Remove the r<2 restriction in the operator-space LDP

Establish a large deviation principle for the covariance-process Markov chain (^2_{N_1}, …, ^{L+1}_{N_L}) on the space of non-negative, symmetric trace-class operators L_1^{+,s} under the activation growth assumption (H2) with r=2, thereby removing the current technical restriction r<2 and characterizing the corresponding good rate function.

Background

The main operator-space LDP (Theorem \ref{thm.main}) is proven under a polynomial growth assumption on the activation function with exponent r<2 in (H2). The authors expect the result to hold for r=2 (linear growth) as well but indicate that addressing the technical difficulties is postponed.

Finite-dimensional analogues with linear-growth activations have been analyzed in recent work, suggesting feasibility; however, carrying this out in the infinite-dimensional trace-class operator setting requires overcoming significant technical hurdles.

References

We expect the same result to hold for r=2 as well, but proving this would require to deal with significant technical details (see for the finite dimensional case) and we postpone it to future work.

— LDP for the covariance process in fully connected neural networks  (2505.08062 - Andreis et al., 12 May 2025) in Section 3.3 (Main.thm), following Theorem \ref{thm.main}