This talk will be an introduction to the methods used in the study of spectral properties of Schroedinger operators with a potential defined via the action of an ergodic transformation. Open problems relating to Lyapunov exponents over a skew shift base will be discussed.
We develop efficient algorithms for estimating low-degree moments of unknown distributions in the presence of adversarial outliers. The guarantees of our algorithms improve in many cases significantly over the best previous ones, obtained in recent works. We also show that the guarantees of our algorithms match information-theoretic lower-bounds for the class of distributions we consider. These better guarantees allow us to give improved algorithms for independent component analysis and learning mixtures of Gaussians in the presence of outliers.
Given a measure preserving dynamical system, a real-valued observable determines a random process (by composing the observable with the iterates of the transformation). An important topic in ergodic theory is the study of the statistical properties of the corresponding sum process.
I will show an explicit construction of a binary error correcting code with relative distance $\frac{1-\epsilon}{2}$ and relative rate $\epsilon^{2+o(1)}$. This comes close to the Gilbert-Varshamov bound that shows such codes with rate $\epsilon^2$ exist, and theLP lower bound that shows rate $\frac{\epsilon^2}{\log \frac{1}{\epsilon}}$ is necessary. Previous explicit constructions had rate about$\epsilon^3$, and this is the first explicit construction to get that close to the Gilbert-Varshamov bound.
This talk will have two parts, on Monday and Tuesday.
I will show an explicit construction of a binary error correcting code with relative distance $\frac{1-\epsilon}{2}$ and relative rate $\epsilon^{2+o(1)}$. This comes close to the Gilbert-Varshamov bound that shows such codes with rate $\epsilon^2$ exist, and theLP lower bound that shows rate $\frac{\epsilon^2}{\log \frac{1}{\epsilon}}$ is necessary. Previous explicit constructions had rate about$\epsilon^3$, and this is the first explicit construction to get that close to the Gilbert-Varshamov bound.
This talk will have two parts, on Monday and Tuesday.