Lower Bounds on Performance of Metric Tree Indexing Schemes for Exact\n Similarity Search in High Dimensions
Bibliographic record
Abstract
Within a mathematically rigorous model, we analyse the curse of\ndimensionality for deterministic exact similarity search in the context of\npopular indexing schemes: metric trees. The datasets $X$ are sampled randomly\nfrom a domain $\\Omega$, equipped with a distance, $\\rho$, and an underlying\nprobability distribution, $\\mu$. While performing an asymptotic analysis, we\nsend the intrinsic dimension $d$ of $\\Omega$ to infinity, and assume that the\nsize of a dataset, $n$, grows superpolynomially yet subexponentially in $d$.\nExact similarity search refers to finding the nearest neighbour in the dataset\n$X$ to a query point $\\omega\\in\\Omega$, where the query points are subject to\nthe same probability distribution $\\mu$ as datapoints. Let $\\mathscr F$ denote\na class of all 1-Lipschitz functions on $\\Omega$ that can be used as decision\nfunctions in constructing a hierarchical metric tree indexing scheme. Suppose\nthe VC dimension of the class of all sets $\\{\\omega\\colon f(\\omega)\\geq a\\}$,\n$a\\in\\R$ is $o(n^{1/4}/\\log^2n)$. (In view of a 1995 result of Goldberg and\nJerrum, even a stronger complexity assumption $d^{O(1)}$ is reasonable.) We\ndeduce the $\\Omega(n^{1/4})$ lower bound on the expected average case\nperformance of hierarchical metric-tree based indexing schemes for exact\nsimilarity search in $(\\Omega,X)$. In paricular, this bound is superpolynomial\nin $d$.\n
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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 | 0.018 | 0.111 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.004 | 0.002 |
| Bibliometrics | 0.003 | 0.006 |
| Science and technology studies | 0.003 | 0.005 |
| Scholarly communication | 0.007 | 0.019 |
| Open science | 0.006 | 0.010 |
| Research integrity | 0.005 | 0.004 |
| Insufficient payload (model declined to judge) | 0.008 | 0.003 |
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".