Neural Collapse Inspired Feature-Classifier Alignment for Few-Shot Class-Incremental Learning
Authors: Yibo Yang
Haobo Yuan
Xiangtai Li
Zhouchen Lin
Philip Torr
Dacheng Tao
Venue: ICLR 2023 notable top 25% (2023)
Keywords: few-shot class-incremental learning
neural collapse

Abstract: Few-shot class-incremental learning (FSCIL) has been a challenging problem as only a few training samples are accessible for each novel class in the new sessions. Finetuning the backbone or adjusting the classifier prototypes trained in the prior sessions would inevitably cause a misalignment between the feature and classifier of old classes, which explains the well-known catastrophic forgetting problem. In this paper, we deal with this misalignment dilemma in FSCIL inspired by the recently discovered phenomenon named neural collapse, which reveals that the last-layer features of the same class will collapse into a vertex, and the vertices of all classes are aligned with the classifier prototypes, which are formed as a simplex equiangular tight frame (ETF). It corresponds to an optimal geometric structure for classification due to the maximized Fisher Discriminant Ratio. We propose a neural collapse inspired framework for FSCIL. A group of classifier prototypes are pre-assigned as a simplex ETF for the whole label space, including the base session and all the incremental sessions. During training, the classifier prototypes are not learnable, and we adopt a novel loss function that drives the features into their corresponding prototypes. Theoretical analysis shows that our method holds the neural collapse optimality and does not break the feature-classifier alignment in an incremental fashion. Experiments on the miniImageNet, CUB-200, and CIFAR-100 datasets demonstrate that our proposed framework outperforms the state-of-the-art performances. Code address: https://github.com/NeuralCollapseApplications/FSCIL 

Published as a conference paper at ICLR 2023
NEURAL COLLAPSE INSPIRED FEATURE-CLASSIFIRR ALIGNMENT FOR FEW-SHOT CLASS INCREMENTAL LEARNING
Yibo Yang",, Haobo Yuan**, Xiangtai Li", Zhouchen Lin Philip Torr", Dacheng Tao
ID Explore Academy "School of Computer Science, Wuhan University
3 National Key Lab of General AI, School of Intelligence Science and Technology. Peking University
d Institute for Artificial Intelligence, Peking University *Peng Cheng Laboratory University Oxford
t: corresponding author *; equal contribution
ABSTRACT
Few-shot class-incremental learning (FSCIL) has been a challenging problem as only few training samples are accessible for each novel class in the new ses- sions. Finetuming the backbone or adjusting the classifier prototypes trained in the prior sessions would inevitably cause misalignment between the feature and classifier of old classes, which explains the well-known catastrophic forgetting problem. In this paper, we deal with this misalignment dilemma in FSCIL in- spired by the recently discovered phenomenon named neural collapse, which re- veals that the last-layer features of the same class will collapse into a vertex, and the vertices of all classes are aligned with the classifier prototypes, which are formed as a simplex equiangular tight frame (ETF). It corresponds to an opti- mal geometric structure for classification due to the maximized Fisher Discrim- inant Ratio. We propose a neural collapse inspired framework for FSCIL. A group of classifier prototypes are pre-assigned a simplex ETF for the whole label space, including the base session and all the incremental sessions. During training, the classifier prototypes are not learnable, and we adopt novel loss function that drives the features into their corresponding prototypes. Theoretical analysis shows that our method holds the neural collapse optimality and does not break the reature-classifer alignment in an incremental fashion. Experiments on the minilmageNNt, CUB-200, and CIFAR- 100 datasets demonstrate that our pro- posed framework outperforms the state-of-the-art performances. Code address: htttpp/s//tttuuoooom///raaaaaaap
INTRODUCTION
Learning incrementally and learning with few-shot data are common in the real-world implementa- tions, and in many applications, such as robotics, the two demands emerge simultaneously. Despite the great success in closed label space, it is still challenging for deep learning model learn new classes continually with only limited samples (LeCun et al. 2015). ). To this end. few-shot class-
incremental learning (FSCIL) was proposed to tackle this problem (Tao al. 2020b).
Compared with few-shot learning Ravi & Larochelle| 2017 Vinyals et al. fers a trained model into new label spaces incrementally. It also ditters from incremental learning (Cauwebeerghs & Poggio 2000; Li & Hoiem 2017 Rebuffi et al. (017)) in that there are only few (usually 5) samples accessible for each new class in the incremental sessions. For each ses- sion's evaluation, the model is required to infer test images coming from all the classes that have been encountered. The hase session of FSCIL contains a large label space and sufficient training samples, while each incremental session only has few novel classes and labeled images. It poses the notorious catastrophic forgetting problem (Goodfellow et al., 2013) because the novel sessions have no access the data of the previous sessions.
2016). FSCIL trans-
Due to the importance and difficulty. FSCIL has attracted much research attention. The initial so- lutions to FSCIL finetune the network new session data with distillation schemes to reduce the forgetting of old classes (To et al., 20206 Dong et al. (02TT. However, the few-shot data in novel sessions can easily induce over-fitting. Following studies favor training backbone network on the

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Base Session
Session
Session
(e) prior studess clacsifir protetypes are learnable
(b) our solaion: classlller prololypes are pre-aesigned and Tixed
curant protorypes
protonypes in but sestion
mita ligneen
Figure 1: A popular choice in prior studies is to evolve the old-class prototypes via delicate design of loss or regularizer to keep them separated from novel-class prototypes, but will cause misalignment. As a comparison, we pre-assign and fix an optimal feature-classifier alignment, and then train model towards the same neural collapse optimality in each session to avoid target conflict.
base session as feature extractor (Zhang et al. 2021 Hersche et al. 2022 Akyürek et al. 2022 For novel sessions, the backbone network is fixed and group of novel-class prototypes (classifier vectors) are learned incrementally. But as shown in Figure T (a), the newly added prototypes may lie close to the old-class prototypes, which impedes the ability to discriminate between the old-class and the novel-class samples in evaluation. As result, adjusting the classifier prototypes is always necessary for two goals: (i) keep a sufficient distance between the old-class and the novel-class prototypes; (ii) prevent the adjusted old-class prototypes from shifting far away from their original positions. However, the two goals rely on sophisticated loss functions or regularizers (Chen & Lee, 2021 Hersche et al. 2022 Akyürek et al. 2022), and are hard to attain simultaneously without qualification. Besides, as shown in Figure (a), there will be a misalignment between the adjusted classifier and the fixed features of old classes. A recent study proposes reserve feature space for novel classes to circumvent their conflict with old classes (Zhou et al. 2002aa) but an optimal feature-classffier alignment hard to be guaranteed with learnable classifier (Pernici al. 2021).
We point out that it is the misalignment dilemma between feature and classifier that causes the catastrophic forgetting problem of old classes. If backbone network is finetuned in novel sessions, the features of old classes will be easily deviated from their classifier prototypes. Alternatively, when backbone network is fixed and group of new prototypes for novel classes are learned incrementally, the adjustment of old-class prototypes will also induce misalignments with their fixed features. In this paper, we pose and study the following question,
"Can we look for and pre-assign an optimal clauur--laassfier alignment such that the model is optimized towards the fixed optimality, so avoids conflict among sessions?"
1.1 MOTIVATIONS AND CONTRIBUTIONS
Neural collapse recently discovered phenomenon that the terminal phase of training (after training error rate), the last-layer features of the same class will collapse into single vertex, and vertices all classes will be aligned with their classifier prototypes and be formed as a simplex equiangular tight frame (ETF) (Papyan et al.) 2020). A simplex ETF is geometric structure of K vectors in Rd 2 K 1. All vectors have the same l2 norm and any pair of two different vectors has an inner product of R-T which corresponds to the largest possible angle of K cquiangular vectors. Particularly when d = K - 1, a simplex ETF reduces to a regular simplex such triangle and tetrahedron. It describes an optimal geometric structure for classification due to the minimized within-class variance and the maximized between-class variance (Martinez & Kak, (001). which 1948) is maximized. Following
indicates that the Fisher Discriminant Ratio (Fisher) 1936 Rao
studies aim to theoretically explain this phenomenon (Fagg al. 2021 Han et al. 2022
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It is expected that imperfect training condition, such as imbalance, cannot induce neural collapse and will cause deteriorated performance (Fang et al. 2021 Yang et al., 2022b). Training in an incremental fashion will also break the neural collapse optimality. Since neural collapse offers us an optimal structure where features and their classifier prototypes are aligned, we can pre-assign such a structure and learn the model towards the optimality. Inspired by this insight, in this paper, initialize group of classifier prototypes W ETF E [dx((K+
simplex ETF for the whole label space, where Ku is the number of classes in the base session and K' is the number of classes in all the incremental sessions. As shown in Figure (b). it serves as the optimization target and keeps fixed throughout all sessions training. We append projection layer after the backhone network and store the mean latent feature of each class output by the backbone in a memory. the training of incremental sessions, we only finetune the projection layer using novel loss function that drives the final features towards their corresponding target prototypes. Without bells and whistles, our method achieves superior performances and relieves the catastrophic forgetting problem.
The contributions of this paper can be summarized as follows:
To relieve the misalignment dilemma in FSCIL,, we propose pre-assign an optimal align- ment inspired by neural collapse as a fixed target throughout the incremental learning. Our model is trained towards the same optimality to avoid optimization conflict among sessions.
We fix the prototypes and apply novel loss function that only finetunes projection layer to drive the output features into their corresponding prototypes. Theoretical and empirical analyses show that our method better holds the neural collapse optimality.
Experiments on minilmageNet, CIFAR-100, and CUB-200 demonstrate that our method is able to surpass the of-tearft performances. In particular, our method achieves an average accuracy improvement of more than 3.5% over a recent strong baseline on both minilmageNet and CIFAR-100.
RELATED WORK
Few-shot class-incremental learning (FSCIL). As variant of dlas-inccremenal learning (CIL)
(Cauwenberghs & Poggio. 2000; Li & Hoiem 2017 Rebuffi et al., (017), FSCIL only has a few
novel-classes and training data in each incremental session (Tao al.] 20206 Dong et al. 2021 2018
which increases the tendency of overfitting on novel classes (Snell et al. 2017 Sung et al.
Both CIL and FSCIL require a delicate balance between well adapting model to novel classes and less forgetting of old classes (Zhao et al. (022). A popular choice is to use meta learning (Yoon et al.) 2020 Chi et al., 2022 Zhou et al. 2022b). Some studies try to make base and incremental sessions compatible via feature (Cheraghian et al. 2021b; Zhou et al. (2022a). augmentation (Peng jet al. 2022), or looking for flat minima (Shi et al. 2021). For training in incremental sessions, the new prototypes for novel classes should be separable from the old-class prototypes. Meanwhile, the adjustment old-class prototypes should not induce large shifts. Current studies widely rely on
evolving the prototypes (Zhang et al., 1021 Zhu et al. (02 or sophisticated designs of loss and
regularizer (Ren et al. [019| Hou et al. 2019 Tao et al. 2000aa Joseph et al.] [2022] Lu et al. [022] Hersche et al. [0022] Chen & Lee| 2021 Akyürek al. 2022] Yang et al. (022a). However, the two
goals have inherent conflict, and a tough effort to balance the loss terms is necessary. In contrast, our method pre-assigns and fixes feature-classifier alignment as an optimality. A model trained towards the same target in all sessions. We only use a single loss without any regularizer.
Neural collapse. Neural collapse describes an elegant geometric structure of the last-layer feature and classifier in a well-trained model (Papyan et al. 2020). It inspires later studies to theoretically explain this phenomenon. Based on a simplified model that only considers the last-layer optimiza- tion, neural collapse is proved to be the global optimality of balanced training with the CE (Weinan & Wojtowytsch 2020 Graf et al. 2021 Lu & Steinerherger, 2020; Fang et al., 2021 Zhu et a 2021b]Ji et al. 2022) and the MSE (Mixon et al.] 2020 |Poggio & Liao| 2020; Zhou et al. 2022c Han et al., 2022: Tirer& & Bruna 2022) loss functions. Recent studies try to induce neural collapse
2022b Zhong et al. 2023) or novel loss in imbalanced training by fixing classifier (Yang et al.
(Xie et al. 2023 Our method is inspired by Yang et al. (2022)), but we apply the classifier in an incremental fashion. Galanti et al. (2022) show that neural collapse is still valid when transferring model into new samples or classes. To the best of our knowledge, we are the first to study FSCIL from the neural collapse perspective, which offers our method sound interpretability.
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3 BACKGROUND
3.1 FEW-SHOT CNASS-INCREMENTAL LEARNING (FSCIL)
In weal-world applications, often needs to adapt model to data coming from new label space with only a few labeled samples. FSCIL trains a model incrementally on a sequence of training datasets {D("), (D.),..,,, D. where D = {( ) D the base session, and the number of incremental sessions. The base session ( usually contains a large label space ( and sufficient training images for each class e E C(O). In each incremental session D(t), I. > O, there are only few labeled images and we have (D(t) pq. where p is the number of classes and q is the number samples per novel class, known as p-way g-shot. The label space C(+) has no overlap with any other session, i.e., C(t) nc(") 0. Vt' + t. For any incremental session > O, we only have access the data in D(t), and the training sets of the previous sessions are available. For evaluation in session f, the test dataset comes from all the encountered classes in the previous and current sessions D i.e. the label space of CO))
Therefore, FSCIL suffers from severe data scarcity and imbalance. It requires model be adapt- able to novel classes, and meanwhile keep the ability on old classes.
3.2 NEURAL COLLAPSE
Neural collapse refers phenomenon the terminal phase of training (after 0 training error rate) on balanced data (Papyan et al., 2020). It reveals a geometric structure formed by the last-layer feature and classifier that can be defined as:
Definition (Simplex Equiangular Tight Frame) A simplex Equiangular Tight Frame (ETF) refers to matrix that composed of K vectors in Rdd and satisfies:
E
where - le, ,ek E Raxk U IR allows a rotation and satisfies U Ix, Ix is the identity matrix, and 1K is an all-ones vector.
All column vectors in E have the same l2 norm and any pair has an inner produce of - K-1 i.c.
1
K
e Ck2
K 1-1 K- Vks, h2 E K],
(2)
where ,x, = when k = 12s and otherwise.
The neural collapse phenomenon includes the following four properties:
(NC1): The last-layer features of the same class will collapse into their within-class mean, i.e., the covariance 2
(*() = Avg, (MA,, - Ma (1u, pax) )"), MA is the feature of sample (k) O, where Ew
i class k, and μ4 is the within-class mean class features;
(NC2): The within-class means of all classes centered by the global mean will converge to the vertices of a simplex ETF defined in Definition [ i.e., fug 1 < k < K satisfy Eq. where Ag (Mu hame-aa)/lll pG)//lmx MGll and μG is the global mean;
(NC3): The within-class means centered by the global mean will be aligned with (parallel to) their corresponding classifier weights, which means the classifier weights will converge to the same sim- plex ETF, i.e.. fa = w /| S ls < K, where ww is the classifier weight class k;
(NC4): When (NC1)-(NC3) hold, the model prediction using logits can be simplified the nearest class centers? i.e., arg max, (H, wx) = arg min; μ 1 Mæll. where (-) is the inner product operator, z is the last-layer feature of sample for prediction.
Neural collapse corresponds to an optimal ceature- alignment for classification due to the maximized Fisher Discriminant Ratio (between-class variance within-class variance).
Different from task-incrementtl lcarning, we do not know which session a test sample comes from. We omit the hias in linear classifier layer for simplicity.
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4 METHOD
Neural collapse tells us an optimal geometric structure for classification problems where the last- layer features and classifier prototype of the same class are aligned, and those of different classes are maximally separated. However, this structure will be broken in imperfect training conditions, 202T Yang al. 2022b). As illustrated in Figure
such imbalanced training data (Fang al. (a), training in an incremental fashion will also break the neural collapse optimality. Inspired by this perspective, what we should do for FSCIL is to keep the neural collapse inspired greatre-classifie alignment sound as possible. Concretely, we adopt fixed classifier and novel loss function as described in Section 4.1 and Section 4.2 respectively. We introduce our framework for FSCII in Section 4.3 Finally, in Section 4.4 we conduct theoretical analysis to show how our method better holds the neural collapse optimality in an incremental fashion.
4.1 ETF CLASSIFIER
Assume that the base session contains label space of Kn classes, each incremental session has p classes, and we have T incremental sessions in total. The whole label space of this FSCIL problem has Ko + K' classes, where K' Tp, i.e., we need to learn model that can recognize samples from Ko + K' classes, We denote backbone network as f, and then we have = f(x, 0y), where E R4 is the output feature of input æ, and f is the backbone network parameters.
A popular choice in current studies learns f and W using the base session data, where W E R Ka is the classifier prototypes for base classes. In incremental sessions > f is fixed as a feature extractor and only W(O E Rddy for novel classes learnable. As shown in Figure (a). one need to adjust (W()).... w(() via sophisticated loss regularizer to ensure separation among these prototypes (Akyürek et al. 2022; Hersche et al., 2022). But it will inevitably introduce mis- alignment between the adjusted prototypes and the fixed features of old classes. It is an underlying reason for the catastrophic forgetting problem (Joseph et al. 2022).
Since neural collapse describes optimal geometric structure of the last-layer feature and classifier, we pre-assign such an optimality by fixing a learnable classifier as the structure instructed by neural collapse. Following Yang al. (2022b). we adopt an ETF classifier that initializes classifier as a simplex ETF and fixes it during training. The difference lies in that we use it in an incremental fashion. Concretely, we randomly initialize classifier prototypes W WETF E (Ko (K))")) by Eq. (1
for the whole label space, i.e.. the union of classes in all session, C(), We have Ko = (C(O)
and K' = Er= (c(i Tp. Then any pair (k1, k2) of classifier prototypes in W WETF satisfies:
1 Ko 1 Ski Ko K"-1" Vki, [1, + =K],
Ko + K'
T W Wk.
(3)
where Wk, and WK, are two column vectors in W EEF. Our ETF classifier ensures that the prototypes of the whole label space have the maximal pair-wise separation. It serves as a fixed target along the incremental training avoid conflict among sessions. We only need to learn model whose output features are aligned with this pre-assigned structure.
4.2 NOT-REGRESSION LOSs
The gradient of cross entropy (CE) loss with respect to the last-layer feature composed of a pull term that drives the feature into its classifier prototype of the same class, and a push term that pushes it away from the prototypes of different classes. As pointed out by Yang et al. (2022b), when the classifier prototypes are fixed as an optimality, the pull term is always accurate towards the solution, and we can drop the push gradient that may be inaccurate, Accordingly, we adopt a novel loss named dot-regression (DR) loss that can be formulated as (Yang et al. 2022b):
C (fr W WETF) = (w (W - 1))² (ETF))
where ça; is the normalized feature, i.e., Pa; = ptt///llaaa || pe = f(x,, yi the label of input æ;, wy, is the fixed prototype in WETF for class y/. and we have |/ = by Eq. (3). The total loss an average over batch of input m.. The gradient of Eq. (4) with respect to fi; takes the form of:
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for the base session
for sessions (0 T)
f
hi
g
normalize
M
(t))
projection layer
W ETF
backbone
Tt
network
e
forward propagation
backward update
(S
fixed ETF classifier prototypes
Figure 2: An illustration of our NC-FSCIL. h, is the intermediate feature from the backbone network
f. fá; is the normalized output feature after the projection layer W WETF is the ETF classifier that contains prototypes of the whole label space and serves as fixed target throughout the incremental training. L denotes the dot-regression loss function. is frozen in the incremental sessions (1 <
S T). A small memory of old-class features is widely adopted prior studies such as Cheraghiann
et al. 2221aa Chen & Lee 2021 Akyürek et al. (2022) and Hersche et al. (2022)
0C/0p, (1 cos (pa,, ye y It is shown that the gradient pulls feature fa; towards the direction of Wy., which a pre-assigned target prototype. Finally, the converged features will be aligned with W WETF. and thus the geometric structure instructed by neural collapse is attained. The theoretical advantage of the DR loss has been proved in Yang et al. (2022b). experiments, we will compare the DR loss with the CE loss to show its effectiveness in FSCIL.
4.3 NC-FSCIL
Based on the ETF classifier and the DR loss, we now introduce our neural collapse inspired frame- work for few-shot class-incremental learning (NC-FSCIL). As shown in Figure composed of two components, a backbone network ff and projection layer The backbone net- work f takes the training data æ; as input, and outputs an intermediate feature h4 The projection layer can be a linear transformation or an MIP block following Hersche et al. (2022); |Png et al. (022). It projects the intermediate feature h, into μı: Finally, we perform an &g normalization on μ1 to get the output feature fú;, i.e.
model is
p4 = wh=g(h,,00 Og), h; f(; Oy),
(5)
where y and Bg denote the parameters of the backbone network and the projection layer, respec- tively. We use the normalized output feature fa, to compute error signal by Eq. 4
In the base session 0, we jointly train both and g using the base session data. The empirical risk to minimize in the hase session can be formulated as:
Σ
C (P WETF..
min B
D(O)
(D.BLEDDO))
where WETF is the pre-assigned ETF classifier introduced in Section 4.1 C is the DR loss as introduced in Section 4.2, and ja, is a function of and g as shown Eq. Spp
In each incremental session 1 < f. < T, we fix the backbone network f as feature extractor, and only finetune the projection layer g. As widely adopted practice in FSCIL studies, small be retained to relieve the overfitting novel
memory of samples classes (Cheraghian et al. 202 Chen & Lee 2021 Akyürek et al. 2022; Hersche et al. 2022). Following Hersche et al. (2022), 2, we only keep memory M of the mean intermediate feature h. for each old class c. Concretely, we have,
features of old classes
M (helc C(0), h. = Avg = c]. 1stšT,
where of has been fixed after the base session. Then we use ((t) as the input of f. and M as the input of g finetune the projection layer g. The empirical risk minimize in incremental sessions

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Published as a conference paper at ICLR 2023
can be formulated as:
min % [D(O] + |M
Σ
L ( W WETF) +
Σ c (ene)
(#..)))ED()
(be.).)EM(())
the number of training samples in session t, and we have |M - |c6)]. Thanks to our pre-assigned alignment, we do not rely on any regularizer in our training.
where pi, and fee are the output features of æ;; and he, respectively, [D(0] |
In the evaluation of session t, predict an input based on the inner product between its output feature fi and the ETF classifier prototypes: arg max; (p. Wa), VI s k S Ko + K'.
4.4 THEORETICAL SUPPORTS
We perform our theoretical work based simplified model that drops the backbone network and only keeps the last-layer features and classifier prototypes as independent optimization variables. This simplification has been widely adopted in prior studies to facilitate analysis (Graf al. 2021 Fang et al. 2021 Zhu et al. 2021b). We investigate the neural collapse optimality of an incremental problem of ' sessions with our ETF classifier. Concretely, consider the following problem,
R(t))
WETF) OstST, m (
min M
NO
8.l. S1,
viSkS K(), 1 isisi n2:
(t) where mg.i / E Rd denotes a feature variable that belongs to the i-th sample of class k in session 1,
ng is number of samples in class k. ((() number of classes in session t, (!) is the number of
K9) samples in session t, i.e,, N(1) Ek=1 Tik, and M( Rdxx(()) denotes a collection of m
(t)
WETF Raxk refers the ETF classifier for the whole label space introduced in Section 4.1
and we have -E -T=O K(t, can be both the cross entropy and the dot regression loss functions.
Theorem Let (()) denote the globul minimizer of Eq. by optimizing the model incrementally
from t = O, and we have M = [M .M E -id No0 When C in Eq. 99 is CE or DR
loss, for any column vector mig in M whose class label is k, we have:
K
|m 1, m.g Wi
K-1 Sg.K
K-1 Ak, E [1, KJ, 1 isis ni:
(10)
where K = Et=o K(9 denotes the total number of classes of the whole label space, 8x = 1 when k K' and otherwise, and wy is the prototype of class k' in WETF-
The proof of Theorem can be found in Appendix A Eq. (1O) indicates that the global minimizer M of Eq. 99) satisfies the neural collapse condition, i.e., features of the same class collapse into single vertex, and the vertices all classes are aligned with WETF as a simplex ETF. It is shown that the feature space is equally separated by prototypes of all classes. More importantly, in problem Eq. (9), the number of classes K(()) among 7 + 1 sessions and the number of samples ng among K classes can be imbalanced, which corresponds to the challenging demand of FSCII..
EXPERIMENTS
In this section, we test method on FSCIL benchmark datasets including minilmageNeet (Rus- sakovsky et al. 2015), CIFAR- 100 (Krizhevsky et al.) 2009), and CUB-200 (Wah et al. 2011), and compare it with state-of-the-ar methods. We also perform ablation studies validate the effects of ETF classifier and DR loss. Finally, we show the feature-cles classr--l structure achieved by our method.
5.1 IMPLEMENTATION DETAILS
Please refer Appendix B for our implementation details.

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Published as a conference paper at ICLR 2023
Table I: Performance of FSCIL in each session on minilmageNet and comparison with other studies. The top rows list class incremental learning and few-shot learning results implemented by Tao et al. ( Zhang et al. (202I) in the FSCIL setting. "Average Acc." is the average accuracy of all sessions. "Final Improv." " calculates the improvement of our method in the last session. indicates that the method saves the within-class feature mean of each class for training or inference.
Methods
Accuracy in each session (
Average
Final
8
Acc.
Improv:
iCaRL (Rebulli al
2017
61.31
46.32
42.94
37.63
30.49
24.00
20.89
18.80
17.21
33.29
+41.1
NCM [oo el all 2019
61.31
47.80
39.30
31.90
25.70
21.40
18.70
17.20
14.17
30.83
+44.14
D-Cosine Vmyala ot a
2016
70.37
65.45
61.41
58.00
54,81
51.89
49.10
47.27
45.63
55.99
+12.68
*TOPIC Tao et 2020b
61.31
50.09
45.17
41.16
37, 37.48
35,52
32.19
29.46
24.42
39.64
+33.89
.DLVV (Chen 1,o0 1121
64.77
59.87
55,93
52.62
49.88
47.55
44.83
43.14
41.84
51.16
+16.47
Sell-promboled (Zhu el ml 202
61.45
63.80
59.53
55.53
$2.50
49.60
46.69
43.79
41.92
52.76
+16.39
CEC 7hang el al. 2022
72.00
66.83
62.97
59.43
56.70
53.73
51.19
49.24
47.63
57.75
+10.68
*LIMIT Zioo 20:22
72.32
57.95
55.07
52.70
50.72
49.19
59.06
* 'Regulatzer (Akyürii
2022
80.37
74.68
69.39
65.51
62.38
59.03
56.36
53.95
51.73
63.71
+6.58
MctaFSCIL CCi al. 2022
72.04
67.94
63.77
60.29
57.58
55.16
52,90
50.79
49.19
58.85
+9.12
*C-FSCIL (Herscho al 2022
76.40
71.14
66.46
63.29
60.42
57.46
54.78
53.11
51.41
61.61
+6.90
Data-free Replay
71.84
67.12
63.21
59.77
57.01
53.95
51.55
49.52
48.21
58.02
+10.10
*ALICE (Peng et il. 2022
80.60
70.60
67.40
64.50
62,50
60.00
57.80
56.80
55.70
63.99
+2.61
*NC-ESCII. (ours)
84.02
76.80
72.00
67.83
66.35
64.04
61.46
59.54
58.31
67.82
Improvement over ALICE
+3.42
+6.20
20
+4.60
+3.33
+3.85
+4.04
+3.66
+2.74
+2.61
+3.83
Table 2: Performance of FSCIL in each session on CIFAR- 100 and comparison with other studies. The top rows list class- insremmtate learning and few-shot learning results implemented by (Tao et al. (020b); Zhang et al. (02T) in the FSCIL setting. "Average Ace," is the average accuracy of all sessions. "Final Improv." calculates the improvement of our method in the last session.
Methods
Accuracy in each session (
Average
Final
2
3
5
6
7
8
Acc.
Improv.
iCaRL. RRhuffi et al. NCM (Hou
2017
64,10
53.28
41.69
34.13
27.93
25.06
20.41
15.48
13.73
32.87
+42.38
64.10
53,05
43.96
36.97
31.61
26.73
21.23
16.78
13.54
34.22
+42.57
D-Cosine Vinyals et
2016
74.55
67.43
63.63
59.55
56.11
53.80
51.68
49.67
47.68
58.23
+8.43
*TOPIC TTo el al
2020b
64.10
55.88
47.07
45.16
40.11
36.38
33.96
31.55
29.37
42,62
+26.74
Self-promesee Zhu alal.
2021a
64.10
65.86
61.36
57.45
53.69
50.75
48.58
45.66
43,25
54,52
+12.86
CEC (Zhing al. 2022
73.07
65.26
61.19
55.57
53.22
$1.34
49.14
59.53
DSN Yang al. 20220
73.00
68.83
64.82
62.64
59.36
56.96
54.04
51.57
50.00
60.14
+6,11
*LIMIT [Zhou al.
20226
73.81
72.09
67.87
63.89
60.70
$7.77
55.67
53.52
$1.23
61.84
+4,88
MetaFSCIL
(Chi 20223
74.50
70.10
66.84
62.77
59.48
56.52
54.36
52.56
49.97
60.79
+6.14
*C-FSCIL Hersehe
2022
77.47
72.40
67.47
63.25
59.84
56.95
54.42
$2.47
50.47
61.64
5.64
Data-free Replay (Liu
2022
74.40
70.20
66.54
62.51
59.71
56.58
54.52
52.39
50.14
60.78
+5.97
*ALICE Peng et hf. 2022
79.00
70.50
67.10
63.40
61.20
59.20
58. 10
56.30
10
63.21
+2.01
*NC-FSCII (ours)
82.52
76.82
73.34
69.68
66.19
62.85
60.96
59.02
56.11
67,50
Improvement over ALICE
+3.52
+6.32
+6.24
+6.28
+4.99
+3.65
+2.86
+2.72
+2.01
+4.29
5.2 PERFORMANCE ON BENCHMARKS
Our experiment results on minImageNet, CIFAR-100 100, and CUB-200 are shown in Table I1 Table 2 and Table g (Appendix C). respectively. We see that our method achieves the best performance in all sessions on both minilmageNer and CIFAR-100 compared with previous studies. ALICE (Peng cet al., (222) is recent study that achieves strong performances challenging baseline, we have an improvement of 2.61% in the last session on minilmageNet, and 2.01% % on CIFAR-100. We achieve an averaged accuracy improvement of more than 3.5% on both minilmageNet and CIFAR- 100. Although we do not surpass ALICE in the last session on CUB-200, still have the best average accuracy among all methods. As shown in the last rows of Table and Table 2 the improvement of our method lasts and even becomes larger in the first several sessions. It indicates that our method is able to hold the superiority and relieve the forgetting of old sessions.
FSCIL. Compared with this
5.3 ABLATION STUDIES
We consider three models to validate the effects of ETF classifier and DR loss. All three models are based on the same framework introduced in Section 4.3 including the backbone network, the projection layer, and the memory module. The first model uses learnable classifier and the CE
8

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Published as a conference paper at ICLR 2023
Table Ablation studies on three datasets to investigate the effects ETF classifier and DR loss. "Learnable+CE" uses learnable classifier and the CE loss; "ETF+CE" adopts our ETF classifier with the CE loss; "ETF+DR" uses both ETF classifier and DR loss. "FINAL" refers to the accuracy of the last session; " "AVERAGE" is the average accuracy of all sessions; "PD" denotes the perfor- mance drop, i.e, the accuracy difference between the first and the last sessions.
miniImageNet AVERAGET
CIFAR-I00 AVERAGET
CUB-200 AVERAGET
Methods
FINALf
PD
FINALÍ
PD
FINAL
PD
Cearnable+CE
50.04
61.30
34,53
52.13
62.68
30.14
50.38
59.58
29.19
ETF+CE
56.66
68.23
28.21
54.42
64.00
27.36
56.83
65.51
23.27
ETF+DR
58.31
67.82
25.71
56.11
67.50
26.41
59.44
67.28
21.01
(a) train (each)
train (accumulate)
(c) (cach)
test (accumulate)
Figure 3: Average cosine similarity between features and classifier prototypes of different classes, i.e, Avgnze {cos Z(me- ma, We)), where my is the within-class mean of class k features, ma denotes the global mean, and War is the classifier prototype of class k. Statistics are performed among classes in each session (a and c), and all encountered classes by current session and d), on train set (a and b) and test set (c and d), for models trained after each session minilmageNet.
loss. which is the most adopted practice. The second model only replaces the classifier with ETF classifier and also uses the CE loss. The third model corresponds to our method using both ETF classifier and DR loss. As shown in Table 3 when fixed ETF classifier used, the final session accuracies are significantly better, and the performance drops get much mitigated. Adopting the DR loss able to further moderately improve the performances. indicates that the success of our method is largely attributed to ETF classifier and DR loss, as they pre-assign neural collapse inspired alignment and drive a model towards the fixed optimality, respectively.
5.4 FEATURE- FEATURE-CLASSIEREE CLASSIFIER STRUCTURE
We check the feature-classifier alignment instructed by neural collapse using our method and "Leamnble+CEE
comparison. As shown in Figure 3, the average cosine similarities between features and classifier prototypes of different classes, i.e.. Avg (cos Z(m; mg wx )} of our method are consistently lower than those of the baseline. Most values of our method are negative and close to O, which in Figure 3b and Figure 3d the average cosine similarities between m mg and wxy (k: * k') among all encountered classes increase fast with session for the baseline method, while ours keep relatively flat. It indicates that the baseline method reduces the feature-classifier margin of different classes as training incrementally, and our method enjoys a stable aligmment. As shown in Figure 4 and Figure [5] also calculate the average cosine similarities between feature and classifier of the same class, i.e., Avga A L(m; - mG, ). and the trace ratio of within-class covariance to between-class covariance, holds the feature-classifier alignment and relieves the forgetting problem.
line with the guidance from neural collapse derived in Eq. (10p. Particularly
These results together support that method better
6 CONCLUSION
In this paper, propose to fix learnable classifier geometric structure instructed by neural collapse for FSCIL. It pre-assigns an optimal feature-classifier alignment as a fixed target throughout incremental training, which avoids optimization conflict among sessions. Accordingly, novel loss function that drives features towards this messisigeed optimality is adopted without any regularizer. Both theoretical and empirical results support that method is able to hold the alignment in incremental fashion, and thus relieve the forgetting problem. In experiments FSCIL, we achieve and even surpass the state-of-the-arr performances three datasets.

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Published as a conference paper at ICLR 2023
ACKNOWLEDGMENTS
Z. Lin was supported by National Key R&D Program of China (20222)016030)); the major key project of PCL., China (No. PCL2021A 12), the NSF China (No. 62276004), Qualcomm, and Project 2020BD006 supported by PKU-Baidu Fund.
STATEMENTS
Ethics Statement. Our study does NOT involve any of the potential issues such as human subject, public health, privacy, fairness, security, etc. All authors of this paper confirm that they adhere the ICLR Code of Ethics.
Reproducibility Statement. For theoretical result Theorem offer the proof in Appendix
A All datasets used in this paper are public and have been cited. Please refer to Appendix B for the dataset descriptions and the implementation details of our experiments. Our source code is released
at https /ttt:://qttttt www.wwww.....uuuu.aaeeee. com http:/hhh....oeeeaaaaooooaoppppa
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A
APPENDIX: PROOF OF THEOREM
Our proof is following Yang et al
2022b).
We consider the problem in Eq. 9
k(t))
min M
WETF)
0 STTST
(()
Am
8.L.
1,
S k K S S nA:
where m Ri E R& denotes a feature variable that belongs to the i-th sample of class in session
ng is number of samples in class k, (()) is number of classes in session t, N is the number of samples in session t, i.e., N( = Ek=1 ng and M E Rdxx denotes collection of m (t)
k.
WETF E Rax K refers to the ETF classifier for the whole label space as introduced in Section 4.1 We have K = Et= ((() and,
K òR.K'
WK
K-1 Wk, E [1,
(11)
where We and Wy are two column vectors in W WTF- From the definition of simplex ETF in Eq. 1.. we have WETF 1x= = Od, where 1g is an all-ones vector in IR and Og is an all-zeros vector Rd. Then we have,
K
W Oa-
(12)
When L is the dot-regression (DR) loss in Eq. 4b. is easy to identify that L 2 0 and the equality (()
holds if and only if w. m
1, v0 I t T. S k K,1 S i ng. Since | =
and
2 < 1, have w m (8)
The equality holds if and only if [|m]
and
(w: k..)) 1. Denote M [ .... M E Rdx N(0) as the global optimality of
Eq. 9 for all sessions 0 < T. For any column vector myg in M, we have,
K
1
|mm,س 1, mf, We
6k.K
Vk, k' [1, 1 1s n
K
K-1
which concludes the proof for DR loss.
When C is the cross-entropy (CE) loss, i.e.
exp(w W m
(0) C m k
WETF) log
(13)
Σラ= expepxppp W
where 0 t S,tST,1SA < K(9, and S i ng. Since the problem separable among T + sessions, only analyze the 1-th session and omit the superscript (t) for simplicity. The objective in Eq. 13 is the sum of an affine function and log-sum-exp functions. When W WETF is fixed, the loss is convex w.E. my.; with convex constraints. So, we can use the KKT condition for its global optimality. Based on Eq. (9) and Eq. B3 we have the Lagrange function.
k(t))
k(t))
log SrowTmeal
AR.
(l 1),
(14)
NO
Σj= exp(xx)) mg.)
Am
where A is the Lagrange multiplier. The gradient with respect to mg, takes the form of:
(1 -1-Pu Pp) Fmg, N70 We
ac
1
Σp, + 2
(15)
NO.
where 1 Ssi nk. 1 S K(0,, and pj the softmax probability of m,, for the j-th class, i.e.,
exp(w> m..))
Pj = E exp W Ff
K
(16)
14

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Published as a conference paper at ICLR 2023
I 1, we have 0 ps ps <1 K. Since || = 1 and ||m
ač
We now solve the equation 0m.,
0. Assume that X = 0, and then we have,
P,w, (1 (1-Pa) pkp W--
(17)
jfk
Since 1 - Pk E Pj and Eq.
11
multiplying we by both sides of Eq. (17). we have,
K- (1 ps) = 0,
(18)
which contradicts with 0 Ph < Then we have the other case Ak,i > 0. Based on the KKT condition, the global optimality mag satisfies that
| 1.
(19)
aė The equation 0m..
= 0 leads to:
Σ P, Wx) + 22(
(20)
J#
Based on Eq. 11.. for any j' # multiplying Wy by both sides of Eq.
20 we have,
K
Ps K
(N An me w = 0.
(21)
Since Ak E [1, K]. Pk > O, we have m w < O. Then for any 11 f2 + k.
Pi exp( w mg,5)
w m
(22)
exp( wT m
w m
The function f(x) exp(x)/ / x is monotonically increasing when 1. So, Eq. (22
indicates that
Py P52:
T W. mes. Vj1.j2 k, W, m...
(23)
and
2N A
m WK (K-1) -
Pj K
bj # k.
(24)
K
Multiplying we by both sides of Eq.
20 we have,
K
pk) (2((1) Xr. an wh 0.
(25)
K
Combing Eq. (24 and Eq. (25). we have,
m k W (K
1) m7 W 0, Vj k
(26)
Based on Py
1-Ps K-
Vj + and Eq. 12 we can rewrite Eq. 20 as:
(1 K We + (1)(1)
(27)
which means that me. is aligned with Wk. i.e. cos / (mmes Wx) Given that || || |mm = (Eq. 11 we have,
1 and
m WE = 1,
and Eq. 26 leads
m W
K
Therefore, for any column vector mag in M. have,
K
1
|m 1, m Wk
8k
Ak, e [], 18i5 nkk
K- K
K
1-1
which concludes the proof for CE loss.
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Table 4: Performance FSCIL in each session on CUB-200 and comparison with other studies. The top rows list class-incremental learning and few-shot learning results implemented by Tao et al. (2020b); Zhang et al.] (202T); [iu et al. (2022): Zhou et al. (2022a) in the FSCIL setting. "Average Acc." is the average accuracy of all sessions, "Final Improv." calculates the improvement of our
method in the last session.
Accuracy in each session (%)
Average
Final
Methods
3
5
6
7
8
9
10
Acc.
Imppov.
ICaRL Rebulli et al.
2017
68.60
52.65
48.61
44.16
36.62
29.52
27.83
26.26
24,01
23.
21.16
36.67
+38.28
Castro al.
2018
68.68
53.63
47.91
44.20
36.
25.93
24.70
23.95
24.15
22.11
36.27
437.33
2019
68.68
57.12
44.21
28.78
26.71
24.62
21.52
20.12
20.06
19.87
32.49
+39.57
classifier
68.47
45.42
35.90
24.24
21.18
17.34
16.20
34.63
+43.24
DoNegCosine
Liu 2020
74.96
70.57
66.62
61.32
60.09
56.06
55.03
52.78
51.50
50.08
48.47
58.86
+10.97
DMD
Zhong
2020
75.35
70.69
66.68
62.34
19.70 76
56.54
14.61
52.62
50.73
49.20
47.60 70
18.73
+11.84
D-Cosine
Viayals et 2016
75.52
70.95
66.46
61.20
60.86
56.88
55.40
53.49
51.94
50.95
49.31
59.36
+10.13
Doeplny
Yin ct al.
2020
75.90
70.21
65.36
60.14
58. 79
88
53.21
51.27
49. 38
47.11
45.67
$7.54
+13.77
TOPIC
Tao et
2020b
68.68
62.49
54.81
49.99
45.25
41.40
38.35
35.36
32.22
28.31
26.28
43.92
+33.16
IDLVO
Chan & Lee 2021
77.37
74.72
70.28
67. 13
65.
63.5 52
62.
61.54
59.04
58. 68
$7.81
65.23
+1.63
20:1a
68.68
61.85
57.43
50.19
43.07
40.17
37.33
49.32
+22.11
Choraghian
2021b
68.78
59.37
59.32
54.96
52.58
19.81
18.09
16.32
44.33
43/43
43.23
51.84
+16.21
CBC
Zhang
3021
75.85
71:94
68.50
63.50
$4.
61.33
LIMI
Zhou el
20222
76.32
74,18
72.68
69.19
68.79
65.64
63.57
62,69 64
61.44
601.44
58.45
66.67
+0.99
MgSvP
/Zhoo et al
202
72.29
70.53
67.00
64.92
62:67
61.89
59.63
59.15
51.73
55.92
54.33
62.37
+5.11
MetaFSCIL Chi el al
2022
75.9
72.41
68.78
64.78
62.96
59.99
58.3
$6.85
$4.78
53.82
52.64
61.92
+6.8
FACT
Zhoo al
31222
75.90
73:23
70.84
66
65. 56
62. 15
61.74
59.83
SW.41
57. 89
56.94
64.43
+2.5
Data- Data- replay
Li et
2022
75.90
72.14
68.64
63.26
62.55 58
59.11
57.82
55.89
54,92
53.58
52.39
61.52
+7.05
ALICE Peng ct
2022
77.10
72:70
70.60
67.20
65.90
63.10
62.90
61.90
60.50
60.60
60.10
65.75
-0.66
NC-FSCIL
75.98
72.30
70.28
68.17
64.43
63.25
60.66
60.01
59.45
67.28
Table 5: A comparison of backbone networks used in different studics.
Methods
minilmageNNet
CIFAR-
CUB-200
TOPIC (Tao al. 2020b)
ResNet-18
ResNet-18
ResNet-18
CEC iZhang et al. 2021
ResNet-18
ResNet-20
ResNet-18
CFSCIL (Hersche cl al. 2022)
ResNet-12
ResNet-12
LIMIT
Zhoo et al. 20226)
ResNet-18
ResNet-20
ResNet-18
ALICE
Peng et al.
2022
ResNet-18 18
ResNet-18
ResNet-18 18
NC-FSCIL (ours)
ResNet-12
ResNet-12
ResNet-18
B APPENDIX: IMPLEMENTATION DETAILS
Datasets. We conduct our experiments three FSCIL benchmark datasets including minilmageNet (Russakovsky et al. 2015), CIFAR-100 (Krizhevsky et al.] (2009), and CUB-200 (Wah et al.) (011]) minilmageNet is variant of ImageNet with an image size of 84 84. It has 100 classes with each class containing 500 images for training and 100 images for testing. CIFAR-100 has the same number of classes and images, and the image size is 32 x 32. CUB-200 a dataset for fine-grained image classification containing 11,788 images of 200 classes in resolution of 224 x 224. There are 5,994 images for training and 5,794 images for testing. We follow the standard experimental settings in FSCIL (Tao et al.) 2020b; Zhang et al. 2021). For both minilmageNet and CIFAR-100) the base session (t = 0) contains 60 classes, and a 5-way 5-shot (5 classes and images per class) problem is adopted for each of the 8 incremental sessions (1 < t < 8). For CUB-200, 100 classes are used in the base session, and there are 10 incremental sessions, each of which is 10-way 5-shot.
Architectures. Prior studies widely adopt ResNet-12, ResNet-18, and ResNet-20 (He et al. 2016) for FSCIL experiments. As shown in Table 5 we compare the backbone networks used in different studies. For minilmageNet and CIFAR- 100, For CUB-200 200, we use ResNet-18 (pre-trained on ImageNet) following other studies. We adopt a two-layer ML.P block as the projection layer following the practice in Peng et al. 2022))
use ResNet-12 following Hersche et al. 2022
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Published as a conference paper at ICLR 2023
Training Details. We adopt the standard data pre proccesssinng and augmentation schemes including random resizing, random flipping, and color jittering (Tao et al. 2020b; Zhang et al. 2021 Peng et al. 2022). We train all models with batchsize of 512 in the base session, and batchsize of 64 (containing new session data and intermediate features train for 500 epochs in the base session, and 100-170 iterations in each incremental session. The initial learning rate is 0.25 for base session, and 0.025 for incremental sessions. On CIFAR-100, we train for 200 epochs in the base session, and 50-200 iterations in each incremental session. The initial learning rate is 0.25 for both base and incremental sessions. On CUB-200, we train for 80 epochs in the base session, and 105-150 iterations in each incremental session. The initial learning rates are 0.025 and 0.05 for base session and incremental sessions, respectively. In all experiments, we adopt a cosine annealing strategy for learning rate, and use SGD with momentum as optimizer. Our code will be publicly available in the final version.
the memory) in each incremental
session. On minilmageNet,
C APPENDIX: MORE RESULTS
Our experimental result on CUB-200 is shown in Table 4 session than most of the baseline methods. Although we do not surpass ALICE in the last session on CUB-200, still have the best average accuracy among all methods.
We achieve better accuracy in the last
We also visualize the average cosine similarities between feature and classifier of the same class, i.e., fcos <(m, mg, W and the trace ratio of within-class covariance to between-class covariance, tr(Eww/tr(EEp)
A higher average [(m;- mG, w&) indicates that feature centers their corresponding classifier prototypes of the same class. As shown in Figure 4 the values of our method are consistently higher than those of the baseline method. Figure 4a and Figure 4d reveal that our method has a better feature-classifier alignment in each session of the incremental training on both train and test sets. When measure on all the encountered classes by each session in Figure and TigureAee the metric for our method does not change obviously after the 4-th session on train set, while the metric for the baseline method keeps decreasing training incrementally. Especially for the base session classes, our method is able to keep the metric stable on both train and test sets after the decline of the first 3-4 sessions, as shown Figure 4c and Figure 4i As a comparison, the baseline method cannot mitigate the deterioration. Given that the base session has the most classes, the performance on base session classes largely decides the final accuracy in the last session for FSCIL. Therefore, the superiority of our method can be attributed to our ability of keeping the
closely aligned with
alignment well for base session classes.
The within-clan class covariance Ew and the between-clase covariance Eg are defined as: (EW) Ew Avga A }, x = Avg,
(og:(mma (ma, m=) ma)"),
and
Avg (m mm) (ma-)))",,
where my.; is the feature of sample in class k my the within-class mean class k features, and mg denotes the global mean of all features. A lower within-class variation with a higher between- class variation corresponds to a better Fisher Discriminant Ratio. As shown in Figure we compare the trace ratio of within-class covariance between-class covariance between our method and the baseline method. We observe similar patterns to Figure 4 Concretely, the trace ratio metric of our method is consistently lower that of baseline. For the base session classes, the metric of our method increases more mildly, which corroborates our ability of maintaining the performance on the old classes, and in line with the indications from Figure SegureJann and Figure
17

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Published as a conference paper at ICLR 2023
Ja
Laavetenct
STF-ORkIGur
3.75
Bust
3357
5.0
0.82-
0.00
Sea
train (each)
(b) train (accumulate)
(c) train (base)
ETT-0110231
CTT+Da 10ur)
LearaateeC
69
0.3
60-
6.5
Sessies
(d) test (each)
(c) lest (accumulate)
(f) test (base)
Figure 4: Average cosine similarity between features and classifier prototypes of the same class, i.e., A Z(me- ma, W.) where mæ is the within-class mean of class k: features, mg denotes the global mean, and wa is the classifier prototype of class k. Statistics are performed among classes each session (a and d), all encountered classes by the current session (b and e), and only the base session classes (c and f), on train set (a, b, c) and test set (d, f), for models trained after each session on minilmageNet.
Liammbie secE
STT+ER (bun)
CisrrableeC
Liarrable-CC
Sensione
Sendinns
Sensions
(a) train (each)
(b) train (accumulate)
(c) train (base)
2.
1.2
2TT+20 (OumH
0.5
ETT-0110usi
1.5
3.
0.5
1.7
LesrmalleCC
319--D4 10
34
0.0
(d) test (each)
(e) test (accumulate)
(l) test (base)
Figure 5: Trace ratio of within-class covariance to between-class covariance, iee, tr(EE)/tr( where Ew is the within-class covariance, and Σ denotes the between-class covariance. Statistics performed among classes in each session and d), all encountered classes by the current session (b and e), and only the base session classes (c and f), on train set (a, b, c) and test set (d, e, f), for models trained after each session on minilmageNet.
18


Reviews:
[{"review_id": "zlTjzMOLrZ1", "reviewer": "Reviewer_mQxd", "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": "", "summary_of_the_paper": "This study involves a few-shot class incremental learning method, inspired by neural collapse inspired feature-classifier alignment. By using fixed prototype vectors to replaced the learned vectors, which have no constraints, it reduces the confusion and collapse caused by incremental classes with few-shot training samples. Experiments on mini-ImageNet and CIFAR-100 validate the effectiveness of the proposed approach. Overall, this is a piece of interesting work, considering an important yet unsolved problem in the machine learning and computer vision areas.", "summary_of_the_review": " While I approve the novel idea of this submission, I am not convinced by the formulation of neural collapse. The performance retain across training session is an important metric, which is missed in the experiments.", "strength_and_weaknesses": "Strength:\n\nThe idea is novel and interesting. The proposed FSCIL method assigns neural collapse inspired feature-classifier alignment as a target and trains a model towards the same optimization in each session to avoid class confusion and conflict. The performance is compare to, if not outperforms, the recently published methods.\n\nWeakness:\n\nThe formulation of neural collapse is not that strict. In my view, the ``neural collapse” appears more like a class confusion in a feature space. The neural collapse, as illustrated in Fig. 1, is also built on the assumption that the feature space is fixed. While as is known, the feature space would change significantly during the model training (either base training or finetuning) procedure. I would like the authors to clarify this point during rebuttal.\n\nThe authors claimed that the proposed approach achieved an average 3.5\\% accuracy improvement over the STOA. But I note in the first session (session 0) in Table 1, the proposed approach has a 3.42\\% accuracy higher than the compared method. While in the last session, it solely outperforms by 3.83\\%. This means that the performance retain is only slightly better. In other words, the proposed approach benefit from a stronger baseline (the first session). ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}, {"review_id": "ziyjld-X0z", "reviewer": "Reviewer_sCMo", "summary": "", "questions": "", "limitations": "", "rating": 6, "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": "", "summary_of_the_paper": "The setting of this paper is few-shot class incremental learning (FSCIL). The key idea is using neural collapse to do FSCIL. This paper use a set of pre-defined classifier prototypes as ETF for the whole label space, and then aligns all classes, including base session's classes and novel classes, to the pre-defined classifier prototypes. What's more, a novel loss function is proposed to drive the features into the corresponding prototypes. Experiments show that the proposed method achieves state-of-the-art performances on three standard benchmarks. \n", "summary_of_the_review": "This paper uses neural collapse for FSCIL and obtain good results. Nonetheless, there is a concern about unfair comparison.\n", "strength_and_weaknesses": "Strength:\n(1) I think the paper is well organized, clearly motivated, and contributes novelty.\n(2) Using Neural Collapse for FSCIL is inspiring, and the proposed loss function to drive the output features to the corresponding prototypes is novel.\n\nWeaknesses:\n(1) Unfair comparison. From Table.5 and Appendix: IMPLEMENTATION DETAILS, this method, and competitors use different network architectures. And even this method use model weights pre-trained on ImageNet on the CUB benchmark. However, all other methods (FACT, LIMIT, etc.) train the model from scratch. \n(2) This method uses memory to save old class prototype features. However, other methods (FACT, LIMIT, etc.) do not save any data. What's more, the feature extractor is fixed after the base phase, saving features is the same as saving raw data.\n", "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": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "gOm21qA5klA", "reviewer": "Reviewer_Smvh", "summary": "", "questions": "", "limitations": "", "rating": 8, "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": "", "summary_of_the_paper": "This paper proposed a new loss function for incremental learning, inspired by the inductive bias of neural collapse. The basic idea is to preallocate a sufficiently large Equiangular Tight Frame (ETF),  so that features from different classes (old and new) will be maximally separated from each other. At the same time, use remembered features from the old classes to fix the representations of the old classes.\n\nThe authors empirically show the effectiveness of the proposed method.", "summary_of_the_review": "Overall, the authors proposed a simple yet effective solution to catastrophic forgetting in incremental learning. I hope that the authors can publish the code for the sake of reproducibility.", "strength_and_weaknesses": "Strength:\n\nThe proposed method is an innovative application of Neural Collapse to incremental learning. The method seems to be effective in the experiments shown.\n\n\n\n", "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": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "_CINWLNT0RG", "reviewer": "Reviewer_NNii", "summary": "", "questions": "", "limitations": "", "rating": 6, "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": "", "summary_of_the_paper": "Inspired by Neural collapse, this paper proposed to ﬁx a learnable classiﬁer as a geometric structure for few-shot incremental learning. \nThe pre-assigned classifier could avoid optimization conﬂict among sessions in the training stage. They also provided a detailed theoretical analysis to show their method could hold the neural collapse optimality.  Experiments on the three datasets achieve the best performance compared with previous methods.", "summary_of_the_review": "I believe this paper is slightly below the acceptance threshold.\n\n\nThe paper presents a novel framework that inspired neural collapse for the FSCIL.  and quantitative experiments show that the proposed method outperforms baseline methods.   But, the overall novelty is incremental. (See the weakness.) \n\nBut if the authors can explain well the weakness I raised, I will consider raising my score.\n\n\n============== Post Rebuttal Update\n\nSorry for the late response. I thank the authors for their diligent efforts in the rebuttal. It was nice to see that the authors could give more detailed intuitions to improve their work and make relevant changes to their paper. Also, based on the other reviewer's comments and the responded of the authors, I have increased my score based on the author's responses.", "strength_and_weaknesses": "Strength: \n\nThe proposed framework that combines neural collapse inspired framework in the few-shot class incremental learning is interesting and inspiring. I also appreciate the extensive experiments (ablation studies and extensions) conducted by the authors.\n\n(1) The proposed neural collapse inspired framework on few-shot incremental learning seems novel.  The pre-assigned classifier could avoid optimization conﬂict in each session in the training stage. Its effectiveness was successfully proved by experiments and theoretical analysis. This is the biggest contribution of this paper.   Although adding an ETF classifier is not so novel, its effectiveness was shown by the experiments. \n\n(2) The experiments in the supplementary material are comprehensive and detailed. \n\nWeakness:\n\nMajor concerns: \n\n(1) The writing needs to be improved (not for the problem of English expression).  e.g.,  The last sentence of the first paragraph. What’s the importance and difficulty of the FSCIL?  The first sentence of the first contribution is completely uninformative and does not show their contribution at all. \n\n(2) In the second paragraph, \"Besides, as shown in Figure 1 (a), there will be a misalignment between the adjusted classiﬁer and the ﬁxed features of old classes.\" How did you know the misalignment between the adjusted classiﬁer and the ﬁxed features of old classes?  and in Figure 1 (a), how did you know the old features and new classes are close?\n\n(3) Apart from the fact that the motivation of neural collapse inspired framework is novel, the novelty of this paper still seems incremental. \nThey only use Yang et al. method for the few-shot incremental learning.  Even the written in sec3.2 is similar to Yang et al. Could you please explain the technical innovation of your approach versus the Yang et al. approach other than in FSCIL?\n\n(4) Since this paper pre-assigned the classifier at the beginning of all classes, even the tested classes,   but in real life, we cannot know how many classes will be tested during the test stage. So how does this method work when there is no way to know the number of classes to test?\n\n(5) It would be better to separate the comparisons between memory-based and non-memory-based methods in Table 1,2.\n\n\n\n\n\n\n \n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "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": {}}]

Author rebuttal:

Decision: Accept: notable-top-25%