  Background:
  Functional magnetic resonance imaging (fMRI) provides non-invasive measures
of neuronal activity using an endogenous Blood Oxygenation-Level Dependent
(BOLD) contrast. This article introduces a nonlinear dimensionality reduction
(Locally Linear Embedding) to extract informative measures of the underlying
neuronal activity from BOLD time-series. The method is validated using the
Leave-One-Out-Cross-Validation (LOOCV) accuracy of classifying psychiatric
diagnoses using resting-state and task-related fMRI.
  Methods:
  Locally Linear Embedding of BOLD time-series (into each voxel's respective
tensor) was used to optimise feature selection. This uses Gau\ss' Principle of
Least Constraint to conserve quantities over both space and time. This
conservation was assessed using LOOCV to greedily select time points in an
incremental fashion on training data that was categorised in terms of
psychiatric diagnoses.
  Findings:
  The embedded fMRI gave highly diagnostic performances (> 80%) on eleven
publicly-available datasets containing healthy controls and patients with
either Schizophrenia, Attention-Deficit Hyperactivity Disorder (ADHD), or
Autism Spectrum Disorder (ASD). Furthermore, unlike the original fMRI data
before or after using Principal Component Analysis (PCA) for artefact
reduction, the embedded fMRI furnished significantly better than chance
classification (defined as the majority class proportion) on ten of eleven
datasets
  Interpretation:
  Locally Linear Embedding appears to be a useful feature extraction procedure
that retains important information about patterns of brain activity
distinguishing among psychiatric cohorts.
