On <i>k</i>-Mer-Based and Maximum Likelihood Estimation Algorithms for Trace Reconstruction
Bibliographic record
Abstract
The goal of the trace reconstruction problem is to recover a string$\mathbf {x}\in \{0,1\}^{n}$given many independenttracesofx, where a trace is a subsequence obtained from deleting bits ofxindependently with some given probability$p\in [0,1$). A recent result of Chase (STOC 2021) shows howxcan be determined (in exponential time) from$\exp ({O}(n^{1/5})\log ^{5} n)$traces. This is the state-of-the-art result on the sample complexity of trace reconstruction. In this paper we consider two kinds of algorithms for the trace reconstruction problem. We first observe that the bound of Chase, which is based on statistics of arbitrary length-ksubsequences, can also be obtained by considering the “k-mer statistics”, i.e., statistics regarding occurrences ofcontiguous k-bit strings (a.k.a,k-mers) in the initial stringx, for$k = 2n^{1/5}$. Mazooji and Shomorony (arXiv.2210.10917) show that such statistics (calledk-mer density map) can be estimated within$\varepsilon $accuracy from$ {\mathrm {poly}} (n, 2^{k}, 1/ {\varepsilon })$traces. We call an algorithm to bek-mer-basedif it reconstructsxgiven estimates of thek-mer density map. Such algorithms essentially capture all the analyses in the worst-case and smoothed-complexity models of the trace reconstruction problem we know of so far. Our first, and technically more involved, result shows that anyk-mer-based algorithm for trace reconstruction must use$\exp (\Omega (n^{1/5} \sqrt {\log n}))$traces, thus establishing the optimality of this number of traces. The analysis of this result also shows that the analysis technique used by Chase (STOC 2021) is essentially tight, and hence new techniques are needed in order to improve the worst-case upper bound. This result is shown by considering an appropriate class of real polynomials, that have been previously studied in the context of trace estimation (De, O’Donnell, Servedio. Annals of Probability 2019; Nazarov, Peres. STOC 2017), and proving that two of these polynomials are very close to each other on an arc in the complex plane. Our proof of the proximity of such polynomials uses new technical ingredients that allow us to focus on just a few coefficients of these polynomials. Our second, simple, result considers the performance of the Maximum Likelihood Estimator (MLE), which specifically picks the source string that has the maximum likelihood to generate the samples (traces). We show that the MLE algorithm uses a nearly optimal number of traces, i.e., up to a factor ofnin the number of samples needed for an optimal algorithm, and show that this factor ofnloss may be necessary under general “model estimation” settings.
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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 | 0.005 | 0.032 |
| Meta-epidemiology (narrow) | 0.003 | 0.002 |
| Meta-epidemiology (broad) | 0.003 | 0.002 |
| Bibliometrics | 0.003 | 0.004 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.005 | 0.004 |
| Research integrity | 0.004 | 0.006 |
| Insufficient payload (model declined to judge) | 0.006 | 0.006 |
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".