[Paper Review] Riemannian Geometry of Optimal Rebalancing in Dynamic Weight Automated Market Makers

Matthew Willetts|arXiv (Cornell University)|Mar 5, 2026

Auction Theory and Applications0 citations

TL;DR

The per-step arbitrage loss in dynamic-weight AMMs is the KL divergence between successive weight vectors, making the Fisher–Rao metric on the weight simplex natural; SLERP in Hellinger coordinates provides the leading-order optimal rebalancing path with a midpoint that matches the AM+GM/normalise heuristic.

ABSTRACT

In Temporal Function Market Making (TFMM), a dynamic weight AMM pool rebalances from initial to final holdings by creating a series of arbitrage opportunities whose total cost depends on the weight trajectory taken. We show that the per-step arbitrage loss is the KL divergence between new and old weight vectors, meaning the Fisher--Rao metric is the natural Riemannian metric on the weight simplex. The loss-minimising interpolation under the leading-order expansion of this KL cost is SLERP (Spherical Linear Interpolation) in the Hellinger coordinates $η_i = \sqrt{w_i}$, i.e. a geodesic on the positive orthant of the unit sphere traversed at constant speed. The SLERP midpoint equals the (AM+GM)/normalise heuristic of prior work (Willetts & Harrington, 2024), so the heuristic lies on the geodesic. This identity holds for any number of tokens and any magnitude of weight change; using this link, all dyadic points on the geodesic can be reached by recursive AM-GM bisection without trigonometric functions. SLERP's relative sub-optimality on the full KL cost is proportional to the squared magnitude of the overall weight change and to $1/f^2$, where $f$ is the number of interpolation steps.

Motivation & Objective

• Motivate and formalize the arbitrage cost of weight rebalancing in dynamic-weight AMMs under constant and stochastic prices.
• Characterize the natural Riemannian structure on the weight simplex via the Fisher–Rao metric and KL divergence.
• Derive SLERP as the leading-order optimal interpolation on the weight simplex and connect it to existing AM+GM/normalise heuristics.
• Show exact SLERP trajectories can be computed without trigonometry via recursive bisection for power-of-two step counts.
• Extend the framework to driftless GBM prices and analyze how LVR exposure modifies the optimal step count.

Proposed method

• Model the per-step arbitrage loss as the KL divergence between successive weight vectors (Theorem 2).
• Use the Fisher–Rao metric as the Riemannian structure on the weight simplex and apply the Hellinger embedding to map to a unit sphere (η = sqrt(w)).
• Prove that the leading-order loss-minimising interpolation is SLERP in Hellinger coordinates (Corollary 5).
• Show the SLERP midpoint equals the (AM+GM)/normalise midpoint exactly (Theorem 6).
• Derive trig-free recursive bisection to compute SLERP trajectories at power-of-two step counts (Corollary 7).
• Extend to stochastic prices under driftless GBM and demonstrate price independence of the retention ratio and path-optimization results (Propositions 1–2; Corollary 0).
• Formulate a modified geodesic problem incorporating LVR as a potential and derive the optimal step count f* by balancing rebalancing cost and LVR exposure (Proposition 0 in §5).

Experimental results

Research questions

• RQ1 What geometric structure underlies the arbitrage cost when rebalancing dynamic-weight AMMs?
• RQ2 Can SLERP in the Hellinger embedding provide the leading-order optimal interpolation for weight changes, and how does it relate to the (AM+GM)/normalise heuristic?
• RQ3 How does price dynamics (driftless GBM) affect the optimal rebalancing path and step count?
• RQ4 What is the impact of LVR exposure on the optimal number of interpolation steps in rebalancing?
• RQ5 Can SLERP trajectories be computed efficiently on-chain without trigonometric functions?

Key findings

• The per-step arbitrage loss is the KL divergence between successive weight vectors, making the Fisher–Rao metric the natural geometry on the weight simplex.
• The leading-order loss-minimising interpolation is SLERP in the Hellinger coordinates, corresponding to a constant-speed geodesic on the positive orthant of the unit sphere.
• The SLERP midpoint exactly matches the (AM+GM)/normalise midpoint for any number of tokens and weight-change magnitude.
• SLERP trajectories at power-of-two step counts can be computed trig-free via recursive bisection.
• SLERP’s sub-optimality on the exact KL cost is bounded by O(Omega^2/f^2) and vanishes as f grows, with practical small gaps observed numerically.
• Under driftless GBM prices, the retention ratio is price-independent, and cross-terms telescope, preserving the core results for the rebalancing cost.
• LVR exposure introduces a finite optimal step count f*, balancing rebalancing cost and volatility exposure; the total cost at optimal f* scales with Omega and bar{ell} (LVR rate).