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Lecture 11

January 24, 2013

Today we started to talk about Matrix Rigidity. We first described the motivating problem from algebraic circuit complexity, namely, the problem of explicitly constructing a matrix, such that, when viewed as a linear transformation, cannot be computed by a linear-circuit with linear size and logarithmic depth simultaneously.

We then discussed matrix rigidity, showed a couple examples and basic facts and proved Valiant’s theorem, which roughly states that if a matrix is rigid enough, it cannot be computed by a circuit as above. In other words, Valiant reduced the computational problem to an algebraic/combinatorial one (which is always great!).

We saw, however, that the most rigid matrices known are not close to being rigid enough so to imply circuit lower bounds. In fact, we showed that the main argument in the construction of many rigid matrices is based on the fact that if any r by r submatrix has full rank, then the number of changes in the matrix must be at least ~(n^2/r)log(n/r), which is the strongest bound we have on the rigidity of any explicit matrix. The proof we saw is due to Shokrollahi, Spielman and Stemann. The key lemma that got into this came from extremal graph theory, and is known as Zarankiewicz Problem, which we proved (following a proof from Bollobas book, Extremal Graph Theory, page 309).

Next lecture will be the last lecture in the course. We will give examples of concrete (modest, though state-of-the-art) rigid matrices, and present an approach by Zeev Dvir for the construction of better rigid matrices.

In this lecture we followed Chapter 13.8 for Stasys Jukna’s great book, and Chapter 2 of a survey by Lokam.

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