**Research problem:**

Bayesian inference with automatic model order selection

**Brief Description:**

Model order selection is a fundamental problem
in signal processing and machine learning. Choosing too small the model size cannot
fit the data well, while choosing too large the model size leads to
overfitting. Traditionally, a regularization term is added to the loss function
to strike a balance between data fitting and the model complexity, with the
regularization parameter tuned to obtain the best performance. However, the
``best'' parameter varies widely across datasets and applications, and may not
be effective for unseen data.

Sparse Bayesian learning provides a tuning-free alternative. The idea is that if a suitable prior distribution is specified, the relative importance of different components in the signal can be learnt. Together with the learnt noise power from the data, one can easily tell what is the proper model order. While the idea is simple, the algorithm derivation is usually not straightforward. In particular, in Bayesian statistics, if we want to estimate a certain parameter, other parameters should be integrated out from the joint posterior distribution. Unfortunately, most of the time, we cannot perform the integration analytically due to the complicated nature of the posterior distribution. To handle this problem, previous Bayesian analysis is mainly based on Monte Carlo statistical methods, such as Markov chain Monte Carlo (MCMC) and Gibbs sampling, where a large number of random samples are generated from the joint distributions and marginalization is approximated by operations on samples. Although sampling methods can approach the true posteriors when the number of samples approaches infinity, it is computationally demanding.

Variational inference is another way to approximate the parameter inference. It seeks a variational distribution that closely approximates the true posterior distribution. Under the commonly used mean-field constraint, the variational distribution will be in a form where marginalization can be easily carried out. This saves the computational complexity significantly, as each update is in closed-form.

We have applied the
variational inference with automatic model order selection to various signal
processing problems. The first one is the iterative joint doubly-selective
channel estimation and data detection in wireless communication systems, where
the model order corresponds to the channel length and Doppler shift. The second
one is the distributed estimation of system state in power grid. The third one
is the subspace identification for channel estimation in massive MIMO systems,
where the model order is the number of paths the signal takes. Recently,
we further extended the Bayesian inference to tensor decompositions, where the
model order is the unknown tensor rank. This latest framework finds
applications in blind CDMA receiver design, face image classification, object
tracking in surveillance video, fluorescence data analysis, and email data
mining.

**Related Publications:**

Data detection in doubly-selective channels:

1. Jingrong Zhou, Jiayin Qin and **Yik****-Chung
Wu**, ``Variational Inference-based Joint Interference Mitigation and OFDM
Equalization under High Mobility,'' * IEEE Signal Processing
Letters*, Vol. 22, no. 11, pp. 1970 - 1974, Nov 2015.

2. Ke Zhong, **Yik****-Chung Wu**, and Shaoqian Li, ``Signal
Detection for OFDM-Based Virtual MIMO Systems under Unknown Doubly Selective
Channels, Multiple Interferences and Phase Noises,"__ __* IEEE Trans. on Wireless Communications*, Vol. 12, no. 10,
pp.5309-5321, Oct 2013.

3. Lanlan He, **Yik****-Chung Wu**, Shaodan Ma, Tung-Sang Ng and
H. Vincent Poor, ``Superimposed
Training Based Channel Estimation and Data Detection for OFDM
Amplify-and-Forward Cooperative Systems under High Mobility," * IEEE Trans. on Signal Processing*, Vol. 60, no. 1, pp. 274-284,
Jan 2012.

4. Lanlan He, Shaodan Ma, **Yik****-Chung Wu**, Yiqing Zhou, Tung-Sang Ng, and H. Vincent Poor, ``Pilot-Aided IQ Imbalance
Compensation for OFDM Systems Operating over Doubly Selective Channels," * IEEE Trans. on Signal Processing*, Vol. 59, no. 5, pp. 2223-2233, May 2011.

Power system state estimation:

- Jian
Du, Shaodan Ma,
**Yik****-Chung Wu**, and H. Vincent Poor, ``Distributed Hybrid Power State Estimation under PMU Sampling Phase Errors,"Vol. 62, no. 16, pp.4052-4063, Aug 2014__IEEE Trans. on Signal Processing,__

Channel estimation in massive MIMO system:

6. Lei
Cheng, Chengwen Xing, and **Yik****-Chung
Wu**, ``Irregular Array Manifold Aided Channel Estimation in Massive
MIMO Communications," * IEEE Journal of Selected Topics in Signal
Processing*, Vol. 13, no. 5, pp. 974-988, Sep 2019.

7.
Lei Cheng, **Yik****-Chung Wu**, Jianzhong
(Charlie) Zhang, and Lingjia Liu, ``Subspace Identification for DOA Estimation
in Massive / Full-dimension MIMO System: Bad Data Mitigation and
Automatic Source Enumeration,'' *IEEE Trans.
on Signal Processing**,* Vol. 63, no. 22, pp. 5897-5909, Nov 2015.

General tensor decompositions:

8. Lei Cheng,
Xueke Tong, Shuai Wang, **Yik****-Chung
Wu**, and H. Vincent Poor, ``Learning Nonnegative
Factors from Tensor Data: Probabilistic Modeling and Inference Algorithm,"
in * IEEE Trans. on Signal Processing*, vol. 68, pp. 1792-1806,
2020, doi: 10.1109/TSP.2020.2975353.

9. Lei Cheng,
**Yik****-Chung Wu**, and H. Vincent Poor, ``Scaling Probabilistic Tensor Canonical
Polyadic Decomposition to Massive Data," * IEEE Trans. on Signal Processing*, Vol. 66, no. 21, pp.
5534-5548, Nov 2018.

10.Lei
Cheng, **Yik****-Chung Wu**, and H. Vincent Poor, ``Probabilistic Tensor Canonical
Polyadic Decomposition with Orthogonal Factors," * IEEE Trans. on Signal Processing*, Vol. 65, no. 3, pp. 663-676, Feb
2017.