# From the simplex to the sphere: Faster constrained optimization using the Hadamard parametrization

arXiv: https://arxiv.org/abs/2112.05273 · HTML: https://mlsanity.com/paper/2112.05273/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Dec 2021.

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

- [Differentiating Through Integer Linear Programs with Quadratic Regularization and Davis-Yin Splitting](https://mlsanity.com/paper/2301.13395/) (2023)
- [Minimizing low-rank models of high-order tensors: Hardness, span, tight relaxation, and applications](https://mlsanity.com/paper/2210.11413/) (2022)
- [On the Linear Convergence of Policy Gradient under Hadamard Parameterization](https://mlsanity.com/paper/2305.19575/) (2023)
- [A space-decoupling framework for optimization on bounded-rank matrices with orthogonally invariant constraints](https://mlsanity.com/paper/2501.13830/) (2025)
- [Two Models for Surface Segmentation using the Total Variation of the Normal Vector](https://mlsanity.com/paper/2412.00445/) (2024)
- [Convex optimization over a probability simplex](https://mlsanity.com/paper/2305.09046/) (2023)
- [Optimization on the Oblique Manifold for Sparse Simplex Constraints via Multiplicative Updates](https://mlsanity.com/paper/2503.24075/) (2025)
- [Entropic Mirror Descent for Linear Systems: Polyak's Stepsize and Implicit Bias](https://mlsanity.com/paper/2505.02614/) (2025)
- [Hadamard-Riemannian Optimization for Margin-Variance Ensemble](https://mlsanity.com/paper/2509.10189/) (2025)
- [Smooth Reparameterizations of Functions on Simplicial Product Spaces: Applications to Probabilistic Tensor Decomposition and Functional Data Registration](https://mlsanity.com/paper/2608.02576/) (2026)

## Abstract (arXiv)

The standard simplex in R^n, also known as the probability simplex, is the set of nonnegative vectors whose entries sum up to 1. They frequently appear as constraints in optimization problems that arise in machine learning, statistics, data science, operations research, and beyond. We convert the standard simplex to the unit sphere and thus transform the corresponding constrained optimization problem into an optimization problem on a simple, smooth manifold. We show that KKT points and strict-saddle points of the minimization problem on the standard simplex all correspond to those of the transformed problem, and vice versa. So, solving one problem is equivalent to solving the other problem. Then, we propose several simple, efficient, and projection-free algorithms using the manifold structure. The equivalence and the proposed algorithm can be extended to optimization problems with unit simplex, weighted probability simplex, or `1-norm sphere constraints. Numerical experiments between the new algorithms and existing ones show the advantages of the new approach