ButterflyMoE: Sub-Linear Ternary Experts via Structured Butterfly Orbits

Linear memory scaling stores $N$ independent expert weight matrices requiring $\mathcal{O}(N \cdot d^2)$ memory, which exceeds edge devices memory budget. Current compression methods like quantization, pruning and low-rank factorization reduce constant factors but leave the scaling bottleneck unresolved. We introduce ButterflyMoE, a method that treats experts not as independent weight matrices but as geometric reorientations of a unified shared quantized substrate. Diversity among experts arises from viewing different angles of shared capacity, not from redundant storage. By applying learned rotations to a shared ternary prototype, each expert yields $\mathcal{O}(d^2 + N \cdot d \log d)$ memory,sub-linear in the number of experts. The key insight: training these rotations with quantization reduces activation outliers and stabilizes extreme low bit training, where static methods collapse. Across language modeling benchmarks, ButterflyMoE achieves 150$\times$ memory reduction at 256 experts with negligible accuracy loss. ButterflyMoE allows multiple experts to fit on edge-constrained devices showing that geometric parameterization breaks linear scaling.

LOOKAT: Lookup-Optimized Key-Attention for Memory-Efficient Transformers

Compressing the KV cache is a required step to deploy large language models on edge devices. Current quantization methods compress storage but fail to reduce bandwidth as attention calculation requires dequantizing keys from INT4/INT8 to FP16 before use. We observe that attention scoring is mathematically equivalent to the inner product similarity search and we can apply some compression techniques from vector databases to compress KV-cache better. We propose LOOKAT, which applies product quantization and asymmetric distance computation, to transformer architecture by decomposing key vectors into subspaces, learning codebooks and computing attention tables via lookup tables. This transforms attention from memory-bound to compute-bound. LOOKAT achieves 64 $\times$ compression at 95.7\% output fidelity and 32 $\times$ compression at 95.0\% fidelity when tested on GPT-2. LOOKAT requires no architecture changes or training while maintaining rank correlation $ρ> 0.95$. Theoretical analysis confirms that rank correlation degrades as $O(d_k/mK)$, with guarantees validated across sequence lengths up to 1024 tokens.