Original: F EW - SHOT L EARNING VIA D IRICHLET T ESSELLATION E NSEMBLE
Revised: F EW - SHOT L EARNING AS C LUSTER - INDUCED V ORONOI D IAGRAMS : A G EOMETRIC A PPROACH

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Few-shot learning (FSL) is the process of rapid generalization from abundant base samples to inadequate novel samples.
Revised: Few-shot learning (FSL) is the process of rapid generalization from abundant base samples to inadequate novel samples.

Original: Despite extensive research in recent years,
Revised: Despite extensive research in recent years,

Original: FSL is still not yet able to generate satisfactory solutions for a wide range of realworld applications.
Revised: FSL is still not yet able to generate satisfactory solutions for a wide range of real-world applications.

Original: To confront this challenge, we study the FSL problem from a geometric point of view in this paper.
Revised: To confront this challenge, we study the FSL problem from a geometric point of view in this paper.

Original: One observation is that the widely embraced ProtoNet model is essentially a Dirichlet Tessellation (Voronoi Diagram) in the feature space.
Revised: One observation is that the widely embraced ProtoNet model is essentially a Voronoi Diagram (VD) in the feature space.

Original: We retroﬁt it by making use of a recent advance in computational geometry called Cluster-induced Voronoi Diagram (CIVD) .
Revised: We retroﬁt it by making use of a recent advance in computational geometry called Cluster-induced Voronoi Diagram (CIVD) .

Original: Starting from the simplest nearest neighbor model, CIVD gradually incorporates cluster-to-point and then cluster-to-cluster relationships for space subdivision, which is used to improve the accuracy and robustness at multiple stages of FSL.
Revised: Starting from the simplest nearest neighbor model, CIVD gradually incorporates cluster-to-point and then cluster-to-cluster relationships for space subdivision, which is used to improve the accuracy and robustness at multiple stages of FSL.

Original: Speciﬁcally, we use CIVD (1) to integrate parametric and nonparametric few-shot classiﬁers; (2) to combine feature representation and surrogate representation; (3) and to leverage feature-level, transformation-level, and geometry-level heterogeneities for a better ensemble.
Revised: Speciﬁcally, we use CIVD (1) to integrate parametric and nonparametric few-shot classiﬁers; (2) to combine feature representation and surrogate representation; (3) and to leverage feature-level, transformation-level, and geometry-level heterogeneities for a better ensemble.

Original: Our CIVD-based workﬂow enables us to achieve new state-of-the-art results on mini-ImageNet, CUB, and tiered-ImagenNet datasets, with ∼ 2% − 5% improvements upon the next best.
Revised: Our CIVD-based workﬂow enables us to achieve new state-of-the-art results on mini-ImageNet, CUB, and tiered-ImagenNet datasets, with ∼ 2% − 5% improvements upon the next best.

Original: To summarize, CIVD provides a mathematically elegant and geometrically interpretable framework that compensates for extreme data insufﬁciency, prevents overﬁtting, and enables fast geometric ensemble for thousands of individual Dirichlet Tessellations.
Revised: To summarize, CIVD provides a mathematically elegant and geometrically interpretable framework that compensates for extreme data insufﬁciency, prevents overﬁtting, and allows for fast geometric ensemble for thousands of individual VD.

Original: These together make FSL stronger.
Revised: These together make FSL stronger.

Original: Our code is publicly available at GitHub.
Revised: Our code is publicly available at GitHub.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Recent years have witnessed a tremendous success of deep learning in a number of data-intensive applications; one critical reason for which is the vast collection of hand-annotated high-quality data, such as the millions of natural images for visual object recognition (Deng et al., 2009), and the large-scale dermatologist-labelled clinical images for skin cancer classiﬁcation (Esteva et al., 2017).
Revised: Recent years have witnessed a tremendous success of deep learning in a number of data-intensive applications; one critical reason for which is the vast collection of hand-annotated high-quality data, such as the millions of natural images for visual object recognition (Deng et al., 2009).

Original: However, in many real-world applications, such large-scale data acquisition might be difﬁcult and comes at a premium, such as in rare disease diagnosis (Yoo et al., 2021), drug responsing (Ma et al., 2021), and precision oncology (Shariﬁ-Noghabi et al., 2020).
Revised: However, in many real-world applications, such large-scale data acquisition might be difﬁcult and comes at a premium, such as in rare disease diagnosis (Yoo et al., 2021) and drug discovery (Ma et al., 2021).

Original: As a consequence, Few-shot Learning (FSL) has recently drawn growing attentions (Wang et al., 2020).
Revised: As a consequence, Few-shot Learning (FSL) has recently drawn growing interests (Wang et al., 2020).

Original: Generally, few-shot learning algorithms can be categorized into two types, namely inductive and transductive , depending on whether estimating the distribution of query samples is allowed.
Revised: Generally, few-shot learning algorithms can be categorized into two types, namely inductive and transductive , depending on whether estimating the distribution of query samples is allowed.

Original: A typical transductive FSL algorithm learns to propagate labels among a larger pool of query samples in a semi-supervised manner (Liu et al., 2019); notwithstanding its normally higher performance, in many real world scenarios a query sample (e.g. patient) also comes individually and is unique, for instance, in personalized pharmacogenomics (Shariﬁ-Noghabi et al., 2020).
Revised: A typical transductive FSL algorithm learns to propagate labels among a larger pool of query samples in a semi-supervised manner (Liu et al., 2019); notwithstanding its normally higher performance, in many real world scenarios a query sample (e.g. patient) also comes individually and is unique, for instance, in personalized pharmacogenomics (Shariﬁ-Noghabi et al., 2020).

Original: Thus, we in this paper adhere to the inductive setting and make on-the-ﬂy prediction for each newly seen sample.
Revised: Thus, we in this paper adhere to the inductive setting and make on-the-ﬂy prediction for each newly seen sample.

Original: Few-shot learning is challenging and substantially different from conventional deep learning, and has been tackled by many researchers from a wide variety of angles.
Revised: Few-shot learning is challenging and substantially different from conventional deep learning, and has been tackled by many researchers from a wide variety of angles.

Original: While there is extensive research on the algorithmic aspects of FSL (see Sec. 2), the understanding, modeling, and exploitation of the data is relatively overlooked.
Revised: Despite the extensive research on the algorithmic aspects of FSL (see Sec. 2), two challenges still pose an obstacle to successful

Original: In this paper, we focus on the following two questions: (1) how to sufﬁciently compensate for the data deﬁciency in FSL?
Revised: FSL: (1) how to sufﬁciently compensate for the data deﬁciency in FSL?

Original: and (2) how to make the most use of the base samples and the pre-trained model?
Revised: and (2) how to make the most use of the base samples and the pre-trained model?

Original: For the ﬁrst question, data augmentation has been a successful approach to expand the size of data, either by Generative Adversarial Networks (GANs) (Goodfellow et al., 2014)
Revised: For the ﬁrst question, data augmentation has been a successful approach to expand the size of data, either by Generative Adversarial Networks (GANs) (Goodfellow et al., 2014)

Original: (Li et al., 2020b;
Revised: (Li et al., 2020b;

Original: Zhang et al., 2018) or by variational autoencoders  (Kingma & Welling, 2014) (Zhang et al., 2019; Chen et al., 2019b).
Revised: Zhang et al., 2018) or by variational autoencoders (VAEs) (Kingma & Welling, 2014) (Zhang et al., 2019; Chen et al., 2019b).

Original: However, in each way, the authenticity of either the augmented data or its feature is not guaranteed, and the out-of-distribution hallucinated samples may hinder the subsequent FSL.
Revised: However, in each way, the authenticity of either the augmented data or the feature is not guaranteed, and the out-of-distribution hallucinated samples may hinder the subsequent FSL.

Original: Recently, Liu et al. (2020b) and Ni et al.
Revised: Recently, Liu et al. (2020b) and Ni et al.

Original: (2021) investigate support-level, query-level, task-level, and shot-level augmentation for meta-learning, but the diversity of FSL models has not been taken into consideration.
Revised: (2021) investigate support-level, querylevel, task-level, and shot-level augmentation for meta-learning, but the diversity of FSL models has not been taken into consideration.

Original: For the second question, Yang et al.
Revised: For the second question, Yang et al.

Original: (2021) borrows the top-2 nearest base classes for each novel sample to recalibrate its distribution and to generate more novel samples.
Revised: (2021) borrows the top-2 nearest base classes for each novel sample to calibrate its distribution and to generate more novel samples.

Original: However, when there is no proximal base class, this calibration may utterly alter the distribution.
Revised: However, when there is no proximal base class, this calibration may utterly alter the distribution.

Original: Another line of work (Sbai et al., 2020; Zhou et al., 2020) learn to select and design base classes for a better discrimination on novel classes, which all introduce extra training burden.
Revised: Another line of work (Sbai et al., 2020; Zhou et al., 2020) learns to select and design base classes for a better discrimination on novel classes, which all introduce extra training burden.

Original: As a matter of fact, we still lack a method that makes full use of the base classes and the pretrained model effectively.
Revised: As a matter of fact, we still lack a method that makes full use of the base classes and the pretrained model effectively.

Original: In this paper, we study the FSL problem from a geometric point of view.
Revised: In this paper, we study the FSL problem from a geometric point of view.

Original: In metric-based FSL, despite being surprisingly simple, the nearest neighbor-like approaches, e.g. ProtoNet (Snell et al., 2017) and SimpleShot (Wang et al., 2019), have achieved remarkable performance that is even better than many sophisticatedly designed methods.
Revised: In metric-based FSL, despite being surprisingly simple, the nearest neighbor-like approaches, e.g. ProtoNet (Snell et al., 2017) and SimpleShot (Wang et al., 2019), have achieved remarkable performance that is even better than many sophisticatedly designed methods.

Original: Geometrically, what a nearest neighbor-based method does, under the hood, is to partition the feature space into a Voronoi Diagram (VD) that is induced by the feature centroids of the novel classes.
Revised: Geometrically, what a nearest neighbor-based method does, under the hood, is to partition the feature space into a Voronoi Diagram (VD) that is induced by the feature centroids of the novel classes.

Original: Although it is highly efﬁcient and simple, Voronoi
Revised: Although it is highly efﬁcient and simple, Voronoi

Original: Diagrams coarsely draw the decision boundary by linear bisectors separating two centers, and may lack the ability to subtly delineate the geometric structure arises in FSL. 
Revised: Diagrams coarsely draw the decision boundary by linear bisectors separating two centers, and may lack the ability to subtly delineate the geometric structure arises in FSL. [Table]

Original: To resolve this issie, we adopt a novel technique called Cluster-induced Voronoi Diagram (CIVD)
Revised: To resolve this issue, we adopt a novel technique called Cluster-induced Voronoi Diagram (CIVD)

Original: (Chen et al., 2013; 2017; Huang & Xu, 2020; Huang et al., 2021), which is a recent breakthrough in computation geometry.
Revised: (Chen et al., 2013; 2017; Huang & Xu, 2020; Huang et al., 2021), which is a recent breakthrough in computation geometry.

Original: CIVD generalizes VD from a point-to-point distance-based diagram to a cluster-to-point inﬂuence-based structure.
Revised: CIVD generalizes VD from a point-to-point distance-based diagram to a cluster-to-point inﬂuence-based structure.

Original: It enables us to determine the dominating region (or Voronoi cell) not only for a point (e.g. a class prototype) but also for a cluster of points, guaranteed to have a (1 + (cid:15) ) -approximation with a nearly linear size of diagram for a wide range of locally dominating inﬂuence functions.
Revised: It enables us to determine the dominating region (or Voronoi cell) not only for a point (e.g. a class prototype) but also for a cluster of points, guaranteed to have a (1 + (cid:15) ) -approximation with a nearly linear size of diagram for a wide range of locally dominating inﬂuence functions.

Original: CIVD provides us a mathematically elegant framework to depict the feature space and draw the decision boundary more precisely than VD without losing the resistance to overﬁtting.
Revised: CIVD provides us a mathematically elegant framework to depict the feature space and draw the decision boundary more precisely than VD without losing the resistance to overﬁtting.

Original: Accordingly, in this paper, we show how CIVD is used to improve multiple stages of FSL and make several contributions as follows.
Revised: Accordingly, in this paper, we show how CIVD is used to improve multiple stages of FSL and make several contributions as follows.

Original: We ﬁrst categorize different types of few-shot classiﬁers as different variants of Voronoi Diagram: nearest neighbor model as VD, linear classiﬁer as Power Diagram, and cosine classiﬁer as spherical Voronoi Diagram.
Revised: We ﬁrst categorize different types of few-shot classiﬁers as different variants of Voronoi Diagram: nearest neighbor model as Voronoi Diagram, linear classiﬁer as Power Diagram, and cosine classiﬁer as spherical Voronoi Diagram (Table 1).

Original: We then unify them via CIVD that enjoys the advantages of multiple models, either parametric or nonparametric.
Revised: We then unify them via CIVD that enjoys the advantages of multiple models, either parametric or nonparametric (denoted as DeepVoro -- ).

Revised: Going from cluster-to-point to cluster-to-cluster inﬂuence, we further propose Cluster-to-cluster Voronoi Diagram (CCVD) , as a natural extension of CIVD.

Original:  Instead of leveraging base classes for distribution calibration  (Yang et al., 2021), we collect the set of similarities to base classes to form a surrogate representation, and integrate it to the VD model. After identifying feature-level, transformation-level, and geometry-level heterogeneities, we use CIVD to integrate a large pool of thousands of conﬁgurations for FSL.This signiﬁcantly compensates for the data insufﬁciency and prevents overﬁtting.
Revised: Based on CCVD, we present DeepVoro which enables fast geometric ensemble of a large pool of thousands of conﬁgurations for FSL. Instead of using base classes for distribution calibration and data augmentation (Yang et al., 2021), we propose a novel surrogate representation , the collection of similarities to base classes, and thus promote DeepVoro to DeepVoro++ that integrates feature-level, transformation-level, and geometry-level heterogeneities in FSL.

Original: Extensive experiments have shown that, although asingle ﬁxed feature extractor is used without independently pretrained or epoch-wise models, our method achieves new state-of-the-art results on all three benchmark datasets including mini-ImageNet, CUB, and tiered-ImageNet, and improves by up to 2 .
Revised: Extensive experiments have shown that, although a ﬁxed feature extractor is used without independently pretrained or epoch-wise models, our method achieves new state-of-the-art results on all three benchmark datasets including mini-ImageNet, CUB, and tiered-ImageNet, and improves by up to 2 .

Original: 18% on 5-shot classiﬁcation, 2 .
Revised: 18% on 5-shot classiﬁcation, 2 .

Original: 53% on 1-shot classiﬁcation, and up to 5 .
Revised: 53% on 1-shot classiﬁcation, and up to 5 .

Original: 55% with different network architectures.
Revised: 55% with different network architectures.

Revised: [Figure]

Original:  R ELATED W ORK
Revised:  R ELATED W ORK

Original: Few-Shot Learning.
Revised: Few-Shot Learning.

Original: There are a number of different lines of research dedicated to FSL.
Revised: There are a number of different lines of research dedicated to FSL.

Original: (1) Metricbased methods employ a certain distance function (cosine distance (Mangla et al., 2020; Xu et al., [Figure]
Revised: (1) Metricbased methods employ a certain distance function (cosine distance (Mangla et al., 2020; Xu et al., 2021), Euclidean distance (Wang et al., 2019;

Original: (See ) 2021), Euclidean distance (Wang et al., 2019; Snell et al., 2017), or Earth Mover’s Distance (Zhang et al., 2020a;b)) to bypass the optimization and avoid possible overﬁtting.
Revised: Snell et al., 2017), or Earth Mover’s Distance (Zhang et al., 2020a;b)) to bypass the optimization and avoid possible overﬁtting.

Original: (2) Optimization-based approaches (Finn et al., 2017) manages to learn a good model initialization that accelerates the optimization in the meta-testing stage.
Revised: (2) Optimization-based approaches (Finn et al., 2017) manages to learn a good model initialization that accelerates the optimization in the meta-testing stage.

Original: (3) Self-supervised-based (Zhang et al., 2021b; Mangla et al., 2020)
Revised: (3) Self-supervised-based (Zhang et al., 2021b; Mangla et al., 2020)

Original: methods incorporate supervision from data itself to learn a robuster feature extractor.
Revised: methods incorporate supervision from data itself to learn a robuster feature extractor.

Original: (4) Ensemble method is another powerful technique that boosting the performance by integrating multiple models.
Revised: (4) Ensemble method is another powerful technique that boosting the performance by integrating multiple models.

Original: For example, Dvornik et al.
Revised: For example, Dvornik et al.

Original: (2019) trains several networks simultaneously and encourages robustness and cooperation among them.
Revised: (2019) trains several networks simultaneously and encourages robustness and cooperation among them.

Original: However, due to the high computation load of training deep models, this ensemble is restricted by the number of networks which is typically < 20 .
Revised: However, due to the high computation load of training deep models, this ensemble is restricted by the number of networks which is typically < 20 .

Original: In Liu et al.
Revised: In Liu et al.

Original: (2020c), instead, the ensemble consists of models learned at each epoch, which, may potentially limit the diversity of ensemble members.
Revised: (2020c), instead, the ensemble consists of models learned at each epoch, which, may potentially limit the diversity of ensemble members.

Original: Geometric Understanding of Deep Learning.
Revised: Geometric Understanding of Deep Learning.

Original: The geometric structure of deep neural networks is ﬁrst hinted at by Raghu et al.
Revised: The geometric structure of deep neural networks is ﬁrst hinted at by Raghu et al.

Original: (2017) who reveals that piecewise linear activations subdivide input space into convex polytopes.
Revised: (2017) who reveals that piecewise linear activations subdivide input space into convex polytopes.

Original: Then, Balestriero et al.
Revised: Then, Balestriero et al.

Original: (2019) points out that the exact structure is a Power Diagram (Aurenhammer, 1987) which is subsequently applied upon recurrent neural network (Wang et al., 2018) and generative model (Balestriero et al., 2020).
Revised: (2019) points out that the exact structure is a Power Diagram (Aurenhammer, 1987) which is subsequently applied upon recurrent neural network (Wang et al., 2018) and generative model (Balestriero et al., 2020).

Original: The Power/Voronoi
Revised: The Power/Voronoi

Original: Diagram subdivision, however, is not necessarily the optimal model for describing feature space.
Revised: Diagram subdivision, however, is not necessarily the optimal model for describing feature space.

Original: Recently, Chen et al.
Revised: Recently, Chen et al.

Original: ; Huang et al.
Revised: ; Huang et al.

Original: (2021) uses an inﬂuence function F ( C , z ) to measure the joint inﬂuence of all objects in C on a query z to build a Cluster-induced Voronoi Diagram (CIVD).
Revised: (2021) uses an inﬂuence function F ( C , z ) to measure the joint inﬂuence of all objects in C on a query z to build a Cluster-induced Voronoi Diagram (CIVD).

Original: In this paper, we utilize CIVD to magnify the expressivity of geometric modeling for FSL.
Revised: In this paper, we utilize CIVD to magnify the expressivity of geometric modeling for FSL.

Original: High dimension, Low Sample Size Data.

Original: The high dimensionality, low sample size (HDLSS) problem is motivated by problems arise in genetics (Zhou & Troyanskaya, 2015) where the number of genes is much larger than the number of patients.

Original: Liu et al.

Original: (2017) is the ﬁrst attempt on HDLSS by deep models that selects subsets of high dimensional features and average over multiple dropouts to alleviate overﬁtting.

Original: The HDLSS problem, to our knowledge, is not well explored in the context of FSL.

Original: By several novel techniques, the HDLSS problem is much better addressed in this paper.

Original:  M ETHODOLOGY 3. P RELIMINARIES
Revised:  M ETHODOLOGY 3. P RELIMINARIES

Original: Few-shot learning aims at discriminating between novel classes C novel with the aid of a larger amount of samples from base classes C base , C novel ∩C base = ∅ .
Revised: Few-shot learning aims at discriminating between novel classes C novel with the aid of a larger amount of samples from base classes C base , C novel ∩C base = ∅ .

Original: The whole learning process usually follows the meta-learning scheme.
Revised: The whole learning process usually follows the meta-learning scheme.

Original: Formally, given a dataset of base classes D = { ( x i , y i ) } , x i ∈ D , y i ∈ C base with D being an arbitrary domain e.g. natural image, a deep neural network z
Revised: Formally, given a dataset of base classes D = { ( x i , y i ) } , x i ∈ D , y i ∈ C base with D being an arbitrary domain e.g. natural image, a deep neural network z

Original: = φ ( x ) , z ∈ R n , which maps from image domain D to feature domain R n , is trained using standard gradient descent algorithm, and after which φ is ﬁxed as a feature extractor.
Revised: = φ ( x ) , z ∈ R n , which maps from image domain D to feature domain R n , is trained using standard gradient descent algorithm, and after which φ is ﬁxed as a feature extractor.

Original: This process is referred to as metatraining stage that squeezes out the commonsense knowledge from D .
Revised: This process is referred to as metatraining stage that squeezes out the commonsense knowledge from D .

Original: For a fair evaluation of the learning performance on a few samples, the meta-testing stage is typically formulated as a series of K -way N -shot tasks (episodes) {T } .
Revised: For a fair evaluation of the learning performance on a few samples, the meta-testing stage is typically formulated as a series of K -way N -shot tasks (episodes) {T } .

Original: Each such episode is further decomposed into a support set S =
Revised: Each such episode is further decomposed into a support set S =

Original: { ( x i , y i ) } K × N i
Revised: { ( x i , y i ) } K × N i

Original: =
Revised: =

Original: 1 , y i ∈ C T and a query set Q = { ( x i , y i ) } K × Q i =1 , y i ∈
Revised: 1 , y i ∈ C T and a query set Q = { ( x i , y i ) } K × Q i =1 , y i ∈

Original: C T , in which the episode classes C T is a randomly sampled subset of C novel with cardinality K , and each class contains only N and Q random samples in the support set and query set, respectively.
Revised: C T , in which the episode classes C T is a randomly sampled subset of C novel with cardinality K , and each class contains only N and Q random samples in the support set and query set, respectively.

Original: For few-shot classiﬁcation, we introduce here two widely used schemes as follows.
Revised: For few-shot classiﬁcation, we introduce here two widely used schemes as follows.

Original: For simplicity, all samples here are from S and Q , without data augmentation applied.
Revised: For simplicity, all samples here are from S and Q , without data augmentation applied.

Original: Nearest Neighbor Classiﬁer (Nonparametric).
Revised: Nearest Neighbor Classiﬁer (Nonparametric).

Original: In Snell et al.
Revised: In Snell et al.

Original: ; Wang et al.
Revised: ; Wang et al.

Original: (2019) etc., a prototype c k is acquired by averaging over all supporting features for a class k ∈ C T :
Revised: (2019) etc., a prototype c k is acquired by averaging over all supporting features for a class k ∈ C T :

Original: [Equation]
Revised: [Equation]

Original: Then each query sample x ∈ Q is classiﬁed by ﬁnding the nearest prototype:
Revised: Then each query sample x ∈ Q is classiﬁed by ﬁnding the nearest prototype:

Original: [Equation]
Revised: [Equation]

Original: arg max k d ( z , c k ) =
Revised: arg max k d ( z , c k ) =

Original: ||
Revised: ||

Original: z − c k || 2 2 , in which we use Euclidean distance for distance metric d .
Revised: z − c k || 22 , in which we use Euclidean distance for distance metric d .

Original: Linear Classiﬁer (Parametric).
Revised: Linear Classiﬁer (Parametric).

Original: Another scheme uses a linear classiﬁer with cross-entropy loss optimized on the supporting samples:
Revised: Another scheme uses a linear classiﬁer with cross-entropy loss optimized on the supporting samples:

Original: [Equation]
Revised: [Equation]

Original: in which W k , b k are the linear weight and bias for class k , and the predicted class for query x ∈ Q is ˆ y = arg max k p ( y | z ; W k , b k ) .
Revised: in which W k , b k are the linear weight and bias for class k , and the predicted class for query x ∈ Q is ˆ y = arg max k p ( y | z ; W k , b k ) .

Original: 3. F EW - SHOT LEARNING AS  D IRICHLET T ESSELLATION
Revised: 3. F EW - SHOT LEARNING AS C LUSTER - INDUCED V ORONOI D IAGRAMS

Original: In this section, we ﬁrst introduce the basic concepts of Dirichlet Tessellation (DT), and then show how parametric/nonparametric classiﬁer heads can be uniﬁed by DT.
Revised: In this section, we ﬁrst introduce the basic concepts of Voronoi Tessellations, and then show how parametric/nonparametric classiﬁer heads can be uniﬁed by VD.

Original: Deﬁnition 3.1 (Dirichlet Tessellation) .
Revised: Deﬁnition 3.1 (Power Diagram and Voronoi Diagram) .

Original: Let Ω = { ω 1 , ..., ω K } be a partition of the space R n , and C = { c 1 , ..., c K } be a set of centers such that ∪ Kr =1 ω r = R n , ∩ Kr =1 ω r = ∅ .
Revised: Let Ω = { ω 1 , ..., ω K } be a partition of the space R n , and C = { c 1 , ..., c K } be a set of centers such that ∪ Kr =1 ω r = R n , ∩ Kr =1 ω r = ∅ .

Original: Additionally, each center is associated with a weight ν r ∈ { ν 1 , ..., ν K } ⊆ R + .
Revised: Additionally, each center is associated with a weight ν r ∈ { ν 1 , ..., ν K } ⊆ R + .

Original: Then the set of pairs { ( ω 1 , c 1 , ν 1 ) , ..., ( ω K , c L , ν K ) } is asectional Dirichlet Tessellation, also known as Power Diagram (PD), where each cell is obtained via ω r = { z ∈ R n : r ( z ) =
Revised: Then the set of pairs { ( ω 1 , c 1 , ν 1 ) , ..., ( ω K , c L , ν K ) } is a Power Diagram (PD), where each cell is obtained via ω r = { z ∈ R n : r ( z ) =

Original: r } , r ∈ { 1 , .., K } , with
Revised: r } , r ∈ { 1 , .., K } , with

Original: [Equation]
Revised: [Equation]

Original: If the weights are equal for all k , i.e. ν k 
Revised: If the weights are equal for all k , i.e. ν k =

Revised: [Equation]

Original: = ν k (cid:48) , ∀ k, k (cid:48) ∈ { 1 , ..., K } , then a PD collapses to a 
Revised: k (cid:48) , ∀ k, k (cid:48) ∈ { 1 , ..., K } , then a PD collapses to a Voronoi

Original: Dirichlet Tessellation, also known as Voronoi Diagram (VD).
Revised: Diagram (VD).

Original: By deﬁnition, it is easy to see that the nearest neighbor classiﬁer naturally partitions the space into K cells with centers { c 1 , ..., c K } .
Revised: By deﬁnition, it is easy to see that the nearest neighbor classiﬁer naturally partitions the space into K cells with centers { c 1 , ..., c K } .

Original: Here we show that the linear classiﬁer is also a Dirichlet Tessellation under a mild condition.
Revised: Here we show that the linear classiﬁer is also a VD under a mild condition.

Original: Lemma 3.1.

Original: The vertical projection from the lower envelope of the hyperplanes { Π k ( z ) : W Tk z + b k } Kk =1 onto the input space R n deﬁnes the cells of a PD.

Original: Theorem 3.1 (Voronoi Diagram Reduction) .
Revised: Theorem 3.1 (Voronoi Diagram Reduction) .

Original: The linear classiﬁer parameterized by W , b partitions the input space R n to a Voronoi Diagram with centers { ˜ c 1 , ..., ˜ c K } given by ˜ c
Revised: The linear classiﬁer parameterized by W , b partitions the input space R n to a Voronoi Diagram with centers { ˜ c 1 , ..., ˜ c K } given by ˜ c

Original: k = 12 W k if b k = − 

Revised: k = 12 W k if b k = −

Original: || W k
Revised: || W k

Original: || 22 , k = 1 , ..., K .
Revised: || 22 , k = 1 , ..., K .

Original: Proof.
Revised: Proof.

Original: See Appendix A for details.
Revised: See Appendix B for details.

Original: 3.2. F ROM V ORONOI D IAGRAM TO C LUSTER - INDUCED V ORONOI D IAGRAM
Revised: 3.2. F ROM V ORONOI D IAGRAM TO C LUSTER - INDUCED V ORONOI D IAGRAM

Original: Now that both nearest neighbor and linear classiﬁer have been uniﬁed by Dirichlet Tessellation, a natural idea is to integrate them together.
Revised: Now that both nearest neighbor and linear classiﬁer have been uniﬁed by VD, a natural idea is to integrate them together.

Original: Cluster-induced Voronoi Diagram (CIVD)
Revised: Cluster-induced Voronoi Diagram (CIVD)

Original: (Chen et al., 2017; Huang et al., 2021) is a generalization of VD which allows multiple centers in a cell, and is successfully used for clinical diagnosis from biomedical images (Wang et al., 2015), providing an ideal tool for the integration of parametric/nonparametric classiﬁer for FSL.
Revised: (Chen et al., 2017; Huang et al., 2021) is a generalization of VD which allows multiple centers in a cell, and is successfully used for clinical diagnosis from biomedical images (Wang et al., 2015), providing an ideal tool for the integration of parametric/nonparametric classiﬁer for FSL.

Original: Formally:
Revised: Formally:

Original: Deﬁnition 3.2 (Cluster-induced Voronoi Diagram (CIVD)
Revised: Deﬁnition 3.2 (Cluster-induced Voronoi Diagram (CIVD)

Original: (Chen et al., 2017; Huang et al., 2021)) .
Revised: (Chen et al., 2017; Huang et al., 2021)) .

Original: Let Ω = { ω 1 , ..., ω K } be a partition of the space R n , and
Revised: Let Ω = { ω 1 , ..., ω K } be a partition of the space R n , and

Original: C =
Revised: C =

Original: {C 1 , ..., C K } be a set (possibly a multiset) of clusters .
Revised: {C 1 , ..., C K } be a set (possibly a multiset) of clusters .

Original: The set of pairs { ( ω 1 , C 1 ) , ..., ( ω K , C K ) } is a Cluster-induced Voronoi Diagram (CIVD) with respect to the inﬂuence function F ( C k , z ) , where each cell is obtained via ω r = { z ∈ R n : r ( z ) =
Revised: The set of pairs { ( ω 1 , C 1 ) , ..., ( ω K , C K ) } is a Cluster-induced Voronoi Diagram (CIVD) with respect to the inﬂuence function F ( C k , z ) , where each cell is obtained via ω r = { z ∈ R n : r ( z ) =

Original: r } , r ∈ { 1 , .., K } , with
Revised: r } , r ∈ { 1 , .., K } , with

Original: [Equation]
Revised: [Equation]

Original: Here C can be either a given set of clusters or even the whole power set of a given point set, and the inﬂuence function is deﬁned as a function over the collection of distances from each member in a cluster C k to a query point z :
Revised: Here C can be either a given set of clusters or even the whole power set of a given point set, and the inﬂuence function is deﬁned as a function over the collection of distances from each member in a cluster C k to a query point z :

Original: Deﬁnition 3.3 (Inﬂuence Function) .
Revised: Deﬁnition 3.3 (Inﬂuence Function) .

Original: The inﬂuence from C k , k ∈ { 1 , ..., K } to z / ∈ C
Revised: The inﬂuence from C k , k ∈ { 1 , ..., K } to z / ∈ C

Original: k is F ( C k , z ) =
Revised: k is F ( C k , z ) =

Original: F ( { d ( c ( i ) k , z ) | c ( i ) k ∈ C k } |C k | i =1 ) .
Revised: F ( { d ( c ( i ) k , z ) | c ( i ) k ∈ C k } |C k | i =1 ) .

Original: In this paper F is assumed to have the following form
Revised: In this paper F is assumed to have the following form

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: The sign function here makes sure that F is a monotonically decreasing function with respect to distance d .
Revised: The sign function here makes sure that F is a monotonically decreasing function with respect to distance d .

Original: The hyperparameter α controls the magnitude of the inﬂuence, for example, in gravity force α = − ( n − 1) in n -dimensional space and in electric force α = − 2 .
Revised: The hyperparameter α controls the magnitude of the inﬂuence, for example, in gravity force α = − ( n − 1) in n -dimensional space and in electric force α = − 2 .

Original: Since the nearest neighbor centers { c k } Kk =1 and the centers introduced by linear classiﬁer { ˜ c k } Kk = are obtained from different schemes and could both be informative, we merge the corresponding centers for a novel class k to be a new cluster C k
Revised: Since the nearest neighbor centers { c k } Kk =1 and the centers introduced by linear classiﬁer { ˜ c k } Kk = are obtained from different schemes and could both be informative, we merge the corresponding centers for a novel class k to be a new cluster C k

Original: = { c k , ˜ c k } , and use the resulting C
Revised: = { c k , ˜ c k } , and use the resulting C

Original: = {C 1 , ..., C K } to establish a CIVD.
Revised: = {C 1 , ..., C K } to establish a CIVD.

Original: In such a way, the ﬁnal partition may enjoy the advantages of both parametric and nonparametric classiﬁer heads.
Revised: In such a way, the ﬁnal partition may enjoy the advantages of both parametric and nonparametric classiﬁer heads.

Revised: We name this approach as DeepVoro -- .

Original: 3. F EW - SHOT C LASSIFICATION VIA S URROGATE R EPRESENTATION
Revised: 3. F EW - SHOT C LASSIFICATION VIA S URROGATE R EPRESENTATION

Original: In nearest neighbor classiﬁer head, the distance from a query feature z to each of the prototypes { c k } Kk =1 is the key discrimination criterion for classiﬁcation.
Revised: In nearest neighbor classiﬁer head, the distance from a query feature z to each of the prototypes { c k } Kk =1 is the key discrimination criterion for classiﬁcation.

Original: We rewrite { d ( z , c k ) }
Revised: We rewrite { d ( z , c k ) }

Original: Kk =1 to be a vector d ∈ R K such that d k =
Revised: Kk =1 to be a vector d ∈ R K such that d k =

Original: d
Revised: d

Original: ( z , c k ) .
Revised: ( z , c k ) .

Original: These distances are acquired by measure the distance between two points in high dimension: z , c k ∈ R n .
Revised: These distances are acquired by measure the distance between two points in high dimension: z , c k ∈ R n .

Original: However, the notorious behavior of high dimension is that the ratio between the nearest and farthest points in a point set P approaches 1 (Aggarwal et al., 2001), making { d ( z , c k ) }
Revised: However, the notorious behavior of high dimension is that the ratio between the nearest and farthest points in a point set P approaches 1 (Aggarwal et al., 2001), making { d ( z , c k ) }

Original: Kk =1 less discriminative for classiﬁcation, especially for FSL problem with sample size N · K (cid:28) n .
Revised: Kk =1 less discriminative for classiﬁcation, especially for FSL problem with sample size N · K (cid:28) n .

Original: Hence, in this paper, we seek for a surrogate representation .
Revised: Hence, in this paper, we seek for a surrogate representation .

Original: In human perception and learning system, similarity among familiar and unfamiliar objects play a key role for object categorization and classiﬁcation (Murray et al., 2002), and it has been experimentally veriﬁed by functional magnetic resonance imaging (fMRI) that a large region in occipitotemporal cortex processes the shape of both meaningful and unfamiliar objects (Op de Beeck et al., 2008).
Revised: In human perception and learning system, similarity among familiar and unfamiliar objects play a key role for object categorization and classiﬁcation (Murray et al., 2002), and it has been experimentally veriﬁed by functional magnetic resonance imaging (fMRI) that a large region in occipitotemporal cortex processes the shape of both meaningful and unfamiliar objects (Op de Beeck et al., 2008).

Original: In our method,intuitively, a connection will be built between each unfamiliar novel class in C novel and each related well-perceived familiar class in C base .
Revised: In our method, a connection will be built between each unfamiliar novel class in C novel and each related well-perceived familiar class in C base .

Original: So the ﬁrst step is to identify the most relevant base classes for a speciﬁc task T .
Revised: So the ﬁrst step is to identify the most relevant base classes for a speciﬁc task T .

Original: Concretely:
Revised: Concretely:

Original: Deﬁnition 3.4 (Surrogate Classes) .
Revised: Deﬁnition 3.4 (Surrogate Classes) .

Original: In episode T , given the set of prototypes { c k } Kk =1 for the support set S and the set of prototypes { c (cid:48) t } |C base | t =1 for the base set D , the surrogate classes for episode classes
Revised: In episode T , given the set of prototypes { c k } Kk =1 for the support set S and the set of prototypes { c (cid:48) t } |C base | t =1 for the base set D , the surrogate classes for episode classes

Original: C T is given as:
Revised: C T is given as:

Original: [Equation]
Revised: [Equation]

Original: (6) in which the top- R function returns R base class indices with smallest distances to c k , and the center for a base class t is given as c (cid:48) t = 
Revised: in which the top- R function returns R base class indices with smallest distances to c k , and the center for a base class t is given as c (cid:48) t = [Equation] =

Revised: [Equation]

Revised: φ

Original: [Equation]
Revised: [Equation]

Revised: x ) .

Original: Here R is a hyperparameter.
Revised: Here R is a hyperparameter.

Original: The rationale behind this selection instead of simply using the whole base classes C base is that, the episode classes C T are only overlapped with a portion of base classes (Zhang et al., 2021a), and discriminative similarities are likely to be overwhelmed by the background signal especially when the number of base classes is large.
Revised: The rationale behind this selection instead of simply using the whole base classes C base is that, the episode classes C T are only overlapped with a portion of base classes (Zhang et al., 2021a), and discriminative similarities are likely to be overwhelmed by the background signal especially when the number of base classes is large.

Original: After the surrogate classes are found, we re-index their feature centers to be { c (cid:48)
Revised: After the surrogate classes are found, we re-index their feature centers to be { c (cid:48)

Original: j } ˜
Revised: j } ˜

Original: Rj =1 , ˜ R ≤ R · K .
Revised: Rj =1 , ˜ R ≤ R · K .

Original: Then, both support centers { c k } Kk =1 and query feature z are represented by the collection of similarities to these surrogate centers:
Revised: Then, both support centers { c k } Kk =1 and query feature z are represented by the collection of similarities to these surrogate centers:

Original: [Equation]
Revised: [Equation]

Original: where d (cid:48) k , d (cid:48) ∈ R ˜
Revised: where d (cid:48) k , d (cid:48) ∈ R ˜

Original: R are the surrogate representation for novel class k and query feature z , respectively.
Revised: R are the surrogate representation for novel class k and query feature z , respectively.

Original: By surrogate representation, the prediction is found through ˆ y = arg max k d ( d (cid:48) , d (cid:48) k ) =
Revised: By surrogate representation, the prediction is found through ˆ y = arg max k d ( d (cid:48) , d (cid:48) k ) =

Original: arg max k || d (cid:48) − d (cid:48)
Revised: arg max k || d (cid:48) − d (cid:48)

Original: k
Revised: k

Original: || 22 .
Revised: || 22 .

Original: This set of discriminative distances is rewritten as d (cid:48)(cid:48) ∈ R K such that d (cid:48)(cid:48) k
Revised: This set of discriminative distances is rewritten as d (cid:48)(cid:48) ∈ R K such that d (cid:48)(cid:48) k

Original: = d ( d (cid:48) , d (cid:48) k ) .
Revised: = d ( d (cid:48) , d (cid:48) k ) .

Original: An illustration of the surrogate representation is shown in Figure 1 on MultiDigitMNIST, a demonstrative dataset.
Revised: An illustration of the surrogate representation is shown in Figure 1 on MultiDigitMNIST, a demonstrative dataset.

Original: Integrating Feature Representation and Surrogate Representation.
Revised: Integrating Feature Representation and Surrogate Representation.

Original: Until now, we have two discriminative systems, i.e., feature-based d ∈ R K and surrogate-based d (cid:48)(cid:48) ∈ R K .
Revised: Until now, we have two discriminative systems, i.e., feature-based d ∈ R K and surrogate-based d (cid:48)(cid:48) ∈ R K .

Original: A natural idea is to combine them to form the following ﬁnal criterion:
Revised: A natural idea is to combine them to form the following ﬁnal criterion:

Original: [Equation]
Revised: [Equation]

Revised: [Equation]

Original: where d and d (cid:48)(cid:48) are normalized by their Manhattan norm, || d
Revised: where d and d (cid:48)(cid:48) are normalized by their Manhattan norm, || d

Original: || 

Revised: ||

Original: =
Revised: =

Original: (cid:80)
Revised: (cid:80)

Original: Kk =1 d k and || d (cid:48)(cid:48)
Revised: Kk =1 d k and || d (cid:48)(cid:48)

Original: ||
Revised: ||

Original: 1 = (cid:80)
Revised: 1 = (cid:80)

Original: Kk =1 d (cid:48)(cid:48) k , respectively [Figure] and β and γ are two hyperparameters adjusting the weights for feature representation and surrogate representation [Figure]
Revised: Kk =1 d (cid:48)(cid:48) k , respectively, and β and γ are two hyperparameters adjusting the weights for feature representation and surrogate representation.

Original: 3. D IRICHLET T ESSELLATION E NSEMBLE
Revised: 3. D EEP V ORO : I NTEGRATING M ULTI - LEVEL H ETEROGENEITY OF FSL

Original: In this section we describe a fast geometric ensemble framework that unites our contributions to multiple stages of FSL. Before the construction and ensemble of geometric structures for the subdivision of the feature space
Revised: In this section we present DeepVoro, a fast geometric ensemble framework that unites our contributions to multiple stages of FSL, and show how it can be promoted to DeepVoro++ by incorporating surrogate representation.

Original: [Figure] we ﬁrst introduce a series of transformations applied on the features, which will increase the polymorphism of the FSL process and thus assist ensemble [Figure] 3.4. P ARAMETERIZED F EATURE T RANSFORMATION
Revised: Compositional Feature Transformation.

Original: It is believed that FSL algorithms favor features with more Gaussian-like distributions, and thus various kinds of transformations are used to improve the normality of feature distribution, including power transformation (Hu et al., 2021), Tukey’s Ladder of Powers Transformation (Yang et al., 2021), and L2 normalization (Wang et al., 2019).
Revised: It is believed that FSL algorithms favor features with more Gaussian-like distributions, and thus various kinds of transformations are used to improve the normality of feature distribution, including power transformation (Hu et al., 2021), Tukey’s Ladder of Powers Transformation (Yang et al., 2021), and L2 normalization (Wang et al., 2019).

Original: While these transformations are normally used independently, here we propose to combine several transformations sequentially in order to enlarge the expressivityand diversity of transformation function.
Revised: While these transformations are normally used independently, here we propose to combine several transformations sequentially in order to enlarge the expressivity of transformation function and to increase the polymorphism of the FSL process.

Original: Speciﬁcally, for a feature vector z , three kinds of transformations are considered: (I) L2 Normalization.
Revised: Speciﬁcally, for a feature vector z , three kinds of transformations are considered: (I) L2 Normalization.

Original: By projection onto the unit sphere in R n , the feature is normalized as: f ( z ) =
Revised: By projection onto the unit sphere in R n , the feature is normalized as: f ( z ) =

Original: z ||
Revised: z ||

Original: z
Revised: z

Original: || 2 .
Revised: || 2 .

Original: (II) Linear Transformation.
Revised: (II) Linear Transformation.

Original: Now since all the features are located on the unit sphere, we then can do scaling and shifting via a linear transformation: g w,b ( z ) =
Revised: Now since all the features are located on the unit sphere, we then can do scaling and shifting via a linear transformation: g w,b ( z ) =

Original: w z + b .
Revised: w z + b .

Original: (III) Tukey’s Ladder of Powers Transformation.
Revised: (III) Tukey’s Ladder of Powers Transformation.

Original: Finally, Tukey’s Ladder of Powers Transformation is applied on the feature: h λ ( z ) = (cid:26) z λ if λ (cid:54)
Revised: Finally, Tukey’s Ladder of Powers Transformation is applied on the feature: h λ ( z ) = (cid:26) z λ if λ (cid:54)

Original: =log( z ) if λ = 0 .
Revised: =log( z ) if λ = 0 .

Original: Transformation Composition.

Original: By the composition of L2 normalization, linear transformation, and Tukey’s Ladder of Powers Transformation, now the transformation function becomes ( h λ ◦ g w,b ◦ f )( z ) parameterized by w, b, λ .
Revised: By the composition of L2 normalization, linear transformation, and Tukey’s Ladder of Powers Transformation, now the transformation function becomes ( h λ ◦ g w,b ◦ f )( z ) parameterized by w, b, λ .

Original: 3.4. M ULTI - LEVEL H ETEROGENEITIES IN FSL
Revised: Multi-level Heterogeneities in FSL.

Original: Now we are ready to articulate the hierarchical heterogeneity existing in different stages of FSL.
Revised: Now we are ready to articulate the hierarchical heterogeneity existing in different stages of FSL.

Original: (I) Feature-level Heterogeneity : Data augmentation has been exhaustively explored for expanding the data size of FSL (Ni et al., 2021), including but not limited to rotation, ﬂipping, cropping, erasing, solarization, color jitter, MixUp (Zhang et al., 2017), etc.
Revised: (I) Feature-level Heterogeneity : Data augmentation has been exhaustively explored for expanding the data size of FSL (Ni et al., 2021), including but not limited to rotation, ﬂipping, cropping, erasing, solarization, color jitter, MixUp (Zhang et al., 2017), etc.

Original: The modiﬁcation of image x will change the position of feature z in the feature space.
Revised: The modiﬁcation of image x will change the position of feature z in the feature space.

Original: We denote all possible translations of image as a set of functions { T } .
Revised: We denote all possible translations of image as a set of functions { T } .

Original: (II) Transformation-level Heterogeneity :
Revised: (II) Transformation-level Heterogeneity :

Original: After obtaining the feature z , a parameterized transformation is applied to it, and the resulting features can be quite different for these parameters (see Figure E.1).
Revised: After obtaining the feature z , a parameterized transformation is applied to it, and the resulting features can be quite different for these parameters (see Figure F.1).

Original: We denote the set of all possible transformations to be { P w,b,λ } .
Revised: We denote the set of all possible transformations to be { P w,b,λ } .

Original: (III) Geometry-level Heterogeneity : Even with the provided feature, the few-shot classiﬁcation model can still be diverse: whether a VD or PD-based model is used, whether the feature or the surrogate representation is adopted, and the setting of R will also change the degree of locality.
Revised: (III) Geometry-level Heterogeneity : Even with the provided feature, the few-shot classiﬁcation model can still be diverse: whether a VD or PD-based model is used, whether the feature or the surrogate representation is adopted, and the setting of R will also change the degree of locality.

Original: We denote all possible models as { M } .
Revised: We denote all possible models as { M } .

Original: 3.4. E NSEMBLE OF D IRICHLET T ESSELLATIONS
Revised: DeepVoro for Fast Geometric Ensemble of VDs.

Original: With the above three-layer heterogeneity, the FSL process can be encapsulated as ( M ◦ P w,b,λ ◦ φ ◦ T )( x ) , and all possible conﬁgurations of a given episode T with a ﬁxed φ is the Cartesian product of these three sets: { T } × { P w,b,λ } × { M } .
Revised: With the above three-layer heterogeneity, the FSL process can be encapsulated as ( M ◦ P w,b,λ ◦ φ ◦ T )( x ) , and all possible conﬁgurations of a given episode T with a ﬁxed φ is the Cartesian product of these three sets: { T }×{ P w,b,λ }×{ M } .

Original: Indeed, when a hold-out validation dataset is available, it can be used to ﬁnd the optimal combination, but by virtue of ensemble learning, multiple models can still contribute positively to FSL (Dvornik et al., 2019).
Revised: Indeed, when a hold-out validation dataset is available, it can be used to ﬁnd the optimal combination, but by virtue of ensemble learning, multiple models can still contribute positively to FSL (Dvornik et al., 2019).

Original: Since the cardinality of the resulting conﬁguration set could be very large, the FSL model M as well as the ensemble algorithm is required to be highly efﬁcient.
Revised: Since the cardinality of the resulting conﬁguration set could be very large, the FSL model M as well as the ensemble algorithm is required to be highly efﬁcient.

Original: The VD is a nonparametric model and no training is needed during the meta-testing stage, making it suitable for fast geometric ensemble.
Revised: The VD is a nonparametric model and no training is needed during the meta-testing stage, making it suitable for fast geometric ensemble.

Original: While CIVD models the cluster-to-point relationship via an inﬂuence function, here we further extend it so that cluster-to-cluster relationship can be considered.
Revised: While CIVD models the cluster-to-point relationship via an inﬂuence function, here we further extend it so that cluster-to-cluster relationship can be considered.

Original: This motivates us to deﬁneCluster-to-cluster
Revised: This motivates us to deﬁne

Original:  Voronoi Diagram (CCVD):
Revised: Cluster-to-cluster Voronoi Diagram (CCVD):

Original: Deﬁnition 3.5 (Cluster-to-cluster Voronoi Diagram) .
Revised: Deﬁnition 3.5 (Cluster-to-cluster Voronoi Diagram) .

Original: Let Ω = { ω 1 , ..., ω K } be a partition of the space R n , and
Revised: Let Ω = { ω 1 , ..., ω K } be a partition of the space R n , and

Original: C =
Revised: C =

Original: {C 1 , ..., C K } be a set of totally ordered sets with the same cardinality L (i.e.
Revised: {C 1 , ..., C K } be a set of totally ordered sets with the same cardinality L (i.e.

Original: |C 1 | =
Revised: |C 1 | =

Original: |C 2 | = ...
Revised: |C 2 | = ...

Original: =
Revised: =

Original: |C K | =
Revised: |C K | =

Original: L ).
Revised: L ).

Original: The set of pairs { ( ω 1 , C 1 ) , ..., ( ω K , C K ) } is a Cluster-to-cluster
Revised: The set of pairs { ( ω 1 , C 1 ) , ..., ( ω K , C K ) } is a Cluster-to-cluster

Original: Voronoi Diagram (CCVD) with respect to the inﬂuence function F ( C k , C ( z )) , and each cell is obtained via ω r = { z ∈ R n : r ( z ) = r } , r ∈ { 1 , .., K } , with
Revised: Voronoi Diagram (CCVD) with respect to the inﬂuence function F ( C k , C ( z )) , and each cell is obtained via ω r = { z ∈ R n : r ( z ) = r } , r ∈ { 1 , .., K } , with

Original: [Equation]
Revised: [Equation]

Original: (9) where C ( z ) is the cluster (also a totally ordered set with cardinality L ) that query point z belongs, which is to say, all points in this cluster (query cluster) will be assigned to the same cell.
Revised: (9) where C ( z ) is the cluster (also a totally ordered set with cardinality L ) that query point z belongs, which is to say, all points in this cluster (query cluster) will be assigned to the same cell.

Original: Similarly, the Inﬂuence Function is deﬁned upon two totally ordered sets C k
Revised: Similarly, the Inﬂuence Function is deﬁned upon two totally ordered sets C k

Original: = { c ( i ) k } Li =1 and
Revised: = { c ( i ) k } Li =1 and

Original: C ( z ) =
Revised: C ( z ) =

Original: { z ( i 
Revised: { z ( i ) } Li

Original: [Equation]
Revised: [Equation]

Original: Li =1 :

Original: [Equation]
Revised: [Equation]

Original: With this deﬁnition, now we are able to streamline our aforementioned novel approaches into a single ensemble model.
Revised: With this deﬁnition, now we are able to streamline our aforementioned novel approaches into a single ensemble model.

Original: Suppose there are totally L possible settings in our conﬁguration pool { T }×{ P w,b,λ }×{ M } , for all conﬁgurations { ρ i } Li =1 , we apply them onto the support set S to generate the K totally ordered clusters {{ c ( ρ i ) k } Li =1 } Kk =1 including each center c ( ρ i ) k acquired through conﬁguration
Revised: Suppose there are totally L possible settings in our conﬁguration pool { T } × { P w,b,λ } × { M } , for all conﬁgurations { ρ i } Li =1 , we apply them onto the support set S to generate the K totally ordered clusters {{ c ( ρ i ) k } Li =1 } Kk =1 including each center c ( ρ i ) k derived through conﬁguration

Original: ρ i , and onto a query sample x to generate the query cluster C ( z ) =
Revised: ρ i , and onto a query sample x to generate the query cluster C ( z ) =

Original: { z ( ρ 1 ) , ..., z ( ρ L ) } , and then plug these two into Deﬁnition 3.5 to construct the ﬁnal Voronoi Diagram.
Revised: { z ( ρ 1 ) , ..., z ( ρ L ) } , and then plug these two into Deﬁnition 3.5 to construct the ﬁnal Voronoi Diagram.

Revised: When only the feature representation is considered in the conﬁguration pool, i.e. ρ i ∈ { T } × { P w,b,λ } , our FSL process is named as DeepVoro; if surrogate representation is also incorporated, i.e. ρ i ∈ { T } × { P w,b,λ } × { M } , DeepVoro is promoted to DeepVoro++ that allows for higher geometric diversity.

Revised: See Appendix A for a summary of the notations and acronyms

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: [Table]
Revised: [Table]

Original: The main goals of our experiments are to: (1) validate the strength of CIVD to integrate parametric and nonparametric classiﬁers and conﬁrm the necessity of Voronoi reduction; (2) investigate how different levels of heterogeneity individually or collaboratively contribute to the overall result, and compare them with the state-of-art method; (3) reanalyze this ensemble when the surrogate representation comes into play, and see how it could ameliorate the extreme shortage of support samples.
Revised: The main goals of our experiments are to: (1) validate the strength of CIVD to integrate parametric and nonparametric classiﬁers and conﬁrm the necessity of Voronoi reduction; (2) investigate how different levels of heterogeneity individually or collaboratively contribute to the overall result, and compare them with the state-of-art method; (3) reanalyze this ensemble when the surrogate representation comes into play, and see how it could ameliorate the extreme shortage of support samples.

Original: See  Appendix C for the detailed descriptions of mini-ImageNet (Vinyals et al., 2016), CUB (Welinder et al., 2010), and tiered-ImageNet (Ren et al., 2018), that are used in this paper.
Revised: See Table 2 for a summary and Appendix D for the detailed descriptions of mini-ImageNet (Vinyals et al., 2016), CUB (Welinder et al., 2010), and tiered-ImageNet (Ren et al., 2018), that are used in this paper.

Original:  Integrating Parametric and Nonparametric Methods via CIVD.
Revised: DeepVoro -- : Integrating Parametric and Nonparametric Methods via CIVD.

Original: To verify our proposed CIVD model for the integration of parameter/nonparametric FSL classiﬁers, we ﬁrst run three standalone models: Logistic Regressions with Power/Voronoi Diagrams as the underlining geometric structure (Power-LR/Voronoi-LR), and vanilla Voronoi Diagram (VD, i.e. nearest neighbor model), and then integrate VD with either Power/Voronoi-LR (see Appendix D for details).
Revised: To verify our proposed CIVD model for the integration of parameter/nonparametric FSL classiﬁers, we ﬁrst run three standalone models: Logistic Regressions with Power/Voronoi Diagrams as the underlining geometric structure (Power-LR/Voronoi-LR), and vanilla Voronoi Diagram (VD, i.e. nearest neighbor model), and then integrate VD with either Power/Voronoi-LR (see Appendix E for details).

Original: Interestingly, VD with the Power-LR has never reached the best result, suggesting that ordinary LR cannot be integrated with VD due to their intrinsic distinct geometric structures.
Revised: Interestingly, VD with the Power-LR has never reached the best result, suggesting that ordinary LR cannot be integrated with VD due to their intrinsic distinct geometric structures.

Original: After the proposed Voronoi reduction (Theorem 3.1), however, VD+Voronoi-LR is able to improve upon both models in most cases, suggesting that CIVD can ideally integrate parameter and nonparametric models for better FSL. 
Revised: After the proposed Voronoi reduction (Theorem 3.1), however, VD+Voronoi-LR is able to improve upon both models in most cases, suggesting that CIVD can ideally integrate parameter and nonparametric models for better FSL. [Table]

Original:  Improving FSL by Hierarchical Heterogeneities.
Revised: DeepVoro: Improving FSL by Hierarchical Heterogeneities.

Original: In this section, we only consider two levels of heterogeneities for ensemble, feature-level and transformation-level ensemble.
Revised: In this section, we only consider two levels of heterogeneities for ensemble: feature-level and transformation-level.

Original: For feature-level ensemble, we utilize three kinds of image augmentations: rotation, ﬂipping, and central cropping summing up to 64 distinct ways of data augmentation (Appendix E).
Revised: For feature-level ensemble, we utilize three kinds of image augmentations: rotation, ﬂipping, and central cropping summing up to 64 distinct ways of data augmentation (Appendix F).

Original: For transformation-level ensemble, we use the proposed compositional transformations with 8 different combinations of λ and b that encourage a diverse feature transformations (Appendix E.1) without loss of accuracy (Figure 2).
Revised: For transformation-level ensemble, we use the proposed compositional transformations with 8 different combinations of λ and b that encourage a diverse feature transformations (Appendix F.1) without loss of accuracy (Figure 2).

Original: The cardinality of the resulting conﬁguration pool becomes 512 and the ensemble result is shown in Table 1, denoted by DeepVoro.
Revised: The size of the resulting conﬁguration pool becomes 512 and DeepVoro’s performance is shown in Table 3.

Original: Clearly, DeepVoro outperforms all previousstate-of-the-art methods especially on 5-way 5-shot FSL.
Revised: Clearly, DeepVoro outperforms all previous methods especially on 5-way 5-shot FSL.

Original: Speciﬁcally, DeepVoro is better than the next best by 2 .
Revised: Speciﬁcally, DeepVoro is better than the next best by 2 .

Original: 18% (than Ni et al.
Revised: 18% (than Ni et al.

Original: (2021)) on mini-ImageNet, by 1 . 47% (than Hu et al.
Revised: on miniImageNet, by 1 . 47% (than Hu et al.

Original: (2021)) on CUB, and by 1 .
Revised: (2021)) on CUB, and by 1 .

Original: 02% (than Yang et al.
Revised: 02% (than Yang et al.

Original: on tiered-ImageNet.
Revised: on tiered-ImageNet.

Original: Note that this is an estimated improvement because not all competitive methods here are tested with the same random seed and the number of episodes.
Revised: Note that this is an estimated improvement because not all competitive methods here are tested with the same random seed and the number of episodes.

Original: More detailed results can be found in Appendix E.
Revised: More detailed results can be found in Appendix F.

Original: By virtue of CIVD and using the simplest VD as the building block, DeepVoro is arguably able to achieve a consistently better result by the ensemble of a massive pool of independent Dirichlet Tessellations.
Revised: By virtue of CCVD and using the simplest VD as the building block, DeepVoro is arguably able to yield a consistently better result by the ensemble of a massive pool of independent VD.

Revised: DeepVoro also exhibits high resistance to outliers, as shown in Figure K.16.

Original:  Further Improvement of FSL via Surrogate Representation.
Revised: DeepVoro++: Further Improvement of FSL via Surrogate Representation.

Original: In surrogate representation, the number of neighbors R for each novel class and the weight β balancing surrogate/feature representations are two hyperparameters.
Revised: In surrogate representation, the number of neighbors R for each novel class and the weight β balancing surrogate/feature representations are two hyperparameters.

Original: With the help of an available validation set, a natural question is that whether the hyperparameter can be found through the optimization on the validation set, which requires a good generalization of the hyperparameters across different novel classes.
Revised: With the help of an available validation set, a natural question is that whether the hyperparameter can be found through the optimization on the validation set, which requires a good generalization of the hyperparameters across different novel classes.

Original: From Figure 3, the accuracy of VD with varying hyperparameter shows a good agreement between testing and validation sets.
Revised: From Figure K.13, the accuracy of VD with varying hyperparameter shows a good agreement between testing and validation sets.

Original: With this in mind, we select 10 combinations of β and R , guided by the validation set, in conjunction with 2 different feature transformations and 64 different image augmentations, adding up to a large pool of 1280 conﬁgurations for ensemble (denoted byDeepVoro++).
Revised: With this in mind, we select 10 combinations of β and R , guided by the validation set, in conjunction with 2 different feature transformations and 64 different image augmentations, adding up to a large pool of 1280 conﬁgurations for ensemble (denoted by

Original:  As shown in Table 1, DeepVoro++ achieves best results for 1-shot FSLon all three datasets — 2 .
Revised: DeepVoro++). As shown in Table 3, DeepVoro++ achieves best results for 1-shot FSL — 2 .

Original: 53% higher than Zhang et al.
Revised: 53% higher than Zhang et al.

Original: (2020b), 2 .
Revised: (2020b), 2 .

Original: 38% higher than Hu et al.
Revised: 38% higher than Hu et al.

Original: (2021), and 1 . 09% higher than Zhang et al.
Revised: (2021), and 1 . 09% higher than Zhang [Table]

Original:  (2020b),  respectively, justifying the efﬁcacy of our surrogate representation.
Revised: [Table] [Figure] et al. (2020b), on three datasets, respectively, justifying the efﬁcacy of our surrogate representation.

Original: See Appendix F for more detailed analysis.
Revised: See Appendix G for more detailed analysis.

Revised: Ablation Experiments and Running Time.

Revised: Table 4 varies the level of heterogeneity (see Table

Revised: F.4 and G.5 for all datasets).

Revised: The average accuracy of VDs without CCVD integration is marked by (cid:93) , and is signiﬁcantly lower than the fully-ﬂedged ensemble.

Revised: Table 5 presents the running time of DeepVoro(++) benchmarked in a 20-core Intel© Core TM i7 CPU with NumPy (v1.20.3), whose efﬁciency is comparable to DC/S2M2 2, even with > 1000 × diversity.

Original:  Experiments with Different Backbones, Meta-training Protocols, and Domains.
Revised: [Table] Experiments with Different Backbones, Meta-training Protocols, and Domains.

Original: Because different feature extraction backbones, meta-training losses, and degree of discrepancy between the source/target domains will all affect the downstream FSL, we here examine the robustness of DeepVoro/DeepVoro++ under a number of different circumstances, and details are shown in Appendices 
Revised: Because different feature extraction backbones, meta-training losses, and degree of discrepancy between the source/target domains will all affect the downstream FSL, we here examine the robustness of DeepVoro/DeepVoro++ under a number of different circumstances, and details are shown in Appendices H, I, J.

Original: G, H, I. Notably, DeepVoro/DeepVoro++ attains the best performance by up to 5 .[Table]
Revised: Notably, DeepVoro/DeepVoro++ attains the best performance by up to 5 .

Original:  and is therefore corroborated as a superior method for FSL, regardless of the backbone, training loss, or domain.[Figure] [Table] [Figure]
Revised: 55% , and is therefore corroborated as a superior method for FSL, regardless of the backbone, training loss, or domain.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: In this paper [Figure] our contribution is threefold.
Revised: In this paper, our contribution is threefold.

Original: We ﬁrst theoretically unify parametric and nonparametric few-shot classiﬁers into a general geometric framework (VD) and show an improved result by virtue of this integration (CIVD).
Revised: We ﬁrst theoretically unify parametric and nonparametric few-shot classiﬁers into a general geometric framework (VD) and show an improved result by virtue of this integration (CIVD).

Original: By extending VD/CIVD to cluster-to-cluster Voronoi Diagram, we propose a fast geometric ensemble method (DeepVoro) that takes into consideration thousands of FSL conﬁgurationsat the same time with high efﬁciency.
Revised: By extending CIVD to CCVD, we present a fast geometric ensemble method (DeepVoro) that takes into consideration thousands of FSL conﬁgurations with high efﬁciency.

Original: To deal with the extreme data insufﬁciency in one-shot learning, we further propose a novel surrogate representation which, when incorporated into the ensemble pipeline, promotes the performance of one-shot learning to a higher level.
Revised: To deal with the extreme data insufﬁciency in one-shot learning, we further propose a novel surrogate representation which, when incorporated into DeepVoro, promotes the performance of one-shot learning to a higher level (DeepVoro++).

Original: Our CIVD-based workﬂow enables ∼ 2% − 5% improvements on popular benchmark datasets, surpassing the state-of-the-art methods with either multi-level data augmentation or multiple meta-trained models. Our code is available at Github.
Revised: In future studies, we plan to extend our geometric approach to meta-learning-based FSL and lifelong FSL.

Original: R EPRODUCIBILITY S TATEMENT
Revised: R EPRODUCIBILITY S TATEMENT

Original: Our code as well as data split, random seeds, hyperparameters, scripts, and everything that is necessary for reproducing all the results in the paper will be released at Github.
Revised: Our code as well as data split, random seeds, hyperparameters, scripts, and everything that is necessary for reproducing all the results in the paper will be released at Github A N OTATIONS AND A CRONYMS [Table] [Table]

Original: A P OWER D IAGRAM S UBDIVISION AND V ORONOI R EDUCTION A. P ROOF OF T HEOREM 3. 
Revised: B P OWER D IAGRAM S UBDIVISION AND V ORONOI R EDUCTION B. P ROOF OF T HEOREM 3.

Revised: Lemma B

Original: Proof.
Revised: [Table] The vertical projection from the lower envelope of the hyperplanes { Π k ( z ) : W T k z + b k } Kk =1 onto the input space R n deﬁnes the cells of a PD [Table] Theorem 3.1 (Voronoi Diagram Reduction).

Revised: The linear classiﬁer parameterized by W , b partitions the input space R n to a Voronoi Diagram with centers { ˜ c 1 , ..., ˜ c K } given by ˜ c

Original: {+k = 12 W k if b k = − 1 
+}
Revised: {+k = 12 W k if b k = − 1 
+}

Revised: ||

Original:  We ﬁrst articulate Lemma 3.1 and ﬁnd the exact relationship between the hyperplane Π k ( z ) and the center of its associated cell in R n .
Revised: W k || 2 2 , k = 1 , ..., K . Proof. We ﬁrst articulate Lemma B.1 and ﬁnd the exact relationship between the hyperplane Π k ( z ) and the center of its associated cell in R n .

Original: By Deﬁnition 3.1, the cell for a point z ∈ R n is found by comparing d ( z , c k ) − ν k for different k , so we deﬁne the power function p ( z , S ) expressing this value
Revised: By Deﬁnition 3.1, the cell for a point z ∈ R n is found by comparing d ( z , c k ) − ν k for different k , so we deﬁne the power function p ( z , S ) expressing this value

Original: [Equation]
Revised: [Equation]

Original: in which S ⊆ R n is a sphere with center u and radius r .
Revised: in which S ⊆ R n is a sphere with center u and radius r .

Original: In fact, the weight ν associated with a center in Deﬁnition 3.1 can be intepreted as the square of the radius r 2 .
Revised: In fact, the weight ν associated with a center in Deﬁnition 3.1 can be intepreted as the square of the radius r 2 .

Original: Next, let U denote a paraboloid y =
Revised: Next, let U denote a paraboloid y =

Original: z 2 , let Π( S ) be the transform that maps sphere S with center u and radius r into hyperplane
Revised: z 2 , let Π( S ) be the transform that maps sphere S with center u and radius r into hyperplane

Original: [Equation]
Revised: [Equation]

Original: It can be proved that Π is a bijective mapping between arbitrary spheres in R n and nonvertical hyperplanes in R n +1 that intersect U (Aurenhammer, 1987).
Revised: It can be proved that Π is a bijective mapping between arbitrary spheres in R n and nonvertical hyperplanes in R n +1 that intersect U (Aurenhammer, 1987).

Original: Further, let z (cid:48) denote the vertical projection of z onto U and z (cid:48)(cid:48) denote its vertical projection onto Π( S ) , then the power function can be written as
Revised: Further, let z (cid:48) denote the vertical projection of z onto U and z (cid:48)(cid:48) denote its vertical projection onto Π( S ) , then the power function can be written as

Original: [Equation]
Revised: [Equation]

Original: 13) which implies the following relationships between a sphere in R n and an associated hyperplane in R n +1 (Lemma 4 in Aurenhammer (1987)): let S 1 and S 2 be nonco-centeric spheres in R n , then the bisector of their Power cells is the vertical projection of Π( S 1 )
Revised: which implies the following relationships between a sphere in R n and an associated hyperplane in R n +1 (Lemma 4 in Aurenhammer (1987)): let S 1 and S 2 be nonco-centeric spheres in R n , then the bisector of their Power cells is the vertical projection of Π( S 1 )

Original: ∩ Π( S 2 ) onto R n .
Revised: ∩ Π( S 2 ) onto R n .

Original: Now, we have a direct relationship between sphere S , and hyperplane Π( S ) , and comparing equation (12) with the hyperplanes used in logistic regression { Π k ( z ) :
Revised: Now, we have a direct relationship between sphere S , and hyperplane Π( S ) , and comparing equation (12) with the hyperplanes used in logistic regression { Π k ( z ) :

Original: W Tk z + b k } Kk =1 gives us
Revised: W Tk z + b k } Kk =1 gives us

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Although there is no guarantee that b k + 14 || W k
Revised: Although there is no guarantee that b k + 14 || W k

Original: || 22 is always positive for an arbitrary logistic regression model, we can impose a constraint on r 2 to keep it be zero during the optimization, which implies
Revised: || 22 is always positive for an arbitrary logistic regression model, we can impose a constraint on r 2 to keep it be zero during the optimization, which implies

Original: [Equation]
Revised: [Equation]

Original: By this way, the radii for all K spheres become identical (all zero).
Revised: By this way, the radii for all K spheres become identical (all zero).

Original: After the optimization of logistic regression model, the centers { 12 W k } Kk =1 will be used for CIVD integration.
Revised: After the optimization of logistic regression model, the centers { 12 W k } Kk =1 will be used for CIVD integration.

Original: B D ETAILS ABOUT THE D EMONSTRATIVE E XAMPLE ON M ULTI D IGIT MNIST D ATASET
Revised: C D ETAILS ABOUT THE D EMONSTRATIVE E XAMPLE ON M ULTI D IGIT MNIST D ATASET

Original: MultiDigitMNIST (Sun, 2019) dataset is created by concatenating two (or three) digits of different classes from MNIST for few-shot image classiﬁcation.
Revised: MultiDigitMNIST (Sun, 2019) dataset is created by concatenating two (or three) digits of different classes from MNIST for few-shot image classiﬁcation.

Original: Here we use DoubleMNIST Datasets (i.e. two digits in an image) consisting of 100 classes (00 to 09), 1000 images of size 64 × 64 × 1 per class, and the classes are further split into 64, 20, and 16 classes for training, testing, and validation, respectively.
Revised: Here we use DoubleMNIST Datasets (i.e. two digits in an image) consisting of 100 classes (00 to 09), 1000 images of size 64 × 64 × 1 per class, and the classes are further split into 64, 20, and 16 classes for training, testing, and validation, respectively.

Original: To better embed into the R 2 space, we pick a ten-classes subset (00, 01, 12, 13, 04, 05, 06, 77, 08, and 09) as the base classes for meta-training, and another ﬁve-class subset (02, 49, 83, 17, and 36) for one episode.
Revised: To better embed into the R 2 space, we pick a ten-classes subset (00, 01, 12, 13, 04, 05, 06, 77, 08, and 09) as the base classes for meta-training, and another ﬁve-class subset (02, 49, 83, 17, and 36) for one episode.

Original: The feature extractor is a 4-layer convolutional network with an additional fully-connected layer for 2D embedding.
Revised: The feature extractor is a 4-layer convolutional network with an additional fully-connected layer for 2D embedding.

Original: In Figure 1 left panel, the Dirichlet Tessellation is obtained by setting the centroid of each base class as the Voronoi center.
Revised: In Figure 1 left panel, the VD is obtained by setting the centroid of each base class as the Voronoi center.

Original: For each novel class, the Voronoi center is simply the 1-shot support sample (Figure 1 central panel).
Revised: For each novel class, the Voronoi center is simply the 1-shot support sample (Figure 1 central panel).

Original: The surrogate representation is computed as the collection of distances from a support/query sample to each of the base classes, as shown in Figure 1 right panel.
Revised: The surrogate representation is computed as the collection of distances from a support/query sample to each of the base classes, as shown in Figure 1 right panel.

Original: Interestingly, the surrogate representations for a novel class, no matter if it is a support sample (dotted line) or a query sample (colored line) generally follow a certain pattern — akin within a class, distinct cross class — make them ideal surrogates for distinguishing between different novel classes.
Revised: Interestingly, the surrogate representations for a novel class, no matter if it is a support sample (dotted line) or a query sample (colored line) generally follow a certain pattern — akin within a class, distinct cross class — make them ideal surrogates for distinguishing between different novel classes.

Original: In our paper, we design a series of algorithms answering multiple questions regarding this surrogate representation: how to select base classes for the calculation of surrogate representation, how to combine it with feature representation, and how to integrate it into the overall ensemble workﬂow.
Revised: In our paper, we design a series of algorithms answering multiple questions regarding this surrogate representation: how to select base classes for the calculation of surrogate representation, how to combine it with feature representation, and how to integrate it into the overall ensemble workﬂow.

Original: C M AIN D ATASETS
Revised: D M AIN D ATASETS

Original: For a fair and thorough comparison with previous works, three widely-adopted benchmark datasets are used throughout this paper.
Revised: For a fair and thorough comparison with previous works, three widely-adopted benchmark datasets are used throughout this paper.

Original: (1) mini-ImageNet (Vinyals et al., 2016) is a shrunk subset of ILSVRC-12 (Russakovsky et al., 2015), consists of 100 classes in which 64 classes for training, 20 classes for testing and 16 classes for validation.
Revised: (1) mini-ImageNet (Vinyals et al. 2016) is a shrunk subset of ILSVRC-12 (Russakovsky et al. , consists of 100 classes in which 64 classes for training, 20 classes for testing and 16 classes for validation.

Original: Each class has 600 images of size 84 × 84 × 3 .
Revised: Each class has 600 images of size 84 × 84 × 3 .

Original: (2) CUB (Welinder et al., 2010) is another benchmark dataset for FSL, especially ﬁne-grained FSL, including 200 species (classes) of birds.
Revised: (2) CUB (Welinder et al., ) is another benchmark dataset for FSL, especially ﬁne-grained FSL, including 200 species (classes) of birds

Original: CUB is an unbalanced dataset with 58 images in average per class, also of size 84 × 84 × 3 .
Revised: CUB is an unbalanced dataset with 58 images in average per class, also of size 84 × 84 × 3 .

Original: We split all classes into 100 base classes, 50 novel classes and 50 validation classes, following previous works (Chen et al., 2019a).
Revised: We split all classes into 100 base classes, 50 novel classes, and 50 validation classes, following previous works (Chen et al., 2019a).

Original: (3) tiered-ImageNet (Ren et al., 2018) is another subset of ILSVRC-12 (Russakovsky et al., 2015)but has
Revised: (3) tiered-ImageNet (Ren et al., 2018) is another subset of ILSVRC-12 (Russakovsky et al., 2015)

Original:  more images, ,165 images in total.
Revised: but has more images, 779,165 images in total

Original: All images are categorized into 351 base classes,  validation  classes
Revised: All images are categorized into 351 base classes, 97 validation classes, and 160 novel classes

Original: and
Revised: The

Original: novel
Revised: number

Original: classes.
Revised: of images in each class is not

Original: The number of images in each class is not always the same, 1281 in average.
Revised: always the same in average.

Original: The image size is also 84 × 84 ×
Revised: The image size is also 84 × 84 ×

Original:  D  I NTEGRATING P ARAMETRIC AND N ONPARAMETRIC M ETHODS VIA CIVD
Revised: E D EEP V ORO -- : I NTEGRATING P ARAMETRIC AND N ONPARAMETRIC M ETHODS VIA CIVD

Original: [Table]
Revised: [Table]

Original: D. E XPERIMENTAL S ETUP AND I MPLEMENTATION D ETAILS
Revised: E. E XPERIMENTAL S ETUP AND I MPLEMENTATION D ETAILS

Original: In this section, we ﬁrst establish three few-shot classiﬁcation models with different underlying geometric structures, two logistic regression (LR) models and one nearest neighbor model: (1) Power
Revised: In this section [Table] we ﬁrst establish three few-shot classiﬁcation models with different underlying geometric structures, two logistic regression (LR) models and one nearest neighbor model: (1) Power

Original: Diagram-based LR (Power-LR), (2) Voronoi Diagram-based LR (Voronoi-LR)
Revised: Diagram-based LR (Power-LR), (2) Voronoi Diagram-based LR (Voronoi-LR)

Original: [Table]
Revised: [Table]

Original: and (3) Voronoi Diagram (VD).
Revised: and (3) Voronoi Diagram (VD).

Original: Then, the main purposes of our analysis are (1) to examine how the performance is affected by the proposed Voronoi Reduction method in Sec. 3.2, and (2) to inspect whether VD can be integrated with Power/Voronoi Diagram-based LRs.
Revised: Then, the main purposes of our analysis are (1) to examine how the performance is affected by the proposed Voronoi Reduction method in Sec. 3.2, and (2) to inspect whether VD can be integrated with Power/Voronoi Diagram-based LRs.

Original: The feature transformation used throughout this section is P w,b,λ with w = 1 . 0 , b = 0 . 0 , λ = 0 .
Revised: The feature transformation used throughout this section is P w,b,λ with w = 1 . 0 , b = 0 . 0 , λ = 0 .

Original: For Power-LR, we train it directly on the transformed K -way N -shot support samples using PyTorch library with an Adam optimizer with batch size at 64 and learning rate at 0.01.
Revised: For Power-LR, we train it directly on the transformed K -way N -shot support samples using PyTorch library with an Adam optimizer with batch size at 64 and learning rate at 0.01.

Original: For Voronoi-LR, the vanilla LR is retroﬁtted as shown in Algorithm 1, in which the bias is given by Theorem 3.1 to make sure that the parameters induce a VD in each iteration.
Revised: For Voronoi-LR, the vanilla LR is retroﬁtted as shown in Algorithm 1, in which the bias is given by Theorem 3.1 to make sure that the parameters induce a VD in each iteration.

Original: In our CIVD model in Deﬁnition 3.2, we use a cluster instead of a single prototype to stand for a novel class.
Revised: In our CIVD model in Deﬁnition 3.2, we use a cluster instead of a single prototype to stand for a novel class.

Original: Here this cluster contains two points, i.e. C k = { c k , ˜ c k } , in which c k is obtained from
Revised: Here this cluster contains two points, i.e. C k = { c k , ˜ c k } , in which c k is obtained from

Original: VD, and ˜ c k is acquired from Power-LR or Voronoi-LR.
Revised: VD, and ˜ c k is acquired from Power-LR or Voronoi-LR.

Original: The question we intend to answer here is that whether Power-LR or Voronoi-LR is the suitable model for the integration.
Revised: The question we intend to answer here is that whether Power-LR or Voronoi-LR is the suitable model for the integration.

Original: Data: Support Set S
Revised: Data: Support Set S

Original: Result: W 1 Initialize W ← W (0) ; 2 for epoch ← 1 , ..., #epoch do b k ← − 14 || W k
Revised: Result: W 1 Initialize W ← W (0) ; 2 for epoch ← 1 , ..., #epoch do b k ← − 14 || W k

Original: ||
Revised: ||

Original: ∀ k
Revised: ∀ k

Original: = 

Revised: =

Original: , ..., K ; (cid:47) Apply Theorem 3.
Revised: , ..., K ; (cid:47) Apply Theorem 3.

Original: Compute loss L ( W , b ) ; (cid:47) forward propagation
Revised: Compute loss L ( W , b ) ; (cid:47) forward propagation

Original: Update W ; (cid:47) backward propagation 6 end 7 return W
Revised: Update W ; (cid:47) backward propagation 6 end 7 return W

Original: [Figure]
Revised: [Figure]

Original: D. R ESULTS
Revised: E. R ESULTS

Original: The results are shown in Table D.1.
Revised: The results are shown in Table E.3.

Original: Interestingly, when integrated with VD, Power-LR never reaches the best result, suggesting that VD and LR are intrinsic different geometric models, and cannot be simply integrated together without additional effort.
Revised: Interestingly, when integrated with VD, Power-LR never reaches the best result, suggesting that VD and LR are intrinsic different geometric models, and cannot be simply integrated together without additional effort.

Original: On mini-ImageNet and tiered-ImageNet datasets, the best results are achieved by either Voronoi-LR or VD+Voronoi-LR, showing that CIVD coupled with the proposed Voronoi reduction can ideally integrate parametric and nonparametric few-shot models.
Revised: On mini-ImageNet and tiered-ImageNet datasets, the best results are achieved by either Voronoi-LR or VD+Voronoi-LR, showing that CIVD coupled with the proposed Voronoi reduction can ideally integrate parametric and nonparametric few-shot models.

Original: Notably, on these two datasets, when Power-LR is reduced to Voronoi-LR, although the number of parameters is decreased ( b is directly given by Theorem 3.1, not involved in the optimization), the performance is always better, for example, increases from 65 .
Revised: Notably, on these two datasets, when Power-LR is reduced to Voronoi-LR, although the number of parameters is decreased ( b is directly given by Theorem 3.1, not involved in the optimization), the performance is always better, for example, increases from 65 .

Original: 45% to 65 .
Revised: 45% to 65 .

Original: 58% on 5-way 1-shot mini-ImageNet data.
Revised: 58% on 5-way 1-shot mini-ImageNet data.

Original: On CUB dataset, results of different models are similar, probably because CUB is a ﬁne-grained dataset and all classes are similar to each other (all birds).
Revised: On CUB dataset, results of different models are similar, probably because CUB is a ﬁne-grained dataset and all classes are similar to each other (all birds).

Original: E I MPROVING FSL VIA H IERARCHICAL H ETEROGENEITIES E. E XPERIMENTAL S ETUP AND I MPLEMENTATION D ETAILS
Revised: F D EEP V ORO : I MPROVING FSL VIA H IERARCHICAL H ETEROGENEITIES F. E XPERIMENTAL S ETUP AND I MPLEMENTATION D ETAILS

Original: In this section we describe feature-level and transformation-level heterogeneities that are used for ensemble in order to improve FSL.
Revised: In this section we describe feature-level and transformation-level heterogeneities that are used for ensemble in order to improve FSL.

Original: See the next section for geometry-level heterogeneity.
Revised: See the next section for geometry-level heterogeneity.

Original: Feature-level heterogeneity.
Revised: Feature-level heterogeneity.

Original: Considering the reproducibility of the methodology, we only employ deterministic data augmentation upon the images without randomness involved.
Revised: Considering the reproducibility of the methodology, we only employ deterministic data augmentation upon the images without randomness involved.

Original: Speciﬁcally, three kinds of data augmentation techniques are used.
Revised: Speciﬁcally, three kinds of data augmentation techniques are used.

Original: (1) Rotation is an important augmentation method widely used in self-supervised learning (Mangla et al., 2020).
Revised: (1) Rotation is an important augmentation method widely used in self-supervised learning (Mangla et al., 2020).

Original: Rotating the original images by 0 °, 90 °, 180 °, and 270 °gives us four ways of augmentation.
Revised: Rotating the original images by 0 °, 90 °, 180 °, and 270 °gives us four ways of augmentation.

Original: (2) After rotation, we can ﬂip the images horizontally, giving rise to additional two choices after each rotation degree.
Revised: (2) After rotation, we can ﬂip the images horizontally, giving rise to additional two choices after each rotation degree.

Original: (3) Central cropping after scaling can alter the resolution and focus area of the image.
Revised: (3) Central cropping after scaling can alter the resolution and focus area of the image.

Original: Scaling the original images to (84+ B ) × (84+ B ) , B increasing from 0 to 70 with step 10 , bringing us eight ways of augmentation.
Revised: Scaling the original images to (84+ B ) × (84+ B ) , B increasing from 0 to 70 with step 10 , bringing us eight ways of augmentation.

Original: Finally, different combinations of the three types result in 64 kinds of augmentation methods (i.e.
Revised: Finally, different combinations of the three types result in 64 kinds of augmentation methods (i.e.

Original: |{ T }| = 64 ).
Revised: |{ T }| = 64 ).

Original: Transformation-level heterogeneity.
Revised: Transformation-level heterogeneity.

Original: In our compositional transformation, the function ( h λ ◦ g w,b ◦ f )( z ) is parameterized by w, b, λ .
Revised: In our compositional transformation, the function ( h λ ◦ g w,b ◦ f )( z ) is parameterized by w, b, λ .

Original: Since g is appended after the L2 normalization f , the vector comes into g is always a unit vector, so we simply set w = 1 .
Revised: Since g is appended after the L2 normalization f , the vector comes into g is always a unit vector, so we simply set w = 1 .

Original: For the different combinations of λ and b , we test different values with either λ = 0 or λ (cid:54) = 0 on the hold-out validation set (as shown in Figure 2 and J.12), and pick top-8 combinations with the best performance on the validation set.
Revised: For the different combinations of λ and b , we test different values with either λ = 0 or λ (cid:54) = 0 on the hold-out validation set (as shown in Figure 2 and K.12), and pick top-8 combinations with the best performance on the validation set.

Original: Ensemble Schemes.
Revised: Ensemble Schemes.

Original: Now, in our conﬁguration pool { T } × { P w,b,λ } , there are 512 possible conﬁgurations { ρ ( i ) } 512 i =1 .
Revised: Now, in our conﬁguration pool { T } × { P w,b,λ } , there are 512 possible conﬁgurations { ρ ( i ) } 512 i =1 .

Original: For each ρ , we apply it on both the testing and the validation sets.
Revised: For each ρ , we apply it on both the testing and the validation sets.

Original: With this large pool of ensemble candidates, how and whether to select a subset { ρ ( i ) } L (cid:48) i =1 ⊆ { ρ ( i ) } 512 i =1 is still a nontrivial problem.
Revised: With this large pool of ensemble candidates, how and whether to select a subset { ρ ( i ) } L (cid:48) i =1 ⊆ { ρ ( i ) } 512 i =1 is still a nontrivial problem.

Original: Here we explore three different schemes.
Revised: Here we explore three different schemes.

Original: (1) Full (vanilla) ensemble .
Revised: (1) Full (vanilla) ensemble .

Original: All candidates in { ρ ( i ) } 512 i =1 are taken into consideration and then plugged into Deﬁnition 3.5 to build the
Revised: All candidates in { ρ ( i ) } 512 i =1 are taken into consideration and then plugged into Deﬁnition 3.5 to build the

Original: CIVD for space partition.
Revised: CIVD for space partition.

Original: (2) Random ensemble .
Revised: (2) Random ensemble .

Original: A randomly selected subset with size L (cid:48) <
Revised: A randomly selected subset with size L (cid:48) <

Original: L is used for ensemble.
Revised: L is used for ensemble.

Original: (3) Guided ensemble .
Revised: (3) Guided ensemble .

Original: We expect the performance for { ρ ( i ) } 

Revised: We expect the performance for { ρ ( i ) }

Original: i =1 on the validation set can be used to guide the selection of { ρ ( i ) } L (cid:48)
Revised: i =1 on the validation set can be used to guide the selection of { ρ ( i ) } L (cid:48)

Original: i =1 from the testing set, provided that there is good correlation between the testing set and the validation set.
Revised: i =1 from the testing set, provided that there is good correlation between the testing set and the validation set.

Original: Speciﬁcally, we rank the conﬁgurations in the validation set with regard to their performance, and add them sequentially into { ρ ( i ) } L (cid:48) i =1 until a maximum ensemble performance is reached on the validation set, then we use this conﬁguration set for the ﬁnal ensemble.
Revised: Speciﬁcally, we rank the conﬁgurations in the validation set with regard to their performance, and add them sequentially into { ρ ( i ) } L (cid:48) i =1 until a maximum ensemble performance is reached on the validation set, then we use this conﬁguration set for the ﬁnal ensemble.

Original: Since VD is nonparametric and fast, we adopt VD as the building block and only use VD for each ρ for the remaining part of the paper.
Revised: Since VD is nonparametric and fast, we adopt VD as the building block and only use VD for each ρ for the remaining part of the paper.

Original: The α value in the inﬂuence function (Deﬁnition 3.3) is set at 1 throughout the paper, for the simplicity of computation.
Revised: The α value in the inﬂuence function (Deﬁnition 3.3) is set at 1 throughout the paper, for the simplicity of computation.

Original: For a fair comparison, we downloaded the trained models 1 used by Mangla et al.
Revised: For a fair comparison, we downloaded the trained models 1 used by Mangla et al.

Original: (2020) and Yang et al.
Revised: (2020) and Yang et al.

Original: The performance of FSL algorithms is typically evaluated by a sequence of independent episodes, so the data split and random seed for the selection of novel classes as well as support/query set in each episode will all lead to different result.
Revised: The performance of FSL algorithms is typically evaluated by a sequence of independent episodes, so the data split and random seed for the selection of novel classes as well as support/query set in each episode will all lead to different result.

Original: To ensure the fairness of our evaluation, DC (Yang et al., 2021), and S2M2 R (Mangla et al., 2020) are reevaluated with the same data split and random seed as DeepVoro.
Revised: To ensure the fairness of our evaluation, DC (Yang et al., 2021), and S2M2 R (Mangla et al., 2020) are reevaluated with the same data split and random seed as DeepVoro.

Original: The results are obtained by running 2000 episodes and the average accuracy as well as 95% conﬁdence intervals are reported.
Revised: The results are obtained by running 2000 episodes and the average accuracy as well as 95% conﬁdence intervals are reported.

Original: E. R ESULTS
Revised: F. R ESULTS

Original: [Table]
Revised: [Table]

Original: Our proposed compositional transformation enlarges the expressivity of the transformation function.
Revised: Our proposed compositional transformation enlarges the expressivity of the transformation function.

Original: When the Tukey’s ladder of powers transformation is used individually, as reported in Yang et al.
Revised: When the Tukey’s ladder of powers transformation is used individually, as reported in Yang et al.

Original: (2021), the optimal λ is not 0 , but if an additional linear transformation g is inserted between f and h , λ = 0 coupled with a proper b can give even better result, as shown in Figure 2 and J.12.
Revised: (2021), the optimal λ is not 0 , but if an additional linear transformation g is inserted between f and h , λ = 0 coupled with a proper b can give even better result, as shown in Figure 2 and K.12.

Original: Importantly, from Figure 2, a combination of λ and b with good performance on the validation set can also produce satisfactory result on the testing set, suggesting that it is possible to optimize the hyperparameters on the validation set and generalize well on the testing set.
Revised: Importantly, from Figure 2, a combination of λ and b with good performance on the validation set can also produce satisfactory result on the testing set, suggesting that it is possible to optimize the hyperparameters on the validation set and generalize well on the testing set.

Original: In terms of the polymorphism induced by various transformations in the feature space, Figure E.1 exhibits the t-SNE visualizations of the original features and the features after three different kinds of transformations, showing that the relative positions of different novel classes is largely changes especially after compositional transformation (as shown in D).
Revised: In terms of the polymorphism induced by various transformations in the feature space, Figure F.1 exhibits the t-SNE visualizations of the original features and the features after three different kinds of transformations, showing that the relative positions of different novel classes is largely changes especially after compositional transformation (as shown in D).

Original: Besides commonly used data augmentation, this transformation provides another level of diversity that may be beneﬁcial to the subsequent ensemble.
Revised: Besides commonly used data augmentation, this transformation provides another level of diversity that may be beneﬁcial to the subsequent ensemble.

Original: The results for different levels of ensemble are shown in Table E.2, in which the number of ensemble member are also indicated.
Revised: The results for different levels of ensemble are shown in Table F.4, in which the number of ensemble member are also indicated.

Original: Although transformation ensemble does not involve any change to the feature, it can largely improve the results for 1-shot FSL, from 65 .
Revised: Although transformation ensemble does not involve any change to the feature, it can largely improve the results for 1-shot FSL, from 65 .

Original: 37% to 66 .
Revised: 37% to 66 .

Original: 45%on mini-ImageNet, from 78 .
Revised: 45% from 78 .

Original: 57% to 80 .
Revised: 57% to 80 .

Original: 98% on CUB, and from 72 . 83% to 74 .
Revised: 98% on CUB, and from 72 . 83% to 74 .

Original: 02% on tiered-ImageNet, respectively, probably because 1-shot FSL is more prone to overﬁtting due to its severe data deﬁciency.
Revised: 02% on tiered-ImageNet, respectively, probably because 1-shot FSL is more prone to overﬁtting due to its severe data deﬁciency.

Original: Featurelevel ensemble, on the other hand, is more important for 5-shot FSL, especially for mini-ImageNet.
Revised: Featurelevel ensemble, on the other hand, is more important for 5-shot FSL, especially for mini-ImageNet.

Original: When combining the two levels together, the number of ensemble members increases to 512 and the performance signiﬁcantly surpasses each individual level.
Revised: When combining the two levels together, the number of ensemble members increases to 512 and the performance signiﬁcantly surpasses each individual level.

Original: On all three datasets, the guided ensemble scheme always achieves the best result for both single-shot and multi-shot cases, showing that the validation set can indeed be used for the guidance of subset selection and our method is robust cross classes in the same domain.
Revised: On all three datasets, the guided ensemble scheme always achieves the best result for both single-shot and multi-shot cases, showing that the validation set can indeed be used for the guidance of subset selection and our method is robust cross classes in the same domain.

Original: When there is no such validation set available, the full ensemble and random ensemble schemes can also give comparable result.
Revised: When there is no such validation set available, the full ensemble and random ensemble schemes can also give comparable result.

Original: To inspect how performance changes with different number of ensemble members, we exhibit the distribution of accuracy at three ensemble levels for mini-ImageNet in Figure E.2 and E.3 , for CUB in Figure E.4 and E.5, and for tiered-ImageNet in Figure E.6 and E.7.
Revised: To inspect how performance changes with different number of ensemble members, we exhibit the distribution of accuracy at three ensemble levels for mini-ImageNet in Figure F.2 and F.3 , for CUB in Figure F.4 and F.5, and for tiered-ImageNet in Figure F.6 and F.7.

Original: Figure (b) in each of them also exhibits the correlation between the testing and validation sets for all 512 conﬁgurations.
Revised: Figure (b) in each of them also exhibits the correlation between the testing and validation sets for all 512 conﬁgurations.

Original: Clearly, better result is often reached when there are more conﬁgurations for the ensemble, validating the efﬁcacy of our method for improving the performance and robustness for better FSL.
Revised: Clearly, better result is often reached when there are more conﬁgurations for the ensemble, validating the efﬁcacy of our method for improving the performance and robustness for better FSL.

Original: Algorithm
Revised: Algorithm

Original: Data: Base classes 
Revised: Data: Base classes D , Support Set S

Original: [Equation]
Revised: = ( x i , y i ) } K × N i =

Original: [Equation] query sample x
Revised: y i ∈ C T , query sample x

Original: Result:
Revised: Result:

Original: ˜ d
Revised: ˜ d

Original: [Equation]
Revised: [Equation]

Original:  (cid:47) 
Revised: ; (cid:47) Extract and

Original: [Table]
Revised: [Table]

Original: and
Revised: feature [Equation] [Equation] [Equation]

Original: || d (cid:48)(cid:48)
Revised: [Equation] [Equation] [Equation]

Original: || 1 ; (cid:47) Compute ﬁnal criterion 26 return ˜ d
Revised: [Equation] [Equation] for ; (cid:47) Compute prototypes of

Original: [Figure]
Revised: [Figure]

Original: [Figure][Figure] [Figure] [Figure]
Revised: [Figure]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: A cc u

Original: [Figure]
Revised: [Figure]

Original: a c y

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: V a li d a t i o n S e t

Original: A

Original: [Figure]
Revised: [Figure]

Original: r
Revised: t

Original: [Figure]
Revised: [Figure]

Original: c y
Revised: o

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: F F URTHER I MPROVEMENT OF FSL VIA S URROGATE R EPRESENTATION F. E XPERIMENTAL S ETUP AND I MPLEMENTATION D ETAILS
Revised: G D EEP V ORO ++: F URTHER I MPROVEMENT OF FSL VIA S URROGATE R EPRESENTATION G. E XPERIMENTAL S ETUP AND I MPLEMENTATION D ETAILS

Original: In this section, we introduce another layer of heterogeneity, that is, geometry-level, that exists in our surrogate representation.
Revised: In this section, we introduce another layer of heterogeneity, that is, geometry-level, that exists in our surrogate representation.

Original: In Deﬁnition 3.4, increasing R will enlarge the degree of locality when searching for the top- R surrogate classes.
Revised: In Deﬁnition 3.4, increasing R will enlarge the degree of locality when searching for the top- R surrogate classes.

Original: In equation (8), if we set γ = 1 then increasing β will make the model rely more on the feature representation and less on the surrogate representation.
Revised: In equation (8) if we set γ = 1 then increasing β will make the model rely more on the feature representation and less on the surrogate representation.

Original: In order to weigh up R and β , we perform a grid search for different combinations of R and β on the validation set, as shown in Figure 3, J.13, and J.14.
Revised: In order to weigh up R and β , we perform a grid search for different combinations of R and β on the validation set, as shown in Figure K.13, K. and K.15.

Original: For each R , we select the β that gives rise to the best result on the validation set, and use this ( R, β ) on the testing set, resulting in 10 such pairs in total.
Revised: For each R , we select the β that gives rise to the best result on the validation set and use this ( R, β ) on the testing set, resulting in 10 such pairs in total.

Original: So there are 10 models in the geometry-level heterogeneity, standing for different degrees of locality.
Revised: So there are 10 models in the geometry-level heterogeneity, standing for different degrees of locality.

Original: In conjunction with feature-level (64 kinds of augmentations) and transformation-level (here only the top-2 best transformations are used) heterogeneities, now there are 1280 different kinds of conﬁgurations in our conﬁguration pool that will be used by the CCVD model.
Revised: In conjunction with feature-level (64 kinds of augmentations) and transformation-level (here only the top-2 best transformations are used) heterogeneities, now there are 1280 different kinds of conﬁgurations in our conﬁguration pool that will be used by the CCVD model.

Original: In conclusion, there are overall 512 + 1280 = 1792 conﬁgurations for a few-shot episode.
Revised: In conclusion, there are overall 512 + 1280 = 1792 conﬁgurations for a few-shot episode.

Original: Generating ∼ensemble candidates is nearly intractable for parametric methods like logistic regression or cosine classiﬁer, which may consume e.g. months for thousands of episodes.
Revised: Generating ∼ensemble candidates is nearly intractable for parametric methods like logistic regression or cosine classiﬁer, which may consume e.g. months for thousands of episodes.

Original: However, the VD model is nonparametric and highly efﬁcient, making it empirically possible to collect all the combinations and integrate them all together via CCVD.
Revised: However, the VD model is nonparametric and highly efﬁcient, making it empirically possible to collect all the combinations and integrate them all together via CCVD.

Original: The complete algorithm for the computation of surrogate representation is shown in Algorithm 2.
Revised: The complete algorithm for the computation of surrogate representation is shown in Algorithm 2.

Original: F. R ESULTS
Revised: G. R ESULTS

Original: The heatmaps for different ( R, β ) pairs on testing/validation sets are shown in Figure 3 for miniImageNet, in Figure J.13 for CUB, and in Figure J.14 for tiered-ImageNet, respectively.
Revised: The heatmaps for different ( R, β ) pairs on testing/validation sets are shown in Figure K.13 for miniImageNet, in Figure K.14 for CUB, and in Figure K.15 for tiered-ImageNet, respectively.

Original: Basically, the testing and validation set follow the same pattern.
Revised: Basically, the testing and validation set follow the same pattern.

Original: When R is small,
Revised: When R is small,

Original: i
Revised: i

Original: [Table]
Revised: [Table]

Original: e. only a small number of base classes are used for surrogate, then a higher weight should be placed on feature representation.
Revised: e. only a small number of base classes are used for surrogate, then a higher weight should be placed on feature representation.

Original: With a ﬁxed β , increasing R beyond a certain threshold will potentially cause a drop in accuracy, probably because the meaningful similarities is likely to be overwhelmed by the signals from the large volume of irrelevant base classes.
Revised: With a ﬁxed β , increasing R beyond a certain threshold will potentially cause a drop in accuracy, probably because the meaningful similarities is likely to be overwhelmed by the signals from the large volume of irrelevant base classes.

Original: [Table]
Revised: [Table]

Original: As shown in Table 1 and F.3, DeepVoro++ further improves upon DeepVoro for 5-way 1-shot FSL by 1 .
Revised: As shown in Table 3 and G.5, DeepVoro++ further improves upon DeepVoro for 5-way 1-shot FSL by 1 .

Original: 82% and 0 .
Revised: 82% and 0 .

Original: 4% on mini-ImageNet and tiered-ImageNet, respectively, and is comparable with
Revised: 4% on mini-ImageNet and tiered-ImageNet, respectively, and is comparable with

Original: DeepVoro on CUB dataset ( 82 .
Revised: DeepVoro on CUB dataset ( 82 .

Original: 95% vs. 82 . 99% ).
Revised: 95% vs. 82 . 99% ).

Original: Notably, on 5-shot FSL, DeepVoro++ usually causes a drop of accuracy from DeepVoro.
Revised: Notably, on 5-shot FSL, DeepVoro++ usually causes a drop of accuracy from DeepVoro.

Original: To inspect the underlying reason for this behavior, we apply VD on 5-way K -shot FSL with K increasing from 1 to 400 and report the average accuracy in Figure F.8.
Revised: To inspect the underlying reason for this behavior, we apply VD on 5-way K -shot FSL with K increasing from 1 to 400 and report the average accuracy in Figure G.8.

Original: It can be observed that, in extreme low-shot learning, i.e. K ∈ [1 , 5] , simply adding one shot makes more prominent contribution to the accuracy, suggesting that the centers obtained from 5-shot samples are much better that those from only 1 sample, so there is no necessity to resort to surrogate representation for multi-shot FSL and we only adopt DeepVoro for 5-shot episodes in the remaining part of this paper.
Revised: It can be observed that, in extreme low-shot learning, i.e. K ∈ [1 , 5] , simply adding one shot makes more prominent contribution to the accuracy, suggesting that the centers obtained from 5-shot samples are much better that those from only 1 sample, so there is no necessity to resort to surrogate representation for multi-shot FSL and we only adopt DeepVoro for 5-shot episodes in the remaining part of this paper.

Original: Ablation study of DeepVoro++ with different levels of ensemble is shown in Table F.3, Figure F.9,
Revised: Ablation study of DeepVoro++ with different levels of ensemble is shown in Table G.5, Figure G.9,

Original: F.10, and F.11.
Revised: G.10, and G.11.

Original: All three layers of heterogeneities collaboratively contribute towards the ﬁnal result.
Revised: All three layers of heterogeneities collaboratively contribute towards the ﬁnal result.

Original: The fully-ﬂedged Dirichlet Tessellation Ensemble establishes new state-of-the-art performance on all three datasets for 1-shot FSL.
Revised: The fully-ﬂedged DeepVoro++ establishes new state-of-the-art performance on all three datasets for 1-shot FSL.

Original: [Figure]
Revised: [Figure]

Original: [Table] [Figure] [Figure]
Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Original: G E XPERIMENTS WITH D IFFERENT B ACKBONES G. I MPLEMENTATION D ETAILS
Revised: H E XPERIMENTS WITH D IFFERENT B ACKBONES H. I MPLEMENTATION D ETAILS

Original: In order to test the robustness of DeepVoro/DeeoVoro++ with various deep learning architectures
Revised: In order to test the robustness of DeepVoro/DeeoVoro++ with various deep learning architectures

Original: [Table]
Revised: [Table]

Original: we downloaded the trained models 2 used by Wang et al.
Revised: we downloaded the trained models 2 used by Wang et al.

Original: We evaluated DC, S2M2 R, DeepVoro, and DeepVoro++ using the same random seed.
Revised: We evaluated DC, S2M2 R, DeepVoro, and DeepVoro++ using the same random seed.

Original: The results are obtained by running 500 episodes and the average accuracy as well as 95% conﬁdence intervals are reported.
Revised: The results are obtained by running 500 episodes and the average accuracy as well as 95% conﬁdence intervals are reported.

Original: G. EXPERIMENTAL RESULTS
Revised: H. EXPERIMENTAL RESULTS

Original: [Table]
Revised: [Table]

Original: On Wide residual networks (WRN-28-10) (Zagoruyko & Komodakis, 2016) [Table] Residual networks (ResNet-10/18/34) (He et al., 2016), Dense convolutional networks (DenseNet-121)(Huang et al., 2017)
Revised: On Wide residual networks (WRN-28-10) (Zagoruyko & Komodakis, 2016), Residual networks (ResNet-10/18/34) (He et al., 2016), Dense convolutional networks (DenseNet-121)

Original: [Table] and MobileNet (Howard et al., 2017), DeepVoro/DeepVoro++ shows a consistent improvement upon DC and S2M2 R.
Revised: (Huang et al., 2017), and MobileNet (Howard et al., 2017), DeepVoro/DeepVoro++ shows a consistent improvement upon DC and S2M2 R.

Original: Excluding DeepVoro/DeepVoro++, there is no such a method that is always better for both 5-shot and 1-shot FSL.
Revised: Excluding DeepVoro/DeepVoro++, there is no such a method that is always better for both 5-shot and 1-shot FSL.

Original: Generally, DC is expert in 1-shot while S2M2 2 favors 5-shot.
Revised: Generally, DC is expert in 1-shot while S2M2 2 favors 5-shot.

Original: According to Table 1, we do not apply DeepVoro++ on 5-shot FSL since DeepVoro usually outperforms DeepVoro++ with more shots available.
Revised: According to Table 3, we do not apply DeepVoro++ on 5-shot FSL since DeepVoro usually outperforms DeepVoro++ with more shots available.

Original: H E XPERIMENTS WITH D IFFERENT T RAINING P ROCEDURES H. E XPERIMENTAL S ETUP AND I MPLEMENTATION D ETAILS
Revised: I E XPERIMENTS WITH D IFFERENT T RAINING P ROCEDURES I. E XPERIMENTAL S ETUP AND I MPLEMENTATION D ETAILS

Original: Our geometric space partition model is built on top of a pretrained feature extractor, and the quality of the feature extractor will signiﬁcantly affect the downstream FSL (Mangla et al., 2020).
Revised: Our geometric space partition model is built on top of a pretrained feature extractor, and the quality of the feature extractor will signiﬁcantly affect the downstream FSL (Mangla et al., 2020).

Original: Here we used another two feature extractors trained with different schemes.
Revised: Here we used another two feature extractors trained with different schemes.

Original: (1) Manifold Mixup training employs an additional Mixup loss that interpolates the data and the label simultaneously and can  help deep neural network generalize better.
Revised: (1) Manifold Mixup training employs an additional Mixup loss that interpolates the data and the label simultaneously and can from help deep neural network generalize better.

Original: (2) Rotation loss is widely used especially in selfsupervised learning in which the network learns to predict the degree by which an image is rotated.
Revised: (2) Rotation loss is widely used especially in selfsupervised learning in which the network learns to predict the degree by which an image is rotated.

Original: We downloaded the corresponding pretrained models used by Mangla et al.
Revised: We downloaded the corresponding pretrained models used by Mangla et al.

Original: (2020) and Yang et al.
Revised: (2020) and Yang et al.

Original: and evaluate the four methods by 500 episodes.
Revised: and evaluate the four methods by 500 episodes.

Original: H. R ESULTS
Revised: I. R ESULTS

Original: [Table]
Revised: [Table]

Original: As shown in Table H.5, DeepVoro/DeepVoro++ achieves best results for both rotation loss and Mixup loss.
Revised: As shown in Table I.7, DeepVoro/DeepVoro++ achieves best results for both rotation loss and Mixup loss.

Original: Interestingly, there is a substantial gap between the two training schemes when they are used out-of-the-box for downstream FSL ( (cid:52) accuracy = 10 .
Revised: Interestingly, there is a substantial gap between the two training schemes when they are used out-of-the-box for downstream FSL ( (cid:52) accuracy = 10 .

Original: 22% for 1-shot and (cid:52) accuracy = 6 .
Revised: 22% for 1-shot and (cid:52) accuracy = 6 .

Original: 52% for 5-shot), but after DeepVoro/DeepVoro++, this gap becomes narrowed ( (cid:52) accuracy = 3 .
Revised: 52% for 5-shot), but after DeepVoro/DeepVoro++, this gap becomes narrowed ( (cid:52) accuracy = 3 .

Original: 98% for 1-shot and (cid:52) accuracy = 2 .
Revised: 98% for 1-shot and (cid:52) accuracy = 2 .

Original: 51% for 5-shot)
Revised: 51% for 5-shot)

Original: , suggesting the strength of DeepVoro/DeepVoro++ to make the most of the pretrained models
Revised: , suggesting the strength of DeepVoro/DeepVoro++ to make the most of the pretrained models

Original: I C ROSS D OMAIN F EW -S HOT L EARNING I. E XPERIMENTAL S ETUP AND I MPLEMENTATION D ETAILS
Revised: J C ROSS D OMAIN F EW -S HOT L EARNING J. E XPERIMENTAL S ETUP AND I MPLEMENTATION D ETAILS

Original: Cross-domain FSL is more challenging than FSL in which base classes and novel classes come from essentially distinct domains.
Revised: Cross-domain FSL is more challenging than FSL in which base classes and novel classes come from essentially distinct domains.

Original: To examine the ability of our method for cross-domain FSL, we apply the feature extractor trained on mini-ImageNet (CUB) on the few-shot data in CUB (mini-ImageNet) for coarse-to-ﬁne
Revised: To examine the ability of our method for cross-domain FSL, we apply the feature extractor trained on mini-ImageNet (CUB) on the few-shot data in CUB (mini-ImageNet) for coarse-to-ﬁne

Original: (ﬁne-to-coarse) domain shifting.
Revised: (ﬁne-to-coarse) domain shifting.

Original: I. R ESULTS
Revised: J. R ESULTS

Original: [Table]
Revised: [Table]

Original: Basically, DeepVoro/DeepVoro++ is more stable than the other two method with a shifting domain, especially on ﬁne-to-coarse FSL (CUB to mini-ImageNet), with an improvement of 6 .
Revised: Basically, DeepVoro/DeepVoro++ is more stable than the other two method with a shifting domain, especially on ﬁne-to-coarse FSL (CUB to mini-ImageNet), with an improvement of 6 .

Original: 68% for 1shot and 6 .
Revised: 68% for 1shot and 6 .

Original: 51% for 5-shot than S2M2 R, and is comparable with DC on coarse-to-ﬁne FSL (miniImageNet to CUB).
Revised: 51% for 5-shot than S2M2 R, and is comparable with DC on coarse-to-ﬁne FSL (miniImageNet to CUB).

Original: J A DDITIONAL F IGURES .
Revised: K A DDITIONAL F IGURES .

Original: [Figure]
Revised: [Figure]

Revised: [Figure]

Revised: [Figure]

Revised: [Table]

Original: [Figure] 
Revised: [Figure] [Table] [Figure] [Table]

Revised: [Figure]

Original: [Table] 
Revised: [Table] [Figure] [Figure] A v e r a g

Original: [Figure]
Revised: e A cc u r a c y S2M2_R DC DeepVoro Episode Accuracy as a Function of Geometric Variance