A filtering technique for Markov chains with applications to spectral embedding

Abstract: Spectral methods have proven to be a highly effective tool in understanding the intrinsic geometry of a high-dimensional data set $\left{x_i \right}{i=1}^{n} \subset \mathbb{R}^d$. The key ingredient is the construction of a Markov chain on the set, where transition probabilities depend on the distance between elements, for example where for every $1 \leq j \leq n$ the probability of going from $x_j$ to $x_i$ is proportional to $$ p{ij} \sim \exp \left( -\frac{1}{\varepsilon}|x_i -x_j|^2_{\ell^2(\mathbb{R}^d)}\right) \qquad \mbox{where}~\varepsilon>0~\mbox{is a free parameter}.$$ We propose a method which increases the self-consistency of such Markov chains before spectral methods are applied. Instead of directly using a Markov transition matrix $P$, we set $p_{ii} = 0$ and rescale, thereby obtaining a transition matrix $P^*$ modeling a non-lazy random walk. We then create a new transition matrix $Q = (q_{ij}){i,j=1}^{n}$ by demanding that for fixed $j$ the quantity $q{ij}$ be proportional to $$ q_{ij} \sim \min((P^*)_{ij}, ((P^*)^2)_{ij}, \dots, ((P^*)^k)_{ij}) \qquad \mbox{where usually}~ k=2.$$ We consider several classical data sets, show that this simple method can increase the efficiency of spectral methods and prove that it can correct randomly introduced errors in the kernel.