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Revisiting Orbital Minimization Method for Neural Operator Decomposition
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Revisits the orbital minimization method (OMM) from computational chemistry, gives a simple linear-algebraic consistency proof, and adapts nested OMM variants to train neural networks to decompose PSD operators across RL Laplacian, Schrödinger, and contrastive learning tasks.
Problem
Approximating top eigenspaces/eigenfunctions of positive semidefinite linear operators with neural networks in a principled, variational, orthogonalization-free way, which existing neural approaches lack due to surrogate losses or architectural constraints.
Approach
Derives the OMM objective (and higher-order OMM-p) from a simple variational argument valid for general PSD matrices, extends it to operators via second-moment matrices, and introduces sequential nesting with partial stop-gradients (and a Sanger-style variant) so ordered eigenfunctions can be learned with a single surrogate loss.
Key results
- On CIFAR-100 with projector, OMM (p=1)+OMM (p=2) reaches 64.77% top-1 versus SimCLR's 66.50%.
- On Laplacian representation learning, NestedOMM performs comparably to ALLO across grid environments without hyperparameter tuning.
- For the 2D hydrogen atom, the Sanger variant of OMM performs on par with LoRA_seq (NeuralSVD), which outperforms SpIN and NeuralEF.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use joint nesting instead of un-nested OMM (p=1) (ablation) | OMM (ResNet-18) · CIFAR-100, with projector | top-1 accuracy (%): 60.02 -> 61.30 | Helped |
| use joint nesting instead of un-nested OMM (p=1) (ablation) | OMM (ResNet-18) · CIFAR-100, DirectCLR (top-64 dim.) | top-1 accuracy (%): 59.83 -> 59.92 | No clear effect |
| use sequential nesting instead of un-nested OMM (p=1) (ablation) | OMM (ResNet-18) · CIFAR-100, with projector | top-1 accuracy (%): 60.02 -> 59.91 | No clear effect |
| use sequential nesting instead of un-nested OMM (p=1) (ablation) | OMM (ResNet-18) · CIFAR-100, DirectCLR (top-64 dim.) | top-1 accuracy (%): 59.83 -> 52.96 | Hurt |
| use OMM with p=2 instead of p=1 (ablation) | OMM (ResNet-18) · CIFAR-100, with projector | top-1 accuracy (%): 60.02 -> 63.92 | Helped |
| use OMM with p=2 instead of p=1 (ablation) | OMM (ResNet-18) · CIFAR-100, DirectCLR (top-64 dim.) | top-1 accuracy (%): 59.83 -> 61.27 | Helped |
| combine OMM p=1 and p=2 objectives rather than use p=1 alone (ablation) | OMM (ResNet-18) · CIFAR-100, with projector | top-1 accuracy (%): 60.02 -> 64.77 | Helped |
| combine OMM p=1 and p=2 objectives rather than use p=1 alone (ablation) | OMM (ResNet-18) · CIFAR-100, DirectCLR (top-64 dim.) | top-1 accuracy (%): 59.83 -> 63.99 | Helped |
| replace OMM (p=1) with SimCLR | ResNet-18 representation learning · CIFAR-100, with projector | top-1 accuracy (%): 60.02 -> 66.50 | Helped |
| re-tune ALLO hyperparameters from the k=11 settings to b=1.0 and α_barrier=0.1 (ablation) | ALLO · Laplacian representation learning, GridMaze-17 | average cosine similarity: 0.8150 -> 0.9798 | Helped |
| re-tune ALLO hyperparameters from the k=11 settings to b=1.0 and α_barrier=0.1 (ablation) | ALLO · Laplacian representation learning, GridMaze-32 | average cosine similarity: 0.3340 -> 0.7359 | Helped |
| re-tune ALLO hyperparameters from the k=11 settings to b=1.0 and α_barrier=0.1 (ablation) | ALLO · Laplacian representation learning, GridRoom-64 | average cosine similarity: 0.3420 -> 0.7417 | Helped |
Surprising findings
- Plain OMM gradient diverged on the 2D hydrogen atom without an identity spectrum shift, while the Sanger variant trained stably.
- Higher-order OMM (p=2) improved CIFAR-100 top-1 accuracy by roughly 4 points over OMM (p=1), and nesting gave no improvement.
Limitations
- OMM applies only to PSD operators; unbounded-spectrum operators require an inverse-operator trick that can be numerically unstable and hyperparameter-sensitive.
- Self-supervised image results do not surpass the SimCLR baseline and are described as preliminary.
- Higher-order OMM cannot be implemented with the partial stop-gradient nesting trick, limiting its use in ordered settings.
Experimental setup
Datasets: CIFAR-100 grid-world RL environments (GridMaze, GridRoom) synthetic 2D hydrogen atom / harmonic oscillator / infinite well operators
Benchmarks: CIFAR-100 linear probe grid-world Laplacian eigenfunctions 2D hydrogen atom eigenfunctions 2D harmonic oscillator 2D infinite well
Metrics: top-1 accuracy top-5 accuracy average cosine similarity per mode relative eigenvalue error angle distance subspace distance
Baselines: ALLO LoRA_seq (NeuralSVD) SpIN NeuralEF SimCLR DirectCLR
Builds on: orbital minimization method Sanger's rule NeuralSVD (low-rank approximation objective with nesting) SimCLR ALLO
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