# Optimal Decay Spectra for Linear Recurrences

arXiv: https://arxiv.org/abs/2604.07658 · HTML: https://mlsanity.com/paper/2604.07658/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Core SSM architectures for language, one of 10 directions in State-space sequence models (S4, Mamba).
- Builds on 25 of its 41 references that are in the map.
- First posted on arXiv 8 Apr 2026.

## Placed in

- State-space sequence models (S4, Mamba) / [Core SSM architectures for language](https://mlsanity.com/ssm/core-ssm-architectures-for-language/)

## Builds on (most cited first, 25 of 25)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality](https://mlsanity.com/paper/2405.21060/) (2024)
- [Train Short, Test Long: Attention with Linear Biases Enables Input Length Extrapolation](https://mlsanity.com/paper/2108.12409/) (2021)
- [RWKV: Reinventing RNNs for the Transformer Era](https://mlsanity.com/paper/2305.13048/) (2023)
- [Simplified State Space Layers for Sequence Modeling](https://mlsanity.com/paper/2208.04933/) (2022)
- [Retentive Network: A Successor to Transformer for Large Language Models](https://mlsanity.com/paper/2307.08621/) (2023)
- [HiPPO: Recurrent Memory with Optimal Polynomial Projections](https://mlsanity.com/paper/2008.07669/) (2020)
- [YaRN: Efficient Context Window Extension of Large Language Models](https://mlsanity.com/paper/2309.00071/) (2023)
- [Gated Delta Networks: Improving Mamba2 with Delta Rule](https://mlsanity.com/paper/2412.06464/) (2024)
- [Gated Linear Attention Transformers with Hardware-Efficient Training](https://mlsanity.com/paper/2312.06635/) (2023)
- [On the Parameterization and Initialization of Diagonal State Space Models](https://mlsanity.com/paper/2206.11893/) (2022)
- [Diagonal State Spaces are as Effective as Structured State Spaces](https://mlsanity.com/paper/2203.14343/) (2022)
- [Griffin: Mixing Gated Linear Recurrences with Local Attention for Efficient Language Models](https://mlsanity.com/paper/2402.19427/) (2024)
- [Zoology: Measuring and Improving Recall in Efficient Language Models](https://mlsanity.com/paper/2312.04927/) (2023)
- [Eagle and Finch: RWKV with Matrix-Valued States and Dynamic Recurrence](https://mlsanity.com/paper/2404.05892/) (2024)
- [RWKV-7 "Goose" with Expressive Dynamic State Evolution](https://mlsanity.com/paper/2503.14456/) (2025)
- [Mamba-3: Improved Sequence Modeling using State Space Principles](https://mlsanity.com/paper/2603.15569/) (2026)
- [DeciMamba: Exploring the Length Extrapolation Potential of Mamba](https://mlsanity.com/paper/2406.14528/) (2024)
- [LongMamba: Enhancing Mamba's Long Context Capabilities via Training-Free Receptive Field Enlargement](https://mlsanity.com/paper/2504.16053/) (2025)
- [Uncovering the Spectral Bias in Diagonal State Space Models](https://mlsanity.com/paper/2508.20441/) (2025)

## Built on by (most cited first, 1 of 1)

- [Key-Value Means: Transformers with Expandable Block-Recurrent Compressed Memory](https://mlsanity.com/paper/2605.09877/) (2026)

## Abstract (arXiv)

Linear recurrent models offer linear-time sequence processing but often suffer from suboptimal long-range memory. We trace this to the decay spectrum: for N channels, random initialization collapses the minimum spectral gap to O(N-2), yielding sub-exponential error exp(-Omega(N/log N)); linear spacing avoids collapse but degrades to exp(-O(N/√T)), practically algebraic over long contexts. We introduce Position-Adaptive Spectral Tapering (PoST), an architecture-agnostic framework combining two mechanisms: (1) Spectral Reparameterization, which structurally enforces geometrically spaced log-decay rates, proven minimax optimal at rate O(exp(-cN/log T)); and (2) Position-Adaptive Scaling, the provably unique mechanism that eliminates the scale mismatch of static spectra (where only Nlog t/log T of N channels are effective at position t) by stretching the spectrum to the actual dependency range, sharpening the rate to O(exp(-cN/log t)). This scaling natively induces fractional invariance: the impulse response becomes scale-free, with channels interpolating between relative and absolute temporal coordinates. PoST integrates into any diagonal linear recurrence without overhead. We instantiate it across Mamba-2, RWKV-7, Gated DeltaNet, Gated Linear Attention, and RetNet. Pre-training at 180M-440M scales shows consistent zero-shot language modeling improvements, significant long-context retrieval gains for Mamba-2 (MQAR and NIAH), and competitive or improved performance across other architectures. Code: https://github.com/SiLifen/PoST.