Linear to multi-linear algebra and systems using tensors
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Bibliographic record
Abstract
In the past few decades, multi-linear algebra also known as tensor algebra has been adapted and employed as a tool for various engineering applications. Recent developments in tensor algebra have indicated that several well-known concepts from linear algebra can be extended to a multi-linear setting with the help of a special form of tensor contracted product, known as the Einstein product. Thus, the tensor contracted product and its properties can be harnessed to define the notions of multi-linear system theory where the input, output signals, and the system are inherently multi-domain or multi-modal. This study provides an overview of tensor algebra tools which can be seen as an extension of linear algebra, at the same time highlighting the differences and advantages that the multi-linear setting brings forth. In particular, the notions of tensor inversion, tensor singular value, and tensor eigenvalue decomposition using the Einstein product are explained. In addition, this study also introduces the notion of contracted convolution for both discrete and continuous multi-linear system tensors. Tensor network representation of various tensor operations is also presented. In addition, application of tensor tools in developing transceiver schemes for multi-domain communication systems, with an example of MIMO CDMA system, is presented. This study provides a foundation for professionals whose research involves multi-domain or multi-modal signals and systems.
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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 it