‘Stretch’ vs ‘slice’ methods for representing three‐way structure via matrix notation
Bibliographic record
Abstract
Abstract A three‐way array must be represented in two‐way form if its structure is to be described and manipulated by means of matrix notation. Historically, two methods, here called ‘array stretching’ and ‘array slicing’, have been used. More recently, however, array slicing has often been overlooked, resulting in a loss of mathematical flexibility. ‘Stretching’ involves matricizing (unfolding) the three‐way array and applying one's mathematical operations to the resulting two‐way matrix; this results in expressions that are often quite useful for parameter estimation but which are relatively long and require practice to interpret properly. ‘Slicing’ involves taking a representative two‐way subarray and applying operations to it; this often gives compact and easily understood expressions but requires the introduction of extra matrix names and becomes awkward if the array is not ‘slicewise regular’. In this paper the advantages of each approach are demonstrated and compared by applying them to a set of models from the Tucker and Parafac families. In addition, we show how slicewise representation can be improved by using (i) angle brackets to eliminate the need for extra diagonal matrices, and (ii) ‘encapsulated summation’ notation to allow representation of array structure that is orderly but not slicewise regular. Copyright © 2002 John Wiley & Sons, Ltd.
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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 | 0.003 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 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".