Holographic Memory for Zero-Shot Compositional Reasoning in Knowledge Graphs

A Mechanistic Study of Where and Why It Fails

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Abstract

Knowledge graph embedding (KGE) models achieve strong single-hop link prediction but cannot answer zero-shot compositional queries. Holographic Reduced Representations (HRR) offer a theoretically appealing candidate through convolution-based binding.

We study two holographic memory variants -- real-valued HRR and phase-only Fourier HRR (FHRR) -- on FB15k-237 over 5 random seeds:

1. Both variants are competitive atomic retrievers (Real HRR MRR 0.358, FHRR MRR 0.350)
2. Both fail at zero-shot composition (accuracy at chance, binomial test p > 0.2)
3. FHRR failure mechanism is phase decorrelation, not modulus collapse

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Results

Single-Hop (Atomic) Link Prediction

Full test set (20,466 queries), filtered metrics, mean +/- std over 5 seeds.

Model	Top-1	MRR	Hits@1	Hits@3	Hits@10
Real HRR (D=1024)	0.158 +/- 0.001	0.358 +/- 0.002	0.267 +/- 0.002	0.392 +/- 0.003	0.540 +/- 0.003
Complex FHRR (D=512)	0.126 +/- 0.001	0.350 +/- 0.021	0.262 +/- 0.017	0.390 +/- 0.024	0.524 +/- 0.028
RotatE (literature)	--	0.338	0.241	0.375	0.533

Two-Hop Zero-Shot Composition

69,855 query pairs, leakage-controlled protocol.

Model	Accuracy	p-value (vs chance)	Result
Real HRR	0.00017 +/- 0.00009	0.22	at chance
Complex FHRR	0.00003 +/- 0.00003	0.85	at chance
Random baseline	0.00007	--	--

Ablation: Effect of Hopfield Cleanup (Real HRR)

Removing the Hopfield cleanup drops atomic Top-1 by ~48%.

Configuration	Atomic Top-1	Zero-Shot
Full model	0.158 +/- 0.001	0.00017 +/- 0.00009
Without Hopfield cleanup	0.081 +/- 0.010	0.00030 +/- 0.00010

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Failure Mechanism

The FHRR failure is phase decorrelation, not modulus collapse:

Probe	What It Tests	Result
A -- Modulus tracking	Does the signal magnitude decay across hops?	No --
B -- Renormalization	Does restoring unit modulus fix it?	No -- accuracy remains 0.0000
C -- Hard cleanup	Does argmax (vs softmax) fix it?	No -- accuracy remains 0.0000
Phase coherence	Is the phase correlated with the target?	No -- cosine similarity = -0.009 (same as random)
Phase error	Per-component error vs ground truth	pi/2 (indistinguishable from uniform)

Why atomic ranking survives: Single-hop readout aggregates similarity across all dimensions, robust to per-component noise. Composition feeds the intermediate into a bind -- a per-component phase operation -- so it requires correct phase per dimension. The cleanup supplies aggregate ranking but not per-component phase coherence.

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Figures

Phase Error Propagation Mean abs phase error at hop 1 and hop 2 sits at uniform limit (pi/2)	Modulus Probe FHRR cleanup forces |z|=1 at every stage magnitude cannot explain the failure
Renormalization + Hard Cleanup Neither intervention recovers accuracy	Beta Temperature Sweep Accuracy flat at chance for all beta

Core Ablation: Removing Hopfield cleanup halves atomic accuracy

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Reproducibility

# Setup
pip install torch numpy scikit-learn matplotlib seaborn tqdm requests wandb

# Run on Modal (A10G GPU)
modal run modal/modal_experiment1.py --seed 42      # single seed
modal run modal/modal_experiment1.py --all-seeds    # 5 seeds parallel

All results: results/final_results.json, results/phase_error_results.json, results/results_inference_ablations.json, results/hop1_probe/ (5 seeds), results/hop2_atomic_probe/ (5 seeds)

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Project Structure

modal/                          Modal A10G experiment scripts
  modal_experiment1.py          Main experiment (5 seeds parallel)
  final_analysis.py             Aggregated analysis
  phase_error_analysis.py       Phase decorrelation measurement
  modal_inference_ablations.py  Ablation + probes + beta sweep
  modal_hop1_probe.py           Hop-1 phase probe across all seeds
  modal_hop2_atomic_probe.py    Hop-2 atomic probe across all seeds
paper/
  holographic_memory_paper_FINAL.tex   LaTeX source (TikZ figures)
  holographic_memory_paper_FINAL.pdf   Compiled PDF
results/                        5-seed experiment outputs (JSON)
  final_results.json            Atomic + zero-shot metrics
  phase_error_results.json      Phase error analysis
  results_inference_ablations.json  Ablation + probes
  hop1_probe/                   Hop-1 phase probe per seed
  hop2_atomic_probe/            Hop-2 atomic probe per seed
figures/                        Publication-quality figures (PNG)
README.md
requirements.txt

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Citation

@misc{holographic-memory-2026,
  title={Holographic Memory for Zero-Shot Compositional Reasoning
         in Knowledge Graphs: A Mechanistic Study of Where and Why It Fails},
  author={Kumar, Randhir},
  year={2026},
  note={Preprint}
}

Acknowledgments

Thanks to Modal for $30 in free GPU compute credits (A10G), which enabled all experiments.

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MIT License