# Inertial Proximal Alternating Linearized Minimization (iPALM) for Nonconvex and Nonsmooth Problems

arXiv: https://arxiv.org/abs/1702.02505 · HTML: https://mlsanity.com/paper/1702.02505/

## Key facts

- Cited by 40 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Feb 2017.

## Built on by (most cited first, 30 of 40)

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- [Stable Deep MRI Reconstruction using Generative Priors](https://mlsanity.com/paper/2210.13834/) (2022)
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- [Explicit Diffusion of Gaussian Mixture Model Based Image Priors](https://mlsanity.com/paper/2302.08411/) (2023)
- [Learned Alternating Minimization Algorithm for Dual-domain Sparse-View CT Reconstruction](https://mlsanity.com/paper/2306.02644/) (2023)
- [Proximal Residual Flows for Bayesian Inverse Problems](https://mlsanity.com/paper/2211.17158/) (2022)
- [Block Alternating Bregman Majorization Minimization with Extrapolation](https://mlsanity.com/paper/2107.04395/) (2021)
- [Deep, Deep Learning with BART](https://mlsanity.com/paper/2202.14005/) (2022)
- [A Momentum Accelerated Algorithm for ReLU-based Nonlinear Matrix Decomposition](https://mlsanity.com/paper/2402.02442/) (2024)
- [An Accelerated Alternating Partial Bregman Algorithm for ReLU-based Matrix Decomposition](https://mlsanity.com/paper/2503.02386/) (2025)
- [Inertial Proximal Deep Learning Alternating Minimization for Efficient Neutral Network Training](https://mlsanity.com/paper/2102.00267/) (2021)
- [A Framework of Inertial Alternating Direction Method of Multipliers for Non-Convex Non-Smooth Optimization](https://mlsanity.com/paper/2102.05433/) (2021)
- [Efficiently Factorizing Boolean Matrices using Proximal Gradient Descent](https://mlsanity.com/paper/2307.07615/) (2023)
- [An Accelerated Block Proximal Framework with Adaptive Momentum for Nonconvex and Nonsmooth Optimization](https://mlsanity.com/paper/2308.12126/) (2023)
- [Learning of Patch-Based Smooth-Plus-Sparse Models for Image Reconstruction](https://mlsanity.com/paper/2412.13070/) (2024)
- [Plug-and-Play Image Restoration with Flow Matching: A Continuous Viewpoint](https://mlsanity.com/paper/2512.04283/) (2025)
- [Structured Sparse Non-negative Matrix Factorization with L20-Norm for scRNA-seq Data Analysis](https://mlsanity.com/paper/2104.13171/) (2021)
- [Joint Non-Linear MRI Inversion with Diffusion Priors](https://mlsanity.com/paper/2310.14842/) (2023)
- [Graph Regularized NMF with L20-norm for Unsupervised Feature Learning](https://mlsanity.com/paper/2403.10910/) (2024)
- [Spatially Coherent Clustering Based on Orthogonal Nonnegative Matrix Factorization](https://mlsanity.com/paper/2104.12289/) (2021)
- [Block Majorization Minimization with Extrapolation and Application to β-NMF](https://mlsanity.com/paper/2401.06646/) (2024)
- [Adaptive Lipschitz-Free Proximal Gradient Methods for Stochastic Composite Nonconvex Optimization](https://mlsanity.com/paper/2502.21099/) (2025)
- [A Triple-Inertial Accelerated Alternating Optimization Method for Deep Learning Training](https://mlsanity.com/paper/2503.08489/) (2025)
- [An Efficient Alternating Algorithm for ReLU-based Symmetric Matrix Decomposition](https://mlsanity.com/paper/2503.16846/) (2025)
- [MotionDPS: Motion-Compensated 3D Brain MRI Reconstruction](https://mlsanity.com/paper/2605.22121/) (2026)
- [Time series forecasting from partial observations via Non-negative Matrix Factorization](https://mlsanity.com/paper/2102.05314/) (2021)
- [A Refined Inertial DC Algorithm for DC Programming](https://mlsanity.com/paper/2104.14750/) (2021)
- [On-the-Fly Rectification for Robust Large-Vocabulary Topic Inference](https://mlsanity.com/paper/2111.06580/) (2021)
- [Unsupervised energy disaggregation via convolutional sparse coding](https://mlsanity.com/paper/2207.09785/) (2022)
- [A Variational Approach for Joint Image Recovery and Feature Extraction Based on Spatially-Varying Generalised Gaussian Models](https://mlsanity.com/paper/2209.01375/) (2022)

## Abstract (arXiv)

In this paper we study nonconvex and nonsmooth optimization problems with semi-algebraic data, where the variables vector is split into several blocks of variables. The problem consists of one smooth function of the entire variables vector and the sum of nonsmooth functions for each block separately. We analyze an inertial version of the Proximal Alternating Linearized Minimization (PALM) algorithm and prove its global convergence to a critical point of the objective function at hand. We illustrate our theoretical findings by presenting numerical experiments on blind image deconvolution, on sparse non-negative matrix factorization and on dictionary learning, which demonstrate the viability and effectiveness of the proposed method.