Most analyses of neural representations focus on trained endpoints, leaving the process by which internal structure forms comparatively unmeasured. We instead treat representation formation itself as an object of study, asking whether the path a representation takes through training reveals information that its endpoint does not. We formalize this idea by viewing representation learning as a geometric trajectory, using Bures geometry to measure movement through representation space and quantum Fisher information to characterize local sensitivity to input factors. We find that learning concentrates local sensitivity on task-relevant factors, and that changing only the prediction target reorganizes this geometry around the corresponding visual factor. More strikingly, representation trajectories become systematically less direct as the learned computation requires higher-order interactions among input factors. This effect persists under matched-performance and training-duration controls and transfers from synthetic vector inputs to residual CNNs on rendered and natural images. Importantly, formation history contains predictive information about the stability of the learned representation: controlling for interaction order, path tortuosity predicts its subsequent drift under continued training and sensitivity to parameter perturbations, whereas endpoint displacement does not. Finally, representation-aware optimization lets us actively shape these trajectories, showing that formation geometry is not only measurable but actionable. Together, these results establish representation formation as a new source of information about learning: its geometry reveals properties of the computation being constructed, predicts subsequent representational stability, and provides a target for intervention during optimization.
Figures & tables
Figure 1: Local and path geometry of representation formation. Left: training produces a trajectory of covariance states in Bures representation space, whose directness is summarized by tortuosity. Right: QFI measures which controlled input-factor perturbations induce the largest local representation movement.
Figure 2: Local geometry tracks task relevance. (a) Learning concentrates QFI on the two relevant synthetic factors. (b) QFI captures task relevance beyond raw activation sensitivity, measured by the representation-Jacobian baseline, without using the labels required by the loss-gradient reference. (c) On dSprites, changing only the prediction target reorganizes QFI around the task-relevant visual factor. Error bars are 95% confidence intervals.
Figure 3: Interaction order shapes representation path complexity. (a) Checkpoint spacings; (b) fixed-output matched-accuracy tasks; (c) dSprites CNNs; and (d) composite CIFAR-10 tasks at approximately matched validation performance.
Figure 4: Metric intervention. (a) Validation-matched paired comparison. (b) RepNG remains more direct than Adam as the shared parameter subspace grows. Error bars are 95% confidence intervals.
Appendix figures & tables11 assets
Supplementary material from the paper’s appendix.
Appendix
k
Test accuracy
LB
DB
τ
1
97.95[97.53,98.37]
3.831[3.618,4.044]
1.272[1.256,1.288]
3.011[2.843,3.179]
2
95.53[94.96,96.10]
6.041[5.699,6.382]
1.228[1.202,1.253]
4.929[4.594,5.263]
3
88.63[79.41,97.85]
8.525[7.103,9.946]
1.192[1.167,1.218]
7.132[6.033,8.230]
Appendix
Table 1: Fixed-output dSprites residual-CNN results over ten paired seeds. Entries are means with 95% Student- t confidence intervals.
Seed
k
LB
DB
τ
1
1
1.431
1.139
1.256
1
2
4.380
1.083
4.045
1
3
7.299
1.057
6.903
2
1
2.110
1.179
1.790
2
2
5.192
1.083
4.794
2
3
7.759
1.119
6.937
Appendix
Table 2: CIFAR-10 path geometry at the selected epoch-level checkpoints. Seeds are anonymized as 1–5.
Figure 6: Robustness. (a) QFI relevance is stable with sample size and increases across hidden layers. (b) The path-complexity ordering persists under three global distances. Error bars are 95% confidence intervals over ten seeds.
Layer
ϵ
n=200
n=500
n=1000
1
2×10−3
0.6257
0.6283
0.6285
2
1×10−3
0.9229
0.9284
0.9323
2
2×10−3
0.9229
0.9284
0.9323
2
5×10−3
0.9229
0.9284
0.9323
Appendix
Table 3: Mean relevant-factor QFI mass over ten seeds. The first row reports hidden layer 1 at the default step; the remaining rows report hidden layer 2.
Transformation
Max logit error
LB
DB
τ
Identity
0
2.729
1.008
2.708
Hidden-unit permutation
<2×10−15
2.729
1.008
2.708
Common orthogonal rotation
<6×10−15
2.729
1.008
2.708
Compensated per-unit rescaling
<2×10−15
2.595
0.937
2.772
Appendix
Table 4: Function-preserving feature-coordinate audit over ten seeds. The same transformation is applied at every checkpoint; LB , DB , and τ are seed means.
Spacing s
k=1
k=2
k=3
k=4
1
1.261
1.607
1.838
2.049
2
1.255
1.601
1.832
2.043
5
1.222
1.563
1.792
2.000
10
1.161
1.477
1.697
1.900
20
1.094
1.351
1.537
1.713
Appendix
Table 5: Mean Bures tortuosity over ten seeds after retaining every s th checkpoint.
Updates
k=1
k=2
k=3
k=4
20
1.178
1.711
1.804
1.970
40
1.209
1.940
1.956
2.133
60
1.225
2.294
2.284
2.597
100
1.245
2.575
2.708
3.428
200
1.263
2.791
3.069
4.590
300
1.270
2.878
3.152
5.142
Appendix
Table 6: Mean tortuosity at matched training budgets over ten paired seeds. Every interaction order receives exactly the listed number of updates.
Stopped at 75% validation accuracy
Fixed 300 updates
k
Stop epoch
LB
DB
τ
LB
DB
τ
1
1.9
0.294
0.290
1.014
1.600
1.259
1.270
2
22.7
1.331
0.775
1.723
3.045
1.061
2.878
3
31.9
1.501
0.803
1.872
3.240
1.028
3.152
4
175.3
3.693
0.889
4.177
4.587
0.893
5.142
Appendix
Table 7: Decomposition of the accuracy-stopped and 300-update parity trajectories. LB is path length and DB is endpoint displacement.
Optimizer
Accuracy (%)
Steps
LB
DB
τ
RepNG
71.43
65.77
1.207
0.636
1.895
Output NG
71.12
31.70
1.426
0.717
1.987
SGD
71.27
62.30
1.549
0.748
2.084
Adam
71.02
47.33
1.945
0.736
2.663
Appendix
Table 8: Optimizer means over 30 paired seeds at unit checkpoint spacing. Accuracy is test accuracy; LB is path length and DB is endpoint displacement.
Method
LB
DB
τ
RepNG
1.679
0.674
2.506
Output NG
2.639
0.753
3.520
Appendix
Table 9: Twenty-seed fixed-budget control with identical maximum update norms.
Outcome
Std. βτ
Cluster p
Perm. p
Continued-training drift
0.87
9.2×10−4
0.0015
Parameter-perturbation drift
0.51
2.4×10−4
0.0010
Appendix
Table 10: Fresh 30-seed representational-stability sweep (120 task–seed runs). Clustered p -values use interaction-order fixed effects; permutation tests shuffle complete task profiles across seeds.