# Cumsum-Composable Phase Transport for Low-Cost Streaming Keyword Spotting

arXiv: https://arxiv.org/abs/2607.20086 · HTML: https://mlsanity.com/paper/2607.20086/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 21 references that are in the map.
- First posted on arXiv 22 Jul 2026.

## Builds on (most cited first, 15 of 15)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [SpecAugment: A Simple Data Augmentation Method for Automatic Speech Recognition](https://mlsanity.com/paper/1904.08779/) (2019)
- [Simplified State Space Layers for Sequence Modeling](https://mlsanity.com/paper/2208.04933/) (2022)
- [Speech Commands: A Dataset for Limited-Vocabulary Speech Recognition](https://mlsanity.com/paper/1804.03209/) (2018)
- [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)
- [Hello Edge: Keyword Spotting on Microcontrollers](https://mlsanity.com/paper/1711.07128/) (2017)
- [LEAF: A Learnable Frontend for Audio Classification](https://mlsanity.com/paper/2101.08596/) (2021)
- [Temporal Convolution for Real-time Keyword Spotting on Mobile Devices](https://mlsanity.com/paper/1904.03814/) (2019)
- [Broadcasted Residual Learning for Efficient Keyword Spotting](https://mlsanity.com/paper/2106.04140/) (2021)
- [Streaming keyword spotting on mobile devices](https://mlsanity.com/paper/2005.06720/) (2020)
- [Latency Control for Keyword Spotting](https://mlsanity.com/paper/2206.07261/) (2022)

## Abstract (arXiv)

State-space sequence models are attractive for streaming speech because they maintain compact recurrent state, but scan-style training kernels can have unfavorable constants for short audio tasks. We study cumsum-composable phase transport, a streaming-native temporal layer for keyword spotting. Each layer projects acoustic frames to complex channels, transports them by learned unitary rotations, accumulates a finite window using prefix differences, and applies a gated residual update. The same prefix representation gives exact batched training with ordinary cumulative sums and exact online inference with one prefix update per frame. Unitary transport is the key constraint: inverse rotations have norm one, keeping prefix terms well conditioned while memory is supplied by windows or block readouts. On Google Speech Commands v2 with 12 labels, mel+cumsum models retain competitive accuracy with compact baselines. The strongest single-seed run reaches 97.3% test accuracy; a 51.6K-parameter tied model also reaches 97.3%, and a 24.8K tied model reaches 96.8% versus 97.1% for a 25.6K MelCNNMaxPool baseline. In a matched cumsum-versus-scan benchmark, cumsum+window gives comparable accuracy, 94.82% versus 94.33%, while training 1.07x faster and reducing single-example latency from 7.09 ms to 5.01 ms on a Tesla T4. These results support cumsum phase transport as a simple low-cost temporal primitive for streaming keyword spotting.