Provable Accuracy Collapse in Embedding-Based Representations under Dimensionality Mismatch

• URL: http://arxiv.org/abs/2605.03346v1
• Date: Tue, 05 May 2026 04:05:34 GMT
• Title: Provable Accuracy Collapse in Embedding-Based Representations under Dimensionality Mismatch
• Abstract summary: We prove sharp dimension-accuracy tradeoffs and identify a fundamental information-theoretic limitation.<n>Unless the embedding dimension $d$ is chosen close to the ground-truth dimension $D$, accuracy undergoes a sudden collapse.
• Score: 5.144873004876024
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Embedding-based representations in Euclidean space $\mathbb{R}^d$ are a cornerstone of modern machine learning, where a major goal is to use the \emph{smallest dimension} that faithfully captures data relations. In this work, we prove sharp dimension--accuracy tradeoffs and identify a fundamental information-theoretic limitation: unless the embedding dimension $d$ is chosen close to the ground-truth dimension $D$, accuracy undergoes a sudden collapse. Our main result shows that this phenomenon arises even in standard contrastive learning settings, where supervision is limited to a set of $m$ anchor--positive--negative triplets $(i,j,k)$ encoding distance comparisons $\mathrm{dist}(i,j) < \mathrm{dist}(i,k)$. Specifically, given triplets realizable by an unknown ground-truth embedding in $D$ dimensions, we prove that there exists constant $c < 1$, such that \emph{every embedding of dimension at most $cD$ violates half of the triplets}, yielding accuracy as low as a trivial one-dimensional solution that ignores the input. We complement our information-theoretic bounds with strong computational hardness results: under the Unique Games Conjecture, even if the given triplets are nearly realizable in $D=1$ dimension, no polynomial-time algorithm -- \textit{regardless of its dimension} -- can achieve accuracy above the trivial $50\%$ baseline.

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