Research map / Learning theory and algorithms / Matrix and tensor methods
Lower and Upper Bounds on the Pseudo-Dimension of Tensor Network Models
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Parameterize the weight tensor with an arbitrary TN structure, with capacity measured by its core parameters and number of vertices | TN regression, classification, and completion models · Arbitrary tensor network structures | VC-dimension / pseudo-dimension upper bound: 2 \params{G}\log(12|V|) | Helped |
| Use the new general TN bound instead of the cited previous bound | Low-rank matrix classifiers · Low-rank matrix classification | VC-dimension upper bound: r(d_1+d_2)\log(r(d_1+d_2)) -> 10r(d_1+d_2) | Helped |
| Use the new general TN bound instead of the cited previous bound | Low-rank matrix completion · Low-rank matrix completion | pseudo-dimension upper bound: r(d_1+d_2)\log\frac{16ed_1}{r} -> 10r(d_1+d_2) | Helped |
| Use the general TN upper bound for rank-one tensors | Rank-one TN models · Rank-one tensors | VC-dimension / pseudo-dimension bound tightness: $(d-1)p $; 2dp\log(12p) | Mixed |
| Use the general TN upper bound for TT and TR tensors | TT/TR models · TT and TR tensors | VC-dimension / pseudo-dimension bound tightness: r^2d; $ 2pr^2d\log(12p) $ | Mixed |
| Use the general TN upper bound for low-rank matrices | Low-rank matrix models · Low-rank matrices | VC-dimension / pseudo-dimension bound tightness: rd; 20rd | Mixed |
| Use rank-r Tucker weights with the general bound | Tucker models · Tucker tensors | parameter/order scaling of the complexity bound: \bigo{(r^p+dpr)\log(p)} | Hurt |
| Use the general arbitrary-structure bound rather than a tree-only covering-number analysis | TN models with cycles, including TR and PEPS · Cyclic TN structures | applicability of the generalization bound | Helped |
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