Matched budgets — CP fits closer, by cancelling

The kernel is layer3.1.conv2.weight from ResNet-18 with torchvision’s ImageNet weights: 256 × 256 × 3 × 3. The fair comparison fixes the number of stored numbers and asks which fits better.

• CP fits more closely than Tucker-2 at every budget. Tucker-2 needs roughly twice the numbers for the same error (cached runs: six budgets from 64× to 2×, 1,000 ALS iterations, three random starts).
• The price: at every budget, on every start, CP’s largest rank-one term is at least four times the kernel’s norm — over two hundred times on one start at 64×. Large terms that nearly cancel are degeneracy: a 1% change to one factor of a term 7 times the kernel’s norm moves the kernel by 7%, which is why fine-tuning can explode.
• Uniqueness runs out: Kruskal’s condition fails past R = 257, so the 4× and 2× fits carry no uniqueness guarantee.

Tucker-2’s orthonormal factors keep its core’s norm at most ‖𝒦‖. Choose CP when the compression ratio is the target and the pipeline controls degeneracy; choose Tucker-2 when a well-conditioned start for fine-tuning matters more.