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The Johnson-Lindenstrauss lemma is one of the corner stone results in dimensionality reduction. It says that given N , for any set of N vectors X ⊂ R n , there exists a mapping f : X → R m such that f (X) preserves all pairwise distances between vectors in X to within (1 ± ε) if m = O(ε −2 lg N ). Much effort has gone into developing fast embedding algorithms, with the Fast Johnson-Lindenstrauss transform of Ailon and Chazelle being one of the most well-known techniques. The current fastest algorithm that yields the optimal m = O(ε −2 lg N ) dimensions has an embedding time of O(n lg n + ε −2 lg 3 N ). An exciting approach towards improving this, due to Hinrichs and Vybíral, is to use a random m × n Toeplitz matrix for the embedding. Using Fast Fourier Transform, the embedding of a vector can then be computed in O(n lg m) time. The big question is of course whether m = O(ε −2 lg N ) dimensions suffice for this technique. If so, this would end a decades long quest to obtain faster and faster Johnson-Lindenstrauss transforms. The current best analysis of the embedding of Hinrichs and Vybíral shows that m = O(ε −2 lg 2 N ) dimensions suffices. The main result of this paper, is a proof that this analysis unfortunately cannot be tightened any further, i.e., there exists a set of N vectors requiring m = Ω(ε −2 lg 2 N ) for the Toeplitz approach to work.

Multiplication is one of the most fundamental computational problems, yet its true complexity remains elusive. The best known upper bound, by Fürer, shows that two n-bit numbers can be multiplied via a boolean circuit of size O(n lg n • 4 lg * n ), where lg * n is the very slowly growing iterated logarithm. In this work, we prove that if a central conjecture in the area of network coding is true, then any constant degree boolean circuit for multiplication must have size Ω(n lg n), thus almost completely settling the complexity of multiplication circuits. We additionally revisit classic conjectures in circuit complexity, due to Valiant, and show that the network coding conjecture also implies one of Valiant's conjectures.

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Johnson-Lindenstrauss Transforms are powerful tools for reducing the dimensionality of data while preserving key characteristics of that data, and they have found use in many fields from machine learning to differential privacy and more. This note explains what they are; it gives an overview of their use and their development since they were introduced in the 1980s; and it provides many references should the reader wish to explore these topics more deeply.The text was previously a main part of the introduction of my PhD thesis [Fre20], but it has been adapted to be self contained and serve as a (hopefully good) starting point for readers interested in the topic.