Linear to multi-linear algebra and systems using tensors
Bibliographic record
Abstract
In past few decades, tensor algebra also known as multi-linear algebra has been developed and customized as a tool to be used for various engineering applications. In particular, with the help of a special form of tensor contracted product, known as the Einstein Product and its properties, many of the known concepts from Linear Algebra could be extended to a multi-linear setting. This enables to define the notions of multi-linear system theory where the input, output signals and the system are multi-domain in nature. This paper provides an overview of tensor algebra tools which can be seen as an extension of linear algebra, at the same time highlighting the difference and advantages that the multi-linear setting brings forth. In particular, the notion of tensor inversion, tensor singular value and tensor Eigenvalue decomposition using the Einstein product is explained. In addition, this paper also introduces the notion of contracted convolution in both discrete and continuous multi-linear system tensors. Tensor Networks representation of various tensor operations is also presented. Also, application of tensor tools in developing transceiver schemes for multi-domain communication systems, with an example of MIMO CDMA systems, is presented. Thus this paper acts as an entry point tutorial for graduate students whose research involves multi-domain or multi-modal signals and systems.
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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 | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.001 | 0.006 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.005 |
| Insufficient payload (model declined to judge) | 0.014 | 0.004 |
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".