# HOT: Hadamard-based Optimized Training

arXiv: https://arxiv.org/abs/2503.21261 · HTML: https://mlsanity.com/paper/2503.21261/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 47 references that are in the map.
- First posted on arXiv 27 Mar 2025.

## Builds on (most cited first, 30 of 34)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [The Cityscapes Dataset for Semantic Urban Scene Understanding](https://mlsanity.com/paper/1604.01685/) (2016)
- [SegFormer: Simple and Efficient Design for Semantic Segmentation with Transformers](https://mlsanity.com/paper/2105.15203/) (2021)
- [HuggingFace's Transformers: State-of-the-art Natural Language Processing](https://mlsanity.com/paper/1910.03771/) (2019)
- [Make-A-Video: Text-to-Video Generation without Text-Video Data](https://mlsanity.com/paper/2209.14792/) (2022)
- [Align your Latents: High-Resolution Video Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2304.08818/) (2023)
- [SmoothQuant: Accurate and Efficient Post-Training Quantization for Large Language Models](https://mlsanity.com/paper/2211.10438/) (2022)
- [EfficientNetV2: Smaller Models and Faster Training](https://mlsanity.com/paper/2104.00298/) (2021)
- [Extending Context Window of Large Language Models via Positional Interpolation](https://mlsanity.com/paper/2306.15595/) (2023)
- [MCVD: Masked Conditional Video Diffusion for Prediction, Generation, and Interpolation](https://mlsanity.com/paper/2205.09853/) (2022)
- [Long-context LLMs Struggle with Long In-context Learning](https://mlsanity.com/paper/2404.02060/) (2024)
- [EfficientFormer: Vision Transformers at MobileNet Speed](https://mlsanity.com/paper/2206.01191/) (2022)
- [LongNet: Scaling Transformers to 1,000,000,000 Tokens](https://mlsanity.com/paper/2307.02486/) (2023)
- [FP8 Formats for Deep Learning](https://mlsanity.com/paper/2209.05433/) (2022)
- [StyleSwin: Transformer-based GAN for High-resolution Image Generation](https://mlsanity.com/paper/2112.10762/) (2021)
- [Training Transformers with 4-bit Integers](https://mlsanity.com/paper/2306.11987/) (2023)
- [A Statistical Framework for Low-bitwidth Training of Deep Neural Networks](https://mlsanity.com/paper/2010.14298/) (2020)
- [NIPQ: Noise proxy-based Integrated Pseudo-Quantization](https://mlsanity.com/paper/2206.00820/) (2022)
- [Accurate Neural Training with 4-bit Matrix Multiplications at Standard Formats](https://mlsanity.com/paper/2112.10769/) (2021)
- [NICE: Noise Injection and Clamping Estimation for Neural Network Quantization](https://mlsanity.com/paper/1810.00162/) (2018)
- [Understanding INT4 Quantization for Transformer Models: Latency Speedup, Composability, and Failure Cases](https://mlsanity.com/paper/2301.12017/) (2023)
- [ResMaster: Mastering High-Resolution Image Generation via Structural and Fine-Grained Guidance](https://mlsanity.com/paper/2406.16476/) (2024)
- [NITI: Training Integer Neural Networks Using Integer-only Arithmetic](https://mlsanity.com/paper/2009.13108/) (2020)

## Built on by (most cited first, 2 of 2)

- [OSCAR: Offline Spectral Covariance-Aware Rotation for 2-bit KV Cache Quantization](https://mlsanity.com/paper/2605.17757/) (2026)
- [AdaHOP: Fast and Accurate Low-Precision Training via Outlier-Pattern-Aware Rotation](https://mlsanity.com/paper/2604.02525/) (2026)

## Abstract (arXiv)

It has become increasingly important to optimize backpropagation to reduce memory usage and computational overhead. Achieving this goal is highly challenging, as multiple objectives must be considered jointly while maintaining training quality. In this paper, we focus on matrix multiplication, which accounts for the largest portion of training costs, and analyze its backpropagation in detail to identify lightweight techniques that offer the best benefits. Based on this analysis, we introduce a novel method, Hadamard-based Optimized Training (HOT). In this approach, we apply Hadamard-based optimizations, such as Hadamard quantization and Hadamard low-rank approximation, selectively and with awareness of the suitability of each optimization for different backward paths. Additionally, we introduce two enhancements: activation buffer compression and layer-wise quantizer selection. Our extensive analysis shows that HOT achieves up to 75% memory savings and a 2.6 times acceleration on real GPUs, with negligible accuracy loss compared to FP32 precision.