• Corpus ID: 4562184

Estimation of Markov Chain via Rank-constrained Likelihood

@article{Li2018EstimationOM,
  title={Estimation of Markov Chain via Rank-constrained Likelihood},
  author={Xudong Li and Mengdi Wang and Anru R. Zhang},
  journal={ArXiv},
  year={2018},
  volume={abs/1804.00795}
}
This paper studies the estimation of low-rank Markov chains from empirical trajectories. We propose a non-convex estimator based on rank-constrained likelihood maximization. Statistical upper bounds are provided for the Kullback-Leiber divergence and the $\ell_2$ risk between the estimator and the true transition matrix. The estimator reveals a compressed state space of the Markov chain. We also develop a novel DC (difference of convex function) programming algorithm to tackle the rank… 

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References

SHOWING 1-10 OF 57 REFERENCES
Model reduction of Markov chains via low-rank approximation
  • Kun Deng, Dayu Huang
  • Computer Science, Mathematics
    2012 American Control Conference (ACC)
  • 2012
TLDR
A nuclear-norm regularized optimization problem is proposed for model reduction for Markov chain models, in which the Kullback-Leibler divergence rate is used to measure the similarity between two Markov chains, and the nuclear norm is use to approximate the rank function.
State Compression of Markov Processes via Empirical Low-Rank Estimation
TLDR
A spectral method is proposed for estimating the frequency and transition matrices, estimating the compressed state spaces, and recovering the state aggregation structure if there is any and upper bounds for the estimation and recovery errors are provided and matching minimax lower bounds are provided.
Optimal Kullback-Leibler Aggregation via Spectral Theory of Markov Chains
TLDR
This paper shows that for a certain relaxation of the bi-partition model reduction problem, the solution is shown to be characterized by an associated eigenvalue problem, closely related to the Markov spectral theory for model reduction.
Minimax Estimation of Discrete Distributions Under $\ell _{1}$ Loss
TLDR
This work provides tight upper and lower bounds on the maximum risk of the empirical distribution, and the minimax risk in regimes where the support size S may grow with the number of observations n, and shows that a hard-thresholding estimator oblivious to the unknown upper bound H, is essentially minimax.
Spectral State Compression of Markov Processes
Model reduction of Markov processes is a basic problem in modeling state-transition systems. Motivated by the state aggregation approach rooted in control theory, we study the statistical state
Restricted strong convexity and weighted matrix completion: Optimal bounds with noise
TLDR
The matrix completion problem under a form of row/column weighted entrywise sampling is considered, including the case of uniformentrywise sampling as a special case, and it is proved that with high probability, it satisfies a forms of restricted strong convexity with respect to weighted Frobenius norm.
Near-optimal stochastic approximation for online principal component estimation
TLDR
A nearly optimal finite-sample error bound is proved for the first time for the online PCA algorithm under the subgaussian assumption, and it is shown that the finite- sample error bound closely matches the minimax information lower bound.
A Majorized Penalty Approach for Calibrating Rank Constrained Correlation Matrix Problems
TLDR
This paper first considers a penalized version of this problem and applies the essential ideas of the majorization method to the penalized problem by solving iteratively a sequence of least squares correlation matrix problems without the rank constraint.
Diffusion Approximations for Online Principal Component Estimation and Global Convergence
TLDR
The diffusion approximation tools are adopted to study the dynamics of Oja's iteration which is an online stochastic gradient method for the principal component analysis and it is shown that the Ojas iteration for the top eigenvector generates a continuous-state discrete-time Markov chain over the unit sphere.
Stochastic DCA for the Large-sum of Non-convex Functions Problem and its Application to Group Variable Selection in Classification
TLDR
A stochastic version of DCA (Difference of Convex functions Algorithm) is presented to solve a class of optimization problems whose objective function is a large sum of nonconveX functions and a regularization term.
...
1
2
3
4
5
...