Research map / Learning theory and algorithms / Matrix and tensor methods
Towards a Zero-One Law for Column Subset Selection
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Apply column subset selection when the loss has approximate monotonicity, an approximate triangle inequality, and an approximate regression algorithm | GeneralFunctionLowRankApprox · General entrywise loss, rank-k | approximation factor: O(k\log k) | Helped |
| Use Huber loss rather than Frobenius or entrywise ℓ1 loss | Low-rank approximation · Synthetic mixture of small Gaussian noise and sparse outliers | Huber loss | Helped |
| Require approximate monotonicity and an approximate triangle inequality for the loss | Column subset selection for general entrywise losses · General entrywise loss | existence of a good small column subset | Helped |
| Use fewer than \sqrt{\log n} selected columns | Column subset selection for Huber loss · Huber loss, rank-1 | approximation factor: o(\sqrt{\log n}) columns; no O(1)-approximation | Hurt |
| Use a jumping loss without an approximate triangle inequality | Column subset selection · Identity matrix with k=\Omega(\log n) | approximation guarantee: rank-k cost 0; every selected-column solution has positive cost | Hurt |
| Use ReLU loss without approximate monotonicity | Column subset selection · Identity matrix | approximation guarantee: best rank-k cost 0; any subset of at most n-1 columns has cost at least 1 | Hurt |
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