On Multilinear Forms: Bias, Correlation, and Tensor Rank
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In this paper, we prove new relations between the bias of multilinear forms, the correlation between multilinear forms and lower degree polynomials, and the rank of tensors over GF(2)= {0,1}. We show the following results for multilinear forms and tensors. 1. Correlation bounds : We show that a random d-linear form has exponentially low correlation with low-degree polynomials. More precisely, for d ≪ 2^o(k), we show that a random d-linear form f(X_1,X_2, ..., X_d) : (GF(2)^k)^d → GF(2) has correlation 2^-k(1-o(1)) with any polynomial of degree at most d/10. This result is proved by giving near-optimal bounds on the bias of random d-linear form, which is in turn proved by giving near-optimal bounds on the probability that a random rank-t d-linear form is identically zero. 2. Tensor-rank vs Bias : We show that if a d-dimensional tensor has small rank, then the bias of the associated d-linear form is large. More precisely, given any d-dimensional tensor T :[k]×... [k]_d times→ GF(2) of rank at most t, the bias of the associated d-linear form f_T(X_1,...,X_d) := ∑_(i_1,...,i_d) ∈ [k]^d T(i_1,i_2,..., i_d) X_1,i_1· X_1,i_2... X_d,i_d is at most (1-1/2^d-1)^t. The above bias vs tensor-rank connection suggests a natural approach to proving nontrivial tensor-rank lower bounds for d=3. In particular, we use this approach to prove that the finite field multiplication tensor has tensor rank at least 3.52 k matching the best known lower bound for any explicit tensor in three dimensions over GF(2).
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