Perturbation Analysis of Neural Collapse
Authors: Tom Tirer
Haoxiang Huang
Jonathan Niles-Weed
Venue: Submitted to ICLR 2023 (2023)
Keywords: deep learning theory
neural collapse

Abstract: Training deep neural networks for classification often includes minimizing the training loss beyond the zero training error point. In this phase of training, a “neural collapse” behavior has been observed: the variability of features (outputs of the penultimate layer) of within-class samples decreases and the mean features of different classes approach a certain tight frame structure. Recent works analyze this behavior via idealized unconstrained features models where all the minimizers exhibit exact collapse. However, with practical networks and datasets, the features typically do not reach exact collapse, e.g., because deep layers cannot arbitrarily modify intermediate features that are far from being collapsed. In this paper, we propose a richer model that can capture this phenomenon by forcing the features to stay in the vicinity of a predefined features matrix (e.g., intermediate features). We explore the model in the small vicinity case via perturbation analysis and establish results that cannot be obtained by the previously studied models. For example, we prove reduction in the within-class variability of the optimized features compared to the predefined input features (via analyzing gradient flow on the “central-path” with minimal assumptions), analyze the minimizers in the near-collapse regime, and provide insights on the effect of regularization hyperparameters on the closeness to collapse. We support our theory with experiments in practical deep learning settings.

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PERTURBATION ANALYSIS OF NEURAL COLLAPSE
Anonymous authors
Paper under double-blind review
ABSTRACT
Training deep neural networks for classification often includes minimizing the training loss beyond the zero training error point. In this phase of training. a "neu- ral collapse" behavior has been observed: the variability of features (outputs of the penultimate layer) of within-class samples decreases and the mean features of different classes approach certain tight frame structure. Recent works analyze this behavior via idealized unconstrained features models where all the minimiz- ers exhibit exact collapse. However, with practical networks and datasets, the features typically do not reach exact collapse. c.g., because deep layers cannot arbitrarily modify intermediate features that are far from being collapsed. In this paper, we propose a richer model that can capture this phenomenon by forcing the features to stay in the vicinity of predefined features matrix (e.g., intermediate features). We explore the model in the small vicinity case via perturbation analy- sis and establish results that cannot be obtained by the previously studied models. For example, we prove reduction in the within-class variability of the optimized features compared to the predefined input features (via analyzing gradient flow the "central-path" with minimal assumptions), analyze the minimizers in the near-collapse regime, and provide insights on the effect of regularization hyperpa- rameters on the closeness to collapse. We support our theory with experiments practical deep learning settings.
INTRODUCTION
Modern classification systems are typically based on deep neural networks (DNNs), whose param- eters are optimized using a large amount of labeled training data. Their training scheme often in- cludes minimizing the training loss beyond the zero training error point (Hoffer et al.) 2017 Ma
cct al. 2018, Belkin et al. 2019). In this terminal phase of training, a "neural collapse" (NC) behav- 1rr has been empirically observed when using either cross-entropy (CE) loss (Papyan et al., 2020) or mean squared error (MSE) loss (Han et al. 2022).
The NC behavior includes several simultancous phenomena that evolve the number of epochs grows. The first phenomenon, dubbed NCI, is decrease in the variability of the features (outputs of the penultimate layer) of training samples from the same class. The second phenomenon, dubbed NC2, is increasing similarity of the structure of the inter-class features' means (after subtracting the global mean) simplex equiangular tight frame (ETF). The third phenomenon, dubbed NC3, is alignment of the last layer's ' weights with the inter-class features' means. A consequence of these phenomena that the classifier's decision rule becomes similar to nearest class center in feature space.
Many recent works attempt to theoretically analyze the NC behavior (Mixon et al., 2020 Lu & |Fang
Steinerberger 2022) Wojtowytsch et al. 202T.Fang et al.) 2021 Zhu et al. 2021 Graf et al. 2021
Ergen & Pilanci| 2021 Ji et 2001 Galanti et al.) [202] Tirer Bruna 2022 Zhou et al. 2022
et al. (022). The mathematical frameworks are almost always based on variants of the unconstrained features model (UFM), proposed by Mixon et al. (222)), which treats the (deepest) features of the training samples as free optimization variables (disconnected from data or intermediate/haalloww features). Typically, in these "idealized" models all the minimizers exhibit "exact collapse" (i.e., their within-class variability is exactly 0 and an exact simplex ETF structure is demonstrated) provided that arbitrary (but nonzero) level of regularization used.
Thrampoulidis et al. 2022;) Yang et al.

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However, the features of DNNs not free optimization variables but outputs of predetermined architectures that get training samples as imput and have parameters (shared by all the samples) that are hard to optimize. Thus, usually, the deepest features demonstrate reduced "NC distance metrics" intermediate layers but do not exhibit be seen in any NC paper that presents empirical
(such as within-class variability) compared to features convergence to exact collapse. Indeed, results, the decrease in the NC metrics is typically finite and stops above zero at some epoch (the margin depends on the dataset complexity, architecture, hyperparameter tuning, etc.).
In this paper, this issue is taken into account by studying model that can force the features to stay in the vicinity of predefined features matrix. By considering the predefined features as intermediate features of DNN, the proposed model allows us to analyze how deep features progress from, or re- late to, shallower features. We explore the model in the small vicinity case via perturbation analysis and establish results that cannot be obtained by the previously studied UFMs. Specifically, we prove reduction in the within-class variability of the optimized features compared to the predefined input features. To obtain this result (for arbitrary input features), class variability along gradient flow on the "central-path" of a UFM with minimal assumptions (i.e., drop the assumptions and modifications of the flow that Han et al. (2022) did to facilitate their analysis). Next, we provide a closed-for formedd-oommmo approximation for the model's minimizer. Then, focusing on the case where the input features matrix is already near collapse (e.g., the penultimate features of a well-trained DNN), we present a fine-grained analysis of our closed-form approximation, which provides insights on the effect of regularization hyperparameters on the closeness to collapse. We support our theory with experiments in practical deep learning settings.
prove monotonic decrease of within-
2 BACKGROUND AND PROBLEM SETUP
Consider a classification task with K classes and n training samples per class. Let us denote by Yı E RK the one-hot vector with I in its k-th entry and by X RP the i-th training sample of the k-th class. DNN-based classifiers can be typically expressed as
D = Who(x) + +b,
where he(-) : IR + IRd (with d 2 K) is the feature mapping that is composed of multiple layers (with learnable parameters 0), and W [wi,..., wx] R xd (wg. denotes the kth row of W) and b E R-K the weights and bias of the last classification layer. The network's parameters θ [W, b, 0} are usually learned by empirical risk minimization
K
n
min
C ((Wh b,) + +R(O),
where L(.,-) is loss function (e.g., CE or MSE and R(-) is a regularization term. Following the work of Mixon et al. (2020), in order to mathematically show the emergence of minimizers with NC structure, most of the theoretical papers have followed the "unconstrained features model" (UFM) approach, where the features (bo()):)) are treated as free optimization
variables (hg,)- Namely, they study problems of the form
K
min (b Kn
Σ 1=1
C (Whe, + b.) R R (W, b, (ba.i))
One such example is the work in (Tirer & Bruna 2022). which considered setting with regularized MSE loss (which shares similarity with models in the matrix factorization literature (Koren et al, Chi et al. 2019), except the assumptions that d 2 K and on the specific structure of Y):
2009
min 1
Aw |/
|WH- Y
H
W.H 2Kn
2K
2Kn
where H = [hi1...... him A1,. h2...... hg.a] dR is the (organized) unconstrained features matrix, Y = Ix @ I RKxKn (where 8 denotes the Kronecker product) its associated one- hot vectors matrix, and Nw and dy positive regularization hyperparameters. It was shown that
Hui & Belkin (222)) have shown that training DNN classifiers with MSE loss is powerful strategy whose nerformance similar training with CR loss.
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all the (global) minimizers of this bias-free UFM exhibit orthogonal collapse, stated in the following theorem
Theorem 2.1 (Theorem 3.1 in (Tierr & Bruna 2022 ). Let d 2 K and define = vdH If c S 1, then any global minimizer (W H)) of Eq. satisfies
h = _... hnnn =: h
Vk E [K],
Aw | 112 =: p= (1- Vo Ay
h hi
h
hgg = =pIx,
w / Xu/Ann Aw hi:
Ak [K].
Ife 1, then Eq. is minimized by (W*, H.) (0,0) 0).
In short, the theorem states that any minimizer (W H)) of Eq. obeys that H" = H ④ 1 for some H E R K and W x H Ho W" x Ig. Is not hard to show that H H = pIk
implies that
H H kolk)
H H- p Ix (Ix (I~
where lig - = collapse to a simplex ETF. From the structure of the problem and the theorem, we see that there infinitely many minimizers of Eq. Indeed, can be deduced from the proof of Theorem 2. in (Tirer & Bruna 2022): Taking any (partial) orthonormal matrix R E jdd (i.e., R R
Σx= k= h H1 is the global mean. Namely, the "mean-subtracted features"
Ix), one can construct a minimizer for Eq. simply by H Vp(Aw, AH & 1 and W / (((Aw,
The existing literature includes other different UFM settings where all the minimizers exhibit NC structures (c.g., see (Lu & Steinerberger, 2022 Wojtowytsch et al. 2021 Zhu et al. 2021 Fang
jet al.) 202T; Thrampoulidis et al.] (022)). However, as discussed in Section ! all the previously studied UFMs are idealized and their results deviate from the situation in practical DNN training.
where the features do not exhibit exact collapse (e.g., since deep layers cannot arbitrarily modify intermediate features that far from being collapsed) and the setting of the hyperparameters affects the distance from NC structure.
In this paper, we consider a different model with the goal of better analyzing the real-world "near collapse" situation where "exact NC" cannot be reached. Motivated by Eq. [ we consider the following model
1 f(W, H:H = 2Kn
dw 2K
A
min W.H
WWH Y +
W
+
H
H 1
2Kn
2Kn
(2)
where Ho RddKn is input features matrix, which is fixed, and B is positive hyperparameter that controls the distance of H from Hg.
Let us discuss the motivation for studying this model. As before, interpret W and H as the final weights and deepest features of the DNN, respectively. Clearly, for Ho = 0 this model reduces Eq. (with | f regularized by Aн + 3). Furthermore, when Ho is nonzero, but already minimizer of Eq. [ (and thus has zero within-clase variability and an orthogonal frame structure), the following statement straightforward.
Corollary 2.2. Let d 2 K, λµ < 1, and let (W* H') be a minimizer of Eq. 7 Then, the minimizer f(W. H, HD) (in Eq. 2 is unique and it is given by (W", H.).
Au (i.c., their Note that the results in Tirer & Bruna 2022) are stated for Aw <- Aw and Au + Kn hyperparameters absorb the factors 1/K and T/Kn that used here). Scaling the terms in the objective according to the number samples, as done in Eq. [ agrees with what is done in practice (c.g., averaging
the squared errors over the minibatch samples rather than summing them). Our scaling also highlights the independence of the minimizers' properties on K and w.
Note that in both Eq. [ and Eq. 2 the minimizer w.r.t W a closed-form function of W' YH (HHT ndwT.) As such, minimizer H' of either objective uniquely implies the associated w' w (H').
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u] we have that the Under review as a conference paper at ICLR 2023
That is, Eq. allows to pick one of the minimizers of Eq. I by Ho and transfer its orthogonal collapse properties, which are stated in Theorem 2.1 to the minimizer of Eq.
However, the usefulness of Eq. comes from exploring cases with nonzero/non-collapsei Ho. In- deed, while Н can be interpreted as the deepest features of a DNN, here we interpret Hn as the features that are obtained in shallower layer. In this case, 1/B can be understood as the complexity of the subnetwork from Ho to H. We are particularly interested in the the large / regime, B > 1, where Hg expresses penultimate features (only one layer before H) that significantly constrain H. In Appendix M we review practical DNNs where the distance between the deepest and penultimate features may be small is even inherently small. In this paper we focus on this large regime, and provide mathematical reasoning for the empirical NC behavior that are not captured by previously studied UFMs, such as proving that the optimized H has smaller within-class variability than Ho. and analyzing how perturbations from collapse of Hg can be mitigated by the minimizer of Eq.
3 DECREASE IN WITHIN-CLASS VARIABILITY
As discussed above, while the features matrix H represents the output a DNN's penultimate layer, the input matrix Ho be interpreted the features of a preceding layer. Several works have presented empirical settings where the within-class variability of the features, measured by some "NC1 metric", decreases across depth (Papyan et al. 2020 Tirer & Bruna 2022] Galanti 2022).
The goal of this section is to prove such phenomenon for the model stated in Eq 2 The theory that provide shows also monotonic decrease of the within-class variability (till exact collapse) along
gradient flow on the "central-path" of the UFM stated in Eq.
Let us begin with several definitions that will be used in this section. For given set of features for each of K classes, {h,,)), we define the per-class and global means as hg - Di=1 hg, and hig == A Σ k= Ei= | he.s. respectively, as well as the mean features matrix H := [i..... hg]
Next, we define the within-class and between-class d covariance matrices
K
Ew (H)
he:: he) hx)
Kn
k=1
K
Σg ( =
(H)
(hg hh - -ha)T.
k=1
The within-class variability collapse (NCI) can be expressed as Ew (H) - while (H) + 0, where the limit takes place with increasing the training epoch, and En 0 filters de- generate cases such as H = 0. Several papers considered in their experiments the metric
1 Tr
W H H)) where Σ// denotes the pseudoinverse of Σn (Papyan et al. 2020 Han
K
al.
2022 Zhu al. 2021b. Yet, we believe that considering the metric
NC) = Tr (Ew (H)) Tr (H)(
more amenable for theoretical analysis while capturing the desired nondegenerate collapse behav-
ior" Indeed, the trace of a covariance matrix equals zero if and only if the covariance matrix is a
zero matrix (this follows from Cov? (Xv?(X,Y) < Var Var(Y))
Recall that the minimizer w.r.t. W in Eq. of H. which is given by W (H) YH equivalent to
(ndd Eq has closed-form expression that is function I2) Thus, the optimization in Eq. is
3
H H1/8 == argmin C(H) +
H Holl
2Kn
H
"The metric Tr Ew Σ was considered in (Han et al.
2022). Yet, to state a result on this metric the
K authors claim (in the proof of Cor. 2) that nonzero eigenvalue of Ww HH / equals the reciprocal of the associated nonzero eigenvalue E//" (HH '''//44 However, this is not correct general (due to the inherent rank deficiency of HH ). For example, for E 2 and HH single nonzero eigenvalue of the former is 5/9 while the single nonzero eigenvalue of the latter is 5.
o
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where ((H) k W (H)H + I Au H
For large B, the minimizer H1/8 can be viewed as a backward/implict gradient descent update from Ho with respect to the loss L. This follows from rewriting the first order optimality condition as
113 Ho KnV
1/B
Observing that for 3 + 00 we have H1/3 + Ho (formally shown in Appendix B the above
equation can be written as dH. gives rise to the gradient flow
1==0 KnV (n where we think of as 8-1. This naturally
dH. Hn,,HH,,,
associated with the UFM in Eq. This means that results this flow can be translated to results on the minimizer Eq. in the large 3 regime. Indeed, in Theorem 3.1 below, we show that NC1 (H) monotonically decreases along this flow, which implies that (H < (Ho) for large enough B (see the statement in Corollary beloow.
Note that flow for an objective that is equivalent to L(H) with Aw = 0 and Xи = 0 has been studied in (Han al., (022). who called the "central path". The motivation for studying such an objective, where the optimization variable W is replaced by the optimal WHHH), comes from the empirical observation in Han al. 2022) that the gap
Y WWH-Yly is rather small (compared to each term) during the optimization process of practical DNNs.
We now state our result for gradient flow on the "central path" (which is proved in Appendix A)..
Theorem 3.1. Assume that Aw >
λи 2 and that Hg is non-collapsed (i.e., Ew (Ho) * 0). Then, along the gradient flow, which is stated in Eq. 4 we have that
(H,,H, strictly decreases along the flow untill it reaches zero.
+ 22n't (r(E decreases along the flow. In particular, when Ah > 0,
Tr(Ew (H,)) decays exponentially.
. t Tr EB (H)) strictly increases along the flow.
Remark. Note that our gradient flow analysis has minimal assumptions. Unlike (Han et al. 2022), our flow does not assume zero global mean (hg = 0), Aw = A = 0 and invertibility of Ew And most importantly, it does not include any engineered renormalization and projection of the gradient, contrary to the previous work. Thus, it is more similar to practical gradient descent optimization of DNNs. Our unmodified flow and minimal assumptions require a different, and more general,
analysis with quite involved computations 5
Not only does Theorem 3.1 state a monotonic decrease toward in the NCI metric, it also provides a separation between the behavior of Tr(Ew) and Tr(Eg) along the flow. A strict separation is observed for λи 0: Tw decreases while Tr(E (EB) increases. As gradient flow is often used as a proxy for analyzing gradient descent with a small step-size (Elkabetz & Cohen (021). if we overlook difference between optimizing the UFM in Eq, joontty w.r.t. W and H and restricting the optimization to the "central path" (W*(H), H), then our theory also provides mathematical reasoning for the experiments on gradient descent in (Tirer & Bruna 2022) that show monotonic decrease in within-class variability.
state
Finally, with our interpretation of as 3-1, the following Corollary is a direct consequence of and the continuity of VL(H) (see Appendix B a formal proof).
Theorem 3.1 and
Corollary 3.2. Assume that Hg is non-collapsed. Then, there exists some constant C = C(H,) > 0 such that for C. we have that NC1(H,))) < (No).
II more detail, all the assumptions in (Han et al. 2022) (including continually renormalization the gradient) lead to the fact that only the singular values (and not the singular vectors) of an "SNR matrix" D 212 1 (H)H vary along their flow. However, since do not make their assumptions, we do not have such matrix whose singular hases fixed along the flow and we need approach the problem in more general way.
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Recall that in the large 3 regime interpret H as features of DNN that are deeper than Hu but such that the architecture between Ho and H is extremely simple (c.g. they are features of adjacent layers) and thus the distance between them is constrained. Under this interpretation, Corollary 3.2 implies that layer-wise optimization of DNN where each time a new layer is added (so that the previous deepest features H 1/8 are considered as the new Ho) will result in gradually depthwise decreasing NC1. An extension of the model in Eq. 2 that will include multiple levels of optimizable parameters may be able to provide similar reasoning to the gradual depthwise decrease in NCI that is observed in practical DNN training, where all the layers are optimized simultaneously.
4 ANALYSIS OF THE
REGIME
In this section, we will explore the behavior of the minimizers of Eq. in the near collapse regime. As stated in Corollary 2.2, if Hg is already collapsed then the minimizer of S also collapsed. This is aligned with the rationale that if we have a DNN that already exhibits collapse at some intermediate layer, we would expect the subsequent layers to maintain this collapse Essentially, we would like to analyze the minimizer of Eq. 2 for Ho that is not already collapsed. Unfortunately. for general non-collapsed Ho it is not likely that the minimizer is amenable for explicit analytical characterization. Yet, the fact that for orthogonally collapsed Ho = H* we get a unique minimizer (W", HH) of Eq. 2 which is still characterized by Theorem 2.1 gives us desirable setting for examining the minimizer of Eq. 2 obbainedd for Hn = H* + HH (with sufficiently small åHo) by exploiting our knowledge on (W H"; Ho H*). Analyzing the near-collapse setting will shed light on the way that the deviation from collapse in the input features is transferred to the optimized features, c.g., the amount of interaction within/bet beetinn/beweennnn classes and the effects of hyperparameters. Such insights can be latter examined empirically beyond the hear-collapse regime.
Let us denote by (W*, ,H') the minimizer of f(W. Ho). We are interested in studying the dependence of &W == W - W and 5H == H H" on HH H, HH H" without the requirement of computing (W ,H*) (that lack analytical expressions). In particular, our focus is on the relation between the features öH and öHo (rather than O' and HHg), both because a minimizer H* uniquely implies the associated W and because important aspects of NC, such as within-class variability decrease (NC1) and inter-class feature structure (NC2), consider the feature mapping rather than the last layer weights.
We begin with establishing such result in the following theorem (which is proved in Appendix C for Ho that is not necessarily collapsed features matrix.
The notation in the theorem is as follows. We use vec(-) denote the column-stack vectorization of a matrix. The derivatives are w.r.t. the vectorized matrices vec(H) and vee(W), For example, H Rdnn stands for the derivative of f w.r.t. vec(H), , and second derivative w.r.t. vee (W)" yields V W E IR xd
Theorem 4.1. Let d K, and some Ho and HHg- Let (W H²) be the minimizer of f(W, H: Ho) (with stated in Eq. 2). Let (W", H²) be the minimizer of f(W, H: Ho H. + 5Hg). Define 8W == W W and öH = H A.. Then, with approximation ac-
curacy of 0(1HH) 11ooi (1HHol?) we have that
B Kn
voc(5H) ~
V - V V V fw Vwf) vec(oH.),
vec(oW)
V Vw -V (V V v Vw
vec()Ho),
Kn
where all the derivatives are evaluated at the point (W*, Hr):H,) In particular, for B >> max(1, A } we have (with additional approximation error of (33-))
(H) [Tank
d Lank
B InK & WT
1 Zz 1 vec( vec(
This also aligned with empirical observations of gradual depthwise collapse in practical DNNs and with Corollary 3.2 at the limit where H, is nearly collapsed.
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where
Z = E + H
TH & IK + Iax (E*T + H & W*), (W'H' voc(ca, (W'H Y)), vec(ca20k, (W'H -)),... veclea.ee T K. (W'H H Y))]
Ex = voo(cs,10k
and Cd, is the standard vector in Id with in its ith entry (similar definition stands for ek,k
Observe that, assuming small approximation error, Theorem 4.1 states the linear operation that trans- forms dHo to äH. We will focus on the large B regime that is stated in Eq. 5 where the matrix inversion can be well approximated. Furthermore, due to the vectorization operation, observe that the linear expression (H)( se Fvec(HHo) has the following block-base representation
voc(oH
F1,1
F1,
vec ( ())
(6)
vec(5H(())
FK,1
FK,
Lvoc(H (K)
where (k) == öHl; dn(k 1) dnK] E Rdxn is the sub-matrix of öH that is composed of the columns associated with the kth class (and similarly for 8H,). Namely, we have that F E Rddk x is composed of blocks of size dn x dn. The diagonal blocks are the "intra-class blocks". Each of them shows the effect of perturbation
certain class in Hg on the features of the same class in H. The off-diagonal blocks are the "inter-class blocks". Each of them shows the effect of perturbation in a certain class Hn on the features of another class H.
Recall that for Ho = H" that is already exactly collapsed, the minimizer of f(; Ho) is also col- lapsed, so H H* in the above theorem. Importantly. in this case the matrix in Eq. transforms deviation from exact collapse in the input features to deviation from exact collapse in the optimized features. Thus, have that stronger attenuation behavior of the blocks of F (e.g., small singular values) implies that the minimizer H" is closer to exact collapse. Based specializing Theorem 4.1 to the near-collapse case, we present in the following theorem (which is proved in Appendix D an exact analysis of singular values of the blocks of F. (The notations Tmaz (-) and mann (-) stand for the largest and smallest singular values of matrix, respectively).
Theorem 4.2. Consider the setting of Theorem 4.1 Ay < (assumed in Theorem 2.1 d > K. max An), and the representation of Eq. that given in Eq. 6 Lel Ho be collapse features matrix (minimizer of Eq. for the same Ah Aw as Eq. 2). Then, for k E [K] with + k we have that k. is full rank, F& rank-1, I, and
0 max (F 1,
0 miin F FR,k) = 1-
0 max (. = 222-11220 v(Hdw).
Remark. In Appendix [D] derive expressions for the complete sin- gular value decomposition of FE, a and F v. Our expressions for the entire spectrum of FR, A reveal its step-wise decreasing shape, vi- sualized in Figure 1 for
100, K = 4, d 10, n 10,Aw v2 and various values of > To keep the paper concise, we state in the above theorem only the results for the maximal and minimal singular values of FR,K. but note that, similarly to 7min (Fx..), almost all sin- gular values decrease λи increases. Even though small portion [IK/dd of the singular values equal (as shown in our analysis in Appendix D)
000-22
0.26
Spechum 1.
Figure 1: The effect of Au on the spectrum FE,K-
still gain insights the attenuation profile since generie perturbations unlikely to concentrate in such an extremely www-dimensional subspace (and, in fact, the singular vectors associated with this subspace do not affect the within-class variability).
From Theorem 4.2 we gain the following insights on the minimizer of Eq.2in the near-collapse and large (3 regime. First, observe that not only do exactly collapsed minimizers have orthogonal features
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for different classes, but also in the near-collapse setting an intra-class block is much more dominant than each inter-class block, as follows from F A.k being rank-1 and Mmaz x.&) < Tmin P Fx For generic perturbations that do not concentrate in specific low-dimensional subspaces this implies that also before/near pure collapse, we have that the deviation from collapse in the features of a certain class is mainly due to deviation from collapse of input (preceding) features of the same class and not those of the K - 1 other classes. (See Appendix [D.] for more details, and note that this also implies preservation of per-class near-collapse). Second, we see that the feature mapping regulariza- tion plays the major role in approaching (near-)collapse behavior. Indeed, increasing Xu decreases the spectral values of the (more dominant) intra-class blocks { (F (contrary to increasing Xw). Recall that reducing the singular values of the blocks of F implies reducing the distance of the min- imizer H' from exact collapse. Third, our result on the inter-class blocks { (.,z hints that the regularization of the last layer's weights (determined by Aw > 0) may still have supportive effect on reaching (near- )collapse behavior by reducing the component of the deviation from collapse that due to
of features of different classes (e.g., when some classes are harder
to be classified then others). In the sequel, we show that the above observations correlate with the NC behavior in practical settings.
EXPERIMENTS
In this section, we translate the insights that are obtained for the model in Eq. 2 to what is observed with practical DNNs and datasets. We evaluate the distance of DNN's features from exact NC using metrics that have been also used in previous works. Despite defining the metric NC1 in Eq.
here we mainly measure within-class variability using NC1 Tr EwEi) where we use the definitions of Section 3 (We use this metric due to its popularity even though it is less amenable for theoretical analysis). We measure the structure of the features using
H 1) (H h)1k) NC2 [( h k) TH boll)) VK-
1
1 (IK
where the simplex ETF is normalized to unit Frobenius norm.
The result of Section 3 provides reasoning to justify depthwise decrease in within-class variability, which has already been empirically demon- strated for end-to-end training in several papers (Papyan et al.] 20200]
NC. and RC
Tirer & Bruna 2022; Galanti 2022) (we present such experiments Appendix E.2). Here we show this behavior also for layer-wise training, which is better represented by our model. We consider the CIFAR-10 dataset and train an MLP with to 10 hidden layers and final classi- fication layer. Each time, we add and train a hidden layer on top of the previous hidden layers, which are maintained fixed. Then we compute the NC1 metries for the deepest features. Due to space limitation, the experimental details are deferred to Appendix [..] E NC, we add more hidden layers on top the previous, which are maintained fixed. Note that theory justifies such decrease for all the layers (the features are not required to be near collapse).
Figure 2: Layer-wise train- ing of MLP on CIFAR-10.
decrease in both NC1 and
Next, we turn to demonstrate correlation of practical NC behavior with the insight gained in Sec- tion 4 that Xи plays a bigger role than dw does in approaching NC. Based on the equivalence of -zgelullaiiottion with weight decay (WD) in gradient-based methods, we can make the analogy of regularizing H in Eq. 2 to WD of the weights of practical DNNs in the feature mapping layers (i.e., excluding the last layer's weights). Importantly, note that this analogy is empirically justified for plain UFMs in (Zhu et al. 2021). Under this analogy, our analysis suggests that, as long as entering the zero training error phase of training is maintained, increasing (resp. decreasing) the WD in the feature mapping layers should decrease (resp. increase) the distance from exact collapse more than increasing (resp. decreasing) the WD in the classification layer. Indeed, we empirically show this behavior below. (More experiments are presented in Appendix E.2) We note that there exists work that empirically"] shows that WD facilitates collapse (Rangamani & Banburski- Faaeei--aay/// 2022). however, they do not examine the WD in feature mapping and classification layers separately.
'Note that the claim in (Rangamani & Banburski-Fabhe larized bras-free MSE loss comes from demanding that H' without subtracting the global mean will be simplex ETF rather than orthogonal frame as shown in Theorem
2022 that NC solution cannot minimize unregu-
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Under review as a conference paper at ICLR 2023
411
Gaaere Natured
132
wil
022
4.0
NC2
1.5
0.2
Sacch
An and
MC2
Figure 3: The effect of modifying the weight decay (WD) on NC metrics for ResNet18 trained on CIFAR-
10. Top: MSE loss without bias; Bottom: CE loss with bias. Observe that modifying the WD in the feature mapping increases the deviation from the baseline more than modifying the WD of the last layer.
We consider the CIFAR-10 dataset and examine how modifying the regularization hyperparameters affects the NC behavior of the widely used ResNet18 (He et al.] (016a) compared to a baseline setting. Specifically, as a baseline hyperparameter setting, we consider one that is used in previous works (Papyan et al. 2020 Zhu et al., 202T]: default PyTorch initialization of the weights, SGD optimizer with learning rate 0.05 that is divided by 10 every 40 epochs, momentum of 0.9, and WD of 5e-4 for all the network's parameters. The modifications include: 1) doubling the WD only for the last (FC) layer; 2) doubling the WD only for feature mapping (conv) layers; 3) zeroing the WD for the last layer; and 4) zeroing the WD for feature mapping layers.
Figure 3 presents the NC1 and NC2 metrics of the (deepest) features for: (Top) MSE loss with no bias in the FC layer (similar to the analyzed model); and (Bottom) CE loss with bias in the FC layer. In all the settings, we reach zero training error at the 40 epoch approximately, The empirical results show that modifying the WD in the feature mapping layers leads to curves with larger deviations from the baseline compared to modifying the last layer's WD, which is aligned with the theory established in Section 4 (i.e., the important role of Xи in attenuating the dominant intra- class perturbations). Reducing (zeroing) the WD in the feature mapping increases the distance from exact NC (i.e., from O value of the metries), while increasing the WD decreases the gap from exact NC, as the theory predicts. The fact that sometimes (c.g.. with CE loss) increasing the WD of the last layer can also decrease the gap from collapse hints that mitigating inter-class interference/corrlaaiion of features in practical deep learning settings is more significant for reaching NC than in our analysis that considers near-collapse regime Yet, both the experiments and the theoretical study show that the regularization of the feature mapping has larger significance in approaching NC.
6 CONCLUSION
The features that are learned by training practical networks on real world datasets typically do not reach NC. In this paper, we addressed this issue by studying model that can foree the features to stay in the vicinity of predefined features matrix. We analyzed it for the small vicinity case and established results that cannot be obtained by the previously studied (idealized) UFMs. We proved reduction in within-class variability of the optimized features compared to the input features (via an- alyzing gradient flow along the "central-path" of a UFM with minimal assumptions, unlike existing literature). We also presented an analysis of the model's minimizer in the near-collapse regime that provides insights on the effect of the regularization hyperparameters on the closeness to collapse, which correlate with the behavior in practical deep learning settings. We believe that our perturba- tion analysis approach, which is based on exploiting our knowledge on exactly collapsed minimizers of UFMs for studying non-collapse cases, can be applied to models other than the one considered in this paper, such models with different loss functions and/or multiple levels of features and/or imbalanced data.
In Appendix F.2] demonstrate the role of Xw in mitigating inter-class interference of features, which identified by our analysis, also empirically with practical DNNs.

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Under review as a conference paper at ICLR 2023
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Processing Systems,
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Under review as a conference paper at ICLR 2023
A PROOF OF THEOREM 3.1
To prove Theorem 3.1 in addition to the within-class and between-class covariance matrices, let us define the total covariance matrix (across all classes) of the non-centered features
K
hg,
Kn k=1 1=1
For convenience we also define the non-centered between-class covariance matrix
1
ΣB( == K k=l
h
We have the decomposition (T(H) Ew (H) + ΣB (H).
Using YH = (Ix + HT = nH and Σr
1 HH Kn
we have that for each feature matrix
H, the optimal weight matrix W (H) is given by
Aw
1 H ( + Aw r)-
W
H
HH
+
1)
Kn
K
K
Next, let us simplify the terms with (H) in L(H):
1
+ Aw
Au
((H)
W (H) XI
|
(H) +
H
2Kn
2K
2Kn
For the first term in L(H), observe that
1
1
1
|W
- YI 2kn
Tr (H)HH W WHH
Tr (W (H)HY
2Kn
Kn
1
Aw
dw
1 Tr
(Ex+ Aw
1
Tr
(Er+
I) - 1-EE
1)- ΣB
I)
+
2K
K
K
K
Aw Tr
(Er + Aw I) ---BBB
1 Tr (Er
dw
+
I)
1
2K
K
2K
K
2'
where in the second equality we used ΣB Au K + Aw I)-1
HH and in the last equality we used (Σr +
=
K
A I) =I
ET
For the second term in C(H), observe that
Aw
Aw Tr (Wo (H)W*(H) (H)
| (H)
2K
2K
Aw
dw
Tr
(Er +
1) 1) -Σ
2K
K
Adding the two terms together,
1 ( - 1
Aw
Aw I) -1
1
W
(H)
Tr
(Er +
'503
2kn
2K
2K
K
2
dy 1)
Tr
(Σy +
I) -1 [Ew +
2K
K
K
2K
where we used (ÊT + Ay 1) Finally, for the third term in L(H) we have
+
Aw K
1) (EE + Aw I).
A
A
Tr (DT)
2kn
2
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To conclude
Tr (Ew + Aw (HH) 2K
1
Aw 1)
1)-1)
dy 2
d - K
D) + K
Tr
Er)
K
2K
Next, we are going to analyze the traces of dEg dEw and dEr df dt
along the flow that is stated in
dt
Eq. which repeated here for the convenience of the reader:
dH, dt KnVC(HH)
In the following lemma, we state the required derivatives.
Lemma A.1. Denote CB = ΣB(Er + K 1) CB == ΣB(Er +
Aw K I- and Cw ==
Ew( (ET + AW I) Along the gradient flow have
dEB (CB(I-OB)
CB) + (I- (1-C1) - 22HH2B
dt K
dEw 1 +
-2x11
dt
K
dET (B-CB-Cw)Cs + Cb -gC)l)
cwr) -2X Er
Proof. We use the notation Ozyi denote the derivative w.r.t. the lth entry of h Then
Oegi BB (er (hg - lnG)T + (hg hh)ee)
Kn
Ouji Ew
(er(hu.) - ha) T + (ha.. - -b)ee))))))))))))))){)))))))-h
Kn
Ozgi ΣT
Kn (ethk + ha.e))),
where e, E Rd is the one-hot vector whose /th entry is one (i.e., a standard basis vector). By the product rule,
))²)
Oesh C(H)
Tr
(OnjiIw)
+
Aw 1)-
+ Tr
(Ew +
Aw
1)0kit
(ET+ +
Aw 11
+
Ah Kn
ef baj
2K
K
2K
K
K
(Ong)Ew) + Aw I) 1)-1
2K
K
Aw I)
dw
-1
dw
Aн ej had
Tr
(Ew +
Eg +
QujLET
Σx +
+
2K
K
K
K
Kn
1
Aw I) (hk.)
dw
Aw
Aw
-1
E +
Ex +
1)
(Ew +
I)
Σx +
bas + dh AHKbES
es.
K2n
K
K
K
Therefore, the gradient of L is given by
VLHH)
(7)
(E
Aw
-
1) H H K
EE
+
Aw
-1
Ew
+
Aw I
Ex +
Aw
-1
H AnKH,
K2n
K
K
K
Next, we compute how each covariance matrix updates along the flow. Let EB (a, b) c ggg
denote the a, b-th entry of ΣB. We further denote C (Σr + Aw I) CB = ΣE ΣΣ +
Aw - = E K
Er + Aw I) Cw = Ew Ex + Aw I) and write CHH = (Laj (L) K
K
where
1
(C(hu.) h4) (I- ( Chu,i + AuKhes).
Laj
K2n
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Using the chain rule, we have that
(E, b)
dh 1 E Ouzi [s ()( (ilkknnkz((H))
Eoun (pa,
dt
dt
RiJ.
ΣΣ ((ea, (r) ha ha) + (ea: hi ha)) (e1, en)) (er, Laji)
Σ
(leas Lay) (eo, ha -ha) + (eas ha (Lij eb)
kjj
ea
T
Σ
Lk. (ha-hh) -h; he LG) eb
kj
K - (( e, e 2
e
Similar computation yields
dEw(a,
(CwCB+ CPCht eb 22HO Ew
K c
(a,b)
K e
1
(I CB
CB C C cw)) e2Hee Σre.
dl
Let TB : + (2Ax(Tr(EB) and Tw : Tr The above lemma suggests that TB strictly increases along the flow, while Tw decreases. Indeed,
dTw 22n dt
d'TrEEw) dt
+ 22 (r(T)(EW))
2
(TCCT(CCWCCL)
K
22nt 2 Tr(E K
Aw K
Aw 1)- S 0,
+
K
The last inequality holds because the trace of the product of two positive semidefinite matrices is always non-negative (e.g. by Von-Neumann's trace inequality). Similarly
dT 2 dt
2A TTC -BCC))
K
2 (ABBTTEEB(Er + Aw -1 ΣB Er + Aw 1)-
K
K
K
2 22nnTT((Dp
Aw I) (Ew
Aw AW 1)-1
(Σ + K
+
1) +
K"
K
K
2
22AN!
Tr<E + Aw 1) - EE + Aw
I) - +
Aw
Tr
Aw -2
K
K
K
K
K
2dw K²2
Aw r) -
2
-) > 0,
K
where the strict inequality again comes from Von-Neumann trace inequality, which ensures that the trace of product of positive definite matrix and non-zero positive semidefinite matrix is positive.
Since NO, Tw WTB. the above computation also shows that NC1 has strictly decrease along the flow.
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B PROOF OF COROLLARY 3.2
Recall that the minimizer 11/ satisfies the first order equation
Kn B
H H1p0 Ho
V
(8)
We first show that 1/9 + Ho as 3 - xc. The following lemma would be helpful.
Lemma B.1. There exists a constant M independent of H. such that
VL(H) S M H|
for any H E RRdxKn Kn
Proof. We bound each term in the expression of V. equation Eq. individuully. For the first term
we have
dw K I) 1-110 (H - H 11 & 1( +
Aw
11 +
- (H K
(H-H@ Dl
K Aw
2K
H
H
se
Dlr S
|
Aw
where || olpp denotes the operator norm and the second inequality is due the fact that each eigen-
value of [E
Aw K
I) -1 is no bigger than K Similarly, Aw
dw K
Aw K 1)
Aw K
Σx +
Ew
ET +
I
H
F
-
K Aw
ST +
dw
Σw + dw I
ΣT
Aw
I
F,
where in the last inequality we used (1/ + Aw -1/21 llop S / K/Aw since every eigenvalue of
K
Er + Au
K I) 1/2 is bounded by VK/Aw. Denote A = (Ew + Aw 1). B = Σr +
K
Aw
R
and
use A Σg B. have
B 1/2 AB
1/2 1/2 oo
(B-1/22A1 (B A/2) T
1lop
=IIB-1/2A (B 1/2 1/2)
|| A lpp
(AA-1/2(A + E A -1/2 1//2)-111m 1 1lop
|I A 1/2 E EB A 1/2)-1111-1100 llop S 1.
Combining the above bounds together, we have obtained for any H E dR
1
3
[L(H)] <
+ Au
H
Kn
Aw
Next, we combine the lemma and the stationary equation Eq.
g to get
nKM
nKM
Holly + nKM
H1/3 - Holf <
H
|1///l
H 1/3
HollF
B
Rearranging, we have the bound
( B
nKM -1)
H 1/3 Holle
Holl
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This implies that Hi
// Hu 3 + Combined with the continuity of VZ((-) and the first
order equation Eq this further implies
H Ho KnVL(Ho).
lim 80000
1/3
Now, by chain rule,
(H (Ho)Ho)
H1/3 Ho
lim
(H.), lim
8-000
1/3
$-000
1/3
(V NC (Ho).
((H)))
(H..
dt c
In the last line, H. denotes the gradient flow iterate defined in Eq. By (the proof of) Theorem 3.1 when Hn is non-collapsed,
d
NC (H.) < 0
dt
t=
must hold. This further implies that there exists some constant C(H,) such that for C we have that (8) NC1 (Ho) < 0.
1/3
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C PROOF OF THEOREM 4.1
Our proof is essentially a perturbation analysis approach that exploits the fact that each of the minimizers is stationary point of its associated objective function. Namely, the mini- mizer of the perturbed problem S(W, H; Ho), i.e. (W", H"), obeys that VS(W', Ho)
Vuf(W*, ; Ho) V ; Ho)
and the minimizer of the unperturbed problem, i.e., (W", H)) where
V V=f(W" H Ho) fwf H H
for brevity we omit the 122 symbol, obeys that V/(W", H"; Ho)
We use these properties in the following first order Taylor approximation of V V/(W", H"; Ho) around (W H"; Ho) (with accuracy of 0[|6H || ||HHol?²) that is given by
VHf(W>, H Ho)
WHf(W* "H; :H) Wwf H.)
VWW, H.; Ho)
V B Vnf(W : Ho) V VH V H Ho)
V W H"; HD)
vec ((H) vcc(6W)
V Ho W
+
veo(oHo).
V W WwffWW H Ho)
H. Vwf H.;
Recall that oH = H H ow: W W and öH, Ho Ho.
Since the two terms in the
first line of Eq. vanish, we get that
vec (6H) vec(
V VFJ V
V V Vw
Ho f V
(oc(6Ho))
(10)
VWS
Ho
Vwfl
where all the derivatives are evaluated at (W", H Ho). which is omitted in order to simplify the presentation. As shown below, in our setting the matrix that is inverted is indeed nonsingular.
We turn now to compute the derivatives. Let us denote == vec(H), w == vee( (), and := voc(Y). Observe that from well known identities on the Kronecker product and the vectorization operation we have
1
WH YI
[T
W)h y112
I( N Ix)w - y11
2Kn
2Kn
2Kn
Therefore, the first order derivatives given by
(Inn @ W (L @ w/h- - + dy h +
B
VHf(W, H: Ho) =
(h vec(H)),,, (11)
Kn
Kn
Kn
Aw
Vw/(W, H; Ho)
(H @ (K) @ 3k)w - y)+ +
w.
Kn
K
Hence,
B
V Ho Kn
VHF
LinK,
Ho
Vw
OKdxdnK.
Plugging these expressions in Eq. 10 and using blockwise matrix inversion gives
.3
vec(HH) ~
V
V V -
"w vec(HHo))
Kn
vec(&W) ~ Kn (V
Vw/)
Vw
VV / V V V (V V
veo(oHo),
which are stated in the theorem, where all the derivatives are evaluated at the point (W*, H. Ho).
Let us state the second order derivatives that appear above. First, one can observe that
1
Ay Lánk +
B
VHUF H"; Ho)
TW" + I.. & W W
LanK,
Kn
Kn
Kn
V Vw/ H' Ho)
1
HH &
Ix +
Aw Ixa-
Kn
K
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As for the mixed partial derivative, applying Vi on Eq. 11 get
V VHf= =
① VHf- = 1 à
Know (IInn & W )r) +
1
(Ikn &
WT) (TTn & Wih- y)
Ow
Kn
Ow
Kn Ow (CIn & W WTr) + 1
1
(Inn & W
dw ((H @ (Ix)w-y)
Kn
1
E(W, H) +
(Inn & (H @ Ik)
Kn
Kn
1
1
Kn E(W H + (H @ W
Kn
wherer == vea( WH- Y) but treated as independent w due to the product rule, and E(W, H)
Zw ((Ikn s W )r). Denoting Wkei = W[k: i). we have that
0
(IInn & W )r) (Inn
W (r) =(Inn & (e,,ek,))))
Ownk
Dwps
-voc(ca,0OK, (WH Y)),
where od is the standard vector in Rd with in its ith entry (similar definition stands for CK.)))
Therefore,
VHVW H;; H:Hoo)
E
(H' & W"),
Kn
Kn
1
1
V Vn/(W H"; Ho)
E
(H
W
Kn
Kn
where Ex E W H*) and E(W, H) E R ddn x is given by
E(W, H)
[vec (vec(ed,1e) e K.1 (WH-
vec.ee,,ee...... (WH- - Y)), vec(eu. (WH Y)),
vec(ee,.ee.... T
(WH Y))].
We focus now on the effect the deviation 5Hg Ho - Ho on the feature learning 5H = H' H'. This requires inverting the dn. x dnK matrix that links oH and oHo, which is quite challenging. Yet, from the derivatives that are stated above we observe the following
B Kn
vec(HH) P0
(VA
- Vw
vec(HHo)
Ah
1 &
Kn
-1 voc(HH.)
Ldnk +
Idnk
B
B
Lank + An
1 W
1 5 vec(HHo) 3
Lank
+ gInk &
where
z' ZE TT' ( ( & I + nAw Lak) -ak) (E + H & W*).
Therefore, under the assumption of B >> m1x(1, An). which is associated with a restrictive link between H, and H, we can use the first-order truncated Neumann series to approximate the matrix inversion (with accuracy of O(3-2)) that is stated in Eq. and repeated here for the convenience of the reader:
Aн Iank
1 I.nK @ T w" + 1/²²) vec()Ho).
(6C
[dnnk
8
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D PROOF OF THEOREM 4.2
In this section we compute the entire spectrum (singular values) for the diagonal blocks ("intra-class blocks") and the off-diagonal blocks ("inter-class blocks") of the block matrix in Eq. 6 To keep the main body of the paper concise, we present in the statement of Theorem 4.2 only the results for man FR,K) and miin (F,,.) of the full rank matrix FR, well men k.kk of the rank-1 matrix F k. (k: + k).
Recall that we consider the (non-degenerate) := <1. Therefore, when Hg H
is minimizer of Eq. (associated with W*), from Corollary 2.2 we have that (W*, H' the minimizer of f(W, H; Ho) is also orthogonally collapsed and characterized by Theorem 2. with
Aw and Aи (independent of K, n, d).). That is, H' H 1, and W H H x W W- x Ix. We also have the following results for the spectral norm of H and W that we denote by 0h and ow respectively:
Aw
Aw
0 = (1 -c)
Aw,
AN
ow (1 c)
λи-
Aw
dw
Observe that these expressions do not depend on the number of samples K,n, Note also that o = vAndw)" c) c) <1 1.
We remind the reader that vec( ~ Fvec(8Ho)) for
An
1 z l Ink 60 W W
F = Link
dnk
3 3
where
== E H W
(HH-T I aaw (E*
+ H WW),
and E = E(W", HH) and E(W, W. H) E RRdkK Kd is defined as
E W. H)
(ea e K.1 (WH- ,),, (WH- Y)), vec(ex2e/, (WH Y)),..
vec(ea..e K. (WH Y))].
where ed is the standard vector in Idd with in its ith entry (similar definition stands for ek,)).
For the collapsed minimizer (W* H"), we know that H' o & 1 and W owRR for some (partial) orthonormal matrix E Rdx (i.e., R R II).
Therefore, we have that W - Y = -I, 21 Ila and that H & W = (1 (1-c) RR. Observe that the alignment of the former expression with the latter (where the locations of the dimensions d and K are swapped) is done using the matrices (ea,*k,a) Indeed, we can write
E Ka,K(-all & 1 & Ia), where Ka,K E Kdxak the permutation matrix that satisfies
K & X2)Kaa X X1
for any X, E ddxd and X, E ERK*K Such matrix Ka, is also known as commutation matrix the matrix theory literature. Another useful property of the commutation matrix that we will frequently use is that
Ka, Y) Y xx
(12)
for any x E Kx and Y E [ddm
Let us extract the k b-th block Zt E Bndxxd of Z". First, observe that
Z" E H W T & Ix + ndw Lak) Iak) -EE H & W*)
BB
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Under review as a conference paper at ICLR 2023
where
A = (f + Aw
B .KK (Ix @ 1 2 Ia) (1-
Denote by } the standard basis vectors in R.K. To extract the k; k-th block of Z', we compute
Z = (ex 2 dd
(eg @ Ian)
1
(B(ek & Lan)) (A & Ix) (B(ex @ lan)),
with
B(ck @ Ida) cK (ek @ 1 & 1.) + (1 (r& & 11 &
= - @ c₂) (1- & R
where in the last line, we have used property Eq. 12 to swap the Kronecker product. Then,
1 ( (-c(1, & La ex) + (1-c) @ R RJ)) T (A (A0x))(-c111 @ +(1 c)(r & RJ))
n
=2
(1 2 A + + - (r (111))) RR
1 (1,11) @ (Argr) + + 11
c1
Let us write [r r2 rx E Ra*k and let rx+,,.., Fa be the orthonormal vectors such that {r. 12
forms orthonormal basis. We know that
RR Aw E
E
rrr
Σ
ryr].
Aw
Aw
j=K+1
Therefore,
E Ar 0² + H
F4
dw
Arg
rgrk A
"7 Aw
We can thus conclude that
Z - n
1 (1,1))
Su (1 c)²
2c(1 c)
RR
rETK
(13)
o2
+ Aw
02 Aw
H
1
When & + k, the off-diagonal block of Zz is given by
2c(1
T (1, 1 69
rgrk
J7 + Aw
which a rank-1 matrix. Since other matrices in do not contribute to the inter-class block, we know that k Z It is well-known that the eigenvalues of Kronecker product of two matrices given by the products of their eigenvalues. We know that (1 has exactly one non-zero eigenvalue, which equals to 1. This implies that
2c(1 -c)
21 (dHdw)
Oman F (M
B(O² Aw)
B
Next, 1et us compute the intra-class block. Setting A = in equation Eq.
13
we get
Z - (1 2
Σ Marar
Hc 1, 1 (r,x,)
1
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Under review as a conference paper at ICLR 2023
where
2
(1 - c)²
2c(1 - c)
-22 AH
+
Xw
0 + Aw
o + Aw
Aw
2
(1 - c)²
AH
= (c² + (1 (1-c)))
for 1 Sis S K and i # k
02
+
Aw
" dw
dw
0²
Mg
AH,
for K < <s
Aw
The intra-class block is therefore given by
d
AH ow B
FAAA = (1
Ind
In
(RR ) + Σ
Mil (1(-1,1,) 2 (r₂17)
B
3 1=1
= (1
Ah
In
2 (r,r,)
2
Σ
In 2 (rar.)
+ 1
Σ
( 8 (rar.)
1
B (r,T,),
1=1
x (1 + Σ
Mi n
1
where
dy
W
1
Yy
A
1
for Siik
(14)
B
3
3
Aw
AH
A
for K < <isd
(15)
B
Let
Jn In and {s:} in be set of orthonormal basis of R" Then, we can further write
1 Σ (s @ (rxx))
FR,
\isys}) @ (rar.) +
B
i=
d
d
ΣΣ /=
Ms (s)) 8r) +
Σ
pr(si 8 (s) &
(16)
3
One can easily verify that {s, @ hesssisssi is orthonormal basis of Rnd So, Eq. 16 gives us the eigendecomposition of FK. The spectral norm of Fx, is therefore given by
1
Oma (F F. = max max (1erl, |A: 3 "n
15:5d
As we consider the large 3 regime, the expressions in both Eq. 14 and Eq. 15 are positive. Observe that for K i d (associated with the wver-parameterization of the model) we have that the eigenvalue associated with the eigenvector (8) @ r;) is given by
d
X; +
1/ =
3
3
1 In perturbation in the direction of Note, though, that due to the Kronecker product with $ = vn
this eigenvector does not affect the variability in the kth class at all. Furthermore, generic/practical perturbations likely correlate with, or have their power spectrum spread over, many compo- nents of the dn dimensional eigenbasis of F k. and not concentrate in an extremely low dimensional K subspace (composed only s F, with K < i d). Thus, we expect these eigenvectors to have small correlation with generic perturbations.
Showing that Taaa (FR..) 1 reduces now to eliminating the option of eigenvalues larger than for 1 < < K. This is equivalent to having that
1
λy
(-1 + (2c (2c- 1) 0.
B
dw
(-1+ (62 + (1 -)) < <0,
3 Aw
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Under review as a conference paper at ICLR 2023
and both are ensured under our assumption model).
vduuw 1 (the non-degenerate case of the
Finally, observing that Eq. 14 is smaller than Eq. 15 and that the second term in Eq. 16 does not include cigenvectors (s, X r, for j 1, we conclude that
1 HY
mnin (FE,k)
3V
Aw
D.1 ADDITIONAL. DISCUSSION ON THE RESULTS OF THE THEOREM
Theorem 4.2 has no restricting assumptions on the number of classes K. The only assumption, which is common in theoretical NC papers and is also what is done in practice is that K, i.c. that the dimension of the features is larger than the number of classes. This means that, regardless of the number of classes, the inter-class (off- diagonal) blocks have rank 1, while the intra-class (diagonal) blocks have full rank (recall that each block is of size dn dr).
Considering the conclusions from Theorem which are stated in Section + if we sum up the max- imal contribution of each of the K - inter-class blocks of certain class, i.e., (K - 1)0 (F)) k.ū then for guaranteeing that this sum is smaller than the minimal contribution of the intra-class block, i... mmi (F (.AA): we may need to assume that >> K. Note that this a reasonable assumption under our large 3 setting. Yet, we believe that the rank difference between the two types of blocks a more important indicator for the dominance of the intra-class blocks, and this property is inde- pendent of the number of classes K. Specifically, since dn K (all the more so, in practice even have n >> K), then for generic perturbations (that uniformly span the entire dn dimensional space) the rank-1 inter-class blocks nullify much of the perturbation contrary the intra-class block (which has full rank). This strengthen our conclusion that the deviation from collapse of each class of the minimizer H dominated by the deviation from collapse of the same class Hg rather than by the deviations of other classes. One thing that should be reminded here is that we analyse the "near-NC" regime, so we assume that the system is already not far from exact NC. Reaching this point in general might become harder when the number of classes grows.
Another point that can be raised regarding the results of Theorem 4.2 that we do not analyze the full matrix F but rather its blocks. In fact, we believe that our analysis, which includes complete spectral analysis for each block separately, is more informative, as clearly distinguishes between properties of intra- and inter-class blocks and provides insights on the roles of the regularization hyperparameters that are aligned with practical DNN training. In contrast, in the large 3 regime we have that F is full rank, which masks the rank-1 property of the inter-class -off-diagonal) blocks. Nevertheless, analyzing the relationship of the full F and its blocks is an interesting a direction for future research.
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E ADDITIONAL EXPERIMENTS AND EXPERIMENTAL DETAILS
E. EXPERIMENTAL. DETAILS FOR THE LAYER- LAYER-WISE EXPERIMENT
In this section, we provide the experimental details for the layer-wise training experiment that is presented in Figure 2 in the main body of the paper.
We train MLP with 10 hidden layers on CIFAR-10 dataset, where each sample is flattened to a 3072x1 vector. Each hidden layer includes 3072 fully connected neurons with default Py PyTorch initialization of the weights, batchnorm, and ReLU nonlinearity. We start with one hidden layer and train the MLP with epochs of Adam with mini-batch size of 256, learning rate of le-4, and CE loss. Then, we compute NC1 metrics for the deepest features. At this point, the first "outer iteration" of the procedure is finished. We fix the parameters in the existing hidden layers, insert a new hidden layer before the final classification layer, and repeat the procedure. Namely, each outer iteration of the procedure we optimize only the deepest hidden layer, which has just been inserted with default PyTorch initialization of the weights, and the final classification layer, which "initialized" with its weights from the previous outer iteration.
Let us provide more details that has led to the implementation decisions that are stated above. We have found that layer-wise training of DNNs (on a practical dataset, e-g., CIFAR-10 that we use here) is significantly harder than end-to-end training in terms of reaching a small training loss value. (Presumably, this is the reason that DNNs are typically trained in end-to-end fashion). Careful configuration of the training procedure was required for reaching considerable low loss (though, still not zero training error) and low NC1 metrics as presented in Figure 2 From our efforts in layer-wise training the 10-layer MLP we observed the following: Adam optimizer worked better than SGD (which is harder tune); Layer-wise minimization with CE loss (rather than MSE loss) has led to lower NC1 metrics; Using no more than 3 epochs per "outer iteration" allowed reaching lower values for the loss and the NCI metrics at the deeper layers. Regarding the latter (i.e., more epochs per outer iteration lead to worse optimization results), when there are only one or two hidden layers then the decrease in the loss and the decrease in the NC1 metries are larger when more epochs are being used. However, when we add in that case more hidden layers, the optimization appears to get stuck at some local minima with higher loss and NC1 metrics compared to what we get with only 3 epochs per outer iteration. As far as we understand, this behavior follows from the (extreme) nonconvexity of the problem.
E.2 MORE EXPERIMENTS ON THE EFFECT OF THE REGULARIZATION HYPERPARAMETERS
In this section, we present more experiments that examine how modifying the regularization hyper- parameters affects the NC behavior of a practical DNN - ResNet18 (He et al.] 2016a) - compared to a baseline setting. Specifically, used in previous works (Papyan et al. 2020 Zhu et al. 2021): default PyTorch initialization of the weights, SGD optimizer with mini-batch size of 256, learning rate of 0.05 that is divided by 10 ev- ery 40 epochs, momentum of 0.9, and weight decay (L2 regularization) of 5e-4 for all the network's parameters.
baseline hyperparameter setting, we consider one that is
The first set of experiments is similar to the experiments in Section 5 These experiments support the insight gained in Section 4 that λи (the regularization of the feature mapping) plays bigger role than Aw (the regularization of the classification layer) does in approaching NC. We compare the NC1 and NC2 metrics (defined in Section (5) of the baseline setting and the following modified settings: 1) doubling the weight decay only for the last (FC) layer; 2) doubling the weight decay only for feature mapping (conv) layers; 3) zeroing the weight decay for the last layer, and 4) zeroing the weight decay for feature mapping layers.
In Figure 4 we consider the MNIST dataset with 3K training samples per class. Figure 4a presents the NC1 and NC2 metrics of the deepest features for MSE loss and no bias in the FC layer. Fig- ures cres|4b]andd 4c the NCI and NC2 metrics of the deepest and intermediate (output of 3 out of the 4 ResBlook) features, respectively, when for CE loss with bias in the FC layer. In all the settings, we reach zero training error at the 40 epoch approximately. In Figure 5 we repeat the experiments with 5K training samples per class. Furthermore, repeating the experiments with 3 different random seeds for initializing the DNN's 's parameters yields similar curves that demonstrate the same trends. In Table T we report the mean and the standard deviation (SD) for the NC metrics computed for the
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Under review as a conference paper at ICLR 2023
Table 1: The effect of modifying the weight decay (WD) on NC metrics for ResNet18 trained on CIFAR-10 and MNIST datasets mean and SD are computed for 3 random seeds. Observe that modifying the WD in the feature mapping increases the deviation from the baseline more than modifying the WD of the last layer.
10, loss
MNIST,
NCT
NC2
NCI
NC2
NCI
NC2
NCT
NC2
0.0061 ::
0.111 d
0.062
0.173 le-2
d
0.072 #
0.004 4:
0.115
WDx2 for W
0.0055 = 36-4
0.101 + Ne-3
0.040 + 26-3
0.161 = 76-3
0,055 # le-2
0.003 + Te-1
0.102 + 3e-3
WDx2
0.0022 =
0.070 = 66-3
0.024 40-3
0.131 96-3
40-4 =
0.048 =
0.002 + 76-5
0.101 + 3o-3
WD=0 for W
0.0188 1: 2e-4
0.101 1: 6
0.104 1: 8e.3
0.195 96-3
1,7e:3 1 1e-4
0.108 4: 2e:2
0.009 1 4e
0.147 1: Se.3
WD=0 for
0.0280 = 36-3
0,226 + 60-3
0.174 = 76-3
0.331 16-2
410-3 2e-3
0.303 = 20-2
0.031 + 4o-4
0.198 : 8e-3
deepest features at the 100 epoch (which is already after the NC metrics reach plateaus) for both the CIFAR-10 and the MNIST datasets.
Similar to previous works, from comparing Figures 4b and 4c (as well as Figures 5b and 5c we see that the NC distance metries are larger in the intermediate features, which correlates with the results for our model in Section Examining all the settings of Figures[aand as well as Table the experiments show the important role the regularization of the feature mapping layers in ap- proaching NC. Namely, modifying the regularization of the feature mapping layers leads to curves with larger deviations from the baseline compared to modifying the last layer's regularization. This aligned with the theory established in Section 4 that links increasing Y reducing the dominant component of the distance from collapse of a class, which is the deviation from collapse of its own features in preceding layers.
The second set of experiments shows the role of Aw in mitigating the interferences between the fea- tures of different classes (such interferences can hinder approaching NC). To visualize such behavior we use a "per-class NC1" metric, defined as
NC = K Tr
Σ (
hg))
Σ/3
Note that the NCI metric, which is defined in Section
5. can be written as
n
1 K
NC
h (g), hy))
Σ
Σ
NC
KK
K
We also use the following metric to measure the alignment of the mean features and the last layer's weights
NC3
W( h)lk)
1
(IK
1 1k) K
1
|| hIl VK-1
where the simplex ETF is normalized to unit Frobenius norm.
In Figure 6a present the NC metrics of the deepest. features of the baseline training scheme on the MNIST dataset with 3K samples per class. The other lines in Figure show the NC metrics for modified training set, where the samples of classes (digits) and are degraded by uniform blur (blur kernel of size 9 x 9) that hardens the distinction between them. Each line corresponds a different value of weight decay for the last layer's parameters. Yet, in all of the settings we reached zero training етrоr at the 40 epoch approximately. The empirical results show that large Aw facilitates reaching reduced NC metrics (closeness NC structure) by reducing the effect ('inter- ference") of the features of the degraded samples on the features of the other classes. This is aligned with the theory that established for our model in Section
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F ADDITIONAL MOTIVATION FOR THE MODEL IN EQ. 2
In the model that we consider in Eq. we interpret H as the deepest features of a DNN and Hg as shallower features of the DNN. In particular, in the large 3 regime that we theoretically analyze in the paper, we interpret Ho as the penultimate features (one layer before H). Even though the relation between and Hg in our model differs from their explicit relation in many practical DNNs, there exist networks where it is very reasonable to assume that the deepest features and the penultimate features are close to each other.
For example, consider the ResNet architecture from (He et al. 2016b), where (under our interpre- tation of H and Hn) the deepest features obey H -Ho Ho). where r(.) denotes residual block. The residual term can potentially be very small if Ho already separates the classes (e.g., has a "near NC" structure). In fact, the popular neural ODE framework (Chen et al. 2018 which is understood as the infinite depth limit of these ResNets, we inherently have that H & Ho- Another example where the concept H Ho inherently holds is deep equilibrium models (DEQ)
(Bai et al. (2019). These practical DNN frameworks provide the rationality for analyzing model. Furthermore, our theoretical results, such as depthwise decrease in the within-class variability, are aligned also with the empirical behavior of DNN architectures beyond the aforementioned examples (e.g., plain MLP),
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15
(a) MSE without bias. Deepest features.
Beatin
4.00
4:5
1903
(b) CE loss with bias, Deepest features.
74
Dmch
CE loss with bias. Intermediate features.
Figure 4: The effect of modifying the weight decay (WD) on NC metrics for ResNetl 8 trained on MNIST with 3K samples per class. Observe that modifying the WD the feature mapping increases the deviation from the baseline more than modifying the WD of the last layer.
Deriie Nestrne
add
May Megan
MSE loss without bias. Deepest features.
4.13
6:0
4.00
20
(b) CE loss with bias, Deepest features.
Scatiter Beier Bectres
1,20
CE loss with bias. Intermediate features.
Figure 5: The effect of modifying the weight decay (WD) on NC metrics for ResNet 8 trained on MNIST with 5K samples per class. Observe that modifying the WD the feature mapping increases the deviation from the baseline more than modifying the WD of the last layer.
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(a) Original samples, WD 5e-4 across layers.
6.1
1000
(b) Samples of classes 4 and 9 blurred, last layer's WD remains 5c-4, The effect of the blurred classes on the NC metrics (avg. and other classes) minor.
(c) Samples of classes 4 and blurred, last layer's WD reduced Se-S. The blurred classes affect the "per-class NC1" other classes and NC metries increase.
nmma
(d) Samples of classes and blurred, last layer has WD. The blurred classes further interfere with other classes and the NC metrics further increase.
Figure 6: The effect of modifying the weight decay (WD) of the last layer's weights NC metrics for ResNet18 trained on MNIST with 3K samples per class where samples from classes and are blurred. Observe that small WD in the last layer increases the effect of the "pre-class NC1" curves of the blurred classes on the other classes, and increases also the other NC metrics.
27


Reviews:
[{"review_id": "yhTwotW_0H", "reviewer": "Reviewer_aJ5E", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper studies neural collapse in DNNs and proposes a modification to the unconstrained features model (UFM) that has been studied by several previous works. In particular, the authors seek to explain a question on the variability of class-wise features (NC1) under the MSE loss. Previous works using the UFM conclude that the features collapse to their class-means, leading to zero variability among the features of examples belonging to a class. Here, the authors, motivated by the fact that in practical neural networks, the last layer features are not truly unconstrained (thus they are a function of the input examples), they propose adding to the UFM a penalty on the features deviating from a fixed features matrix. For this model, the authors theoretically prove that an NC1 metric decreases monotonically along the gradient flow path. Further, with the so-called perturbation analysis, approximate values of the deviation in the optimal feature are given when the fixed feature deviates by a small amount. This result leads to some useful insights on the sensitivity of the optimal features under near-collapse to regularization hyperparameters. Some empirical evidence is provided that correlates with certain insights gained from the theoretical study.", "summary_of_the_review": "Overall I think the topic is relevant, the author's idea of modeling non-perfect NC is interesting and the paper is contributing something new to the recent literature on neural collapse. I have some question about the actual implications of the result, the practical relevance of the model and the experiments as discussed above. My first impression is positive and looking forward to the discussion.", "strength_and_weaknesses": "Strengths:\n-----------------------------------------------\n\n1. The motivating hypothesis is interesting- whether the widely used UFM is sufficient for addressing the NC1 behavior in DNNs\n2. The authors propose a simple modification to the UFM to penalize deviation from a fixed feature matrix. A justification on its usefulness is provided by connecting the effect to that of finite depth neural networks not permitting the last layer features to wander freely. This can prove to be a useful model for relaxing the infinite approximation capability of neural networks that is proposed as a heuristic justification for studying the unconstrained features model.\n3. The authors prove two main properties of the optimum of the proposed modified UFM under MSE loss: the monotonic reduction in NC1, effect of regularization on the deviation. These are consistent with observed behavior in DNN training.\n\nWeaknesses:\n-----------------------------------------------\n\n1. It is not clear that the hypothesis of NC1 not being 0 is completely justified- can this be proved in a simple model, where when SGD is performed, the “final” solution does not exhibit exact collapse? Is the experimental observation strong enough to eliminate other effects such as not being deep in the terminal phase of training? For example, NC1 is Fig. 3 top row has a value less than 1e-2. For practical purposes it might be sufficient to say that the features have collapsed. Are there significant and specific consequences of the metric not being much smaller?\n2. I think more extensive and careful experimentation can make a stronger case for the proposed model to be useful as a strictly better model than the simple UFM. \n3. On the layer-wise experiment: this is an interesting study. However, from the details in the appendix, it seems like each layer is trained for only 3 epochs- if run longer, the layer would have likely achieved smaller NC1, taking the setting closer to near-collapse. Any specific reason for stopping at 3 epochs? Also, what is the reason for using Adam rather than SGD for this experiment? Perhaps it is better to show this result with MSE loss rather than CE to be consistent with the theoretical study.\n4. On the effect of regularization, how consistent are the empirical observations on different architectures and datasets? Is the effect of needing to train for more epochs when using a smaller regularization taken into account in the interpretation? Specifically, when using 0 WD, typically the training needs to be much longer. Results shown in Fig 3 will be more convincing if the training length aspect is eliminated.\n\nMiscellaneous queries, comments:\n\n1. Theorem 4.1: the result is up to O(norm squared of deviation in optimal solutions with and without perturbation). However, this applies to the statement approximating the actual deviation in the optimal solutions. Is this reasonable?\n2. Just before the statement of Cor 2.2, H0 is stated as “already collapsed”. It is better to clarify that this H0 is in fact the ETF/OF solution itself in addition to being collapsed.\n\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": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}, {"review_id": "bLxNrQki1-S", "reviewer": "Reviewer_1Pqf", "summary": "", "questions": "", "limitations": "", "rating": 3, "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 authors analyze properties of a sparsity regularized matrix factorization where one of the factor matrices $H$ is further regularized to be close to a given matrix $H_0$:\n\\begin{align} \\min_{W,H}f(W,H) = \\lVert Y-WH \\rVert^2 +\\lambda_W\\lVert W\\rVert^2 +\\lambda_H \\lVert H\\rVert^2 + \\beta \\lVert H-H_0\\rVert\\end{align}\nThe properties of the solvers of the above objective shall give insight into properties of neural collapse, describing the phenomenon that neural networks tend to resemble nearest centroid-classifiers after training, where the class centroid is computed in the penultimate layer. The matrix $Y\\in\\\\{ 0,1 \\\\}^{K\\times Kn}$ indicates here the one-hot encoded targets for $K$ classes and $n$ samples per class. $H\\in\\mathbb{R}^{d\\times Kn}$ is in this case related to the $d$-dimensional output of the penultimate layer and the matrix $H_0$ indicates architectural restrictions on what kind of representations can be learned. The matrix $W$ is in return related to the weight matrix connecting the penultimate layer with the last layer.\n\nThe main result (Thm 1) indicates (approximately) that when $\\lambda_W>0$ the within-class scatter decreases and the between-class scatter increases with $\\beta\\rightarrow 0$.\nExperiments evaluate how the weight decay on specific layers influences the neural collapse behavior on Cifar 10.", "summary_of_the_review": "The authors provide support to the hypothesis that sparsity in the learned representations of neural networks, given by the output of the penultimate layer, induces/strengthen the behavior of neural collapse. The provided theory is only loosely connected to the reality of neural network training and is also not novel in the light of matrix factorization theory. Experiments are also just loosely simulating the sparsity in learned representations, which makes the results overall not very convincing.\n\n# After Rebuttal Thoughts\nI did not find the rebuttal very convincing. The theoretical analysis is still weak. Even when the network is trained with MSE loss, the objective function would not be the considered objective, but softmax would be applied on the matrix product. Generally, the nonlinear parts of the objective and the learned function, represented by $H$ are neglected. The analysis of the considered objective does still not deliver more knowledge than what is known from matrix factorization theory. The use of gradient flow is not a contribution in itself, it doesn't really fit here (see my criticism on approximations). The addition of the penalty term is emphasized as a contribution, but the anlysis focuses on the case where the weight of the penalty term goes to zero. Hence, I keep my score and I strongly encourage to have a look into the connection of NMF with clustering methods to improve further work.", "strength_and_weaknesses": "The main weakness of this paper and the referenced paper on which the authors build (Tirer&Bruna, 2022) is that most of the results and contributions already have been studied in the field of matrix factorization. This whole field is entirely neglected and properties of the objective solvers are derived over approximations of approximations, that just as well follow directly from the theory of matrix factorization (using maybe even a more realistic explanation, see below). Considering this, the contributions of this paper seem thin. \n\nI think that the introduced neural collapse score is intuitive and that it could be useful. I also think that there is some insight gained from the experiments confirming that sparsity in the learned transformed feature representation induces neural collapse. Although I have some doubts whether the experiments actually correspond to the theory.\n\n# Assumptions and Approximations\nThere are quite a few assumptions that make the derived theory less impactful. First of all, analyzing the matrix factorization model as a substitute for neural network optimization is very doubtful. After all, the targets (class labels) are not directly approximated by a linear function, but by the softmax of a linear function. Then, cross-entropy is typically used instead of MSE. In addition, the authors make the assumption that each class has exactly $n$ samples. To make the $l_2$-norm regularized matrix factorization model of Tirer & Bruna more \"realistic\", the authors introduce the regularization term $\\lVert H-H_0\\rVert^2$. This is an approximation for the fact that $H$ can not be chosen freely, but is instead a function of the input, given by the neural network's representations learned as the output of the penultimate layer. So, this penalty term is quite far from what's happening in reality and I also don't see how the provided analysis of this objective can actually provide insight.\n\nTo prove Theorem 3.1, the gradient flow is analyzed of, again, an approxiation $\\frac{dH_t}{dt} = -Kn\\nabla\\mathcal{L}(H_t)$. In fact, we only have   $\\frac{H_t-H_0}{t} = -Kn\\nabla\\mathcal{L}(H_t)$. \n\n# The Results from a Matrix Factorization Perspective\nI think that the most interesting result of this paper is the analysis of the within and between class scatter in dependence of the sparsity penalization of $H$. The theorems in this paper don't show exact properties but rely on multiple approximations. The approximate result can be also directly derived from the theory of matrix factorization. Considering that we actually analyze now $\\lVert Y-WH\\rVert$, then  after the penultimate layer, most architectures use a ReLU activation. Hence, $H$ is nonnegative. If we restrict $H$ to be a one-hot encoded matrix, that is $H\\in\\\\{0,1\\\\}^{d\\times Kn}$ and $\\lVert H_{\\cdot i}\\rVert=1$, then the objective $\\min_{W,H}\\lVert Y-WH\\rVert$ is _equivalent_ to $k$-means on the matrix $Y$, where $W$ indicates the centroids. Hence, trivially, this objective would minimize the within-class scatter and maximize the between-class scatter. A similar result can however be achieved if we introduce a sparsity constraint on $H$. the nonnegative, sparse matrix $H$ indicates in this case a fuzzy clustering, where the nonzero elements in $H_{k j}$ indicate a degree with which the  sample $x_j$ belongs to class $k$. The higher the sparsity regularization weight, the fewer classes are selected for a sample to belong to, and the closer we get to the $k$-means clustering.\n\n# Experiments\nThe experiment results are depicted in Figure 3. A network is trained layer-wise on Cifar 10. The sparsity constraint on $H$ is simulated by a weight decay on all weights between layers, except for the last one. This is however not the same as sparsity in the learned representations given by $H$, which is the output of the penultimate layer. The experiments indicate that a higher weight decay on \"H\" increases the neural collapse metric on Cifar-10, which is an interesting result, but it only loosely connects to the theory.\n\n## References\nUdell, Madeleine et al. “Generalized Low Rank Models.” Found. Trends Mach. Learn. 9 (2016): 1-118. (see in particular \"2.2 quadratically regularized PCA\" and \"3.2 Examples\" )", "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": "Uymh8sjMSdF", "reviewer": "Reviewer_uexe", "summary": "", "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": "", "summary_of_the_paper": "This paper focuses on the understanding of neural collapse (NC) phenomena of the learned last-layer features and classifiers observed in deep learning classifiers. Most of the existing work provides analysis under the so-called unconstrained features model where the features are viewed as free optimization variables. Motivated by the fact that dee layers cannot arbitrarily modify intermediate features that are far from being collapsed, this paper proposes a slightly different model by forcing the features to stay in the vicinity of predefined features. This is achieved by adding a regularizer to the existing unconstrained features model. The authors provide a perturbation analysis for the new problem with an additional penalty on the distance to predefined features and show that within-class variability of the output features is reduced compared to the predefined input features. ", "summary_of_the_review": "This paper studies a clear and important question about approximate neural collapse observed with practical networks. However, there is a lack of sufficient description of why the proposed approach can be used to explain the approximate neural collapse. For example, it is not clear to me what the term $||H - H_0||_F^2$ represents in a practical network.  It also appears that the proposed approach can not capture the observed phenomena in practice.", "strength_and_weaknesses": "## Strength:\n- This paper provides a new model based on the existing unconstrained features model for understanding neural collapse. \n- The new model could potentially provide a further justification for neural collapse observed in practical networks.\n- The authors provide analysis based on gradient flow and perturbation analysis for the global solutions of the new model. The analysis shows a reduction in the within-class variability of the output features compared to the predefined input features. \n## Weakness: \n- The major concern is the rationality behind the proposed new model. In particular, the new model, as shown in eq. (2), is the same as the existing unconstrained features model but with an additional regularizer on the distance to predefined features $H_0$. Why this model can approximate what happened in practice? What do these predefined features $H_0$ represent? If these $H_0$ represent the features before the last-layer features $H$, then in practice, $H$ is obtained as a linear transformation (and plus nonlinear functions) of $H_0$, which may be far away as simply adding regularizer on the distance between $H$ and $H_0$. In any case, more discussions should be provided to support this new model. \n- The analysis in Section 3 uses a new measure for neural collapse, $\\tilde NC_1$, and shows the decrease of $\\tilde NC_1$. But the experimental results are plotted in terms of $NC_1$. Does a decrease of $\\tilde NC_1$ imply a decrease of $NC_1$?\n- The results in Section 3 only show a decrease in neural collapse, but do not provide a specific amount of decrease. \n- The results in Section 4 assume $\\beta$ is much larger than 1, which implies that the learned features will be very close to $H_0$. The main result in Theorem 4.1 also suggests this, as eq. (5) implies that the collapsing gap is approximately preserved, i.e.,  $\\delta H \\approx \\delta H_0$. This is not close to what we observed in practice as in Figure 2, where the features become collapse very quickly across layers.   ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "F6jpl8DphzN", "reviewer": "Reviewer_8Wp5", "summary": "", "questions": "", "limitations": "", "rating": 5, "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 presents a new model for neural collapse (NC) which penalizes the degree to which the learned features diverge from some initial value. This model, it is argued, more closely aligns with the evolution of features from one layer to the next in a neural network. This modification to the unconstrained features model, coupled with the assumption that the weight vector is optimal with respect to the features, yields a gradient flow which also exhibits neural collapse. Finally, a perturbation analysis is applied to the model when the previous-layer features are not collapsed. The principal take-away from this analysis is that it is intra-class variation, rather than inter-class variation, of the previous-layer features which drives the deviation from collapse of the output features. ", "summary_of_the_review": "This paper presents an interesting perspective on neural collapse by studying the behaviour of networks that are near, but do not precisely exhibit, feature collapse. The theoretical model used in this analysis comes closer to capturing the limited ability of neural networks to arbitrarily change the feature representation, but still makes strong assumptions on e.g. the optimality of the weights $W$ over the course of training and the evolution in the value of the initial earlier layer features. The perturbation analysis results are intriguing and provide some intuition for how features may exhibit slow convergence to total collapse, but the closed-form solution obtained by these results is quite unwieldly and difficult to interpret except through analysis of sub-matrices. Overall, I think this paper makes a step towards studying neural collapse in a more realistic regime, but there remains a wide gap between the theoretical model studied here and practical settings which limits the significance of the contribution.\n", "strength_and_weaknesses": "Strengths\n\t\n- The model studied in this paper is more realistic than previous works that have completely unconstrained features.\n- The paper is clearly written.\n- The modified trace term measuring neural collapse defined in Section 3 is well justified, and I appreciated the comparison against prior works to highlight the necessity of $\\widetilde{NC}_1$ in Theorem 3.1 \n- The perturbation analysis in Section 4 provides a nice intuition for how non-collapsed features in early layers of the network can slow down convergence to NC in later layers.\n- The experiments are easy to interpret and provide a sensible evaluation of the theoretical predictions.\n- The visualization of the operator $F$ studied in Theorems 4.1 and 4.2 is helpful to ground the discussion around these results.\n\nWeaknesses/Questions:\n- I’m unsure how realistic the gradient flow in (4) is as a model for feature learning in neural networks.\nIn moving from the penalty on the distance $\\|H - H_0\\|$ to the gradient flow, the analysis seems to implicitly change the interpretation of $H_0$ as the output of a previous layer to the value of the current features.\n- Further, this requires that the weights always be optimal with respect to the current features. This assumption seems unlikely to be true in neural networks, and it’s not clear to me how the approximation error and resulting dynamics attained by suboptimal weights would relate to the “central path” described in this paper. \n- The conclusions from Theorem 4.2 seem to implicitly assume a relatively small number of classes in order for the contribution of the inter-class blocks of $\\mathbf{F}$ on the off-diagonal to be negligible. I would be interested in seeing (at least in the appendix) a discussion of how the relative contribution of inter- and intra-class interactions vary as a function of the number of classes. In particular, how do the spectra of the blocks translate to the spectrum of the overall linear operator $F$? While I appreciate the empirical example in Figure 1, I would have liked to see a more detailed theoretical analysis of the relationship between the sub-blocks and the overall matrix.\n\nMinor errata:\n“Values” is misspelled twice as “vlaues”.\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": "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": {}}]

Author rebuttal:

Decision: Reject