# Low-Interaction-Rank Learning: Unifying Multiplicative Dual-Encoder Heads

arXiv: https://arxiv.org/abs/2608.11661 · HTML: https://mlsanity.com/paper/2608.11661/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 29 references that are in the map.
- First posted on arXiv 12 Aug 2026.

## Builds on (most cited first, 15 of 15)

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- [Barlow Twins: Self-Supervised Learning via Redundancy Reduction](https://mlsanity.com/paper/2103.03230/) (2021)
- [VICReg: Variance-Invariance-Covariance Regularization for Self-Supervised Learning](https://mlsanity.com/paper/2105.04906/) (2021)
- [Embedding Entities and Relations for Learning and Inference in Knowledge Bases](https://mlsanity.com/paper/1412.6575/) (2014)
- [Complex Embeddings for Simple Link Prediction](https://mlsanity.com/paper/1606.06357/) (2016)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Successor Features for Transfer in Reinforcement Learning](https://mlsanity.com/paper/1606.05312/) (2016)
- [Whitening for Self-Supervised Representation Learning](https://mlsanity.com/paper/2007.06346/) (2020)
- [A vector-contraction inequality for Rademacher complexities](https://mlsanity.com/paper/1605.00251/) (2016)
- [Bilinear Bandits with Low-rank Structure](https://mlsanity.com/paper/1901.02470/) (2019)
- [CrossFi: A Cross Domain Wi-Fi Sensing Framework Based on Siamese Network](https://mlsanity.com/paper/2408.10919/) (2024)
- [Triple-BERT: Do We Really Need MARL for Order Dispatch on Ride-Sharing Platforms?](https://mlsanity.com/paper/2510.03257/) (2025)

## Abstract (arXiv)

A multiplicative dual-encoder network computes a real-valued output for a pair of inputs as the inner product of their separate encodings. This architecture has been developed independently in operator learning, bipartite matching, contrastive vision-language models, retrieval, and other areas, yet no unified theory guides the basic design decisions: how many interaction modes to represent, how to normalize the encoders, and when the architecture should be avoided. We provide such a foundation by introducing the class of functions of low interaction rank, a class whose intrinsic complexity is measured by its interaction spectrum. Within this framework, approximation error decomposes into a spectral truncation term and an encoder-realization term; sample complexity is governed by the sum of the two encoder complexities rather than their product; and a usability criterion based on spectral decay determines when the architecture can succeed. The same framework exposes a central identifiability problem: the encoders are defined only up to a linear gauge symmetry that leaves the learned coordinates arbitrary. We show that normalization is gauge fixing and that whitening pins the interaction modes up to permutation and sign, thereby explaining the uninterpretability of contrastive dimensions and providing a constructive remedy. Experiments on synthetic kernels, operator learning, and CLIP models validate the theoretical predictions: spectral decay rates match the predicted scaling, whitening recovers the true modes, and independently trained CLIP models are related by a single rotation which, after removal by whitening, exposes interpretable concept axes. The code of this paper is provided at https://github.com/RS2002/Mul-Net .