Neural Information Processing Systems http://nips.cc/

In this paper, we provide a theoretical understanding of word embedding and its dimensionality. Motivated by the unitary-invariance of word embedding, we propose the Pairwise Inner Product (PIP) loss, a novel metric on the dissimilarity between word embeddings. Using techniques from matrix perturbation theory, we reveal a fundamental bias-variance trade-off in dimensionality selection for word embeddings. This bias-variance trade-off sheds light on many empirical observations which were previously unexplained, for example the existence of an optimal dimensionality. Moreover, new insights and discoveries, like when and how word embeddings are robust to over-fitting, are revealed. By optimizing over the bias-variance trade-off of the PIP loss, we can explicitly answer the open question of dimensionality selection for word embedding.

We thank all the reviewers for their thoughtful feedback. We will clarify these points. R2 and R3 had concerns about the amount of content we deferred to the appendix. In Appendix B.4, we discuss a variant of the embedding reconstruction error applicable to R2 asked about our question answering results in Section 2.3. We use the DrQA model [5], as described in Section 4. R3 proposed an idea to use non-uniform quantization to further improve the performance of quantized embeddings.

In this work a Pairwise Inner Product Loss is developed, motivated by unitary invariance of word embeddings. It then investigates theoretically the relationship between word embedding dimensionality to robustness under different singular exponents, and relates it to bias/variance tradeoff. The discovered relationships are used to define a criterion for a word embedding dimensionality selection procedure, which is empirically validated on 3 intrinsic evaluation tasks. The PIP loss technique is well motivated, clear, and easy to understand. It would be interesting to see this technique applied in other contexts and for other NLP tasks. The paper is clearly written, well motivated, and sections follow naturally.

In this paper, we provide a theoretical understanding of word embedding and its dimensionality. Motivated by the unitary-invariance of word embedding, we propose the Pairwise Inner Product (PIP) loss, a novel metric on the dissimilarity between word embeddings. Using techniques from matrix perturbation theory, we reveal a fundamental bias-variance trade-off in dimensionality selection for word embeddings. This bias-variance trade-off sheds light on many empirical observations which were previously unexplained, for example the existence of an optimal dimensionality. Moreover, new insights and discoveries, like when and how word embeddings are robust to over-fitting, are revealed. By optimizing over the bias-variance trade-off of the PIP loss, we can explicitly answer the open question of dimensionality selection for word embedding.

In this paper, we provide a theoretical understanding of word embedding and its dimensionality. Motivated by the unitary-invariance of word embedding, we propose the Pairwise Inner Product (PIP) loss, a novel metric on the dissimilarity between word embeddings. Using techniques from matrix perturbation theory, we reveal a fundamental bias-variance trade-off in dimensionality selection for word embeddings. This bias-variance trade-off sheds light on many empirical observations which were previously unexplained, for example the existence of an optimal dimensionality. Moreover, new insights and discoveries, like when and how word embeddings are robust to over-fitting, are revealed. By optimizing over the bias-variance trade-off of the PIP loss, we can explicitly answer the open question of dimensionality selection for word embedding.

In this paper, we provide a theoretical understanding of word embedding and its dimensionality. Motivated by the unitary-invariance of word embedding, we propose the Pairwise Inner Product (PIP) loss, a novel metric on the dissimilarity between word embeddings. Using techniques from matrix perturbation theory, we reveal a fundamental bias-variance trade-off in dimensionality selection for word embeddings. This bias-variance trade-off sheds light on many empirical observations which were previously unexplained, for example the existence of an optimal dimensionality. Moreover, new insights and discoveries, like when and how word embeddings are robust to over-fitting, are revealed. By optimizing over the bias-variance trade-off of the PIP loss, we can explicitly answer the open question of dimensionality selection for word embedding.

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.