We introduce the Locally Linear Latent Variable Model (LL-LVM), a probabilistic model for non-linear manifold discovery that describes a joint distribution over observations, their manifold coordinates and locally linear maps conditioned on a set of neighbourhood relationships. The model allows straightforward variational optimisation of the posterior distribution on coordinates and locally lin...

We introduce the Locally Linear Latent Variable Model (LL-LVM), a probabilistic model for non-linear manifold discovery that describes a joint distribution over observations, their manifold coordinates and locally linear maps conditioned on a set of neighbourhood relationships. The model allows straightforward variational optimisation of the posterior distribution on coordinates and locally lin...

We consider the ferromagnetic Ising model on a sequence of graphs Gn converging locally weakly to a rooted random tree. Generalizing [Probab. Theory Related Fields 152 (2012) 31–51], under an appropriate “continuity” property, we show that the Ising measures on these graphs converge locally weakly to a measure, which is obtained by first picking a random tree, and then the symmetric mixture of ...

Many real-life applications consider nominal categorical predictor variables that have a hierarchical structure, e.g. economic activity data in Official Statistics. In this paper, we focus on linear regression models built the presence of type variables, and study consolidation their categories to better tradeoff between interpretability fit model data. We propose so-called Tree based Linear Re...

There are many methods in identification developing every day. But identification of dynamic systems has still remained a complex open problem. One of the new effective methods of identification in nonlinear problems is identification with Neurofuzzy approach. Compared with classic neural network and wavelet network this method is faster and more accurate, demonstrated with an example in this p...

Model-based trees are used to find subgroups in data which differ with respect to model parameters. In some applications it is natural to keep some parameters fixed globally for all observations while asking if and how other parameters vary across the subgroups. Existing implementations of model-based trees can only deal with the scenario where all parameters depend on the subgroups. We propose...

We give constructions of probabilistically checkable proofs (PCPs) of length n · polylog n proving satisfiability of circuits of size n that can verified by querying polylog n bits of the proof. We also give analogous constructions of locally testable codes (LTCs) mapping n information bits to n · polylog n bit long codewords that are testable with polylog n queries. Our constructions rely on n...