Learning Circular Hidden Quantum Markov Models: A Tensor Network Approach

In this paper, we propose circular Hidden Quantum Markov Models (c-HQMMs), which can be applied for modeling temporal data in quantum datasets (with classical datasets as a special case). We show that c-HQMMs are equivalent to a constrained tensor network (more precisely, circular Local Purified State with positive-semidefinite decomposition) model. This equivalence enables us to provide an efficient learning model for c-HQMMs. The proposed learning approach is evaluated on six real datasets and demonstrates the advantage of c-HQMMs on multiple datasets as compared to HQMMs, circular HMMs, and HMMs.

[1]  Byron Boots,et al.  Quantum Tensor Networks, Stochastic Processes, and Weighted Automata , 2020, AISTATS.

[2]  Byron Boots,et al.  Expressiveness and Learning of Hidden Quantum Markov Models , 2019, AISTATS.

[3]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[4]  Jens Eisert,et al.  Expressive power of tensor-network factorizations for probabilistic modeling, with applications from hidden Markov models to quantum machine learning , 2019, NeurIPS.

[5]  R. Sarpong,et al.  Bio-inspired synthesis of xishacorenes A, B, and C, and a new congener from fuscol† †Electronic supplementary information (ESI) available. See DOI: 10.1039/c9sc02572c , 2019, Chemical science.

[6]  Byron Boots,et al.  Learning Quantum Graphical Models using Constrained Gradient Descent on the Stiefel Manifold , 2019, ArXiv.

[7]  Román Orús,et al.  Tensor networks for complex quantum systems , 2018, Nature Reviews Physics.

[8]  Sertac Karaman,et al.  On algorithms for and computing with the tensor ring decomposition , 2018, Numer. Linear Algebra Appl..

[9]  Yifan Sun,et al.  Wide Compression: Tensor Ring Nets , 2018, 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition.

[10]  Byron Boots,et al.  Learning Hidden Quantum Markov Models , 2017, AISTATS.

[11]  V. Aggarwal,et al.  Efficient Low Rank Tensor Ring Completion , 2017, 2017 IEEE International Conference on Computer Vision (ICCV).

[12]  A. Cichocki,et al.  Learning Efficient Tensor Representations with Ring Structure Networks , 2017, ICLR.

[13]  Liqing Zhang,et al.  Tensor Ring Decomposition , 2016, ArXiv.

[14]  David J. Schwab,et al.  Supervised Learning with Quantum-Inspired Tensor Networks , 2016, ArXiv.

[15]  Almut Beige,et al.  Hidden Quantum Markov Models and Open Quantum Systems with Instantaneous Feedback , 2014, 1406.5847.

[16]  Lieven De Lathauwer,et al.  Canonical Polyadic Decomposition of Third-Order Tensors: Reduction to Generalized Eigenvalue Decomposition , 2013, SIAM J. Matrix Anal. Appl..

[17]  Roman Orus,et al.  A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States , 2013, 1306.2164.

[18]  J. Biamonte,et al.  Quantum Techniques in Stochastic Mechanics , 2012, 1209.3632.

[19]  Ivan Oseledets,et al.  Tensor-Train Decomposition , 2011, SIAM J. Sci. Comput..

[20]  K. Wiesner,et al.  Hidden Quantum Markov Models and non-adaptive read-out of many-body states , 2010, 1002.2337.

[21]  W. Zucchini,et al.  Hidden Markov Models for Time Series: An Introduction Using R , 2009 .

[22]  A. Munk,et al.  Hidden Markov models for circular and linear-circular time series , 2006, Environmental and Ecological Statistics.

[23]  Ismail Shahin,et al.  Enhancing speaker identification performance under the shouted talking condition using second-order circular hidden Markov models , 2006, Speech Commun..

[24]  David Haussler,et al.  Combining phylogenetic and hidden Markov models in biosequence analysis , 2003, RECOMB '03.

[25]  Kevin Barraclough,et al.  I and i , 2001, BMJ : British Medical Journal.

[26]  Fatos T. Yarman-Vural,et al.  A shape descriptor based on circular hidden Markov model , 2000, Proceedings 15th International Conference on Pattern Recognition. ICPR-2000.

[27]  Aaron F. Bobick,et al.  Parametric Hidden Markov Models for Gesture Recognition , 1999, IEEE Trans. Pattern Anal. Mach. Intell..

[28]  Monson H. Hayes,et al.  Hidden Markov models for face recognition , 1998, Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP '98 (Cat. No.98CH36181).

[29]  D. Haussler,et al.  Hidden Markov models in computational biology. Applications to protein modeling. , 1993, Journal of molecular biology.

[30]  Biing-Hwang Juang,et al.  Hidden Markov Models for Speech Recognition , 1991 .

[31]  G.G. Cano,et al.  An approach to cardiac arrhythmia analysis using hidden Markov models , 1990, IEEE Transactions on Biomedical Engineering.

[32]  Y.-C. Zheng,et al.  Text-dependent speaker identification using circular hidden Markov models , 1988, ICASSP-88., International Conference on Acoustics, Speech, and Signal Processing.

[33]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[34]  Quantum Tensor Networks , 2019 .

[35]  W. Hager,et al.  and s , 2019, Shallow Water Hydraulics.

[36]  Pieter Lagrou.  States , 2019, Europe’s Postwar Periods 1989, 1945, 1918.

[37]  Rogemar S. Mamon,et al.  Hidden Markov Models In Finance , 2007 .

[38]  Jinhai Cai,et al.  Image Retrieval Using Circular Hidden Markov Models with a Garbage State , 2007 .

[39]  Gilbert Ritschard,et al.  Classification de parcours de vie à l'aide de l'optimal matching , 2007 .

[40]  Valeria De Fonzo,et al.  Hidden Markov Models in Bioinformatics , 2007 .

[41]  Jon C. Dattorro,et al.  Convex Optimization & Euclidean Distance Geometry , 2004 .

[42]  L. Rabiner,et al.  An introduction to hidden Markov models , 1986, IEEE ASSP Magazine.

[43]  K. Kraus,et al.  States, effects, and operations : fundamental notions of quantum theory : lectures in mathematical physics at the University of Texas at Austin , 1983 .