The RBM mutual-information lower bound is fixed by the marginals, not by a learning principle

A recent review [1] (Sec. 3.2 and Fig. 4 there, after Ref. [34] of the review) characterizes a restricted Boltzmann machine (RBM) learner by two numbers for each visible–hidden pair. The first is the covariance , which is read from the imaginary part of an out-of-time-order correlator. The second is the mutual information . The pair is shown to lie between analytic bounds and . Trained RBMs sit on the lower bound for every system size and field ratio of transverse-field Ising drivers, and the review reads this as a learning principle: the network would use the least mutual information compatible with the covariance. We point out that a pair of units is fixed by three numbers, the two marginals , and . In those variables, is exactly the case , and it is the minimum of over the marginals at fixed . is the perfectly correlated pair with . The transverse-field Ising driver is -symmetric, so a symmetric RBM has zero marginals and sits on by construction. We test the consequence that could have failed. With exact enumeration ( visible spins, ), a longitudinal field of – moves the same learner off by up to bits, in step with the measured marginals. In the ordered phase, a symmetry-broken and a symmetric RBM of the same target sit off and on respectively. The OTOC reading of is unaffected; we suggest reporting the marginals alongside .