Sum of squared singular values

Context of VAMPnets:

Measures the correlation between a state at time $t$ and $t + τ$ (Lag time).

1. The Covariance Matrices

$C_{00}$ (Input Covariance): Measures how the features at time t correlate with themselves. It captures the variance and redundancy in the input batch.
$\overset{ˉ}{C}_{01}$ (Cross-Covariance): The most important term. It measures the correlation between the state at time t and the state at time t+τ. This captures the kinetics (how the system evolves over time). See Temporal Autocorrelation.
$C_{11}$ (Output Covariance): Measures how the features at time t+τ correlate with themselves.
$\overset{ˉ}{C}$ (Mean-free): The bar notation indicates that the mean has been subtracted from the data. This centers the data at zero, which is why the +1 term is added later (to account for the stationary constant).

$C_{00}^{- 1/2}$ & $C_{11}^{- 1/2}$ : The $C_{00}$ and $C_{11}$ terms are raised to $- 1/2$ to perform a geometric operation called whitening. This removes linear correlations within the input and output batches. Without this, the network might learn redundant features (e.g., two neurons learning the exact same slow process). Whitening forces every neuron to learn a unique kinetic process.

The Trivial Eigenvalue: The Koopman operator always has one trivial eigenfunction with an eigenvalue of λ=1. This corresponds to the constant function (the fact that probability mass is conserved and the system exists).
Accounting for it: Since the covariance matrices were calculated on “mean-free” (centered) data, this constant component was mathematically removed. The +1 adds it back to the score so that R^2 correctly estimates the sum of all squared singular values, including the trivial one.