Research map / Learning theory and algorithms / Matrix and tensor methods
Geometry of low-rank and SPD structure
Studies rank, random matrix geometry and positive-definite manifolds in network training and learning.
71 accepted papers at ICML, NeurIPS, ICLR, CVPR and AAAI (2016–2026) · an approach in Matrix and tensor methods, keyword cluster “matrices · matrix completion · rank matrix”
State History Tried Checklist Preprints Who Export
Where it stands
Not enough benchmark data Fewer than four papers report the same benchmark, so there is no record to follow.
11 papers in 2023–24 and 27 in 2025–26 (at ICML, NeurIPS, ICLR, CVPR and AAAI); its share of the topic grew, from 6.4% to 14.5%.
How this is decided
A benchmark is tracked when at least four papers of this approach report their own method on the same dataset and metric in at least two different years. Its record is the best number reported up to 2024 compared with the best reported in 2025–26: still improving means at least 1% better, flat means at most 0.2% better with at least two papers from 2025–26 trying. Numbers on a different scale from the rest (a fraction among percentages) and far outliers are left out. Results come from different papers with different setups, so they are signals, not head-to-head comparisons. The share of papers is counted at ICML, NeurIPS, ICLR, CVPR and AAAI.
How it got here
Most cited paper of each year
- 2016 · Analysis of Deep Neural Networks with Extended Data Jacobian Matrix (ICML 2016 · 17 citations)
- 2017 · Practical Gauss-Newton Optimisation for Deep Learning (ICML 2017 · 104 citations)
- 2018 · Deep Models of Interactions Across Sets (ICML 2018 · 73 citations)
- 2019 · Learning Grid Cells as Vector Representation of Self-Position Coupled with Matrix Representation of Self-Motion (ICLR 2019 · 12 citations)
- 2020 · Implicit Regularization in Deep Learning May Not Be Explainable by Norms (NeurIPS 2020 · 61 citations)
- 2021 · A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups (ICML 2021 · 25 citations)
- 2022 · Fully-Connected Network on Noncompact Symmetric Space and Ridgelet Transform based on Helgason-Fourier Analysis (ICML 2022 · 3 citations)
- 2023 · Implicit Bias of Large Depth Networks: a Notion of Rank for Nonlinear Functions (ICLR 2023 · 2 citations)
- 2024 · Trade-Offs of Diagonal Fisher Information Matrix Estimators (NeurIPS 2024 · 5 citations)
- 2025 · Learning to Normalize on the SPD Manifold under Bures-Wasserstein Geometry (CVPR 2025 · 6 citations)
- 2026 · Why Deep Jacobian Spectra Separate: Depth-Induced Scaling and Singular-Vector Alignment (ICML 2026 · 0 citations)
Earliest papers here
- Analysis of Deep Neural Networks with Extended Data Jacobian Matrix (ICML 2016 · 17 citations)
- High resolution neural connectivity from incomplete tracing data using nonnegative spline regression (NeurIPS 2016 · 4 citations)
- A Random Matrix Approach to Echo-State Neural Networks (ICML 2016 · 2 citations)
What has been tried
48 papers here report 343 measured changes (a component added, removed or swapped, compared with a baseline): helped 65%, no clear effect 13%, hurt 17%, mixed 5%. Papers mostly publish changes that helped, so "helped" is over-represented.
Ablations in recent papers
| What was changed | Measured on | Result | Effect | Paper |
|---|---|---|---|---|
| Use θ=0.5 rather than θ=1 | HDM05 · accuracy | 62.01 -> 69.10 | Helped | Learning to Normalize on the SPD Manifold under Bures-Wasserstein Geometry (CVPR 2025 · 6 citations) |
| use Pow-EMLR instead of Pow-TMLR | Cars · Top-5 accuracy (%) | 59.51 -> 82.04 | Helped | Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry (ICLR 2025 · 1 citation) |
| Use a global observable instead of local observables | State approximation (global observable) · objective-function variance | Hurt | On the Trainability and Classical Simulability of Learning Matrix Product States Variationally (AAAI 2025 · 0 citations) | |
| add the attention module | BCIC-IV-2a · accuracy | 73.04±6.34 -> 78.08±4.8 | Helped | Neural networks on Symmetric Spaces of Noncompact Type (ICLR 2025 · 0 citations) |
| Add nesting to the LoRA objective | 2D hydrogen atom, 16 eigenmodes · subspace distance | 3.56\times 10^{-4}{ {± 6.60\times 10^{-5}}} -> 2.12\times 10^{-4}{ {± 2.09\times 10^{-5}}} | Helped | Operator SVD with Neural Networks via Nested Low-Rank Approximation (ICML 2024 · 0 citations) |
What reviewers will expect
Datasets and baselines named in the 19 papers of this approach from 2025–2026 that have a full-text summary. Leaving out the common ones is a frequent review question.
Datasets
- CIFAR-10 · 4 of 19 papers
- MNIST · 2 of 19 papers
- MAMEM-SSVEP-II · 2 of 19 papers
- HDM05 · 2 of 19 papers
Baselines
- SPDNet-BN · 2 of 19 papers
- EEG-TCNet · 2 of 19 papers
- MBEEGSE · 2 of 19 papers
Recent preprints
Loading recent arXiv preprints…
Who publishes on it
Institutions on the most papers of this approach since 2023 (of 38 papers; from OpenAlex and the papers' affiliations).
- Massachusetts Institute of Technology · 4 papers
- University of California, Berkeley · 3 papers
- New York University · 2 papers
- University of Melbourne · 2 papers
- University of Trento · 2 papers
Use it
Download BibTeX of 38 key papersOpen the topic in the interactive map
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Related approaches
Same keyword cluster
- Low-rank approximation theory (142 papers)
- Matrix completion and factorization (118 papers)
- Randomized sketching for matrices (96 papers)
- Structured low-rank learning (87 papers)
Elsewhere in the topic
- Nonconvex low-rank and sparse optimization (112 papers)
- Sparse regression with guarantees (94 papers)
- High-dimensional estimation theory (89 papers)
- Signal recovery and phase retrieval (66 papers)
- Sparse and robust PCA (66 papers)
- Tensor network decompositions (45 papers)
- Low-rank tensor completion (45 papers)
- Randomized tensor decomposition algorithms (43 papers)
Benchmark numbers, ablations and baseline numbers are extracted by AI from each paper's full text and can be wrong; the paper is the authority. Data: results index of 2026-11-02, citation counts from OpenAlex (2026-10-09).