A filtering technique for Markov chains with applications to spectral   embedding
  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. 
Stefan Steinerberger
cs.DM