Revisiting High-Order Tensor Singular Value Decomposition From Basic Element Perspective
Bibliographic record
Abstract
Recently, tensor singular value decomposition (t-SVD), based on the tensor-tensor product (t-product), has become a powerful tool for processing third-order tensor data. However, constrained by the fact that the basic element is the fiber (i.e., vector) in the t-product, higher-order tensor data (i.e., order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d>3$</tex-math></inline-formula>) are usually unfolded into third-order tensors to satisfy the classical t-product setting, which leads to the destroying of high-dimensional structure. By revisiting the basic element in the t-product, we suggest a generalized t-product called element-based tensor-tensor product (elt-product) as an alternative of the classic t-product, where the basic element is a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(d-2)$</tex-math></inline-formula>th-order tensor instead of a vector. The benefit of the elt-product is that it can better preserve high-dimensional structures and that it can explore more complex interactions via higher-order convolution instead of first-order convolution in classic t-product. Starting from the elt-product, we develop new tensor-SVD and low-rank tensor metrics (e.g., rank and nuclear norm). Equipped with the suggested metrics, we present a tensor completion model for high-order tensor data and prove the exact recovery guarantees. To harness the resulting nonconvex optimization problem, we apply an alternating direction method of the multiplier (ADMM) algorithm with a theoretical convergence guarantee. Extensive experimental results on the simulated and real-world data (color videos, light-field images, light-field videos, and traffic data) demonstrate the superiority of the proposed model against the state-of-the-art baseline models.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".