Research map / Learning theory and algorithms / Matrix and tensor methods
Cost-efficient Gaussian tensor network embeddings for tensor-structured inputs
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the proposed Gaussian TN embedding instead of recursive leverage score sampling | CP-ALS · Low-rank CP decomposition with \(s\gg R^{1.5}\) and \(\epsilon = \Theta(1)\) | per-iteration cost: The per-iteration cost is \(\Theta(NR\epsilon/\delta)=\Omega(NR)\) times better | Helped |
| Use the proposed Gaussian TN embedding instead of recursive leverage score sampling | CP-ALS · High-rank CP decomposition with \(R\gg s\) | per-iteration cost: The cost ratio ... is \(\Tilde{\Theta}(N^{1.5}\delta/(\epsilon^5R^{0.5}))\) | Mixed |
| Use TN embedding rather than tree or tensor train embeddings | Tensor train input sketching · Tensor train inputs with rank below 32 | FLOPs | Helped |
| Use tree or tensor train embedding rather than TN embedding | Tensor train input sketching · Tensor train inputs with rank at least 32 | FLOPs | No clear effect |
| Use TN, tree, or tensor train embedding rather than Khatri-Rao product embedding | Kronecker product input sketching · Increasing tensor order | sketch size | Helped |
| Use TN embedding rather than tree or tensor train embedding | Kronecker product input sketching · Kronecker product inputs | FLOPs | Helped |
| Use the proposed sketching method rather than tree embedding | Kronecker product input sketching · Kronecker product inputs | asymptotic sketching cost: \(\Theta(\sum_{j= 1}^{N} s_jm + Nm^{2.5})\) vs. \(\Theta(\sum_{j= 1}^{N} s_jm + Nm^{3})\) | Helped |
| Use the existing tensor train embedding for randomized rounding | Tensor train rounding · Tensor train rounding | asymptotic sketching cost: \(\Theta(NsR^2m)\), matching the lower bound \(\Omega(NsR^2m)\) | Helped |
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