Learning in High Dimension Always Amounts to Extrapolation

arxiv:2110.09485

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Data, training, optimization · 2021-10-18

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2. Part 2

Contents

• Learning in High Dimension Always Amounts to Extrapolation
• Abstract
• 1 Introduction
• Definition 1 .
• Theorem 1 ( Bárány and Füredi, (1988) ) .
• 2 Interpolation is Doomed by the Curse of Dimensionality
• 2.1 The Role of the Intrinsic, Ambient and Convex Hull Dimensions
• 2.2 Real Datasets and Embeddings are no Exception
• 3 Theoretical Quantification of Interpolation Probabilities
• Definition 2 (Convex position problem) .
• Theorem 2 ( Valtr, (1995) ; Valtr, (1996) ) .
• Theorem 3 ( Buchta, (1986) ) .
• Theorem 4 ( Kabluchko and Zaporozhets, (2020) ) .
• 4 Conclusion