Intrinsic Information-Theoretic Analysis of ReLU Nets

Johan Mylius-Kroken; Elisabeth Wetzer; Ali Ramezani-Kebrya; Robert Jenssen; Kristoffer Knutsen Wickstrøm

stat.MLOct 27, 2026

Intrinsic Information-Theoretic Analysis of ReLU Nets

Authors: Johan Mylius-Kroken, Elisabeth Wetzer, Ali Ramezani-Kebrya, Robert Jenssen, Kristoffer Knutsen Wickstrøm

Organizations: UiT The Arctic University of Norway · University of Oslo · Norwegian Computing Center · University of Copenhagen

Abstract

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 information $I(Y; Π)$ , 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 $\mathcal{X}\subset\mathbb{R}^{2}$ into convex linear regions; black lines are the projections of the layer’s neuron hyperplanes onto $\mathcal{X}$ . A single region $\omega$ 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 $\omega$ itself is split. The network is affine on every region of this refinement. (c) The same region $\omega$ is uniquely identified by its activation pattern $\pi_{\omega}=(\pi^{(1)}_{\omega},\pi^{(2)}_{\omega})=((1,0,1,0),(0,1,0,0))$ — 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	$\rho<0.05$ ( $n=41$ )	$\rho>0.3$ ( $n=56$ )	$\rho>0.9$ ( $n=19$ )
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

\rho

, over the

138

(dataset, architecture, PCA dimension, layer) cells of both studies;

n

: number of cells. Small corrections at

\rho>0.9

mean the corrected value stays at the

H(Y)

ceiling. Full table: Table 4 .

Figure 2: Pointwise calibration of

\hat{I}(Y;\Omega_{\mathcal{D}})

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=|\mathcal{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

\pm 1\sigma

across seeds and matching architectures. Solid:

\hat{I}(Y;\Omega_{\mathcal{D}})

. 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:

\hat{I}

(grey) and

\hat{I}_{\mathrm{func}}(\varepsilon)

(light to dark blue); error bars

\pm 1\sigma

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

\varepsilon\in[0.3,0.5]

pulls it back below

H(Y)

except at widths

25

and

50

Figure 5:

\rho_{\mathrm{func}}(\varepsilon)

on PCA-reduced MNIST of dimension

d

at the last hidden layer and training epoch. Lower

\rho_{\mathrm{func}}

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

\sim 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:

\rho_{\mathrm{func}}^{\,l}(\varepsilon)

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

\varepsilon

, mean

\pm 1\sigma

across architectures and seeds.

\rho_{\mathrm{func}}^{\,1}=1

structurally on every panel; for

\varepsilon>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

n_{\omega}

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

n_{\omega}=1

is the singleton-sample fraction of Table 2 ; curves that reach

1

at small

n_{\omega}

belong to saturated partitions (

\rho\to 1

Dataset	Architecture	Regions	$\rho$	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:

|\Omega_{\mathcal{D}}|

; 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

\rho

at the last hidden layer on MNIST (

N=10\,000

), per architecture and PCA dimension

d

. The calibration networks of Study 1 are

[3,3,3]

[5,5,5]

and

[7,7,7]

d=10

$\rho$ range	Estimator	Value	Correction	Seed std	Gap to $H(Y)$
$\rho<0.05$ ( $n=41$ )	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
$\rho>0.3$ ( $n=56$ )	Plug-in	2.586	0.000	0.025	0.146

Appendix

Table 4: Bias corrections grouped by occupancy

\rho

(

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	$\rho$ held-out	$\rho$ in-sample	$\hat{I}$ held-out	$\hat{I}$ in-sample	$\Delta\hat{I}$
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

\Delta\hat{I}

: 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.	$\hat{I}$	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.	$\rho$	$\hat{I}$	$\hat{I}_{\mathrm{func}}$	$\rho_{\mathrm{func}}$
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.

\hat{I}_{\mathrm{func}}

and

\rho_{\mathrm{func}}

\varepsilon=0.3

. MI in bits;

H(Y)=2.639

(Composite) and

0.949

(WBC).

Layers	$\varepsilon$	Cells	Median	95th pct.	Max	Max / $R$	Max $\Delta\hat{I}^{\mathrm{MM}}_{\mathrm{func}}$
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

\varepsilon

. Cells: (network, layer) pairs. Median, 95th percentile and maximum of the spread of the quotient size; max

/R

: largest spread relative to

|\Omega_{\mathcal{D}}|

; 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

\pm 1\sigma

over five seeds.

Figure 10: WBC: test accuracy (solid) and test loss (dashed) of the six Study-1 architectures, mean

\pm 1\sigma

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]

[50,50,50]

are the Study-2 networks at every

d

(

[7,7,7]

d=10

also enters Study 1). Mean

\pm 1\sigma

over five seeds.

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Characterizing the Discrete Geometry of ReLU Networks

Blake B. Gaines, Jinbo Bi

Department of Computer Science University of Connecticut

Neural Representation Geometry ReLU Neural Networks Neural Network Expressivity

Open paper AskPDF arXiv

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.

Abstract

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