Multilinear Dynamical Systems for Tensor Time Series
Data in the sciences frequently occur as sequences of multidimensional arrays called tensors. How can hidden, evolving trends in such data be extracted while preserving the tensor structure? The model that is traditionally used is the linear dynamical system (LDS) with Gaussian noise, which treats the latent state and observation at each time slice as a vector. We present the multilinear dynamical system (MLDS) for modeling tensor time series and an expectation-maximization (EM) algorithm to estimate the parameters. The MLDS models each tensor observation in the time series as the multilinear projection of the corresponding member of a sequence of latent tensors. The latent tensors are again evolving with respect to a multilinear projection. Compared to the LDS with an equal number of parameters, the MLDS achieves higher prediction accuracy and marginal likelihood for both artificial and real datasets.
This paper proposes a tensorial LDS model called MLDS. 
      As such, the model is a restriction of the general Kalman filter model 
      with state transition and observation operators having a certain Kronecker 
      product factorization. The model can be understood intuitively as a 
      sequence of core tensors that form a Markov Chain, and the observed tensor 
      at each time slice as well as the core tensor at the next time slice are 
      obtained by Tucker models with tensor factors C and A respectively. The 
      term multilinear in the title is misleading in this respect as the model 
      is linear. After setting up the notation and background, authors 
      derive the posterior distribution of hidden tensors given the observed 
      tensor in Sec. 4, Lemma 2. This is direct generalization of standard 
      marginalization and conditioning results of multivariate Gaussians. What 
      is not discussed is the form of the state covariance and whether it admits 
      a useful factorization (compare with matrix Gaussians where the covariance 
      is given as a Kronecker Product). I suspect that this model would not 
      admit such a structured covariance but I would have welcomed at least a 
      discussion of this issue because calculating the covariance would be the 
      computational bottleneck with large Z, not much mentioned here and 
      somewhat swept under the carpet. The ideas are quite analogous to 
      theory of linear dynamical systems (LDS) but the tensor notation is 
      unevitably quite heavy, yet the authors made a pretty good job in keeping 
      the notation intuitive by drawing analogies with the vector case. Perhaps 
      not surprisingly, the paper concludes that by preserving the linear 
      structure, MLDS is more effective than a plain LDS where the tensors are 
      simply vectorized. However, the paper does not much detail on inference 
      (filtering/smoothing recursions) and instead focuses on parameter 
      inference via EM. Another issue I would have seen discussed is 
      identifiability and the justification of the Tucker transition model. The 
      LDS states are identifiable upto similarity transform, what is the case 
      for the case discussed here? For the latter point, whilst the observation 
      model is clear, it would be good at least to justify the Tucker transition 
      model; does this have a physical meaning? Otherwise I would suspect taking 
      an unconstrained transition operator A or the other extreme of taking A as 
      the identity (e.g., a drifting core tensor) are equally viable. 
      Sec. 5.2 develops an EM algorithm to for parameter estimation. 
      This section is rather short and many details are omitted. The classical 
      EM algorithm for learning an LDS derives update rules for each parameter, 
      including noise covariances. In the paper, although authors state that, 
      ‘MLDS inherits the E-step of the LDS and the M-step of the LDS for all 
      parameters except A and C’, it is not clear what they mean and it is not 
      possible to see the update rules of noise covariances at one-step. 
      Furthermore, authors do not explicitly state the update rules for A and C 
      and merely derive the gradient. In the Eq. (11), authors use \phi and 
      \varphi which are defined with expectations. In the LDS, these 
      expectations require to find smoothed estimates (in addition to filtered 
      estimates). The paper nowhere mentions the smoothing recursions explicitly 
      and the reader has to simply redrive the results from scratch. From the 
      description in the paper it is not clear that how these expectations are 
      computed making the work hard to reproduce. Despite all the small 
      glitches, I find the model interesting and well executed enough deserving 
      publication. I am surprised that the literature does not contain 
      special cases of this model such as 'Matrix variate Kalman Filter'. 
      Short comments Abstract and introduction should be more 
      clear and state the precise results about inference and learning 
      algorithms that are obtained in this paper. Experimental section 
      is good. If authors think to resubmit this paper, they should focus on the 
      Section 4 and Section 5. The statement in the related work section 
      page 3, line 122 is not entirely correct as GCTF is not limited to CP but 
      can accomodate arbitrary factorization forms and fairly general noise 
      models. The statement in the conclusion "permitting us to learn 
      all but the transition and projection operators A and C of the MLDS 
      via the Kalman filter" is vague (and actually incorrect as KF provides 
      the filtered state estimates and covariances) The citation to [18] 
      looks off-topic and can either be omitted or must be elaborated more to 
      stress why the non existence may be an issue.