By Satyanarayana V. Lokam
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Extra info for Complexity Lower Bounds using Linear Algebra (Foundations and Trends in Theoretical Computer Science)
We thus get a 2r × (n − 2rR/n) submatrix B that contains no changes and hence is a submatrix of H. By deﬁnition of R, this submatrix must have rank at most r. 6, we get r ≥ rank(B) ≥ 2r(n − 2rR/n)/n, since B 2F is exactly the number entries in B. Rearranging this inequality, we get R ≥ n2 /4r. 1. Kashin and Razborov  also use spectral methods to prove an Ω(n2 /r) lower bound on the rigidity of a generalized Hadamard matrix. The essential claim in their paper is that a random k × k submatrix of a generalized Hadamard matrix has rank Ω(k).
4) and . Morgenstern used determinantal arguments to prove an Ω(n log n) lower bound on the bounded coeﬃcient complexity of the Fourier transform. We will give a more general proof of the statement based on . We will also give a proof due to Pudl´ ak which also uses bounds on the determinant, but in a diﬀerent way from Morgenstern. His proof has the nice feature that it gives the strongest bounds known to date on constant depth circuits as well. The main lemma for his result is the following.
The proof of Morgenstern, discussed later, however, gives a much better lower bound of log det(A)/ log 2c with no assumptions on the depth. Proof. A depth-d synchronous circuit can be viewed as consisting of d layers of intermediate nodes with i nodes on level i (inputs at level 0, outputs at level d). The edges from level i to level i + 1 deﬁne an i × i+1 matrix Ai+1 , where 0 = d = n. We thus have A = A1 · · · Ad . 14 to this factorization of A. Now, note that if si is the number of edges from level i − 1 to level i, then, Ai 2F ≤ si c2 .
Complexity Lower Bounds using Linear Algebra (Foundations and Trends in Theoretical Computer Science) by Satyanarayana V. Lokam