# Shape of Memory: a Geometric Analysis of Machine Unlearning in Second-Order Optimizers

arXiv: https://arxiv.org/abs/2604.23046 · HTML: https://mlsanity.com/paper/2604.23046/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Machine unlearning, one of 12 directions in LLM safety and security.
- Builds on 7 of its 10 references that are in the map.
- First posted on arXiv 24 Apr 2026.

## Placed in

- LLM safety and security / [Machine unlearning](https://mlsanity.com/safety/machine-unlearning/)

## Builds on (most cited first, 7 of 7)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Machine Unlearning](https://mlsanity.com/paper/1912.03817/) (2019)
- [Certified Data Removal from Machine Learning Models](https://mlsanity.com/paper/1911.03030/) (2019)
- [Making AI Forget You: Data Deletion in Machine Learning](https://mlsanity.com/paper/1907.05012/) (2019)
- [Remember What You Want to Forget: Algorithms for Machine Unlearning](https://mlsanity.com/paper/2103.03279/) (2021)
- [When Machine Unlearning Jeopardizes Privacy](https://mlsanity.com/paper/2005.02205/) (2020)
- [Global Convergence of Online Limited Memory BFGS](https://mlsanity.com/paper/1409.2045/) (2014)

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

- [Form and Function: Machine Unlearning as a Problem of Misaligned States](https://mlsanity.com/paper/2605.17590/) (2026)

## Abstract (arXiv)

We argue that current definitions of machine unlearning are underspecified for second-order optimizers. We compare first-order and second-order learners for their ability to handle the data deletion task with varying degrees of eigendecomposition to mimic the loss model memory. While both first and second-order methods realign with the ideal counterfactul in terms of performance and gradient, the second-order optimizer shows significant volatility in the optimizer state. This indicates residual information, supposedly deleted, that isn't detectable by first-order analysis. Various eigendecay treatments show that stability and information loss is regained only under controlled state pertubation where geometric information (or memory) is erased.