Research map / Learning theory and algorithms / Matrix and tensor methods
Neural Networks Perform Sufficient Dimension Reduction
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Shows that a feedforward network with a rank-regularized first layer (linear map B with orthonormal columns) recovers the central mean subspace, with population unbiasedness and sample consistency results, and evaluates it against classical SDR methods in simulations and on Seoul weather data.
Problem
Sufficient dimension reduction (estimating the central mean subspace) in regression, and whether neural networks inherently learn the low-dimensional linear structure B_0 in their first layer.
Approach
Imposes rank regularization by constraining the first layer to a linear map B with orthonormal columns feeding a ReLU network trained by least squares; proves the first layer spans the central mean subspace at population level and that the sample estimator is consistent, and extends the idea to central subspace via a kernel-based loss.
Key results
- In simulation, the NN method achieved the lowest subspace error in several settings, e.g., Setting 3 with n=1000 gave mean 0.075 versus 0.924 for MAVE and 1.652 for SIR.
- On the Seoul weather data, NN with rank regularization gave mean testing error 0.602 (std 0.043) versus 0.774 for vanilla NN and 0.743 for MAVE over 20 replicates.
- In simple settings 1 and 2 the NN method performed similarly to classical methods, with SAVE best in setting 2.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use rank-regularized NN instead of MAVE | Setting 1 SDR · Setting 1, n=100, p=10; n=200, p=30; n=300, p=30 | ||π_B̂−π_B₀||_F (mean): 0.160 -> 0.135; 0.214 -> 0.276; 0.130 -> 0.120 | Mixed |
| Use rank-regularized NN instead of SAVE | Setting 2 SDR · Setting 2, σ=0.1, 0.2, 0.5 | ||π_B̂−π_B₀||_F (mean): 0.319 -> 0.296; 0.338 -> 0.628; 0.367 -> 1.197 | Mixed |
| Use rank-regularized NN instead of the strongest classical baseline | Setting 3 SDR · Setting 3, n=200, 500, 1000 | ||π_B̂−π_B₀||_F (mean): 1.246 -> 0.639; 1.076 -> 0.248; 0.554 -> 0.075 | Helped |
| Use rank-regularized NN instead of the strongest classical baseline | Setting 4 SDR · Setting 4, n=1000, d=4; n=1500, d=6; n=2000, d=8 | ||π_B̂−π_B₀||_F (mean): 0.293 -> 0.127; 0.415 -> 0.144; 0.344 -> 0.140 | Helped |
| Add rank regularization to a vanilla neural network (ablation) | Neural network regression on Seoul weather · Seoul weather test data | testing error (mean): 0.774 -> 0.602 | Helped |
| Use rank-regularized NN instead of MAVE-based regression | Seoul weather regression · Seoul weather test data | testing error (mean): 0.743 -> 0.602 | Helped |
Surprising findings
- The paper argues the separate first-stage initialization used in prior neural SDR work may be unnecessary, since the first layer alone recovers the subspace in a toy example.
Limitations
- Theory assumes the intrinsic dimension d is known; the unknown-d case is only sketched in the supplement.
- Consistency theorem requires depth growing with n and fixed width, and the rate may not be optimal.
- Central subspace extension is only heuristic with no theoretical guarantees or substantial experiments.
Experimental setup
Datasets: simulated regression settings 1-4 Seoul weather bias-correction dataset (UCI)
Benchmarks: simulated SDR settings 1-4 Seoul weather temperature prediction (UCI)
Metrics: Frobenius distance between projection matrices testing mean squared error
Baselines: SIR SAVE PHD MAVE GKDR vanilla neural network
Builds on: sufficient mean dimension reduction MAVE information bottleneck
Closest papers by meaning
- Enhancing Sufficient Dimension Reduction via Hellinger Correlation (ICML 2024 · similarity 0.87)
- Deep Principal Support Vector Machines for Nonlinear Sufficient Dimension Reduction (ICML 2025 · similarity 0.86)
- Deep linear networks for regression are implicitly regularized towards flat minima (NeurIPS 2024 · similarity 0.86)
- Generalization Bounds for Rank-sparse Neural Networks (NeurIPS 2025 · similarity 0.85)
- Stable Minima of ReLU Neural Networks Suffer from the Curse of Dimensionality: The Neural Shattering Phenomenon (NeurIPS 2025 · similarity 0.85)
- Sharp Generalization for Nonparametric Regression by Over-Parameterized Neural Networks: A Distribution-Free Analysis in Spherical Covariate (ICML 2025 · similarity 0.85)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.