Entropic Affinities: Properties and Efficient Numerical Computation

Gaussian affinities are commonly used in graph-based methods such as spectral clustering or nonlinear embedding. Hinton and Roweis (2003) introduced a way to set the scale individually for each point so that it has a distribution over neighbors with a desired perplexity, or effective number of neighbors. This gives very good affinities that adapt locally to the data but are harder to compute. We study the mathematical properties of these “entropic affinities” and show that they implicitly define a continuously differentiable function in the input space and give bounds for it. We then devise a fast algorithm to compute the widths and affinities, based on robustified, quickly convergent root-finding methods combined with a tree- or density-based initialization scheme that exploits the slowly-varying behavior of this function. This algorithm is nearly optimal and much more accurate and fast than the existing bisection-based approach, particularly with large datasets, as we show with image and text data.