# Positive semidefinite rank

arXiv: https://arxiv.org/abs/1407.4095 · HTML: https://mlsanity.com/paper/1407.4095/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jul 2014.

## Built on by (most cited first, 10 of 10)

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- [A Non-commutative Extension of Lee-Seung's Algorithm for Positive Semidefinite Factorizations](https://mlsanity.com/paper/2106.00293/) (2021)
- [Manifold-based Algorithms for the Hadamard Decomposition](https://mlsanity.com/paper/2605.28980/) (2026)
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- [Multiplicative updates for symmetric-cone factorizations](https://mlsanity.com/paper/2108.00740/) (2021)
- [On the Complexity of Low-Rank Matrix Signing and Entrywise Power Matrix Factorization](https://mlsanity.com/paper/2607.04875/) (2026)
- [Quantum Coordination Advantages in AI State-Tracking Tasks: Semantic Compilation and Latent Memory](https://mlsanity.com/paper/2608.11066/) (2026)

## Abstract (arXiv)

Let M be a p-by-q matrix with nonnegative entries. The positive semidefinite rank (psd rank) of M is the smallest integer k for which there exist positive semidefinite matrices Ai, Bj of size k × k such that Mij = trace(Ai Bj). The psd rank has many appealing geometric interpretations, including semidefinite representations of polyhedra and information-theoretic applications. In this paper we develop and survey the main mathematical properties of psd rank, including its geometry, relationships with other rank notions, and computational and algorithmic aspects.