{"id":"99d507c5-7714-4086-b3fb-9404c6b70e18","arxiv_id":"2608.12828","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A finite-horizon feedback law built from projected one-dimensional optimal transport maps steers distributions, with Gaussian terminal convergence, randomized-to-average convergence, and exact finite-step realization for controllable linear systems.","lead":"The paper builds feedback controllers that steer a probability distribution to a target using only one-dimensional optimal transport along random directions, avoiding full-dimensional transport maps. It proves that the averaged controller decreases a sliced Wasserstein distance and that the randomized controller converges to this average as the sampling period vanishes, with extensions to linear systems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3's convergence rests on Assumption 2, a Lipschitz stability condition that fails for the atomic target laws used in the paper's own numerical experiments.","rationale":"The reader's CONDITIONAL verdict is appropriate, but I would elevate the unverified Assumption 2 over the covariance lower bound as the principal reason. The covariance lower bound in Theorem 1 is an explicit hypothesis with a clearly stated scope, and Corollary 1 proves such a bound for the λ_SW Gaussian flow; although the general case is not covered, the theorem is honest about its conditions. Assumption 2, by contrast, is the load-bearing regularity condition for the convergence of the randomized controller, which is the algorithm actually implemented. It is not merely unverified; it is incompatible with atomic target laws, exactly what the numerical experiments use. The theoretical framework of Section 4 assumes continuous, sufficiently regular targets, but Section 7 implements fixed particle targets, so the advertised connection between the discrete controller and the continuous averaged flow is not established in the demonstrated regime. The issue is fixable, for example by smoothing the empirical target or by proving a different convergence bound for atomic targets, so the paper is not unsalvageable. I therefore recommend keeping the CONDITIONAL verdict unchanged.","tokens_in":26260,"tokens_out":16375,"duration_ms":173853,"concrete_test":"For the fixed 10^4-particle target in Example 1, compute ḡ[ρ0](x) on a fine grid and evaluate the finite-difference ratio |ḡ(x)−ḡ(y)|/|x−y| at points x,y separated by the particle discretization. Since the empirical target is atomic, the projected one-dimensional OT maps are step functions, so the ratio diverges as |x−y|→0 on a set of positive measure; this directly falsifies Assumption 2 in the exact regime where the randomized controller is tested.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central bridge between the implementable randomized controller and the analyzed averaged flow is Theorem 3, whose proof requires Assumption 2: the averaged sliced field ḡ[ρ](x) must be Lipschitz in x and in the law along the generated random and averaged flows. This is not a mild regularity condition. The field is built from one-dimensional optimal transport maps into the target's projections, and those maps are discontinuous whenever the target law has atoms. The numerical experiments in Section 7 use exactly such atomic targets (fixed particle sets). For an atomic ρ1, each one-dimensional map T^θ_ρ is a step function, so gθ[ρ] and hence ḡ[ρ] have jump discontinuities on sets of positive ρ-mass; no finite L can satisfy the Lipschitz inequality in Assumption 2. Thus Theorem 3 does not apply to the implemented algorithm, and the paper neither states this restriction nor offers a convergence analysis for atomic targets. The advertised O(h) convergence of the randomized controller to the averaged sliced flow is therefore not established for the cases actually tested.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a finite-horizon feedback framework for steering the probability law of a linear dynamical system between prescribed endpoint distributions, using only one-dimensional optimal transport maps along random projections. For the single-integrator dynamics, averaging the directional controllers over the sphere yields a deterministic sliced feedback. Proposition 1 derives a dissipation identity for the sliced Wasserstein distance; Proposition 2 shows that this averaged feedback is affine and preserves Gaussianity; Theorem 1 gives terminal convergence under a uniform covariance lower bound; Theorem 2 constructs a law-dependent gain with linear decay and explicit energy bounds; and Theorem 3 proves O(h) convergence of the randomized controller to the averaged flow under Assumptions 1 and 2. Sections 5 and 6 extend the construction to fully actuated and general controllable linear systems, and Section 7 presents numerical experiments for Gaussian mixtures and image color transfer.","tokens_in":26446,"tokens_out":11806,"duration_ms":134429,"significance":"The paper is a genuinely useful contribution to distribution steering: it replaces full-dimensional transport maps with projected one-dimensional maps, provides a clean dissipation identity, gives an exact affine Gaussian flow with explicit decay and energy bounds, and develops realizations over linear dynamics. The appendix proofs are detailed, and the paper is honest about some limitations, notably in Section 6.2 where it explicitly states that partition refinement of the finite-step realization need not converge. The main caveat is that the central randomized-convergence theorem rests on a stability assumption that is not verified for any non-Gaussian class or for the atomic empirical targets actually used in the numerical experiments; a revision should either verify that assumption in a relevant class or restrict the statement of Theorem 3 accordingly.","major_comments":[{"comment":"","section":"Section 4, Assumption 2 and Theorem 3"},{"comment":"","section":"Section 3.4, Theorem 1 and Proposition 2"},{"comment":"","section":"Section 6, Proposition 6 and Algorithm 1"}],"minor_comments":[{"comment":"","section":"Section 7.1"},{"comment":"","section":"Section 7.2, Figure 6"},{"comment":"","section":"Notation, Definition 1"},{"comment":"","section":"Corollary 1"}],"recommendation":"major_revision","confidential_remarks":"The core dissipation and Gaussian-flow analysis appear sound and useful, and the paper is clearly written. The main obstacle is that Theorem 3, advertised as the link between the randomized controller and the averaged flow, is conditional on Assumption 2, which is not verified for the atomic empirical targets used in the paper's own experiments. A revision that either verifies Assumption 2 for a relevant class or explicitly restricts the convergence claim would be acceptable. I also recommend that the editorial decision weigh whether the finite-step realization section overclaims 'distribution steering' for general controllable systems given that no terminal convergence is proved there."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. The genuinely new part is the finite-horizon feedback realization: interpreting one-dimensional projected OT maps as terminal conditions, deriving the minimum-energy single-direction controller, and then showing the averaged flow has the descent and Gaussian properties. The Gaussian analysis is careful — the covariance equation, the affine feedback, the energy bounds with the chi(t) cancellation factor, and the reachability-normalized realization for linear systems. Proposition 1's dissipation identity and the energy characterization look solid, and the appendix proofs are coherent. I found no circular steps.\n\nThe soft spots are real but not fatal. Theorem 1 assumes Sigma(t) >= alpha I on [0,1), while Proposition 2 only proves strict positivity. That's an extra condition the paper doesn't verify, though the Gaussian corollary later produces such an alpha by continuity at the endpoint. More importantly, Theorem 3's O(h) convergence relies on Assumption 2, a Lipschitz stability condition on the averaged sliced field along the generated flows. For the atomic target laws used in the numerical experiments — fixed particle sets — the one-dimensional OT maps are step functions, and the averaged field need not be Lipschitz, so Assumption 2 isn't justified. The paper never states this restriction, and the convergence result is therefore not established for the cases actually tested. That's a gap in the bridge between the analyzed averaged flow and the implementable randomized controller, exactly where the paper claims its main methodological link.\n\nThe numerical work is illustrative: no code or data, no error bars, and the runtime comparison compares one exact map against a full 60-step sliced run. That's a minor issue.\n\nOverall, the central Gaussian and linear-system results appear correct and are the paper's real value. The convergence theorem needs either a verification of Assumption 2 for a concrete class or an explicit statement that it holds only for absolutely continuous targets with appropriate regularity, with the atomic case treated separately. I'd send this to peer review with a conditional verdict — the authors can close the gap without changing the main contribution. I'd cite the Gaussian steering results in my own work.","headline":"Solid sliced-OT control framework with a real gap: Theorem 3 doesn't cover the atomic targets used in its own numerics, but the Gaussian and linear-system results stand.","tokens_in":26963,"tokens_out":2875,"would_cite":true,"duration_ms":32483,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49Q22","93B05","93C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Sliced optimal transport can be realized as feedback laws that steer a system's law to a prescribed target using only one-dimensional projections.","keywords":["sliced optimal transport","distribution steering","feedback control","sliced Wasserstein distance","Gaussian preservation","continuity equation","reachability Gramian","randomized sampling"],"falsifier":"Run the averaged sliced feedback on two Gaussians in $\\mathbb{R}^2$ with initial covariance $10^{-6}I$ and target covariance $I$, using $\\lambda(t)=(1-t)^{-1}$, and monitor the smallest eigenvalue of $\\Sigma(t)$; if it dips below any fixed $\\alpha>0$ before $t=1$ and the sliced Wasserstein distance stops following the exponential bound (19), the uniform lower-bound hypothesis is the active obstruction. Independently, on the three-component Gaussian mixture of Section 7, estimate $\\mathbb{E}[W_2^2(\\rho^h_k,\\rho_{t_k})]$ for shrinking $h$; if the ratio to $h$ diverges, Assumption 2 does not hold for that case and the $O(h)$ convergence theorem does not apply.","tokens_in":26041,"feed_emoji":"🎯","tokens_out":15754,"duration_ms":136222,"temperature":0.7,"pith_summary":"This paper tries to establish that distribution steering—driving the probability law of a dynamical system from one prescribed distribution to another—can be carried out with sliced optimal transport, using only one-dimensional projections of the current and target laws. The key move is to read each projected optimal transport map as a directional terminal condition, lift its minimum-energy realization back to the ambient state space, and average over all projection directions. For single-integrator dynamics the resulting averaged feedback makes the sliced Wasserstein distance to the target non-increasing; for Gaussian endpoint laws the feedback is affine, preserves Gaussianity, and under a uniform covariance lower-bound condition drives the mean and covariance to their target values. The paper also proves that the randomized single-direction iteration converges to this averaged flow with expected squared Wasserstein error $O(h)$ as the sampling period vanishes, and extends the construction to linear systems through reachability-normalized coordinates and local controllability Gramians. A reader should care because this offers a sample-friendly, feedback-realizable alternative to full-dimensional optimal transport for steering distributions.","feed_headline":"One-dimensional transport maps can steer distributions to target law","feed_subtitle":"Averaging projected transport shifts makes the sliced Wasserstein distance non-increasing.","key_machinery":"The load-bearing object is the sliced discrepancy field $g_t(x)=\\int_{S^{n-1}}(\\theta^{\\top}x-T^{\\theta}_t(\\theta^{\\top}x))\\,\\theta\\,\\sigma(d\\theta)$, computed from one-dimensional monotone optimal transport maps $T^{\\theta}_t$ between projected current and projected target laws. Its averaged square $D(t)=\\int\\|g_t(x)\\|^2\\,\\rho_t(dx)$ appears in the dissipation identity, and its directional versions define the randomized update. In the Gaussian case the field specializes to the matrix function $H(\\Sigma)=\\int_{S^{n-1}}\\alpha(\\Sigma,\\theta)\\theta\\theta^{\\top}\\sigma(d\\theta)-\\frac{1}{n}I$, where $\\alpha(\\Sigma,\\theta)=\\sqrt{\\theta^{\\top}\\Sigma_1\\theta/\\theta^{\\top}\\Sigma\\theta}$; the covariance equation $\\dot\\Sigma=K\\Sigma+\\Sigma K^{\\top}$ with $K=\\lambda H$ preserves Gaussianity and controls terminal convergence. A law-dependent gain $\\lambda_{\\mathrm{SW}}(t)=n/(\\chi(t)(1-t))$ compensates directional cancellation and produces linear decay of the sliced Wasserstein distance. For linear systems, reachability-normalized coordinates $z=LF_t x$ and local controllability Gramians $G_k$ translate the sliced velocity into an exact finite-step input.","core_discovery":"At each sampling instant, choose a direction $\\theta\\in S^{n-1}$, compute the monotone optimal transport map $T^{\\theta}_t$ between the projected current law and the projected target law, and solve a scalar minimum-energy problem that sends $\\theta^{\\top}x$ to $T^{\\theta}_t(\\theta^{\\top}x)$ over the remaining horizon. Lifting that scalar control back to $\\mathbb{R}^n$ gives a single-direction controller; averaging it over all directions with the spherical measure yields the deterministic averaged sliced feedback $v(t,x)=-\\lambda(t)g_t(x)$, where $g_t(x)$ is the sliced discrepancy field. The paper establishes that, along this feedback and the continuity equation, the sliced Wasserstein distance obeys $\\frac{d}{dt}\\mathrm{SW}_2^2(\\rho_t,\\rho_1)=-2\\lambda(t)\\int\\|g_t\\|^2\\,d\\rho_t$, so it never increases. For Gaussian endpoints the field is affine, the flow preserves Gaussianity, and with $\\int_0^1\\lambda=\\infty$ plus $\\Sigma(t)\\succeq\\alpha I$ the mean and covariance converge to the prescribed terminal values, giving exponential decay of the sliced Wasserstein distance. The randomized sampled-direction iteration is shown to converge to the averaged sliced flow in expectation with squared Wasserstein error $O(h)$ as the sampling period $h\\to 0$.","pith_inferences":["An implication the paper leaves implicit is that the covariance lower-bound condition in Theorem 1 is not implied by Proposition 2's strict positivity; one could test Gaussian examples with a near-degenerate initial covariance, where the exponential-decay guarantee may fail even though the mean still converges.","The $O(h)$ convergence proof treats projection directions as i.i.d.; using stratified or antipodal direction sampling should reduce the centered fluctuation variance and might remove the $1/n$ factor in the sliced-Wasserstein bound, a testable modification of the algorithm.","For the finite-step realization, shrinking the partition can make the local Gramians ill-conditioned even when they stay positive definite; the paper explicitly notes no finite-energy continuous-time limit should be expected, which suggests a practical trade-off between step count and numerical conditioning that is not quantified.","One could extend the same 'directional terminal condition' reading to nonlinear or stochastic dynamics by re-solving the projected endpoint problem in a moving horizon; the paper lists this direction implicitly in its conclusion."],"forward_implications":["The sliced controller reduces the per-iteration cost of distribution steering from solving an $N\\times N$ assignment problem to sorting $N_{\\mathrm{dir}}$ one-dimensional projections: $O(K N_{\\mathrm{dir}} N\\log N)$ operations and $O(N_{\\mathrm{dir}} N)$ memory over the full horizon.","Whenever the Gaussian terminal-convergence conditions hold, the same affine feedback steers the mean and covariance of any law with finite second moment to the prescribed values, not only Gaussian laws.","For uniformly fully actuated linear systems, the physical control energy is sandwiched between $E_z/q_+$ and $E_z/q_-$, where $E_z$ is the kinetic energy of the transformed sliced flow and $q_-$, $q_+$ are uniform actuation bounds.","For general controllable systems, applying the local-Gramian input (50) reproduces the virtual sliced iteration exactly at the sampling instants, so discretization error in the law appears only through the choice of sliced update map, not through the linear dynamics.","With the gain $\\lambda_{\\mathrm{SW}}$, the sliced Wasserstein distance decays exactly linearly, $r(t)=(1-t)r(0)$, and the total control energy obeys $n\\,\\mathrm{SW}_2^2(\\rho_0,\\rho_1)\\le E_u(0,1)\\le (n/\\chi)\\,\\mathrm{SW}_2^2(\\rho_0,\\rho_1)$ whenever $\\chi(t)\\ge\\chi>0$."],"supporting_citations":[{"why":"supplies the metric Wasserstein absolute-continuity and continuity-equation theorems used to differentiate sliced Wasserstein distance along the flow.","marker":"[AGS08]"},{"why":"supplies the Benamou–Brenier dynamical formulation that gives the velocity-field-as-control viewpoint the sliced controller is built on.","marker":"[BB00]"},{"why":"provides the one-dimensional quantile representation and the affine form of optimal transport maps between Gaussians used in every sliced update.","marker":"[PC19]"},{"why":"provides the reachability-Gramian realization of optimal transport over linear dynamics that Section 5 extends to sliced feedback.","marker":"[CGP17]"},{"why":"provides the projection-based iterative distribution transfer procedure that the randomized sliced controller is compared with and generalizes.","marker":"[PKD07]"},{"why":"is the conference version for Gaussian marginals that the present paper extends to weak solutions, randomized convergence, and general linear realizations.","marker":"[ID26]"}],"fun_headline_variants":["Sliced optimal transport feedback steers distributions to target","Averaged projected maps yield control for distribution steering","Randomized single-direction control steers laws via sliced transport","Projection-based feedback forces distribution matching","Sliced transport controller shapes state distributions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the direction-averaged force field stays Lipschitz stable along both the random and averaged flows and that the Gaussian covariance remains uniformly bounded away from zero on the whole horizon; the paper states these as assumptions without verifying them for concrete distributions, and the terminal-convergence and $O(h)$-rate claims collapse if either fails.","fun_headline_variants_meta":{"raw":{"variants":["Sliced optimal transport feedback steers distributions to target","Averaged projected maps yield control for distribution steering","Randomized single-direction control steers laws via sliced transport","Projection-based feedback forces distribution matching","Sliced transport controller shapes state distributions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001109,"raw_usage":{"total_tokens":4686,"prompt_tokens":1077,"completion_tokens":3609,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":693,"completion_tokens_details":{"reasoning_tokens":3536}},"tokens_in":693,"tokens_out":3609,"duration_ms":23702,"temperature":1.0,"reasoning_tokens":3536,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:25:56.931836+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the averaged sliced feedback on two Gaussians in $\\mathbb{R}^2$ with initial covariance $10^{-6}I$ and target covariance $I$, using $\\lambda(t)=(1-t)^{-1}$, and monitor the smallest eigenvalue of $\\Sigma(t)$; if it dips below any fixed $\\alpha>0$ before $t=1$ and the sliced Wasserstein distance stops following the exponential bound (19), the uniform lower-bound hypothesis is the active obstruction. Independently, on the three-component Gaussian mixture of Section 7, estimate $\\mathbb{E}[W_2^2(\\rho^h_k,\\rho_{t_k})]$ for shrinking $h$; if the ratio to $h$ diverges, Assumption 2 does not hold for that case and the $O(h)$ convergence theorem does not apply.","supporting_citations":[],"review_version":1}