The striking ability of unsupervised word translation has been demonstrated recently with the help of low-dimensional word vectors / pretraining, which is used by all successful methods and assumed to be necessary. We test and challenge this assumption by developing a method that can also make use of high dimensional signal. Freed from the limits of low dimensions, we show that relying on low-dimensional vectors and their incidental properties miss out on better denoising methods and signals in high dimensions, thus stunting the potential of the data. Our results show that unsupervised translation can be achieved more easily and robustly than previously thought -- less than 80MB and minutes of CPU time is required to achieve over 50\% accuracy for English to Finnish, Hungarian, and Chinese translations when trained in the same domain; even under domain mismatch, the method still works fully unsupervised on English NewsCrawl to Chinese Wikipedia and English Europarl to Spanish Wikipedia, among others. These results challenge prevailing assumptions on the necessity and superiority of low-dimensional vectors and show that the higher dimension signal can be used rather than thrown away.