This is a short and self-contained example that illustrates how to fit canonical polyadic (CP) tensor decompositions to multi-trial neural data.
A very common and generic experimental design in neuroscience is to record the activity of many neurons across repeated behavioral trials. Let's say we record the activity of N neurons at T time points in each trial, and that there are K total trials. A natural way to represent this data is a three-dimensional data array with dimensions N x T x K. Higher-order arrays like this are called tensors.
We would like to find a compact and interpretable description of this multi-trial dataset. This goal is often called dimensionality reduction, and involves reducing the measured dimensionality of the data (which can easily involve hundreds of neurons, and hundreds of trials given current experimental technologies) to a handful of latent factors. Principal Components Analysis (PCA) is a classic technique for dimensionality reduction (click here for a shameless plug).
CP decomposition extends PCA to higher-order tensors. In fact, PCA is CP decomposition on a matrix (i.e. a second-order tensor). As described above, multi-trial data is naturally represented as a third-order tensor. Applying CP decomposition to this tensor produces low-dimensional factors for within-trial as well as across-trial changes in neural activity.
CP decomposition is an attractive technique both because it is conceptually simple (each trial is modeled as a linear combination of latent factors) and because it has some subtle advantages (the optimal model is unique, whereas the factors identified by PCA can be rotated arbitrarily without affecting reconstruction error).
- Some notes on PCA, CP decomposition, and Demixed PCA
- Bader & Kolda (2009). Tensor Decompositions and Applications. SIAM Review.
- See
/matlabfor a tutorial on fitting tensor decompositions with Sandia's TensorToolbox - See
/pythonfor an overview of the tensorly Python package - Julia toolboxes are coming soon
Other toolboxes include TensorLab (MATLAB) and scikit-tensor (Python).
Please get in touch if you have any questions/comments.