# The Alternating Descent Conditional Gradient Method for Sparse Inverse Problems

arXiv: https://arxiv.org/abs/1507.01562 · HTML: https://mlsanity.com/paper/1507.01562/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Jul 2015.

## Built on by (most cited first, 11 of 11)

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- [Thinned Mean Field Langevin Dynamics](https://mlsanity.com/paper/2605.28589/) (2026)
- [Fenchel-Young Duality Gaps: Certified Early Stopping for Regularized Inverse Problems](https://mlsanity.com/paper/2609.17629/) (2026)
- [SparseAlign: A Super-Resolution Algorithm for Automatic Marker Localization and Deformation Estimation in Cryo-Electron Tomography](https://mlsanity.com/paper/2201.08706/) (2022)
- [Simultaneous off-the-grid learning of mixtures issued from a continuous dictionary](https://mlsanity.com/paper/2210.16311/) (2022)
- [Gridless 2D Recovery of Lines using the Sliding Frank-Wolfe Algorithm](https://mlsanity.com/paper/2403.11649/) (2024)
- [Effective regions and kernels in continuous sparse regularisation, with application to sketched mixtures](https://mlsanity.com/paper/2507.08444/) (2025)

## Abstract (arXiv)

We propose a variant of the classical conditional gradient method for sparse inverse problems with differentiable measurement models. Such models arise in many practical problems including superresolution, time-series modeling, and matrix completion. Our algorithm combines nonconvex and convex optimization techniques: we propose global conditional gradient steps alternating with nonconvex local search exploiting the differentiable measurement model. This hybridization gives the theoretical global optimality guarantees and stopping conditions of convex optimization along with the performance and modeling flexibility associated with nonconvex optimization. Our experiments demonstrate that our technique achieves state-of-the-art results in several applications.