Distributional Stability of Tangent-Linearized Gaussian Inference on Smooth Manifolds

Distrib utional Stability of T angent-Linearized Gaussian Infer ence on Smooth Manif olds Junghoon Seo, Hakjin Lee, Jaehoon Sim Abstract — Gaussian inference on smooth manif olds is central to robotics, b ut exact marginalization and conditioning ar e generally non-Gaussian and geometry-dependent. W e study tangent-linearized Gaussian inference and derive explicit non- asymptotic W 2 stability bounds for projection marginalization and surface-measure conditioning . The bounds separate local second-order geometric distortion from nonlocal tail leakage and, f or Gaussian inputs, yield closed-f orm diagnostics fr om ( µ, Σ) and curvatur e/reach surrogates. Circle and planar - pushing experiments validate the predicted calibration transition near p ∥ Σ ∥ op /R ≈ 1 / 6 and indicate that normal-direction uncertainty is the dominant failur e mode when locality breaks. These diagnostics provide practical triggers for switching from single-chart linearization to multi-chart or sample-based manifold infer ence. I . I N TR O D U C T I O N Probabilistic inference on smooth manifolds is fundamental in robotics whenev er state spaces or constraints are intrinsi- cally nonlinear (e.g., pose/Lie groups [ 1 ], contact sets [ 2 , 3 ], and kinematic manifolds [ 4 ]). In these settings, estimators must return both manifold-valued states and calibrated un- certainty for planning, control, and sensor fusion [ 1 ]. T wo recurring operations are marginalization (eliminating nuisance directions) and conditioning (enforcing constraints). Although both are classical in Euclidean spaces, their manifold counter- parts depend on local geometry and probability-mass locality and are not captured by linear-subspace theory alone [ 5 ]. For positi ve-codimension manifolds, { X ∈ M} has zero ambient measure; conditioning must therefore be defined by restricting the ambient density to M and renormalizing with respect to surface measure, rather than by ev ent conditioning in R n . Recently , Guo et al. (2025) [ 6 ], the winner of the 2025 IEEE ICRA Best Conference Paper A ward, derived explicit Gaussian identities for linear constraint manifolds and advo- cated a linearize–infer–retract workflo w for smooth manifolds: linearize at ˜ µ ∈ M , apply affine-manifold identities on T ˜ µ M , and map the result back to M using a retraction or chart. In robotics estimation, their framework enables tight, geometry- consistent uncertainty extraction for constrained inference by performing closed-form marginalization/conditioning on tangent-space surrogates and retracting back to the manifold. This vie wpoint connects to constrained cov ariance extraction in factor-graph systems [ 7 ], projection-based directional un- certainty models on circles and spheres [ 8 , 9 ], and geometry- aw are filtering on Lie groups/manifolds [ 10 , 11 , 12 ]. Y et the question remains: when is a single tangent linearization reliable at the distribution level? The authors are with AI Robot T eam, PIT IN Corp., South K orea. { sjh,hj,simjeh } @pitin-ev.com In this paper , we formalize reliability as a distributional stability question: how close are the exact manifold-induced laws to the laws produced by tangent linearization and retrac- tion? W e quantify this discrepancy using the 2 -W asserstein distance W 2 , since it metrizes weak conv ergence with second moments and therefore directly controls mean and cov ariance errors [ 13 , 14 ]. Concretely , for an ambient Gaussian X ∼ N ( µ, Σ) and a C 2 embedded submanifold M ⊂ R n , we take as exact targets the two canonical ways of pushing mass onto M : • Marginalization via projection: P marg := g # N ( µ, Σ) where g is (locally) the Euclidean metric projection onto M . • Conditioning via surface measure: P cond with density proportional to the ambient Gaussian density restricted to M , i.e., dP cond ∝ φ µ, Σ d V ol M . Giv en a linearization point ˜ µ ∈ M , its surrogate replaces M by its tangent space T ˜ µ M , applies the corresponding affine-manifold identities, and then maps the resulting law back to M via a retraction or chart [ 6 , 15 ]. Our contributions are threefold: • W e deriv e explicit non-asymptotic W 2 stability bounds for projection marginalization and surface-measure condition- ing, decomposing error into local second-order geometric distortion and nonlocal tail leakage. • W e specialize these bounds to Gaussian inputs, yielding computable diagnostics from ( µ, Σ) together with local curvature/reach surrogates. • W e validate the predicted regimes in circle and planar- pushing e xperiments, including anisotropy , offset, and directional-noise stress tests. I I . P R E L I M I N A R I E S W e work in the Euclidean space ( R n , ⟨· , ·⟩ ) with norm ∥ x ∥ := p ⟨ x, x ⟩ . For r > 0 and x ∈ R n , let B ( x, r ) := { z ∈ R n : ∥ z − x ∥ ≤ r } and B ( r ) := B (0 , r ) . For a matrix A , ∥ A ∥ op and ∥ A ∥ F denote the operator and Frobenius norms, respectiv ely . W e write 1 A for the indicator of an event A . For a measurable map h : R n → R m and a probability measure P on R n , the pushforward is h # P , defined by ( h # P )( B ) = P ( h − 1 ( B )) for Borel B ⊂ R m . If X has law L ( X ) = P , then L ( h ( X )) = h # P . W asser stein distance: Let P 2 ( R n ) be the set of Borel probability measures on R n with finite second moment. For P , Q ∈ P 2 ( R n ) , the 2 -W asserstein distance [ 13 , 14 ] is W 2 ( P , Q ) := inf π ∈ Π( P,Q )  Z R n × R n ∥ x − y ∥ 2 dπ ( x, y )  1 / 2 , (1) where Π( P , Q ) denotes the set of couplings of ( P , Q ) . Equiv alently , W 2 ( P , Q ) = inf n  E ∥ X − Y ∥ 2  1 / 2 : L ( X ) = P , L ( Y ) = Q o . (2) Hence, for any coupling ( X , Y ) , W 2 ( P , Q ) ≤  E ∥ X − Y ∥ 2  1 / 2 . (3) If h : R n → R m is L -Lipschitz, W 2 ( h # P , h # Q ) ≤ L W 2 ( P , Q ) . (4) This metric directly serves our goal: deciding when tangent- linearized manifold inference giv es reliable first- and second- order uncertainty summaries. For P , Q ∈ P 2 ( R n ) with means µ P , µ Q and uncentered second moments M 2 ( P ) := E [ X X ⊤ ] , M 2 ( Q ) := E [ Y Y ⊤ ] , optimal coupling and Cauchy– Schwarz yield ∥ µ P − µ Q ∥ ≤ W 2 ( P , Q ) , (5) ∥ M 2 ( P ) − M 2 ( Q ) ∥ F ≤ q 2  m 2 ( P ) + m 2 ( Q )  W 2 ( P , Q ) . (6) Here m 2 ( ν ) := E Z ∼ ν ∥ Z ∥ 2 for ν ∈ P 2 ( R n ) . Therefore, W 2 controls both location and second-order discrepancy up to explicit scale factors [ 16 ], so we state our main stability results in W 2 as practical calibration criteria for both marginalization and conditioning. Embedded submanifolds, curvatur e, and r each: Let M ⊂ R n be a C 2 embedded submanifold of dimension d . F or y ∈ M , denote by T y M its tangent space and by N y M = ( T y M ) ⊥ its normal space. Let Π T y M and Π N y M be the corresponding orthogonal projectors. Write I I y : T y M × T y M → N y M for the second fundamental form. The r each of a closed set S ⊂ R n , reac h( S ) , is the largest ρ such that ev ery point within distance < ρ has a unique nearest point in S . If reac h( M ) ≥ ρ > 0 , the metric projection g is well-defined on the tube T ρ ( M ) := { x : dist( x, M ) < ρ } [ 17 ]. Retractions and charts: Fix ˜ µ ∈ M and write T := T ˜ µ M . A (local) retraction at ˜ µ is a C 2 map R ˜ µ : B T ( r ) → M satisfying R ˜ µ (0) = ˜ µ and D R ˜ µ (0) = Id T . W e use a quadratic accuracy bound [ 15 ]: there exists κ R ≥ 0 such that ∥ R ˜ µ ( v ) − ( ˜ µ + v ) ∥ ≤ κ R 2 ∥ v ∥ 2 , ∥ v ∥ ≤ r. (7) More generally , for a chart Ψ : B r ⊂ R d → M with Ψ(0) = ˜ µ , the induced volume satisfies d V ol M (Ψ( v )) = J ( v ) dv [ 18 ]. A. Marginalization and conditioning onto manifolds Let X ∈ R n hav e law P with density p , and let M = { x ∈ R n : f ( x ) = 0 } be an m -dimensional smooth manifold defined by a regular map f : R n → R n − m . Following [ 6 ], there are two canonical ways to push probability onto M : (i) mar ginalization by pr ojection : choose g M : R n → M and set P marg := ( g M ) # P ; (8) (ii) conditioning : restrict and renormalize, giving dP cond ( x ) = Z − 1 M p ( x ) d V ol M ( x ) . (9) For co dim( M ) > 0 , this is not ev ent conditioning on { X ∈ M} (which has measure zero under the ambient law); it is a new probability measure obtained by restricting the density to M and normalizing with respect to V ol M . For an affine linear manifold M lin := { x ∈ R n : S ⊤ x = c } with full-rank S ∈ R n × m , m < n , let N span n ull( S ⊤ ) and define Π := N ( N ⊤ N ) − 1 N ⊤ . For X ∼ N ( µ, Σ) , marginalization and conditioning admit closed forms [ 6 ]: µ marg = Π µ + x 0 , Σ marg = ΠΣΠ ⊤ , (10) Σ cond = N  N ⊤ Σ − 1 N  − 1 N ⊤ , µ cond = x 0 + Σ cond Σ − 1 ( µ − x 0 ) , (11) where x 0 := S ( S ⊤ S ) − 1 c ∈ M lin . B. Linearization principle for smooth manifolds The closed-form identities (10) – (11) hold when the con- straint manifold is an affine subspace of R n . T o extend them to a smooth nonlinear submanifold M ⊂ R n , we adopt a standard tangent-plane surr ogate : select a linearization point ˜ µ ∈ M —typically the Euclidean projection of the ambient mean onto M or a current estimate—and replace M locally by its first-order approximation at ˜ µ [ 6 ]. Specifically , let M = { x : f ( x ) = 0 } be a regular level set for a smooth f : R n → R n − d with rank D f ( ˜ µ ) = n − d . The tangent space at ˜ µ then admits the affine representation T ˜ µ M = { x ∈ R n : S ⊤ x = c } , S := D f ( ˜ µ ) ⊤ , c := S ⊤ ˜ µ, where N := null( S ⊤ ) spans tangent directions and Π := N ( N ⊤ N ) − 1 N ⊤ is the orthogonal projector onto T ˜ µ M . Applying (10) or (11) in this affine setting yields a de generate Gaussian on T ˜ µ M , constituting a first-order approximation to the on-manifold distribution. Because tangent-space distributions do not lie on M itself, the resulting Gaussian is mapped back to M via a retraction R ˜ µ or an explicit chart Ψ , producing the pushforward b P := ( R ˜ µ ) # P T (or b P := Ψ # Q T in local coordinates), where P T is the Gaussian on T ˜ µ M . In what follows, we adopt this tangent–retraction construction as the baseline and deriv e distribution-le vel error bounds that quantify when a single linearization suffices. I I I . W 2 S T A B I L I T Y F O R G AU S S I A N M A R G I N A L I Z A T I O N Let M ⊂ R n be a C 2 embedded submanifold and let X ∼ N ( µ, Σ) with Σ ≻ 0 . With g : R n → M denoting a measurable projection map (locally , the metric projection), let Y := g ( X ) and P marg := g # N ( µ, Σ) as in (8) . This section quantifies, in W 2 , the discrepancy between P marg and a tangent–retraction surrogate. Given a linearization point ˜ µ ∈ M , write T := T ˜ µ M with orthogonal projector Π T and define ˆ g ( x ) := ¯ R  Π T ( x − ˜ µ )  , b Y := ˆ g ( X ) , b P marg := L ( b Y ) . 4 3 2 1 0 1 2 3 4 1 0 1 2 ( a ) S t a b l e : 1 E x a c t p r o j . g T a n g e n t a p p r o x . g r T 4 3 2 1 0 1 2 3 4 A r c - l e n g t h o n 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Density W 2 0 . 0 3 4 5 = 0 . 1 2 , = 0 . 7 = 0 . 0 9 E x a c t P m a r g A p p r o x P m a r g Discrepancy 6 4 2 0 2 4 6 3 2 1 0 1 2 3 4 5 ( b ) U n s t a b l e : 1 E x a c t p r o j . g T a n g e n t a p p r o x . g r T 6 4 2 0 2 4 6 A r c - l e n g t h o n 0.00 0.05 0.10 0.15 0.20 0.25 0.30 W 2 0 . 7 5 0 1 = 0 . 5 0 , = 1 . 8 = 0 . 9 0 E x a c t P m a r g A p p r o x P m a r g Discrepancy Fig. 1: W 2 stability of marginalizing a Gaussian onto a circular arc. Exact metric projection g vs. the tangent–retraction approximation ˆ g . where ¯ R : T → M is any measurable extension of a local retraction R ˜ µ : B T ( r ) → M outside B T ( r ) . The retraction satisfies the quadratic accuracy bound (7) . The main estimate below separates a local second-order geometric error from a tail contribution outside the localization radius r . Geometric locality: Assume reac h( M ) ≥ ρ > 0 and fix r ∈ (0 , ρ/ 2) . Under this assumption, g coincides with the unique metric projection on the tube T ρ ( M ) . Assume a local curvature bound on M ∩ B ( ˜ µ, 2 r ) : sup y ∈M∩ B ( ˜ µ, 2 r ) ∥ I I y ∥ op ≤ κ. (12) Theorem III.1 (W 2 stability of marginalization under tangen- t–retraction) . There exists a constant C loc > 0 (depending only on dimensionless tube/curvatur e mar gins, e.g. r /ρ and r κ , and on the ambient dimension) such that, with A := {∥ X − ˜ µ ∥ ≤ r } and ε := P ( A c ) , W 2  P marg , b P marg  ≤ C loc ( κ + κ R )  E  ∥ X − ˜ µ ∥ 4 1 A   1 / 2 + C tail ε 1 / 4 , (13) wher e C tail :=  E ∥ Y − ˜ µ ∥ 4  1 / 4 +  E ∥ b Y − ˜ µ ∥ 4  1 / 4 < ∞ . Remark III.2 (Practical calibration of C loc ) . The pr oof writes C loc as a combination of deterministic local geometry constants (tube-Lipschitz and C 2 graph constants), hence it is independent of ( µ, Σ) . A practical recipe is to fix ( r /ρ, r κ ) , sample local test points x j ∈ B ( ˜ µ, r ) \ { ˜ µ } , evaluate ˆ c j := ∥ g ( x j ) − R ˜ µ (Π T ( x j − ˜ µ )) ∥ ( κ + κ R ) ∥ x j − ˜ µ ∥ 2 , and set C loc to a high quantile of { ˆ c j } times a safety factor . This makes conservativeness explicit and r epr oducible. Pr oof of Theor em III.1. W e couple Y = g ( X ) and b Y = ˆ g ( X ) through the same Gaussian input X , and we estimate E ∥ Y − b Y ∥ 2 by separating a local region (where geometry is second-order accurate) from its complement. The determinis- tic part is a local comparison between metric projection and the tangent–retraction map. Lemma III.3 (Local quadratic comparison of projection and tangent–retraction) . Under the geometric locality assumptions (Equation (12) ), let R ˜ µ : B T ( r ) → M be a C 2 r etraction with (7) . Then there exists C loc > 0 such that for every x with ∥ x − ˜ µ ∥ ≤ r ,   g ( x ) − R ˜ µ  Π T ( x − ˜ µ )    ≤ C loc ( κ + κ R ) ∥ x − ˜ µ ∥ 2 . (14) Pr oof. Fix x with d := ∥ x − ˜ µ ∥ ≤ r , and define y := g ( x ) ∈ M and v := Π T ( x − ˜ µ ) ∈ T . Since r < ρ , we hav e x ∈ T ρ ( M ) , so y is well-defined. For sets with reach at least ρ , the metric projection is Lipschitz on T t ( M ) for any t < ρ : Lip  g | T t ( M )  ≤ ρ ρ − t . (15) T aking t = r and using g ( ˜ µ ) = ˜ µ giv es ∥ y − ˜ µ ∥ ≤ ρ ρ − r d ≤ 2 d , hence y ∈ M ∩ B ( ˜ µ, 2 r ) and (12) applies at y . Standard C 2 graph estimates on M ∩ B ( ˜ µ, 2 r ) (see, e.g., [ 19 , Lem. 3.2, Lem. 3.5] and [ 20 , Lem. 5.4, Prop. 6.1–6.2])) provide constants C ht , C ang > 0 such that for all z ∈ M ∩ B ( ˜ µ, 2 r ) , ∥ Π T ⊥ ( z − ˜ µ ) ∥ ≤ C ht κ 2 ∥ z − ˜ µ ∥ 2 , (16) ∥ Π T z M − Π T ∥ op ≤ C ang κ ∥ z − ˜ µ ∥ . (17) Because y = g ( x ) is a metric projection, Π T y M ( x − y ) = 0 , so Π T ( x − y ) = (Π T − Π T y M )( x − y ) . Applying (17) at z = y , together with ∥ x − y ∥ ≤ d and ∥ y − ˜ µ ∥ ≤ 2 d , yields ∥ v − Π T ( y − ˜ µ ) ∥ ≤ 2 C ang κ d 2 . (18) Like wise, (16) at z = y gives ∥ Π T ⊥ ( y − ˜ µ ) ∥ ≤ 2 C ht κ d 2 . (19) Decomposing y − ( ˜ µ + v ) = Π T ⊥ ( y − ˜ µ ) + (Π T ( y − ˜ µ ) − v ) and combining (18) with (19), we obtain ∥ y − ( ˜ µ + v ) ∥ ≤ C pro j κ d 2 , C pro j := 2( C ht + C ang ) . (20) Finally , (7) and ∥ v ∥ ≤ d imply ∥ R ˜ µ ( v ) − ( ˜ µ + v ) ∥ ≤ κ R 2 d 2 , and the triangle inequality between y , ˜ µ + v , and R ˜ µ ( v ) gi ves (14). Returning to the random setting, the shared-input coupling and (3) giv e W 2  L ( Y ) , L ( b Y )  ≤  E ∥ Y − b Y ∥ 2  1 / 2 . (21) Let A := {∥ X − ˜ µ ∥ ≤ r } . Splitting the second moment ov er A and A c , E ∥ Y − b Y ∥ 2 := E  ∥ Y − b Y ∥ 2 1 A  + E  ∥ Y − b Y ∥ 2 1 A c  . (22) On A , Lemma III.3 yields ∥ Y − b Y ∥ ≤ C loc ( κ + κ R ) ∥ X − ˜ µ ∥ 2 , therefore  E  ∥ Y − b Y ∥ 2 1 A   1 / 2 ≤ C loc ( κ + κ R ) ×  E  ∥ X − ˜ µ ∥ 4 1 A   1 / 2 . (23) On A c , by the triangle inequality , ∥ Y − b Y ∥ ≤ ∥ Y − ˜ µ ∥ + ∥ b Y − ˜ µ ∥ . Setting Z := ∥ Y − ˜ µ ∥ + ∥ b Y − ˜ µ ∥ , we get ∥ Y − b Y ∥ 2 1 A c ≤ Z 2 1 A c , and H ¨ older yields  E  ∥ Y − b Y ∥ 2 1 A c   1 / 2 ≤  E [ Z 4 ]  1 / 4 ε 1 / 4 . (24) By Minkowski’ s inequality in L 4 ,  E [ Z 4 ]  1 / 4 ≤  E ∥ Y − ˜ µ ∥ 4  1 / 4 +  E ∥ b Y − ˜ µ ∥ 4  1 / 4 =: C tail , so  E  ∥ Y − b Y ∥ 2 1 A c   1 / 2 ≤ C tail ε 1 / 4 . (25) Combining (21), (22), (23), and (25) yields (13). Gaussian specialization: Set δ := µ − ˜ µ and write X − ˜ µ = δ + ξ with ξ ∼ N (0 , Σ) . Then the untruncated fourth moment in (13) has the closed form E ∥ X − ˜ µ ∥ 4 = ∥ δ ∥ 4 + 2 ∥ δ ∥ 2 tr Σ +4 δ ⊤ Σ δ + (tr Σ) 2 + 2 ∥ Σ ∥ 2 F . (26) On the localization ev ent A := {∥ X − ˜ µ ∥ ≤ r } , the localized fourth moment obeys the explicit bound E  ∥ X − ˜ µ ∥ 4 1 A  ≤ r 4 , and can be sharpened by ev aluating the CDF of the noncentral quadratic form ∥ X − ˜ µ ∥ 2 (noncentral χ 2 in the isotropic case). For the tail probability ε := P ( A c ) , let λ max := ∥ Σ ∥ op and Z ∼ N (0 , I n ) . Using P ( ∥ Z ∥ ≥ √ n + t ) ≤ e − t 2 / 2 [ 21 ] yields, for any r > ∥ δ ∥ , ε = P ( ∥ X − ˜ µ ∥ > r ) ≤ exp  − 1 2  r − ∥ δ ∥ √ λ max − √ n  2 +  . (27) Equiv alently , choosing r = ∥ δ ∥ + p λ max ( √ n + t ) = ⇒ ε ≤ e − t 2 / 2 . Finally , the constant C tail in (13) can be made explicit under mild global extensions. For example, on the tube T r ( M ) with r < ρ , the metric projection is Lipschitz with Lip( g ) ≤ ρ/ ( ρ − r ) , hence  E ∥ Y − ˜ µ ∥ 4  1 / 4 ≤  ρ ρ − r  E ∥ X − ˜ µ ∥ 4  1 / 4 . Moreov er , the retraction accuracy (7) implies (for v = Π T ( X − ˜ µ ) ) ∥ ˆ g ( X ) − ˜ µ ∥ = ∥ ¯ R ( v ) − ˜ µ ∥ ≤ ∥ v ∥ + κ R 2 ∥ v ∥ 2 ≤ ∥ X − ˜ µ ∥ + κ R 2 ∥ X − ˜ µ ∥ 2 , 4 3 2 1 0 1 2 3 4 1 0 1 2 ( a ) S t a b l e : 1 E x a c t P c o n d C h a r t a p p r o x . P c o n d r T 4 3 2 1 0 1 2 3 4 A r c - l e n g t h o n 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Density W 2 0 . 0 7 5 7 = 0 . 1 2 , = 0 . 7 = 0 . 0 9 E x a c t P c o n d A p p r o x P c o n d Discrepancy 6 4 2 0 2 4 6 3 2 1 0 1 2 3 4 5 ( b ) U n s t a b l e : 1 E x a c t P c o n d C h a r t a p p r o x . P c o n d r T 6 4 2 0 2 4 6 A r c - l e n g t h o n 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 W 2 0 . 8 2 0 9 = 0 . 5 0 , = 1 . 8 = 0 . 9 0 E x a c t P c o n d A p p r o x P c o n d Discrepancy Fig. 2: W 2 stability of conditioning a Gaussian onto a circular arc. Surface-measure conditioning vs. a tangent-plane chart approximation. so E ∥ ˆ g ( X ) − ˜ µ ∥ 4 < ∞ and can be bounded by Gaussian moments of order up to 8 when a fully explicit C tail is needed. Substituting these quantities into (13) giv es a computable (conservati ve) criterion in terms of ( µ, Σ) and locality parameters ( ρ, κ, κ R , r ) . Interpr etation and numerical illustration: Equation (13) decomposes the error into a local geometric term and a tail- leakage term. Inside A = {∥ X − ˜ µ ∥ ≤ r } , the mismatch between g and ˆ g is second order , scaling as ( κ + κ R ) ∥ X − ˜ µ ∥ 2 (Lemma III.3). For Gaussian inputs, the effecti ve spread scale is ∥ δ ∥ + √ λ max √ n , so the local term is roughly ( κ + κ R )( ∥ δ ∥ + √ λ max √ n ) 2 . The tail term is controlled by ε = P ( ∥ X − ˜ µ ∥ > r ) and decays exponentially once r ≳ ∥ δ ∥ + √ λ max ( √ n + t ) , but can dominate when reach forces small r . Hence, r must balance probability-mass cov erage against local geometric v alidity . Figure 1 supports this prediction: in local regimes, the exact and approximate laws nearly coincide; beyond the transition, linearization becomes miscalibrated and typically overconfident. I V . W 2 S T A B I L I T Y F O R G AU S S I A N C O N D I T I O N I N G Let M ⊂ R n be a C 2 embedded submanifold of dimension d and X ∼ N ( µ, Σ) with density φ µ, Σ . This section quanti- fies, in W 2 , the discrepancy between the exact conditioned law P cond from (9) (with φ = φ µ, Σ ) and an approximation that first conditions on T ˜ µ M and then maps back to M through a local chart. T ang ent-plane refer ence law: Fix ˜ µ ∈ M and let T := T ˜ µ M as in Section III. Let N ∈ R n × d hav e orthonormal columns spanning T and define the affine parametrization x lin ( v ) = ˜ µ + N v for v ∈ R d . Let Ω := Σ − 1 denote the precision matrix, and recall δ = µ − ˜ µ from Section III. Define the probability measure Q T on R d by dQ T ( v ) := φ µ, Σ ( ˜ µ + N v ) Z T dv , Z T := Z R d φ µ, Σ ( ˜ µ + N v ) dv ∈ (0 , ∞ ) . (28) Equiv alently , Q T = N ( m T , Σ T ) with Σ T = ( N ⊤ Ω N ) − 1 , m T = Σ T N ⊤ Ω δ. (29) Chart model and appr oximation: Fix r > 0 and write B r := { v ∈ R d : | v | ≤ r } . Assume Ψ : B r → M is a C 2 embedding onto M r := Ψ( B r ) with Ψ(0) = ˜ µ , D Ψ(0) = N , and Lip(Ψ) ≤ L Ψ on B r . Assume the second-order residual bound   Ψ( v ) − ( ˜ µ + N v )   ≤ κ Ψ 2 | v | 2 , v ∈ B r , (30) and a quadratic control on the Jacobian factor J ( v ) defined by d V ol M (Ψ( v )) = J ( v ) dv : | log J ( v ) | ≤ κ J 2 | v | 2 , v ∈ B r , (31) Let ¯ Ψ : R d → M be a measurable extension of Ψ such that ¯ Ψ( v ) ∈ M r iff v ∈ B r (e.g., map v / ∈ B r to any fixed point in M \ M r ), and define the approximate conditioned law b P cond := ¯ Ψ # Q T . (32) T ail quantities: Define ε P := P cond ( M \ M r ) , ε Q := Q T ( R d \ B r ) , and the corresponding tail second moments (w .r .t. ˜ µ ) τ P ( r ) :=  E  ∥ Y − ˜ µ ∥ 2 1 { Y / ∈M r }   1 / 2 , ( Y ∼ P cond ) , τ b P ( r ) :=  E  ∥ b Y − ˜ µ ∥ 2 1 { b Y / ∈M r }   1 / 2 , ( b Y ∼ b P cond ) . Define the curvature/v olume mismatch exponent η r := ∥ Ω ∥ κ Ψ 2 ( ∥ δ ∥ + r ) r 2 + ∥ Ω ∥ κ 2 Ψ 8 r 4 + κ J 2 r 2 . (33) Theorem IV .1 (W 2 stability of surface-measure conditioning under chart linearization) . Under the assumptions above, W 2  P cond , b P cond  ≤ τ P ( r ) + τ b P ( r ) + L Ψ r  √ ε P + √ ε Q  + 2 L Ψ r p tanh( η r ) . (34) Lemma IV .2 (T runcation bound) . Let P be a pr obability measur e on a subset of R m with finite second moment. Let A be measurable with P ( A ) = 1 − ε ∈ (0 , 1] , and let P ( · | A ) be the conditional law . Fix any x 0 ∈ A and define R A := sup x ∈ A ∥ x − x 0 ∥ (possibly + ∞ ). Then W 2  P , P ( · | A )  ≤  E  ∥ X − x 0 ∥ 2 1 A c   1 / 2 + R A √ ε, ( X ∼ P ) . Pr oof. Define the auxiliary measure ˜ P := (1 − ε ) P ( · | A ) + ε δ x 0 . Couple P with ˜ P by keeping the mass on A fixed and sending all A c mass to x 0 . By (3), this gives W 2 ( P , ˜ P ) ≤  E  ∥ X − x 0 ∥ 2 1 A c   1 / 2 . Next, couple ˜ P to P ( · | A ) by moving the extra point mass ε δ x 0 into A according to P ( · | A ) ; the cost is at most R A √ ε . The claim follows by the triangle inequality . Lemma IV .3 (Diameter–TV control of W 2 on bounded support) . Let p, q be pr obability measures supported on a set S ⊂ R d with diam( S ) := sup u,v ∈ S ∥ u − v ∥ < ∞ . Then W 2 ( p, q ) ≤ diam( S ) p TV( p, q ) , TV( p, q ) := 1 2 Z | dp − dq | . Pr oof. Let γ be a maximal coupling of ( p, q ) so that γ { ( U, V ) : U  = V } = TV( p, q ) [ 22 ]. On { U = V } the cost is 0 , and on { U  = V } we hav e ∥ U − V ∥ ≤ diam( S ) . Thus E γ ∥ U − V ∥ 2 ≤ diam( S ) 2 TV( p, q ) , and the claim follows by infimizing ov er couplings. Lemma IV .4 (TV control from a likelihood-ratio bound) . Let p, q be pr obability measur es with p ≪ q and dp dq ∈ [Λ − 1 , Λ] q -a.e . , for some Λ ≥ 1 . Then TV( p, q ) ≤ Λ − 1 Λ + 1 . Pr oof. Let h := dp/dq , a := Λ − 1 , and b := Λ . Then TV( p, q ) = 1 2 E q [ | h − 1 | ] , E q [ h ] = 1 , h ∈ [ a, b ] q -a.e. Since x 7→ | x − 1 | is con ve x on [ a, b ] , for x ∈ [ a, b ] , | x − 1 | ≤ b − x b − a | a − 1 | + x − a b − a | b − 1 | . Applying this with x = h and taking E q , E q [ | h − 1 | ] ≤ b − E q [ h ] b − a | a − 1 | + E q [ h ] − a b − a | b − 1 | = 2(1 − a )( b − 1) b − a . Hence TV( p, q ) ≤ (1 − a )( b − 1) b − a = Λ − 1 Λ + 1 . Pr oof of Theor em IV .1. Recall P cond from (9) , Q T from (28) , and b P cond = ¯ Ψ # Q T from (32) . Let M r = Ψ( B r ) , and define truncations P ( r ) cond := P cond ( · | M r ) , b P ( r ) cond := b P cond ( · | M r ) . W e first separate tail and interior contributions. By the triangle inequality , W 2 ( P cond , b P cond ) ≤ W 2 ( P cond , P ( r ) cond ) + W 2 ( P ( r ) cond , b P ( r ) cond ) + W 2 ( b P ( r ) cond , b P cond ) . (35) The two outer terms are tail terms. Applying Lemma IV .2 with P = P cond , A = M r , and x 0 = ˜ µ ∈ M r giv es W 2 ( P cond , P ( r ) cond ) ≤ τ P ( r ) + L Ψ r √ ε P , (36) since sup x ∈M r ∥ x − ˜ µ ∥ ≤ L Ψ r . Applying the same lemma to b P cond yields W 2 ( b P ( r ) cond , b P cond ) ≤ τ b P ( r ) + L Ψ r √ ε Q , (37) because b P cond ( M \ M r ) = Q T ( | V | > r ) = ε Q . For the interior term, use that Ψ : B r → M r is bijectiv e and pull both truncated laws back to B r . Define unnormalized densities ˜ p ( v ) := φ µ, Σ (Ψ( v )) J ( v ) , ˜ q ( v ) := φ µ, Σ ( ˜ µ + N v ) . Let p r := ˜ p/ R B r ˜ p and q r := ˜ q/ R B r ˜ q . Then P ( r ) cond = Ψ # p r and b P ( r ) cond = Ψ # q r , so by (4) W 2 ( P ( r ) cond , b P ( r ) cond ) ≤ L Ψ W 2 ( p r , q r ) . (38) Since p r , q r are supported on B r and diam( B r ) ≤ 2 r , Lemma IV .3 gives W 2 ( p r , q r ) ≤ 2 r p TV( p r , q r ) , hence W 2 ( P ( r ) cond , b P ( r ) cond ) ≤ 2 L Ψ r p TV( p r , q r ) . (39) It remains to bound TV( p r , q r ) . Write x ( v ) := ˜ µ + N v and e ( v ) := Ψ( v ) − x ( v ) . By (30) , ∥ e ( v ) ∥ ≤ κ Ψ 2 | v | 2 ≤ κ Ψ 2 r 2 on B r . Since log φ µ, Σ ( x ) = const − 1 2 ( x − µ ) ⊤ Ω( x − µ ) ,   log φ µ, Σ (Ψ( v )) − log φ µ, Σ ( x ( v ))   ≤ ∥ Ω ∥ κ Ψ 2 ( ∥ δ ∥ + r ) r 2 + ∥ Ω ∥ κ 2 Ψ 8 r 4 . T ogether with (31) , this yields | log ˜ p ( v ) − log ˜ q ( v ) | ≤ η r on B r . Hence e − η r ˜ q ≤ ˜ p ≤ e η r ˜ q on B r , and after normalization p r ( v ) q r ( v ) ∈ [ e − 2 η r , e 2 η r ] . Lemma IV .4 then gi ves TV( p r , q r ) ≤ e 2 η r − 1 e 2 η r + 1 = tanh( η r ) . (40) Combining (39) and (40), W 2 ( P ( r ) cond , b P ( r ) cond ) ≤ 2 L Ψ r p tanh( η r ) . (41) Finally , inserting (36) , (41) , and (37) into (35) giv es (34) . Gaussian specialization: Recall Q T = N ( m T , Σ T ) on R d from (29) . Define ε Q := Q T ( R d \ B r ) = P ( ∥ V ∥ > r ) for V ∼ Q T . As in (27) , Gaussian concentration gives (with λ max ,T := ∥ Σ T ∥ op ) ε Q ≤ exp − 1 2  r − ∥ m T ∥ p λ max ,T − √ d  2 + ! . The truncation moment term τ b P ( r ) can be bounded in terms of Q T and chart Lipschitzness: since ∥ Ψ( v ) − ˜ µ ∥ ≤ L Ψ ∥ v ∥ on B r and b P cond = ¯ Ψ # Q T , τ b P ( r ) :=  E  ∥ ˆ Y − ˜ µ ∥ 2 1 { ˆ Y / ∈M r }   1 / 2 ≤ L Ψ  E  ∥ V ∥ 2 1 {∥ V ∥ >r }   1 / 2 . Here ˆ Y denotes a sample from b P cond . A con venient fully explicit bound follows from Cauchy–Schwarz: E  ∥ V ∥ 2 1 {∥ V ∥ >r }  ≤  E ∥ V ∥ 4  1 / 2 ε 1 / 2 Q , where E ∥ V ∥ 4 has the same closed form as (26) with ( δ, Σ) replaced by ( m T , Σ T ) : E ∥ V ∥ 4 = ∥ m T ∥ 4 + 2 ∥ m T ∥ 2 tr Σ T + 4 m ⊤ T Σ T m T + (tr Σ T ) 2 + 2 ∥ Σ T ∥ 2 F . Thus τ b P ( r ) admits an explicit upper bound in terms of ( m T , Σ T , L Ψ , r ) and ε Q . The remaining tail quantities ( ε P , τ P ( r )) are geometric (they quantify how much of the true conditioned law lies outside the chart image M r ). Interpr etation and numerical illustration: Equation (34) splits conditioning error into (i) tail/truncation penalties and (ii) interior in-chart distortion. The tail terms τ P ( r ) + τ b P ( r ) + L Ψ r ( √ ε P + √ ε Q ) measure mass outside M r = Ψ( B r ) under the true law P cond and the approximation b P cond . The interior term 2 L Ψ r p tanh( η r ) captures chart-induced likelihood distortion, where η r com- bines second-order chart residual and Jacobian distortion weighted by precision ∥ Ω ∥ . Thus, high-confidence regimes can amplify small geometric errors ev en when truncation is mild. For affine manifolds with exact charts ( κ Ψ = κ J = 0 ), η r = 0 and only truncation remains. Figure 2 illustrates this transition: near-local regimes show close agreement, whereas larger spread introduces mode/variance mismatch and then tail-dominated failure. V . N U M E R I C A L E X P E R I M E N T S A. 2-D cir cle benchmark Goals and setup: W e consider M = { x ∈ R 2 : ∥ x ∥ = R } with curvature κ = 1 /R , reach ρ = R , and X ∼ N ( µ, Σ) where µ = ( R + δ, 0) . The baseline uses isotropic covariance Σ = σ 2 I 2 with δ = 0 . 2 . W e then add two stress tests to assess robustness: (i) anisotropic cov ariance, Σ = diag ( σ 2 n , σ 2 t ) in the normal/tangential frame, and (ii) offset sweeps in δ /R . These tests probe the computable diagnostics in Theorem III.1, especially the roles of p ∥ Σ ∥ op and the tail-leakage term ε . For marginalization, the exact map is g ( x ) = R x/ ∥ x ∥ , and the approximation is tangent projection plus normalization at ˜ µ = g ( µ ) = ( R , 0) . For conditioning, we compare surface- measure conditioning on M against tangent-line conditioning followed by retraction. Quantitative sweep: Figure 3 sweeps σ /R ∈ [0 . 02 , 1 . 2] for R ∈ { 0 . 5 , 1 , 2 } and reports (i) variance ratio ϱ = V ar lin ( θ ) / V ar exact ( θ ) , (ii) coverage of a nominal 95% inter- v al, and (iii) normalized theoretical/empirical W 2 diagnostics for marginalization. Panel (a) sho ws ϱ crossing below 1 near σ /R = 1 / 6 ; panel (b) shows corresponding coverage degra- dation outside that regime; panel (c) uses p ∥ Σ ∥ op /R and compares empirical W 2 proxies against tightened theoretical W 2 bounds with data-calibrated C tail (90% quantile pre-fit plus a strict upper-en velope safeguard: C tail = 0 . 137 for R = 0 . 5 , 0 . 165 for R = 1 , and 0 . 183 for R = 2 ), so the bound remains abov e the empirical proxy while preserving the failure-onset trend. Anisotr opic Σ and offset sweeps: Figure 4 extends the circle benchmark along two complementary axes. Panel (a) v aries anisotropy η := σ n /σ t ∈ { 0 . 5 , 1 , 2 , 4 } and replots cal- ibration against the spectral scale p ∥ Σ ∥ op /R . The transition remains aligned across anisotropy levels, supporting ∥ Σ ∥ op       / R         = V a r l i n / V a r e x a c t    < 1       / R = 1 / 6          / R                         o p / R                  W 2                     W 2                     W 2             W 2                 W 2         R = 0 . 5 R = 1 R = 2 Fig. 3: Circle sweep ( δ = 0 . 2 ) over σ /R : (a) variance ratio ϱ = V ar lin / V ar exact , (b) realized 95% coverage, and (c) normalized diagnostics on p ∥ Σ ∥ op /R : tightened theoretical W 2 bound (data-calibrated C tail with strict upper-en velope safeguard) and empirical W 2 proxy . Dashed line marks σ /R = 1 / 6 ; colors indicate R ∈ { 0 . 5 , 1 , 2 } .    o p / R                                                                = n / t = 0 . 5 = n / t = 1 = n / t = 2 = n / t = 4        / R                                              = 2  o p / R = 0 . 1 0 o p / R = 0 . 1 8 o p / R = 0 . 2 8        / R                    o p / R = 0 . 1 0 o p / R = 0 . 1 8 o p / R = 0 . 2 8 Fig. 4: Circle generality stress tests: (a) anisotropic sweep ( η = σ n /σ t ) versus p ∥ Σ ∥ op /R , (b) tail proxy ε versus offset δ /R , and (c) coverage degradation versus δ /R . as an effecti ve diagnostic variable in the bound. Panels (b)– (c) fix anisotropy and sweep δ /R : as of fset increases, the tail proxy ε rises rapidly and coverage degrades ev en at fixed curv ature/noise scale, rev ealing a second failure axis beyond curvature-only effects. This behavior matches the decomposition in Theorem III.1, where local distortion and nonlocal leakage contribute separately . B. Planar pushing: diagnostics in a contact-rich task Setup and covariance models: W e use the standard 2-D planar-pushing benchmark from constrained GTSAM/InCOpt to ev aluate cov ariance extraction under hard contact con- straints [ 2 , 3 ]. A rectangular box state x k = ( p x,k , p y ,k , θ k ) ∈ R 2 × S 1 is pushed by a circular probe whose center follows a kno wn deterministic path; the probe is assumed to remain in contact with the box boundary at ev ery step, encoded as a per- pose contact factor c k ( x k ) = 0 . Odometry provides relative- pose measurements with wrapped-angle noise, u k = x k +1 − x k + ε k , ε k ∼ N (0 , Q ) , Q = diag(0 . 03 2 , 0 . 03 2 , 0 . 01 2 ) . W e use r p = 0 . 1 m, ( w , h ) = (0 . 2 , 1 . 0) m, and n steps = 50 . The constrained mean is obtained from constrained least squares: min x 1: n n − 1 X k =0 ∥ r k ( x ) ∥ 2 Q − 1 s.t. c k ( x k ) = 0 , ∀ k , with r k ( x ) := x k +1 − ( x k + u k ) . Linearizing about the opti- mized trajectory , we form the (unconstrained) Gauss–Newton information F ≈ H ⊤ W − 1 H with W = blkdiag( Q, . . . , Q ) , gi ving Σ unc = F − 1 , and compute the constrained covariance by conditioning onto the linearized contact constraints: Σ con = N ( N ⊤ F N ) − 1 N ⊤ , where N ∈ null( S ⊤ ) spans the tangent directions of the constraint manifold. Aggr egate covariance r eduction: Across 49 optimized timesteps, equality constraints are satisfied to numerical preci- sion. T able I reports representativ e timesteps: the constrained T ABLE I: Quantitativ e cov ariance metrics for planar pushing at timesteps tr(Σ xy ) λ max (Σ xy ) Red. ∥ ∆Σ xy ∥ F k Unc. Con. Unc. Con. (%) 1 0.0018 0.0009 0.0009 0.0009 50.5 0.0009 13 0.0234 0.0105 0.0117 0.0105 55.3 0.0118 25 0.0450 0.0190 0.0225 0.0190 57.8 0.0228 37 0.0666 0.0278 0.0333 0.0278 58.2 0.0337 49 0.0882 0.0377 0.0441 0.0377 57.3 0.0446 xy -marginal covariance trace is reduced by 50 . 5% at k = 1 and 57 . 3% at k = 49 , with a peak reduction of 58 . 2% at k = 37 . These v alues establish a trajectory-le vel baseline before the Monte Carlo calibration diagnostics. T rajectory-wide diagnostics: T o mov e beyond single- pose checks, we ev aluate diagnostics at every timestep. Let c k ( x ) = 0 be the contact constraint at step k , linearized at ˆ x k . With N k spanning null  ∇ c k ( ˆ x k ) ⊤  , we define ˆ κ k :=   N ⊤ k ∇ 2 c k ( ˆ x k ) N k   op ∥∇ c k ( ˆ x k ) ∥ , ˆ ρ k := 1 max( ˆ κ k , ϵ κ ) , as curvature/reach proxies ( ϵ κ = 10 − 8 for numerical stability), and the uncertainty spread s k := q ∥ Σ unc ,k ∥ op . At each k , we compare tangent-plane cov ariance Σ lin ,k := Π k Σ unc ,k Π ⊤ k against Monte Carlo cov ariance Σ MC ,k after exact nonlinear projection, via ϱ MC k := tr(Σ MC ,xy ,k ) tr(Σ lin ,xy ,k ) , ∆ k := ∥ Σ MC ,xy ,k − Σ lin ,xy ,k ∥ F ∥ Σ lin ,xy ,k ∥ F . Figure 5 sho ws a non-uniform failure profile over time. Larger mismatch co-occurs with larger locality-pressure indicators ( s k / ˆ ρ k ), consistent with the tail-leakage interpretation in Theorem III.1. The largest mismatch occurs at k = 13 ( ϱ MC 13 = 2 . 162 , locality index s 13 / ˆ ρ 13 = 0 . 119 ); the median trajectory-wide mismatch is median( ϱ MC k ) = 1 . 557 with median relativ e Frobenius discrepancy median(∆ k ) = 0 . 525 . Dir ectional uncertainty stress test: Isotropic scaling Σ 7→ α 2 Σ obscures direction-specific failures. At the terminal pose k ⋆ , with local unit normal ˆ n , define A ( α n , α t ) := α t ( I − ˆ n ˆ n ⊤ ) + α n ˆ n ˆ n ⊤ and Σ( α n , α t ) := A Σ unc ,k ⋆ A ⊤ . W e compare isotropic ( α, α ) , normal-only ( α, 1) , and tangential- only (1 , α ) sweeps. Figure 6 shows a clear asymmetry for α ≥ 1 : normal-direction inflation causes the lar gest calibration and covariance mismatch. V I . D I S C U S S I O N Practical validity test for r obotics estimation: The bounds can be implemented as a runtime gate in factor-graph pipelines. At each linearization point, compute the curvature- load indicator s c := κ p ∥ Σ ∥ op and offset indicator s δ := δ /ρ . For marginalization, evaluate (26) and (27) with ( ρ, κ, r ) to obtain local-distortion and tail-leakage terms in (13) . For conditioning, combine computable ( ε Q , τ b P ( r ) , η r ) with sample-based or certified estimates of ( ε P , τ P ( r )) . Single- chart tangent-linearized inference is retained only when the predicted bound is below an application tolerance and both      k               k o p                           k                         k                 k             k                 k o p                k o p / k            k        t r ( M C ) / t r ( )      Fig. 5: T rajectory-wide planar-pushing diagnostics: (a) curvature/reach proxies ( ˆ κ k , ˆ ρ k ) , (b) spread s k and locality indicator s k / ˆ ρ k , (c) Monte Carlo mismatch ( ϱ MC k , ∆ k ) .                     t r ( M C ) / t r ( )                                M C F / F                        o p /    Fig. 6: Directional scaling at the terminal planar-pushing pose: isotropic, normal-only , and tangential-only covariance inflation. indicators remain in a calibrated safe range. Otherwise, the solver escalates in stages to iterated relinearization [ 23 ], multi-chart updates [ 24 , 25 ], and finally sample-based infer- ence [ 26 , 27 ] when tail leakage or shape mismatch persists. Limitations: The constants are conservati ve because the proofs use general coupling/concentration inequalities rather than manifold-specific transport constructions, so theorem- le vel bounds can ov erestimate empirical error in low-curv ature regimes. The analysis also assumes C 2 manifolds with positiv e reach and does not cover corners, self-intersections, or singular geometries. Extending guarantees to piecewise- smooth contact manifolds and iterativ e multi-chart updates remains an important direction for future work. V I I . 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