# Elastic Selective Spectral Hybrids for Train-Once, Export-Many Budgeted Inference

arXiv: https://arxiv.org/abs/2609.32486 · HTML: https://mlsanity.com/paper/2609.32486/

## Key facts

- No citing papers in the map yet (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 21 of its 72 references that are in the map.
- First posted on arXiv 26 Sep 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, 21 of 21)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [Longformer: The Long-Document Transformer](https://mlsanity.com/paper/2004.05150/) (2020)
- [Mistral 7B](https://mlsanity.com/paper/2310.06825/) (2023)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [The Pile: An 800GB Dataset of Diverse Text for Language Modeling](https://mlsanity.com/paper/2101.00027/) (2020)
- [GLU Variants Improve Transformer](https://mlsanity.com/paper/2002.05202/) (2020)
- [Compressive Transformers for Long-Range Sequence Modelling](https://mlsanity.com/paper/1911.05507/) (2019)
- [Slimmable Neural Networks](https://mlsanity.com/paper/1812.08928/) (2018)
- [Matryoshka Representation Learning](https://mlsanity.com/paper/2205.13147/) (2022)
- [Zamba: A Compact 7B SSM Hybrid Model](https://mlsanity.com/paper/2405.16712/) (2024)
- [Language Models Enable Simple Systems for Generating Structured Views of Heterogeneous Data Lakes](https://mlsanity.com/paper/2304.09433/) (2023)
- [Just read twice: closing the recall gap for recurrent language models](https://mlsanity.com/paper/2407.05483/) (2024)
- [Spectral State Space Models](https://mlsanity.com/paper/2312.06837/) (2023)
- [Flash STU: Fast Spectral Transform Units](https://mlsanity.com/paper/2409.10489/) (2024)
- [MatMamba: A Matryoshka State Space Model](https://mlsanity.com/paper/2410.06718/) (2024)
- [Training Matryoshka Mixture-of-Experts for Elastic Inference-Time Expert Utilization](https://mlsanity.com/paper/2509.26520/) (2025)
- [Nemotron Elastic: Towards Efficient Many-in-One Reasoning LLMs](https://mlsanity.com/paper/2511.16664/) (2025)
- [Star Elastic: Many-in-One Reasoning LLMs with Efficient Budget Control](https://mlsanity.com/paper/2605.07182/) (2026)
- [Elastic Spectral State Space Models for Train-Once Budgeted Inference](https://mlsanity.com/paper/2601.22488/) (2026)

## Abstract (arXiv)

Deploying a language model under different computing and latency budgets calls for compact models with different quality-cost trade-offs. Towards this end, elastic spectral state space models provide ordered and temporally decomposed channels that can be truncated, but they are associated with linear time-invariant filters that cannot selectively preserve relevant past information or forget irrelevant information as the context evolves. To address this, we introduce the Elastic Selective Spectral Hybrid (ESSH), which realizes each Hankel spectral channel as an independent recurrent unit using fitted damped rotation modes. It also features an input-dependent decay and write/read gates that make temporal retention and state update input-dependent while preserving channel-wise truncation and a structured recurrence for efficient execution. ESSH combines these selective spectral mixers with sliding-window attention and jointly trains multiple capacities by reducing spectral-channel count and feed-forward width at different rates through a two-rate capacity map with full-model distillation. The resulting models support chunked parallel training and fused recurrent decoding while avoiding computation for discarded channels. At full capacity, ESSH achieves language-modeling quality comparable to similarly sized independently trained models, while smaller exports exhibit a smooth quality-cost trade-off. We validate the effectiveness of the proposed framework using language understanding, retrieval, cross-domain text, and DNA experiments, by assessing quality retention and the trade-off against independently trained and elastic baselines. At the 1.53B model configuration, fused batch-one decoding takes 1.37 ms per token on a B300, providing a 2.14-2.80x speedup over the tested Mamba-2 and Mamba-3 implementations and 3.03x over Transformer++ at matched parameter counts.