# Joint Graph Learning and Model Fitting in Laplacian Regularized Stratified Models

arXiv: https://arxiv.org/abs/2305.02573 · HTML: https://mlsanity.com/paper/2305.02573/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 38 references that are in the map.
- First posted on arXiv 4 May 2023.

## Builds on (most cited first, 17 of 17)

- [Federated Multi-Task Learning](https://mlsanity.com/paper/1705.10467/) (2017)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [How to learn a graph from smooth signals](https://mlsanity.com/paper/1601.02513/) (2016)
- [Laplacian Regularized Few-Shot Learning](https://mlsanity.com/paper/2006.15486/) (2020)
- [A Unified Framework for Structured Graph Learning via Spectral Constraints](https://mlsanity.com/paper/1904.09792/) (2019)
- [Analysis of p-Laplacian Regularization in Semi-Supervised Learning](https://mlsanity.com/paper/1707.06213/) (2017)
- [Adaptive and Robust Multi-Task Learning](https://mlsanity.com/paper/2202.05250/) (2022)
- [Structured Graph Learning for Scalable Subspace Clustering: From Single-view to Multi-view](https://mlsanity.com/paper/2102.07943/) (2021)
- [Large Scale Graph Learning from Smooth Signals](https://mlsanity.com/paper/1710.05654/) (2017)
- [A New Look and Convergence Rate of Federated Multi-Task Learning with Laplacian Regularization](https://mlsanity.com/paper/2102.07148/) (2021)
- [Overcoming the curse of dimensionality with Laplacian regularization in semi-supervised learning](https://mlsanity.com/paper/2009.04324/) (2020)
- [Kernel-based Graph Learning from Smooth Signals: A Functional Viewpoint](https://mlsanity.com/paper/2008.10065/) (2020)
- [A Distributed Method for Fitting Laplacian Regularized Stratified Models](https://mlsanity.com/paper/1904.12017/) (2019)
- [Least Squares Auto-Tuning](https://mlsanity.com/paper/1904.05460/) (2019)
- [Fitting Laplacian Regularized Stratified Gaussian Models](https://mlsanity.com/paper/2005.01752/) (2020)
- [Adaptive Data Fusion for Multi-task Non-smooth Optimization](https://mlsanity.com/paper/2210.12334/) (2022)
- [Portfolio Construction Using Stratified Models](https://mlsanity.com/paper/2101.04113/) (2021)

## Built on by (most cited first, 1 of 1)

- [Graph-Regularized Sparse Autoencoders for LLM Safety Steering](https://mlsanity.com/paper/2512.06655/) (2025)

## Abstract (arXiv)

Laplacian regularized stratified models (LRSM) are models that utilize the explicit or implicit network structure of the sub-problems as defined by the categorical features called strata (e.g., age, region, time, forecast horizon, etc.), and draw upon data from neighboring strata to enhance the parameter learning of each sub-problem. They have been widely applied in machine learning and signal processing problems, including but not limited to time series forecasting, representation learning, graph clustering, max-margin classification, and general few-shot learning. Nevertheless, existing works on LRSM have either assumed a known graph or are restricted to specific applications. In this paper, we start by showing the importance and sensitivity of graph weights in LRSM, and provably show that the sensitivity can be arbitrarily large when the parameter scales and sample sizes are heavily imbalanced across nodes. We then propose a generic approach to jointly learn the graph while fitting the model parameters by solving a single optimization problem. We interpret the proposed formulation from both a graph connectivity viewpoint and an end-to-end Bayesian perspective, and propose an efficient algorithm to solve the problem. Convergence guarantees of the proposed optimization algorithm is also provided despite the lack of global strongly smoothness of the Laplacian regularization term typically required in the existing literature, which may be of independent interest. Finally, we illustrate the efficiency of our approach compared to existing methods by various real-world numerical examples.