Decoupling Dynamical Richness from Representation Learning: Towards Practical Measurement
Authors: Yoonsoo Nam
Nayara Fonseca
Seok Hyeong Lee
Chris Mingard
Niclas Alexander Göring
Ouns El Harzli
Abdurrahman Hadi Erturk
Soufiane Hayou
Ard A. Louis
Venue: ICLR 2026 Poster (2026)
Keywords: training dynamics
representation learning
lazy/rich regime
neural collapse
grokking
kernel methods
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: Practical method to quantify dynamical richness independent of performance.

Abstract: Dynamic feature transformation (the rich regime) does not always align with predictive performance (better representation), yet accuracy is often used as a proxy for richness, limiting analysis of their relationship. We propose a computationally efficient, performance-independent metric of richness grounded in the low-rank bias of rich dynamics, which recovers neural collapse as a special case. The metric is empirically more stable than existing alternatives and captures known lazy-to-rich transitions (e.g., grokking) without relying on accuracy. We further use it to examine how training factors (e.g., learning rate) relate to richness, confirming recognized assumptions and highlighting new observations (e.g., batch normalization promote rich dynamics). An eigendecomposition-based visualization is also introduced to support interpretability, together providing a diagnostic tool for studying the relationship between training factors, dynamics, and representations.

Published as a conference paper at ICLR 2026
DECOUPLING DYNAMICAL RICHNESS FROM REPRE- SENTATION LEARNING: TOWARDS PRACTICAL MEA- SUREMENT
Yoonsoo Nam', Nayara Fonseca', Seok Hyeong Lee², Chris Mingard', Niclas Goring', Ouns El Harzli', Abdurrahman Hadi Erturk', Soufiane Hayou', Ard A. Louis! 'xford University. 'Scoul National University, 'Johns Hopkins University
ABSTRACT
Dynamic feature transformation (the rich regime) does not always align with pre- dictive performance (hetter representation), yet accuracy is often used a proxy for richness, limiting analysis of their relationship. We propose a computationally efficient, irfformance-indeeeddeen metric of richness grounded in the low-rank bias of rich dynamics, which recovers neural collapse as a special case. The metric is empirically more stable than existing alternatives and captures known lazy-to- rich transitions (e.g., grokking) without relying on accuracy. We further use it to examine how training factors (e.g.. learning rate) relate richness, confirming recognized assumptions and highlighting new observations (e.g., batch normal- ization promotes rich dynamics). An eigendecompositionn based visualization also introduced to support interpretability, together providing diagnostic tool for studying the relationship between training factors, dynamics, and representations.
INTRODUCTION
In machine learning, feature learning often viewed through two complementary perspectives: improvement of representations and non-linear training dynamics. The representation perspective emphasizes feature quality how well it supports downstream tasks like classification and promotes generalization (Bengio et al. 2013). The dynamics perspective often called rich regime in rich versus lazy training (Chizat et al. 2019)- concerns the dynamic transformation of features beyond linear models. While dynamical richness frequently correlates with representation usefulness, rich dynamics reflect a preference (inductive bias) toward certain solutions, without necessarily benefiting all tasks (Geiger et al. 2020 Göring et al. 2025). For example, dynamic feature learning can impair the performance on an image classification task (Fig. D.
The representation perspective of feature learning central to deep learning's 's success remains poorly understood. However, its dynamic aspects, often shaped in practice by initialization and optimization, well understood in the context of layerwise linear models (e.g., linear networks). Greedy dynamics with low-rank biases (Saxe et al., (2014) Mixon et al., 2020) explains neural collapse (Papyan al. 2020) in vision tasks; delayed saturation dynamics (Nam al.
(2024) has been linked
to emergence (Brown et al.) (220)) and scaling laws (Kaplan et al. (2020) in large language models; and initialization-deeendde
dynamics (Kumar al.] 2024] Kunin al. 2024 Lyu al., 2024
offers insights into grokking (Power et al., (022) in arithmetic tasks. See Nam et al. (2025 for
comprehensive overview.
To better understand the link between rich dynamics and representation improvement, we need independent metrics for dynamics and representations. However, prior dynamics metrics, while insightful, are not optimized for practical richness measurement (Section 4.1). In this paper, we propose a dynamical richness metric that compares the activations before and after the last layer. The metric (1) generalizes neural collapse, (2) is computationally cheap, (3) is performance-inddeeedennt and (4) operates in the function space. We demonstrate that our metric empirically tracks dynamical richness without referencing performance, and assess its robustness against existing metrics. Finally, we experiment our metric, with complementary visualization, across various setups, highlighting its potential as tool for uncovering empirical insights.

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Published as a conference paper at ICLR 2026
Training Set
Test Set
Correct Label Encoding
Incomrett I abel Encoding
Full backprop (Rich)
Last layer
a
4
only (Lazy)
4
Train acc. f
100%
98.7%
Test acc.
10.0%
74.4%
DRR +
0.0087
0.63
(a) MNIST with label encoding
(b) Performance and richness metric
full backprop (10.0 (10.0%), PER 8.7e-3]
last layer only (74.4%), DLR [6.3e-1]
1.00
10-1
0.75
0.75
0.50
0.50
10
0.25
0.25
0.00
0.00
10-5
10a
10¹
10a
10a
10a
(i) Target projection
(ii) Self projection
(iii) Eigenvalue
(c) Visualization of representation quality (i) and our metric (ii,ii)) using the eigendccomposition
Figure 1: Rich dynamics * better generalization. We trained 4-layer MLP on label-encoded
MNIST. (a): The first 10 pixels are encoded with true labels in training and random labels in testing;
(b): A full backpropagation model (rich) biases toward the encodings and generalizes poorly, while last-layer-only-trainee model (lazy) relies on the full image and generalizes better. Our low-rank-basee metric DRR (Eq. (4) quantifies the dynamical richness (DLR E [0, where is richest) independent of the performance. (c): Complementary visualization method (Eq. (5): (i) Cumulative contribution of last-layer features in expressing the target function - top 10 features are irrelevant in the rich model. (ii) Contribution the learned function the rich model uses only the top 10 features, while the lazy model uses all. (iii) Relative feature norms rich model concentrates on the top 10; lazy model decays more gradually. Test accuracies and ID LR values shown in parentheses and square brackets, respectively. See Section 5.1 for details, and Section [ background and motivation.
both the encoding and the image serve valid features for training.
Contributions:
We introduce the Minimum projection operator TMP (Definition and define the dynamical low-rank measure PLR (Eq. (4). lightweight and performance-independenl metric.
We show that DLR reduces to neural collapse special case, connecting formulation to a well-studied phenomenon while extending it to settings without labels.
We empirically confirm that DL captures known lazy-to-rice transitions such grokking and target downscaling (Table 2. aligning with theoretical expectations while giving direct evaluation without referencing performance. We also empirically assess its robustness against alternatives.
We demonstrate the utility of our metric by showing how various training factors (e.g.. architecture and learning rate) relate richness and performance (Table (3). Our metric explicitly confirms prior assumptions and uncovers new observations, such as batch nor- malization shifting VGG-16 on CIFAR- 100 from lazy rich dynamics. We further adapt eigendecomposition to visualize the metric for improved interpretability.
SETUP AND BACKGROUND
For analytical tractability, we use MSE loss and supervised settings where target function entries are orthogonal and isotropic (e.g., balanced C-way classification regression with scalar output).
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Notations. For the input space X, we train on samples from 9. the underlying true distribution on the imput space X. The width of the last layer (or the linear model) is The dimension of the output function (and the output of the neural network) is C. Denote f and f as the learned
function (at any point in training) and the larget function, respectively. We use bra-ket notation
(flg) (f)o((())g(4)) to mean the inner product over the input distribution (Section B See
Section for glossary.
Empirical details. Full experimental details, link to the source code, and statistical significance for the tables are provided in Section All error bars represent one standard deviation.
Feature kernel operator. We define the feature map cb X - Rp as the map from the input to
the post-activations of the penultimate layer. For 1 S k S p, denote the h feature as the th entry (x) of ((x) = [ (x), @ (x(x)]. We define the feature kernel operator T (x) 1. (2)) generalization of self-correlation matrix in the function space, as
Σ (ele))((e)((()))
Tif(2z) = Eggg
T= Σ
22) (Pal.
where (x)) is the Hilbert space of square-integrable functions on X' (with distribution g). The second expression is T in bra-ket notation (Section B). Using Mercer's theorem (Mercer 1909). we
can decompose the operator into eigenfunctions and eigenvalues
Pk lex) (eal,
T(ex] = PkCK,
(es (ei) = 5kt,
(2)
where Pk E RR is a non-negative eigenvalue and ex X + R is the orthonormal eigenfunction.
The operator T and the eigenfunctions 6 play a key role in kernel regression with feature map (i.e. K(x, )') = T (Tx())) The linear model (kernel) has inductive bias toward eigenfunctions
corresponding to larger eigenvalues (Bordelon et al. 2020 Jacot et al.) 2020: Spigler et al.) 2020 Canatar et al. [022] Cui al. 2021 El Harzli al. 2024), and generalizes better when eigenfunctions with larger corresponding eigenvalues also better describe the target function (larger (ex (ex]/")) also
known as the task-model alignment (Bordelon et al. 2020). See Section Ceationnoorr foor gentle introduction.
Rich dynamics. Chizat et al. (0119) introduced rich dynamics as deviations from the exponential saturation observed in linear models. Subsequent works (Kunin et al.) 2024 Dominé et al. 2025
Nam et al. 2025) identified these dynamics as sigmoidal saturation or amplifying dynamics arising
from gradient descent in layerwise models - and connected them to phenomena like neural collapse
(Mixon et al. 2020), feature emergence (Nam et al. 2024), and grokking (Kunin et al. 2024). See Nam et al. (2255) for comprehensive overview.
Low-rank bias and neural collapse. Low-rank hidden representations naturally emerge in the rich regime, as shown by studies across linear networks (Saxe et al. 2014 Ji & Tel Telgarsky 2019; Arora et al. 2019a; Lampinen & Ganguli 2019 Gidel et al. 2019 Tarmoun et al. 2021), unconstrained feature models (Mixon et al. 2020 Fang et al., 2021). and matrix factorization (Arora et al.) 2019b]
Li et 2020). These works find that gradient dynamics decouple into minimal number of modes, governed by the rank of the input-output correlation matrix (often that of the output C),, causing gradients to concentrate on those modes and produce low-rank representations. In practice, the low-rank bias is observed as neural collapse (Papyan et al., (2020), which consists of four conditions NC1-4 stated in Section D Although often associated with improved representations, neural collapse does not consistently imply better generalization (Zhu et al. [202T: Nguyen et al. [2022 Hui et al.)
2022 (Su et al. 2023). remaining largely a dynamical phenomenon. See| Kothapalli et al. 2022) for an overview.
Change in NTK as measure of rich dynamics. Beyond performance, changes in the Neural tangent kernel (NTK, Jacot et al. (2018) ) before and after training are among the most commonly used proxies for rich dynamics (Chizat et al. 2019; Atanasov et al. 2021; Kumar et al.) 2024). Although theoretically foll-founded, NTK-based measures scale with the total number of parameters and are computationally infeasible even for moderate vision models. For this reason, we omit NTK from our analysis and we focus only the feature kernel of the last layer (Eq. (1).
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3 THE RICHNESS MEASURE DER (MINIMUM PROJECTION ALIGNMENT)
We introduce a dynamical richness measure that exploits the low-rank bias of rich dynamics and show that it reduces to neural collapse special case.
3.1 Low RANK BIAS RICHNESS MEASURE
We define the learned function space H = fp:n(fi....... Sc} where fa X' + IR is the kth entry of the network's learned function (at any given time). In ideal rich dynamics scenario, only the minimal number of features are learned throughout the dynamics and are sufficient to express (linearly span) the learned function space. Leveraging this idea, we define the minimal projection operator TMP.
Definition 1 (Minimum Projection (MP) operator). For a neural network with learned function f and features (x), the corresponding T is an MP-operator TMP if can be expressed as
Turlu] = an (1lu) 1 + a2Px(u) for all u E (2),,
(3)
where a1, 02 > 0, 1 E (X) is a constant functionn, and Pa : (2) + (x() is the orthogonal projection in (2)) onto H.
Ignoring the constant function (setting a 0). whose discussion is deferred to Section D the TMP is (up to a constant scale) a projection operator Pa that removes all components orthogonal to H. If T is TMP, the last-layer features span only C-dimensional space that matches the learned function space, reflecting the low-rank structure characteristic of rich dynamics. We thus define the low rank measure DLR as the similarity between the T- defined by the current features and the MP-operator TMP defined by the current learned function as a metric for richness:
:= = 1n:=--CKAAL CC TMP),
(4)
where CKA is the centered kernel alignment (Kornblith et al. 2019) with bounded value in [0. 1]. Because C'A is normalized and uses centered (zero-mean) alignment, the metric remains consistent for any 01 and 02 (Section B.3) We subtract the CKA measure from 1 so that lower values indicate richer dynamics, consistent with more widely used metrics of richness. The novelty of our metric DI. lies in defining the minimum projection operator TMP and comparing it to T to quantify dynamical richness, not in using well-established CKA. Notably, CKA is primarily used to compare the NTK before and after training (Chizat et al. 2019 Kumar et al.
2024 Baratin et al. (02T).
Intuition. Our metric (Eq. (4) compares the activations before (T, at most rank p) and after (TMP, rank C) the last layer. In rich dynamics, low-rank bias dictates that only the minimal necessary C out of p features are learned and used before the final layer (DLæ 0). In this case, the final layer performs no additional processing because earlier layers have already completed the task. Conversely, an excess of features indicates bottleneck - such as limited expressivity in earlier layers - that prevents the full manifestation of low-rank bias.
Time complexity. The algorithm is highly efficient. For samples (from either train or test), it requires n forward passes to record activations before and after the final layer, producing matrices of size p and C, where is the last-layer width and C' the number of classes. DLR is then computed in O(npC). Since typically n > p >> C, setting n z3 O(p) suffices, reducing complexity to O(p²C), C). In standard models with p 10 this is far cheaper than NTK-based methods, which scale quadratically with the total number of parameters. See Section for details.
3.2 CONNECTION TO NEURAL COLLAPSE
Suppose that the empirical distribution coincides with the true distribution and the neural network perfectly classifies all labels with one-hot vectors. We show that if T is an MP-operator, then the NC1 and NC2 conditions (Section D of neural collapse hold. Let dagger denote pseudo inverse, A; the set of datapoints with class label i, so the ith learned function fo coincides with the one-hot indicator function for A, Following Papyan et al. (2020), we define the tth class mean vector as
/4 = Eze A, (x)) and the global mean vector as J == C-1 E Pi.
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Proposition 1. I is MP-operaton then the NC1 condition (collapse of within-class variability) E holds, where inter-class covariance matrix E.b and intra-class covariance matrix Ew
C Σi (Ns ( p and Ew E EzEa: [ (x)(((x) p.)²
Proposition 2. If is an MP-operator, then the NC2 condition (convergence of features to simplex equiangular tight frame) holds:
(nz: - (us - dig C
See Section for proofs and further discussions.
Although the ideal criteria of T is T implies neural collapse criteria as a special case, they differ in general: we measure how well features (random variables or functions) express the learned function, while neural collapse concerns how training feature vectors represent class mean vectors. Based on the function space, measure extends to test data, enables feature quality assessment (Eq. (5)), and is more empirically robust (Tables [ and 2). extending beyond neural collapse (see Section
4 EXPERIMENTS
Here, we share our empirical results. We first compare our metric with existing metrics of richness, then we confirm that our measure empirically tracks known lazy-to-rich transitions. We further conduct experiments various training setups to explore the relationship between training factors, rich dynamics, and performance (better representations).
4.1 COMPARISON TO PRIOR MEASURES OF RICH DYNAMICS
Three alternative metrics for richness measures (1) similarity to the initial kernel, Sinit
(miit Kiaaned E [0, 1] Yang & Hu 2021), (2) the parameter norm | (yuu et al. 2024). and (3) class separation from neural collapse, NC1 Tr TrEE,Ew Ew) (Papyan et al. 2020 Stevens et al. 2002 He Su 2023 Xu & Liu| 2023] Súkeník et al. 2024), where Σ, and Ew are the inter- and intra-class covariances (Proposition D.. Prior metrics depend on the initial kernel, parameter norms, or class labels, which can constrain their use as independent measures of richness.
Table Richness for weight decay
Table Richness metrics for target downscaling
Epoch
0 (init)
200
a
10
2 2-
10a
Train Acc.t
10.0%
10.2%
Train Acc.1
100 %
100%
100%
Test Acc.f
9.85%
10.0%
Test Acc.1
92.7%
92.4%
88.3%
DiRd
0.59
1.0
DEnk
4.9 1.9-00-2 10-2
1.1 10-1
10-11--
Syntth
1.0
0.20
Smntt
10-2
4.1
5.2 10-2
|
3.1 3.1
2.2
|
3.4 10
3.2 10
3.2 10
NCW
1.2- 10
1.--10-14
NC1,
2.3
3.2 103
8.1
ssole an extreme case of training an MLP on MNIST with large weight decay and negligible learning rate. Here, the dynamics are dominated by 2 weight decay, with little meaningful learning. While existing metrics can sometimes misinterpret this as rich behavior (smaller after training), our span-based measure correctly identifies the lack of dynamical richness (bigger after training).
Table 2 presents more practical setting, where we tune laziness via target downscaling (i.c., y/ a). Prior works (Chizat al. 2019 Geiger et al. (020)) show that scaling targets by factor a induces lazier training where larger a implies greater laziness. A good metric should capture this.
Our measure aligns with while all other measures misalign with laziness. By not relying initial kernel, weight norm, or labels, our metric shows greater robustness this setup. See Fig. in Section for visualization of the dynamics in Table
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Kernel distance from initialization (Sit))) Kernel deviation Yang & Hu
202 albeit often
the comparison NTKs, is a common measure rich dynamics (Chizat et al. measures the deviation from the initial kernel, not how the kemel changed (e.g. weight decay alone changes the kernel in T. While theoretically appealing for near-lazy models, the metric can be less accurate in deep, rich-training regimes where most practical training occurs.
2019). However,
Parameter norm ()) Smaller parameter norms often correlate with rich dynamics in practice, similar to how the authors of Lyu et al. (2024) used it in their study of grokking. However, small weights promote rich dynamics (Kumar al. 2024 |ttanaovv al., 2024] Nam et al. [2255] Saxe ct al. [2014] 2019), not the other way around. In fact, rich dynamics can occur for larger initialization
(Braun al.
2022
2025) and depends on broader factors such as layer imbalances Dominé et al.
(Kunin et al.
2024 Dominé et al.
2025 Nam et al. 2025
Neural collapse measure (NC1). Separation-based metrics measure how training samples deviate
relative class boundaries (aayyn et al., 2020; Stevens al. 2002; He & Su 2023 Xu & Liu 2023] Sūkenik et al., (024). While most similar to our metric, they are unbounded, sensitive to output scaling, and can be empirically ill-conditioned. As shown Table these issues can lead to dramatic value shifts and hinder interpretability.
Low-rank measure (D. Our proposed metric evaluates the alignment between features and the learned function, achieving its optimum when they span the same space isotropically (i.e., a scaled projection operator). Our metric exploits the low-rank bias of rich dynamics through alignment and normalized between [0, 1]. Crucially, it does so without relying on class labels, accuracy, or the independent measure of richness.
initial kernel, making the measure more appealing
4.2
EMPIRICAL FINDINGS REGARDING TRAINING FACTORS
Table Richness measure and performance on various setups
Task
Architecture
Condition
Train acc.
Test acc.
DLR
Figure
Mod
2-layer
Step 200) (before grokking)
100%
5.2%
0.51
Fig. 12
97
transformer
Step 3000 (after grokking)
100%
99.8%
0.11
learning rate 0.005
100%
66.3%
0.053
CIFAR-
ResNet18
learning rate = 0.05
100%
78.3%
0.025
Fig.
100
learning rate 0.2
100%
74.5%
0.039
CIFAR- ResNet18
weight decay = (
100%
93.5%
0.05
Fig. 13
10
weight decay = 10
100%
94.1%
0.015
weight decay 10 10-3
100%
94.8%
0.003
CIFAR- ResNet18
This experiment compares
100%
94.8%
0.026
Fig.
10
MLP
architectures only.
99.8%
55.4%
0.48
label shuffling
100%
95.0%
0.031
CIFAR- ResNet18
10% label shuffling
100%
66.1% %
0.042
Fig. 15
10
full label shuffling
100%
9.5%
0.034
full backpropagation
100%
99.1%
0.043
MNIST CNN
Fig. 16
last layer only training
99.7%
96.8%
0.51
CIFAR- VGG-16
without batch normalization
99.5%
21.7%
0.66
Fig.
100
with batch normalization 100%
72.0%
0.073
Table 3 demonstrates the practical usefulness of our cerformance-independie metric, summarizing the correlation between training factor, performance, and richness in various setups. The visualization of most experiments provided in Section
The first row empirically confirms that metric is richness metric, explicitly capturing the known lazy-te moay-to-ric to-rich transition of grokking (Kumar et al. 2024; Kunin et al.] 2024 Lyu et al. 2024 Nam
al. 2025).
Rows two to four confirm the assumptions that the optimal learning rate n2nd row), the weight decay (3rd row), and the architecture h4th row) achieve high performance through rich dynamics (Ginsburg

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Published as a conference paper at ICLR 2026
al. 2018 Liu al. 2022 He et al. 2016). Indeed, our metric makes this link explicit by having smallest value for the optimal setting. Although such relationships are broadly recognized, they are typically only addressed implicitly.
The fifth and sixth rows, like Fig. T demonstrate that rich dynamics do not strictly correlate with performance. R CIFAR-10 10 retains rich dynamics when the labels are shuffled, and convolutional neural network (CNN) on MNIST can achieve similar performance through both lazy and rich dynamics.
The last row reveals a new observation: VGG-16 on CIFAR-100 is lazy without batch normalization but rich with accompanied by significant performance gap. While the performance effect of batch normalization is empirically established, its underlying role remains debated. Our metric helps clarify this by reframing generalization in terms of the more tractable problem of rich dynamics.
+
data augmentation
data augmentation
+
Gaussian Process
100
100
100
test: errorr
test loss
DLR
10-1
10-1
10-2
10-1
102
103
10
102
103
10*
102
103
104
n
Test
(b) Test loss
(c) Richness
Figure 2: Learning curve and feature learning metric. (a): Learning curves of ResNet18 8 on CIFAR-10. Both error (a) and loss (b) learning curves show transition a faster-decaying power law with additional data near z 10a, correlating with the shift in decay of the richness measure D.n (c). This agrees with theoretical study on phase transition (Rubin et al., 2024) that sufficiently large number of data points is critical for rich dynamics
promising observation toward better understanding feature learning dynamics. A linear model (Gaussian process) was plotted in (a.b) to highlight the transition into faster-decaying learning curve.
VISUALIZATION METHODS
While metrics provide quantitative summaries, component-wise visualization aids interpretation. For instance, how significant is the contrast between VGG-16 trained with versus without batch normalization (Fig. 3 Or why does small change in DER correspond to large performance differences under varying learning rates (Fig. 4 Does the CIFAR-10 MLP begin in a lazy regime, or transition into it during training (Fig. 14 in Section ( To address such questions, we introduce complementary visualization based on widely used eigendecomposition of kernel (Eq. (I))
5.1 VISUALIZATION THROUGH DECOMPOSED FEATURES
Bengio et al. (2013) described feature learning as the process of learning better representations for a downstream model, such as the classifier. Building this idea, view the earlier layers, represented by the feature map X - IR, as providing improved features for the final linear layer. Combined with our --dependent richness metric, this motivates visualization method that extends the linear model analysis of T. See Section [ for related works and a gentle overview.
Our visualization quantifies three aspects (cumulative quality (k),, cumulative utilization πT(k), and relative eigenvalue pk/ (p)) of the last-layer features through the eigenfunctions eg of T:
Px (ex])? dim(H()
(ex P (P(la)) dim(H)
11 (k)
Σ
(ii) ni(k)
Σ
(iii) pa/P1s (5)

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where 712 span fi. ................................ fC) and H fi.n(fi..... fc} the target and learned function spaces, and PH: ( + H is the projection operator onto the space H. All cumulative measures lie in [0, 11, reflecting how well the top k features span the respective space. Notably, II (k) is the cumulative power in Canatar & Pehlevan (2022), quantifying the contribution of the top features in expressing the target.
These three measures together capture complementary views of feature learning. The cumulative quality (IT) (k)) reflects how well the features align with the task. The utilization (TT(k) indicates how many features are used by the final layer, while the relative eigenvalues (pk/ (P) show their relative magnitudes or importance. The latter two (íi and pk/p1) decompose the deviation from DzR = condition by visualizing how many features are used and how many are significant (non-negligible).
5.2 CALCULATING THE VISUALIZATION METRICS
To calculate the eigenvalues and eigenfunctions (and thus the quality, utilization, and intensity) for feature map & X + IRP, need the true input distribution q. which is inaccessible. However, we can use Nyström method (Baker & Taylor 1979 Williams & Rasmussen 2006) for a sufficiently large sample size of p approximate the eigenfunctions and eigenvalues of interest. We can create p X p empirical self-covariance matrix Σ from n samples of (x)). By diagonalizing Σ = USUT, we can obtain the empirical eigenvectors and eigenvalues, which can be used to approximate the true eigenvalues and eigenfunctions (Algorithm I:
Pk
Sk
ex(x) ((x)² Vsk
(6)
Algorithm 1 Empirical eigenfunctions and eigenvalues
1: Φb Drr (Prepare a feature map & and D,y: n samples of training set)
2: ø x (Drr (forward transform the input samples to feature vectors)
3: U,S,U² <SV SVDd)"" (perform singular value decomposition)
4: Ui,,,,,..,, + U (find the column vectors of U)
5: for k ,.
6: Pk + Sk (approximate eigenvalues)
7: (x(x)) t ((x)TTux /////E (approximate eigenfunctions)
8: end for
Functions not vectors. Note that eigenfunctions e/,s defined beyond the training samples used to compute ó and differ from the eigenvectors of the empirical self-covariance matrix. This enables evaluate feature quality via inner products with the target function on the test set rather than the training set, capability not available for empirical eigenvectors. For validity of approximation and dependency on sample size, see Section]] As cigendecomposition arises naturally in study of kernels, similar visualizations have appeared in prior works (e.g., (Canatar et al. 2021 see Section G for details.
5.3 VISUALIZATION RESULTS
Interpreting visualization In perfectly rich dynamics (c.g., blue line in Fig. (ii), showing that only the first C features are used by the last layer. Additionally. P2 R PC and PC PC+1 in (iii), indicating that the features have only C' significant dimensions. The first eigenvalue, corresponding to a constant function, is ignored because of the centering in CKA (see Section D). If the performance is high, (C) in (i), showing that first C features well express the target. For the lazy regime, more eigenfunction will be used (slower capping in (ii) and the feature occupy higher dimensions (slower decay (ii)).
ii(C) 1 in
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batchnorm (72.0%), 7.3e-2]
+
na batchnorm (21.7%). 6.6e-1]
1.00
1.00
10-1
0.75
0.75
0.50
0.50
10-3
0.25
0.00
0.00
10-5
50
150
100
200
10a
10
102
10a
10*
(i) Target projection
(ii) Self projection
(iii Eigenvalue
Figure 3: Visualization of VGG16 on CIFAR-100 with and without batch normalization. We visualize the last row of Table where batch normalization shifts the model from the lazy to the rich regime. The eigenvalue distribution (iii) highlights this difference: with batch normalization, only 100 features are significant, whereas without the eigenvalues decay slowly.
1r=0.2 (74.5%), [3.9--2]
Ir=0.05 (78.3%), 2.5e-2) + Ir=0.005 (66.3%), [5.3e-2]
1.0
10-1
0.7
0.9
(T(()
0.8
10-3
0.5
0.7
10-5
50
100
150
50
100
150
10"
10
102
10a
10a
k
Target projection
(ii) Self projection
(ii) Eigenvalue
Figure 4: Visualization on the role of learning rate. We visualize the 2nd row of Table where the learning rates are varied (up to training instability) for ResNet18 on CIFAR-100, The second column (ii) shows that smallest learning models uses minimal 100 eigenfunctions, indicating a lazier dynamics.
significantly more eigenfunctions (features), while other
epoch 0 (1%)
+
epoch 1 (15%)
epoch (34%) + epoch 30 (55%)
+
epoch 400 (78%)
1.0
Q3 (mnrr(Q)
1.00
(mazz (Q.)
1.00
0.75
0.75
0.50
0.50
0.5
0.25
0.25
0.00
0.00
50
100
150
200
50
100
150
200
50
100
k 200
150
(i) Target Projection
(ii) Self Projection
(iii) Eigenvalue
Figure 5: Correlation among dynamic of feature quality, utilization, and intensity. We show individual metrics (e.g., (k) (k) IT 1)) instead of cumulative metrics (IT*(k) and fi(k) at different epochs for ResNet18 on CIFAR-100 normalized for better presentation. Larger intensity features exhibit higher quality and utilization during training.

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Figs. and 4 iillustrate how visualization complements the metric. In Fig. 3 the batch-norn included model uses only 100 features (eigenfunctions) in (ii) and shows clear low-rank structure in (ii). The batch normalization lacking model uses significantly more feature (4096 in total) in (ii) and show power-law distribution in (iii): indicating significantly lazy dynamics. In Fig. Ii.. the model with the smallest learning rate uses 100 features to express the learned function, unlike models with larger learning rates, suggesting lazier dynamics and poorer performance.
Fig. reveals another novel pattern in feature learning dynamics: feature quality correlates with feature intensity during training, with larger features improving faster. While it is expected that generalizing model the rich regime obtains few high-quality features after training, the correlation between quality and intensity during training has not been previously observed or studied. Interestingly, differs from the silent alignment effect (Atanasov et al. 202T), where, in the dynamics of linear networks in rich regime, quality precedes intensity.
6 DISCUSSION AND CONCLUSION
We introduced a performance-independeee metric for dynamical richness, DLn, which reduces to neural collapse as a special case (Propositions) and and empirically agrees with prior studies on lazy-to-rich transitions (Tables 2 and 3).
Table 3 and Fig. 2 provide examples of how the metric can be used for the analysis. We confirm establish ishd assumptions explicitly with an independent metric, while potentially challenging a few (Fig.) We also reveal new findings (Fig. 3 offering a direction for theoretical research.
Moreover, we provide complementary visualization to extend these results for additional information, also uncovering novel insights such as the alignment between representation quality and feature intensity (Fig.)
Interpretation of neural collapse. We share our secondary findings on neural collapse. The functional utility of neural collapse remains an open question, particularly regarding its debated link to generalization (Hui et al. 2022, Kothapalli et al.) 2022 Zhu et al. 202I). As established in Propositions I and 2] our metric mathematically tied to neural collapse. Because our metric quantifies representational richness, this suggests neural collapse may serve fundamentally as an indicator of rich feature dynamics rather than generalization.
Intermediate layers. While our metric achieves computational efficiency by focusing exclusively on the last layer, this design inherently omits the dynamics of intermediate layers. In Section || we share our exploration of the intermediate layers and the justification of our choice. Theoretically, kernel-based framework assumes linearity between features and the learned function - premise and empirically,
less applicable intermediate layers subject non-linear transformations intermediate layers yield negligible additional insight into dynamical richness. We attribute this to the last layer exhibiting the strongest low-rank bias, view supported by the comparative rarity of deep neural collapse Rangamani et al. (003) and the shallow nature of unlearning Jung et al. (2025). We leave the extension of this framework deep feature learning dynamics for future work.
Isotropic target function. The current form of T is limited to orthogonal and isotropic target functions. While this covers most classification tasks, a more general setup would be preferable. However, imbalanced label remains an active area of research in neural collapse (Fang al. 2021 Yang et al. [2022 Hong & Ling 2023 Yan et al.] (2024] Gio et al. 2024), and we leave the extension for future work. See Section D.4 for further discussion.
Conclusion. By offering lightweight, robust, and
metric on rich dynam- ies, we aim to lay the groundwork for future theoretical studies on their connection to representation learning. More broadly, we see this work as diagnostic tool toward bridging empirical observations on representation learning (e.g., rule of thumb) and theoretical understanding of rich dynamics. In future work, we plan to extend beyond the current focus on balanced tasks.
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REPRODUCIBILITY STATEMENT
As mentioned in the main text, full experimental details, link to the source code, and statistical significance for the tables are provided in Section H All enror bars represent standard deviation.
The algorithm for approximating the metric and visualization is provided Section E
ACKNOWLEDGMENT
We thank Charles London, Zohar Ringel, and Shuofeng Zhang for their helpful comments. NF acknowledges the UKRI support through the Horizon Europe guarantee Marie Sklodowska-Curie grant (EP/X036820/1). SL was supported by the National Research Foundation of Korea (NRF) grants No. 20.20020RI0 1 A5A10161 1016126 and EPSRC iCASE grant with IBM (grant number EP/SS13842/1). SH gratefully acknowledges partial support from NSF grants DMS-2209977
CM acknowledges funding from an
2015341, 20241842, NSF grant 2023505 on Collaborative Research: Foundations of Data Science Institute (FODSI), the NSF and the Simons Foundation for the Collaboration the Theoretical Foundations of Deep Learning through awards and 814639, and NSF grant MC2378 to the Institute for Artificial CyberThreat Intelligence and OperatioN (ACTION). The authors would also acknowledge support from His Majesty's Government in the development of this research.
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A GLOSSARY
Symbol.
Name
Definition
Ref
Class count
The number of classes.
Section
P
Layer width
The width of the last layer
Section
Sample count
The number of training samples Space of inputs
Section
X
Input space
Section
Input distribution
The probability distribution that generates samples in the input space.
Section
Feature map
A map from input to post-activation of the penultimate layer (the activations fed to the last layer).
Section
((x)
(Last layer)
A p-dimensional random variable or the post-activation of penultimate layer for q.
Section
Features
T
(Feature kernel)
A map from Hilbert space to Hilbert space that depends on (x),, I ~ q (typically the last layer features)
Eq.
integral operator
TMP
Minimum projec- tion operator
A special set of operators that depends the learned function f.
Definition
ok
4hh eigenfunction 4 eigenvalue
The eigenfunction of
Eq.
Pk
The eigenvalue of
Eq.
f
Learned function
A C-dimensional vector output function f X' + RRC expressed by the neural network.
Section
f
Target function
A C-dimensional vector output function f X - R
Section
with correct labels. The output is always one-hot vector (up to a scaling constant).
Target function
Space linearly spanned by the entries of the target func- tion.
Section
space
H
Learned function space
Space linearly spanned by the entries of the learned func- tion
Section
Pи
Projection opera- tor
Projection operator onto H. If spanfer, c,....,......................... ep). where is are orthonormal, the projection operator is given as Px = >h_1 les) (ex)- Target downscaling prefactor: y - y/a
Section
a
Downscale constant
Section
E,
Inter-class covari- ance matrix
Covariance of class mean vectors for the training set.
Eq.
34
Ew
Intra-class covari- ance matrix
Covariance of training feature vectors within given class.
Eq.
33
CKA(-))
Centered kernel alignment
Alignment measure between two matrices operators.
Eq.
It is normalized between [0, 1] and ignores the mean (for matrices) or constant function (for operators)
||0l|
Parameter
Norm of all parameters in the model. It was used rich- ness measure with smaller value meaning richer dynamics. Our proposed metric of dynamical richness. Smaller is richer.
Section 4.1
Der
low rank metric
Eq. 44
Smit
Kernel deviation
The CKA measure between the learned and initial kernel.
Section 4.1 Proposition
NC1
Neural collapse metric
The trace of Σ Ew measuring the training feature vectors variance compared to class boundaries.
(T))))
Cumulative qual- ity
Measure how well the first eigenfunctions span the target function space.
Eq.
5
ft(k)
Cumulative
Measure how well the first eigenfunctions span the learned function space.
Eq.
utilization
(*())
Quality
Per-feature quality IT (k) II" (k=1) where IT IT*(0)
Fig.
((k)
Utilization
Per-feature utilization or Tī(k) πī where IT(0)
Fig.
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B TECHNICAL SUPPLEMENTARY MATERIAL
Here, we introduce technical terms used in the main text and the following appendices.
B. BRA-KET NOTATION
In physics, bra-ket notations are widely used to express the inner product in function space. In our paper, we use them to avoid overload of expectations and to clarify that we are using functions. Note that we use bra-ket notation for the expectation over the true input distribution only:
(f () E
The notation is also useful for expressing operators such as T:
TV Σ (x) (0xlf) 2 (0..f)).....))
..... 2p (0pl5)]
Because T maps vector function f X + RP to a vector function, T[f] : - R2 is also a vector function with
TD](x) ( (z) (oilf) ...., 2x(x) (0,l5)), where (Pxlf) are scalars and 4x X R are functions.
B.2 FEATURES ARE RANDOM VARIABLES AND (WELL-BEHAAEED RANDOM VARIABLES FORM HILBERT SPACE
In machine learning, textbooks often overlook the mathematical distinction between feature maps and features, treating them as interchangeable. However, they are fundamentally different.
A feature map is a function f X' + V, defined independently of any distribution. A feature, by contrast, is a random variable induced by applying feature map to inputs drawn from a distribution q over X.
Features are therefore distribution-dependeer applying the same feature map to different input distributions yields different features. For instance, a fixed neural network defines feature map, but the resulting features - such as last-layer activations - will differ between MNIST and Fashion- MNIST due to changes in the input distribution. The distinction also applies to kernel operators T : + H. (distribution dependent) and kernels K X' x X + (distribution independent).
The distinction becomes more important when we wish to discuss Hilbert space. A Hilbert space requires an inner product between functions (feature maps), which depends on the un- derlying distribution. For example, sin(z) and sin are orthogonal for q undf 2x] where 1-20 sin(x) (in(2x)de = O, but not for unit Gaussian distribution = 1) where _ sin(x) sin(2x)) /2 + 0.
For an underlying probability distribution and set of functions (feature maps), form a Hilbert space. common example of a Hilbert space is the set solutions expressed by linear model such
(z) W1 + Wyr + ugg...
Assuming the features [ar x2 3 are linearly independent input distribution q, they form 3-dimensional Hilbert space.
B.3 CENTERED KERNEL ALIGNMENT
The centered kernel alignment (CK A) Kornblith et al. 2019) measures the similarity between two
matrices or operators, and are commonly used for tracking the evolution NTK (Baratin et al. 2021
Lou et al. 2022): :
B) = Tr(((A)c(B))
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where c(A) ( (1)) is centering operator, where is identity, [1] is the constant function or constant vector, and - is Frobenius norm. The centering operator removes the constant shift. The analogy is measuring the similarity of two multivariate Gaussian random variables by comparing their covariance matrices E E[x]) E[X])²] instead of their autocorrelation matrices E[X x
For T. if the learned function space H. contains the constant function which is empirically true throughout the training for all our experiments and always true if it perfectly fits the training samples becomes an orthogonal projection operator on the constant complement H. and independent of the values a1 and dz
(Section D) c(Tmp)//l<(TMr))
B.4 EFFECTIVE DIMENSION
covariance matrix or an integral operator may be fully ranked, but their eigenvalues may decay fast (e.g., by power-law). The small eigendirections are often negligible, and researchers use various effective dimensions measure the number of significant dimensions. In this appendix, we will use the exponent of entropy (Hill 1973) to measure the effective dimension.
For matrix an operator with positive eigenvalues (PI.-.., .Ppl. the effective dimension is
P
Pt
exp
(8)
1== Pj
Note that the effective dimension, the exponential Shannon entropy, when the eigenvalues have d equal non-zero entries. A slower decay of entries of (in non-increasing order of entries) generally yields higher effective dimension than a vector with a faster decay.
The effective dimension of a matrix (operator) can be interpreted as the number of linearly independent vectors (functions) needed to effectively describe the matrix (operator). This is similar in spirit to the principal component analysis (PCA) (Abdi & Williams 2010) in that only the directions with significant variance are considered.
As discussed in Section D. the first eigenfunction is always constant function, which is irrelevant for our measure PLR and dynamical interpretation. Because the first eigenvalue is often much larger than other significant eigenvalues,
Dess(p) 1 1 exp
Σ 1=2
In
Pt
(9)
ΣØ=
2
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C LINEAR MODELS, FEATURES, AND THEIR INDUCTIVE BIAS
Here, we will introduce the recent findings on the inductive bias of linear models and demonstrate the significance of cigenvalues and eigenfunctions of T. Linear regression or kernel regression is
rare case where its inductive bias is analytically calculable (Bordelon et al. 2020 Jacot et al. 2020
Spigler et al., 2020; Canatar al. 2021 Cui et al.
2021
El Harzli al. 2024 Simon et al. 2023).
For a given feature map Φ X R or kernel K
I, x
((x)) ), the model is expressed as ((x)
the p dimensional (p can be infitnite) linear model
f(:;; w)
Σ
ux (xr).
(10)
The objective of (ridgeless) regression minimize the following empirical loss
Lemp (f) Σ - + lim A||w||?
(11)
X-
where (1()) denotes the 8th sample in the training set. The second term is the regularization which we take the limit so the solution unique (pseudo inverse solution) for an overparameterized setup (n <))
C.1 EIGENFUNCTIONS ORTHONORMALIZED FEATURES
In many cases, (x), 22(x).... ..... (x(x)] are not orthonormal. For example, the features [xr x²] of f(x) wix W22 for q = Uniform 00, 1] are not orthonormal:
(x|x²) = L/ dx #0. 0,
(0|2) = [.
x 2 dx
(x9127) #1,
x'dxx dx 1.
(12)
The diagonalization into the eigenfunctions returns the orthonormal basis of the hypothesis (Hilbert) space H. = span @1 (x), (z).). ... & (x)), more formally known as Reproducing Kernel Hilbert Space (RKHS). The eigenvalue now shows the norm of the feature along the direction of eigenfunction
as the features { (Ir), (2)) ... px))) are not normalized (Fig. (6)
v P222
[wi, W2]
f = [0,, 0z]
T 1
vpic1
Figure 6: Diagonalization of features Since the features [ (I)),,, D2(x), L Pp(2) span vec- tor space, each function 4x : X - R can be represented as a vector. The overlap among L 's (non-zero linear correlation) and differing norms of d result in diagonalized features [VPies, ""2",,""",, to have distinct intensities (norms) PI, Pp-
The transformation between the eigenfunction basis and the feature basis is
$ (x: Σ (psees()),,
Oi
(Txlez)
(13)
/Pj
where O E RPXP an orthogonal matrix (follows trivially from Eq. be reparameterized in the eigenfunction basis
2
Using Eq.
13 the model
f(x)
w. (x()
Σ On (L))nen(x).
(14)
Note that because = Ow, the norm of the parameters conserved (i.e. |00| = [|w]])). indicating that weighted basis [yPre1, - pppep] instead of normalized basis [e1, 62, be used to represent the 'intensity' or norm of the features [ (z), (.)))
(2,, -..-.. epl must
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C.2 INDUCTIVE BIAS TOWARD LARGE FEATURES FOR MINIMUM NORM SOLUTION
For overparameterized linear models, the inductive bias determines the returned function as many functions can express the training Here, brief the linear models' inductive bias toward larger
(significant) features or eigenfunctions (Bordelon et al.) 2020; Jacot et al. 2020 Spigler et al. 2020
Canatar al. 2021; Cui et 2021 El Harzli al. 2024] Simon al. 2023).
The inductive bias is quantified through learnability La (Simnn et al.
2023 or the expectation ratio
between the learned coefficient and true coefficient for eg:
(flec)
Pk
Pk
Li = Eigign
(f*lex)
+
where
+ n.
(15)
The learnabilities always to the number of training datapoints and constant is the constant that satisfies the equality.
We can understand the inductive bias from the (vanishing) regularization on L norm (Allwl in Eq. ( and parameterization in Eq. (14). If different eigenfunctions can equally express the training set, eigenfunction with larger eigenvalue often requires smaller coefficient 0g express the samples.
Because | = | expressing the data with larger (eigenvalue) cigenfunctions minimizes the norm, creating an inductive bias toward larger features.
Instead of the derivation found in the references (Bordelon et al. 2020 Jacot et al. 2020 Spigler
al., 2020; Canatar et al. [202]] Cui al., 2021 El Harzli et al. 2024)
which requires random
matrix theory or replica trick we show an example of the inductive bias in Fig. 7
10a
q=-4 4.0
2-
10-2
10
10:4
10a
10
0.0
0.2
0.4
0.6
0.8
1.0
(a) Eigenvalues
(b) Bias by the cigenvalues
Figure Different inductive biases by the eigenvalues (intensity). Two 10-parameter linear models are trained on 4 datapoints (overparameterized) with gradient flow. Both linear models use sin basis functions such that f(x) Σ wok" /2 sin(2nkz)) space - but differ in the cigenvalues with a = -4 (blue) and a = 4 (orange), leading to different inductive biases. (a): The blue model has greater intensity for lower sever-frequecc sin functions, while the orange model shows the opposite.
spanning identical function
(b): The learned functions show that the blue model used lower-frequency functions while the orange used frequen-r functions to express the training samples.
Fig. shows that two linear models with same hypothesis space but different eigenvalues learn dramatically different functions. The blue model has large eigenvalues for low-frequency functions and fits the datapoints using lowfrrequency functions. The orange model with opposite eigenvalue distribution fits the training set with high-frequency functions a clear example of an inductive bias toward large (intensity) features.
In the references, the generalization loss is
La -- I
(1-L
L2) (a
(16)
The equation formalizes the task-model alignment: the alignment between large features (large thus large T.x) and high quality (large ))) leads to better generalization.
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C.3 DYNAMICAL INDUCTIVE BIAS TOWARD LARGE FEATURES
Here, we review how gradient descent introduces the dynamical inductive bias toward larger features. For linear model with feature map and kernel integral operator T. the gradient flow dynamics under MSE loss with infinite training datapoints and zero initialization becomes
df di
-VV-
(17)
The derivation is trivial with
df dl
p
dwn
dc Owk
x Σ dl
(2(x')
Σ
2x(x')
(18)
df
= --E, (f(z) - -x))
(x)
2x(2')
(19)
d'v
E2
(ʃ(x) - (x))) Σ (x(x))
Dx(e')
(20)
-f - f',,
(21)
where in the second line, we used Eq. (T on infinite datapoints so Lemp C. In the last line, we used the definition of integral operator (Eq.
Using the eigenvalues Pa and eigenfunctions es the kernel integral operator T, we obtain the dynamics for each (exlf) by taking inner products with eg on both sides of Eq. (17
d (exls) En dl
(ex|T[f - f1)) (ex(z') (a') di
En
(22)
(23)
=-p(((ea(f) - (exls*)),
(24)
where use the definition of eigenfunction in the last line (Eq, (2)). Expanding f in the eigenfunction basis and plugging in Eq. (22). the function ʃ at time / is a sum of p independent modes that saturate faster for larger eigenvalues:
f())l:
Σ 1=1
(25)
where we assumed is zero function at initialization. Eq. (25) shows that gradient flow decouples the dynamics of linear models into p modes, where each mode corresponds to the evolution of (exlf) having saturation speed governed by Pk thus the dynamical inductive bias toward larger eigenfunction.
C.4 APPLICATION TO OUR METHOD
Here, we detail the intuition of our visualization methods. As described in Bengio et al. (2013). feature learning (of representation) is providing better features for a simpler learner. The analogy allows us describe a neural network as (1) feature map & providing better features for (2) the last layer- the simple learner (Fig. 8). Yet, better features for the last layer is ill-defined.
The last layer interacts with rest the network only through the features. While the exact dynamics differs from linear models, the last layer maintains the dynamical inductive bias (Section C.3 toward larger features. Furthermore, the quality (fexlf"), utilization ((ex))), and intensity Pk formalisms readily extend to the last layer features, motivating our visualization method.
In the rich regime, features evolve during training, and Fig. 9 al-c) demonstrates example of change in features for 4-layer MLP trained to fit the Heaviside step function. The eigenfunctions are difficult to visualize in practice the input space is high dimensional. Fig. d- shows our visualization, capturing the evolution of features by measuring properties natural to linear regression.
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last lnnear layer
featurey piee
(())) = Osy/psey(a)
fealure map
input data
+ X P
APP
class
larget f
Feature Metrics
Quality cos a
Utilisation cos Intensity: Pa
class C
learned
layer
Figure 8: A neural network decomposed into feature map and a linear last layer. (left):An abstract diagram depicting a DNN architecture as a combination of the feature map Φ : X + R from the input space to the post activations of the penultimate layer, and final linear classifier for C-way classification. Most neural networks share this abstract structure and mainly vary in how they create their feature maps. (right): Illustration of the visualization methods in function space.
f
61
E2
63
8
1
0
-2
-1.0
-0.5
0.0
1.0
-1.
-0.5
0.0
1.0
-1.0
-0.5
0.0
(a) epoch: 0
(b) epoch:
(c) epoch: 100
epoch
epoch 10
epoch 100
1.0
1.0
10"
0.9
0.9
P1
0.8
0.8
0.7
0.7
10-6
10
15
20
10
15
20
0
10
15
20
(d) Target Projection
(c) Self Projection
(1) Eigenvalue
Figure 9: Toy model demonstrating feature learning. A 4-layer MLP with width 1000 and scalar input and output is trained to learn the Heaviside step function f" over the domain [-1, 1]. (a): The first three eigenfunctions (dashed) are shown at initialization, resembling orthogonal polynomials. (b,c): During training, features evolve such that single feature (red) fits both the target and learned functions (grey and green). (d,e,f): Our visualization metrics defined in Eq. (5) at different epochs. (d) and (e) show that fewer features express the target/learned function, and (f) shows that only few features are significant after training.
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D
THE RELATIONSHIP BETWEEN THE MP-OPERATOR TMP AND NEURAL COLLAPSE
(NC) refers to state of DNN when the feature vectors of the training set and last layer weights form a symmetric and clustered structure at the terminal phase of training (TPT) when trained past the point where the training error vanishes. The structure has been studied mainly in the Unconstrained Feature Model (UFM) (Mixon et al. 2020) and has sparked theoretical studies. See Kothapalli et (022) for a review.
Neural collapse PPapynn al. 2020
NC is defined by the emergence of four interconnected phenomena upon TPT: NCI) collapse of within-class variability, NC2) convergence of features to a rigid simplex equiangular tight frame (ETF) structure, NC3) alignment of the last layer and features, and NC4) simplified decision by nearest class:
1. Within class variance tends to 0
EfE +0,
(26)
2. Convergence to simplex ETF
(us (T) (KL -b)
Cō.y -1 C
(27)
(Au - (2) - 12)12
3. Convergence to self duality
W2
14 F
(14 1l O,
(28)
keg 2
4. Simplification to nearest class center
arg mggmax,4(6c + b, - arg arg min (1(()) 14ll2
(29)
D.1 PROOFS OF PROPOSITION AND
For completeness, restate the conditions. We assume the true distribution q equals the empirical distribution of the training set - the expectation Eg (and bra-ket notation) is over the training samples. We assume balanced classification and define A, the set of training samples with class label i. We assume the learned function is a perfectly classifying indicator function, giving
0 : & A
fx(x)
(30)
1
: A
The feature class mean vector for label i
C Σ Φ(x) = C (j.(o)
#4 = Eze a, [D(x)]
(31)
n
zEA,
Note that (f (b) Ro is vector with (f(() 1(f( ,....: (fi b.)) The global mean vector is
Σ 14 ( Σ = (110)
(32)
C
The feature intra (within)-class covariance Σw E Roxp is
Ew
Σ Eres, [((x) - -
-x) m
(33)
The feature inter (between)-class covariance Eg E RRX is
Σg = 1 C 0
5Cun (m4 p) - -
(34)
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Lemma 1. For the conditions given above. iff (x'')" " Ezzy' [+(xf((x)) is TMP, then
(xx)² (m= pt) (2((f((zz) -).
(355
Proof. By the definition of indicator functions equal partitions,
(filf)) -
1
Σ 2m
1 (1lf)) C
(36)
It follows that Cf2 is orthogonal to 1:
(1 - 1) = 0. We can express the function (xx') -T)
(37)
(p)))) :2 - R in terms of
((x') =0<''² (
(011) = E. -)]
(38)
(39)
- - 1]('',,
(40)
where used the definition of the means (Eqs. (31 and 32 in the first line and the definition of T
(Eq. (T) in the last line. As T is TAP,, we obtain
((a')² (mu - (b) TMr(Ch-
(41)
= ==(((f(z') 1),
(42)
where we used Eq. 37 and the definition of TMP (Definition Corollary 1. In the setup of Lemma
we have
p (Hg - (i) = aC(i 1).
Proof. From the definition of class mean (Eq. 3 and Lemma
1 (ps p.) Eg
ci)(z')(('') (m: (
(43)
E (Cf)(z")as(Cfita) -
(44)
ac((0 - 1),
(45)
where used that E. (f) === C-
Proposition 1. I is an MP -peoperator, then the NC1 condition (collapse of within-clas class variability)
DiE 0 holds, where inter-class covariance matrix E., and intra-class covariance matrix Ew are
C
Mi = (m3 -m) - p.tTT and Ew Σ Eres, [(((z) - pz)((()) - p1)
1=1
Proof. We will show that Tr(E,Ew) O. which automatically proves the proposition as covariance matrices are positive semi-definite. We have
Tr(E,Ew) = Tr Σ(e (p) - 11)" Σ EgE (0(x) - ())((()) - 14)²)
(46)
Σ ErEA ((0(z) - -1) - -m))
(47)
Σ ETEA ag (Cff(z) 1 Cd.s + +1)
(48)
(B(0) - -515 )r.
= c3c² Σ EVEA,
(49)
1
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where we used linearity of trace, sum, expectation in the first line and Lemma in the second line.
Expanding Eq. (49).
Tr(ExEw) a²
Σ
EREA, (1-28%) (1=201))f(()) 8ig)
(50)
1,
a2. Σ
(ō:g 2)tz)
(51)
i,j
= 0,
(52)
where we used that learned functions are indicator functions in the second line.
Proposition 2. I is an MP-operator, then the NC2 condition (convergence of features to simplex equiangular tight frame) holds:
(1z - -
diz T.
Proof. From the definition of global mean (Fq.
(32) and Lemma 1
n² (m4 - (3) = Eg (0xx') - [i)
(53)
= Eg (22(Cf((x') - 1)]
(54)
= O.
(55)
Using Corollary I and Eq. (55).
(14-1)) p)) Ca2(ó1 )
(56)
D.2 THE FIRST FIGENFUNCTION, THE CONSTANT FUNCTION, THE ANALOG OF GLOBAL MRAN VECTOR IN SIMPLEX ETF
Here, we discuss the constant function, which, by definition of TMr (Definition Tp. is the largest eigenfunction if 'T is Tur. In all experiments except Fig. 9 the largest eigenfunction is the constant function throughout training, with (1(ei) > 0.95. For our visualization, we ignore the first eigenvalue from analysis as the constant function is removed in CKA (Eq. 77)
In the NC2 condition, the k-simplex ETF structure a set of & orthogonal vectors projected on dimensional space along the complement of the global mean vector. For example, projecting [0, 1], [0, 1, 0j. and [1, 0, 0) onto the orthogonal complement of 1 3 1.l (the global mean vector) gives three vertices [- 3: 3: ] 1 3. -1) 13. 3.
1
- E which forms an (ordinary plane) equilateral triangle.
Our operator TMr (Definition [ has analogous structure, where it is (scaled) projection operator (02 P2)) except along the constant function. The C'AAA A (Eq. (7)) also measures the alignment except along the mean direction (for matrices) or the constant direction (for operators). This allows neural networks to have any arbitrary eigenvalue for the constant function, yet perfectly align with DLR 0.
When T is TMP, the definition of TMP (Definition) I trivially shows that the largest eigenfunction is the constant function. A peculiar observation is that the first eigenfunction of T remains the constant function ((1 (e1) 0.95) throughout training all our experiments (except Fig. 9). This pattern also appears in hierarchical datasets (Saxe et al. 2022), suggesting that natural data may consistently prioritize the constant function as the dominant mode warranting further investigation.
D.3 GENERALITY OF OUR METRIC BEYOND NEURAL COLLAPSE
As discussed in the main text, metrics defined in terms of functions (random variables) while neural collapse is defined by the feature vectors the training set. Working in the function space, we measure the quality Eq. (5) which is inaccessible from formalism using training feature vectors.
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Another important difference is that focus on the learned function instead of mean class label vectors. This allows more independence beyond neural collapse, allowing the method to be applied on broader set of tasks: tasks with orthogonal and isotropic target functions form a broader class than balanced classification tasks. Regression task with scalar output an example.
In Fig. T0. train ResNet18 MNIST regression task with scalar output (f* X - IR), where a digit is correctly predicted if 0.5 < f()) < + 0.5. Unlike C-way classification, one of
which typically uses C' features, the regression model reaches the rich regime with just which is constant. Neural collapse formalism, focusing on class labels, does not readily extend to regression tasks, demonstrating that our function-based formalism more general, indepednenn and applicable beyond classification.
regression (98.7%), 4.9e-2]
classification (99.7%), 4.4e-3]
1.00
1.00
10-1
0.75
0.75
TT(A)
0.50
0.50
E 10-3
0.25
0.00
0.00
10-5
5
15
20
10
20
10a
102
(a) Target projection
(b) Self projection
(c) Eigenvalue
Figure 10: Rich dynamies for regression problem with scalar output. ResNet18 was trained on MNIST via regression (bluc) and classification (orange). For regression, the target/learned functions are scalar output functions, and an image z with digit correctly classified if i 0.5 fxx < i 0.5. It can be seen that both models are in the rich regime, using the minimum number of features.
Finally, the independence from the class labels allows another benefit of being applicable during training. Neural collapse is defined only during the terminal stage of training and assumes 100% classification. measure without perfect classification can lead to unstable values, typically rising from the pscudo-inverse of Eg (See also Section(H). The NC1 measure shows dramatic change in logarithmic scale for experiments Fig. [[aabb) while our measure is more robust to such diffcrences.
D.4 ISOTROPICITY ASSUMPTION
As stated at the beginning of Section 2 and in the limitation paragraph, paper's scope is isotropic outputs. This is slightly weaker assumption compared to prevailing standards in neural collapse literature (Papyan al. 2020 Mixon et al., 2020 Han et al. 2022] Jacot al.) 20255) as discussed above. Because simplex ETF structures rely significantly on the symmetry, proposing plausible extension for imbalanced tasks is active area of research on its own (Fang al. 202 Yang et al. 2022 Yan et al. 2024| Gao et al. 2024). For example, imbalanced targets suffer from the minority collapse where multiple classes collapse to a single vector for finite dataset (Fang et al.) 2021). This becomes particularly a problem for long-tailed target (e.g., natural language dataset), which is also an active area of research (Gao et al. 2024).
As another challenge, the eigenvalue distribution of features now depends many factors such as weight decay. Assume diagonal last layer with p C. Assuming anisotropic target fī (x) (zex(z), the intensity dş either come from the | (x)]l or the last layer weight wx. In other words,
as (0)l))xxx)))
(57)
For strong weight decay, the weights (wi,..., Wp become more isotropic and anisotropy are absorbed in [l (x) For two-layer diagonal linear networks in rich regime with isotropic input (xg ~ (0, 1)), We x (x)1 where anisotropy distributed evenly between the weights and features. Since metric compares ((x) to f (x)), these differences, though yielding the same output, returns different levels of dynamical richness. But then, what representation shapes are favored by rich dynamics? While symmetry from an isotropic target and normalized CKA have let bypass this issue, introducing anisotropy and factors such as weight decay complicates the dynamics and requires more concise understanding of the rich dynamics.
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E CALCULATING THE METRIC AND VISUALIZATION IN PRACTICE
E.
METRIC
In the main text, we used function and operator formalism. In practice, we must use finite approxima- tion. We can use the CKA equation Eq. 7
CKA(A, = Tr(((A)((B)
(58)
([)))
but feature matrices for A and B instead of operators. Let A E RRxx be feature vectors (of the penultimate layer) of n samples and /3/² E Rex= be output vectors (of the network) for samples. We can then approximate CKA by inserting A = (A4/3)² A1/2 and B = (BL/2)² B1/2 into the above equation. Because we already have square root matrices, the computation cost of CKA is (rppc).
Recall that n can be any number, and can be sampled from both train and test set. As and are at most rank p and respectively, can expect (ppcc computation. In an author's laptop, it can be computed less than a minute for all experiments. For NTK, p is no longer the width of the last layer, but total number of parameters, quickly becoming infeasible.
suffices in practice. Because 10 in many models,
E.2 VISUALIZATION
Time complexity. The algorithm is computationally efficient in practice. The main cost in comput- ing Eq. (6) is the SVD, typically with time complexity Opp²). A good approximation requires only n (p) datapoints- than the last layer width. Performing SVD on é instead of dØ yields U and VS with O(p²) cost. Because last layers are typically narrow (c.g., p ze 10²) and the algorithm requires just = (p) forward passes, the overall computation is lightweight. In author's laptop, the computation for all experiments takes less than a few minutes.
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F EXAMPLES OF OUR VISUALIZATION METHOD
In this section, we report additional findings made with our visualization tools.
o= 10-1 (92.7%), DLR = 4.9e-2]
a=2-10" (92.4%), Die =1 1.le-11
a a-2-101 (88.3%), DER - 5.6e-1]
1.0
1.0
10
0.8
11(30
0.8
10
0.6
0.6
10-1
10
30
20
10a
10a
10a
(i) Target projection
(ii) Self projection
(iil) Eigenvalue
Figure 11: Lazy/rich transition in target downscaling, illustrating how lazier dynamics (ii) use more features and (iii) show slower decay of eigenvalues (ignoring Pl of constant function, see Section
epoch 200 (5.2%).. [5.1e-1]
epoch 3000 (99. (9)). 1.le-1]
1.00
1.00
10-1
0.75
0.75
0.50
0.50
& 10
0.25
0.25
0.00
10->
100
200
120
160
200
100
200
(1) Target Projection
(ii) Self Projection
(ii) Eigenvalue
Figure 12: Lazy/rich transition in grokking. A 2-layer transformer is trained on the modular division task. The inset the middle shows the training and test accuracies, where green and red vertical lines indicate before and after grokking (steps 200 and 3000). Our metric in square brackets shows the transition into the rich regime, while our visualization shows clear difference the use of features.
Low rank bias. Table 4 shows that low-rank inductive bias is strong in vision tasks, where the number of significant features always equals C' (the number orthogonal functions). See Eq. (9) in Section B.4 the (p) expression. We also observe that our measure DIR R is small in these setups.
Table 4: Metrics of ResNet18 and VGG16 trained on image datasets
model
dataset
Dess(p)
Dess(
Dr.R
test accuracy (%)
ResNet18
MNIST
10.1
10.0
4.4 10
99.7
VGG16
MNIST
10.0
10.0
1.8 10-2
99.4
ResNet18
CIFAR10
10.4
10.0
30-3
94.8
VGG16
CIFAR10
10.0
10.0
1.-
93.3
ResNet18
CIFARI00
99.9
99.7
10-2
78.3
VGG16
CIFARI00
95.8
99.4
7.4.10-2
71.9
Weight decay is not the source of inductive bias low-rank representation and smaller DLR- However, the model without any weight decay already shows sufficiently low-rank representations. suggesting that dynamical inductive bias, not the weight decay is the main driving cause of low-rank representations.
Fig. 13 shows that weight decay indeed aids
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wd=le-3 (94.8%), 3.1e-3]
wd=1e-4 (94.1%). [1.5e-2]
wd (93.5%). 5.0e-2
1.0
10
0.9
Ti(1))
0.9
0.8
10-3
0.7
0.8
10-3
100
102
10a
(i) Target projection
(ii) Self projection
(iii) Eigenvalue
Figure 13: Effect of weight decay rich dynamics. ResNet18 trained CIFARIO with varying weight-decay (shown relative value to the fixed learning rate 0.05). Larger weight decay leads to richer dynamics with smaller DLR, but the dynamics is already rich without the weight decay.
init (10.5%), (..ee11]
epoch (49.7%). 2.8e-1]
epach (75.6%). 99.0e-2]
epoch 200 (94.8%), 2.6e
1.00
1.00
0.75
0.75
0.50
0.50
3
0.25
0.00
0.00
10-1
20
(i) Target Projection
(ii) Self Projection
(fii) Eigenvalue
(a) ResNet18 CIFARIO
init (9.9%). 6.1e-1]
epoch 335).
[4.2e=1]
epech 30 (55.7%). 3.5e-1]
epoch 200 (55.4%)) 4.8e-1]
1.00
1.00
10
0.75
0.75
0.50
0.50
0.25
0.25
15
20
10
15
20
10
15
20
Target Projection
(ii) Self Projection
(iii) Eigenvalue
(b) MLP CIFAR10
Figure 14: Role of architecture in the richness of the dynamics. ResNet18 8 (a) and 4- MLP of width 512 (b) are trained on CIFAR10, and their metries are shown different epochs. ResNet18 concentrates the metrics on the first 10 features after just one epoch, which persists until the end of training. contrast, MLP shows a less dramatic concentration on the first 10 features and deviates from the rich dynamics around epoch 30 when it begins to more features.
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Role of architecture. As studied in Saxe et al. 2022), the architecture and dataset pair can influence the performance even when dynamics are greedy (race toward shared representation) In Fig. [1] examine the dynamics of ResNet18 and 4-layer MLP trained CIFAR- 10. In Fig. 14 a), the
training dynamics restricted to the first 10 significant features, consistent with theoretical work in linear neural networks. In Fig. 14),), the training is mainly focused the first 10 features, but observe that more features are used as training progresses, more similar to the dynamics of linear models. We speculate that the lack of ability to feature learn in the earlier layers leads the of additional features and thus leads to lazier training dynamics.
showed example where rich dynamics led to poorer performance. Fig. 16 showsws a similar experiment in which both rich and lazy models achieve near-perfect performance (> 95% test accuracy). Our visualization clearly shows the different usages of features even when the performances are similar.
Rich dynamics * generalization. In Fig,
Rich dynamics without underlying data structure. Fig. 1.5 illustrates that rich dynamics can occur independently of representation enhancement or underlying data structure. We trained ResNet on CIFAR-10 with varying levels of label shuffling and observed that even with fully randomized labels, the model enters the rich regime. This suggests that the dynamical low-rank bias is strong enough to collapse expressive networks into minimal representations, consistent with prior observations of neural collapse under random labeling (Zhu et al. 2021 Hui et al. 2022).
no shuffle (95.0%). 3.1e-3]
10% shuffle (66.1%). [422e-2]
random labels (9.5%), 3.4e-2)
1.00
1.00
10-1
0.75
0.50
0.50
10-3
0.25
0.25
0.00
0.00
10-5
5
10
15
20
5
10
15
20
10a
10
102
10a
(i) Target projection
(it) Self projection
(ii) Eigenvalue
Figure 15: ResNet18 on 10a CIFAR-10 datapoints with shuffled labels. Models are trained with 0% (blue), 10% (orange), and 100% (green) label shuffling. As shown in the square brackets, all cxhibit rich dynamics with DLR < 0.1. Visualizations in (ii) and (iii) confirm the use of the top 10 significant features, but varying feature qualities (i) lead to varying test accuracies. shown in parentheses.
Last layer only (96.8%), 5.1e-1]
Backpropagation (99.1%),, 4.3e-2]
1.00
1.00
10
0.75
[T(K)
0.50
0.50
0.25
0.25
0.00
10-5
10a
101
10a
10a
109
10
1
10a
10"
101
10a
10a
(a) Target projection
(b) Self projection
(c) Eigenvalue
Figure 16: Generalization does not imply rich regime. A CNN with width p 1024 is trained on the full MNIST dataset using two dynamics: last-layer-only training (lazy) and full backpropagation (rich). Although both dynamics achieve test accuracy above 95%. they lead to dramatically different use/significance of features: the metrics concentrate on the first 10 features for the rich regime, and they more evenly spread for the lazy regime.
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Cross entropy loss. Fig. 17 shows that our method extends to models trained on cross-entropy loss, while the mathematical interpretation is less straightforward.
init (10.5%). 55.6e-1]
(45.3%), 4.0e-1]
epech (78.8%), 2.6e-1]
epoch 200 (95.1%), 2.9e-2]
1.00
1.00
10
0.50
0.50
10"3
0.25
0.25
10
10
15
20
10
15
20
10
15
20
Target Projection
(it) Self Projection
(ii) Eigenvalue
Figure 17: Training with cross-entropy loss. We apply our visualization method to a ResNet18
trained on CIFAR-10 with cross-entropy loss. Similar the MSE-trained model, it achieves small value of LR and shows concentration of feature usage and significance See Fig. 2[a) for comparison. The notable difference in the speed difference among the first C' eigenfunctions. In MSE, the first C eigenfunctions show nearly isotropic growth from epoch 1, while for CE, there is a notable difference in (T(k) and depending on until epoch 5. This may arise from the non-linearity coming from Cross entropy. Insets display all 512 features rather than just the top 25.
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G RELATED WORKS ON VISUALIZATION
Figendecomposition has long been a standard tool in science and engineering (Buckley Xu
1990 Amadei al. 1993 Vrugt Wittkowski 2020), demonstrating its value for visualization. Building on this, related empirical works have appeared (Raghu et al., 2017: Morcos et al. 2018
Korbliih et al. 2019). For example, Singular vector canonical correlation analysis (Raghu al. 2017) resembles eigendecomposition but focuses on performance along leading directions rather than on the number of features used to represent the learned function (i.e., lacking IT). (Morcos) al. 2018 Kornblith et al., 2019) compare different networks, whereas our focus is on before- and after-last-layer activations. Importantly, these approaches were not developed visualization tools for analyzing rich dynamics.
Similarly, visualizations related to neural collapse use eigendecomposition (Papyan 2020). but are inherently class-dependent, whereas our metric and visualization are glass-independent. See Section D.3 and Fig.
T0
The closest line of work is kernel-based visualization (Bordelon et al. 2020 Canatar et al. 2021
Simon et al. 2023), especially (Canatar & Pehlevan, 2022). However, their visualization focuses on cumu from performance, providing a complementary perspective. Additionally, the rich dynamics in the literature refers to change in NTK, not the collapse (low-rank bias) of feature kernel.
cumulative latve power and the eigenvalue spectrum, whereas ours additionally separates dynamics
While visualization is part of contribution, main contribution lies in introducing Din as met- ric of richness and presenting supporting empirical results. Visualization serves as a complementary tool illustrate the dynamics and facilitate future studies.
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H
EMPIRICAL DETAILS
Here, we share information on our practical setups and their statistical significance. The code can be found
H.1 STATISTICAL SIGNIFICANCE
We reproduce tables from the main text but with the addition of one standard deviation values. In the main text, all error bars in the plots represent standard deviation over least three runs.
Table 5: Statistical significance of Figure 1(b)
Full backprop (Rich)
Last layer only (Lazy)
Train acc.
100 (+00%
98.7 (=:011) %
Test acc. f
10.0 (+4.3 10-2)%
74.4 (+0.19) %
DLR
8.7 10 (#4.0 10 10-4)
6.3 10 (#3.22 10-3)
Table 6: Statistical significance of Table
Epoch
0 (init)
200
Train Acc.
10.0 (+0.8)%
10.2(=0.27)%
Test Acc.f
9.85 (=1.0)%
10.0(=0.33)%
D DLR
0.59 (#1.6 10-2)
1.0 (16.4-
Sinith
1.0 (=0)
0.20 (#1.1 10-2)
3.1 -10 (#3.5 - 10a)
2.2-10 (+2.5 10
NC1,
1.2 10 (+7.2 -)P)
7.5 10-14 (+1.5- 10-14)
Tr(Ew)
15.6 (+1.4)
7.1 10 1+1.1.10-11
Table Statistical significance of Table
2 10-1
2 100
2-10' 10a
Train Acc.t
100 (+00%
100 (+0)%
100 (+0)%
Test Acc.f
92.7 (2..441-11--2) 10-20%
92.4 (=5.2- 10-2)%
88.3 (#3.7 10-1)%
D DERV
4.9 10-2 (+4.4 10-3)
1.1 11-110-1 -199.-10099
5.6-10-2 (#6.1 10-2)
Smark
10-2 (#2.1 10-3)
4.1 10-2 (+4.5 10-3)
5.2-10-2 (#1.8 10 10-2)
3.4 10 -
3.2 10 (#4.4)
3.2 10 (+2.0)
NCW
2.3 10 (#4.1 109)
3.2 10 (#4.3 10
8.1 112 (#1.4 10²)
Tr(Ew)4
2.0 (+0025)
3.1 (+2.3 10-2)
1.2 10-1 (+1.2 10-2)
H.2 DISCUSSION ON NCI MEASURE
In Tables 6 and the NC1 measure is numerically unstable due to the pseudo-inverss of Eg. For stability, we used 2 Rs (E, + 10-47)t. In addition NCI, we consider a similar measure Tr <<Ew>> Papyan et al. (2020) for collapse of within class variability, which is also reported in Tables and The measure Tr(Ew) also shows a similar trend to NC1.
H.3 DATASET DETAILS
We publicly available datasets, including MNIST Deng (2012) and CIFAR-10/100 et al. (2009) from PyTorch Paszke et al. (2019). and mod p division task from //tt://///iiii com comm/taddykoker//rokking
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Table Statistical significance of Table
Task
Architecture
Condition
DLR
Mod
2-layer
Step 200 (before grokking)
0.51 (+0.04)
97
transformer
Step 3000 (after grokking)
0.11 (#0.01)
learning rate 0.005
0.053 (#0.01)
CIFAR-
ResNet18
learning rate 0..5
0.025 (#0.004)
100
learning rate 0.2
0.039 (#0.007)
CIFAR-
weight decay 0
0.05 (#0.003)
ResNet18
10
weight decay 10
0.015 (#0.003)
weight decay 10-3
0.003 (+0.001)
CIFAR-
ResNet18
0.026 (10.003)
MI.P
0.48 (+0.07)
label shuffling
0.031 (#0.004)
CIFAR-
ResNet18
10% label shuffling
0.042 (+0.003)
10
full label shuffling
0.034 (#0.004)
full backpropagation
0.043 (+0.005)
MNIST CNN
last layer only training
0.51 (#0.008)
CIFAR-
without batch normalization
0.66 (+0.01)
VGG-16
100
with batch normalization
0.073 (=0.002)
H.4 MODEL DETAILS
Our model implementations are based on publicly available code assets, including VGG16 from PyTorch Paszke al. (2019), ResNet18 from Nakkiran al. Nakkiran et (202I). and the modular division task from ht :
grokking
As described in the main text, we only trained our models on MSE loss where the targets are one-hot vectors (up to scaling constant). The constant a is used to scale the output + y/a.
Table 9; Dataset details
Figure
dataset
training
a
batch size
sample count
Fig.
Encoded MNIST
60,000 (all)
1/3
128
(Section H.4.1
Table
MNIST
60,000 (all)
128
Fig.
CIFAR-10
1/3
128
Fig.
C1FAR-100
50,000 (all)
1/10
128
Fig.
CIFAR-100
50,000 (all)
1/10
128
Fig.
MNIST
1,000
128
Fig. 12
mod-p division
4,656
512
Fig.
15
CIFAR-10
10,000
1/3
128
H.4.1 ENCODED MNIST
In Fig. [ we encoded the labels as one-hot vectors on the first 10 pixels of the MNIST dataset. For the training set, encoded the true labels, but for the test set, encoded random labels.
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H.4.2 CIFAR-10/100
For CIFAR-10/100, we use standard augmentation using random crop 32 with padding and random horizontal flip with probability 0.5. We also use standard normalization with mean [0.4914, 0.4822, 0.4465] and standard deviation [0.2023, 0.1994, 0.2010] for each channel.
Table 10: Training details
Figure
optimizer
learning rate
momentum beta
weight decay
epochs
learning rate scheduling
Fig.
Adam
1 10-3
[0.9, 0.999)
100
None
Table
SGD
1-10-6
10
200
None
Fig.
SGD
5-10-2
0.9
5 5-10-5
200
x0.2 per 60 epochs
Fig.
SGD
1-10-2
0.9
1.10-
400
None
Fig.
SGD
1--0-
0.9
5.10-5
400
x0,2 per 60 epochs
Fig.
Adam
1.10-3
[0.9, 0.999)
0
100
None None
Fig.
12
Adam
1-10-3
[0.9, 0.98]
4000
Fig.
15
SGD
5 10-
0.9
50
200
x0.2 per 60 epochs
H.5 COMPUTE RESOURCES
The models ran on GPU cluster containing RTX 1080 (8GB), RTX 2080 (8GB), RTX3060 (12 GB), and RTX3090 (24GB). The typical time to train model is 2 hours, but it varies from 10 minutes to hours depending on the experiment. The evaluation metrics take less than minutes and may need up to 2GB of CPU memory.
H.6
USE OF LLMS
Large language models (LLMs) were used to polish writing to make the paragraphs more concise.
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I INTERMEDIATE LAYER FEATURES
Because our paper focuses on a practical measure of dynamical richness rather than explaining the rich dynamics themselves, restricted our study to the computationally efficient last-layer features. Nevertheless, to encourage future work on rich dynamics using our approach, we include preliminary explorations of intermediate layers.
We can generalize our metric and visualization by replacing 7 in Eq. ① with that of the 1 layer:
PL
T
(1)
(59)
where ((1) is the 1th layer's feature map and pr is width. When the output the feature map $ is multidimensional (e.g. the output from 2D convolutional module), the output is flattened to vector (similarly to Montavon et al. 2011 Alain & Bengio (2016); Raghu et al. (0077))
However, several challenges arise when considering features from earlier layers. First, the inner product (correlation) between intermediate-laye features no longer reflects their alignment with the output (or with TM since these features undergo nonlinear transformations before reaching the last layer. This limits the interpretability of CKA, II", and íi, all of which rely feature inner products. Still, we can predict how linearly correlated these features are with the output or TMP.
expensive. Whereas last-layer features typically narrow (~ 109), intermediate layers are often much wider, and our methods scale quadratically with width. Larger width also leads finite-size effect from the dataset, which we discuss in Section
Second, computation becomes
With these caveats in mind, we present the behavior of intermediate-layer features. Because the metric lacks clear theoretical interpretation in this setting, we provide only visualizations. Fig. T8 shows ResNet18 8 trained on CIFAR 100, which exhibits rich dynamics under our metric (Table 3). While we observe depth-wise transition from using many weakly aligned features to using fewer but more aligned ones, it is difficult to discern lazy/rich dynamics from the intermediate layers alone. The effect of learning rate shown in Fig. but the intermediate layers again exhibit no additional information compared to the last layer.
Fig. 20 corresponding to Fig. 3 shows the lazy-to-rich transition in VGG16 with and without batch normalization. With batch normalization, the model follows the use of many weakly aligned features to the use of fewer but more aligned ones depth. Without batch normalization, we observe no comparable shift. Moreover, for the model without batch normalization, layers 3-6 show little variation, suggesting that the convolutional layers learn minimal feature structure and that training primarily affects the final fully connected layers only. Yet, the difference between lazy and rich dynamics most pronounced for the last layer.
layers:
last
10a
1.00
1.00
0.75
0.75
0.50
10-4
0.25
0.25
0.00
0.00
100
101
102
10a
100
10:
102
10a
10"
101
102
10a
(a) Target Projection
(b) Self Projection
(c) Eigenvalue
Figure 18: ResNet18 on CIFAR100. Colors represent the features of the intermediate and the last layer. See Fig. [2 for the definition of the layers. As the layers go deeper, we observe transition from many less correlated features fewer more correlated features. The purple line ends early as the last layer only has 512 features.
As shown in Figs. 19 and 20 the low-rank bias most pronounced in the last layer, enabling our computationally efficient richness measure in the main text. This motivates the hypothesis that the
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layers:
3
4
last
(r=0.))74%)
Ir:0.))778%)
0r=0. (..00)))6))
1.00
1.00
10
0.75
0.75
(1)
II(A))
Pui01
0.50
0.50
10-3
0.25
0.25
0.00
0.00
10-5
100
10
102
10a
109
10
10a
10a
k
(i) Target projection
(it) Self projection
(ii) Eigenvalue
Figure 19: Effect of learning rate for ResNet18 on CIFAR100. The models corresponds to Fig.
layers:
5
last
bn(72%)
no bn(22%)
1.00
1.00
10
0.75
0.75
IT'(K))
11(e)
Pai0
0.50
0.50
10-3
0.25
0.00
0.00
10-5
10"
10
102
10a
10a
10a
10a
k
(a) Target projection
(b) Self projection
(c) Eigenvalue
Figure 20: Effect of batch normalization in VGG trained CIFAR100. This is the intermediate values for Fig. 3 Solid and dashed line indicate the model with and without batch normalization, respectively.
last layer consistently exhibits the strongest low-rank bias. To test this, we examined the greedy layerwise training method (Ivakhnenko & Lapa 1965). which trains each layer by appending an auxiliary classifier, freezing the layer after training, and then repeating the process for subsequent layers.
Fig. [T shows that greedy layerwise training, compared to full backpropagation, causes feature collapse in earlier layers (e.g., layer 2), indicating that the last layer carries the strongest low-rank inductive bias. This supports our focus on last-layer features for lazy/rich detection. Under greedy layerwise training, feature quality also stagnates after layer 2, leading to poorer overall performance. This observation suggests that such early collapse dynamics may be suboptimal, warranting further theoretical study.
layers:
last
backprop(944g) -. greedy Abyerwise882%)
1.00
1.00
10
0.75
0.75
(0(0))
0.50
10-3
0.25
0.25
0.00
0.00
10-5
10
101
102
10a
10"
101
102
10a
10a
10a
102
10a
(a) Target projection
(b) Self projection
(c) Eigenvalue
Figure 21: Greedy layerwise training leads collapse of features in earlier layers. In contrast full backpropagation (solid), greedy layerwise training (dashed) leads to more pronounced alignment lower for earlier layers.
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1.1
LAYER DETAILS
Fig. shows the layers defined in this section. The notation k x Conv( refers to the 2d convolutional module with kernel of size k x k, and arguments within parenthesis refer to the input channel, output channel, stride, and padding respectively. Linear(:..) refers to the fully connected module with the arguments referring input width and output width respectively. MaxPooll.....) refers to max pooling with the arguments as kernel size, stride, and padding respectively. Adaptive Avgpool(:.) refers to adaptive average pooling where arguments are the size of the image after pooling. The curved arrow refers to the skip connection (identity), and any modules that appear in the arrow are applied. BN refers to batch normalization and (BN, ReLU) refers to the application of batch normalization and ReLU in respective order.
X
S0 2013
C
Sreed
(a) ResNet
(b) VGG16
Figure 22: Architectures and layers The diagrams show the CNN architectures used in the study and their layers. (a): ResNet18 architecture. The size of feature space (post-activation) for each layer 65536, 32768, 16384, and 512 respectively. (b): VGG16 architecture. The feature space size for each layer is 16384, 8192, 4096, 2048, 512, and 4096. Note that the penultimate layer only consists of linear modules (fully connected) and cineaiicasssies.
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J ACCURACY OF APPROXIMATION METHODS
In Section 5.1 we proposed three vector metrics for visualization: quality Q", utilization Q and eigenvalues (or intensity) p. These measures require the true input distribution g. which we cannot access in practice. The approximation method for these measures was described in Section E the Nyström method [aker & Taylor (1979) for approximating 6x- In this section,
provide an empirical example where the true eigenfunctions and eigenvalues are known to assess the validity of Nyström method in approximating eigenvalues and eigenfunctions. We also discuss why quality and utility were presented in a cumulative manner (II" and IT) while the eigenvalues (pt) are presented directly.
J.1 THE NYSTRÖM METHOD
As discussed in Section B the Nyström method diagonalizes the empirical feature matrix ((X) E RPxR to approximate the eigenfunctions eg and eigenvalues Pk. This is more explicitly written in lines 6 and 7 of Algorithm[]
Pb - kk
ex(x) +- (
(60)
where and are the left orthogonal matrix and singular values obtained from singular value decomposition ((X).
J.1.1 EXPERIMENT SETUP FOR TESTING NYSTROM METHOD
The accuracy of the Nyström method depends on the accuracy of singular values (sg) and left eigenvectors (ug) of random matrix (X))). To empirically assess the accuracy of the method, we will use p independent Gaussian random variables to sample our feature matrix. power-law distribution with exponent will be used as the eigenvalues:
(x) NO, Diag((Pt, ,Pp))) Pk
(61)
Gaussian random variables serve as valid features since the features are random variables (æ: ~ is random variable, and eg is fixed map, thus ex(x), q is also random variable) and are orthogonal to other eigenfunctions:
ex (x) = Vps
(x))).
(eele)) = 8:j.
(62)
Figenfunctions can also be expressed with Oę as
cx(x) = (x) ok/ VPb
(63)
where O3 one-hot vector with its hth entry equal to 1 and all other entries (i.c. o =
-0.... 1,,10..... O)). Indeed can check that ek is the eigenfunction of T with eigenvalue
Pk
T
(2)) ok
/ (x))
(x')=
(00=/00 ((aa)(a())
D(x)Fon on g(x))d
(64)
Pk
Vph
(a')
(65)
=Pick:
(66)
where in the second line, we used that of a one-hot vector and all entries are independent.
J.1.2 ACCURACY MEASURES
With the underlying features and eigenvalues defined, propose measures for calculating the accuracy of approximated values obtained via Algorithm [
The error of eigenvalue is measured square distance between Pk (true value) and (X) (approxi- mation) normalized by Px:
(8
(X)
Ex
(67)
P
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Note that Ex is expectation over all possible features matrices ((X).
The error of cigenfunction is measured as minus the inner product between uy (X) E IP and og:
Ex [u[ox] ox
(68)
The inner product uz ok is proportional to the inner product between 6k
approx and ex:
ex) = /
((x)) (u(x))x(()) T
((x))o ok q(r)dr
(e
approz
(69)
(x
VPk
(x))
(()))) ((z)((z)) (()ddx ok
(70)
(X)
VPL.
un (x
Diag(p),... -pp)
pp) Ok VPk
(71)
(X)
(x)XXTo
(72)
sx())vP
where the denominator is not of interest as we are not particularly interested in the norm of the approz ek
J.1.3 RESULTS
In Fig. 23 we generate feature matrix containing 10, 000 datapoints and p 1000 features with Gaussian random variables. We use the Nyström method to approximate the eigenvalues and eigen- functions and assess their accuracy with the metrics in Eq. (67 and Eq. (68). We observe in Fig. 23 a)
that eigenvalues are approximated precisely except for eigenvalues with k = In Fig. 23 b). we observe that the accuracy of the eigenfunctions correlates with the respective eigenvalues: the larger the eigenvalue (compared to the median eigenvalue), the better the accuracy. In the main text, we are often interested only in the first few eigenfunctions with large eigenvalues; Nyström method is sufficiently accurate for our needs.
conclude that the
Ps:
1.00
1.00
1.00
14k- SVOC4)
0.75
Varlplp
0.50
0.50
0.50
0.25
0.25
0.25
0.00
0.00
0.00
100
10a
102
10a
10a
101
100
101
10a
k:
(a) eigenvalue
(b) eigenfunction
(c) function space error
Figure 23: Nyström upproximation for different eigenvalue distributions A feature matrix was sampled from Gaussian random variables and then approximated with Nyström method. Power law accurately
eigenvalue spectra with different decaying exponents were tested. (a) The eigenvalues predicted except for the tail eigenvalues with & R p. (b) The eigenfunctions are accurate only when the eigenvalues large compared the median eigenvalue. (c) The function space error below 0.25 for all k.
J.2 FUNCTION SPACE APPROXIMATION
The Nyström method's prediction accuracy degrades for smaller eigenvalues. The spanned space
of (eproro ,g approz] however, often accurately approximates that of le,, , ex]; the larger error of the cx approb is often due to the cumulation of error predicting e aphroo for < k. Fig. 24
illustrates example where [u1, U2 uz] span the same space as [01, 02, 0z), and ū, and u2 (thus e, and 62) are well approximated the errors in predicting u, and U2 accumulate to the error of predicting ug, and results in poorer approximation for ug.
41

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Published as a conference paper at ICLR 2026
V Vague
v 010
V
V
Figure 24: Approximation different eigenvalue distributions
We can use the Gaussian random variable setup to check the error of the function space approx-
imation. To measure the similarity between span[[e)," , and approximated function space
[eppproe ,, ,ep measure in Eq. ().
eapproz], we can use k-SvCCA (Raghu et al. 2017) or the cumulative projection
d:
h
(eslepprron opprony?
(73)
(=1j=1
Since express the inner product between true and approximated eigenfunctions with uy and og (Eq. (69 we can express the metric in terms of uy and og:
oo
(74)
where subtracted the measure from the measure will be when there is error. In Fig. 23 we observe that the function space can be approximated more accurately in comparison to individual eigenfunctions in Fig. bb::); This is why often present the quality and utility in cumulative measure IIo or IT.
J.3 ERROR IN PROJECTION MEASURES
We now discuss the error in the projection measures like II", The projection measure (Eq. 5 consists of the sum of squares of inner products between eigenfunctions and function interest (target or expressed). The Riemann sum over the test set Ste approximates the inner product - similar how test loss approximates generalization loss.
(ex, f Je 1
(() Tx (x(e)o(d)dd Ix f1()) Extrs ex(x01)2 ExWEs.. f
(Enves, es (z((
~
(75)
Using the Riemann sum, measure the correlation between spanned spaces (Eq, 5)) Using the vector representation of eigenfunction in the test set e; eppproa (test) E R.aee and the vector representation of function of interest (Xteat),, we can approximate the quantities of Eq.
opproo (a05)) Eig C, Σ, E
f((x01)
(eali)) Ti(EF"FET) j=1==1
FúJ
(76)
Σ F3
where z00) are iterated over the test set. If the rows of E E R.x and CC are orthogonal, then Eq. (6) is smaller than equal to 1: consistent with the definition of Eq. (
However, Eq. 76 susceptible to error when: rows of are not orthogonal and when p RJ lltest- We will discuss how we handled such problems in the paper.
J.3.1 NON ORTHOGONAL. FIGENFUNCTIONS
As discussed earlier, approximated eigenfunctions are less accurate when their eigenvalues are smaller; they may even be orthogonal in g the empirical distribution of the test set. The inner
42

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Published as a conference paper at ICLR 2026
product between two approximated eigenfunctions is
opprom
(pprror)
" (()()="" (x),(()))
(77)
8;8j
x
u?
(78)
8,88
ul Juj.
(79)
8,85
where the Fisher information matrix J E IRexp Karakida et al. 2021 is defined as Jx ((x) (x)""()))) (z){(r)dr Eq. (77) equals Kronecker delta if only if [1,-.. Upl are the eigenvec- tors of J with eigenvalues [st ...... .g). Since we obtained u, from the empirical feature matrix of the training set, the approximated eigenfunctions, especially those with small eigenvalues, are often not orthogonal q (and the test set).
To handle such an issue, we use QR decomposition on F [e] @ppprox (Xtest).".. eapproz (Xtest))
ipxxisett before applying Eq. (6)): because QR decomposition orthogonalizes the rows; preserves the span the first & vectors; handles overlapping vector between rows by conserving larger eigenvalue eigenvectors (eigenfunctions) which more accurate.
In Fig. 25 we plot the cumulative measure with and without QR decomposition for ResNet18 on CIFARIO. We observe that IT" and fl grow beyond 1 without QR decomposition, indicating that approximated eigenfunctions
not orthogonal; The inconsistent behaviour is removed when QR decomposition applied. Note that the deviation between QR and no-QR plots emerges when the eigenvalues sufficiently small (larger error approximating the eigenfunction).
layers:
2
last
QR (3c0mposed(34%)
base (no (R)(94%)
1.00
1.00
10-1
0.75
0.50
0.50
10-3
0.25
0.25
0.00
0.00
10
10
10a
102
10a
10a
10
10a
102
10a
104
10
101
102
10a
104
(i) Target projection
(ii) Sclf projection
(ii) Eigenvalue
Figure 25: Comparing approximation methods for the projection measures. ResNet18 on CIFARIO. For (a) and (b), the measures are calculated with (solid) and without (dashed) QR- decomposition. Note that QR decomposition removes the inconsistent behavior of the cumulative projection measures (e.g. larger than 1). With QR decomposition, the projection measure for layers with 2 1 consistently equals when Nitett 10a
J.3.2 FINITE SIZE EFFECT
For two sets of basis functions, increasing the number of bases to a large but finite value does not guarantee any overlap between the function spaces: because functions intinite-dimensional vectors. This is not the case for two mnite-dimensional matrices: If p Neest in Eq. (76) and E has full rank, then any F will spanned by the rows of E.
In Fig. 25 a.),, we observe that IT* (Ttes))) and TT(ttt are equal to for layers with p > Mestt- This does not indicate that intermediate features can express the target/expressed function; It is achieved because Thsest orthogonal vectors (E) can always span Nsesst dimensional vector space (and thus the row space of F)).
43


Reviews:
[{"review_id": "BT0id3HGQP", "reviewer": "Reviewer_GFrJ", "summary": "In this paper, the authors develop the ***low-rank measure*** \\$\\mathcal{D}_{LR}\\$ to measure the richness of the representations learned by a neural network, independently of its performance. Using it, and comparing it with the metrics from the existing literature, they can efficiently measure whether a model is in the *lazy* or *rich* regime.\n\nThe authors first introduce \\$\\mathcal{D}_{LR}\\$ theoretically, basing it on the concept of low-rank bias where in rich dynamics, the rank of the features learned by the model before the final linear layer should be similar to the dimensionality of the output, indicating well-separated classes. The metric itself is simple, as it is based on the CKA between the representations after the penultimate and the ultimate layer. It is therefore also inexpensive to compute.\n\nThey then compare their metric with other ones from the rich dynamics literature. Across two different test cases, they demonstrate it better illustrates rich-vs-lazy dynamics than other metrics. They further test their metric on realistic use-cases, showing with different examples how it represents richness, and where it is related (or not) to model performance.\n\nThe last part of the paper focuses on visualisations to explain shifts in the learning dynamics of neural networks. The authors separate three different metrics through eigendecomposition of kernel: cumulative quality, cumulative utilisation and relative eigenvalue. They demonstrate how they relate to rich dynamics.", "questions": "Q1: It is not immediately clear to me how exactly the metrics introduced in Section 5 relate to the *low-rank metric*. They are derived from \\$\\mathcal{T}\\$’s eigendecomposition and the learned/target functions, but I have difficulties directly linking them to \\$\\mathcal{D}_{LR}\\$.\n\nQ2: The authors mention two main limitations: focus on last-layer dynamics, and validity constrained to orthogonal and isotropic target functions. Regarding the second point: the authors mentions that “while this covers most classification tasks, a more general setup would be preferable”. Could the authors please explain in what cases their method would not work, and/or would not be theoretically justified, and give some examples? Also, are there practical scenarios where this limitation could lead to misleading conclusions if the metric is applied without caution? If relevant, including such details in the paper could help practitioners understand when to avoid or adapt the use of this metric in their work.", "limitations": "", "rating": 8, "confidence": 2, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "I think that this paper is a strong paper, which proposes a new metric for dynamic richness and demonstrates how it works better than other examples from recent literature.\n\nS1: The *low-rank metric* is simple to understand and inexpensive to compute. The authors give good intuition on how it works, why it works, and bounds on computation complexity.\n\nS2: Theoretical foundations for the *low-rank metric* look solid. Although the authors make some assumptions to make calculations tractable, formulations are very general and encompass a large portion of neural networks.\n\nS3: Empirical evidence covers the validity of the metric, comparison to existing other metrics from the literature, and practical use cases. The authors further provide statistical significance results. I think it strongly supports the main claims of the paper. Furthermore, the authors propose visualisation methods for increased interpretability of their results.\n\nS4: The literature review seems extensive, and most claims are backed through proofs or citations. This work is well contextualised within the existing literature.\n\nS5: I strongly commend the authors’ work on making their paper clear and easy to read. Writing and presentation are of high quality. It allows readers to more easily understand the theoretical framework around the development of the *low-rank metric*.", "weaknesses": "I have not found major weaknesses in this paper. The following points are either minor or nitpicks.\n\nW1: I feel like the authors should find a clearer name and/or an acronym to designate the *low-rank metric*, which is often designated as “the metric” or as its mathematical notation \\$\\mathcal{D}_{LR}\\$ throughout the paper. Such a name/acronym would make it easier to designate and reference in text, discussions and future work that will rely on it.", "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": "", "extra_scores": {}}, {"review_id": "4Gbi009MEJ", "reviewer": "Reviewer_ztAm", "summary": "This paper proposes DLR (Dynamical Low-Rank measure), a computationally efficient and performance-independent metric for quantifying dynamical richness in neural networks. The metric is grounded in the low-rank bias characteristic of rich dynamics and compares activations before and after the last layer via a functional kernel operator. The authors show that DLR reduces to neural collapse as a special case and empirically validate that it captures known lazy-to-rich transitions (e.g., grokking) without relying on accuracy. They further introduce an eigendecomposition-based visualization tool to enhance interpretability.", "questions": "- Typos: 049 \"a dynamical richness (metric?) that\", 373 \"Our visualizes (visualizations)\"\n- How does the metric behave with different loss functions beyond MSE? The cross-entropy results are mentioned but not thoroughly analyzed.\n- Metric sensitivity/ablation: More detail on estimation stability (sample size n for Nyström, dependence on width p, class count C) and hyperparameter sensitivity would help practitioners choose sampling parameters confidently.\n- The batch normalization finding (last row of Table 3) is correlational. The paper doesn't establish that batch norm causes rich dynamics or explain the mechanism.\n- The paper doesn't provide clear guidance on what intermediate DLR values mean.\n- Interpretability link not quantified: The eigendecomposition visualization is qualitative; quantitative metrics (e.g., subspace alignment or effective rank) could have strengthened claims of interpretability.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- Important problem: Decoupling dynamical richness from representation quality addresses a fundamental issue in understanding neural network training. The observation that rich dynamics ≠ better generalization (Figure 1) is compelling.\n- Computationally efficient: The O(p²C) complexity is a significant improvement over NTK-based methods, making the metric practical for modern architectures.\n- Strong theoretical grounding: The connection to neural collapse (Propositions 1 and 2) provides solid theoretical foundation, while extending applicability beyond classification (e.g., regression with scalar output).\n- Performance-independent: Unlike existing metrics (Sinit, parameter norm, NC1), DLR doesn't rely on accuracy, initial kernel, or class labels, making it more robust (Tables 1 and 2).", "weaknesses": "- Limited scope: The current formulation only applies to orthogonal and isotropic target functions. While this covers many classification tasks, the restriction is significant. The authors acknowledge this but don't provide a clear path to generalization.\n- Empirical validation: While the authors demonstrate DLR's utility across diverse settings (grokking, learning rate variations, weight decay, batch norm) the experiments conducted are of small scale and somewhat artificial. It would be interesting to see this applied to more modern scenarios, i.e. bigger models and more difficult datasets.\n- Last-layer focus: By only examining last-layer features, the metric misses dynamics in earlier layers. While this is a trade-off for efficiency, it's unclear how much information is lost.\n- Limited theoretical analysis: Beyond the connection to neural collapse, there's limited theoretical characterization of when DLR accurately reflects \"richness\" or what DLR values imply about learning.\n- Comparisons could be broader: NTK-based measures are set aside due to computational cost; although justified, a scaled-down NTK comparison on smaller models (which the authors already use) would strengthen claims that DLR is a better proxy rather than merely cheaper.\n- Clarity: While the method is well motivated theoretically and relative to prior work, the paper is occasionally difficult to follow. The bra-ket notation and operator formalism, while mathematically precise, may limit accessibility for a broader audience. The empirical approximations (Appendix E) are much clearer and could be introduced earlier to improve intuition. At times, the paper presents symbols and equations with insufficient context or motivation. Furthermore, the term dynamical richness is used somewhat loosely throughout; a dedicated subsection defining and contrasting rich vs lazy regimes would help.\n- Unexplored temporal dynamics: The paper treats DLR at convergence or snapshot points, but since it is dynamical, a time-series analysis (e.g., how DLR evolves over training) would provide stronger evidence of what “rich” dynamics actually look like.", "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": "", "extra_scores": {}}, {"review_id": "TMhYO85reC", "reviewer": "Reviewer_Czxk", "summary": "The paper proposes a performance-independent metric for dynamical richness (DLR) that compares the feature kernel built from penultimate-layer activations with a “minimum projection” operator determined by the network’s learned function. Intuitively, truly rich dynamics should learn only the minimal features needed to span the learned function space; DLR quantifies the gap to this ideal and is normalized in [0,1] (lower is richer). The authors show that when the feature operator attains this minimum form, neural collapse conditions follow as a special case, linking the metric to established phenomena. Empirically, DLR is lightweight to compute and tracks lazy-to-rich transitions across MLPs/CNNs/Transformers, and an eigendecomposition-based visualization complements the scalar score.", "questions": "1. Can DLR be generalized to non-isotropic/imbalanced tasks (e.g., via class-reweighted inner products or whitening), and does the neural-collapse link still hold?\n2. What happens if you compute DLR per layer (or per block) and aggregate? This could localize where richness emerges and inform architecture design.\n3. Can you theoretically connect decreasing DLR with transition conditions in lazy-to-rich analyses/grokking, beyond empirical alignment?\n4. Why CKA over alternatives (e.g., centered HSIC variants)? Any cases where CKA’s centering removes signal that matters for DLR?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. Originality. Recasts richness as low-rank alignment between features and the learned function via a principled MP-operator; this is a fresh angle relative to NTK-deviation or label-based collapse metrics. The reduction to neural collapse provides a clean conceptual bridge.\n2. Quality. The metric is simple, normalized, and computationally cheap, enabling use on standard vision models. Comparative experiments are thoughtful and reveal expected patterns.\n3. Clarity. The paper is well structured and the three-panel eigendecomposition plots (quality/utilization/eigenvalues) are helpful.\n4. Significance. A lightweight, performance-independent diagnostic for dynamics could become a standard tool, much as CKA did for representational comparisons.", "weaknesses": "1. DLR inspects only the final-layer features; rich dynamics might manifest earlier and be attenuated by a constrained head. Consider a hierarchical variant (layer-wise DLR or block-DLR) and show whether conclusions persist will be helpful.\n2. The technical background is not much sufficient for readers to capture the essence. More straightforward interpretation and introduction are helpful.\n3. The MP-operator and some guarantees rely on orthogonal/isotropic targets and supervised, one-hot settings; this narrows immediate applicability (e.g., class imbalance, multilabel, regression, self-supervised).\n4. Broader settings (imbalance/SSL), deeper ablations (layer-wise, NTK-feasible baselines), and stronger analysis of assumptions would raise confidence and impact.\n5. Writing/format polish. Minor issues distract: e.g., “batch nomralization” typo in Table 3; occasional spacing/notation inconsistencies around figures/equations; some insufficient definitions/clarifications on heavy-weight mathematical notations..", "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": "", "extra_scores": {}}, {"review_id": "pIj1LgdzSc", "reviewer": "Reviewer_cwga", "summary": "This paper introduces a new metric for quantifying rich (feature) learning. The authors motivate this by distinguishing between two perspectives on feature learning: a representation perspective (focus on usefulness of features for performance/generalization) and a dynamics perspective (focus on the transformation of features). They show that although it is often associated with representational usefulness, rich learning doesn’t necessarily lead to generalization and instead reflects an inductive bias towards certain solutions. Thus, the paper focuses on dynamic feature learning, introducing a computationally-efficient and performance-independent metric that effectively compares the network activations before and after the last layer. They show that this metric generalizes neural collapse, a phenomenon associated with rich learning. By comparing to prior measures of rich dynamics, they illustrate special cases where their metric captures dynamical richness, while others do not. They also conduct experiments across several different models and datasets, demonstrating how different training conditions affect performance and richness separately. Finally, the authors provide complementary visualization methods, make it easier to see the low-rank feature bias of rich dynamics.", "questions": "1. Would the metric still work if the target function is full-rank?\n2. Would be any principled way to applying the same approach to other layers in the network? And are there concrete settings where the comparison between the last two layers is a limiting assumption?\n\nSuggestion: there are a few typos but nothing major", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "The presentation and clarity of the paper is excellent. It is well-written, clear, and accessible. The paper is well-motivated, with a comprehensive introduction and background. \n\nThe soundness of the paper is also excellent. The authors take care to support each of their claims with ample evidence and solid experiments/methodology. The paper identifies an important distinction in perspectives on feature learning (dynamics- versus representation-focused) that challenges implicit assumptions about rich learning necessarily leading to improved performance/generalization and provides experiments to support this. The metric introduced has good potential for studying feature learning, as it is efficient to compute and captures rich dynamics in settings where prior metrics fail. The authors provide well-thought experiments to demonstrate the utility of their metric and support their central claims, showing that it captures grokking, neural collapse, and other known feature learning phenomenon. They also provide helpful visualization methods, which they show the utility of in interpreting different network behavior. The paper offers a good contribution to the ICLR community and I recommend it for acceptance. It was a pleasure to read and I thank the authors for their solid work.", "weaknesses": "To clarify, I think these points are relatively specific (possibly only applying to certain edge cases) and that the work as a whole still stands well despite some potential limitations.\n\nOne potential weakness of the metric introduced is that it’s limited in its generality (to orthogonal and isotropic target functions). However, this is properly acknowledged by the authors and the metric still captures many classification tasks. \n\nAs the authors also state, their metric depends on a comparison between the last two layers. Thus, two networks with different feature learning behavior in earlier layers may appear identical if their final two layers are similar. I’m wondering if there would be any principled way to applying the same approach to other layers in the network? And are there concrete settings where the comparison between the last two layers is a major limiting assumption?\n\nAs far as I understand, the metric is based on the low-rank bias of rich dynamics. I know that this might be rare in practice, but would the metric still work if the target function is full-rank? Is the assumption of low-rank dynamics potentially limiting?\n\nThe authors state that in Figure 5 “feature quality correlates with feature intensity during training, with larger features improving faster… the correlation between quality and intensity during training has not been previously observed or studied.” Isn’t this expected, as it’s known that large features (singular values) are learned first (Saxe et al., 2014)? Or, do you mean that eigenvector alignment is occurring simultaneously with singular vector scaling? In Atanasov et al., 2022, they show that for anisotropic data, “the NTK must necessarily change its eigenvectors when the loss is significantly decreasing, destroying the silent alignment phenomenon.” I’m a bit skeptical of the statement that this correlation has not been observed or studied.", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)