Provable Accuracy Collapse in Embedding-Based Representations under Dimensionality Mismatch

Dionysis Arvanitakis, Vaggos Chatziafratis, Yiyuan Luo
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:3891-3904, 2026.

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 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 every embedding of dimension at most $cD$ violates almost 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—regardless of its dimension—can achieve accuracy above the trivial 50% baseline.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-arvanitakis26a, title = {Provable Accuracy Collapse in Embedding-Based Representations under Dimensionality Mismatch}, author = {Arvanitakis, Dionysis and Chatziafratis, Vaggos and Luo, Yiyuan}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {3891--3904}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/arvanitakis26a/arvanitakis26a.pdf}, url = {https://proceedings.mlr.press/v306/arvanitakis26a.html}, 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 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 every embedding of dimension at most $cD$ violates almost 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—regardless of its dimension—can achieve accuracy above the trivial 50% baseline.} }
Endnote
%0 Conference Paper %T Provable Accuracy Collapse in Embedding-Based Representations under Dimensionality Mismatch %A Dionysis Arvanitakis %A Vaggos Chatziafratis %A Yiyuan Luo %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-arvanitakis26a %I PMLR %P 3891--3904 %U https://proceedings.mlr.press/v306/arvanitakis26a.html %V 306 %X Embedding-based representations in Euclidean space $\mathbb{R}^d$ are a cornerstone of modern machine learning, where a major goal is to use the 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 every embedding of dimension at most $cD$ violates almost 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—regardless of its dimension—can achieve accuracy above the trivial 50% baseline.
APA
Arvanitakis, D., Chatziafratis, V. & Luo, Y.. (2026). Provable Accuracy Collapse in Embedding-Based Representations under Dimensionality Mismatch. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:3891-3904 Available from https://proceedings.mlr.press/v306/arvanitakis26a.html.

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