# Estimating the Effective Rank of Vision Transformers via Low-Rank Factorization

arXiv: https://arxiv.org/abs/2512.00792 · HTML: https://mlsanity.com/paper/2512.00792/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 8 references that are in the map.
- First posted on arXiv 30 Nov 2025.

## Builds on (most cited first, 6 of 6)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Deep Double Descent: Where Bigger Models and More Data Hurt](https://mlsanity.com/paper/1912.02292/) (2019)
- [The Low-Rank Simplicity Bias in Deep Networks](https://mlsanity.com/paper/2103.10427/) (2021)
- [A Token is Worth over 1,000 Tokens: Efficient Knowledge Distillation through Low-Rank Clone](https://mlsanity.com/paper/2505.12781/) (2025)

## Built on by (most cited first, 1 of 1)

- [What Does Model Growth Really Add? Functional Capacity Beyond Parameter Count](https://mlsanity.com/paper/2607.14571/) (2026)

## Abstract (arXiv)

Deep networks are heavily over-parameterized, yet their learned representations often admit low-rank structure. We introduce a framework for estimating a model's intrinsic dimensionality by treating learned representations as projections onto a low-rank subspace of the model's full capacity. Our approach: train a full-rank teacher, factorize its weights at multiple ranks, and train each factorized student via distillation to measure performance as a function of rank. We define effective rank as a region, not a point: the smallest contiguous set of ranks for which the student reaches 85-95% of teacher accuracy. To stabilize estimates, we fit accuracy vs. rank with a monotone PCHIP interpolant and identify crossings of the normalized curve. We also define the effective knee as the rank maximizing perpendicular distance between the smoothed accuracy curve and its endpoint secant; an intrinsic indicator of where marginal gains concentrate. On ViT-B/32 fine-tuned on CIFAR-100 (one seed, due to compute constraints), factorizing linear blocks and training with distillation yields an effective-rank region of approximately [16, 34] and an effective knee at r* ~ 31. At rank 32, the student attains 69.46% top-1 accuracy vs. 73.35% for the teacher (~94.7% of baseline) while achieving substantial parameter compression. We provide a framework to estimate effective-rank regions and knees across architectures and datasets, offering a practical tool for characterizing the intrinsic dimensionality of deep models.