# Geodesics of learned representations

arXiv: https://arxiv.org/abs/1511.06394 · HTML: https://mlsanity.com/paper/1511.06394/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Nov 2015.

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

- [DreamSim: Learning New Dimensions of Human Visual Similarity using Synthetic Data](https://mlsanity.com/paper/2306.09344/) (2023)
- [No-Reference Image Quality Assessment via Transformers, Relative Ranking, and Self-Consistency](https://mlsanity.com/paper/2108.06858/) (2021)
- [Shift-tolerant Perceptual Similarity Metric](https://mlsanity.com/paper/2207.13686/) (2022)
- [Locally Adaptive Structure and Texture Similarity for Image Quality Assessment](https://mlsanity.com/paper/2110.08521/) (2021)
- [Similarity and Quality Metrics for MR Image-To-Image Translation](https://mlsanity.com/paper/2405.08431/) (2024)
- [Neural Population Geometry Reveals the Role of Stochasticity in Robust Perception](https://mlsanity.com/paper/2111.06979/) (2021)
- [Incorporating Semi-Supervised and Positive-Unlabeled Learning for Boosting Full Reference Image Quality Assessment](https://mlsanity.com/paper/2204.08763/) (2022)
- [How Convolutional Neural Networks Deal with Aliasing](https://mlsanity.com/paper/2102.07757/) (2021)
- [On convex decision regions in deep network representations](https://mlsanity.com/paper/2305.17154/) (2023)
- [Learning the solution operator of two-dimensional incompressible Navier-Stokes equations using physics-aware convolutional neural networks](https://mlsanity.com/paper/2308.02137/) (2023)
- [Input Space Mode Connectivity in Deep Neural Networks](https://mlsanity.com/paper/2409.05800/) (2024)
- [GEOPHYS: The Geometry of Physical Plausibility](https://mlsanity.com/paper/2606.20707/) (2026)
- [Content-Diverse Comparisons improve IQA](https://mlsanity.com/paper/2211.05215/) (2022)
- [Stretching Beyond the Obvious: A Gradient-Free Framework to Unveil the Hidden Landscape of Visual Invariance](https://mlsanity.com/paper/2506.17040/) (2025)

## Abstract (arXiv)

We develop a new method for visualizing and refining the invariances of learned representations. Specifically, we test for a general form of invariance, linearization, in which the action of a transformation is confined to a low-dimensional subspace. Given two reference images (typically, differing by some transformation), we synthesize a sequence of images lying on a path between them that is of minimal length in the space of the representation (a "representational geodesic"). If the transformation relating the two reference images is linearized by the representation, this sequence should follow the gradual evolution of this transformation. We use this method to assess the invariance properties of a state-of-the-art image classification network and find that geodesics generated for image pairs differing by translation, rotation, and dilation do not evolve according to their associated transformations. Our method also suggests a remedy for these failures, and following this prescription, we show that the modified representation is able to linearize a variety of geometric image transformations.