# Linear, or Non-Linear, That is the Question!

arXiv: https://arxiv.org/abs/2111.07265 · HTML: https://mlsanity.com/paper/2111.07265/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 39 references that are in the map.
- First posted on arXiv 14 Nov 2021.

## Builds on (most cited first, 22 of 22)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [LightGCN: Simplifying and Powering Graph Convolution Network for Recommendation](https://mlsanity.com/paper/2002.02126/) (2020)
- [Graph Convolutional Neural Networks for Web-Scale Recommender Systems](https://mlsanity.com/paper/1806.01973/) (2018)
- [Neural Collaborative Filtering](https://mlsanity.com/paper/1708.05031/) (2017)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Neural Graph Collaborative Filtering](https://mlsanity.com/paper/1905.08108/) (2019)
- [LINE: Large-scale Information Network Embedding](https://mlsanity.com/paper/1503.03578/) (2015)
- [Measuring and Relieving the Over-smoothing Problem for Graph Neural Networks from the Topological View](https://mlsanity.com/paper/1909.03211/) (2019)
- [Ups and Downs: Modeling the Visual Evolution of Fashion Trends with One-Class Collaborative Filtering](https://mlsanity.com/paper/1602.01585/) (2016)
- [FBNet: Hardware-Aware Efficient ConvNet Design via Differentiable Neural Architecture Search](https://mlsanity.com/paper/1812.03443/) (2018)
- [A* Sampling](https://mlsanity.com/paper/1411.0030/) (2014)
- [Revisiting Graph based Collaborative Filtering: A Linear Residual Graph Convolutional Network Approach](https://mlsanity.com/paper/2001.10167/) (2020)
- [Collaborative Deep Learning for Recommender Systems](https://mlsanity.com/paper/1409.2944/) (2014)
- [Modeling User Exposure in Recommendation](https://mlsanity.com/paper/1510.07025/) (2015)
- [DSNAS: Direct Neural Architecture Search without Parameter Retraining](https://mlsanity.com/paper/2002.09128/) (2020)
- [MiLeNAS: Efficient Neural Architecture Search via Mixed-Level Reformulation](https://mlsanity.com/paper/2003.12238/) (2020)
- [Collaborative Memory Network for Recommendation Systems](https://mlsanity.com/paper/1804.10862/) (2018)
- [LT-OCF: Learnable-Time ODE-based Collaborative Filtering](https://mlsanity.com/paper/2108.06208/) (2021)
- [MEGAN: Mixture of Experts of Generative Adversarial Networks for Multimodal Image Generation](https://mlsanity.com/paper/1805.02481/) (2018)

## Built on by (most cited first, 19 of 19)

- [Blurring-Sharpening Process Models for Collaborative Filtering](https://mlsanity.com/paper/2211.09324/) (2022)
- [An Attentive Inductive Bias for Sequential Recommendation beyond the Self-Attention](https://mlsanity.com/paper/2312.10325/) (2023)
- [Infinite Recommendation Networks: A Data-Centric Approach](https://mlsanity.com/paper/2206.02626/) (2022)
- [A Novel Approach to Temporal QoS Estimation via Extended Kalman Filter-Incorporated Latent Feature Analysis](https://mlsanity.com/paper/2606.23010/) (2026)
- [It's Enough: Relaxing Diagonal Constraints in Linear Autoencoders for Recommendation](https://mlsanity.com/paper/2305.12922/) (2023)
- [Graph Spectral Filtering with Chebyshev Interpolation for Recommendation](https://mlsanity.com/paper/2505.00552/) (2025)
- [Towards Fair Graph Anomaly Detection: Problem, Benchmark Datasets, and Evaluation](https://mlsanity.com/paper/2402.15988/) (2024)
- [SCONE: A Novel Stochastic Sampling to Generate Contrastive Views and Hard Negative Samples for Recommendation](https://mlsanity.com/paper/2405.00287/) (2024)
- [SVD-AE: Simple Autoencoders for Collaborative Filtering](https://mlsanity.com/paper/2405.04746/) (2024)
- [MONET: Modality-Embracing Graph Convolutional Network and Target-Aware Attention for Multimedia Recommendation](https://mlsanity.com/paper/2312.09511/) (2023)
- [RDGCL: Reaction-Diffusion Graph Contrastive Learning for Recommendation](https://mlsanity.com/paper/2312.16563/) (2023)
- [QoS-Aware Graph Contrastive Learning for Web Service Recommendation](https://mlsanity.com/paper/2401.03162/) (2024)
- [Hgformer: Hyperbolic Graph Transformer for Recommendation](https://mlsanity.com/paper/2502.15693/) (2024)
- [Modeling Reference-dependent Choices with Graph Neural Networks](https://mlsanity.com/paper/2408.11302/) (2024)
- [Fair Augmentation for Graph Collaborative Filtering](https://mlsanity.com/paper/2408.12208/) (2024)
- [CAPER: Enhancing Career Trajectory Prediction using Temporal Knowledge Graph and Ternary Relationship](https://mlsanity.com/paper/2408.15620/) (2024)
- [Variational Autoencoder-Based Approach to Latent Feature Analysis on Efficient Representation of Power Load Monitoring Data](https://mlsanity.com/paper/2506.08698/) (2025)
- [TV-Rec: Time-Variant Convolutional Filter for Sequential Recommendation](https://mlsanity.com/paper/2510.25259/) (2025)
- [Denoised Recommendation Model with Collaborative Signal Decoupling](https://mlsanity.com/paper/2511.04237/) (2025)

## Abstract (arXiv)

There were fierce debates on whether the non-linear embedding propagation of GCNs is appropriate to GCN-based recommender systems. It was recently found that the linear embedding propagation shows better accuracy than the non-linear embedding propagation. Since this phenomenon was discovered especially in recommender systems, it is required that we carefully analyze the linearity and non-linearity issue. In this work, therefore, we revisit the issues of i) which of the linear or non-linear propagation is better and ii) which factors of users/items decide the linearity/non-linearity of the embedding propagation. We propose a novel Hybrid Method of Linear and non-linEar collaborative filTering method (HMLET, pronounced as Hamlet). In our design, there exist both linear and non-linear propagation steps, when processing each user or item node, and our gating module chooses one of them, which results in a hybrid model of the linear and non-linear GCN-based collaborative filtering (CF). The proposed model yields the best accuracy in three public benchmark datasets. Moreover, we classify users/items into the following three classes depending on our gating modules' selections: Full-Non-Linearity (FNL), Partial-Non-Linearity (PNL), and Full-Linearity (FL). We found that there exist strong correlations between nodes' centrality and their class membership, i.e., important user/item nodes exhibit more preferences towards the non-linearity during the propagation steps. To our knowledge, we are the first who design a hybrid method and report the correlation between the graph centrality and the linearity/non-linearity of nodes. All HMLET codes and datasets are available at: https://github.com/qbxlvnf11/HMLET.