Paper:
On the Dimensionality of 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 biasvariance trade-off of the PIP loss, we can explicitly answer the open question of dimensionality selection for word embedding.

Review:
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. Only section 5.2 would benefit from some more details (which are deferred to the appendix, presumably due to space constraints), and the Figures are very small. To improve the latter point, I recommend increasing font sizes of axis labels and ticks, as well as legend, and line width.  The work is original and provides a new theoretical view on the robustness of word embeddings. Perhaps the title "On Word Embedding Dimensionality" is a bit too broad as the work chooses a definite route, even though it reveals connections between several existing approaches.  A potential criticism of this work is that it empirically only investigates word embedding dimensionality using intrinsic evaluation (though on 3 tasks), but not extrinsically / in a downstream NLP task. This is even though understanding the effect of word embedding dimensionality on downstream models and applications is one of the major motivations laid out in the Introduction.   Second, it is acknowledged that empirical validation carries "the risk of selecting sub-optimal dimensionalities" (line 37), but a similar caveat is missing for the proposed method "We can explicitly answer the open question of dimensionality selection for word embedding" (line 10-11). It should be kept in mind that also the proposed method (even though theoretically motivated) also only minimises an _expected_ quantity, and that it is also prone to the same risk of choosing sub-optimal dimensionality as a purely empirical selection.   Related to the previous point - another potential criticism is that the PIP Loss metric does not take into account word frequency. Less frequent words will have embeddings with more variance than frequent words. But by Zipf's Law, most words are rare, and giving all words the same contribution in the PIP loss will result in increased variance of the PIP loss when word embeddings are fitted on different corpora. This is speculative, but perhaps it is worth down-weighting infrequent word dimensions in the PIP loss? To investigate this possible phenomenon one could fit vectors either other datasets than text8, or different portions of text8, and observe confidence intervals.  Minor comments: - Line 227: could you explain why the square root term is dominated by (\lambda_k - \lambda_{k-1})^{-1} - Could you expand on line 166 (third line)? - Line 227 (end) -- should the exponent be -2? - to avoid misinterpretation, one could be more explicit in which exact term is plotted in Figure 1a. - line 254: the relative gap to the optimal dimension would be more informative - line 250: Are the chosen dimensionalities really exactly the optimal ones also empirically? It would be better if this was more concrete: dimensionalities chosen with PIP loss minimisation are generally in line with empirical validation, but not exactly identical? - line 267: could you elaborate on this?