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Fix r, 1 ≤ r ≤ n. Let Rign−r (L1 , . . , Lk ) =: R. We will fix a (by a probabilistic argument) to satisfy the following two properties: √ (1) ∀i, 1 ≤ i ≤ k,, |Li (a)| ≤ R 2 ln k + 4. (2) For this a, the bounded coefficient linear circuit complexity of x → Ax, where A is the circulant matrix defined above is at least (1/2)(n − r) log n − cn for some constant c. 21. The bounded coefficient bilinear complexity of convolution is at least (1/12)n log n − O(n log log n). To prove the existence of a satisfying (1) and (2), let V ⊆ Cn be an n/2-dimensional subspace achieving Rign/2 (L1 , .

18 with the following result cf. ([12, 4, Ex. 41]). An integer is square-free if it is not divisible by the square of any prime number. 20. The square roots of all positive square-free integers are linearly independent over Q. In particular, for distinct primes √ √ p1 , . . , pm , [Q( p1 , . . , pm ) : Q] = 2m . The next rigidity lower bound uses the Generalized SS-dimension. 21. Let Z := e2πi/pjk 1≤j,k≤n , where pjk are the first n2 distinct primes. Then, for 0 ≤ r ≤ n, we have RZ (r) ≥ n(n − 9r), assuming n is sufficiently large.

Lk ) =: R. We will fix a (by a probabilistic argument) to satisfy the following two properties: √ (1) ∀i, 1 ≤ i ≤ k,, |Li (a)| ≤ R 2 ln k + 4. (2) For this a, the bounded coefficient linear circuit complexity of x → Ax, where A is the circulant matrix defined above is at least (1/2)(n − r) log n − cn for some constant c. 21. The bounded coefficient bilinear complexity of convolution is at least (1/12)n log n − O(n log log n). To prove the existence of a satisfying (1) and (2), let V ⊆ Cn be an n/2-dimensional subspace achieving Rign/2 (L1 , .

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Complexity Lower Bounds using Linear Algebra (Foundations and Trends in Theoretical Computer Science) by Satyanarayana V. Lokam


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