Training Neural Networks by Optimizing Neuron Positions

Alexandru Vasilache; Juergen Becker; Laura Erb; Nicola Toschi; Tommaso Boccato

REVIEW 3 major objections 5 minor 16 references

Training Neural Networks by Optimizing Neuron Positions

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Training networks by moving neuron positions instead of learning weights cuts parameters from $O(n^2)$ to $O(n)$ and keeps MNIST accuracy past 80% pruning

desk verdict A genuinely new weight-parameterization idea with honest reporting, but the paper's central 'competitive performance' claim is undermined by a parameter-matched baseline that matches counts while badly mismatching capacity. read the letter →

arxiv 2506.13410 v1 pith:4XLP7H4V submitted 2025-06-16 cs.LG

Laura Erb , Tommaso Boccato , Alexandru Vasilache , Juergen Becker , Nicola Toschi This is my paper · ORCID

classification cs.LG

keywords neuronpositionoptimizationspatialembeddingsdistance-dependentweightsparameterefficiencyspikingneuralnetworksnetworkpruningMNIST

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes training neural networks by optimizing neuron positions in Euclidean space rather than learning connection weights directly. Each synaptic weight is defined by the inverse-distance rule $w_{ij}=1/\|p_i-p_j\|_2$, so a full weight matrix is replaced by a few coordinates per neuron, cutting the parameter count from $O(n^2)$ to $O(n)$. The authors test this on MNIST with multi-layer perceptrons and spiking neural networks: the spiking version beats a conventional spiking baseline with the same parameter count, and both variants keep accuracy when more than 80% of connections are pruned. The MLP versions remain less accurate than conventional MLPs even at matched parameter counts, a gap the paper attributes to the coupling that distance-derived weights impose.

What carries the argument

The machinery is the wiring rule of Eq. (1), $w_{ij}=1/\|p_i-p_j\|_2$, combined with a fixed per-layer z-coordinate and a learnable continuous inhibition mask that assigns each neuron an excitatory or inhibitory sign. Because every weight in a fully connected layer is a function of only the two endpoint positions, backpropagation moves points instead of scalars; a layer with $n_l$ neurons is stored with $O(n_l)$ position parameters rather than $O(n_l n_{l+1})$ weights. A single step of backpropagation therefore updates all connections incident to a neuron at once, which is the parameter saving and also the main source of weight interdependence.

What would settle it

Re-run the pruning experiments with a baseline that has the same 2,048 hidden neurons as the 3D MLP but whose per-connection weights are free and whose total trainable parameter count is forced to about 10,500, for instance by weight sharing across connections; if this width-matched free-weight baseline matches or outperforms the 3D MLP at every pruning level, the inverse-distance wiring rule is not what produces the observed pruning robustness.

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Extended reading notes

Core claim

The central discovery is that a distance-geometry wiring rule can carry most of the learning burden: instead of storing and updating an $O(n^2)$ weight matrix, the network stores $d-1$ coordinates per neuron plus a bias and an inhibition sign, computes weights as $w_{ij}=1/\|p_i-p_j\|_2$, and learns by moving points. On MNIST, spatially embedded MLPs and SNNs reach competitive accuracy, and both retain accuracy under magnitude-based pruning above 80% sparsity; at 95% post-training pruning the 3D MLP even beats a wide baseline with far more parameters. The same experiments show that relaxing the layer structure and raising embedding dimensionality improve accuracy, and that dense MLP accuracy stays below conventional MLPs, which the authors explain by the restricted set of weight configurations reachable through positions.

Load-bearing premise

The evaluation assumes that matching a conventional network only by total parameter count is a fair comparison: the 3D MLP has 2,048 hidden neurons and about 1.63 million effective connections but only 10,576 trainable parameters, while the parameter-matched MLP Baseline 14 has 14 hidden neurons and about 11,116 weights; if this premise is wrong, the claims of competitive performance and pruning superiority are not established.

Editorial extensions

If this is right

A fully connected network can be deployed with weights reconstructed from stored positions, so connection memory falls from $O(n^2)$ to $O(n)$, a direct benefit for edge-device and real-time inference.
Magnitude-based pruning has a simple geometric reading: removing the smallest weights is the same as cutting the longest physical connections, so pruning can be specified in space rather than in weight space.
The framework composes with other compression techniques, such as quantization and pruning, and can be dropped into existing MLP and SNN architectures without changing the forward-pass structure.
Spiking versions outperform their parameter-matched conventional baseline, indicating that temporal spike timing can compensate for reduced weight flexibility; the same may hold for other temporally coded or recurrent models.
Allowing richer geometry such as learned z-coordinates and higher-dimensional embedding spaces monotonically improves MNIST accuracy, so the geometry itself is a tunable resource for trading parameters against expressiveness.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An editorial extension: the inverse-distance rule is one instance of a broader prior, monotone weight decay with spatial separation; testing other monotone parameterizations such as Gaussian or exponential kernels of distance would show whether the pruning robustness comes from the geometric prior or from the specific $1/d$ form.
An editorial extension: the overlap of learned z-distributions across layers suggests the rigid layer code is not essential, so a fully unconstrained position embedding plus a routing mechanism might improve accuracy further, at the cost of feedforward ordering.
An editorial extension: because each position update changes many weights, the effective number of independent degrees of freedom in a spatially embedded layer is far below its raw parameter count; measuring the rank of the Jacobian of the weight matrix with respect to positions would quantify that gap directly.
An editorial extension: the same position-to-weight map transfers naturally to graph neural networks, where node coordinates could be learned jointly with edge weights and the inverse-distance rule would supply an inductive bias aligned with spatially structured graphs.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Here is my read. The actual new thing is the wiring rule as a weight parameterization: neurons embedded in Euclidean space, weights set to the inverse distance, and training optimized positions plus an inhibitory/excitatory sign per neuron. That is a real extension of the earlier spatial-embedding line from Achterberg, Stiso, Jacobs, and others, where positions were fixed and mainly regularized. The presentation is clean, and the O(n) parameter count for fixed embedding dimension is arithmetically correct, with the caveat that the number of connections and the inference cost stay O(n^2).

Credit where it is due: the experiments are transparent, the numbers in Tables 1 and 2 and the figures are internally consistent, and the Discussion explicitly admits that the spatial MLPs underperform conventional MLPs. The SNN result against the 14-neuron baseline at least shows one positive data point. The z-coordinate relaxation and the higher-dimensional embedding ablations are sensible probes of the idea.

The soft spots are real, and they are mostly in the abstract's wording. 'Competitive with conventional architectures' is not supported by the paper's own tables: the 3D MLP gets 0.9217 versus 0.9429 for the parameter-matched 14-neuron MLP, and Figure 5 shows every spatial MLP below its parameter-matched baseline. The baseline itself is the issue: it matches total parameter count by shrinking the hidden layer to 14 neurons, so the comparison is parameter-matched but capacity-mismatched. The spatial model has 2,048 hidden neurons and about 1.6 million weights; the 14-neuron baseline has 14 hidden neurons and about 11,000 weights. At 80% pruning the spatial model still has roughly 325,000 nonzero weights, so 'same conditions' is misleading. The one truly interesting result is the 95% post-training pruning point, where the 3D MLP beats the width-matched 2,048-neuron baseline; that should have been the headline, not the weak parameter-count comparison.

There are a few smaller omissions: no code, no initialization details for the x,y coordinates, and only MNIST with one seed set. The citation pattern is fine; the biological motivation is heavily cited and the prior spatial-embedding work is acknowledged.

Bottom line: this is a paper a serious editor should send to peer review. The idea deserves scrutiny and the experiments can be fixed with matched-capacity baselines, width-matched comparisons, and at least one more dataset. I would take it to a reading group if your group works on parameter-efficient architectures, but I would not yet cite it as evidence for the competitive-performance claim.

Referee Report

3 major / 5 minor

Summary. The paper proposes a neural architecture in which each neuron is assigned a position in Euclidean space and connection weights are computed as the inverse of the Euclidean distance between connected neurons. Instead of learning individual weights, the model optimizes neuron positions (plus a small number of auxiliary parameters such as layer distances and inhibition values). The authors argue that this reduces the parameter complexity from O(n^2) to O(n) and provides a biologically motivated inductive bias. They validate the approach on MNIST for both MLPs and fully connected SNNs, comparing against conventional baselines matched either by neuron count or by parameter count. They also report pruning experiments and an extension in which z-coordinates are learned, as well as embeddings in higher-dimensional spaces. The main claim is that spatially embedded networks are competitive with conventional architectures and maintain accuracy at high pruning rates.

Significance. The core idea is interesting and potentially useful: if a weight matrix can be replaced by a distance-dependent wiring rule, then a compact parameterization with a clear geometric interpretation becomes possible. The experiments are transparent, report multiple random seeds, and use an external benchmark, which is a strength. However, the current evidence does not establish the central claim of competitive performance: the paper's own MLP results consistently show the spatial models underperforming conventional baselines with comparable parameter counts but far fewer neurons, and the pruning advantage rests on comparisons that confound total parameter count with the number of effective connections. The significance of the work therefore depends on whether the claims can be appropriately reframed and whether the pruning robustness can be demonstrated under genuinely matched conditions.

major comments (3)

[Abstract and Section 3.1, Table 1, Fig. 5] The abstract claims that spatially embedded networks 'achieve a performance competitive with conventional architectures,' but the paper's own experiments contradict this for every MLP configuration tested. In Table 1, the 3D MLP reaches 0.9217 accuracy, well below MLP Baseline 14 with 0.9429 and MLP Baseline 2048 with 0.9745. Table 3 shows the Relaxed 3D MLP at 0.9337 versus MLP Baseline 17 at 0.9507, and Fig. 5 shows every higher-dimensional spatial MLP below its corresponding parameter-matched baseline. Section 4 explicitly concedes that spatially embedded MLPs 'were consistently outperformed by conventional MLPs.' The only MLP result that favors the spatial model is the single 95%-pruning point in Fig. 3a. The central empirical claim should either be substantially weakened or supported by additional evidence beyond this one point.
[Section 3.2, Fig. 3 and Table 1] The pruning comparison conflates parameter count with capacity and with the number of effective connections. The 3D MLP has 2,048 hidden neurons and 1,626,112 weights, while MLP Baseline 14 has 14 hidden neurons and 11,116 weights. When 80% of the 3D MLP's weights are pruned, roughly 325,000 weights remain, which is about 30 times the baseline's entire network. The abstract's phrase 'outperforming traditional networks with the same number of parameters under similar conditions' is therefore not an accurate description of what Fig. 3 shows. The width-matched control, MLP Baseline 2048, outperforms the 3D MLP at most pruning levels. A fair assessment of pruning robustness would need to compare networks with matched numbers of remaining connections, or otherwise control for the large difference in network width.
[Section 2, Eq. (1)] Equation (1) defines the synaptic weight as the inverse of the Euclidean distance between connected neurons, but this quantity is undefined when the distance is zero. The paper does not specify how neuron positions are initialized, or what constraint, regularization, or projection step prevents two connected neurons from occupying the same position during optimization. Since positions are updated by gradient descent, a collision is not only possible but also not addressed by any stated mechanism. The model definition is therefore incomplete without a minimum-distance safeguard or an explicit justification for why collisions cannot occur.

minor comments (5)

[Abstract and Section 4] The abstract says spatially embedded models 'outperform traditional networks with the same number of parameters under similar conditions,' but Section 4 states the opposite for MLPs. This inconsistency should be resolved by aligning the abstract with the actual results.
[Section 2, Eqs. (3) and (4)] The parameter count in Eq. (4) includes positions for the input layer, but the text does not explain why input neurons need learnable positions. Clarifying the role of input positions would make the parameter-counting convention easier to interpret.
[Section 3.1, Tables 1 and 2] The learning rates differ between the spatial models (0.005) and the baselines (0.001). Since no learning-rate sweep is reported, it is unclear whether the performance gap could be reduced by more careful hyperparameter tuning of the spatial models.
[Section 3.2] The paper describes pruning the longest connections as mimicking the brain's tendency to minimize wiring length, but removing long connections is not the same as actively minimizing total wiring length. The biological analogy would be more accurate if framed as a heuristic consistent with wiring minimization.
[Section 5] The conclusion mentions 'Recurrent Neural Networks' as a possible extension in the introduction, but the experiments only cover MLPs and SNNs. The paper would benefit from stating this scope limitation explicitly in the relevant sections.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation chain found; all results are empirical comparisons on an external benchmark.

full rationale

This paper does not derive a predicted constant or falsifiable quantity from fitted inputs; rather, it defines a parameterization and reports measured accuracies on MNIST. Equation (1) defines each synaptic weight as the inverse Euclidean distance between neuron positions, and Equation (4) counts position coordinates, biases, and inhibition values as parameters. These are definitions and parameter-counting identities, not predictions fitted to data. The experimental comparisons use an external benchmark and baselines constructed by the authors, which raises a question of evaluation fairness but does not make any claim equivalent to its own inputs by construction. Self-citations to Boccato et al. appear only as general related work on network topologies and are not load-bearing for the proposed method. No quoted passage exhibits a reduction of a claimed result to a fitted parameter or to a self-cited uniqueness theorem. Therefore the appropriate finding is no significant circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities, forces, or conserved quantities. The spatial embedding, inhibition mask, and distance-based weighting are parameterizations and architectural choices rather than invented entities. The main load-bearing choices are the inverse-distance rule and the fixed-layer structure, and both are stated as postulates rather than derived.

free parameters (4)

Per-neuron spatial coordinates (x,y) in each layer = learned from data; values not reported
These are the primary trainable parameters; weights are deterministic functions of them. The architecture's O(n) parameter claim depends on treating these as the only connection-related parameters.
Learnable layer distances = two distances for the 3-layer 3D MLP and 3D SNN; not individually reported
Relaxing the fixed unit layer distance improves accuracy from 0.9018 to 0.9217, so this small set of fitted scalars is load-bearing for the final results.
Per-neuron inhibition values = learned continuous values; not reported
A sigmoid-transformed inhibition value per neuron determines excitatory versus inhibitory sign of outputs, effectively recovering signed weights that pure distance weighting would not allow.
Learnable z-coordinates in the Relaxed 3D MLP = learned per neuron; not reported
Relaxing fixed layer z-coordinates adds one parameter per neuron and improves accuracy to 0.9337; this variant appears in Section 3.3.

assumptions (5)

domain assumption Synaptic weight between two neurons is exactly the inverse Euclidean distance between their positions (Eq. 1).
This is the central architectural postulate; no derivation is given beyond an analogy to conductance (Eq. 2), and it determines the model's expressiveness.
domain assumption Neurons are organized into layers by fixing the z-coordinate to the layer index, preserving feedforward connectivity.
This constrains positions to two degrees of freedom per neuron in the base model; Section 2, Fig. 1.
domain assumption The sign of a neuron's contribution is controlled by a continuous inhibition value passed through a steep sigmoid; positions alone cannot produce negative weights.
Since distances are positive, negative weights must come from this separate mechanism; Section 2.
ad hoc to paper No two connected neurons occupy the same position, since Eq. (1) diverges at zero distance; the paper does not specify an initialization or regularization that guarantees this.
Colliding positions would produce infinite weights; the paper does not discuss this degeneracy.
domain assumption MNIST on three-layer networks is a sufficient probe of the architecture's general properties.
All conclusions about competitiveness and pruning robustness come from one dataset and one depth; the paper states that extension to deeper networks and diverse datasets is future work.

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Cite this review

Pith. "Pith review of Training Neural Networks by Optimizing Neuron Positions." pith.science (2026). https://pith.science/paper/4XLP7H4V

@misc{pith2026250613410,
  author       = {Pith},
  title        = {Pith review of: Training Neural Networks by Optimizing Neuron Positions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4XLP7H4V}},
  note         = {Machine review of arXiv:2506.13410}
}

read the original abstract

The high computational complexity and increasing parameter counts of deep neural networks pose significant challenges for deployment in resource-constrained environments, such as edge devices or real-time systems. To address this, we propose a parameter-efficient neural architecture where neurons are embedded in Euclidean space. During training, their positions are optimized and synaptic weights are determined as the inverse of the spatial distance between connected neurons. These distance-dependent wiring rules replace traditional learnable weight matrices and significantly reduce the number of parameters while introducing a biologically inspired inductive bias: connection strength decreases with spatial distance, reflecting the brain's embedding in three-dimensional space where connections tend to minimize wiring length. We validate this approach for both multi-layer perceptrons and spiking neural networks. Through a series of experiments, we demonstrate that these spatially embedded neural networks achieve a performance competitive with conventional architectures on the MNIST dataset. Additionally, the models maintain performance even at pruning rates exceeding 80% sparsity, outperforming traditional networks with the same number of parameters under similar conditions. Finally, the spatial embedding framework offers an intuitive visualization of the network structure.

Figures

Figures reproduced from arXiv: 2506.13410 by the authors.

**Figure 2.** Accuracy comparison of the (a) 3D MLP and (b) 3D SNN (with 2,048 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Accuracy comparison of the 3D MLP (with 2,048 hidden neurons) and the [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: (a)Accuracy comparison of the Relaxed 3D MLPs with learned z [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: Accuracy comparison of MLPs with 2,048 hidden neurons embedded in [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: The final neuron positions of a spatially embedded MLP with three layers. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

Review history (2 revisions) →

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