low-rank-gluing-compositionality

name	low-rank-gluing-compositionality
description	Mathematical framework for compositional computation in inhibition-dominated threshold-linear networks via low-rank gluing rules. Proves how structural modularity enables functional compositionality - component subnetworks' fixed points determine global network dynamics. Activation: compositionality, threshold-linear networks, TLN, fixed points, modular network, network assembly, gluing rules, low-rank coupling, inhibition-dominated, neural circuit design, computational primitives
metadata	{"arxiv_id":"2606.07336","published":"2026-06-05","authors":"Juliana Londono Alvarez","category":"computational neuroscience, neural dynamics, network theory"}
license	Complete terms in LICENSE.txt

Low-Rank Gluing Compositionality in Threshold-Linear Networks

Mathematical framework proving how modular network structure supports compositional computation via specific low-rank couplings between subnetworks.

Core Theory

Compositional Computation Problem

Brains generate complex behaviors from stable structures via compositionality - decomposing tasks into reusable primitives. This skill provides the mathematical proof for how structural modularity enables functional compositionality.

Key Question: How do component subnetworks combine to preserve their individual computations while forming new global behaviors?

Threshold-Linear Networks (TLNs)

Inhibition-dominated TLNs model biological neural circuits:

dx/dt = -x + [Wx + b]₊

where:

• x: neuron activity vector
• W: weight matrix (inhibition-dominated means negative diagonal)
• b: external input
• [·]₊: ReLU nonlinearity (threshold-linear)

Fixed points satisfy: x* = [Wx* + b]₊

Low-Rank Gluing Rules

Novel contribution: Introduces low-rank gluings - a class of modular network assembly where:

1. Component subnetworks have arbitrary internal connectivity W₁, W₂, ...
2. Inter-network couplings are low-rank matrices C_{ij} = UV^T (rank << dimension)
3. Global fixed points = composition of component fixed points

Theorem: If subnetworks have fixed points x₁*, x₂*, ..., then the glued network's fixed points are precisely combinations of these component fixed points, provided the gluing satisfies specific rank conditions.

Methodology

When to Use This Framework

Use when:

• Designing modular neural circuits with compositional properties
• Analyzing how network architecture enables task decomposition
• Proving stability/computation preservation in network assembly
• Understanding fixed point structure in recurrent networks

Design Procedure

1. Define component subnetworks:

   • Specify individual weight matrices W_i (inhibition-dominated)
   • Compute their fixed points x_i* (solve x = [Wx + b]₊)
   • Verify component computations are distinct primitives

2. Design gluing couplings:

   • Choose coupling rank r (typically r = 1 or r = 2)
   • Construct C_{ij} = U_{ij} V_{ij}^T where:
     • U_{ij}: output pattern from network i
     • V_{ij}: input pattern to network j
   • Ensure coupling respects fixed point structure

3. Assemble global network:

   W_global = [W₁   C₁₂  C₁₃]
              [C₂₁  W₂   C₂₃]
              [C₃₁  C₃₂  W₃ ]
   

4. Verify compositional fixed points:

   • Solve global fixed point equations
   • Confirm global solutions = combinations of component solutions
   • Check inhibition dominance preserved

Mathematical Tools

• Fixed point analysis: Solve nonlinear systems via iterative methods or convex relaxations
• Low-rank matrix theory: Use singular value decomposition to constrain coupling structure
• Lyapunov stability: Prove attractor stability via spectral analysis

Implementation Patterns

Pattern 1: Primitive Composition

# Define two primitives (e.g., "reach" and "grasp")
W_reach = inhibition_dominated_network(n=50, bias=b_reach)
W_grasp = inhibition_dominated_network(n=30, bias=b_grasp)

# Compute individual fixed points
x_reach_star = solve_fixed_point(W_reach)
x_grasp_star = solve_fixed_point(W_grasp)

# Glue with rank-1 coupling
U = output_pattern(x_reach_star)  # What "reach" produces
V = input_pattern_for_grasp()     # What "grasp" needs
C = U @ V.T  # Rank-1 gluing

# Assemble "reach+grasp" network
W_combined = block_matrix([[W_reach, C], [C.T, W_grasp]])

# Verify: x_combined_star = [x_reach_star, x_grasp_star]

Pattern 2: Hierarchical Composition

Nested gluings enable hierarchical task decomposition:

Primitive A → (A+B) → (A+B+C) → Full behavior
   ↑           ↑           ↑
 Rank-1      Rank-2      Rank-3
 gluings     gluings     gluings

Key Results

1. Fixed Point Preservation: Component fixed points survive gluing - no destruction of primitive computations

2. Compositional Emergence: Global fixed points = union of component fixed points + new combinations (when couplings create novel task-specific states)

3. Stability Guarantee: Low-rank gluings preserve inhibition dominance → stable attractors

4. Efficiency: Small coupling rank (r << n) minimizes connection cost while enabling composition

Pitfalls

• Rank too high: High-rank couplings destroy fixed point structure → non-compositional dynamics
• Excitatory coupling violates inhibition dominance: Ensure couplings don't destabilize attractors
• Component fixed points unstable: Must verify individual network stability before gluing
• Mismatched primitive semantics: Gluing requires semantic compatibility (output of A must be valid input to B)

Applications

• Motor control: Compose reach + grasp + manipulate primitives
• Language processing: Compose phoneme → syllable → word primitives
• Reasoning: Compose inference steps via gluing rules
• Circuit design: Build compositional neuromorphic processors

Extensions

• Learnable gluings: Train coupling matrices via gradient descent
• Dynamic gluings: Time-varying couplings for sequential composition
• Multi-scale composition: Glue at neuron, population, and network levels

References

• Original paper: arXiv:2606.07336 (Londono Alvarez, 2026)
• Related: Hopfield networks, reservoir computing, fixed point theory
• See also: cortico-cerebellar-modularity-rnn (cerebellar compositional architecture)

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Activation: compositionality, fixed points, threshold-linear networks, TLN, modular networks, low-rank coupling, network assembly, gluing rules, inhibition-dominated networks, computational primitives, neural circuit design