SU(d) Spectral Associative Memory
Implementation & benchmark of arXiv:2608.08226 — High-Capacity Generalized Hopfield Networks
Source paper: High-Capacity Generalized Hopfield Networks arXiv:2608.08226 ⓘ · analyzed Sep 1, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Replace vector-valued Hopfield neurons by SU(d)-valued latent states and construct Hebbian couplings from matrix memories. Recall is performed by iterating toward the dominant eigenmode of the induced lifted coupling operator, with each iterate projected back onto SU(d); the larger matrix representation should reduce random crosstalk and increase critical memory capacity.
Formulas
Mathematical statement
Let U_i in SU(d) be the state of neuron i, M_i^mu in SU(d) the mu-th memory at neuron i, and P the number of stored memories. Define the Frobenius inner product by <A,B> = Re Tr(A dagger B). The lifted Hebbian operator is J_ij(X) = N^(-1) sum_mu M_i^mu (M_j^mu) dagger X. The energy is E(U) = -0.5 sum_ij <U_i, J_ij(U_j)>. In the lifted representation, the coupling has a signal-plus-noise form J = J_signal + J_noise. Retrieval is expected when the signal eigenvalue lambda_s exceeds the upper noise spectral edge lambda_plus(J_noise). After a linear update, return each matrix to SU(d) with the unitary polar projection Pi(X) = Q / det(Q)^(1/d), where Q = X (X dagger X)^(-1/2).
Implementation notes
Integration point: implement an associative-memory layer whose hidden state is a batch of N complex d by d matrices. Initialize the state from a corrupted cue and store P matrix memories. Avoid explicitly forming the full N by N operator. Pseudocode: set U_i to the cue; for each iteration compute X_i = N^(-1) sum over j and mu of M_i^mu (M_j^mu) dagger U_j; compute the polar factor Q_i = X_i (X_i dagger X_i + epsilon I)^(-1/2); normalize its determinant and assign U_i = Q_i. Track the overlap O_mu = N^(-1) sum_i Re Tr(U_i dagger M_i^mu) / d. The Hebbian action and polar projection are computed exactly; estimate the spectral boundary with power iteration or Lanczos on the matrix-free operator, and estimate the noise edge using shuffled memories or a no-signal realization. First cheap experiment: use N = 256, random SU(2), SU(3), and SU(4) memories, vary alpha = P/N, corrupt cues by random local unitary rotations, and compare with vector Hopfield and continuous Hopfield baselines at similar parameter count. The quantitative prediction is a sharp retrieval transition when the top eigenvalue outlier crosses the measured noise edge; critical alpha should increase with d and SU(3) should outperform the vector baseline.
Verification
Stage 1 · Toy mechanism gate: Failed ✗
Stage 2 · Mechanism transferred to benchmark: Not tested
Stage 2 · Practical benchmark result: Not run
Methodology: Toy-system gate first; the benchmark stage runs only after a pass. How verification works
Stage 1 — Mechanism check agent confidence 9/10
Built a matrix-free SU(d) associative-memory layer with Hebbian action, SU(d) polar projection, overlap tracking, lifted spectral diagnostics, and a matched normalized vector baseline. The projection check passed with maximum unitarity error below 7.1e-8 and determinant error below 3.6e-8; for P=1, the predicted signal eigenvalue 1.0 matched to numerical precision (residuals 1.2e-16–4.0e-16). However, the dimension-scaling prediction failed: observed signal/noise gaps were 0.138 (d=2), 0.101 (d=3), and -0.053 (d=4), while retrieval never exceeded the 0.75 success threshold at any tested load or dimension and did not beat the vector baseline.
- Agent confidence
- 9/10
- Baseline
- Vector Hopfield final target overlap: 0.094–0.052 (d=2), 0.046–0.024 (d=3), 0.026–0.018 (d=4) across alpha=0.125–1.0.
- Idea
- SU(d) final target overlap: 0.357–0.174 (d=2), 0.304–0.165 (d=3), 0.219–0.101 (d=4); no tested condition reached the 0.75 retrieval-success threshold. Spectral gaps versus shuffled reference: d=2 0.138, d=3 0.101, d=4 -0.053.
Limitations: Only small finite-size CPU experiments were tested, with one realization per retrieval cell, N=64, alpha up to 1, d in {2,3,4}, and 20% site-replacement corruption. No large-N capacity estimate, repeated statistical trials, power-iteration implementation, GPU benchmark, or comparison to the paper's Lie-algebra/LLG dynamics was performed.
How to run: python3 su_memory_experiment.py
Verdict computed by deterministic test code from paired-seed statistics — not by the language model.
Artifacts
- report.md 1.7 KB View
- results.json 5.0 KB View Raw
- su_memory_experiment.py 5.3 KB View Raw