Jensen’s Inequality for Separately Convex Noncommutative Functions
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Abstract Classically, Jensen’s Inequality asserts that if $X$ is a compact convex set, and $f:K\to {\mathbb {R}}$ is a convex function, then for any probability measure $\mu $ on $K$, that $f(\text {bar}(\mu ))\le \int f\; \text {d}\mu $, where $\text {bar}(\mu )$ is the barycenter of $\mu $. Recently, Davidson and Kennedy proved a noncommutative (“nc”) version of Jensen’s inequality that applies to nc convex functions, which take matrix values, with probability measures replaced by ucp maps. In the classical case, if $f$ is only a separately convex function, then $f$ still satisfies the Jensen inequality for any probability measure that is a product measure. We prove a noncommutative Jensen inequality for functions that are separately nc convex in each variable. The inequality holds for a large class of ucp maps that satisfy a noncommutative analogue of Fubini’s theorem. This class of ucp maps includes any free product of ucp maps built from Boca’s theorem, or any ucp map that is conditionally free in the free-probabilistic sense of Młotkowski. As an application to free probability, we obtain some operator inequalities for conditionally free ucp maps applied to free semicircular families.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.004 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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