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 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) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.004 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.002 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.000 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".