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Tools for Tensor Decomposition.

## Tensortools

TensorTools is a bare bones Python package for fitting and visualizing canonical polyadic (CP) tensor decompositions of higher-order data arrays. I originally developed this library for applications in neuroscience (Williams et al., 2018), but the code could be helpful in other domains.

## Installation

From the command line run:

pip install git+https://github.com/ahwillia/tensortools


(You will need to have git installed for this command to work.)

Alternatively you can download the source code and install locally by running:

git clone https://github.com/ahwillia/tensortools
cd tensortools
pip3 install -e .


## Quick Start

Here's how to perform a parameter sweep over 1 - 9 components, and plot the reconstruction error and similarity diagnostics as a function of the model rank (these diagnostics are described in Williams et al., 2018). The snippet also uses plot_factors(...) to plot the factors extracted by one of the models in the ensemble.

The method "ncp_hals" fits a nonnegative tensor decomposition, other methods are "ncp_bcd" (also nonnegative) and "cp_als" (unconstrained decomposition). See the tensortools/optimize/ folder for the implementation of these algorithms.

import tensortools as tt

data = # ... specify a numpy array holding the tensor you wish to fit

# Fit an ensemble of models, 4 random replicates / optimization runs per model rank
ensemble = tt.Ensemble(fit_method="ncp_hals")
ensemble.fit(data, ranks=range(1, 9), replicates=4)

fig, axes = plt.subplots(1, 2)
tt.plot_objective(ensemble, ax=axes[0])   # plot reconstruction error as a function of num components.
tt.plot_similarity(ensemble, ax=axes[1])  # plot model similarity as a function of num components.
fig.tight_layout()

# Plot the low-d factors for an example model, e.g. rank-2, first optimization run / replicate.
num_components = 2
replicate = 0
tt.plot_factors(ensemble.factors(num_components)[replicate])  # plot the low-d factors

plt.show()


Check out the scripts in the examples/ folder for other short demos.

## Time-Shifted Tensor Decompositions

This repo contains a module tensortools.cpwarp which allows for time-shifted tensor decompositions of 3d-arrays. The motivation behind this model and some of its implementaional details are laid out in the following set of notes.

Alex H. Williams (2020). Combining tensor decomposition and time warping models for multi-neuronal spike train analysis. bioRxiv. 2020.03.02.974014

A very similar model was previously proposed by Mørup et al. (2008). Also see Sorokin et al. (2020) for an application of this model to neural data. To fit this model, check out the script in examples/shift_cpd.py, which should reproduce Figure 4 from the Williams (2020) paper.

The important function to call is fit_shifted_cp(), like below:

fit_shifted_cp(
data,
rank,
boundary="wrap",
n_restarts=5,
max_shift_axis0=0.1,
max_shift_axis1=None,
min_iter=10,
max_iter=100,
u_nonneg=True,
v_nonneg=True,
tol=1e-4,
patience=5,
mask=None,
)

"""
Fits a time-shifted CP decomposition to 3d-array data. The model parameters
are three factor matrices {u, v, w} and two sets of shift parameters {u_s, v_s}.

u.shape == (rank, data.shape[0])
v.shape == (rank, data.shape[1])
w.shape == (rank, data.shape[2])
u_s.shape == (rank, data.shape[0])
v_s.shape == (rank, data.shape[1])

The element data[i, j, k] is approximated by:

sum_r ( u[r, i] * v[r, j] * w[r, t + u_s[r, i] + v_s[r, i]] )

Note that if the shift parameters are zero (u_s == v_s == 0), this is the typical
CP tensor decomposition.

Parameters
----------
data : ndarray
Data tensor.
rank : int
Number of components.
init_u : ndarray or None
Initial guess for factor matrix u.
init_v : ndarray or None
Initial guess for factor matrix v.
init_w : ndarray or None
Initial guess for factor matrix w.
max_shift_axis0 : float or None
Maximum absolute value for u_s, expressed
as a fraction on the interval (0, 0.5].
If None, then all u_s shifts are set to zero.
max_shift_axis1 : float
Maximum absolute value for v_s, expressed
as a fraction on the interval (0, 0.5].
If None, then all v_s shifts are set to zero.
u_nonneg : bool
If True, the factor matrix u is constrained
to be nonnegative.
v_nonneg : bool
If True, the factor matrix v is constrained
to be nonnegative.
boundary : str
If equal to "wrap" the shifting along axis=2
has a periodic boundary condition. Otherwise
the behavior is similar to "edge" mode in the
numpy.pad() function.
min_iter : int
Minimum number of iterations before stopping.
max_iter : int
Maximum number of iterations before giving up.
tol : float
Convergence tolerance
patience : int
Number of iterations to wait between convergence
checks.
mask : ndarray of booleans or None
Specifies missing data, and can be used for
cross-validation.
"""


I hope to upload a more detailed tutorial soon; until then, please refer to the papers cited above and reach out to me by email if you are interested in further details.

## Citation

If you found this resource useful, please consider citing this paper.

@ARTICLE{Williams2018,
title    = "Unsupervised Discovery of Demixed, {Low-Dimensional} Neural
Dynamics across Multiple Timescales through Tensor Component
Analysis",
author   = "Williams, Alex H and Kim, Tony Hyun and Wang, Forea and Vyas,
Saurabh and Ryu, Stephen I and Shenoy, Krishna V and Schnitzer,
Mark and Kolda, Tamara G and Ganguli, Surya",
journal  = "Neuron",
volume   =  98,
number   =  6,
pages    = "1099--1115.e8",
month    =  jun,
year     =  2018,
}


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