Lagrangian Optimization of Two-Description Scalar Quantizers
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Bibliographic record
Abstract
In this paper, we study the problem of optimal design of balanced two-description fixed-rate scalar quantizer (2DSQ) under the constraint of convex codecells. Using a graph-based approach to model the problem, we show that the minimum expected distortion of the 2DSQ is a convex function of the number of codecells in the side quantizers. This property allows the problem to be solved by Lagrangian minimization for which the optimal Lagrangian multiplier exists. Given a trial multiplier, we exploit a monotonicity of the objective function, and develop a simple and fast dynamic programming technique to solve the parameterized problem. To further improve the algorithm efficiency, we propose an RD-guided search strategy to find the optimal Lagrangian multiplier. In our experiments on distributions of interest for signal compression applications the proposed algorithm improves the speed of the fastest algorithm so far, by a factor of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> /log <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> ), where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> is the number of codecells in each side quantizer. We also assess the impact on the optimality of the convex codecell constraint. Using a published performance analysis of 2DSQ at high rates, we show that asymptotically this constraint does not preclude optimality for <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> distortion measure, when channels have a higher than 0.12 loss rate.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.005 |
| 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