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How Much Restricted Isometry is Needed In Nonconvex Matrix Recovery?
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use an RIP measurement operator with δ=1/2 that admits a spurious local minimum | Randomly initialized SGD for matrix sensing · Matrix sensing counterexample (n=2, r=1) | SGD failure rate: 12.1±0.3% | Hurt |
| Initialize SGD randomly rather than at the spurious local minimum | SGD on the engineered bad instance · Engineered bad instance (n=12, r=2, δ=0.975) | SGD success rate: 100% | Helped |
| Use an RIP measurement operator with δ=1/2 that admits a spurious local minimum | Randomly initialized SGD for matrix sensing · Engineered good instance (n=12, r=1, δ=1/2) | SGD failure rate: 5.90±2.24% | Hurt |
| Initialize SGD at the spurious local minimum instead of randomly | SGD on the engineered bad instance · Engineered bad instance (n=12, r=2, δ=0.975) | SGD recovery error: >100% | Hurt |
| Allow spurious local minima in the objective landscape | Randomly initialized SGD for matrix sensing · Engineered bad instance (n=12, r=2, δ=0.975) | SGD success rate: 100% | No clear effect |
| Allow spurious local minima in the objective landscape | Randomly initialized SGD for matrix sensing · Engineered good instance (n=12, r=1, δ=1/2) | SGD failure rate: 5.90±2.24% | Hurt |
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