Research map / Learning theory and algorithms / Matrix and tensor methods
Randomized Sparse Matrix Compression for Large-Scale Constrained Optimization in Cancer Radiotherapy
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Proposes Randomized Minor-value Rectification (RMR), a matrix sparsification method that keeps large dose-matrix entries and redistributes the row-wise sum of small entries via randomized sampling, to speed up radiotherapy treatment-plan optimization.
Problem
Solving large-scale constrained quadratic optimization for radiotherapy beam intensities requires sparsifying the huge dose influence matrix, but the common naive thresholding degrades treatment quality, and existing randomized sparsifiers have sample counts scaling poorly at high sparsity.
Approach
Deterministically keep entries above a threshold and, for each row, randomly resample the sum of the discarded small entries (ℓ1-proportional sampling with ceil(Σi/ϵ) samples), yielding an unbiased sparse sketch with nonzero count growing as 1/τ and theoretical feasibility and optimality gap bounds.
Key results
- At 98% sparsification on 10 lung patients, RMR achieves much lower optimality gaps (e.g., ~9-32% for RMR vs ~100-350% for AHK06 and BKKS21 in Table 7) and near-zero feasibility gaps, while AHK06 gives lower ℓ2 error (e.g., 8.64% vs 9.32% for patient 1).
- For patient 1 to reach 35% optimality gap, RMR needs 98.90% sparsification and solves the optimization in 57 s versus 200 s for AHK06 at 95.05% sparsification.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use RMR instead of AHK06 at a fixed 98% sparsification level | Dose influence matrix for radiotherapy optimization · Lung patients (10 patients) | Optimality gap | Helped |
| Use RMR instead of AHK06 at a fixed 98% sparsification level | Dose influence matrix for radiotherapy optimization · Lung patients (10 patients) | Relative ℓ2-norm error | Hurt |
| Use RMR instead of AHK06 at a fixed 98% sparsification level | Dose influence matrix for radiotherapy optimization · Lung patients (10 patients) | Algorithm runtime | Hurt |
| Replace naive threshold sparsification with randomized sparsification | Dose influence matrix for radiotherapy optimization · Radiotherapy optimization | Relative ℓ2-norm error | Helped |
| Use RMR instead of AHK06 to create a sparse matrix for patient 1 | Radiotherapy optimization · Lung patient 1 | Optimization runtime: 200 seconds vs. 57 seconds | Helped |
| Use RMR instead of AHK06 for prostate patients | Dose influence matrix for radiotherapy optimization · Prostate patients (5 patients) | Algorithm runtime | Helped |
| Use RMR instead of AHK06 for prostate patients at high sparsity | Dose influence matrix for radiotherapy optimization · Prostate patients (high sparsity, such as 99%) | Relative ℓ2-norm error | Hurt |
| Replace the full dose influence matrix with an RMR sparse matrix at 98% sparsity | Radiotherapy optimization · Lung patients (10 patients) | Optimization runtime: 2826 -> 87.25; 3212 -> 122.5; 2147 -> 45.84; 1277 -> 53.30; 870 -> 32.17; 1447 -> 58.25; 1492 -> 81.47; 714 -> 29.18; 83 | Helped |
Surprising findings
- AHK06 and BKKS21 achieve lower spectral-norm error than RMR, yet RMR yields substantially better feasibility and optimality gaps, so the norm is not a sufficient proxy for treatment quality.
- On prostate patients the naive approach appears to outperform AHK06, suggesting prior sparsifiers do not consistently beat naive thresholding in this application.
Limitations
- Theoretical feasibility and optimality bounds are proven only for the quadratic problem (Eq. 2), not the general formulation.
- Clinical benefit of the improved plans is not comprehensively evaluated; only dose maps and DVHs are shown qualitatively.
- RMR is slower than AHK06 in sparsification runtime, and AHK06 and BKKS21 attain lower spectral error.
Experimental setup
Datasets: PortPy lung patient dose influence matrices (10 patients) prostate patient dose influence matrices (5 patients, not public)
Benchmarks: PortPy lung cancer treatment planning data prostate cancer patient data
Metrics: relative spectral-norm error relative optimality gap feasibility gap relative dose discrepancy sparsification runtime optimization runtime
Baselines: Naive thresholding AHK06 DZ11 AKL13 BKKS21
Builds on: Arora et al. sparsification (AHK06) Drineas and Zouzias (DZ11) Achlioptas et al. (AKL13) Braverman et al. (BKKS21) matrix Bernstein inequality
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