  The particle filter (PF) is a powerful inference tool widely used to estimate
the filtering distribution in non-linear and/or non-Gaussian problems. To
overcome the curse of dimensionality of PF, the block PF (BPF) inserts a
blocking step to partition the state space into several subspaces or blocks of
smaller dimension so that the correction and resampling steps can be performed
independently on each subspace. Using blocks of small size reduces the variance
of the filtering distribution estimate, but in turn the correlation between
blocks is broken and a bias is introduced.
  When the dependence relationships between state variables are unknown, it is
not obvious to decide how to split the state space into blocks and a
significant error overhead may arise from a poor choice of partitioning. In
this paper, we formulate the partitioning problem in the BPF as a clustering
problem and we propose a state space partitioning method based on spectral
clustering (SC). We design a generalized BPF algorithm that contains two new
steps: (i) estimation of the state vector correlation matrix from predicted
particles, (ii) SC using this estimate as the similarity matrix to determine an
appropriate partition. In addition, a constraint is imposed on the maximal
cluster size to prevent SC from providing too large blocks. We show that the
proposed method can bring together in the same blocks the most correlated state
variables while successfully escaping the curse of dimensionality.
