(a) Linear discriminant analysis maps the firing rates of individual neurons onto a latent component that allows us to decode a task parameter of interest. Shades of grey inside each neuron show the proportion of variance due to the various task parameters (e.g. stimulus, decision, and time), illustrating mixed selectivity. In contrast, the LDA component is maximally demixed. (b) At any moment in time, the population firing rate of neurons is represented by a point in the -dimensional space; here . Each trial is represented by a trajectory in this space. Colors indicate different stimuli and dot sizes represent time. The LDA component for stimulus is given by the projection onto the LDA axis (black line); projections of all points are shown along this line. All three stimuli are clearly separated, but their geometrical relation to each other is lost. (c) Principal component analysis linearly maps the firing rates into a few principal components such that a second linear transformation can reconstruct the original firing rates. (d) The same set of points as in (b) is projected onto the first PCA axis. However, the stimuli are no longer separated. Rather, the points along the PCA axis have complex dependencies on stimulus and time (mixed selectivity). The PCA axis minimizes the distances between the original points and their projections. (e) Demixed principal component analysis also compresses and decompresses the firing rates through two linear transformations. However, here the transformations are found by both minimizing the reconstruction error and enforcing a demixing constraint on the latent variables. (f) The same set of points as in (b) projected onto the first dPCA decoder axis. The three stimuli are clearly separated (as in LDA), but some information about the relative distances between classes is preserved as well (as in PCA). (g) The same data as in (b) linearly decomposed into the time effect, the stimulus effect, and the noise. (h) The dPCA projection from (f) has to be mapped onto a different axis, given by the dPCA encoder, in order to reconstruct the stimulus class means (large colored circles). The decoder and encoder axes together minimize the reconstruction error between the original data and the stimulus class means.