Mixed matrix factorization: a novel algorithm for the extraction of kinematic-muscular synergies.

Synergistic models have been employed to investigate motor coordination separately in the muscular and kinematic domains. However, the relationship between muscle synergies, constrained to be non-negative, and kinematic synergies, whose elements can be positive and negative, has received limited attention. Existing algorithms for extracting synergies from combined kinematic and muscular data either do not enforce non-negativity constraints or separate non-negative variables into positive and negative components. We propose a mixed matrix factorization (MMF) algorithm based on a gradient descent update rule that overcomes these limitations. It allows to directly assess the relationship between kinematic and muscle activity variables, by enforcing the non-negativity constrain on a subset of variables. We validated the algorithm on simulated kinematic-muscular data generated from known spatial synergies and temporal coefficients, by evaluating the similarity between extracted and ground truth synergies and temporal coefficients when the data are corrupted by different noise levels. We also compared the performance of MMF to that of non-negative matrix factorization applied to separate positive and negative components (NMFpn). Finally, we factorized kinematic and electromyographic data collected during upper-limb movements to demonstrate the potential of the algorithm. MMF achieved almost perfect reconstruction on noiseless simulated data. It performed better than NMFpn in recovering the correct spatial synergies and temporal coefficients with noisy simulated data. It also allowed to correctly select the original number of ground truth synergies. We showed meaningful applicability to real data; MMF can also be applied to any multivariate data that contain both non-negative and unconstrained variables. NEW & NOTEWORTHY The mixed matrix factorization (MMF) is a novel method for extracting kinematic-muscular synergies. The previous state of the art algorithm (NMFpn) factorizes separately positive and rectified negative waveforms; the MMF instead employs a gradient descent method to factorize mixed kinematic unconstrained data and muscular non-negative data. MMF achieves perfect reconstruction on noiseless data, improving the NMFpn. MMF shows promising applicability on real data.