Mutual information (MI) is fundamental to representation-learning analysis in deep networks, but in high dimensions MI estimators depend on hyperparameters whose bias often dominates the signal. We show that continuous piecewise-linear (CPWL) networks, including ReLU networks, admit an intrinsic information-theoretic analysis that avoids this pitfall: their inherent partition of the input space into convex linear regions assigns to every input a discrete region label Π alongside the continuous representation T and label Y. We introduce a decomposition of the mutual information I(Y;T) into interpretable geometric terms. We show that the dominant term, the routing informationI(Y;Π), admits an estimable plug-in estimator, that in the well-sampled regime, approximates a population lower bound on the joint information I(Y;Π,T). When the partition saturates, a functional-equivalence quotient collapses regions implementing the same linear operator and restores informativeness; this extension has a single bounded, dimensionless parameter, which is reported through the quotient's compression ratio. The routing estimator agrees strongly with three established MI baselines applied to the same networks and data. The functional quotient recovers informative estimates in the regime bordering the well-sampled and under-sampled regime. We provide a framework for analysing the internal information geometry of CPWL networks, grounded in their inherent piecewise-linear structure rather than in external discretisation.
Figures & tables
Figure 1: From geometry to combinatorics in a ReLU network. (a) The first layer partitions the input space X⊂R2 into convex linear regions; black lines are the projections of the layer’s neuron hyperplanes onto X . A single region ω is highlighted in yellow. (b) Composing the second layer refines the partition: each new hyperplane subdivides regions of (a), so the partition through layer 2 is the common refinement of the per-layer partitions and ω itself is split. The network is affine on every region of this refinement. (c) The same region ω is uniquely identified by its activation pattern πω=(πω(1),πω(2))=((1,0,1,0),(0,1,0,0)) — the layer-wise binarisation of the pre-activations (green: active, white: inactive). The bijection between regions and activation patterns is the bridge between the geometric and combinatorial views of CPWL networks that our framework exploits for information-theoretic analysis.
Estimator
ρ<0.05 ( n=41 )
ρ>0.3 ( n=56 )
ρ>0.9 ( n=19 )
Miller–Madow
0.114
3.739
5.403
Grassberger
0.029
0.067
0.006
Chao–Shen
0.032
0.139
0.054
Chao–Wang–Jost
0.032
0.185
0.065
Table 1: Mean bias correction (plug-in minus corrected estimate, bits) by occupancy ρ , over the 138 (dataset, architecture, PCA dimension, layer) cells of both studies; n : number of cells. Small corrections at ρ>0.9 mean the corrected value stays at the H(Y) ceiling. Full table: Table 4 .
Figure 2: Pointwise calibration of I^(Y;ΩD) against three MI baselines: binning and k -means with the Miller–Madow correction, and the k -nearest-neighbour estimator KSG (last epoch, last hidden layer, held-out points). There is one marker per network (architecture and seed) on Composite (blue), WBC (orange), and MNIST FC (green). Pearson r pooled across all three datasets: r=0.911 (binning, K=8 ), r=0.922 ( k -means, K=∣Y∣ ), r=0.892 (KSG, k=3 ). WBC networks with KSG near zero: Sec. G.5 .
Figure 3: Layerwise bits at the last training epoch, mean ±1σ across seeds and matching architectures. Solid: I^(Y;ΩD) . Dashed: the three baselines (binning and k -means with the Miller–Madow correction, KSG). Horizontal dash-dot: H(Y) . The Composite and WBC panels pool the three five-hidden-layer architectures; the MNIST FC panel pools width 5 and 7 networks.
Figure 4: Routing MI on PCA-reduced MNIST (last layer, last epoch) as a function of input dimension d , for four 3-hidden-layer architectures. Bars: I^ (grey) and I^func(ε) (light to dark blue); error bars ±1σ over 5 seeds. Dashed line: H(Y)=3.319 . As d grows the raw bar approaches H(Y) for the wider architectures (saturation); the functional quotient at ε∈[0.3,0.5] pulls it back below H(Y) except at widths 25 and 50 .
Figure 5: ρfunc(ε) on PCA-reduced MNIST of dimension d at the last hidden layer and training epoch. Lower ρfunc means more aggressive merging.
Appendix figures & tables14 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 6: A realisation of the seven-class 2D Composite dataset after min-max scaling to [−1,1]2 . Classes 0–1 (moons) and 2–3 (circles) each carry 20% of the mass; classes 4–6 (blobs) each carry ∼6.7% . The two circles enclose the moons and the blobs lie outside the circles; no two generators overlap. This gives the heterogeneous routing geometry analysed in Secs. 5.1 and 5.2 .
Figure 7: ρfuncl(ε) as a function of hidden-layer depth on the calibration networks (the five-layer architectures of Composite and WBC, and [5,5,5] and [7,7,7] on MNIST FC), one curve per ε , mean ±1σ across architectures and seeds. ρfunc1=1 structurally on every panel; for ε>0 the mean ratio decreases with depth.
Figure 8: Distribution of region sizes at the deepest layer: the fraction of points that lie in a region holding at most nω points. Colour is the hidden width, line style the depth; MNIST at PCA dimension 10 . Held-out points, last epoch, five seeds pooled. The value at nω=1 is the singleton-sample fraction of Table 2 ; curves that reach 1 at small nω belong to saturated partitions ( ρ→1 ).
Dataset
Architecture
Regions
ρ
Singleton regions
Singleton samples
Composite
[5,5,5]
59.2
0.030
0.123
0.004
[5,5,5,5,5]
108.2
0.054
0.138
0.009
[9,9,9]
128.4
0.064
0.133
0.009
[9,9,9,9,9]
294.2
0.147
0.291
0.044
[25,25,25]
607.4
0.304
0.438
0.133
[25,25,25,25,25]
822.2
0.411
0.540
0.222
Appendix
Table 2: Occupancy at the deepest layer, Composite ( N=2,000 ) and WBC ( N=114 ), held-out points, mean over seeds. Regions: ∣ΩD∣ ; singleton regions: fraction of regions holding one point; singleton samples: fraction of points alone in their region.
Architecture
d=2
d=3
d=4
d=5
d=10
d=15
d=20
[3,3,3]
–
–
–
–
0.004
–
–
[5,5,5]
–
–
–
–
0.030
–
–
[7,7,7]
0.012
0.028
0.047
0.082
0.122
0.125
0.103
[15,15,15]
0.049
0.195
0.342
0.518
0.743
0.774
0.783
[25,25,25]
0.124
0.452
0.734
0.874
0.954
0.965
0.979
[50,50,50]
0.317
0.826
0.960
0.987
0.998
0.999
1.000
Appendix
Table 3: Occupancy ρ at the last hidden layer on MNIST ( N=10000 ), per architecture and PCA dimension d . The calibration networks of Study 1 are [3,3,3] , [5,5,5] and [7,7,7] at d=10 .
ρ range
Estimator
Value
Correction
Seed std
Gap to H(Y)
ρ<0.05 ( n=41 )
Plug-in
1.439
0.000
0.148
1.698
Miller–Madow
1.325
0.114
0.131
1.812
Grassberger
1.410
0.029
0.144
1.727
Chao–Shen
1.407
0.032
0.143
1.730
Chao–Wang–Jost
1.407
0.032
0.143
1.730
ρ>0.3 ( n=56 )
Plug-in
2.586
0.000
0.025
0.146
Appendix
Table 4: Bias corrections grouped by occupancy ρ ( n : number of cells). Value: corrected estimate; correction: plug-in minus corrected value; seed std: standard deviation over the five seeds; gap to H(Y) : H(Y) minus the corrected value. All in bits, averaged over cells.
Dataset
Architecture
ρ held-out
ρ in-sample
I^ held-out
I^ in-sample
ΔI^
Composite
[5,5,5]
0.030
0.007
1.949
1.939
− 0.010
[5,5,5,5,5]
0.054
0.013
2.261
2.245
− 0.015
[9,9,9]
0.064
0.016
2.188
2.174
− 0.014
[9,9,9,9,9]
0.147
0.043
2.550
2.538
− 0.012
[25,25,25]
0.304
0.104
2.535
2.515
− 0.020
[25,25,25,25,25]
0.411
0.163
2.620
2.609
− 0.011
Appendix
Table 5: Held-out versus in-sample routing estimates at the deepest layer (mean over seeds, bits). Held-out / in-sample N : Composite 2,000 / 10,000 , WBC 114 / 569 . ΔI^ : in-sample minus held-out.
Baseline
Setting
Composite + WBC ( n=60 )
MNIST FC ( n=15 )
plug-in
MM
plug-in
MM
Binning ( K )
2
0.923
0.946
0.877
0.875
4
0.966
0.973
0.758
0.728
8 ∗
0.981
0.945
0.807
0.690
16
0.982
0.852
0.835
− 0.570
30
0.975
0.655
0.824
− 0.888
Appendix
Table 6: Pearson r between the routing estimator and each baseline setting, one point per network at the last hidden layer. MM: Miller–Madow-corrected. KSG is not a plug-in estimator and has a single column. ∗ The setting used in Fig. 2 .
Architecture
Seed
Test acc.
I^
Binning
KSG raw
KSG
Tied points
[5,5,5,5,5]
101
98.2%
0.791
0.736
−0.469
0.000
41 (36%)
[5,5,5,5,5]
103
97.4%
0.758
0.795
0.148
0.148
26 (23%)
[5,5,5,5,5]
104
63.2%
0.392
0.217
−3.070
0.000
89 (78%)
[5,5,5,5,5]
105
97.4%
0.911
0.576
−1.743
0.000
64 (56%)
[5,5,5]
104
96.5%
0.779
0.720
−1.161
0.000
53 (46%)
Appendix
Table 7: WBC networks on which KSG collapses (last hidden layer, last epoch, held-out points, N=114 ). Binning: Miller–Madow, K=8 . KSG raw: the estimate before clipping at zero. Tied points: size of the largest group of points with identical representation, and its share of N . MI in bits.
Dataset
Architecture
Labels
Test acc.
Train-label acc.
ρ
I^
I^func
ρfunc
Composite
[5,5,5]
clean
97.9%
–
0.030
1.949
1.829
0.765
[5,5,5]
permuted
19.6%
20.8%
0.022
1.411
1.096
0.656
[25,25,25]
clean
99.5%
–
0.304
2.535
2.362
0.377
[25,25,25]
permuted
15.9%
22.5%
0.245
2.433
2.329
0.645
WBC
[5,5,5]
clean
97.4%
–
0.184
0.767
0.700
0.680
[5,5,5]
permuted
62.5%
62.8%
0.211
0.469
0.295
0.709
Appendix
Table 8: Clean versus label-permuted training (deepest layer, held-out points, true labels, mean over seeds). Test acc.: accuracy on the true test labels; train-label acc.: accuracy on the (permuted) labels the network was trained on. I^func and ρfunc at ε=0.3 . MI in bits; H(Y)=2.639 (Composite) and 0.949 (WBC).
Layers
ε
Cells
Median
95th pct.
Max
Max / R
Max ΔI^funcMM
all layers
0.1
690
0
9.0
17
6.7%
0.053
0.3
690
2
38.5
84
9.4%
0.148
0.5
690
4
43.0
83
18.8%
0.348
1
690
1
19.0
47
14.3%
0.474
deepest layer
0.1
210
0
11.6
17
6.2%
0.053
0.3
210
11
45.5
60
6.5%
0.100
Appendix
Table 9: Spread of the functional quotient over visiting orders, per ε . Cells: (network, layer) pairs. Median, 95th percentile and maximum of the spread of the quotient size; max /R : largest spread relative to ∣ΩD∣ ; last column: largest spread of the Miller–Madow quotient MI (bits).
Figure 9: Composite: test accuracy (solid) and test loss (dashed) of the six Study-1 architectures, mean ±1σ over five seeds.
Figure 10: WBC: test accuracy (solid) and test loss (dashed) of the six Study-1 architectures, mean ±1σ over five seeds.
Figure 11: MNIST: test accuracy (top two rows) and test loss (bottom two rows), one colour per PCA dimension d . [3,3,3] and [5,5,5] are the Study-1 networks at d=10 ; [7,7,7] to [50,50,50] are the Study-2 networks at every d ( [7,7,7] at d=10 also enters Study 1). Mean ±1σ over five seeds.
Rectified linear unit (ReLU) neural networks use piecewise-linear activations to study approximation, learning dynamics, expressivity, and verification.
It is well established that ReLU networks define continuous piecewise-linear functions, and that their linear regions are polyhedra in the input space. These regions form a complex that fully partitions the input space. The way these regions fit together is fundamental to the behavior of the network, as nonlinearities occur only at the boundaries where these regions connect. However, relatively little is known about the geometry of these complexes beyond bounds on the total number of regions, and calculating the complex exactly is intractable for most networks. In this work, we prove new theoretical results about these complexes that hold for all fully-connected ReLU networks, specifically about their connectivity graphs in which nodes correspond to regions and edges exist between each pair of regions connected by a face. We find that the average degree of this graph is upper bounded by twice the input dimension regardless of the width and depth of the network, and that the diameter of this graph has an upper bound that does not depend on input dimension, despite the number of regions increasing exponentially with input dimension. We corroborate our findings through experiments with networks trained on both synthetic and real-world data, which provide additional insight into the geometry of ReLU networks. Code to reproduce our results can be found at https://github.com/bl-ake/ICLR-2026.
Blake B. Gaines, Jinbo Bi
Department of Computer Science University of Connecticut
We study the realization map of deep ReLU networks, focusing on when a function determines its parameters up to scaling and permutation. To analyze hidden redundancies beyond these standard symmetries, we introduce a framework based on weighted polyhedral complexes. Our main result shows that for every architecture whose input and hidden layers have width at least two, there exists an open set of identifiable parameters. This implies that the functional dimension of every such architecture is exactly the number of parameters minus the number of hidden neurons. We further show that minimal functional representations can still have non-trivial parameter redundancies. Finally, we establish a generic depth hierarchy, whereby for an open set of parameters the realized function cannot be represented generically by any shallower network.
Moritz Grillo, Guido Montúfar
Max Planck Institute for Mathematics in the Sciences, Leipzig · Departments of Mathematics and Statistics & Data Science, UCLA
Deep neural networks transform input data into latent representations that support a wide range of downstream tasks. These representations can be characterized along information-theoretic and geometric dimensions, but their relationship remains poorly understood. A central open question is whether low mutual information (MI) between inputs and representations necessarily implies geometrically compressed latent spaces and vice versa. We investigate this question using class-wise clustering as a measure of geometric compression and theoretically sound MI estimation in conditional entropy bottleneck (CEB) networks and continuous dropout networks. We evaluate the interplay between MI, geometric compression, and generalization on classification tasks under controlled noise injection schemes. Our findings show that low MI does not reliably correspond to geometric compression, and that the connection between the two is more nuanced than often assumed. Indeed, our experiments reveal a negative and nonlinear relationship that can reverse when varying training setup. Our results put forward a hypothesis that generalization acts as a potential confounder in this connection rather than being their direct consequence.
Linara Adilova, Henning Petzka, Asja Fischer +1
Research Center Trustworthy Data Science and Security, Dortmund · Faculty of Computer Science, Ruhr University Bochum · Signal Processing and Speech Communication Laboratory, Graz University of Technology +2