Graph Signal Processing Meets Mamba2: Adaptive Filter Bank via Delta Modulation

We present Harmonic, a hierarchical state space model (SSM) for language modeling. The architecture stacks three recurrent levels at progressively slower timescales; each level receives the prediction error of the level below as input, rather than its raw hidden state. On enwiki8 with equal token budgets, Harmonic outperforms a comparable Transformer (28M params) by +1.4% at 1K tokens, +6.7% at 8K tokens, and +11.4% at 32K tokens (bpt, lower is better). It also outperforms Mamba at every tested length by 0.7--1.8%. At 64K tokens, both Mamba and Transformer run out of memory on an 80GB H100; Harmonic trains successfully, reaching 6.169 bpt. Results replicate on WikiText-103 (H-TF gap +1.7% to +7.2% across 1K--32K). At 1B parameter scale, replacing all attention layers in TinyLlama 1.1B with HarmonicBlock eliminates the RoPE positional encoding limit: the resulting Hallamonic model maintains stable loss across sequence lengths 1K--8K on two independent clean benchmarks (Lambada and fineweb-edu held-out), while TinyLlama degrades catastrophically past its 2K-token RoPE limit (gap: +9.4 bpt at seq=8K on Lambada). Compute is O(L) per forward pass vs. O(L^2) for attention. Logs: https://github.com/Omibranch/harmonic-logs.

Manifold-Constrained Hyper-Connections (mHC) introduce a stability-motivated variant of multi stream residual mixing by constraining residual stream mixing matrices to the manifold of doubly stochastic matrices via Sinkhorn-Knopp projection. In his work, we study whether mHC-style constrained multi-stream residual topology transfers effectively to state space model (SSM) language modeling. We implement a static mHC mechanism around an SSM block by expanding the residual stream into multiple parallel streams, aggregating streams into a single SSM input through simplex-constrained pre-mixing, scattering the SSM output back to streams through simplex-constrained post-mixing, and applying Sinkhorn-projected residual stream mixing at each layer. We further introduce stream-specialized adapters that add lightweight stream-specific capacity through a shared bottleneck with per-stream scaling, applied both before stream aggregation and after the SSM output prior to scattering. We evaluate baseline single-stream SSM, static mHC SSM, and mHC SSM with adapters on WikiText-2 using identical training settings and report checkpoint-based validation loss, perplexity, throughput, and peak GPU memory. Under the reported fair checkpoint evaluation, static mHC improves validation loss from 6.3507 to 6.2448 and reduces perplexity from 572.91 to 515.35, while mHC with adapters further improves validation loss to 6.1353 and perplexity to 461.88. These gains are accompanied by modest throughput reductions from 1025.52 to 964.81 and 938.90 tokens per second, and increased peak memory from 2365 MB to 2568 MB and 3092 MB. The results suggest that mHC-inspired constrained multi-stream residual mixing can yield measurable quality improvements in SSM language models and that stream-specialized adapter capacity can further enhance performance with predictable efficiency tradeoffs.

While parameter-efficient fine-tuning (PEFT) typically targets attention projectors, its efficacy for tasks requiring sequential state accumulation remains under-explored. We examine if PEFT for such tasks can benefit from state space model (SSMs) adapters, and if MLP blocks are better injection sites. We introduce Hankel Reduced order Model (HRM) adapter, an SSM-based residual module initialized via Balanced Truncation of empirical Hankel Grammians. By leveraging the time-invariance of the system matrix $\bar{A}$, HRM enables an exact FFT-based parallel scan, achieving computational parity with LoRA across all context lengths. In iso-parametric evaluations on Mistral-7B (8.4M trainable parameters), HRM outperforms LoRA variants on LongBench tasks, including QuALITY (+34.8% relative accuracy) and QMSum (+71.6% relative ROUGE-1). HRM further demonstrates consistent superiority across 18 configurations of synthetic state-tracking (DFA, Parity) and character-level language modeling (enwik8). Gate analysis reveals that HRM adapters effectively learn to modulate recurrence, providing a robust architectural alternative to low-rank adaptation for long-context sequence modeling.