# Projection onto the probability simplex: An efficient algorithm with a simple proof, and an application

arXiv: https://arxiv.org/abs/1309.1541 · HTML: https://mlsanity.com/paper/1309.1541/

## Key facts

- Cited by 67 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Sep 2013.

## Built on by (most cited first, 30 of 67)

- [Dirichlet Flow Matching with Applications to DNA Sequence Design](https://mlsanity.com/paper/2402.05841/) (2024)
- [Antipodes of Label Differential Privacy: PATE and ALIBI](https://mlsanity.com/paper/2106.03408/) (2021)
- [Gradient play in stochastic games: stationary points, convergence, and sample complexity](https://mlsanity.com/paper/2106.00198/) (2021)
- [Direction-oriented Multi-objective Learning: Simple and Provable Stochastic Algorithms](https://mlsanity.com/paper/2305.18409/) (2023)
- [Full-Atom Peptide Design based on Multi-modal Flow Matching](https://mlsanity.com/paper/2406.00735/) (2024)
- [A Novel Framework for Policy Mirror Descent with General Parameterization and Linear Convergence](https://mlsanity.com/paper/2301.13139/) (2023)
- [Policy Gradient Algorithms for Robust MDPs with Non-Rectangular Uncertainty Sets](https://mlsanity.com/paper/2305.19004/) (2023)
- [The Max-Min Formulation of Multi-Objective Reinforcement Learning: From Theory to a Model-Free Algorithm](https://mlsanity.com/paper/2406.07826/) (2024)
- [Fast and High-Quality Blind Multi-Spectral Image Pansharpening](https://mlsanity.com/paper/2103.09943/) (2021)
- [The Hitchhiker's Guide to Prior-Shift Adaptation](https://mlsanity.com/paper/2106.11695/) (2021)
- [Robust Group Synchronization via Quadratic Programming](https://mlsanity.com/paper/2206.08994/) (2022)
- [Inaccurate Label Distribution Learning](https://mlsanity.com/paper/2302.13000/) (2023)
- [Foundations of Top-k Decoding For Language Models](https://mlsanity.com/paper/2505.19371/) (2025)
- [Hotspot-Driven Peptide Design via Multi-Fragment Autoregressive Extension](https://mlsanity.com/paper/2411.18463/) (2024)
- [Birds of a Feather Flock Together: A Close Look at Cooperation Emergence via Multi-Agent RL](https://mlsanity.com/paper/2104.11455/) (2021)
- [Optimal and Robust Category-level Perception: Object Pose and Shape Estimation from 2D and 3D Semantic Keypoints](https://mlsanity.com/paper/2206.12498/) (2022)
- [Instance-weighted Central Similarity for Multi-label Image Retrieval](https://mlsanity.com/paper/2108.05274/) (2021)
- [BanditQ: Fair Bandits with Guaranteed Rewards](https://mlsanity.com/paper/2304.05219/) (2023)
- [Bypass Back-propagation: Optimization-based Structural Pruning for Large Language Models via Policy Gradient](https://mlsanity.com/paper/2406.10576/) (2024)
- [Neural Probabilistic Circuits: Enabling Compositional and Interpretable Predictions through Logical Reasoning](https://mlsanity.com/paper/2501.07021/) (2025)
- [Balanced Rate-Distortion Optimization in Learned Image Compression](https://mlsanity.com/paper/2502.20161/) (2025)
- [Learning Globally Smooth Functions on Manifolds](https://mlsanity.com/paper/2210.00301/) (2022)
- [Exact Solution to Data-Driven Inverse Optimization of MILPs in Finite Time via Gradient-Based Methods](https://mlsanity.com/paper/2405.14273/) (2024)
- [On the Robustness of Randomized Ensembles to Adversarial Perturbations](https://mlsanity.com/paper/2302.01375/) (2023)
- [A proof of convergence of inverse reinforcement learning for multi-objective optimization](https://mlsanity.com/paper/2305.06137/) (2023)
- [Unveiling the Role of Message Passing in Dual-Privacy Preservation on GNNs](https://mlsanity.com/paper/2308.13513/) (2023)
- [Primal Methods for Variational Inequality Problems with Functional Constraints](https://mlsanity.com/paper/2403.12859/) (2024)
- [ShortListing Model: A Streamlined SimplexDiffusion for Discrete Variable Generation](https://mlsanity.com/paper/2508.17345/) (2025)
- [Maximum Risk Minimization with Random Forests](https://mlsanity.com/paper/2512.10445/) (2025)
- [Policy Gradient Algorithms in Average-Reward Multichain MDPs](https://mlsanity.com/paper/2602.18003/) (2026)

## Abstract (arXiv)

We provide an elementary proof of a simple, efficient algorithm for computing the Euclidean projection of a point onto the probability simplex. We also show an application in Laplacian K-modes clustering.