A Random Matrix Approach to Low-Multilinear-Rank Tensor Approximation
Authors: Hugo Lebeau, Florent Chatelain, Romain Couillet
JMLR 2025 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | Figure 2 plots, for an order-3 tensor, as a function of the signal-to-noise ratio (SNR) ω = P 2 F/σN, the alignments between the singular subspace of the signal P spanned by X(ℓ) and the dominant singular subspace of the observation T spanned by ˆU (ℓ). Solid curves are the alignments predicted by Theorem 9 while dotted curves are empirical alignments computed on a 100 200 300 tensor with signal-rank (3, 4, 5). If the SNR ω is too small, there is no alignment, meaning that truncated MLSVD fails to recover P the signal is masked by the noise. When it exceeds a critical value (see Theorem 9 and Section 3.2 for details), a phase transition phenomenon occurs6: the alignment starts to grow i.e., truncated MLSVD now partially recovers P and converges to 1 as ω + . |
| Researcher Affiliation | Academia | Hugo Lebeau EMAIL Université Grenoble Alpes, CNRS, Inria, Grenoble INP, LIG Grenoble, 38000, France Florent Chatelain EMAIL Université Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab Grenoble, 38000, France Romain Couillet EMAIL Université Grenoble Alpes, CNRS, Inria, Grenoble INP, LIG Grenoble, 38000, France |
| Pseudocode | Yes | Algorithm 1: Higher-Order Orthogonal Iteration (De Lathauwer et al., 2000a) for ℓ= 1, . . . , d do U (ℓ) 0 rℓdominant left singular vectors of T (ℓ) for ℓ= 1, . . . , d do U (ℓ) t+1 rℓdominant left singular vectors of T (ℓ) ℓ =ℓU (ℓ ) t until convergence at t = T GHOOI T(U (1) T , . . . , U (d) T ) |
| Open Source Code | No | The paper does not contain any explicit statement about releasing source code or provide links to a code repository. |
| Open Datasets | No | The paper analyzes a 'general spiked tensor model' and its experiments are based on simulations using generated data. For example, 'Experimental setting: d = 3, (n1, n2, n3) = (100, 200, 300), N = n1 + n2 + n3 and (r1, r2, r3) = (3, 4, 5)'. It does not refer to any external or publicly available datasets. |
| Dataset Splits | No | The paper's experiments are based on a 'general spiked tensor model' with 'additive Gaussian noise tensor N whose entries are independent N(0, 1) random variables'. The data is generated based on specified parameters (e.g., tensor dimensions), not split from a pre-existing dataset. Therefore, traditional training/test/validation splits are not applicable or provided. |
| Hardware Specification | No | The paper does not contain any specific details about the hardware (e.g., GPU, CPU models, memory) used to run the simulations or experiments. |
| Software Dependencies | No | The paper mentions 'MATLAB toolbox Tensorlab (Vervliet et al., 2016)' as an example of a tool for CPD, but it does not state that this specific software with a version number was used for the implementation or experiments described in the paper, nor does it list any other software dependencies with version numbers. |
| Experiment Setup | Yes | Experimental setting: d = 3, (n1, n2, n3) = (100, 200, 300), N = n1 + n2 + n3 and (r1, r2, r3) = (3, 4, 5). Figure 3: ... Experimental setting: d = 3, (n1, n2, n3) = (300, 500, 700), N = n1 + n2 + n3, (r1, r2, r3) = (3, 4, 5) and P 2 F/σN = 15. Figure 4: ... Experimental setting: d = 3, (n1 = 6), N = n1 + n2 + n3 and (r1, r2, r3) = (3, 4, 5). |