# Echo: KV-Cache-Free Associative Recall with Spectral Koopman Operators

arXiv: https://arxiv.org/abs/2605.06997 · HTML: https://mlsanity.com/paper/2605.06997/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 28 of its 39 references that are in the map.
- First posted on arXiv 7 May 2026.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)
- LLM agents and tool use / [Agent memory systems](https://mlsanity.com/agents/agent-memory-systems/)
- Efficient LLM inference and serving / [KV cache optimization](https://mlsanity.com/inference/kv-cache-optimization/)

## Builds on (most cited first, 28 of 28)

- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [PIQA: Reasoning about Physical Commonsense in Natural Language](https://mlsanity.com/paper/1911.11641/) (2019)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [GLU Variants Improve Transformer](https://mlsanity.com/paper/2002.05202/) (2020)
- [Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality](https://mlsanity.com/paper/2405.21060/) (2024)
- [The FineWeb Datasets: Decanting the Web for the Finest Text Data at Scale](https://mlsanity.com/paper/2406.17557/) (2024)
- [The LAMBADA dataset: Word prediction requiring a broad discourse context](https://mlsanity.com/paper/1606.06031/) (2016)
- [SnapKV: LLM Knows What You are Looking for Before Generation](https://mlsanity.com/paper/2404.14469/) (2024)
- [H2O: Heavy-Hitter Oracle for Efficient Generative Inference of Large Language Models](https://mlsanity.com/paper/2306.14048/) (2023)
- [Simplified State Space Layers for Sequence Modeling](https://mlsanity.com/paper/2208.04933/) (2022)
- [What learning algorithm is in-context learning? Investigations with linear models](https://mlsanity.com/paper/2211.15661/) (2022)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Jamba: A Hybrid Transformer-Mamba Language Model](https://mlsanity.com/paper/2403.19887/) (2024)
- [Parallelizing Linear Transformers with the Delta Rule over Sequence Length](https://mlsanity.com/paper/2406.06484/) (2024)
- [Diagonal State Spaces are as Effective as Structured State Spaces](https://mlsanity.com/paper/2203.14343/) (2022)
- [Repeat After Me: Transformers are Better than State Space Models at Copying](https://mlsanity.com/paper/2402.01032/) (2024)
- [WinoGrande: An Adversarial Winograd Schema Challenge at Scale](https://mlsanity.com/paper/1907.10641/) (2019)
- [Zoology: Measuring and Improving Recall in Efficient Language Models](https://mlsanity.com/paper/2312.04927/) (2023)
- [One Step of Gradient Descent is Provably the Optimal In-Context Learner with One Layer of Linear Self-Attention](https://mlsanity.com/paper/2307.03576/) (2023)
- [Mamba-3: Improved Sequence Modeling using State Space Principles](https://mlsanity.com/paper/2603.15569/) (2026)
- [MesaNet: Sequence Modeling by Locally Optimal Test-Time Training](https://mlsanity.com/paper/2506.05233/) (2025)
- [Inference-Time Hyper-Scaling with KV Cache Compression](https://mlsanity.com/paper/2506.05345/) (2025)
- [Lost in Backpropagation: The LM Head is a Gradient Bottleneck](https://mlsanity.com/paper/2603.10145/) (2026)
- [Preconditioned DeltaNet: Curvature-aware Sequence Modeling for Linear Recurrences](https://mlsanity.com/paper/2604.21100/) (2026)
- [Training Dynamics of Softmax Self-Attention: Fast Global Convergence via Preconditioning](https://mlsanity.com/paper/2603.01514/) (2026)

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

- [Anatomy of Associative Recall in Fixed-State Recurrences: A Matched-State Decomposition, an Interference Wall, and a Curriculum That Breaks It](https://mlsanity.com/paper/2609.16183/) (2026)

## Abstract (arXiv)

Long chain-of-thought reasoning and agentic tool-calling produce traces spanning tens of thousands of tokens, yet Transformer KV caches grow linearly with sequence length, creating a memory bottleneck on commodity hardware. State-space models offer constant-memory recurrence but suffer a memory cliff: retrieval accuracy collapses once the gap between a stored fact and its query exceeds the effective horizon of the recurrent state. We introduce Echo, a KV-cache-free associative recall architecture built around Spectral Koopman Attention (SKA); a drop-in replacement for attention layers that augments SSM blocks with a closed-form dynamical operator whose sufficient statistics are accumulated in constant memory with no KV cache. Echo fits a spectral linear system to the key and value history via kernel ridge regression and retrieves through a learned power-iterated filter, all from O(r2) streaming state where r is a small projection rank. On the Multi-Query Associative Recall benchmark, a pure Mamba-2 SSM fails to exceed chance accuracy (~3%) across all gap lengths and KV-pair counts, while at the 50M parameter scale SKA-augmented models achieve 100% retrieval accuracy on every configuration tested, including distractor gaps of 4,096 tokens with 32 KV pairs. Across five additional transfer benchmarks including needle-in-a-haystack, tool-trace, and multi-hop retrieval, SKA consistently outperforms both pure SSM and SSM+Attention hybrids while maintaining constant inference memory. Ablations confirm that the spectral operator, not the prefix masking strategy, drives the retrieval gain.