# Multiplicative updates for symmetric-cone factorizations

arXiv: https://arxiv.org/abs/2108.00740 · HTML: https://mlsanity.com/paper/2108.00740/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 35 references that are in the map.
- First posted on arXiv 2 Aug 2021.

## Builds on (most cited first, 7 of 7)

- [Positive semidefinite rank](https://mlsanity.com/paper/1407.4095/) (2014)
- [Positive Semidefinite Matrix Factorization: A Connection with Phase Retrieval and Affine Rank Minimization](https://mlsanity.com/paper/2007.12364/) (2020)
- [On representing the positive semidefinite cone using the second-order cone](https://mlsanity.com/paper/1610.04901/) (2016)
- [Algorithms for Positive Semidefinite Factorization](https://mlsanity.com/paper/1707.07953/) (2017)
- [A Non-commutative Extension of Lee-Seung's Algorithm for Positive Semidefinite Factorizations](https://mlsanity.com/paper/2106.00293/) (2021)
- [Worst-Case Results For Positive Semidefinite Rank](https://mlsanity.com/paper/1305.4600/) (2013)
- [Computing approximate PSD factorizations](https://mlsanity.com/paper/1602.07351/) (2016)

## Abstract (arXiv)

Given a matrix X∈ ℝm× n_+ with non-negative entries, the cone factorization problem over a cone Ksubseteq ℝk concerns computing  a1,ldots, am  subseteq K and  b1,ldots, bn  subseteq~K^* belonging to its dual so that Xij = langle ai, bj rangle for all i∈ [m], j∈ [n]. Cone factorizations are fundamental to mathematical optimization as they allow us to express convex bodies as feasible regions of linear conic programs. In this paper, we introduce and analyze the symmetric-cone multiplicative update (SCMU) algorithm for computing cone factorizations when K is symmetric; i.e., it is self-dual and homogeneous. Symmetric cones are of central interest in mathematical optimization as they provide a common language for studying linear optimization over the nonnegative orthant (linear programs), over the second-order cone (second order cone programs), and over the cone of positive semidefinite matrices (semidefinite programs). The SCMU algorithm is multiplicative in the sense that the iterates are updated by applying a meticulously chosen automorphism of the cone computed using a generalization of the geometric mean to symmetric cones. Using an extension of Lieb's concavity theorem and von Neumann's trace inequality to symmetric cones, we show that the squared loss objective is non-decreasing along the trajectories of the SCMU algorithm. Specialized to the nonnegative orthant, the SCMU algorithm corresponds to the seminal algorithm by Lee and Seung for computing Nonnegative Matrix Factorizations.