Temporal alignment and latent Gaussian process factor inference in population spike trains
We introduce a novel scalable approach to identifying common latent structure in neural population spike-trains, which allows for variability both in the trajectory and in the rate of progression of the underlying computation. Our approach is based on shared latent Gaussian processes (GPs) which are combined linearly, as in the Gaussian Process Factor Analysis (GPFA) algorithm. We extend GPFA to handle unbinned spike-train data by incorporating a continuous time point-process likelihood model, achieving scalability with a sparse variational approximation. Shared variability is separated into terms that express condition dependence, as well as trial-to-trial variation in trajectories. Finally, we introduce a nested GP formulation to capture variability in the rate of evolution along the trajectory. We show that the new method learns to recover latent trajectories in synthetic data, and can accurately identify the trial-to-trial timing of movement-related parameters from motor cortical data without any supervision.
-- Paper Summary   This paper tackles the problem of identifying patterns in spiking population activity of neurons by considering latent structure and trajectories. This is a particularly challenging problem due to potential time misalignment between multiple trials. Additionally, the various time series are generally assumed to be independent while there are also issues with tractability due to linear algebraic operations involving a TxT kernel matrix. This work tackles the latter issue by introducing the sparse variational approximation within the GPFA model, and addresses shared latent structure across trials (or even partitions of trials) by exploiting a nested GP architecture.   -- Originality + Significance  The variational sparse approximation to Gaussian process inference has become the go-to approach for solving tractability issues with GPs without compromising on performance. Although the approximation in itself has been used and applied extensively in other papers, to the best of my knowledge, this paper is the first in applying it to the GPFA model in particular.  In combination with the point process likelihood being considered, the resulting model is sufficiently different from existing approximate models based on sparse VI while also incorporating additional elements (explored in Section 4 of the paper).  A recent paper discussing simultaneous modelling of data and temporal alignment, ‘Gaussian Process Latent Variable Alignment Learning (Kazlauskaite et al)’, seems to be related in terms of scope. In fact, the generalisation to GPLVMs is listed in the concluding remarks of the paper under review. Although there is a clear distinction between the two papers, particularly with regards to the likelihood considered, any similarity to this work should be duly discussed.  It might be interesting to draw a connection to papers such as ‘Collaborative Multi-output Gaussian processes (Nguyen & Bonilla)’, where inducing points are shared across multiple latent functions. It is not clear whether the inducing points considered in this paper are unique to each latent function or shared between them, although I believe it’s the former. If not, this could possibly contribute towards further improving scalability.   -- Quality  The model is clearly detailed in Sections 3 and 4. As highlighted earlier, the sparse variational approximation is fairly standard; however, the authors do not limit themselves to the base approximation, but also consider several extensions which make the paper more complete and well-rounded.   Once all aspects of the model are described, I encourage the authors to summarise the overall complexity of the model with regards to how many trials, conditions, time points etc are considered in a given experiment.   -- Writing and Clarity  The paper is very well-written and a pleasure to read - I did not spot any typos while reading it. The figures are well-captioned and easy to interpret. Corresponding discussion of the figures in the text is also appropriate.  Some minor comments: - While the ‘macaque’ species might be well-known in the biology domain, I would suggest adding a footnote or small parenthesis explaining the term first. It is currently introduced in L57 without much context. - Similarly, it would be good to explain what is ‘innovations noise’ (L198)  - ‘Where’ in L121 could be changed to lower case  The content provided in the supplementary material complements the main text with more detailed proofs and figures. I found the balance between the two to be just right.   -- Overall Recommendation  This is a well-written paper with clearly established goals that are both individually and jointly investigated in this paper.  The contributions themselves are fairly modest and application-specific; nonetheless, state-of-the-art results are obtained in the problem setting being investigated, while the majority of the modelling choices are sufficiently general to be of interest to the broader NIPS community.  ------------  -- Post-review  Thank you for replying to my queries, in particular for expanding on the relation of this work to the indicated papers and detailing the overall complexity of the method. I believe that incorporating this discussion into the main paper (along with other minor fixes) will only elevate what is already a very good piece of work!