# On Embeddings for Numerical Features in Tabular Deep Learning

arXiv: https://arxiv.org/abs/2203.05556 · HTML: https://mlsanity.com/paper/2203.05556/

## Key facts

- Cited by 186 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 58 references that are in the map.
- First posted on arXiv 10 Mar 2022.

## Builds on (most cited first, 27 of 27)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Revisiting Deep Learning Models for Tabular Data](https://mlsanity.com/paper/2106.11959/) (2021)
- [TabNet: Attentive Interpretable Tabular Learning](https://mlsanity.com/paper/1908.07442/) (2019)
- [Self-Normalizing Neural Networks](https://mlsanity.com/paper/1706.02515/) (2017)
- [Tabular Data: Deep Learning is Not All You Need](https://mlsanity.com/paper/2106.03253/) (2021)
- [OpenML: networked science in machine learning](https://mlsanity.com/paper/1407.7722/) (2014)
- [Hopfield Networks is All You Need](https://mlsanity.com/paper/2008.02217/) (2020)
- [TabTransformer: Tabular Data Modeling Using Contextual Embeddings](https://mlsanity.com/paper/2012.06678/) (2020)
- [Deep & Cross Network for Ad Click Predictions](https://mlsanity.com/paper/1708.05123/) (2017)
- [AutoInt: Automatic Feature Interaction Learning via Self-Attentive Neural Networks](https://mlsanity.com/paper/1810.11921/) (2018)
- [SAINT: Improved Neural Networks for Tabular Data via Row Attention and Contrastive Pre-Training](https://mlsanity.com/paper/2106.01342/) (2021)
- [Neural Oblivious Decision Ensembles for Deep Learning on Tabular Data](https://mlsanity.com/paper/1909.06312/) (2019)
- [Neural Importance Sampling](https://mlsanity.com/paper/1808.03856/) (2018)
- [Well-tuned Simple Nets Excel on Tabular Datasets](https://mlsanity.com/paper/2106.11189/) (2021)
- [Bayesian Optimization is Superior to Random Search for Machine Learning Hyperparameter Tuning: Analysis of the Black-Box Optimization Challenge 2020](https://mlsanity.com/paper/2104.10201/) (2021)
- [Introducing LETOR 4.0 Datasets](https://mlsanity.com/paper/1306.2597/) (2013)
- [Self-Attention Between Datapoints: Going Beyond Individual Input-Output Pairs in Deep Learning](https://mlsanity.com/paper/2106.02584/) (2021)
- [Learnable Fourier Features for Multi-Dimensional Spatial Positional Encoding](https://mlsanity.com/paper/2106.02795/) (2021)
- [The Tree Ensemble Layer: Differentiability meets Conditional Computation](https://mlsanity.com/paper/2002.07772/) (2020)
- [Gradient Boosting Neural Networks: GrowNet](https://mlsanity.com/paper/2002.07971/) (2020)
- [An Embedding Learning Framework for Numerical Features in CTR Prediction](https://mlsanity.com/paper/2012.08986/) (2020)

## Built on by (most cited first, 30 of 186)

- [TabICL: A Tabular Foundation Model for In-Context Learning on Large Data](https://mlsanity.com/paper/2502.05564/) (2025)
- [TabLLM: Few-shot Classification of Tabular Data with Large Language Models](https://mlsanity.com/paper/2210.10723/) (2022)
- [Why do tree-based models still outperform deep learning on tabular data?](https://mlsanity.com/paper/2207.08815/) (2022)
- [TabDPT: Scaling Tabular Foundation Models on Real Data](https://mlsanity.com/paper/2410.18164/) (2024)
- [TabM: Advancing Tabular Deep Learning with Parameter-Efficient Ensembling](https://mlsanity.com/paper/2410.24210/) (2024)
- [LLM-Rubric: A Multidimensional, Calibrated Approach to Automated Evaluation of Natural Language Texts](https://mlsanity.com/paper/2501.00274/) (2024)
- [CARTE: Pretraining and Transfer for Tabular Learning](https://mlsanity.com/paper/2402.16785/) (2024)
- [TabR: Tabular Deep Learning Meets Nearest Neighbors in 2023](https://mlsanity.com/paper/2307.14338/) (2023)
- [Making Pre-trained Language Models Great on Tabular Prediction](https://mlsanity.com/paper/2403.01841/) (2024)
- [Relational Deep Learning: Graph Representation Learning on Relational Databases](https://mlsanity.com/paper/2312.04615/) (2023)
- [Generative Table Pre-training Empowers Models for Tabular Prediction](https://mlsanity.com/paper/2305.09696/) (2023)
- [Transfer Learning with Deep Tabular Models](https://mlsanity.com/paper/2206.15306/) (2022)
- [OpenFE: Automated Feature Generation with Expert-level Performance](https://mlsanity.com/paper/2211.12507/) (2022)
- [TALENT: A Tabular Analytics and Learning Toolbox](https://mlsanity.com/paper/2407.04057/) (2024)
- [Griffin: Towards a Graph-Centric Relational Database Foundation Model](https://mlsanity.com/paper/2505.05568/) (2025)
- [Revisiting Pretraining Objectives for Tabular Deep Learning](https://mlsanity.com/paper/2207.03208/) (2022)
- [T2G-Former: Organizing Tabular Features into Relation Graphs Promotes Heterogeneous Feature Interaction](https://mlsanity.com/paper/2211.16887/) (2022)
- [A Closer Look at TabPFN v2: Understanding Its Strengths and Extending Its Capabilities](https://mlsanity.com/paper/2502.17361/) (2025)
- [ReMasker: Imputing Tabular Data with Masked Autoencoding](https://mlsanity.com/paper/2309.13793/) (2023)
- [TabPFN Unleashed: A Scalable and Effective Solution to Tabular Classification Problems](https://mlsanity.com/paper/2502.02527/) (2025)
- [Representation Learning for Tabular Data: A Comprehensive Survey](https://mlsanity.com/paper/2504.16109/) (2025)
- [ExcelFormer: A neural network surpassing GBDTs on tabular data](https://mlsanity.com/paper/2301.02819/) (2023)
- [Mixture of In-Context Prompters for Tabular PFNs](https://mlsanity.com/paper/2405.16156/) (2024)
- [TabReD: Analyzing Pitfalls and Filling the Gaps in Tabular Deep Learning Benchmarks](https://mlsanity.com/paper/2406.19380/) (2024)
- [TANGOS: Regularizing Tabular Neural Networks through Gradient Orthogonalization and Specialization](https://mlsanity.com/paper/2303.05506/) (2023)
- [PyTorch Frame: A Modular Framework for Multi-Modal Tabular Learning](https://mlsanity.com/paper/2404.00776/) (2024)
- [Revisiting Nearest Neighbor for Tabular Data: A Deep Tabular Baseline Two Decades Later](https://mlsanity.com/paper/2407.03257/) (2024)
- [UniTabE: A Universal Pretraining Protocol for Tabular Foundation Model in Data Science](https://mlsanity.com/paper/2307.09249/) (2023)
- [A Performance-Driven Benchmark for Feature Selection in Tabular Deep Learning](https://mlsanity.com/paper/2311.05877/) (2023)
- [Relational Deep Learning: Challenges, Foundations and Next-Generation Architectures](https://mlsanity.com/paper/2506.16654/) (2025)

## Abstract (arXiv)

Recently, Transformer-like deep architectures have shown strong performance on tabular data problems. Unlike traditional models, e.g., MLP, these architectures map scalar values of numerical features to high-dimensional embeddings before mixing them in the main backbone. In this work, we argue that embeddings for numerical features are an underexplored degree of freedom in tabular DL, which allows constructing more powerful DL models and competing with GBDT on some traditionally GBDT-friendly benchmarks. We start by describing two conceptually different approaches to building embedding modules: the first one is based on a piecewise linear encoding of scalar values, and the second one utilizes periodic activations. Then, we empirically demonstrate that these two approaches can lead to significant performance boosts compared to the embeddings based on conventional blocks such as linear layers and ReLU activations. Importantly, we also show that embedding numerical features is beneficial for many backbones, not only for Transformers. Specifically, after proper embeddings, simple MLP-like models can perform on par with the attention-based architectures. Overall, we highlight embeddings for numerical features as an important design aspect with good potential for further improvements in tabular DL.