What are the optimal oracle bounds for rounding convex bodies to isotropic position?

Isotropic rounding means centering and flattening variance

Isotropic rounding is the problem of finding an affine change of coordinates that centers a distribution and makes its variance comparable in every direction. For a full-dimensional probability measure with mean m and covariance A, the measure is isotropic when m = 0 and A equals the identity matrix I_n. The paper asks for an approximate version with fixed constants: a pair (T, z), with T invertible, rounds a measure if Y = T(X − z) satisfies (1/2)I_n ⪯ Cov(Y) ⪯ 2I_n and ‖E Y‖₂ ≤ 1/2.

For a convex body K, the target is its uniform probability measure. The algorithm is given only a membership oracle, which answers whether a queried point lies in K, plus radii 0 < r ≤ R with rB₂ⁿ ⊆ K ⊆ RB₂ⁿ. This normalization matters because the query bounds depend on the geometry of the input, and the paper measures its guarantees in terms of the dimension n and the aspect ratio R/r.

A randomized algorithm rounds any promised body in Õ(n³) queries

Theorem 1.2 states that there is a randomized membership-oracle algorithm that, given rB₂ⁿ ⊆ K ⊆ RB₂ⁿ, returns a pair (T, z) satisfying the near-isotropic conditions with probability at least 5/6 using Õ(n³) queries. Here Õ(f) denotes O(f log^C(2 + nR/r)) for some absolute constant C, so the bound may carry a fixed power of a logarithm in the dimension and aspect ratio.

The same Õ(n³) upper bound extends to logconcave distributions given by a convex-potential evaluation oracle, a known ball in a level set, and a second-moment bound. The paper states the precise assumptions for that extension in Theorem 4.1.

The sampler improves the covariance dependence to a square root

The key lemma behind the upper bound concerns sampling rather than rounding directly. If B(0,1) ⊆ K and Cov(Unif(K)) ⪯ κI_n with κ ≥ 1, then approximate uniform sampling from an initial density bounded by a constant times the uniform density uses Õ(n²√κ) expected membership queries. The guarantee is any prescribed total variation error ε ∈ (0, 1/2), as stated in Theorem 4.2.

This is the paper's main sampling improvement. For a body containing the unit ball, the standard ball walk uses Õ(n²κ) queries per sample from a constant-warm start when A ⪯ κI, so the new bound replaces the linear dependence on κ with a square-root dependence. When the body is nearly isotropic, κ = O(1), and the sampler cost becomes quadratic in n.

Reflected kinetic dynamics supply the sampling guarantee

The paper proves the sampler bound by simulating reflected kinetic dynamics, using the analysis of Eberle and Lörler. The implementation uses membership queries, and the resulting sampler is then combined with the ideas of Jia, Laddha, Lee, and Vempala for rounding well-rounded bodies to obtain the Õ(n³)-query rounding algorithm.

The combination matters because the rounding algorithm does not assume that a target stays well-rounded after cooling. Instead, the Gaussian-cooling algorithm rounds each new target under a weaker moment condition. The paper also reports that the algorithm in Section 4 was obtained by prompting GPT 6 Astra to further improve the manuscript around mid September 2026, and that the authors are investigating how this happened because the route skips most steps in earlier versions and could not be duplicated shortly afterward by direct prompting or by running GPT work on the statement.

A matching lower bound holds even at polynomial aspect ratio

Theorem 1.3 states that for all sufficiently large n, there are known radii r_n and R_n with R_n/r_n = n^{O(1)} such that every randomized algorithm satisfying the near-isotropic conditions with probability at least 2/3 on every promised body requires Ω̃(n³) membership queries in worst-case expectation. Section 5 proves the explicit Ω(n³/log² n) bound and gives the radii in Theorem 5.1.

The lower-bound construction uses polars of Gaussian polytopes with a small common translation. The bodies have covariance nearly equal to a scalar matrix, but their centroid points in an unknown direction. A successful rounding map must locate this centroid, which reveals the hidden direction to constant accuracy. Recovering that direction requires Ω(n) mutual information, while the answers to q adaptive queries contain at most O(q log² n / n²) mutual information, giving q = Ω(n³/log² n). Because the upper-bound algorithm succeeds with probability 5/6 > 2/3, the lower bound applies to its success guarantee as well.

Doubling an ellipsoid can break the covariance guarantee

The paper disproves an ellipsoid-growth conjecture from previous work on this topic. Jia, Laddha, Lee, and Vempala conjectured that doubling an ellipsoid whose intersection with the body is nearly isotropic would leave the intersection well-rounded. The paper gives a symmetric counterexample in Section B.

The construction shows that even for a symmetric body, the intersection can start as isotropic, with covariance trace equal to n, and after the enlargement the trace of the covariance is Ω(n²). For sufficiently large n, the enlarged intersection has covariance not ⪯ 2I_n, so it no longer satisfies the near-isotropic definition in those coordinates. The counterexample also disproves the weaker trace bound conjectured in that earlier work, which asked whether tr A_{K∩2E} = Õ(n) when A_{K∩E} = I_n. The paper notes that this rules out the proposed way of taking larger ellipsoid-growth steps.

Volume estimation inherits the cubic query bound

The rounding result feeds directly into volume estimation. Cousins and Vempala give a membership-oracle algorithm that, for a body containing the unit ball with second moment O(n), estimates its volume to relative error ε ∈ (0, 1) using Õ(n³ε^{−2}) queries, with any fixed constant success probability. Corollary 4.7 checks these geometric assumptions after applying the rounding map.

Combining the two gives the same Õ(n³ε^{−2}) query bound for every convex body satisfying rB₂ⁿ ⊆ K ⊆ RB₂ⁿ. The paper contrasts this with the Ω̃(n²) membership-query lower bound of Rademacher and Vempala for randomized volume approximation with sufficiently small constant relative error, even on parallelotopes with polynomial aspect ratio. The gap between the cubic upper bound and the quadratic lower bound is therefore not closed by this work.

The cubic bound is the paper's central comparison

The most decision-relevant result is the pair of matching bounds: Õ(n³) queries suffice for rounding any convex body with known radii, and Ω̃(n³) queries are necessary even when R/r = n^{O(1)}. Previous rounding algorithms distinguished general inputs from well-rounded bodies. Jia, Laddha, Lee, and Vempala gave a nearly cubic algorithm for rounding a well-rounded convex body and a general rounding bound of Õ(n^{7/2}) using polylogarithmic bounds for the Kannan–Lovász–Simonovits constant. Kook and Vempala obtained Õ(n^{7/2}) rounding for general logconcave distributions.

The improvement from n^{7/2} to n³ comes from the sampler's square-root dependence on the covariance bound, which is what allows the Gaussian-cooling algorithm to round each new target under a weaker moment condition rather than assuming it remains well-rounded after cooling. The counterexample to ellipsoid growth is what rules out the alternative route of taking larger ellipsoid-growth steps. Readers comparing rounding algorithms should therefore track two separate quantities: the query bound for the rounding step itself, and the covariance dependence of the sampler that the rounding step relies on. The paper's contribution is to improve the second and to show that the first cannot be improved beyond polylogarithmic factors.

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