In this paper, we introduce a new estimator for the emission densities of a nonparametric hidden Markov model. It is adaptive and minimax with respect to each state's regularityas opposed to globally minimax estimators, which adapt to the worst regularity among the emission densities. Our method is based on Goldenshluger and Lepski's methodology. It is computationally efficient and only requires a family of preliminary estimators, without any restriction on the type of estimators considered. We present two such estimators that allow to reach minimax rates up to a logarithmic term: a spectral estimator and a least squares estimator. We show how to calibrate it in practice and assess its performance on simulations and on real data.
This paper considers the deconvolution problem in the case where the target signal is multidimensional and no information is known about the noise distribution. More precisely, no assumption is made on the noise distribution and no samples are available to estimate it: the deconvolution problem is solved based only on the corrupted signal observations. We establish the identifiability of the model up to translation when the signal has a Laplace transform with an exponential growth smaller than 2 and when it can be decomposed into two dependent components. Then, we propose an estimator of the probability density function of the signal without any assumption on the noise distribution. As this estimator depends of the lightness of the tail of the signal distribution which is usually unknown, a model selection procedure is proposed to obtain an adaptive estimator in this parameter with the same rate of convergence as the estimator with a known tail parameter. Finally, we establish a lower bound on the minimax rate of convergence that matches the upper bound.Corollary 3. Assume that there exists ρ < 2 such that R (1) and R (1) for all z 1 ∈ C d1 and z 2 ∈ C d1 , and Φ R (1) can not be identically zero since Φ R (1) (0) = 1. We then apply Theorem 1.
In this paper, we consider a bivariate process (Xt, Yt) t∈Z which, conditionally on a signal (Wt) t∈Z , is a hidden Markov model whose transition and emission kernels depend on (Wt) t∈Z. The resulting process (Xt, Yt, Wt) t∈Z is referred to as an input-output hidden Markov model or hidden Markov model with external signals. We prove that this model is identifiable and that the associated maximum likelihood estimator is consistent. Introducing an Expectation Maximization-based algorithm, we train and evaluate the performance of this model in several frameworks. In addition to learning dependencies between (Xt, Yt) t∈Z and (Wt) t∈Z , our approach based on hidden Markov models with external signals also outperforms state-of-the-art algorithms on real-world fashion sequences.
We introduce a new general identifiable framework for principled disentanglement referred to as Structured Nonlinear Independent Component Analysis (SNICA). Our contribution is to extend the identifiability theory of deep generative models for a very broad class of structured models. While previous works have shown identifiability for specific classes of time-series models, our theorems extend this to more general temporal structures as well as to models with more complex structures such as spatial dependencies. In particular, we establish the major result that identifiability for this framework holds even in the presence of noise of unknown distribution. The SNICA setting therefore subsumes all the existing nonlinear ICA models for time-series and also allows for new much richer identifiable models. Finally, as an example of our framework's flexibility, we introduce the first nonlinear ICA model for time-series that combines the following very useful properties: it accounts for both nonstationarity and autocorrelation in a fully unsupervised setting; performs dimensionality reduction; models hidden states; and enables principled estimation and inference by variational maximum-likelihood.
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