Public defence in Automation and Control Engineering, M.Sc. Adrien Corenflos

The title of the thesis: Computationally efficient statistical inference in Markovian models 

Thesis defender: Adrien Corenflos
Opponent: Prof. Omiros Papaspiliopoulos, Bocconi University, Italy
Custos: Prof. Simo Särkkä, Aalto University School of Electrical Engineering, Department of Electrical Engineering and Automation

State-space models, or more generally Feynman-Kac models, are ubiquitous in today's world where they are used to represent diverse physical/engineering phenomena ranging from geolocalisation and weather prediction to more artificial constructions like large language or denoising diffusion models in machine learning and artificial intelligence. They link unknown latent dynamics to observed noisy and partial signals on the system's state. These models have been studied since the middle of the 20th century, and two main families of inference methods have emerged: Kalman-approximated and sequential Monte Carlo (SMC) solutions. 

While these provide gold standards, they still present many issues: 

1. Both methods are inherently sequential, whereby the observed data points are processed one by one. 
2. Kalman approximations are not easily controlled for their error, and may fail dramatically (and silently) when the model at hand is far from linear, where these are exact. 
3. SMC methods suffer from differentiability issues, restricting the use of traditional gradient-based parameter estimation methods. 
4. SMC methods suffer from a well-documented curse of dimensionality, making their use restricted. 

Overall, the thesis provides new computational tools that solve these issues either independently or together, in particular in the presence of parallel hardware, which is traditionally left as an afterthought in literature. Throughout, we try to highlight the statistical-computational tradeoffs that may arise from using statistically suboptimal but computationally superior methods. 

In detail, the first article in the thesis is concerned with solving the problem of non-differentiability in particle filtering, relying on optimal transport maps that are only computationally practical on GPU. The second article parallelises extended and sigma-points Kalman approximations, relying on a well-known Gauss-Newton equivalent procedure. The third one applies techniques similar to the latter article to perform Gaussian process regression in logarithmic time. The fourth discusses parallelisation techniques for SMC methods. The fifth is concerned with implementing gradient-based MCMC samplers, in combination with articles 2,4 to solve the curse of dimension while retaining parallel computations. The sixth discusses a new form of Kalman-like approximations, based on Wassertein geodesics. The final article extends the fifth, providing a unifying framework between SMC and Langevin MCMC.

Keywords: State-space models, Markov chain Monte Carlo, Sequential Monte Carlo, Variational inference, Parallelisation, Differentiable programming

Thesis available for public display 10 days prior to the defence at: https://aaltodoc.aalto.fi/doc_public/eonly/riiputus/

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Doctoral theses in the School of Electrical Engineering: https://aaltodoc.aalto.fi/handle/123456789/53