Notice: The reproducibility variables underlying each score are classified using an automated LLM-based pipeline, validated against a manually labeled dataset. LLM-based classification introduces uncertainty and potential bias; scores should be interpreted as estimates. Full accuracy metrics and methodology are described in [1].
Scaling and Scalability: Provable Nonconvex Low-Rank Tensor Estimation from Incomplete Measurements
Authors: Tian Tong, Cong Ma, Ashley Prater-Bennette, Erin Tripp, Yuejie Chi
JMLR 2022 | Venue PDF | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | In this section, we provide numerical experiments to corroborate our theoretical findings, with the codes available at https://github.com/Titan-Tong/Scaled GD. The simulations are performed in Matlab with a 3.6 GHz Intel Xeon Gold 6244 CPU. We illustrate the numerical performance of Scaled GD for tensor completion to corroborate our findings, especially its computational advantage over the regularized GD algorithm Han et al. (2020) that is closest to our design. Their algorithm was originally proposed for tensor regression, nevertheless, it naturally applies to tensor completion and exhibits similar results. Since the scaled projection does not visibly impact the performance, we implement Scaled GD without performing the projection. Also, we empirically find that the regularization used in Han et al. (2020) has no visible benefits, hence we implement GD without the regularization. |
| Researcher Affiliation | Collaboration | Tian Tong EMAIL Department of Electrical and Computer Engineering Carnegie Mellon University Pittsburgh, PA 15213, USA. Cong Ma EMAIL Department of Statistics University of Chicago Chicago IL 60637, USA. Ashley Prater-Bennette EMAIL Air Force Research Laboratory Rome, NY 13441, USA. Erin Tripp EMAIL Air Force Research Laboratory Rome, NY 13441, USA. Yuejie Chi EMAIL Department of Electrical and Computer Engineering Carnegie Mellon University Pittsburgh, PA 15213, USA. |
| Pseudocode | Yes | Algorithm 1 Scaled GD for low-rank tensor completion. Algorithm 2 Scaled GD for low-rank tensor regression |
| Open Source Code | Yes | In this section, we provide numerical experiments to corroborate our theoretical findings, with the codes available at https://github.com/Titan-Tong/Scaled GD. |
| Open Datasets | No | We construct the ground truth tensor X = (U , V , W ) S by generating U , V and W as random orthonormal matrices, and the core tensor S composed of i.i.d. standard Gaussian entries, i.e. S (j1, j2, j3) N(0, 1) for 1 jk r, k = 1, 2, 3. |
| Dataset Splits | No | The paper uses synthetically generated data and describes sampling observations (e.g., 'Each entry of the tensor is observed i.i.d. with probability p (0, 1].') rather than predefined training/test/validation splits from a publicly available dataset. |
| Hardware Specification | Yes | The simulations are performed in Matlab with a 3.6 GHz Intel Xeon Gold 6244 CPU. |
| Software Dependencies | No | The paper states that 'The simulations are performed in Matlab', but does not specify a version number for Matlab or any other software libraries used. |
| Experiment Setup | Yes | Throughout the experiments, we used the ground truth value σmax(X ) in running (30), while in practice, this parameter needs to estimated; to put it differently, the step size of GD is not scale-invariant, whereas the step size of Scaled GD is. To ensure the ground truth tensor X = (U , V , W ) S has a prescribed condition number κ, we generate the core tensor S Rr r r according to S (j1, j2, j3) = σj1/ r if j1 +j2 +j3 0 (mod r) and 0 otherwise, where {σj1}1 j1 r take values spaced equally from 1 to 1/κ. It then follows that σmax(X ) = 1, σmin(X ) = 1/κ, and the condition number of X is exactly κ. Figure 3 illustrates the convergence speed of Scaled GD and GD under different step sizes, where we plot the relative error after at most 80 iterations (the algorithm is terminated if the relative error exceeds 102 following an excessive step size). It can be seen that Scaled GD outperforms GD quite significantly even when the step size of GD is optimized for its performance. Hence, we will fix η = 0.3 for the rest of the comparisons for both Scaled GD and GD without hurting the conclusions. |