (Intel I7-13700F). Table 7
demonstrates the inference time cost for 1000 few-shot tasks. From the table,
our method exhibits an evident advantage in inference time in comparison with
the compared methods. It is believed because: (1) Both $\alpha$-TIM and PUTM
need to optimize the learnable parameters during inference, (2) ProtoLP
requires the construction of graph iteratively and has more complex iteration
procedure. In contrast, the proposed PSLP involves no learnable parameters.
The inference speed is fast and converges after only a few iterations.

##  5 Conclusion

In this paper, we integrate the advantages of prototype-based and label
propagation-based methods and develop an efficient and robust transductive FSL
method, termed Prototype-based Soft-label Propagation (PSLP). Our proposed
PSLP propagates the labels of all samples in which the soft labels of query
samples are assigned by prototypes. By alternatively updating the prototypes
with propagated labels and assigning new soft labels to query samples for
propagation, the prototypes and labels are progressively estimated. The soft-
label propagation adaptively adjusts the propagation strength of each class,
which is robust to address the class-imbalanced setting problem. Moreover, we
further propose the joint message passing to utilize the global information of
all samples to refine feature representation and query-support graph
structure. Notably, the proposed method is parameter-free and efficient,
facilitating the end-to-end inference on the CPU. Our method achieves
competitive results compared to the state-of-the-art methods under various
settings.

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