On Geometric Asymmetry and Information in Sequential Dimension Reduction

Referee: [Abstract] the central claim that the initial two-step upper bound on conditional mutual information characterizes retention across the entire sequence requires a proof that the Haar-measure parameterization remains valid and sequence-independent once each projection alters the observed body's geometry and symmetry group; the provided text supplies no sketch, error analysis, or verification that the bound is non-vacuous.

Authors: We agree that an explicit proof sketch is necessary to rigorously establish that the two-step bound extends to the full sequence while remaining sequence-independent. In the revised manuscript, we will include a dedicated subsection deriving the extension using the Markov property, showing that the conditional mutual information I(X_{k+1}; X_k | X_0) is bounded by the Haar measure on the equivalence class defined by the current observed body X_k. This parameterization updates at each step but preserves the form of the bound without introducing new terms. We will also add a brief error analysis and numerical examples on polytopes to verify that the bound is non-vacuous. revision: yes

Referee: [Derivation of the bound] the upper bound is defined in terms of the Haar measure over projection spaces that produce identical observed bodies; if this measure is itself fitted or chosen to match observed data rather than derived from first principles, the information-retention claim becomes partly circular by construction.

Authors: The Haar measure is not fitted to data but is the unique (up to scaling) invariant probability measure on the orthogonal group O(n) induced by the left-invariant Haar measure, restricted to the stabilizer of the observed body under the group action. This construction follows directly from the definition of the projection map and the geometry of the convex body, independent of any empirical observations. We will revise the derivation section to explicitly state this first-principles derivation and contrast it with data-driven approaches to eliminate any perception of circularity. revision: partial

Referee: [Extension to n steps] the argument analyzes projection spaces under the symmetry group of the initial body only; it is unclear whether the two-step bound automatically carries over without additional terms that depend on the reduced symmetry of intermediate bodies, undermining the claim that the initial bound suffices for the whole chain.

Authors: The symmetry group of intermediate bodies is indeed a subgroup of the original symmetry group. However, because the process is Markovian, the bound at each step is conditioned on the current body, allowing the Haar measure to be taken with respect to the current symmetry group without affecting the overall characterization. The initial bound provides the general form that applies recursively. To address this concern, we will expand the extension section with a recursive argument that explicitly incorporates the evolving symmetry groups, demonstrating that the bound remains valid and sufficient for the entire chain. revision: yes