# Compositional Objectives: Learning Structure in Structure

arXiv: https://arxiv.org/abs/2609.32566 · HTML: https://mlsanity.com/paper/2609.32566/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 37 references that are in the map.
- First posted on arXiv 26 Sep 2026.

## Builds on (most cited first, 21 of 21)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Masked Autoencoders Are Scalable Vision Learners](https://mlsanity.com/paper/2111.06377/) (2021)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Barlow Twins: Self-Supervised Learning via Redundancy Reduction](https://mlsanity.com/paper/2103.03230/) (2021)
- [EuroSAT: A Novel Dataset and Deep Learning Benchmark for Land Use and Land Cover Classification](https://mlsanity.com/paper/1709.00029/) (2017)
- [CLEVR: A Diagnostic Dataset for Compositional Language and Elementary Visual Reasoning](https://mlsanity.com/paper/1612.06890/) (2016)
- [VICReg: Variance-Invariance-Covariance Regularization for Self-Supervised Learning](https://mlsanity.com/paper/2105.04906/) (2021)
- [Object-Centric Learning with Slot Attention](https://mlsanity.com/paper/2006.15055/) (2020)
- [What Makes for Good Views for Contrastive Learning?](https://mlsanity.com/paper/2005.10243/) (2020)
- [Dynamic Routing Between Capsules](https://mlsanity.com/paper/1710.09829/) (2017)
- [Understanding Dimensional Collapse in Contrastive Self-supervised Learning](https://mlsanity.com/paper/2110.09348/) (2021)
- [On Mutual Information Maximization for Representation Learning](https://mlsanity.com/paper/1907.13625/) (2019)
- [Large-Scale Study of Curiosity-Driven Learning](https://mlsanity.com/paper/1808.04355/) (2018)
- [A Theory of Usable Information Under Computational Constraints](https://mlsanity.com/paper/2002.10689/) (2020)
- [RankMe: Assessing the downstream performance of pretrained self-supervised representations by their rank](https://mlsanity.com/paper/2210.02885/) (2022)
- [Learning Diverse and Discriminative Representations via the Principle of Maximal Coding Rate Reduction](https://mlsanity.com/paper/2006.08558/) (2020)
- [How to represent part-whole hierarchies in a neural network](https://mlsanity.com/paper/2102.12627/) (2021)
- [SpatialSense: An Adversarially Crowdsourced Benchmark for Spatial Relation Recognition](https://mlsanity.com/paper/1908.02660/) (2019)
- [From Entropy to Epiplexity: Rethinking Information for Computationally Bounded Intelligence](https://mlsanity.com/paper/2601.03220/) (2026)
- [Intelligence from Learnable Novelty](https://mlsanity.com/paper/2607.18433/) (2026)

## Abstract (arXiv)

Intelligence is defined in many ways. One of these definitions defines intelligence as the pursuit of learnable novelty. However, learnable novelty can be meaningless without the ability to compose the learned structures to take action and achieve goals. Learnable novelty builds on epiplexity, which is a way to measure learnable structure in data through a bounded observer. In this paper, we investigate a closed-form spectral approximation to compute epiplexity. We use a fixed-trace constraint and find that the epiplexity objective prefers a more uniform distribution of spectral mass rather than concentrating it in a small number of directions. However, a representation may spread information across many directions without organizing that information into features useful for a particular task. To address this gap, we propose a compositional objective whose observer measures the relationships between the parts and interactions of an image. We compare it with the original spectral objective given only the masked parts. In our ImageNet training runs, the spectral objective with masked parts produces an almost maximally spread representation while achieving the strongest frozen-feature classification performance on most evaluations, more than doubling the linear-probe accuracy of the whole-image baseline. Across multiple image benchmarks, changing what the observer sees matters more than adding relation and interaction tokens. At the same time, our prediction-oriented compositional objective produces substantially better held-out observer prediction but relatively weaker classification, revealing that spectral diversity, predictability, and downstream utility are distinct properties. These results suggest that the usefulness of spectral spreading depends not only on how much structure is preserved, but on which relationships the observer makes available to the objective.