# Binary embeddings with structured hashed projections

arXiv: https://arxiv.org/abs/1511.05212 · HTML: https://mlsanity.com/paper/1511.05212/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Nov 2015.

## Built on by (most cited first, 3 of 3)

- [On the Expressive Power of Self-Attention Matrices](https://mlsanity.com/paper/2106.03764/) (2021)
- [LLC: Accurate, Multi-purpose Learnt Low-dimensional Binary Codes](https://mlsanity.com/paper/2106.01487/) (2021)
- [Rediscovering Hashed Random Projections for Efficient Quantization of Contextualized Sentence Embeddings](https://mlsanity.com/paper/2304.02481/) (2023)

## Abstract (arXiv)

We consider the hashing mechanism for constructing binary embeddings, that involves pseudo-random projections followed by nonlinear (sign function) mappings. The pseudo-random projection is described by a matrix, where not all entries are independent random variables but instead a fixed "budget of randomness" is distributed across the matrix. Such matrices can be efficiently stored in sub-quadratic or even linear space, provide reduction in randomness usage (i.e. number of required random values), and very often lead to computational speed ups. We prove several theoretical results showing that projections via various structured matrices followed by nonlinear mappings accurately preserve the angular distance between input high-dimensional vectors. To the best of our knowledge, these results are the first that give theoretical ground for the use of general structured matrices in the nonlinear setting. In particular, they generalize previous extensions of the Johnson-Lindenstrauss lemma and prove the plausibility of the approach that was so far only heuristically confirmed for some special structured matrices. Consequently, we show that many structured matrices can be used as an efficient information compression mechanism. Our findings build a better understanding of certain deep architectures, which contain randomly weighted and untrained layers, and yet achieve high performance on different learning tasks. We empirically verify our theoretical findings and show the dependence of learning via structured hashed projections on the performance of neural network as well as nearest neighbor classifier.