PIP Distance: A Unitary-invariant Metric for Understanding Functionality and Dimensionality of Vector Embeddings
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In this paper, we present a theoretical framework for understanding vector embedding, a fundamental building block of many deep learning models, especially in NLP. We discover a natural unitary-invariance in vector embeddings, which is required by the distributional hypothesis. This unitary-invariance states the fact that two embeddings are essentially equivalent if one can be obtained from the other by performing a relative-geometry preserving transformation, for example a rotation. This idea leads to the Pairwise Inner Product (PIP) loss, a natural unitary-invariant metric for the distance between two embeddings. We demonstrate that the PIP loss captures the difference in functionality between embeddings. By formulating the embedding training process as matrix factorization under noise, we reveal a fundamental bias-variance tradeoff in dimensionality selection. With tools from perturbation and stability theory, we provide an upper bound on the PIP loss using the signal spectrum and noise variance, both of which can be readily inferred from data. Our framework sheds light on many empirical phenomena, including the existence of an optimal dimension, and the robustness of embeddings against over-parametrization. The bias-variance tradeoff of PIP loss explicitly answers the fundamental open problem of dimensionality selection for vector embeddings.
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