arXiv subjects

Joona Karjalainen

Publications and source records attributed to Joona Karjalainen.

7 recordsLinked to original sources

arXiv2026

On the Forgetting of Particle Filters

We study the forgetting properties of the particle filter when its state - the collection of particles - is regarded as a Markov chain. Under a strong mixing assumption on the particle filter's underlying Feynman-Kac model, we find that the particle filter is exponentially mixing, and forgets its initial state in $O(\log N )$ 'time', where $N$ is the number of particles and time refers to the number of particle filter algorithm steps, each comprising a selection (or resampling) and mutation (or prediction) operation. We present an example which shows that this rate is optimal. In contrast to our result, available results to date are extremely conservative, suggesting $O(α^N)$ time steps are needed, for some $α>1$, for the particle filter to forget its initialisation. We also study the conditional particle filter (CPF) and extend our forgetting result to this context. We establish a similar conclusion, namely, CPF is exponentially mixing and forgets its initial state in $O(\log N)$ time. To support this analysis, we establish new time-uniform $L^p$ error estimates for CPF, which can be of independent interest. We also establish new propagation-of-chaos type results using our proof techniques, discuss implications to couplings of particle filters and an application to processing out-of-sequence measurements.

math.PR↗

arXiv2025

Mixing time of the conditional backward sampling particle filter

The conditional backward sampling particle filter (CBPF) is a powerful Markov chain Monte Carlo sampler for general state space hidden Markov model (HMM) smoothing. It was proposed as an improvement over the conditional particle filter (CPF), which has an $O(T^2)$ complexity under a general `strong' mixing assumption, where $T$ is the time horizon. Empirical evidence of the superiority of the CBPF over the CPF has never been theoretically quantified. We show that the CBPF has $O(T \log T)$ time complexity under strong mixing: its mixing time is upper bounded by $O(\log T)$, for any sufficiently large number of particles $N$ independent of $T$. This $O(\log T)$ mixing time is optimal. To prove our main result, we introduce a novel coupling of two CBPFs, which employs a maximal coupling of two particle systems at each time instant. The coupling is implementable and we use it to construct unbiased, finite variance, estimates of functionals which have arbitrary dependence on the latent state's path, with a total expected cost of $O(T \log T)$. We use this to construct unbiased estimates of the HMM's score function, and also investigate other couplings which can exhibit improved behaviour. We demonstrate our methods on financial and calcium imaging applications.

stat.CO↗

arXiv2024

Clique and cycle frequencies in a sparse random graph model with overlapping communities

A statistical network model with overlapping communities can be generated as a superposition of mutually independent random graphs of varying size. The model is parameterized by the number of nodes, the number of communities, and the joint distribution of the community size and the edge probability. This model admits sparse parameter regimes with power-law limiting degree distributions and non-vanishing clustering coefficients. This article presents large-scale approximations of clique and cycle frequencies for graph samples generated by the model, which are valid for regimes with unbounded numbers of overlapping communities. Our results reveal the growth rates of these subgraph frequencies and show that their theoretical densities can be reliably estimated from data.

math.PR↗

arXiv2022

Normal and stable approximation to subgraph counts in superpositions of Bernoulli random graphs

The clustering property of complex networks indicates the abundance of small dense subgraphs in otherwise sparse networks. For a community-affiliation network defined by a superposition of Bernoulli random graphs, which has a nonvanishing global clustering coefficient and a power-law degree distribution, we establish normal and $α$--stable approximations to the number of small cliques, cycles and more general $2$-connected subgraphs.

math.PR↗

arXiv2021

Assortativity and bidegree distributions on Bernoulli random graph superpositions

A probabilistic generative network model with $n$ nodes and $m$ overlapping layers is obtained as a superposition of $m$ mutually independent Bernoulli random graphs of varying size and strength. When $n$ and $m$ are large and of the same order of magnitude, the model admits a sparse limiting regime with a tunable power-law degree distribution and nonvanishing clustering coefficient. In this article we prove an asymptotic formula for the joint degree distribution of adjacent nodes. This yields a simple analytical formula for the model assortativity, and opens up ways to analyze rank correlation coefficients suitable for random graphs with heavy-tailed degree distributions. We also study the effects of power laws on the asymptotic joint degree distributions.

math.PR↗

arXiv2018

Moment-based parameter estimation in binomial random intersection graph models

Binomial random intersection graphs can be used as parsimonious statistical models of large and sparse networks, with one parameter for the average degree and another for transitivity, the tendency of neighbours of a node to be connected. This paper discusses the estimation of these parameters from a single observed instance of the graph, using moment estimators based on observed degrees and frequencies of 2-stars and triangles. The observed data set is assumed to be a subgraph induced by a set of $n_0$ nodes sampled from the full set of $n$ nodes. We prove the consistency of the proposed estimators by showing that the relative estimation error is small with high probability for $n_0 \gg n^{2/3} \gg 1$. As a byproduct, our analysis confirms that the empirical transitivity coefficient of the graph is with high probability close to the theoretical clustering coefficient of the model.

cs.SI↗

arXiv2018

Parameter estimators of random intersection graphs with thinned communities

This paper studies a statistical network model generated by a large number of randomly sized overlapping communities, where any pair of nodes sharing a community is linked with probability $q$ via the community. In the special case with $q=1$ the model reduces to a random intersection graph which is known to generate high levels of transitivity also in the sparse context. The parameter $q$ adds a degree of freedom and leads to a parsimonious and analytically tractable network model with tunable density, transitivity, and degree fluctuations. We prove that the parameters of this model can be consistently estimated in the large and sparse limiting regime using moment estimators based on partially observed densities of links, 2-stars, and triangles.

math.PR↗