Measuring and exploiting data-model fit with sheaves
Thursday, May 24, 2018 - 1:30pm - 2:00pm
Because sheaves model consistency relationships between local data, they are easily assembled from detailed models of systems. Being topological in nature, sheaves mediate local-to-global inference. By incorporating local geometry from the start, the global fit between local data and models can be quantified, which supports robust inferences about missing or inaccurate data. The utility of this approach is not merely its intellectual cohesion; it also yields performant algorithms. Why? Because statistical robustness requires geometry, and statistical inference is fundamentally a balance between local data and global models. Time-honored statistical techniques that fit data to models are projections of these geometric sheaf algorithms. But since sheaf theory encourages abstraction, sheaves may provide topological invariants that govern model selection, the bias-variance tradeoff, and ultimately the problem of overfitting. This talk will provide an overview of this connection and a review of new and open problems.