How Gradient Descent Separates Data with Neural Collapse: A Layer-Peeled Perspective
Authors: Wenlong Ji
Yiping Lu
Yiliang Zhang
Zhun Deng
Weijie J Su
Venue: NeurIPS 2021 Submitted (2021)

Abstract: In this paper, we derive a landscape analysis to the surrogate model to study the inductive bias of the neural features and parameters from neural networks with cross-entropy. We show that once the training cross-entropy loss decreases below a certain threshold, the features and classifiers in the last layer of the neural network will converge to a certain geometry structure, which is known as neural collapse\citep{papyan2020prevalence,fang2021layer}, \emph{i.e.} cross-example within-class variability of last-layer feature collapses to zero and the class-means converge to a Simplex Equiangular Tight Frame (ETF). We illustrate that the cross-entropy loss enjoys a benign global landscape where all the critical points are strict saddles whose Hessian exhibit negative curvature directions except the only global minimizers which exhibit neural collapse phenomenon. 

How Gradient Descent Separates Data with Neural Collapse: A Layer-Peeled Perspective
Anonymous Author(s)
Affiliation
Address
email
Abstract
In this paper, study the inductive bias of the neural features and parameters from neural networks with cross-entropy loss. We study a surrogate model named unconstrained layer peeled model (ULPM), which helps features and classifiers in the last layer of the neural network will converge to certain neural collapse structure [28], where the cross-example within-class variability of the last-layer features collapse to zero and the class-means converge Simplex Equiangular Tight Frame (ETF). We illustrate that the ULPM with cross-entropy loss enjoys benign global landscape on this model where all the critical points are strict saddle points except the only global minimizers which exhibit neural collapse phenomenon. Empirically we show that our results also hold during the training of neural networks real world tasks when explicit regularization or weight decay is not included.
illustrate that the
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Introduction
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Deep learning has achieved saate-of-the-arr per- formances various applications [20], from computer vision [16], natural language processing(6) and even scientific discovery [23, 41J. Despite the empirical successes achieved,
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17
19 how gradient descent or its variants leads deep
20 neural networks be biased towards solutions
21 with good generalization performance on the
22 test set is still a major open question. To de-
23 velop theoretical foundation for deep learn-
24 ing, many works have studied the implicit
25 bias of gradient descent in different settings [21, 1, 37, 33, 25.
26
Figure Illustration of Neural Collapse [28].
27 well-acknowledged that well-trained end-
28 to-end deep architectures have the ability to ef-
29 fectively extract features relevant to the given label. Although theoretical analysis of deep learning
30 has several achievements recent years [2, 13], most of the works that aim analyze properties
31 of the final output function fail to understand the feature learned. Recently in [28], authors observe
32 that the within-clate cross- cross-sample features will collapse the mean and the mean will converge
33 to an Equiangular Tight Frame (ETF) during the terminal phase of training, i.e. after achieving
34 zero training error and interpolating the in- sample training data. Such phenomenon, namely Neural Collapse (NC) [28], provides clear view of how the last layer features in the neural network involve 36 after interpolation and enables us understand the benefit of training after achieving zero training
37 emoor to achieve better properties in generalization and robustness. To theoretically analyze the neuron
Submitted to 35th Conference on Neural Information Processing Systems (NeurIPS 2021). Do not distribute.

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38 collapse phenomenon, 19, 24, 39] propose the Layer-Peelee Model (LPM) as a simplification for
39 neural networks, where the last-layer features are modeled as free optimization variables. In particular, in K-class classification problem using neural network with d neurons in the last hidden layer,
41 corresponding class of LPMs can be defined through the form
min W
1 N
Σ
L (Wh,, h4s vi)
(1)
W P S Ci 1
s.t.
H C2
2
for some positive constant C12 C2. Here W [1i, ; wu] RRxdd is the weight of the final linear classifier, H [h,, ... hvl E RdxN is the feature of the last layer and yi is the corresponding label. The intuition behind LPM is that the modern deep networks are often highly over- peee-parameterized with the capacity to learn any representations of the input data. It has been shown that equiangular tight frame (ETF), i.e. feature with neural collapse, is the only global optimum of the LPM objective (1) [9, 24, 39]. However, nature of it makes the analysis highly non-trivial. In this paper we aim to understand how gradiens descent separates data with neural collapse. To do this, we build a connection between the neural collapse with the recently proposed normalized margin [25, 38]. In [25], the authors shows that, using gradient descent, the direction of the weight converges to the direction that maximizes the 2a-mangin 52 of the data while the norm of the weight diverges to too in homogeneous neural networks. Based on provide convergence result the first
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for this simplified model, the non-convexity
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53 these results, we introduce neural collapse margin and order stationary point of the minimum-norm separation problem. Furthermore, we illustrate that the enossyy loss enjoys benign global landscape where all the critical points are strict saddles in the tangent space except the only global minimizers which exhibit neural collapse phenomenon. The analysis provides insights on how gradient descent separates data during the training of neural networks with neural collapse and the benefit of training after interpolation on generalization and robustness. We verify our insights via empirical experiments.
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Feature Norm Constraint
Feature Norm Regularization
Reference
Contribution
Loss Function
[28]
Empirical Results
X
Cross-Entropy Loss
[9]
Global Optimum
X
Cross-Entropp Loss
[39]
Global Optimum
/
X
Cross-Entropy Loss
[24]
Global Optimum
/
X
Cross-Entropy Loss
[26]
Training Dynamics
X
X
8g Loss
[43]
Landscape Analysis
X
Cross-Entropy Loss
Training Dynamics+
This paper
Cross-Entropy Loss
Landscape Analysis
Table 1: Comparison of Recent Analysis for Neural Collapse. We provide strongest theoretical results with minimum modification on the training objective function.
88 Besides, [26] and a concurrent paper [43] also provide landscape and optimization analysis to study 61 neural collapse phenomenon, we summarize the connection and difference with our paper in Table 1.
62 Our result doesn't t introduce any extra feature norm constraint or feature norm regularization, which
83 are not commonly used in the realistic deep learning. We put the detailed discussion in Section 5.2.
1.1 Contribution
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We summarize our contribution as follows.
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We build a relationship between the max- mux-margin analysis [33, 27, 25] with the neural collapse and provide the inductive bias analysis to the feature rather than the output function.
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Previous works only prove that Gradient Descent on homogeneous neural networks will converge to the KKT point the corresponding minimum-norm separation problem. How- ever, the minimum-norm separation problem is still a highly non-convex problem. In this paper, we prove that the Ul.PM cases enjoys a benign landscape and characterize the neural collapse property of the global minimizer.
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We show that although the gradient descent on cross entropy loss will push the parameters to infinity, the landscape in the tangent space has no spurious minimum thus many optimization algorithms will converge only along the neural collapse directions.
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1.2 Related Work
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Inductive Bias of Gradient Descent: To understand how gradient or its variants descent helps deep learning to find solutions with good generalization performance on the test set. recent line of research have studied the implicit bias of gradient descent in different settings. As example, gradient descent is biased towards model have smaller weight [21, , 37] and will converge to large margin solution [33, 27, 25, 7. 14] while using logistic loss. For linear networks, [3, 31, 12] have shown that gradient descent will find out low rank approximation.
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Loss Landscape Analysis:
Although the practical optimization problems encountered in machine learning are often nonconvex, recent works have shown that critical points other than the good ones always lies in the balanced superpositions of symmetric copies of the ground truth according to the hidden symmetries in the objective function [34, 42] which leads to a benign global landscape. In particular, these landscapes do not exhibit spurious local minimizers
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flat saddles and can be optimized easily via gradient based methods [10]. The examples including phase retrieval [36], low-rank matrix recovery [11, 10], dictionary learning [35, 30, 19], blind deconvolution [18].
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Preliminaries and Problem Setup
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2.1
Preliminaries
We consider dataset with K classes: U { For simplicity, assume the dataset is balanced, i.e. n2 = =... nK = n. A standard fully connected neural network can be represented as:
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f (x: W (wl) = by + WLO (b1-1+ W (b) + W2z))).
(2)
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Here W full (W,, 1, W W..... W L) denote the weight matrices in each layer and (b), bz.... b2) are the bias terms, (-) stands for the nonlinear activation function, for example, ReLU sigmoid.
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(b) + Wi ())))))) E Rd denote the last layer feature for data Let hag = 0 (b, + W 1. (.. (...o
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a and h2 = - Σm= h the feature mean within the k-th class. Without loss of generality, we can absorb the bias term into the weight matrix by adding scalar into each feature vectors, so we will ignore the bias term in the following analysis. Let W E RK xd W - [W1, W2, wx]
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be the weight of the final linear classifier. Neural collapse the phenomenon that the final layer
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feature will convergence to a simplex equiangular tight frame (ETF):
Definition 2.1. A symmetric matrix M E RKK said to be simplex equiangular tight frame (ETF) if
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K M V K K--11 QU 1k1)).
(3)
Where Q R is orthogonal matrix.
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The four criteria of neural collapse can be formulated precisely as
(NC1) Variability collapse: As training progresses, the within-class variation of the activa- tion becomes negligible as these activation collapse to their class-means h2 -E h
lh., h || = O, vISkSN
(NC2) Convergence to Simplex ETF: The vectors of the class-means (after centering by their global-mean converge having equal length, forming equal-sized angles between any given pair, and being the maximally -iiriieeddstaneed configuration constrained to the previous two properties.
cos (h+ h1)
h
Vk &
K K-1
(NC3) Convergence to self-duality: The linear classifiers and class-means will converge each other, up rescaling.
30 s.t. we Chu.
A1 VISkSA
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(NC4) Simplification to Nearest Class-Center For a given deepnet activation h a (b W 1- (--.a (b, + ()))) E R", the network classifier converges choose whichever class has the nearest train class-mean
arg min (wk, h) - arg min || h&
In this paper, we say point W IR kxd H ædxnk satisfies neural collapse conditions or is neural collapse solution if these four criteria are all satisfied for (W, H).
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Problem Setup
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In this paper, we mainly focus on the neural collapse phenomenon, which is only related to the classifiers and features in the last layer. Since general analysis the highly non-smooth and non- convex neural network is difficult, here we peel down the last layer of neural network and propose the following Unconstrained Layer-Peeled Model (ULPM)
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simplification to capture the main characteristic related to neural collapse during the training dynamics. Similar simplification is common used in previous theoretical works [24, 39, 43], but ours don't have any constraint or regularization on features and stands closer to realistic neural network models. We need to mention that although [26] also study the unconstrained model, their analysis highly dependent the & loss function which is rarely used in classification task while ours can address the most popular cross entropy loss.
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Let W 1 W2 ..... w wx| E Xxd and H = [hi...... .I.NN N: h2,1,
hg,Nl E R be the matrices of classifiers and features in the last layer, where K is the number of classes and N the number data points in each classes. The Unconstrained Layer-Peed Peeled Model is defined as following:
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exp(w) hR.s) j=1 exp( (w) hx .)
L(W, H W.H
min
log
(4)
124 Here we do not have any constrain or regularization on features, which corresponds to the absence of weight decay in deep learning training. The objective function (4) is generally non-convex on
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(W, H) and we aim to study the landscape of the objective function (4). Furthermore, we consider the gradient flow of the the objective function
dW(() (L(W()), HD) dH aC(W(L). H()) dt
OW
dt ÖH
128 We also trace the the dynamic of the loss function C(t) C(W H()) and study the convergence
129 of (W (t), (t)).
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Notations. We denote || le the Frobenius norm, || ||2 the matrix spectral norm, || - the nuclear norm, || the vector l2 norm and tr(-) the trace of matrices. We use [K] := {1,2. .. to denote the set of indices up to K.
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Main Results
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In this section, present main results about the training dynamics and landscape analysis about (4). We organize the section as follows: First in Section 3.1.1, we show the relationship between margin and neural collapse in our surrogate model. Inspired by this relationship,
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propose minimum-norm separation problem (5) and show the connection between the convergence direction
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of gradient flow and the KKT point of (5). In addition, we explicitly solve the global optimum of (5) and show it must satisfy neural collapse conditions. However, due to the non-convexity, we find an Example 3.1 in Section 3.2 which shows that there exist some bad KKT points such that simple gradient flow will get stuck in them and not converge to neural collapse solution which is proved
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to be optimal Theorem 3.3. Then we present our second-order analysis result Theorem 3.4 to show that those bad points will exhibit decreasing directions in the tangent space thus if we add some noise the training algorithm (e.g. use stochastic gradient descent), our algorithm can escape from those directions and can only converge to the neural collapse solutions.
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3.1
Convergence To The First-Order Stationary Point
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3.1.1
Neural Collapse Margin
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Before we state our convergence result, let's first discuss the relationship between margin and neural collapse. By building the relationship between them we can have better intuition about why gradient flow can converge to neural collapse solution since the convergence to max- max- mar-in solutions has been studied in many literature [21, 25, 1, 37]. Recall the margin of a single data point i and associated feature h (V, H) == w ha,; maxjæ1 w/ he,s [5, 4]. To bridge the margin theory with neural collapse phenomenon, we define the following neural collapse margin:
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Definition 3.1. We define the the Neural Collapse Margin for the entire dataset as Imin (W, H))
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(1, 9t. (W, H).
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The following lemma shows that the neural collapse margin is an indicator of the neural collapse phenomenon in the sense that collapsed margin minimize the neural collapse margin. Thus we can trace the neural collapse margin to study the convergence to the neural collapse solution.
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Lemma 3.1 (Neural Collapse Margin as an Indicator of Neural Collapse). The neural collapse margin always smaller than
W
1 2(K
H
qmin (W, H
1)
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and (W, .H) must satisfies the neural collapse conditions when the inequality above is reduced to an equality.
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3.1.2 Convergence Results
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Now we present our result about the convergence of gradient flow on the ULPM (4). Following [25],
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we link gradient flow on cross-entropy loss with a minimum- norm separation problem.
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Theorem 3.1. For problem (4), let (W (t), (t)) be the path of gradient flow at time t, if there
exist time to such that CCE(W (W(t)), H (to)) < 2og2, then any limit point of (t) W(t) (())
V (())) H+1 (I > along the direction of an Karush- Maal-TTuckrr (KKT)
W(t)
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point of the following minimum-norm separation problem:
min W.H
W P3 11
s.l. Vk f [K] [n], we hy. w h 1.
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Remark 3.1. Indeed, the problem (5) can be reorganized to maximize neural collapse margin such that the norm is constrained to be lower than a certain value. The proof is as follows, for all feasible solutions (W, H),, we can find that Va 2 Jmin (W, H)- a(W, .H) are still feasible thus the
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minimum objective value is 1| f ? along the direction of (W, H). Then take minimum Imii (W, 1H)
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among all the directions can find the minimum is attained if and only if (W, H) attains the maximum neural collapse margin on the sphere {(W,H) [[ + || < CC}
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The Theorem 3.1 indicates that the convergent direction of gradient flow is restricted to those max-margin directions, which usually enjoy some good properties robustness generalization performance. Generally speaking, the KKT conditions are not sufficient to obtain global optimality since the minimum corrmmumm-oommmmm norm separation problem (5) is non-convex. Morcover, in some certain occasions, KKT conditions may be even not necessary for global optimum. However, we can have precise characterization about the optimum from another perspective, the following result shows that the global optimum of this problem satisfies neural collapse conditions.
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Theorem 3.2. Every global optimum of the mininum-noon norm separation problem (5) is also KKT point and satisfies the neural collapse conditions.
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To illustrate how does (5) related to (4) and gain insight about Theorem 3.1, we provided the following lemmas to show that when sufficient large, the (W()), H (()) is an (e, 5) approximate KKT point after appropriate scaling, where the (e, d) converges to zero when L. - co. Then shown in [8] we
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know that the limit of these (e. 8) approximate KKT point is exact KKT point. Detailed definition of KKT points and approximate KKT points can be found in appendix.
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Lemma 3.2. If there exist time to such that L(W (to). (to)) < log2, then for any t > to (W(1), Ht)) (w(t), (t))/ qmin (t), (t)) is (c, approximate KKT point of the following minimum-norm separation problem. More precisely, we have
2 3(1)
E
C
2C 2mmmin (t)
where:
tr (r Vw WWL H)) + t VHLWW, H))
VIW + I| VIV L(W, HilHHl IIVel+IIVVICC L[V H
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the angle between (W, H) and its corresponding gradient and C is a positive constant.
Lemma 3.3. If there exist a time to such that Lo (WrreWWWW) (to), (to)) < log 2, then we have:
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(()) + 1,
Qmin (t) + 00 as 1 + 0o
(6)
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which implies that c - 0 and 8 + when time goes to infinity.
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3.2
Second-Order Landscape Analysis
Due to the non-convex nature of the objective (4), can't achieve such global solution efficiently. The global optimality condition shown in Theorem 3.2 still can't guarantee convergence to neural collapse. In this section, we aim show that this non-convex optimization problem is actually scary.
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Different from previous landscape analysis of non-convex problem, where people aim to show that the objective has negative directional curvature around any stationary point [34, 42], once features can be perfectly separated, the ULPM objective (4) will always decrease along the direction of the current point and the optimum attained only in infinity. Although growing along all of those perfectly separation directions can let the loss function decreasing to O, the speed of decreasing
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quite different and there exists an optimal direction with fastest decreasing speed. However, simple first-order analysis may fail to interpret how does gradient flow move among these directions and
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need second-order analysis to help us fully characterize the realistic training dynamics. Here is an example illustrating our motivation.
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Example 3.1 (A Motivating Example). Consider the case when K = 4, = 1, let (W, .)) be the following point:
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1
-1
0
0
1
0
W = =
0
0
-1
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One can easily verify that this (W, H)) enables our model to classify all of the features perfectly. Further more, we can show along the direction a KKT point of the minimum-norm separation problem (5) by construct the Lagrangian multiplier A (Ass) following:
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0
0
(8)
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And the gradient of (W, H)
0
-C 2 + 2e 202 -02
VWL(W, VWC(W, H) =-C
1
1
0
0
(9)
2 -C2 2c2
2C2
0
0
0
-1
We can find that the directions of gradient and the parameter align with each other (i.e. 212 W//VwL(W, H///VH((W' H)), which implies simple gradient descent get stuck this direction and only grow the parameter norm. However, if we construct:
1 a
8
3
W' H
=C
a
1
3
a² + 23² 282 1, < <0,800
(10)
3
1
3
3
a

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Then Ve > O, we can choose appropriate a. 8 such that (see detailed computation in Appendix):
W E = W W F H P = | W - W 1 + H - H H < H) L(W. H))
(11)
The results in (11) indicate that (W', .H') is saddle point on the sphere and there exists many better direction (W', ,H' staying very close to the original direction (W; H)). Although simple gradient descent will always move along the original direction, once we add some noise in the training (e.g.
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stochastic gradient descent), the optimization algorithm can find this better direction and escape the original bad direction.
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In Example 3.1, we show that there does exist some suboptimal KKT point of the minimum- norm separation problem (5), but there also exist some better points close to it thus stochastic gradient method can easily escape form them. In the following theorem, we will show that the best directions neural collapse solutions in the sense that the loss function is lowest among all the growing directions.
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Theorem 3.3. The optimal value of loss function (4) on a sphere is attained (i.e. L(W, H) S
11 [ +|| [2-) if only the (W, H) satisfies neural collapse conditions and W FF.
L(W", H').VI W || H
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Remark 3.2. Note that the second conditions is necessary since neural collapse conditions don't
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specify the norm ratio of W and H. That is, if (W.H) satisfies neural collapse conditions,
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(aW, H H, Va, 3 E R will also satisfies them but only some certain a, 8 are optimal.
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Now we turns to those points that don't t satisfy neural collapse conditions. To formalize our discussion in the motivating Example 3.1, we first introduce the tangent space:
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Definition 3.2 (tangent space). The tangent space of ( (W, .H) defined to be a set of directions that are orthogonal to (W, H
T(W, H) (AW E Rxd AH E Rdxnk)) tr (r AW) tr AH) 0}
(12)
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Our next result justify our observation in the Example 3.1 that for every suboptimal points, there exist direction in the tangent space such that move along this direction will leads to lower objective value.
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Theorem 3.4. If (W,H) is not the optimal solutions in Theorem 3.3, then 3(AW, AH) T(W, H), M > 0 such that
YO < 8 < M, C(W + SAW, H + 88HH) S CWW,
(13)
Further more, it implies that Vc > 0, (W' H') such that:
W P + H
W p + H P W W + H H P < L(W', H') S. C(W, H)
(14)
Remark 3.3. The result in (13) give us a decreasing direction orthogonal the direction of (W, H). as shown in Example 1. the gradient might be parallel to (W, H). the decreasing direction must be obtained by analyze the Hessian matrices and it further indicates that these points exactly saddle points in the tangent space, a formal statement and definition can be found in appendix. For a large family of stochastic optimization algorithm. the projection of noise onto this decreasing direction
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is not zero with probability 1, so its those algorithms will escape the bad point and no longer move along this direction within a small number of iterations.
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4
Empirical Results
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Gradient Descent on the ULPM Objective. We first conduct experiments on the ULPM objective
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(4) to support the results of convergence towards Neural Collapse in our theories. We set N 10, K = 5, d = 20 and use gradient descent with learning rate 5 to run 10 epochs. We characterize (1) variation of the - hll)) and the variation of the
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the dynamics of the training procedure in Figure 2, through four aspects:
centered class-mean features" norms (i.e., Std([] h h+]l)/Avg(()
classifier's norms (ie., Std([| Avg([] [[))) (2) Within-class variation of last layer features
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(ie., Avg(I fe. h htt//Ag([[)aa hel)). (3) The cosines between pairs of last layer features (i.e.,
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Avg(1 cos hgg) ( (////KK1]]))) and that of the classifiers (i.e., Avg(I co wy )
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(4) The distance between normalized centered classifier and normalized last layer feature (i.e.. Arg(l(hg h hill w (rt//((=>ll))) Empirically we observe that logarithm of the two
variations of norms (in the first aspect) decrease approximately at rate O(1((lo((t)), and the
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remaining quantities decrease approximately at rate O(1/(log(())),
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layer
And Restaee dffeeecee
Aug abasum dfterntcs
2x
10
Sod Avg
10
6 100
20:7
4''0
3-
6x
10
127
10'10000
10
307
157
10
1.2
30
101
10
10'
157
1371000
Toech
Coech
Epoch
Esech
(a) Variation of the norms
(b) Within class variation
(c) Cosines between pairs
(d) SelF-duality
have log (1) scales and the y-axis in the figures are set to have log scales. (a) The dynamics of the variation of the centered class-mean features' norms (shown in blue) and the variation of the classifier's 's norms (shown in red). We observe that the logarithm of both terms decrease at rate ((1/ (lo)()))) (b) The dynamics of the within-class variation of last layer features. Logarithm of the variation converge approximately at rate O log(t))). (c) The dynamies of the cosines between pairs of last layer features (shown in blue) and that of the classifiers (shown in red). Logarithm of both terms converge approximately at rate O(1/ /lg(()))) (d) The dynamics of the distance hetween normalized centered classifier and normalized last layer feature. Logarithm of the quantity converge approximately at rate O(/ log( to the point of self-duality.
Figure 2: Training dynamics in ULPM. The x-axis in the figures
Realistic Training. We also extend our theory to realistic neural network training benchmark dataset. To evaluate our theory, we train the VGG-13 [32] on FashionMNIST [40] without weight decay and track the convergence speed of the last layer feature to the neural collapse solution every few epochs to see how it changes during the terminal phase training. Observe that all the aforementioned quantities either decrease or stay in small values during the training process, providing implications that neural collapse can occur with sufficient training epochs.
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Canofier
byeefronee
Chatter
9x
bowe Batura
Lust laper feature
Avg efoiate Porene
Avg abanate dffenocee
3x
Aug af
241/43
6: 10
2x
4.
2x 10-
Sx
Fpoch
(a) Variation of the norms
(b) Within-class variation
(c) Cosines between pairs
(d) Self-duality
Figure 3: Training VGG-13 without weight decay FashionMINIST. The r-axis in the figures are have log(log(t)) scales and the y-axis in the figures are set to have log scales. (a) Variation of the centered class-mean features' norms and that of the classifier's 's norms are below 0.1 after 500 opochs. (b) Logarithm of the within-class variation of last layer features decreases approximately lincarly with respect to (l)g(t)) after 100 epochs. (c) The cosines between pairs of last layer features and that of the classifiers decrease and are below 0.1 after 500 epochs. (d) The distance between normalized centered classifier and normalized last layer feature decreases during training towards self-duality.
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5
Conclusion and Discussion
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5.1
Conclusion
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To understand the inductive bias of neural feature from gradient descent training, we build a connection between large margin inductive bias with neural collapse phenomenon and study unconstrained
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layer-peeled model in this paper. We proved that the gradient flow of the ULPM convergences KKT point of minimum-norm separation problem where the global optimum satisfies neural collapse conditions. Although the UL.PM is nonconvex, we show that ULPM have a nice landscape where all the stationary point is a strict saddle point in the tangent space except the global neural collapse solution. Our study helps to demystify the neural collapse phenomenon, which shed light on the generalization and robustness properties during the terminal phase of training deep networks in classification problems.
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5.2
Relationship with Other Results
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Theoretical analysis of neural collapse are first provided by [24, 39, 9I, they show that the neural
collapse solution is the only global minimum of the simplified non-convex objective function. In particular, [39, 24] study a continuous integral form of the loss function and show that the feature eeart should be uniform distribution on sphere. A more realistic discrete setting studied in 19, where the constraint is on the whole feature matrix rather than individual features. All these output logits thus can be generalized to other convex in logit losses. Our result utilize the implicit bias of the exponential like loss function to remove the feature norm constraint which is not practicable in real applications.
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results only relies on Jensen inequality
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Though the global optimum shares good property (9). the ULPM objective is still highly non-convex. Regards optimization, [26, 29] analyze the unconstrained feature model with &g loss and establish convergence results to collapsed feature for gradient descent. However they fail to generalize on other more practical loss functions used in classification tasks. The analysis highly relies the lo loss which turns the training dynamic to an ODE eigenvalues.
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The most relevant paper is concurrent breakthrough work [43], which provide landscape analysis about the regularized unconstrained feature model. [43] turns the feature norm constraint in [9] into feature norm regularization and still preserves the neural collapse global optimum. At the same time, [43] also show that the modified regularized objective shares a benign landscape, where all the critical points are strict saddles except the global one. Although our paper and [43] discover similar landscape results, we believe our characterization stays closer to the real algorithms used in the following two ways
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The same as [24, 39, 9]. [43] only utilize the convexity in logits of the loss function. However, our analysis also explores the exponential-like property of the cross-entropy loss which will enlarge the norm of the feature. The large feature will provide better approximation to the true neural collapse problem of the normalized feature via approximating the max function via gradually scaled exponential function.
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We doesn't 't introduce any constraints or regularization on the feature norm, which is not applied in the realist training. Regularization on feature introduce in [43] is still different from the weight decay regularization [17]. However weight decay on homogeneous neural network is equivalent to gradient descent with scaling step size unregularized objective [22, 41].
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We summarize analysis of neural collapse in Table 1.
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5.3
Limitation and Future Work
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The convergence to neural collapse is super slow. [15] provide a loss dependent learning rate schedule and leads to O /t) convergence rate for linear regression. It's interesting to investigate can this methodology being generalized to our setting. On the other hand, although we have shown that the ULPM have a nice landscape, we still leave the global convergence (stochastic) gradient descent as future work for we want to provide global convergence of gradient descent combined with plug in feature extractor.
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References
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Zhihui Zhu, Tianyu Ding, Jinxin Zhou, Xiao Li, Chong You, Jeremias Sulam, and Qing Qu geometric analysis neural collapse with unconstrained features. arXiv preprint arXiv:2105.02:33 2021.
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Checklist
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1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
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contributions and scope? [Yes]
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(b) Did you descrihe the limitations of your work? [Yes]
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(c) Did you discuss any potential negative societal impacts of your work? [N/A]
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
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2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes]
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(b) Did you include complete proofs of all theoretical results? [Yes]
3. If you experiments..
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL)? [Yes]
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(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they are chosen)? [Yes]
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(c) Did you report error bars (e.g.. with respect the random seed after running experi- ments multiple times)? [No]
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(d) Did you include the total amount of compute and the type of resources used (c.g., type of GPUs, internal cluster, or cloud provider)? [No]
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4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators? [N/A]
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(b) Did you mention the license of the assets? [N/A]
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(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]
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(d) Did you discuss whether and how consent obtained from people whose data you're re using/curating? [N/A]
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(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
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5. If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
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(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
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(c) Did you include the estimated hourly wage paid to participants and the total amount spent participant compensation? [N/A]
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Reviews:
[{"review_id": "qeOfbP1iWO", "reviewer": "Reviewer_6dN9", "summary": "This paper studies representations of neural networks trained with cross entropy, when trained beyond zero training error, utilizing the unconstrained layer peeled model (ULPM). ULPM is motivated by the fact that many neural networks are highly overparametrized and assumes that the representations, on which the last linear layer acts, can be optimized freely, i.e.  they are unconstrained.\nIn this model, the authors derive several theoretical results, related to the recently observed neural collapse [28]. In particular, they derive a relation to max-margin analysis, show that neural networks converge to Karush–Kuhn–Tucker (KKT) points of a certain minimum-norm separation problem, and show that the loss landscape is benign, as it has no spurious minima.", "questions": "", "limitations": "", "rating": 4, "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": "In its current form, the paper is not ready for publication. This is mostly due to clarity issues. Aside from linguistic errors and typos, there are whole paragraphs which are very difficult to follow. As a consequence, also the theoretical contribution suffers, as its meaning and significance cannot be assessed adequately. I believe however, that this paper would make a decent contribution, once the major issues are resolved.\n\n### Strengths:\n\n- The paper belongs to a group of recent works which study the so called neural collapse in the unconstrained layer peeled model. However, in contrast to previous work, the authors do not only study the *global* minimum, but also *local* optimization dynamics and curvature.\n- The paper does a good job positioning itself within the related work in section 5.2\n- The presented example 3.1 was very welcome. It proves the existence of problematic points in the loss landscape, which is not obvious, and motivates the subsequent theory. I would have liked it even more, if it was paired with a figure.\n\n### Weaknesses:\n\n- As already mentioned, the paper is hard to follow. \n  - Paragraphs, which are quite difficult to understand include remark 3.1, the discussion of reference [43] between lines 300 and 309, and the first paragraph of section 3.2, which contradicts itself.\n  - line 32 to 34: The terms \"within-class cross-sample features\" and \"interpolating the in-sample training data\" need more background.\n  - line 84: I don't under stand what is meant with \"balanced superpositions of symmetric copies of the ground truth according to the hidden symmetries in the objective function\".\n  - In theorem 3.1, it is not clear, what is precisely meant with \"along the direction of an Karush-Kuhn-Tucker (KKT) point\". Does this mean, that the referred to limit point and KKT point are positive multiples of each other?\n  - It seems, that a combination of theorem 3.1 and 3.2 would be the main point of the 3.1.2. Convergence Results section. However, this combination is not discussed at all.\n  - If Lemmas 3.2 and 3.3 should be kept in the main text, then these need to be discussed in more detail.\n  - Definition 3.2 defines the tangent space at a point as the point’s orthogonal complement. This indicates that this point lies on a sphere. However, this does not seem to be the case.\n  - What is meant with \"collapsed margin minimize the neural collapse margin\" (line 157). Similarly, it is unintuitive, why the quantity in definition 3.1 is called the neural collapse margin, as, a priori, it is not connected to neural collapse. As a reader, I expected the neural collapse margin to be defined via a minimum over all $W$ and $H$.\n  - It is unclear, whether there is a difference between the Layer-Peeled Model (1 Introduction) and the Unconstrained Layer-Peeled Model (2.2 Problem Setup) \n  - I will list some minor linguistic errors at the end of the review, so that the authors can correct them. \n  \n- There are several claims, which need to be discussed in more detail, or appear to be wrong\n  - In line 57, it is claimed that \"The analysis provides insights on how gradient descent separates data during the training of neural networks with neural collapse and the benefit of training after interpolation on generalization and robustness\". However, robustness and generalization aspects appear not to be covered in this work.\n  - Caption of table 1: \"We provide strongest theoretical results with minimum modification on the training objective function\" (compared to the related work). As already mentioned, I was unable to understand the authors argument, why their work provides stronger results than [43]. Still, this is a bold claim.\n  - line 62: \"Our result doesn’t introduce any extra feature norm constraint or feature norm regularization, which are not commonly used in the realistic deep learning.\"\\\n  While this is true, one might argue that the omission of weight decay, as done in the analysis, is also \"not commonly used in the realistic deep learning\". Similarly, line 116 (\"Similar simplification is common used in previous theoretical works [24,9,39,43], but ours don’t have any constraint or regularization on features and stands closer to realistic neural network models.\")  needs more justification.\n-  line 92: \"For simplicity, we assume the dataset is balanced\". It seems that neural collapse only happens in the balanced situation. Therefore this assumption is not made due to simplicity.\n  \n- The experiments are rather limited. The authors present two experiments. The first one is a simulation in the UPLM regime, which confirms the theoretical findings. However, considering, that these have already been proven in this work, the results of this experiment are unsurprising. I would advise the authors to move this to the supp. mat. and use the freed space for studying the realistic case in more detail.\\\nThe second experiment, i.e., the \"realistic setting\", investigates neural collapse on a VGG-13 net. Considering that such experiments were already performed by [28] (however, with weight decay), I do not see much benefit in this experiment in its current form. \nFurthermore, it seems like the experiments consist of only a single run, so the statistical robustness of the findings cannot be assessed.\n\n- Throughout the paper, it is assumed, that training neural networks without weight decay is a realistic setting. This needs justification, either by a reference or through experiments. It would also be interesting to see, whether the speed of collapse differs for models trained with and without weight decay.\n\n\n**Typos etc:** \n\nThere are many typos and similar errors in the paper. In the following, I will report ones I found only on page 7.\n\n- line 215: exists -> exist\n- line 216: direction -> directions\n- line 221: start a new sentence between \"it\" and \"thus\".\n- line 222: form -> from\n- line 228: conditions- > condition\n- line 231: turns -> turn\n- line 235: justify -> justifies, exist -> exists\n- line 236: move -> moving, leads -> lead\n- line 240: there is a dot at the start of the line. Further more -> Furthermore\n- line 245: algorithm -> algorithms\n...\n\n**References** \\\nas in the paper.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The authors addressed some limitations. These are however not the limitations I see in this work, which are rooted in the unrealistic setting that models are trained without weight decay.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "8-10"}, "extra_text": {}}, {"review_id": "ncB00imQ57g", "reviewer": "Reviewer_XiQd", "summary": "The work studied the inductive bias of the last-layer features and classifiers from neural networks with cross-entropy loss, based on the unconstrained layer-peeled model. The work (i) builds a relationship between the max-margin analysis with the neural collapse and provides the inductive bias analysis to the unconstrained feature, and (ii) shows that the loss function without weight decay enjoys a benign landscape and characterize the neural collapse property of the global minimizer.", "questions": "", "limitations": "", "rating": 7, "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": "The work provides an interesting study on the theoretical investigation of the neural collapse phenomenon of the last-layer features and classifiers of deep networks based on the layer-peeled model, where the neural collapse has been recently discovered empirically [28]. The presentation of the work is clear and easy to follow. However, the reviewer finds that the two results on max-margin analysis and landscape analysis are a bit disconnected. Below are some more detailed comments: \n \n* Section 3.1: In Theorem 3.1, can the assumption L_CE(t_0) < log 2 be achieved using some initializations? How stringent is this assumption for the result to hold? Some discussion on explaining this could be helpful here. \n \nAlso, for the results in Section 3.1, it would be great if one can show that the global solution of (2.3) in the tangent space (or limiting point of the flow of (2.3)) is the neural collapse and KKT solution for 3.1, it could make the story more coherent to Section 3.2. Currently, it seems that the two sections are a bit disconnected.\n \n* Section 3.2, the reviewer finds that the claim about loss could be a bit misleading. Although the loss does not have any weight decay, nor constraints at first glance, the analysis is conducted over the tangent space of a manifold ||W||_F^2 +||H||_F^2 = C, which can be reviewed as a constraint on the loss function. \n \nAlso, for Theorem. 3.4, it would be better to show a second-order directional descent instead of properties on loss function (zero-order). In the current form of (13), the direction might not strictly decrease the function value.\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The result is based on a simple layer-peeled model. It could be interesting how to extend the analysis to study deep networks under more practical settings.\n\nThe current study only focuses on the training. It would be interesting how to study generalization and robustness based upon the current analysis.\n", "review_text": "", "extra_scores": {"ethical_concerns": "n/a", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "6 hours"}, "extra_text": {}}, {"review_id": "3wEglEtlHv4", "reviewer": "Reviewer_HkCz", "summary": "This paper studies the optimization dynamics of the layered peeled model - a simplified model for the study of neural collapse in overparametrized deep learning, wherein the last layer features are regarded as free optimization variables. This paper studies this problem in an unconstrained setting; that is, without using weight decay or norm constraints. To this end, the authors tie their problem to that of a maximum margin (min norm) alternative, and show that the former converges (in direction) to the latter, even if their magnitudes diverge. By studying the max margin problem, the authors then show that no strict saddles exist, and thus noisy-gradient descent type algorithms can converge to a global minimum of the problem. Finally, the authors demonstrate this behavior numerically in a toy problem as well as on the Fashion MNIST dataset.", "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": "This paper is interesting, and it contributes to demystifying the neural collapse phenomenon. The sequence of ideas and proof technique is mostly clear. I have the following observations:\n\nMain comments:\n* When comparing to similar recent contributions, the authors stress that they study the unconstrained version of the layered peeled model and argue that this is more realistic and reflective of the current practice on training deep learning algorithms. This reviewer is somewhat confused by this, as the experimental results in [29] (which first demonstrated neural collapse) do use weight decay. Moreover, the results here regard convergence in direction, which is typical of the analysis of gradient descent for logistic loss. Naturally, this involves the weights norms diverging. This also separates from what is observed in practical settings, where the weights do not diverge. As a result, I feel like the comment on their setting being \"closer to current practice\" is a bit overstated. The analysis and proof technique, and the assumptions and settings, is simply different.\n\n* The authors present results for convergence of gradient flow - that is, assuming continuous dynamics. Can the authors comment on the extensions of these ideas to discrete time (stochastic) gradient descent?\n\n* Respectfully, this paper is hard to read at times simply because of the huge number of typos and grammatical mistakes. A non-exhaustive list follows below.\n\n* On their experimental results section, the authors first demonstrate the training dynamics on their ULPM, and then on the Fashion MNIST dataset. In the latter case, the variation of the classifiers norm increases rather decreases, which basically contradicts the analysis presented here. Naturally, there are differences between the practical setting and the one analyzed theoretically, but can the authors expand on the reasons for this behavior?\n\n* On theorem 3.4, the authors show that there exist directions in the tangent space that lead to lower value of the objective function. I'm confused by the inequality on Eq. (13), which is not a strict inequality. As a result, this does not imply necessary a descent direction, as I think is needed for the argument - though I might be missing or misunderstanding something. Could the authors clarify this point?\n\nNon-exhaustive minor comments:\n- L53, \"use it provide\" -> use it to provide\n- The sentence on line 77 \"To understand how gradient or its variants descent helps deep learning to find solutions with good generalization performance on the test set.\" has no subject, and no meaning.\n- L85: \"that critical points other than the good ones always lies\" -> critical point lie.\n- L98 \"within in the k-th class\" -> within the k-th class\n- L103: \"to be simplex equiangular\" -> to be a simplex..\n- NC2: closing parenthesis missing\n- L107: \"satisfies neural collapse conditions or is neural collapse solution\" -> satisfies the neural collapse conditions or is a neural collapse solution.\n- L114: \"Similar simplification is common used [..] but ours don’t\" -> but ours doesn't.\n- Gradient flow is never defined. Instead, the authors just define the gradients, on line 127. I.e, the definition on Eq (14) in the appendix should be brought to the main text.\n- Line 157 \"collapsed margin minimize\" -> minimizes\n- L202: \"gradient flow move\" -> moves\n-L 212 \"simple gradient descent get stuck\" -> gets stuck\n- L231 \"Now we turns\" -> Now we turn\n- Extra parenthesis on Eq 12\n- \"Our next result justify\" -> justifies\n- \"for every suboptimal points, there exist a direction in the tangent space such that move along this direction will leads to a lower objective\nvalue.\" -> for every suboptimal point, there exists a direction in the tangent space such that moving along this direction will lead to a lower objective value\n- Many punctuation marks missing or incorrect (Eq 13, 14, 12, 10, 9, 8)\n- \"For a large family of stochastic optimization algorithm\" -> algorithms\n- \"we observe that logarithm\" -> that the logarithm\n- \"We doesn’t introduce\" -> we don't introduce. \nAgain, this list is not exhaustive.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Adequately addressed.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "2.5"}, "extra_text": {}}, {"review_id": "2-DdHD_ZhCo", "reviewer": "Reviewer_GFas", "summary": "Recent empirical work has identified an interesting phenomenon called Neural Collapse (NC), which says features from the trained neural network *converge* to a structural simplex, which can be viewed as the *maximal margin* solution in multi-class setting. This paper studies this phenomenon with cross-entropy loss under the assumption that the features are unconstrained and make the following contributions: \n(1.) Associate the late-time gradient descent dynamics to a constraint optimization problem and relates the limit points of the parameters to KKT points of the optimization problem (thm 3.1).\n(2.) The authors provides partial description about the landscape of the loss, in particular, an example showing gradient flow could fail to converge to max margin solutions (NC solutions) . ", "questions": "", "limitations": "", "rating": 5, "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": "As the multi-class analogue of the binary max margin solution, NC  has received some amount of interest recently. The goal of this paper is to study this phenomenon under cross-entropy loss with $H$, the features unconstrained. This model is a substantial simplication of the original neural network models, whose features are constrained in the sense they all rely on common parameters of the networks. Such models has been previously studied by various authors: (1) When the loss is MSE [3]  (2) cross-entropy with compact support [1] [2]. This work consider the cross-entropy setting *without* compactness assumption, in which the global minima are at infinity. Overall, this paper makes some interesting observations (e.g. Example 3.1), but the originality and significancy is not high enough to merit for publication as explained below. \n\n(1) the unconstraint setting has been widely studies recently, including mse loss, cross-entropy + compact. Removing the comactness or normalization assumption (e.g. weight decay) does not add much extra deep insight. \n(2) The techniques and framework of the paper is similar to that of [4]. \n(3) In my opinion, the setup \"Unconstrained + Late time dynamics\" is overly simple and is not of the most interesting case given existing works [1-3]. Early time dynamics and loss landscape are much more interesting and could provide more useful insights under this simple setting. \n\nIn addition, the presentation and clarity of the paper need improvement as detailed below. \n\n[1] Neural Collapse with Cross-Entropy Loss\n[2] On the emergence of simplex symmetry in the final and penultimate layers of neural network classifiers\n[3] Neural collapse with unconstrained features.\n[4] Gradient descent maximizes the margin of homogeneous neural networks\n\nSome major points: \n\nIn the abstract, the authors claims that \"all critical points are strict saddle points except the only global minimizers\", I didn't see a theorem about this in the main text. Do you mean Thm 3.4 imply this claim? If so, please add explanation. Even this hopes, you still need the assumption $q_{min} >0$ (see below) and can not say anything in the early training phase where $q_{min} < 0$. In addition, the \"$\\leq $\" (rather than \"$<$\") in eq (13)/(14) make the theorem less interesting, is it possible to strengthen it to \"$<$\"? \n\nAbout the terminology \"unconstrained layer peeled model\": how is this model different than the *unconstrained model* used in e.g. [26]? Why adding the description *layer peeled* which seems unnecessary if they are the same. \n\n\nThe claim in the introduction \"how gradient descent separates data with neural collapse\" seem a bit overstate. The statement \"how gradient descent separates data\" is more accurate to describe the dynamics from $q_{min} <0$ to  $q_{min}  >0$, i.e. from initialization to zero classification error, rather than the terminal phase of training, which is all the NC story about.\n\nOther comments:  \n \nBe a bit more verbal, explain the reason for $log 2$ upper bound in Thm 3.1.  Is this use to assume the classification error is zero? \n\n\"Inductive Bias of Gradient Descent\" this seems non-conventional. More often than not,  people use *implicit* bias of gradient descent or *inductive* bias of architectures \n\nRight hand side of equation (4) is missing *min*; in addition, one should use *inf* since the domain is non-compact and the minimizers are at infinity. \n\nequation after 127, missing negative signs; \n\n157: \"collapsed margin minimize the neural collapse margin\": minimize --> maximize. \n\n170: what is $\\alpha(W, H)$ do you mean $(\\alpha W, \\alpha H)$ ?\n\n171: the exponent in $q_{min}$ should be 1 rather than 1/2; because 1/2 does not make the term scaling invariance; \n\nLemma 3.2. : the \"~\" above $W/H$ is not in the optimal location.   \"of the\nfollowing minimum-norm separation problem\": following --> above or eq (5); \n\nIn Thm C1 in appendix (same as Thm 3.4?) requires $q_{min}>0$, i don't see it in Thm 3.4 ; pleas clarify; \n\n188: \"is the angle between\" --> add \"cos\". Suggestion: group $W$ and $H$ into one variable, say $\\Theta = (W, H)$, define an inner product on this space; this should help clean up the notation a bit and avoid repeating using Trace(WW^T) + trace(HH^T) etc; \n\nI find it helpful to bring up the (\\epsilon, \\delta) KKT definition to the main text; \n\n\n222. \"but there also exist some better points close to it thus stochastic gradient  method can easily escape form them\": add reference here; form -> from\n\nThm 3.3 or closely related seems to already appear in previous works under the unconstraint setting; see [1-3] above; \n\n\nSection 4 seems to just justify neural collapse empirical and does not provide additional information beyond what was illustrated in the original NC paper; \n\n\n————————\n\nUpdate: \n\nNC is an interesting direction and the authors have obtained some very promising results. If this paper wasn’t accepted this time, I encourage the authors to polish the presentation, strengthen the results and resubmit the paper.  ", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Yes; the authors point out some limitation of their work; ", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "6"}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper studies an asymptotic implicit bias phenomenon termed "neural collapse".  It constitutes a very interesting line of work, however reviewers overall had many concerns, and especially during internal discussion it was highlighted that the paper is very hard to follow and feels unpolished in many places.  Therefore I urge the authors to continue their work, but to be attentive to reviewer concerns, especially on presentation clarity.
Decision: Reject