Few-shot Learning via Dirichlet Tessellation Ensemble
Authors: Chunwei Ma
Ziyun Huang
Mingchen Gao
Jinhui Xu
Venue: ICLR 2022 Poster (2022)
Keywords: Few-shot Learning
Computational Geometry
Dirichlet Tessellation
Voronoi Diagram
Ensemble Learning

Abstract: Few-shot learning (FSL) is the process of rapid generalization from abundant base samples to inadequate novel samples. Despite extensive research in recent years, FSL is still not yet able to generate satisfactory solutions for a wide range of real-world applications. To confront this challenge, we study the FSL problem from a geometric point of view in this paper. One observation is that the widely embraced ProtoNet model is essentially a Voronoi Diagram (VD) in the feature space. We retrofit it by making use of a recent advance in computational geometry called Cluster-induced Voronoi Diagram (CIVD). 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. 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. Our CIVD-based workflow enables us to achieve new state-of-the-art results on mini-ImageNet, CUB, and tiered-ImagenNet datasets, with ${\sim}2\%{-}5\%$ improvements upon the next best. 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. These together make FSL stronger.

Published as a conference paper at ICLR 2022
FEW-SHOT LEARNING AS CLUSTER-INDUCED VORONOI DIAGRAMS: A GEOMETRIC APPROACH
Chunwei Ma', Ziyun Huang", Mingchen
Xu!
Department of Computer Science and Engineering, University at Buffalo CComuuter Science and Software Engineering, Penn State Erie {f (chunweim, mgao8, jinhuij @buffalo. edu {zxh201 @psu edu
ABSTRACT
Few-shot learning (FSL) the process of rapid generalization from abundant base samples to inadequate novel samples. Despite extensive research in recent years, FSL is still not yet able to generate satisfactory solutions for a wide range of real-world applications. To confront this challenge, we study the FSL problem from a geometric point view in this paper. One observation that the widely embraced ProtoNet model is essentially a Voronoi Diagram (VD) in the feature space. We retrofit it by making use of a recent advance in computational geom- etry called
Voronoi Diagram (CIVD). Starting from the simplest nearest neighbor model, CIVD gradually incorporates cluster-to-pont to-po and then cluster-t--cluser relationships for space subdivision, which is used improve the accuracy and robustness at multiple stages of FSL. Specifically, we use CIVD (I) integrate parametric and nonparametric few-shot classifiers; (2) to combine fea- ture representation and surrogate representation; (3) and to leverage feature-level. transformation-level. and geometry-leve heterogeneities for better ensemble. Our CIVD-based based workflow enables us to achieve new state-of-the-art results on miti-ImageNNet, CUB, and tiered- Imrgenne datasets, with ~2%-5% improve- ments upon the next best. To summarize, CIVD provides mathematically el- egant and geometrically interpretable framework that compensates for extreme data insufficiency, prevents overfitting, and allows for fast geometric ensemble for thousands of individual VD. These together make FSL stronger.
INTRODUCTION
Recent years have witnessed 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). However, in many real-world applications, such large-scale data acquisition might be difficult and comes at premium, such as in rare disease diagnosis (Yoo et al. 2021) and drug discovery (Ma et al. 2021b
2018.. a consequence, Few-shot Learning (FSL) has recently drawn growing interests (Wang
et al. 2020).
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. A typical transductive FSL algorithm learns to propagate labels among a larger pool of query samples in semi-supervised manner (Liu al. 2019); notwithstanding its normally higher performance, in many real world scenarios a query sample (c.g. patient) also comes individually and is unique, for instance, in personalized pharmacogenomics
et al.) 2020 Thus, in this paper adhere to the inductive setting and make on-the-fly prediction for each newly seen sample.
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. Despite the extensive research
All four authors corresponding authors.

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Published as a conference paper at ICLR 2022
on the algorithmic aspects of FSL (see Sec. 21. two challenges still pose obstacle to successful FSL: (1) how to sufficiently compensate for the data deficiency in FSL? and (2) how to make the most use of the base samples and the pre-trained model?
For the first question, data augmentation has been a successful approach to expand the size of data, either by Generative Adversarial Networks (GANs) (Goodfellow al.] 2014) (Li et al. 2020b Zhang et al. 2018) or by variational autoencoders (VAEs) (Kingma & Welling 2014) (Zhang et al.
2019 Chen et al. 2019b). However, in each way, the authenticity of either the augmented data or the feature is not guaranteed, and the out-of-distribution hallucinated samples (Ma et al., 2019) may hinder the subsequent FSL. Recently, [Lu al. (2020b) and et al. (202T) 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. For the second question, Yang et al. (202T) borrows the top-2 nearest base classes for each novel sample to calibrate its distribution and to generate more novel samples. However, when there no proximal base class, this calibration may utterly alter the distribution. Another line of work (Sbai et al. 2020 Zhou et al. (220)) learns to select and design base classes for a better discrimination on novel classes, which all introduce extra training burden. As matter of fact, we still lack method that makes full use of the base classes and the pretrained model effectively.
In this paper, we study the FSL problem from a geometric point of view. 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. 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. Although it is highly efficient and simple, Voronoi 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.
To resolve this issue, we adopt a novel tech- nique called Cluster-induced Voronoi Di- agram (CIVD) (Chen et al., [2013] 2017] Huang & Xu 2020; Huang et al. 2021), which is a recent breakthrough in computa- tion geometry. CIVD generalizes VD from a point-to-point distance- based diagram to cluster-to-poin influence-based structure.
Table 1: The underlying geometric structures for var- ious FSL methods.
Method
Geometric Structure
ProtoNet (Snell
2017
Voronoi Diagram spherical VD
S2M2.R Mangla la ct al. 2020
DC (Yang et al. (021))
Power Diagram
Deep Voro-- (ours)
CIVD
Deep DreePooro/Deep/orgg Voro++ (ours)
CCVD
It enables us to determine the dominating region (or Voronoi cell) not only for a point (e.g. class prototype) but also for cluster of points,
guaranteed to have a (1 + e) papproximationn with a nearly linear size of diagram for a wide range of locally dominating influence functions.
CIVD provides us mathematically elegant framework to depict the feature space and draw the decision boundary more precisely than VD without losing the resistance to overfitting.
Accordingly, in this paper, we show how CIVD is used to improve multiple stages of FSL and make several contributions as follows.
1. 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 D. We then unify them via CIVD that enjoys the advantages of multiple models, either parametric or nonparametric (denoted as DeepVoro--).
2. 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. Based on CCVD, we present Deep which enables fast geometric ensemble of a large pool of thousands of configurations for FSL.
3. Instead of using base classes for distribution calibration and data augmentation YYang et al., 2021), we propose novel surrogate representation, the collection of similarities to base classes, and thus promote Deep Voro to DeepVoro++ that integrates feature-level, transformation-levell and geometry-level heterogeneities in FSL.
Extensive experiments have shown that, although a fixed feature extractor is used without inde- pendently pretrained epoch-wise models, our method achieves new state-of-the-art results all
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Base Class Voronoi Diagram
Novel Class Voronol Diagram
Surrogate Representation
08
02
04
05
20-
20
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00
c-19,
01
06
36
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Figure 1: Schematic illustrations of Voronoi Diagram (VD) and surrogate representation on Multi- DigitMNIST dataset (Sun 2019). Left and central panels demonstrate the VD of base classes and novel classes (5-way I-shot) in I2 respectively. The colored squares stand for the 1-shot support samples. In the right panel, for each support sample, the surrogate representation (dotted line) ex- hibits a unique pattern which those of the query samples (colored lines) also follow. (See Appendix foor details.)
three benchmark datasets including mini-ImageNet, CUB, and tiered- ImageNet, and improves by up 28% 5-shot classification,, 2.53% on 1-shot classification, and up 5.55% with different network architectures.
2
RELATED WORK
Few-Shot Learning, There are a number of different lines of research dedicated to FSL. (1) Metric- based methods employ a certain distance function (cosine distance (Mangla et al. 2020 Xu al.
2021
Euclidean distance (Wang et al. 2019 Snell et al., 2017), or Earth Mover's ' Distance (Zhang al. 2020a b)) to bypass the optimization and avoid possible overfitting. (2) Optimization-
approaches (Finn et al.) (017)) manages to learn good model initialization that accelerates the op- (Zhang et al., 2021b; Mangla et al.) 2020) methods incorporate supervision from data itself to learn robuster feature extractor. (4) En- semble method is another powerful technique that boosting the performance by integrating multiple models (Ma al. 2021a). For example. Dvornik al. (2019) trains several networks simultancously and encourages robustness and cooperation among them. However, due to the high computation load of training deep models, this ensemble is restricted by the number of networks which is typically <20. In Liu et al.] (2020c), instead, the ensemble consists models learned at each epoch, which, may potentially limit the diversity of ensemble members.
timization the meta-testing stage. (3) Self-supervised-barl
Geometric Understanding of Deep Learning. The geometric structure of deep neural networks first hinted at by Raghu al. (2017) who reveals that piecewise linear activations subdivide in- put space into convex polytopes. Then, Balestricro et al. (2019) points out that the exact structure Power Diagram (Aurenhammer 1987) which is subsequently applied upon recurrent neural 2018) and generative model (Balestriero et al. 2020). The Power/Vorone Diagram subdivision, however, is necessarily the optimal model for describing feature space. Recently, Chen et al.
network (Wang et al.
2013 2017); Huang al. (2021) uses influence function F(C, 2) to mea- sure the joint influence of all objects on a query z to build a Cluster-induced Voronoi Diagram (CIVD). In this paper, we utilize CIVD to magnify the expressivity of geometric modeling for FSL.
METHODOLOGY
3.1 PRELIMINARIES
Few-shot learning aims at discriminating between novel classes Chovel with the aid of a larger amount of samples from base classes Chese guovel Cbbba (.. The whole learning process usually follows the

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meta-learning scheme. Formally, given dataset of base classes D {(x,, 8:)) E D. Chase with D being an arbitrary domain c.g. natural image, a deep neural network 2 = Φ p E R", which maps from image domain D to feature domain R", is trained using standard gradient descent algorithm, and after which & is fixed as a feature extractor. This process is referred to as meta- training stage that squeezes out the commonsense knowledge from D.
For a fair evaluation of the learning performance on a few samples, the meta-testing stage is typically formulated series of K-way N-shot tasks (episodes) {T}. Each such episode is further decom- posed into support set S = {( ) KxN Di E CT and query set = {( Di) J xx2 De E CT, in which the episode classes C* is a randomly sampled subset of Crovel with cardinality K, and each class contains only N and Q random samples in the support set and query set, respectively. For few-shot classification, we introduce here two widely used schemes as follows. For simplicity, all samples here are from S and Q, without data augmentation applied.
Nearest Neighbor Classifier (Nonparametric) Snell al. (2017): Wang et al. (2019) etc., prototype C₁ acquired by averaging over all supporting features for a class E C²:
1 N
Ck = E o(x)
Then each query sample T E Q is classified by finding the nearest prototype: 9 = arg mingd(z,,c,) || 12, in which we use Euclidean distance for distance metric d.
Linear Classifier (Parametric). Another scheme uses a linear classifier with cross-entropy loss optimized on the supporting samples:
exp ((x:) by) C(W,b, b) = Emnn)s - (explvl((a) W.b) Emmn)es - log Σ exp W ((x) + bx)
(2)
in which WR, by are the linear weight and bias for class k, and the predicted class for query E Q ÿ arg max, p(y|z; WR. bk).
3.2 FEW-SHOT LEARNING AS CLUSTER-INDUCEE VORONOI DIAGRAMS
In this section, first introduce the basic concepts of Voronoi Tessellations, and then show how parametric/nonpaaametiii classifier heads can be unified by VD.
Definition 3.1 (Power Diagram and Voronoi Diagram). Let Q = {wi, (w,,... wK} be a partition of the space RR, and C {c1, .... CK) } be set of centers such that U Wi R Rn.nk S 14, W. Additionally, each center is associated with a weight Vr E {Vi, 1/k C R Then the set of Power Diagram (PD), where each cell is obtained via wir z E R. r(()) = r E (1,.., K}, with
pairs { (Wi, C1, M1) .... (WK, Ct, "k)}
r(z) arg min d(z,cx)² Vk. kE(L....K)
(3)
the weights are equal for all k, i.e. v = un, Vk, k' E (1,.., K}. then a PD collapses to a Voronoi Diagram (VD).
By definition, it is easy to see that the nearest neighbor classifier naturally partitions the space into K cells with centers (CI...., ]. Here we show that the linear classifier is also a VD under a mild condition.
Theorem 3.1 (Voronoi Diagram Reduction), The linear classifier parameterized by W, b partitions the input space R" to a Voronoi Diagram with centers (či,.., čK given by čN = 1/2 Wy by = - W,
|1, 1,..., K.
Proof. See Appendix B for details.
3.2.1 FROM VORONOT DIAGRAM TO CLUSTER-INDUCHE VORONOT DIAGRAM
Now that both nearest neighbor and linear classifier have been unified by VD, a natural idea is to integrate them together. Cluster-induced Voronoi Diagram (CIVD) (Chen et al. 2017 Huang al.)
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2021 is a generalization of VD which allows multiple centers in cell, and is successfully used for clinical diagnosis from biomedical images (Wang et al. 2013), providing an ideal tool for the integration of parametticcnooppaaaetiii classifier for FSL. Formally:
Definition 3.2 (Cluster-induced Voronoi Diagram (CIVD) (Chen et al. 2017 Huang et al. 2021
Let Ω = (wi,..., wk) be a partition of the space IR", and = [CI:.., CK be set (possibly multiset) of clusters. The set of pairs (W1, C1), (uk,CK)) is Cluster-inducee Voronoi Diagram
(CIVD) with respect to the influence function F(Ck; z), where each cell is obtained via wi- {z E
R=" : r(=) r,. {I,.., K,, with
r(z) = arg max F(Ck; ke K))
(4)
Here C can be either given set of clusters 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 Cs to a query point
Definition 3.3 (Influence Function). The influence from Cg2 k {I...., K} to C2 is F(C,, z) F((d(c 2) E ( this paper F assumed to have the following form
F(Ck, 2) sign(a) Σ Call d(ck 2))"
(5)
The sign function here makes sure that F is monotonically decreasing function with respect to distance d. The hyperparameter a controls the magnitude of the influence, for example, in gravity force = (n - 1) in n-dimenniioal space and in electric force = -2
Since the nearest neighbor centers {c=] and the centers introduced by linear classifier (cx )= obtained from different schemes and could both be informative, we merge the corresponding centers for novel class to be a new cluster C2 = {ck, Cx ). and use the resulting C = [C1,..., Cx to establish CIVD. In such way, the final partition may enjoy the advantages of both parametric and nonparametric classifier heads. We name this approach as Deep Voro--
3.3 FEW-SHOT CLASSIFICATION VIA SURROGATE REPRESENTATION
In nearest neighbor classifier head, the distance from a query feature each of the prototypes (cx 1KK k==1 is the key discrimination criterion for classification. We rewrite (d(2, cx)) k== to be vec- tor d E R such that d2 = d(z, cx). These distances are acquired by measure the distance between two points in high dimension: z, Cj R". However, the notorious behavior of high dimension is that the ratio between the nearest and farthest points in point set P approaches 1 (Aggarwal et al.) 2001 making (d(z, Cx) { less discriminative for classification, especially for FSL problem with sample size N K < n. Hence, in this paper, we seek for a surrogate representation.
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 experimen- tally verified by functional magnetic resonance imaging (IMRI) that a large region in occipitotem- poral cortex processes the shape of both meaningful and unfamiliar objects (Op de Beeck et al. 2008). In our method, a connection will be built between each unfamiliar novel class in Covvel and each related well-perceived familiar class in Chase So the first step to identify the most relevant base classes for a specific task T. Concretely:
Definition 3.4 (Surrogate Classes). episode T. given the set of prototypes { for the support set S and the set of prototypes |ct0a for the base set D, the surrogate classes for episode classes CT is given as:
Cuurogglee
U Top-R d(cks c/)
k=1 EE ((
in which the top- R function returns R base class indices with smallest distances to Cg. and the center for a base class is given as ci [(=.)) (()) Here R is a hyperparameter.
The rationale behind this selection instead of simply using the whole base classes Chase is that, the episode classes CT are only overlapped with portion of base classes (Zhang al. 2021a). and
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discriminative similarities are likely to be overwhelmed by the background signal especially when the number of base classes is large. After the surrogate classes are found, we re-index their feature
centers to be { R < R - K. Then, both support centers { k= and query feature are represented by the collection of similarities to these surrogate centers:
d' = (d(ce, ci),-,, de, 2n ) k. 1,...,K K,
d' = (d(z,c)),,..) d(z,,,,(,,,,
where d/, d' RR are the surrogate representation for novel class i and query feature 2, respec- tively. By surrogate representation, the prediction is found through ÿ = arg d(di d%) arg mingl d' dx l1 This set of discriminative distances is rewritten as d" E RK such that d' d(d', d.). An illustration of the surrogate representation is shown in Figure on Multi- DigitMNIST, a demonstrative dataset.
Integrating Feature Representation and Surrogate Representation. Until now, we have two discriminative systems, i.e., feature-based d E R and surrogate-based d" E R.K A natural idea is to combine them to form the following final criterion:
d"
(8)
dd'lli'
where d and d' are normalized by their Manhattan norm, ||d]| Σm
k- d2 and ||d'l|
Ek= I.
respectively, and 3 and 7 are two hyperparameters adjusting the weights for feature representation and surrogate representation.
3.4 VORO: INTEGRATING MULTI-LEVEL HETEROGENEITY OF FSL
In this section we present DeepVoro, a fast geometric ensemble framework that unites our contribu- tions to multiple stages of FSL., and show how it can be promoted to DeepVoro++ by incorporating surrogate representation.
Compositional Feature Transformation. 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. (02T). Tukey's 's Ladder of Powers Transformation (Yang et al., 202I), and L2 normalization (Wang et al. 2019). While these transformations normally used independently, here we propose to combine several transforma- tions sequentially in order to enlarge the expressivity of transformation function and to increase the polymorphism of the FSL. process. Specifically, for feature vector z, three kinds of transforma- tions are considered: (I) L2 Normalization. By projection onto the unit sphere in I", the feature is normalized as: f(z) = TE (II) Linear Transformation. Now since all the features are located on the unit sphere, we then can do scaling and shifting via a linear transformation: fwe,(z)) = wz + b. (III) Tukey's Ladder of Powers Transformation. Finally, Tukey's Ladder of Powers Transformation
if X + 0 A = 0" By the composition of L2 normaliza- tion, linear transformation, and Tukey's Ladder of Powers Transformation, now the transformation function becomes (fx Sw.b& o (z) parameterized by w, A.
is applied on the feature: hx(=)
log(z)
Multi-level Heterogeneities in FSL. Now are ready to articulate the hierarchical heterogeneity existing in different stages of FSL. (I) Feature-level Heterogeneity: Data augmentation has been exhaustively explored for expanding the data size of FSL (Ni al. (021). including but limited to rotation, flipping. cropping, erasing, solarization, color jitter, MixUp (Zhang et al.) 2017), etc. the feature space. We denote set of functions (T). (II) Transformation-level Heterogeneity: After obtaining the feature 2, a parameterized transformation is applied to it, and the resulting features can be quite different for these parameters (see Figure F.I). We denote the set of all possible transformations to be (.w.b.A.. (III) Geometry-level Heterogeneity: Even with the provided feature, the fhw-shot classification model can still be diverse: whether VD or PD-based model is used, whether the feature or the surrogate representation is adopted, and the setting of will also change the degree of locality. We denote all possible models (M).
The modification of image æ will change the position of feature
all possible translations of image

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Deep Voro for Fast Geometric Ensemble of VDs. With the above three-layer heterogeneity, the FSL process can be encapsulated as episode T with fixed ó is the Cartesian product of these three sets: {T] (P., x {M 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 al.) 2019). Since the cardinality of the resulting configuration set could be very large, the FSL model M as well as the ensemble algorithm required be highly efficient. The VD a nonparametric model and no training is needed during the meta-testing stage, making it suitable for fast geometric ensemble. While CIVD models the cluster-to- point relationship via an influence function, here we further extend it so that cluster-to-cluste relationship can be considered. This motivates us to define Cluster-to-cluster Voronoi Diagram (CCVD):
(2), and all possible configurations of given
Definition 3.5 (Cluster-to-cluste Voronoi Diagram). Let Ω {wi,,.., be a partition of the space R" and C = {C,, .... Ck) } be a set of totally ordered sets with the same cardinality L (i.e. |C|| = C2 = ... = [CK] L). The set of pairs {[w]. Cy)..., (WK, Cx)]
} is a Cluster-t-ccluster
Voronoi Diagram (CCVD) with respect to the influence function FC2,, C(z)), and each cell is ob-
tained via w, {2 E IR"" r(2) = r), E (1,.,, K), with
max F(Cx,C(z)) r rz)) kE(
where C(2) is the cluster (also a totally ordered set with cardinality L) that query point = be- longs, which is to say, all points in this cluster (query cluster) will be assigned to the same cell. Similarly, the Influence Function is defined upon two totally ordered sets Ck- = (ck ( C(z) = 1
limi and
F(C,, sign(a) El
(10)
With this definition, now we are able to streamline our aforementioned novel approaches into a single ensemble model. Suppose there are totally L possible settings in our configuration pool {T} X (Pu.b.)) x {M). for all configurations {p:)--1 we apply them onto the support set S to generate the K totally ordered clusters ([ configuration Pt, and onto a query sample æ to generate the query cluster C(x) ) = ((((1), x 2(PL) },
including each center ck derived through
and then plug these two into Definition 3.5 to construct the final Voronoi Diagram.
When only the feature representation is considered in the configuration pool, i.e. P: E (T) x (Pu,b,)): }. our FSL process is named as Voeo, if surrogate representation is also incorporated, ie. Ps E {T} x (Pu.s.}} x {M). Deep is promoted to
that allows for higher geometric diversity. See Appendix A for summary of the notations and acronyms
4 EXPERIMENTS
The main goals of our exper- iments are to: (1) validate the strength of CIVD to integrate parametric and nonparamet- ric classifiers and confirm the necessity of Voronoi reduc-
Table 2: Summarization of the datasets used in the paper.
Datasets
Base classes
Novel classes
Image size
Images per class
MultDigiMMNIST miin--maaeeNet CUB
20
x 64
1030
20
84
600
100
50
84
x 84 x
44 ~ 60
Sered-ImageNer
tion; (2) investigate how different levels of heterogeneity individually or collaboratively contribute to the overall result, and compare them with the state-of-ar 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. 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.
Deep Voro-- Integrating Parametric and Nonparametric Methods via CIVD. To verify our proposed CIVD model for the integration of parameter/nooppaaamett FSL classifiers, we first run three standalone models: Logistic Regressions with Power/Voronol Diagrams as the underlining ge- ometric structure Nloeer-R/Vorrnoi-LL
and vanilla Voronoi Diagram (VD, i.e. nearest neighbor model), and then integrate VD with either Nower/Voronoi. -RR (see Appendix for details). Interest- ingly, VD with the Power-IR has never reached the best result, suggesting that ordinary LR cannot
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be integrated with VD due to their intrinsic distinct geometric structures. After the proposed Voronoi reduction (Theorem B.T), 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 3: The 5-way few-shot accuracy (in %) with 95% confidence intervals of Deep and DeepVoro++ compared against the oute-oo-the-assd art results on three benchmark datasets. 4I The results of DC and S2M2.K R are reproduced based on ppen-soureed implementations using the same random seed with VeepV
Methods
Nnii--ImageNet
CUB
liered- Gareed-imaeNee
Sway Ishot
5way 5chot
Sway
Ishot
5way Sshot
5way Ishot
5way Schoo
MAML (Fina al.
2017
54.69 0.89
66.62 1. 0.83
71.29 1. 0.95
80.33 1 0.70
51.67 1 1.81
70.30 0.08
Meta-SGD (La et 2017
50.47 1: 1.87
64.03 1: 0.94
53.34 1: 0.97
67.59 1 0.82
Meta Varianoe Trinsser Fark ct
2020
67.67 = 0.70
80.33 = 0.61
MetaGAN ZZhnn al. 2018
52.71 0.64
68.63 + 0.67
Enceder Schwwir al 2018
59.9
69.7
69.8
82.6
Matching
Vinvals ala
76.32
73.49
0.89
68.50
0.71
Prototypical NaT Nmell ct mi: 2017
54.16 0:8.2
73.68 + 0.65
72.99
+: 0.88
86.64 + 0.51
65.65 + 0.92
83.40 0.65
Baseline++ Chen
57.53
72.99
70.40 0.81
0.78
60.98
75,93 0.17
Viriational Few-s 2ian 41
2019
61.23:1:0 0.26
77.69 + 0.17
TriNet CChen 2019
58.12
76.92 - 0.69
69.61 0.46
84.10 + 0.35
LEOO Ruag alr 2015
61.76 0.08
77.59 +: 0.12
68.22 + 0:22
78.27 + 0.16
66.33 + 0.05
81.44 0.09
DCO Lee
2019
62.64 0.6411.0.61
78.63 1 0.46
65,99 1 0.72
81,56 0.53
Nogaltie- Cosine LLu 0t
2020a
63.85:0003.
81.57 1: 0,56
72.66 0.85
89.40 : 0.43
MTL (Wang et al 2023
0.22
71.72 0.09
67.11 = 0.12
83.69 0.02
Constetaaiion (irrral
2021
64.89
79.95 + 0.17
AFIIN (Liet al. Arzzhh
62.38 1. 0.72
78.16 1 0.56
70.53 1 1.01
83.95 1 0.63
AM3+TRAML
67.10
79.54
E3BM [Lu et al 30
63.80 1. 0.40
80,29 1 0.25
71.20 1: 0.40
85.30 0.30
SimpleShor Waig
2019
81.50
71.32 1
0.15
R2-D2 (Liu Srai 20,006
81.96 = 0,32
Rohust-artte
0.62
81.19 + 0.43
70.44 + 0.32
85.43 0.21
IEPT (7hang et al. 202216
67.05 4: 0:444
82,90 +: 0.30
72.24 +: 0.50
86.73 0.34
MELR [Ter 2021
67.40 0.44
83,40 1 0.28
70.26 0.50
85.01 1. 0,32
72.14
1: 0.51
87.01 0.35
S2M2 RT MMagaa orai. 2020
64.65 0.45
83.20 = 0,30
80.14 = 0.45
90.99 - 0.23
68.12 = 0.52
0.34 86.71
M-SVM- M+----- (1 m
2021
0.32
84.57 = 0.21
DeepEMD
Zhang et al_ 20200
66.91+00.88
82.41 +
75.65 0.83
0.50
71.16
0.58
DeeplEMD ZZiing e al. 2020
68.77 1177..0.222 0.29
84.13 1 0.53
79.27
1 0.29
89.80
0.51
74.29 1 0.32
86.98 0.60
DC9 (Yang
67.79
1:
0.46
90.77
74.24
0.31
PT+NCM (Hu al. 2021
65.35 33 + 0.20
83.87 =: 0.13
80.37 = 0.20
91.15
033::0000 0.10
69.96 + 0.22
86.45 1:45:000 0.15
Vorp Voroo+
69.48 0.45
86.75 + 0.28
82.99 + 0.43
92.62
0.22
74.98 + 0.48
89.40 +:00:000 0.29
71.30 0.46
85.40 + 0.30
82.95
0.43
91.21 + 0.23
75.38 + 0.48
87.25 + 0.33
Deep Voro: Improving FSL by Hierarchical Heterogeneities. In this section, we only consider two levels of heterogeneities for ensemble: feature-level and transformation-lewt 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). For transformation-leve en- semble, we use the proposed compositional transformations with 8 difffeent combinations of À and b that encourage a diverse feature transformations (Appendix F.I) without loss of accuracy (Fig- ure 2) The size of the resulting configuration pool becomes 512 and Deep 's performance is shown in Table 3 Clearly, Deep outperforms all previous methods especially on 5-way 5-shot FSL. Specifically. Voro is better than the next best by 2.18% (than [Ni et al. (2021) I) on mini- ImageNet, by 1.47% (than Hu et al. (2021)) on CUB, and by 1.02% (than Yang et al. (2021) ) on Imrred-lmageNet. 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. More detailed results can be found in Appendix F By virtue of CCVD and using the simplest VD as the building block, Voro is arguably able to yield a consistently better result by the ensemble of a massive pool of independent VD. VeepVoro also exhibits high resistance to outliers, as shown in 16g
Deep Voro++: Further Improvement of FSL via Surrogate Representation. In surrogate rep- resentation, the number of neighbors R for each novel class and the weight 33 balancing surro- gate/feature representations are two hyperparameters. With the help of an available validation set, 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. From Figure K.13 the accuracy of VD with varying hyperparameter shows good agree- ment between testing and validation sets. With this in mind, we select 10 combinations of /3 and R, guided by the validation set, in conjunction with 2 different feature transformations and 64 different image augmentations, adding up to large pool of 1280 configurations for ensemble (denoted by Voropor)++).. As shown in Table 3 Deep Voro++ achieves best results for 1-shot FSI. 2.53%
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Published as a conference paper at ICLR 2022
Table 4: Verp ablation experiments with feature(Feat. haaersoomnmmttnntaaa//eeeeentnet level heterogeneities on mini- mini-ImageNet 5-way few-shot dataset. L denotes the size of configuration pool, i.c. the number of ensemble members. +These lines show the average VD accuracy without CCVD integration.
Methods
Geometric Structures
Feat.
Trans.
Geo.
L 5-way 1-shot
5-way 5-shot
tunable parameters:
rotation etc.
w. A
B,
DeepVoro--
CIVD
X
x
X
65.85 + 0.43
84.66 + 0.29
X
11
66.92
84.64 0,30
X
X
0.44 66.45
84.55 + 0.29
Vorp
CCVD
X
X
67.88 0.45 69.48
86.39 0.29
64
8
X
512
0.45
86.75 + 0.28
X
14
68.68 0.46
84.28 1 0.31
CCVD w/ surrogate representation
2
10
68.38 + 0.46
83.27 - 0.31
DecpVoro+4
X
70.95 0.46 71.30
84.77 0.30
2
10
1280
85.40 - 0.30
A. minni-masstie S-eay S-ence
0. mins4magelet S-tay 1-deet
CLn Svaay S-hor
cLB Traay 14hhe
Figure 2: The 5-way few- shot accuracy of VD with different À and mini- ImageNet and CUB Datasets.
37-
70-
50-
78-
64
83-
92-
Texting
60-
Texting
Teating
70
Teatrs
120
A
00
b 1A
higher than Zhang et al. (2020b), 2.38% higher than Hu et al. 2021), and 1.09% higher than Zhang et al. (2020b), on three datasets, respectively, justifying the efficacy of our surrogate representation. See Appendix for more detailed analysis.
Ablation Experiments and Running Time. Table 4 varies the level of heterogencity (see Table F.4 and G.3 for all datasets). The average accuracy VDs without CCVD integration marked by and is significantly lower than the fully-fledged ensemble. Table presents the running time of Voro(++)+++ benchmarked in a 20-core Intel@ Core TM i7 CPU with NumPy (v1.20.3), whose efficiency is comparable to DC/S2M2.2, 2, even with > 1000x diversity.
Experiments with Different Backbones, Meta-training Protocols, and Domains. Because different feature ex- traction backbones, meta-training losses, and degree of discrepancy between the source/target domains will all af- fect the downstream FSL, we here examine robustness of Deep Vorp/Depp Voro++ under number of different cir- cumstances, and details are shown in Appendices H J Notably, Deep Vore/Deep Vewp/Deeeoooo+ Voro++ attains the best perfor- mance by up to 5.55%. and therefore corroborated as superior method for FSL, regardless of the backbone, training loss, or domain.
Table Running time comparison.
Methods
Time (min)
DC
88. 29
S2M2.R
33.89
#ensemble members:
0.05
DeepVoro
25.67
0.14
Voro+o
1280
179.05
CONCLUSION
In this paper, our contribution is threefold. We first theoretically unify parametric and nonparamet- ric few-shot classifiers into general geometric framework (VD) and show an improved result by virtue this integration (CIVD). By extending CIVD to CCVD, we present a fast geometric en- semble method (Deep Voro) that takes into consideration thousands of FSL configurations with high efficiency. 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++) In future studies, we plan to extend our geo- metric approach meta-learning-basse FSL and lifelong FSI.

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Published as a conference paper at ICLR 2022
ACKNOWLEDGMENTS
This research was supported in part by NSF through grant IIS-1910492.
REPRODUCIBILITY STATEMENT
Our code well as data split, random seeds, hyperparameters, scripts for reproducing the results in the paper are available at http://ggtttb..oomoorrepreeeee/s
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A NOTATIONS AND ACRONYMS
Table A.I: Notations and acronyms for VD, PD, CIVD, and CCVD, four geometric structures dis- cussed in the paper.
Geometric Structures
Acronyms
Notations
Description
Voronoi Diagram
Ck
center for Voronoi cell w2, & {1,... K}
dominating region for center R E fl,,,, K}
c
center for Power wy. k {I,.,K}
Power Diagram
PD
1%
weight center ce & [I,.,, dominating region for center ck, {l,.., K)
wik
C
cluster as the "center" for a CIVD cell Wi, & (1..., K}
Cluster-inducod Voronoi Diagram
dominating region for cluster C.
CIVD
influence function F F(C,, x) from cluster C₁ to query point z
a
magnitude of the influence
C₁
cluster us the "center" for CCVD cell wk, (I,... K} dominating region for cluster C₁
Cluster-to-cluster
CCVD
C(=)
the cluster that query point belongs
Voronoi Diagram
influence function F (Cg, C(z)) from Cj to query cluster C(z)
magnitude of the influence
Table A.2: Summary and comparison of geometric structures, centers, tunable parameters, and numbers of tunable parameters (denoted by #) for DeepVoro- DeepVoro, and Deep Voro+++ Parameters for feature-level, transformatiio-lleelll and geometry-level heterogeneity are shown
in yellow blue and red respectively. See Sec. F for implementation details. *Here PD is reduced to VD by Theorem 3.1] #For every À (or R), the b (or 3) value with the highest validation accuracy is introduced into the configuration pool.
Geometric
Methods
Structures
Centers
Tunable Param.
Description
Cs [ck.
Voro--
CIVD
6 from VI)
2 from PDt
angle rotation flipping scaling & cropping
C₁ = tel ( A (T) (Pa.o)
Vorp
CCVD
ar b
scale factor in lineur transformation shift factor linear transformation exponent in powers transformation
A
2
#configurations L 512
angle of rotation flipping or not scaling & cropping
w
scale factor in linear transformation shift factor in lineur transformation exponent powers transformation the number of top-R neurest buse prototypes for novel prototype weight for surrogate representation weight for feature representation
C (e dahe A: E (T) x (Ponat x (M}
Voro++ CCVD
2
#configurations 1250
B POWER DIAGRAM SUBDIVISION AND VORONOT REDUCTION
B. PROOF THEOREM 3.1
Lemma B.1. The vertical projection from the lower envelope of the hyperplanes (TI,(z) WT + by bi onto the input space IR" defines the cells of PD.
Theorem 3.1 (Voronoi Diagram Reduction). The linear classifier parameterized by W, b partitions the input space 8" Voronoi Diagram with centers {č1, ..., ČK) given by č₁ W. if by
W Wt 1,... K.
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Proof. We first articulate Lemma B.1 and find the exact relationship between the hyperplane (x) and the center of its associated cell in R". By Definition 3.1 the cell for a point 2 E R" is found by comparing d(z, cx 24 for different k so define the power function p(z, S) expressing this value
2z, S) (z (z= -²
(11)
in which 5 C IR"h is a sphere with center u and radius r. In fact, the weight v associated with a center in Definition 3.1 can be intepreted the square of the radius r². Next, let U denote a paraboloid
y = 2², let (5) be the transform that maps sphere S with center u and radius r into hyperplane
(T()) : y 22 u u
(12)
can be proved that IT is bijective mapping between arbitrary spheres in R and nonvertical hyperplanes in 1=+1 that intersect U (Aurenhammer 1987). Further, let æ' denote the vertical projection of z onto U and z" denote its vertical projection onto II(S), then the power function can be written as
p(z, 5) = d(2,
=(2,
(13)
which implies the following relationships between sphere in 1 and an associated hyperplane in Ra+1 (Lemma 4 in Aurenhammer (1987)): ) let S₁ and S2 be nonco-centeric spheres in R", then the
bisector of their Power cells is the vertical projection of (I1s (S22) onto R". Now, we have direct relationship between sphere 8, and hyperplane II,.., and comparing equation (12) with the
hyperplanes used in logistic regression (T) W z by == gives us
K
1
u WR
2
(14)
1
, = by + 1
Wx
Although there is no guarantee that 6y + W gression model, we can impose a constraint on 2 to keep it be zero during the optimization, which implies
Wlll [2 is always positive for an arbitrary logistic re-
by
(15)
By this way, the radii for all K spheres become identical (all zero). After the optimization of logistic regression model, the centers { H will be used for CIVD integration.
C DETAILS ABOUT THE DEMONSTRATIVE EXAMPLE ON MULTIDIGIMNIST DATASET
MultiDigitMNIST (Sun 2019) dataset is created by concatenating two (or three) digits of different classes from MNIST for few-shot image classification. Here we use DoubleMNIST Datasets (i.c. two digits in image) consisting of 100 classes (00 to 09), 1000 images of size 64 x 64 x per class, and the classes are further split into 64, 20, and 16 classes for training, testing, and validation, respectively. To better embed into the R space, we pick 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. The feature extractor a 4-layer convolutional network with an additional fully-connected layer for 2D embedding. In Figure I left panel, the VD is obtained by setting the centroid of each base class as the Voronoi center. For each novel class, the Voronoi center simply the 1-shot support sample (Figure [ the collection of distances from a support/query sample each of the base classes, as shown in Figure I right panel. Interestingly, the surrogate representations for 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. 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 with feature representation, and how to integrate into the overall ensemble workflow.
central panel). The surrogate representation is computed as
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D MAIN DATASETS
For fair and thorough comparison with previous works, three widely-adopted benchmark datasets are used throughout this paper.
(1) mini-ImageNNet (Vinyals al. 2016) is 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. Each class has 600 images of size 84 x 84 x 3.
(2) CUB (Welinder et al. 2010) is another benchmark dataset for FSL, especially fine-grained FSL, including 200 species (classes) of birds. CUB an unbalanced dataset with 58 images in average per class, also of size 84 x 84 x 3. We split all classes into 100 base classes, 50 novel classes, and 50 validation classes, following previous works (Chen et al. 2019a).
(3) tiered-ImaggeNet (Ren et al. 2018) is another subset of ILSVRC-12 (Russakovsky et al. 2015) but has more images, 779,16s images in total. All images are categorized into 351 base classes,
97 validation classes, and 160 novel classes. The number of images in each class is not always the
same, 1281 in average. The image size also 84 84
E DEEPVORO--; INTEGRATING PARAMETRIC AND NONPARAMETRIC METHODS VIA CIVD
Table E.3: Cluster-induced Voronoi Diagram (CIVD) for the integration of parametric Logistic Re-
gression (LR) and nonparametric nearest neighbor (i.e. Voronoi Diagram, VD) methods. The results from S2M2_R and DC are also included in this table but excluded for comparison. Best result is marked in bold.
Methods
mini-ematenet
CUB
5-way I-shot
5-way 5-shot
5-way 1-shot
5-way 5-shot
5-way 1-shot
5-way 5-shot
S2M2 R DC
64.65 0.45
83.20 + 0.30
80.14 + 0.45
90.99
0.23
68.12 + 0.52
86.71
0.34
67.79
83.69 i 0.31
79.93
0.46
90.77 0.24
74.24 0.50
88.38
0.31
Power-LR
65.45 0.44
84.47 = 0.29
79.66
0.44
91.62 0.22
73.57 0.48
89.07
0.29
Voronoi-LR
65.58 0.44
84.51 1 0.29
79.63 0.44
91.61 0.22
73.65 1 0.48
89.15
0.29
VD
65.37 0.44
84.37 + 0.29
78.57 0.44
91.31 + 0.23
72.83 + 0.49
88.58 + 0.29
CIVD-based Voro-
VD + Power-LR
65.63 0.44
84.25 + 0.30
79.52 + 0.43
91.52 + 0.22
73.68 + 0.48
88.71 + 0.29
VD + Voronoi-LR
65.85 + 0.43
84.66 + 0.29
79.40 = 0.44
91.57 = 0.22
73.78 + 0.48
89.02 + 0.29
E. EXPERIMENTAL. SETUP AND IMPLEMENTATION DETAILS
In this section, we first establish three few-shot classification models with different underlying ge- ometric structures, two logistic regression (LR) models and one nearest neighbor model: (1) Power Diagram-bass based LR (Power-LR), (2) Voronoi Diagram-based LR (Voronoi-LR), and (3) Voronoi Di- agram (VD). 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/Voronol Diagram- LRs.
The feature transformation used throughout this section is Pur.. with 1.0, = 0.0. X = 0.5. For Power-LR, we train it directly on the transformed K K-way --shot support samples using PyTorch library with an Adam optimizer with batch size at 64 and learning rate at 0.01. For Voronoi-LR, the vanilla LR is retrofitted as shown in Algorithm| [ in which the bias given by Theorem 3.1 to make sure that the parameters induce a VD in each iteration.
In our CIVD model Definition 3.2 we use a cluster instead of a single prototype to stand for a novel class. Here this cluster contains two points, i.e. C₁ = {Ck, Cx), in which C₁ is obtained from VD, and č is acquired from Power-LR or Voronoi-LR. The question we intend to answer here is that whether Power-LR Voronoi-I.R is the suitable model for the integration.
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Algorithm 1: Voronoi Diagram-based Logistic Regression.
Data: Support Set
Result: W
Initialize W W
for epoch f- 1, L..., #epoch do
by f = vk = 1,... K;
Apply Theorem 3.1 forward propagation backward propagation
Compute loss L(W,0) b)
Update W
end
return W
Transformation
L2 Normalization
Power Transtomaation
Log Transtormation
80
40
60
60
40-
40
20
20
20
-20
-20-
-40
.40
-60
-40
-20
-40
-20
20
-100
-50
40
60
Figure F.1: The t-SNE visualizations of (A) original features, (B) L2 normalization, (C) Tukey's Ladder of Powers Transformation with = 0.5, and (D) compositional transformation with A 1,b 0.04 of novel classes from Amai-emaeNNet dataset.
E.2 RESULTS
The results are shown in Table E.3 the best result, suggesting that VD and LR are intrinsic different geometric models, and cannot be simply integrated together without additional effort. On mini-ImageNet and tiered- 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 few-shot models. Notably, on these two datasets, when Power-LR reduced to Voronoi-LR, al- though 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 65.45% to 65.58% on 5-way 1-shot mini-ImageNer data. On CUB dataset, results of different models similar, prob- ably because CUB is a fine-grained dataset and all classes are similar to each other (all birds).
Interestingly, when integrated with VD, Power-LR never reaches
ideally integrate parametric and nonparametric
F DEEPVORO: IMPROVING FSL VIA HIERARCHICAL HETEROGENEITIES
F.1 EXPERIMENTAL SETUP AND IMPLEMENTATION DETAILS
In this section we describe feature-level and transformation-levee heterogeneities that are used for ensemble in order to improve FSL. See the next section for geometry-level heterogeneity.
Feature-level heterogeneity. Considering the reproducibility of the methodology, we only employ deterministic data augmentation upon the images without randomness involved. Specifically, three kinds of data augmentation techniques are used. (1) Rotation is an important augmentation method widely used in self-supervised learning (Mangla al. 2020). Rotating the original images by 0%, 90". 180", and 270eg us four ways of augmentation. (2) After rotation, we can flip the images horizontally, giving rise to additional two choices after each rotation degree. (3) Central cropping after scaling can alter the resolution and focus area of the image. Scaling the original images to (84+B) x (84- B), B increasing from 0 70 with step 10, bringing eight ways of augmentation.
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Finally, different combinations of the three types result in 64 kinds of augmentation methods (i.e. [[H = 64).
heterogeneity. In our compositional transformation, the function (hx g ,6 (z) is parameterized by w, b, X. Since g is appended after the L2 normalization f. the vector comes into g is always unit vector, so we simply set w 1. For the different combinations of À and b, we test different values with either À = 0 or À o O on the hold-out validation set (as shown in Figure 2 and 1.12). and pick top-8 combinations with the best performance on the validation set.
Transformation-leve
Ensemble Schemes. Now, in our configuration pool (T} x (PuLb.)) there are 512 possible con- figurations {o(*) )am. For each p. we apply it on both the testing and the validation sets. With this large pool of ensemble candidates, how and whether to select subset {p(0) S (o(0) a nontrivial problem. Here we explore three different schemes. (1) Full (vanilla) ensemble. All ]i=]
]i= is still
candidates in {p(t) ) are taken into consideration and then plugged into Definition 3.5 to build the CIVD for space partition. (2) Random ensemble. A randomly selected subset with size I'' < L. is fi=1 on the validation ]i=1 from the testing set, provided that there is good correlation between the testing set and the validation set. Specifically, we rank the configurations in the validation set with regard to their performance, and add them sequentially into {p(0) // until a maximum ensemble performance is reached on the validation set, then use this configuration set for the final ensemble. Since VD is nonparametric and fast, we adopt VD as the building block and only use VD for each p for the remaining part of the paper. The value in the influence function (Definition 3.3 is set at throughout the paper, for the simplicity of computation.
used for ensemble. (3) Guided ensemble. We expect the performance for {500)
set can be used to guide the selection of {
For fair comparison, we downloaded the trained model used by Mangla al. 2020 and Yang it al. (,02T) The performance of FSL algorithms is typically evaluated by a sequence of inde- pendent 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. To ensure the fairness of our eval- uation, DC (Yang et al. 2021), and S2M2 (Mangla et al. 2020) are reevaluated with the same data split and random seed as Deep Voro. The results are obtained by running 2000 episodes and the average accuracy as well as 95% confidence intervals are reported.
F.2 RESULTS
Table F.4: Ablation study of Deep Voro's 's performance with different levels of ensemble. The number of ensemble members are given in parentheses.
Methods
Feature-teve Trunsformaton--vel
mini- ImageeNt
CUB
tiered tiered- Imaggeeet
I-shot
S-hhot
I-shor
5-bhot
I-shot
S-toor
No Ensembii
65.37 0.44
84.37 = 0.20
78:57
0:44
91.31 + 0.23
72.83
+ 0.49
88.58 + 0125
Vanilla Ensemble
66.45
0.44
84.55
0.29
80.98
0.41
91.47
1 0.22
74.02
0.49
88.90
0.29
Vanilla Ensemble (6-)
67.88
0.29
77.30
0.43
91.26
73.74
0.49
W8.67
0129
Vanilla Ensemble
20.23
79.00
74.51
0.45
Random Linsemble 512)
69.30
80.40
0.43
91.94
74.64 0.45
89.15
Guided Essemble (512)
69.48
0.28
82.99
0.43
74.98 - 0.48
89.40
Our proposed compositional transformation enlarges the expressivity of the transformation function. When the Tukey's 's ladder of powers transformation is used individually, as reported in YYngg et al.
20211, the optimal À is not O. but if an additional linear transformation g is inserted between f and h, A 0 coupled with a proper b can give even better result, as shown in Figure 2 and K. Impor- tantly, from Figure a combination of À and b with good performance on the validation set can also produce satisfactory result on the testing set, suggesting that is possible to optimize the hyperpa- rameters on the validation set and generalize well on the testing set. In terms of the polymorphism induced by various transformations in the feature space, Figure F. exhibits the t-SNÉ visualiza- tions 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). Besides commonly used data augmentation, this transformation provides another level of diversity that may be beneficial to the subsequent ensemble.
The results for different levels of ensemble are shown in Table F.4 in which the number of ensemble member are also indicated. Although transformation ensemble does not involve any change to the feature, it can largely improve the results for 1-shot FSL, from 37% 66.4%% on ImageNe,
downloaded from httpp://ggthbbb.ocooo/m/////////
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from 78.37% to 80.98% CUB, and from 72.83% to 74.02% tiered- ImageNet, respectively, probably because 1-shot FSL is more prone to overfitting due to its severe data deficiency. Feature- level ensemble, the other hand, is more important for 5-shot FSL, especially for mini-lmagell When combining the two levels together, the number of ensemble members increases to 512 and the performance significantly surpasses each individual level. 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 indeed be used for the guidance of subset selection and our method is robust cross classes in the same domain. When there is no such validation set available, the full ensemble and random ensemble schemes can also give comparable result.
To inspect how performance changes with different number of ensemble members, we exhibit the distribution accuracy at three ensemble levels for mini-ImageNet Figure F:2 and F3 for CUB in Figure[F.4] and and for tiered- ImageNeet in Figure [-6] and F.7 Figure (b) in each of them also exhibits the correlation between the testing and validation sets for all 512 configurations. Clearly, better result is often reached when there more configurations for the ensemble, validating the efficacy of our method for improving the performance and robustness for better FSL.
Algorithm 2: VD with Surrogate Representation for Episode T.
Data: Base classes D, Support Set {( wi) RXN Di CT, -uery sample æ
Result: d
D' (Pw.b, o T) S & (Pabb. o 7 (S);
Extract and transform feature
+ (Pwnb,) (
for + I,,. Chare
< Compute prototypes of base classes
do
ci f (('')) Ex'ED) x==
end
for
<- k;
a Compute prototypes from support samples
do
10
C₁ 1/4 Ez'SS z1:
11
dk + ckz,ck)
12 end
13
Csurrogate f- 0:
14 for ..., K;
Find surrogate classes
15 do
16
Churrogate @surrogate
Top-R d(ckk c))
(E(I....,(Coo)) (roul)
17 end
18 R [surrogate]:
19 d' < (d(z, cG):.... d(z, cg)):
Compute surrogate representation for query sample
20 for k t- K:
Compute surrogate representations for support samples
21 do
d' (d(ck, d)),.., d(ck, cg))
23
d' d(d', d,)
end
25 d +-
d
d"
4 Compute final criterion
|
26 return
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mini-ImageNNt 5-way 5-shot
mini-lmageNeet transformation-level Ensemble
805
84.60
Tretaformal Esserblle
Full Ensemble
38.0
Dasp/roro
leve teertbl
84.55
Accuracy
84.50
87.0
Valtdatio Set
84.45
86.5
36.0
Number of Ensemble Members
02
53
85
Testing Set Accuracy
(a) Transformation-lvvel Ensemble
(b) Testing/Validatt Sets Correlation
05.5
Ersentle
05.0
20 2433333
Number Mumbers
(c) Feature-leve Ensemble 5-way 5-shot mini- Ima--emeeeNet Dataset
09.0
05.0
Randem
200
420
5to
Numbor Ensemble Members
(d) Voro on 5-way 5-ohot mini- mmiee-lmaeeNet Dataset
Figure F2: Three levels ensemble and the correlation between testing and validation sets with different configurations in the configuration pool.
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minl-Image 5-way 1-shot
mini-lmageNNt transformation-evell Ensemble
66.8
72
Tretatormal Ensembie leve
Dosptore
66.6
66.4
Accuracy
66.2
Valtiation
67-
Number of Ensemble Members
63
55
65
67
60
Testing Siit Accuracy
(a) Transformation-level Ensemble
(b) Testing/Validatte Sets Correlation
m
67.
100
66.0
Ersentle
65.0
2022222033311 24 Mumbers
Number
(c) Feature-leve Ensemble 5-way 1-shot mini- Imageeet Dataset
Randem
100
400
sto
Numbor Ensemble Members
(d) Voro on 5-way 1-shot mini- mmiee-lmaeeNet Dataset
Figure F3: Three levels ensemble and the correlation between testing and validation sets with different configurations in the configuration pool.
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CUB 5-way
CUB transformation-leve Ensemble
Tretatormal Exsembble leve teentbl
91.6
Full Ensemble
DeapVors
90
91.4
Accuracy
91.2
91.0
Validiation
90.8
82-
1
Number of Ensemble Members
02
84
80
92
Testing Set Accuracy
(a) Transformation-level Ensemble
(b) Testing/Validation Sets Correlation
21
Feature-Ievel Ensemble on 5-way 5-shot CUB Dataset
Accurcoy
8
Randem
102
300
500
Number Ensemble Nombor
(d) Voro on 5-way 5-shot CUB Dataset
Figure E4: Three levels ensemble and the correlation between testing and validation sets with different configurations in the configuration pool.
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CUB 5-way 1-shot
81.50
CUB transformation-lvel Ensemble
825
Tretatormal Encemblc
Full Ensemble
leve Ensenbie
81.25
Doaptore
81.00
77.5
Accuracy
75.0
80.50
72.5
80.25
80.00
67.
79.75
65.0
Number of Ensemble Members
52.5
00.5
57.
70.0
72.5
75.0
77.5
80.0
82.5
Testing Set Accuracy
(a) Transformation-level Ensemble
(b) Destinn/Validation Sets Correlation
Emnemble
21 23:32333334333
(c) Feature-Ievel Ensemble on 5-way 1-shot CUB Dataset
02.
Ensembee truamile
200
400
5t0
Numbor Ensemble Members
(d) Voro on 5-way 1-shot CUB Dataset
Figure F5:: Three levels ensemble and the correlation between testing and validation sets with different configurations in the configuration pool.
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tiered-ImageNet 5-way 5-shot
Niered-lmageNett trenssormation-levell Ensemble
Tretaformat Exsemble
88.95
87
leve toursbll
Desp/roro
88.90
88.85
Accuracy
88.80
85
Validation
68.75
88.70
88.65
Number of Ensemble Members
85
86
00
90
Testing Sat Accuracy
(a) Transformation-level Ensemble
(b) Testing/Validatt Sets Correlation
00.5
88.0
06.5
Ersentle
Number
Mumbers
(c) Feature-level Ensemble on 5-way 5-shot tiered Ieeee-ee Dataset
07
Random
trsemble
102
300
500
Number Ensemble Nombor
(d) Voro on 5-way shoh tiered- ImageNet Dataset
Figure F6: Three levels ensemble and the correlation between testing and validation sets with different configurations in the configuration pool.
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tiered-ImageNet 1-shat
Jered-lmageNNl leens Ensemble
73
Tretaformal Exsemble
leve Ensemble
Dasp/roro
74.0
71
73.8
Accuracy
69
73.6
Valitation
73.4
73.2
Number of Ensemble Members
50
71
72
73
74
Testing Sett Accuracy
(a) Transformation-level Ensemble
(h) Testing/Validation Sets Correlation
73.
73.0
71.0
Ersentle
Number Mumbers
(c) Feature-level Ensemble on 5-way 1-shot tiered -mageeet Dataset
72
trsemble
102
300
500
Number Ensemble Nombor
(d) Voro on 5-way 1 shot tiered- ImageNet Dataset
Figure F7: Three levels ensemble and the correlation between testing and validation sets with different configurations in the configuration pool.
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Effect the number of shots on mini-ImageNet dataset
20 235
200:91.22%
400: 51 11.55%
15. 68.9199
10: 6.06
7:95-95%
3:844050%
Accuracy
80 -
20.12%
11%
100
300
Number of Shots
Figure G.8: The accuracy of VD with increasing number of shots on mini-ImageNet dataset.
G DEEPVORO+4: FURTHER IMPROVEMENT OF FSL VIA SURROGATE REPRESENTATION
G.1 EXPERIMENTAL SETUP AND IMPLEMENTATION DETAILS
In this section, we introduce another layer of heterogeneity, that is, geometry-level. that exists in our surrogate representation. In Definition 3.4 increasing R will enlarge the degree of locality when searching for the top- surrogate classes. In equation (8). if we set 7 then increasing will make the model rely more on the feature representation and less the surrogate representation. In R and on the
order to weigh up R and B. we perform grid search for different combinations validation set, shown in Figure K.13 K.14, and K.15 For each R, we select the 3 that gives rise to the best result on the validation set, and use this (H, on the testing set, resulting in 10 such pairs in total. So there of locality. In conjunction with feature-level (64 kinds of augmentations) and transformatton-leew (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. In conclu- sion, there are overall 512 1280 1792 configurations for few-shot episode. Generating 1800 ensemble candidates is nearly intractable for parametric methods like logistic regression or cosine classifier, which may consume e.g. months for thousands of episodes. However, the VD model nonparametric and highly efficient, making it empirically possible to collect all the combinations and integrate them all together via CCVD. The complete algorithm for the computation surrogate representation is shown in Algorithm 2
10 models in the geometry-leve heterogeneity, standing for different degrees
G.2 RESULTS
The heatmaps for different (R, 9)) pairs on testing/validation sets are shown in Figure 1.13 for mini- ImageNet, in Figure K. K.144for CUB, and Figure hereddImageNet respectively. Basically, the testing and validation set follow the same pattern. When Rt is small, i.e. only small number of base classes are used for surrogate, then a higher weight should be placed on feature representation. With fixed B, increasing R beyond certain threshold will potentially cause drop in accuracy, probably because the meaningful similarities is likely to be overwhelmed by the signals from the large volume of irrlevvnt base classes.
Table G.5: Ablation study of Voro++'s performance with different levels of ensemble. The number of ensemble members are given in parentheses.
Methods
Feature-level Transfommation-eed
Geometry-level mini- ImageNet
CUB
lireed-ImageNet
No Ensemble
65.37 0.44
78.57 + 0.44
72,83 : 0.49
Vanilla Ensemble (20)
68.38 0.46
80.70
0.45
74.48
0.50
Vanilla Ensemble (64)
70.95 0.46
81.04
+ 0.44
74,87
: 0.49
Vanilla Ensemble (1280)
Random Ensemble (1280)
71.34 0.46
81.98 + 0.43
75.07
0.48
Guided Ensemble (1280)
71.30 0.46
82.95
0.43
75.38
0.48
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Voro++ further improves upon Deep Voro for 5-way 1-shot FSL by 1.82% and 0.4% on mini- Imag-em and tiered-ImageNet, respectively, and is comparable with 82.99%). Notably, on 5-shot FSL. Deep DeepVoro++ usually causes drop accuracy from Voro To inspect the underlying reason for this behavior, we apply VD 5-way K-shot FSL with K increasing from 400 and report the average accuracy in Figure G.8 It can be observed that, in extreme low-shot learning, i.c. K E [1, 5], simply adding one shot makes more prominent contribution the accuracy, suggesting that the centers obtained from 5-shot samples are much better that those from only sample, so there is no necessity to resort surrogate representation for multi-shot FSL and we only adopt VeeoVoro for 5-shot episodes in the remaining part of this paper.
As shown in Table 3 and G.5
Vero CUB dataset (82.95%
Ablation study of Deep Voro++ with different levels of ensemble is shown in Table G.5 Figure G.9 G.10 and(G.T) All three layers of heterogeneities collaboratively contribute towards the final resulr. The fully-ffeee -flddeedd Deep Voro++ establishes new state-of-the-art performance on all three datasets for 1-shot FSL.
Surrogate Representation
mini-Image Net 5-way 1-shot
Surrogate Representation
78.0
mini-ImageNett wwssfomationggemmmttyyeeee
Ensemble
68.6
77.-
Full Ensemble
68.4
72.0
765
Accuracy
68.2
76.0
68.0
Valto
755
67.8
75.0
245
feature level Ensemble
Transiormation lesn Ensenble
Deep/iomes
6 91011223141522111811922
74.0
Number
Members
Testing Set Accurscy
(a) Transformation-level Ensemble
(b) Testing/Validaan Sets Correlation
Surragate Represontation Enserei
400
8
Ersentle
22 Number Ensemble Nembers
(c) Feature-level Ensemble 5-way 1-shot mini--IageNee Dataset
Surrogate Resriscntation
71.5
71.0
70.5
70.0
68.5
400
600
800
1030
1200
Ensemble Members
(d) Deep on 5-way 1-shot mini mini-Immnede ImageNet Dataset
Figure G.9: Three levels of ensemble and the correlation between testing and validation sets with different configurations in the configuration pool.
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Reprasentation CUB 5-way 1-shot
Surrogate Regresentation on CUB transtormattnnggemmettt--tt eevel Ensemble
250
Tretatormal Geoneryy leve Ensembi
leve Ensemble
$2.5
DonpVors
30.0
77.5
Accuracy
80.5
75.0
725
80.0
700
79.5
675
2 4
5
8 1011121314151617181833
Number of Ensemble Members
63.0
675
70.0
72.5
75.0
77.5
85.0
82.5
Testing Selt Accuracy
(a) Transformation-level Ensemble
(h) Testing/Validation Sets Correlation
Surrogate Representatinn
62.5
80.0
72.5
Ersentle
21
Feature-Ievel Ensemble on 5-way 1-shot CUB Dataset
trsemble
72
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Number Ensemble Nombors
(d) Voro+t on 5-way 1-shot CUB Dataset
Figure 10: Three levels of ensemble and the correlation between testing and validation sets with different configurations in the configuration pool.
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Reprasentation
tiered-ImageNet 5.way 1-shot
Surrogate Regreseniatiin
74
tiered-imageNet taanstormatengggmmeeryyyyy Ensemble
Trenatormal Geaneryy leve Ensembic
leve Ensemble
DeepVors
74.55
Accuracy
74.50
439
71-
Valicattion
70-
74.40
2 4 5
8 101112131411661111192
Number of Ensemble Members
74
75
Testing Sett Accuracy
(a) Transformation-level Ensemble
(h) Testing/Validation Sets Correlation
Surrogate Representotion
Ersentle
(c) Feature-level Ensemble on 5-way 1-shot tiered Imageeee Dataset
Resresentation
05.0
24.0
Ensembee truemiie
73.0
400
800
1020
1200
Numbor Ensemble Members
(d) DeepVoro++ on 5-way --shot tiered ImageNet Dataset
Figure G.1I: Three levels of ensemble and the correlation between testing and validation sets with different configurations in the configuration pool.
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H EXPERIMENTS WITH DIFFERENT BACKBONES
H.1 IMPLEMENTATION DETAILS
In order to test the robustness of DeepVoro//Deeo Voro/DeeoVoooot with various deep learning architectures, we downloaded the trained model used by Wang et al. (2019). We evaluated DC, S2M2 R, Deep Voro, and Deep Voro++ using the same random seed. The results are obtained by running 500 episodes and the average accuracy as well as 95% confidence intervals are reported.
H.2 EXPERIMENTAL RESULTS
Table H.6: Comparison of FSL algorithms with different network architectures. WRN-28-10g was trained with rotation loss and MixUp loss (Mangla et al. 2020 instead of using ordinary softmax loss (WRN-28-10)).
Methods
WRN-28-103
WRN-28-10;
1-shot
5-shot
1-shot
5-shot
DC
67.79 + 0.45
83.69 + 0.31
62.09 + 0.95
78.47 + 0.67
S2M2_R
64.65
0.45
83.20 + 0.30
61.11 + 0.92
79.83 - 0.64
VerpVoro
69.48 0.45
86.75 + 0.28
62.26 + 0.94
82.02 + 0.63
Voro++
71.30 0.46
65.01 + 0.98
DenseNet-121
ResNet-34
1-shot
5-shot
1-shot
5-shot
DC
62.68 + 0.96
79.96 1 0.60
59.10 + 0.90
74.95 1 0.67
S2M2.R
60.33 + 0.92
80.33 1 0.62
58.92 1 0.92
77.99 1 0.64
Deep Voro
60.66 + 0.91
82.25 1 0.59
61.61 + 0.92
81.81 + 0.60
DeepVoross Voro++
65.18 0.95
64.65 + 0.96
ResNet-18
ResNet-10
1-shot
5-shot
1-shot
5-shot
DC
60.20 + 0.96
75.59 + 0.69
59.01 + 0.92
74.27 + 0.69
S2M2_R
59.57 = 0.93
78.69 + 0.69
57.59 + 0.92
77.10 + 0.67
DeepVore
61.50 0.93
81.58 + 0.64
58.34
1 0.93
79.05 + 0.63
DeepVoro+>
64.79 + 0.97
61.75 + 0.95
MobileNet
Conv-4
I-shot
5-shot
I-shot
5-shot
DC
59.41 = 0.91
76.07 + 0.66
49.32 + 0.87
62.89 + 0.71
S2M2.R
58.36 0.93
76.75 + 0.68
45.19 + 0.87
64.56 + 0.74
Voro
60.91 0.93
80.14 + 0.65
48.47 + 0.86
65,86 + 0.73
Vero++
63.37 0.95
52.15 + 0.98
On Wide residual networks (WRN-28-L) (Zagoruyko & Komodakis 2016). Residual networks
(ResNet- 10/18/34)) (He al. 2016). Dense convolutional networks (DenseNet-122 (Huang al. 2017]. and MobileNet (Howard et al. 2017). DeepVoro/eeepp Voro++ shows a consistent improve- ment upon DC and S2M2.R. Excluding Voro/DDe/eeep Voro++, there is no such method that is always better for both 5-shot and I-shot FSL. Generally, DC is expert in I-shot while S2M2.2 favors 5-shot. According to Table we do not apply Deep Voro++ on 5-shot FSL since Deep usually outperforms Deep Voro++ with more shots available.
EXPERIMENTS WITH DIFFERENT TRAINING PROCEDURES
I.1 EXPERIMENTAL SETUP AND IMPLEMENTATION DETAILS
Our geometric space partition model built on top a pretrained feature extractor, and the quality of the feature extractor will significantly affect the downstream FSL (Mangla et al.) 2020). Here we used another two feature extractors trained with different schemes. (I) Manifold Mixup training employs additional Mixup loss that interpolates the data and the label simultaneously and can
downloaded from http:::/gitthbb..cm//ieeeeeaaap
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Published as a conference paper at ICLR 2022
help deep neural network generalize better. (2) Rotation loss is widely used especially in self- supervised learning in which the network learns to predict the degree by which an image is rotated. We downloaded the corresponding pretrained models used by Mangla al. (2020) and Yang et al. 2021 and evaluate the four methods by 500 episodes.
1.2
RESULTS
Table L.7: Comparison of performance with different meta- vreta-training procedures.
Methods
Self-supervision w/ Rotation Loss
Manifold Mixup
1-shot
5-shot
1-shot
5-shot
DC
66.43 0.86
82.61 0.62
62.61 0.90
78.62 0.68
S2M2.R
58.33 + 0.96
79.26 + 0.66
48.11 + 0.96
72.74 + 0.74
DeepVoro
68.80 + 0.86
85.70 + 0.58
65.00 + 0.93
83.19 + 0.65
Voro++
69.23 + 0.89
65.25 + 0.93
achieves best results for both rotation loss and Mixup loss. Interestingly. there is substantial gap between the two training schemes when they are used out-of-the-box for downstream FSL (Aaccuracy 10.22% for 1-shot and Zaccuracy 6.52% for 5-shot), but after VeeeooooDDeeeoooeee 1-shot and Laccuracy 2.51% % for 5-shot), suggesting the strength of DeepVoro/Deep to make the most of the pretrained models
As shown in Table 1.7 VoepVoro/Dee
this gap becomes narrowed (Aaccuracy = 3.98% for
J CROSS DOMAIN FEW-SHOT LEARNING
J.1 EXPERIMENTAL SETUP AND IMPLEMENTATION DETAILS
Cross-domain FSL more challenging than FSL in which base classes and novel classes come from essentially distinct domains. To examine the ability of our method for cross-domain FSL, apply the feature extractor trained on mini-Image.N (CUB) on the few-shot data CUB (mini-ImageNet) for coarse-to-fie to-se-to-fine (fine-to-coarse) domain shifting.
J.2 RESULTS
Table J.8: Comparison of performance on cross-domain FSL.
Methods
CUB mini- ImageNe
mini-ImageNet CUB
I-shot
5-shot
I-shot
5-shot
DC
46.25 0.93
62.99 + 0.81
54.64 + 0.87
72.83 + 0.71
S2M2.R
41.15 = 0.84
58.09 1 0.79
49.01 i 0.88
69.99 1 0.71
Voro
46.15 0.90
64.60 + 0.80
49.03 = 0.87
72.30 + 0.74
Voro++
47.83 0.97
54.88 + 0.92
Basically, Deep Vorto//DDeeetott is more stable than the other two method with shifting domain, especially on fine-to-coarss FSL (CUB to mini-ImageNNt), with an improvement of 6.68% for 1- shot and 6.51% % for 5-shot than S2M2_R R, and is comparable with DC on coarse-to-fine FSL (mini- ImageNet to CUB).
K ADDITIONAL FIGURES
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tiered-ImageNeet 5-way 5-shot
tiered-imageNet 5-way 1-shot
89
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Validttio Testing
Validation Testing
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0.10
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b
Figure K. 12: The 5-way few-shot accuracy of VD with different À and 6 values on tiered- Beree-lmaaeNet datasets.
mini-lmageNet 5-way 1-shot Test Set
mini-ImageNet 5-way 1-shot Va Validation Set
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57.01
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R
R
Figure K.13: The 5-way 1-shot accuracy with different and R values on mini-ImageNet testing (left) and validation (right) datasets.
CUB 5-way 1-shot Test Set
5-way 1-shot Validation
0.5
50:13
00.18
77.54
77.70
1.5
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60.
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90.4
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00 46
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00
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80.00 02
77.59
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77.97
77 77.95
7.0
R
Figure K.
The 5-way 1-shot accuracy with different and R values CUB testing (left) and validation (right) datasets.
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tiered-ImageNet 5-way 1-shot Test Set
biered-lmageNet 5-way 1-shat Validation Set
75.34
72.59
21.42
10.57
70.24
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72.83
72.54
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12.25
22-12
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75.01
72.88
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71.45
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71:25
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12.52
78.99
73.21
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72.42
7211
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2.5
73,02
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13.72
73.66
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13.57
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73.47
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71.49
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7355
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73.45
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93.02
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70.36
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73.55
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70.34
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70:41
2
4
6
8
10
7
10
R
Figure K.15: The 5-way 1-shot accuracy with different B and R values on tiered-ImageNer testing (left) and validation (right) datasets.
Episode Accuracy as a Function of Geometric Variance
S2M2_R
DC
DeepVoro
90
Average Accuracy
85
80
75
70
1.4
1.5
1.6
1.8 Geometric Variance of Support Set
1.7
Figure K.16: Outlier Analysis. In order to investigate the resistance outlier for various methods, we here define Geometric Variance (GV) as reflection of the possibility that a support set contains outlier, due the difficulty of inferring auttof-distribution sample from merely Formally, for support set S {(z,, y/ ) KxN its Geometric Variance is defined GV(S)
samples.
K DE (3) Eie(L..., ('EEL..,,)JJ(LI...... ||2; - 2ý measuring the average point-to-point distance in this support set. The larger GV is, with higher probability contains an outlier. For every episode in 2000 episodes from 5-way 5-shot mini-ImageNet data, GV is computed as well the episode accuracy. As shown in Figure very high GV causes a significant decrease of episode accuracy, but method Deep Voro is more resistant the presence of outliers.
33


Reviews:
[{"review_id": "_4v0c2J6qek", "reviewer": "Reviewer_Qc9m", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Strengths\n1.\tThe new geometric point of view of FSL is novel and interesting. This paper makes a bridge between computational geometry and FSL. \n2.\tThis approach achieves the new state-of-the-art performance over three standard datasets, including mini-ImageNet, CUB, and tiered-ImagenNet. This demonstrates its effectiveness. \n\nWeaknesses\n\n1.\tThe ablation study seems to be not enough. What is the performance of the proposed method if the ensemble strategy is removed? The reviewer did not find the corresponding experimental results. \n2.\tUsually, the ensemble strategy is time-cost. What is the computation time cost of the proposed methods, especially for Deep Voro++. This method with 1280 configurations seems to require a large amount of computation time.\n", "summary_of_the_paper": "This paper provides a new geometric point of view for few-shot learning (FSL). In this view, the widely used ProtoNet can be regarded as a Dirichlet Tessellation (Voronoi Diagram)\nin the feature space. Furthermore, the authors use the recent Cluster-induced Voronoi Diagram (CIVD) for FSL and propose an ensemble approach to achieve a stronger FSL model. Extensive experimental results on three standard benchmarks demonstrate the effectiveness of the proposed method.", "summary_of_the_review": "The main idea is novel and interesting, and the experimental results demonstrate the superior performance of the proposed method. However, this paper lacks some ablation studies and time comparisons. In summary, the reviewer thinks this paper is marginally above the acceptance threshold.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "XiypVc1lsQJ", "reviewer": "Reviewer_uC53", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The paper is written in a way that makes it extremely hard to read. The text is not sufficiently self-contained to follow the line of arguments, terms and methods are not always well-defined, and many side-tracks with questionable relevance are considered. Most importantly, the evaluated methods DeepVoro(++) are never defined in the method part.\n\nTerminology is sometimes used in unusual ways, e.g. I would identify data augmentation rather with what is said under point 3 in section 1 than GANs and VAEs in the top of the page.\n\nFigure 1 is confusing. Usually, when adding new classes, old ones are not supposed to be forgotten (cmp. left and center panel). \n\nSome equations contain trivial errors, e.g. the optimization below (1) should be a minimization of distance. \n\nSome presentations confuse unnecessarily, e.g. (3) where an offset/bias for the distance is introduced, but it is called weight. The text below uses Dirichlet tessellation as a special case of PD, whereas before (3) they are identified.\n\nThe datasets used for the comparisons should be described in the main paper, not the supplementary material.\n\nThe overall results look impressive, but the paper needs a restructuring such that the following obvious questions are answered: How exactly are the methods DeepVoro(++) defined, what is their baseline and what modifications have been made? How are these modifications motivated from theory or experiments? Which of these modifications or theoretical insights are the main contributions of the paper?\n\nIn order to address these questions, include the dataset descriptions and the most relevant ablative results, section 3 needs to be streamlined and less relevant theoretical considerations need to be removed or moved to the supplementary material.", "summary_of_the_paper": "The paper proposes a CIVD-based approach to few-shot learning. CIVD, cluster-induced Voronoi diagrams, are a known technique that is used to categorize / describe different types of few-shot classifiers. In the experiment section DeepVoro(++) is shown to perform superior to other methods on three datasets.", "summary_of_the_review": "The manuscript is technically overloaded but not self-contained. Important descriptions, in particular of the proposed methods, are missing. The experimental results look impressive, but it remains unclear what exactly has been used in the DeepVoro(++) methods.\n\nPost-rebuttal:\nmost concerns have been addressed and the correctness has improved. The assessment has been raised accordingly.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "8l9dUoDkQ2", "reviewer": "Reviewer_bCqa", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Strengths \n- Mathematical formulation of FS approaches in terms of VD terminology\n- Link to CIVD which allows ensembling different FS classifiers by means of a means of an influence functions that allows multistep\n- Extensive quantitative results on mini-ImageNet, CUB, tiered_ImageNet\n- Paper is well-written\n\nWeaknesses\n- Mathematical derivation of FS as a tessellation is provided for nearest neighbor and linear classifiers, it would be helpful to understand if all FS classifiers have a VD explanation, or there are some classes of FS approaches that are not VD\n- The methodology section could be clarified if it has sections named after different contributions (for easy access)\n- Lack of explanation of DeepVoro and DeepVoro++\n- Lack of implementation details section\n- Lack of discussion how the ensemble approach using CIVD is related to meta-learning techniques\n\nSmall correction:\n- Typo: “To resolve this issie” -> To resolve this issue\n\nOverall, the paper presents a mathematical link between VD and FS approaches, and shows how to use a multi-set VD diagram (CIVD) to unify multiple FS classifiers together. It is well written, and I found it easy to understand the main points of the paper. My main concern is the lack of details provided about the implementation, as I can see this approach would be useful in many future FS research techniques. It would also be valuable to add a discussion of how outliers would affect the method, computational complexity of the proposed technique. Can it be used for other tasks, such as segmentation?\n", "summary_of_the_paper": "The paper presents an approach for few-shot (FS) learning using Voronoi Diagrams (VD). In particular, it relates the objectives of existing FS approaches to VD, and shows how Cluster-induced Voronoi Diagrams (CIVD), a variant of VD that allows multiple centers in a cell, can be used in for FS learning ensemble method (DeepVoro). Extensive quantitative evaluations show improvements over prior work  on three public datasets. \n", "summary_of_the_review": "Mathematical link is presented between FS classification and voronoi tessellation, description of a novel ensemble FS method based on  CIVD. Paper is well written and conclusions are throughly validated. Paper needs clarifications about implementation details, and discussion of limitations, effect of outliers and generalization to other tasks.\n", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "1tS9lBgCg77", "reviewer": "Reviewer_JRwP", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Positive:\n- The use of the Cluster-induced Voronoi Diagram and its variant introduced in this paper are novel to FSL (to the best of my knowledge).\n- The resulting voronoi-diagram formulation is geometrically elegant, and allows the integration of various classifiers, heterogeneities and types of feature representations.\n- The results appear to be state of the art (at least relative to the compared works) and standard datasets used by the community.\n- The ablation study appears to be a good first order approximation of what I'd expect the authors to verify.\n\nNegative:\n- The paper is a bit difficult to read. I'd attribute this primarily to the extended use of the appendix. I agree that proofs can be included primarily in the appendix, but I do not particularly appreciate that much of the ablation results and tables are included only in the appendix.\n- Beyond the geometric point of view of FSL (which I do appreciate), to my understanding, the paper proposes a relatively straightforward aggregation of method/features/representations for FSL. Furthermore, the classifiers it integrates and the various heterogeneities are not novel, and neither is the cluster-induced Voronoi Diagram (which itself is a relatively straightforward generalization of voronoi diagrams).", "summary_of_the_paper": "The authors introduce the use of Cluster-induced Voronoi Diagrams to few shot classification, and show that it can be used to combine feature and surrogate representations with various types of few-shot classifiers (eg nearest neighbour, linear classifier and cosine) and various types of heterogeneities (eg feature level, transformation level and geometry level) into a single, coherent, mathematical formulation. ", "summary_of_the_review": "I overall like the paper and appreciate the geometric formalism and results. That being said, from a mathematical point of view, I am unsure about the amount of novelty the paper would have had, had the same approaches for integration been used without the overarching geometrical point of view.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper analyzes popular metric-based few-shot learning (FSL) methods from the perspective of computational geometry. Namely, viewing prototypical networks as Voronoi diagrams (VDs). This lends itself to incorporating extensions based on the recently proposed CIVD that allows for multiple centers per cell. The paper then discusses various aspects of the FSL pipeline (data augmentation, feature transformations, geometries and representations), referred to as heterogeneities, that can be efficiently ensembled via a cluster-to-cluster VD (CCVD). The resulting model produces state-of-the-art results.

Initial concerns from the reviewers pointed to a potential lack of novelty (since it can be seen as applying existing ideas to FSL), lack of self-containment in the main paper, weak positioning in the context of other FSL methods (which ones can be interpreted under the VD framework) and a potentially impractical computational complexity. The discussion period settled these issues, with the paper receiving several updates, and the reviewers all ended up recommending acceptance.

Personally, I would like to see an addition to Figure 1 with the resulting decision boundaries from CIVD and CCVD. I think that this would greatly improve the ability of the reader to reason about the approach intuitively. Also, as a minor comment, I think that the argmax below eqs 1 and 7 should either be an argmin, or the distances should be negated. Otherwise, I think this is a valuable contribution to the FSL literature.
Decision: Accept (Poster)