Technical Progress Update: AGEP PyTorch Pilot for Multi-Head Attention Diagnostics
Status: Initial implementation and synthetic-case diagnostics completed; end-to-end LLM fine-tuning speedups and energy-efficiency claims remain to be tested.
I have implemented an audit-ready pilot PyTorch hook API, AGEPAttentionHookV2, that attaches to standard nn.MultiheadAttention layers and computes algebraic-geometric and empirical-process diagnostics for rank-adaptive attention.
The interactive Google Colab notebook provides a reproducible starting point for examining per-head rank structure, local Schur complements, and truncation distortion bounds in a controlled multi-head attention experiment.
Reproducible pilot (Google Colab):
https://colab.research.google.com/drive/1ueLY8TygEimHR1ulUx23t5pyUhiS_V_x?usp=sharing
What the pilot currently demonstrates
Direct Parameter & Covariance Inspection: A working prototype hook that inspects internal per-head projection blocks (\bm{W_Q^{(h)}, W_K^{(h)}, W_V^{(h)} \in \mathbb{R}^{d_k \times E}}) from module.in_proj_weight alongside empirical token covariance spectra (\bm{\hat{\Sigma}_X}) from forward activations.
Explicit Separation of Capacity Baselines:
1. JMLR Polynomial Score Surrogate (\bm{d_k \times d_k} subspace, \bm{d_k=4, r=1}): Confirms machine-precision vanishing of the skew-symmetric quadratic form (\bm{\max \vert{}x^\top \mathrm{skew}(A) x\vert{} \sim 10^{-8}}), tightening the low-rank score+value capacity exponent from \bm{D_r^{\mathrm{gen}} = r(2d_k-r)+d_k^2 = 23} to \bm{D_r^{\mathrm{sym}} = \frac{r(2d_k-r+1)}{2}+d_k^2 = 20} (13.04% cut vs. generic low-rank; 37.50% cut vs. unconstrained \bm{2d_k^2 = 32}).
2. IEEE TIT Linear Projection Manifold (\bm{d_{\mathrm{out}} \times d_{\mathrm{in}}} projection, \bm{4 \times 16, s=1}): Quantifies the manifold effective dimension \bm{p_{\mathrm{eff}}(s) = s(d_{\mathrm{out}}+d_{\mathrm{in}}-s) = 19} against the ambient projection dimension \bm{d_{\mathrm{out}} d_{\mathrm{in}} = 64} (70.3125% reduction in ambient degrees of freedom).
Controlled Pruning Verification (Collapsed vs. Full-Rank Control):
When Head #0 is synthetically restricted to rank 1 (\bm{\sum_{j>1}\sigma_j(W)^2 \sim 10^{-16}}, \bm{K_{\mathrm{NF}} \sim 10^{-14}}), rank-1 SVD truncation yields a floating-point output deviation at machine precision (\bm{\Vert{}Y_{\mathrm{before}} - Y_{\mathrm{after}}\Vert{}_F / \Vert{}Y_{\mathrm{before}}\Vert{}_F \sim 10^{-7}} in FP32).
Conversely, applying the same rank-1 truncation to full-rank Head #1 (\bm{K_{\mathrm{NF}} \approx 0.14}) produces significant output distortion (relative error \bm{> 10^{-1}}), confirming that the diagnostic correctly distinguishes prunable heads from active full-rank heads.
Delineation of Conditional Theoretical RLCT: Clarifies that the Real Log Canonical Threshold \bm{\lambda_{\mathrm{NF}} = \frac{1+d_k^2}{2} = 8.50} (\bm{d_k=4}) is an analytical closed-form value derived under monoidal desingularization of the determinantal normal form \bm{K_{\mathrm{NF}}(x,y,z,w,v) = (xy-zw)^2 + \Vert{}v\Vert{}_2^2}, which governs local asymptotics conditionally when a head approaches the singular locus (\bm{K_{\mathrm{NF}} \to 0}).
What remains to be established
The current pilot is a synthetic-case verification of the mathematical diagnostics; it is not yet evidence of general lossless pruning in pre-trained LLMs, end-to-end training speedups, or measured GPU energy savings. The behavior of approximate rank decay (\bm{\sigma_{j>s} > 0}) during real gradient-based fine-tuning requires empirical validation on standard benchmarks.
Next milestone (Months 1–4 Target with Initial Funding)
With initial funding, I will:
1. Validate the hook diagnostics against independent numerical reference checks across FP32/FP64 precision and varying condition numbers.
2. Quantify the accuracy–distortion trade-off on pre-trained small-scale Transformer layers where singular values decay gradually rather than vanishing exactly.
3. Implement a controlled benchmark comparing rank-adaptive updates against standard fixed-rank LoRA, tracking task performance, FLOPs, wall-clock time, peak VRAM, and hardware power draw (nvidia-smi / pyNVML telemetry).
The goal of this project is to build a bridge between algebraic-geometric theory and practical methods for energy-efficient LLM fine-tuning.
