Algebra 2 [Lecture notes] by Ben Smith

By Ben Smith

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We can make this more precise as follows: first note that polynomials with small formulas, such as f (xij ) = x11 x22 · · · xnn are easy to compute. Let b ⊂ M atn×n denote the subset of upper triangular matrices, and G(detn ) the symmetry group of the determinant. Observe that detn |b is just the polynomial f above, and that G(detn ) can move any matrix into b relatively cheaply. This gives a geometric proof that the determinant is easy to compute. One can ask: what other polynomials are easy to compute?

Comparing the two formulas, the difference between the permanent and the determinant is “just” a matter of a few minus signs. For example, one can convert the permanent of a 2 × 2 matrix to a determinant by simply changing the sign of one of the off-diagonal entries. Initially there was hope that one could do something similar in general, but these hopes were quickly dashed. The next idea was to write the permanent of an m × m matrix as the determinant of a larger matrix. Using the sequence (detn ), Valiant defined an algebraic analog of the class P, denoted VPws , as follows: a sequence of polynomials (pm ) where pm is a homogeneous polynomial of degree d(m) in v(m) variables, with d(m), v(m) polynomial functions of m, is in the class VPws if dc(pm ) = O(mc ) for some constant c.

1 ⎜ .. ⎟ r ⎜ .. 2) ⎝ . ⎠ = ⎝ . ⎠ x (t) + · · · + ⎝ . ⎠ x (t) + ⎝ . 3) y = Ax + v. Here A is a fixed m × r matrix, v = v(t) ∈ Rm is a vector-valued function representing the noise, and x(t) = (x1 (t), . . 3. Tensor decomposition 15 sources. ) The v i (t) are assumed (i) to be independent random variables which are independent of the xj (t), (ii) E{v i } = 0, and (iii) the moments E{v i1 · · · v ip } are bounded by a small constant. One would like to recover x(t), plus the matrix A, from knowledge of the function y(t) alone.

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