# Breaking the Softmax Bottleneck via Learnable Monotonic Pointwise Non-linearities

arXiv: https://arxiv.org/abs/1902.08077 · HTML: https://mlsanity.com/paper/1902.08077/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Feb 2019.

## Built on by (most cited first, 14 of 14)

- [RankMe: Assessing the downstream performance of pretrained self-supervised representations by their rank](https://mlsanity.com/paper/2210.02885/) (2022)
- [Contrastive and Non-Contrastive Self-Supervised Learning Recover Global and Local Spectral Embedding Methods](https://mlsanity.com/paper/2205.11508/) (2022)
- [Closing the Curious Case of Neural Text Degeneration](https://mlsanity.com/paper/2310.01693/) (2023)
- [Low-Rank Softmax Can Have Unargmaxable Classes in Theory but Rarely in Practice](https://mlsanity.com/paper/2203.06462/) (2022)
- [Addressing Token Uniformity in Transformers via Singular Value Transformation](https://mlsanity.com/paper/2208.11790/) (2022)
- [Taming the Sigmoid Bottleneck: Provably Argmaxable Sparse Multi-Label Classification](https://mlsanity.com/paper/2310.10443/) (2023)
- [Sequences of Logits Reveal the Low Rank Structure of Language Models](https://mlsanity.com/paper/2510.24966/) (2025)
- [Lost in Backpropagation: The LM Head is a Gradient Bottleneck](https://mlsanity.com/paper/2603.10145/) (2026)
- [Breaking the Softmax Bottleneck for Sequential Recommender Systems with Dropout and Decoupling](https://mlsanity.com/paper/2110.05409/) (2021)
- [On the Low-Rank Parametrization of Reward Models for Controlled Language Generation](https://mlsanity.com/paper/2407.04615/) (2024)
- [On the Theoretical Limitations of Embedding-based Link Prediction](https://mlsanity.com/paper/2506.22271/) (2025)
- [Explicit Word Density Estimation for Language Modelling](https://mlsanity.com/paper/2406.10256/) (2024)
- [The Gradient That Top-k Distillation Never Computes](https://mlsanity.com/paper/2610.15589/) (2026)
- [Copy-Paste Language Models](https://mlsanity.com/paper/2610.19247/) (2026)

## Abstract (arXiv)

The Softmax function on top of a final linear layer is the de facto method to output probability distributions in neural networks. In many applications such as language models or text generation, this model has to produce distributions over large output vocabularies. Recently, this has been shown to have limited representational capacity due to its connection with the rank bottleneck in matrix factorization. However, little is known about the limitations of Linear-Softmax for quantities of practical interest such as cross entropy or mode estimation, a direction that we explore here. As an efficient and effective solution to alleviate this issue, we propose to learn parametric monotonic functions on top of the logits. We theoretically investigate the rank increasing capabilities of such monotonic functions. Empirically, our method improves in two different quality metrics over the traditional Linear-Softmax layer in synthetic and real language model experiments, adding little time or memory overhead, while being comparable to the more computationally expensive mixture of Softmaxes.