Tensor Least Angle Regression for Sparse Representations of Multidimensional Signals
Bibliographic record
Abstract
Sparse signal representations have gained much interest recently in both signal processing and statistical communities. Compared to orthogonal matching pursuit (OMP) and basis pursuit, which solve the [Formula: see text] and [Formula: see text] constrained sparse least-squares problems, respectively, least angle regression (LARS) is a computationally efficient method to solve both problems for all critical values of the regularization parameter [Formula: see text]. However, all of these methods are not suitable for solving large multidimensional sparse least-squares problems, as they would require extensive computational power and memory. An earlier generalization of OMP, known as Kronecker-OMP, was developed to solve the [Formula: see text] problem for large multidimensional sparse least-squares problems. However, its memory usage and computation time increase quickly with the number of problem dimensions and iterations. In this letter, we develop a generalization of LARS, tensor least angle regression (T-LARS) that could efficiently solve either large [Formula: see text] or large [Formula: see text] constrained multidimensional, sparse, least-squares problems (underdetermined or overdetermined) for all critical values of the regularization parameter [Formula: see text] and with lower computational complexity and memory usage than Kronecker-OMP. To demonstrate the validity and performance of our T-LARS algorithm, we used it to successfully obtain different sparse representations of two relatively large 3D brain images, using fixed and learned separable overcomplete dictionaries, by solving both [Formula: see text] and [Formula: see text] constrained sparse least-squares problems. Our numerical experiments demonstrate that our T-LARS algorithm is significantly faster (46 to 70 times) than Kronecker-OMP in obtaining [Formula: see text]-sparse solutions for multilinear leastsquares problems. However, the [Formula: see text]-sparse solutions obtained using Kronecker-OMP always have a slightly lower residual error (1.55% to 2.25%) than ones obtained by T-LARS. Therefore, T-LARS could be an important tool for numerous multidimensional biomedical signal processing applications.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.002 |
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 source (direct Gemma or distilled Codex), 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".