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[Submitted on 12 Sep 2026] Title:When the Gradient Sees Rank: Provable Necessity, Causal Recruitment, and Composition in Trained Matrix Memories View a PDF of the paper titled When the Gradient Sees Rank: Provable Necessity, Causal Recruitment, and Composition in Trained Matrix Memories, by Samuel Larson View PDF HTML (experimental) Abstract:Can gradient-based training learn the rank needed to store and compose associations in a matrix memory? In our earlier study, we used a matrix-augmented reasoner on a task that admits a rank-1 solution, leaving this question open. We train matrix memories on $K$ fresh key-value bindings whose exact linear recovery requires $\mathrm{rank}(Z) \geq K$. A fixed linear readout queries a single matrix state without access to the original bindings. Experiments measure recovery by cosine similarity greater than 0.9, a threshold distinct from mathematical equality. Learned effective rank increases with $K$ across the tested grid (Spearman $\rho = 1.0$ at $d = 16$). Training-time rank caps produce a recovery transition near $k = K$: at $d = 8$, $K = 4$, rank 3 gives at most 0.0004 recovery and rank 4 gives 0.97. Four of five seeds retain at least 0.9996 recovery through 21-fold self-application of the trained operator. On the entity subspace, the learned operator has effective rank close to $K$ and approximates the ideal cycle. For the single converged seed capped below $K$, a calculation using the entity-subspace operator and ideal cycle predicts the measured cosine within 0.008 through seven applications. Extending training resolves several initial failures, but recovery still declines at larger matrix dimensions with encoder width fixed. Comments: 6 pages, 2 figures. Code and archived results: this https URL Subjects: Machine Learning (cs.LG) Cite as: arXiv:2609.17594 [cs.LG] (or arXiv:2609.17594v1 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2609.17594 arXiv-issued DOI via DataCite Submission history From: Samuel Larson [view email] [v1] Sat, 12 Sep 2026 04:44:45 UTC (51 KB) Full-text links: Access Paper: View a PDF of the paper titled When the Gradient Sees Rank: Provable Necessity, Causal Recruitment, and Composition in Trained Matrix Memories, by Samuel Larson View PDF HTML (experimental) TeX Source view license Current browse context: cs.LG new | recent | 2026-09 Change to browse by: cs References & Citations NASA ADS Google Scholar Semantic Scholar Loading... Data provided by: Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) IArxiv recommender toggle IArxiv Recommender (What is IArxiv?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)