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role in analyzing negligible terms. After the power iteration and projection in the algorithm without sample splitting in Section 3.2, for anyj= 1,2,3, we know that bUjcontains the top- rjeigenvectors of bTunbs j PbU(1) j+2⊗ PbU(1) j+1 bTunbs⊤ j . Consequently, bUjbU⊤ jis the spectral projector for the top- rjleft ei... | https://arxiv.org/abs/2501.16223v1 |
PUj+1 PUj+1×j+2PUj+2⊥ PbUj+2− PUj+2 PUj+2 F ≤∥A∥F·" σ4 ξ λ4σ4 r3p1/2log(p)3/2 n2+ ∆·pRlog(p) n2+ ∆2·p3/2Rlog(p) n2+ ∆5·p2 n2!# (159) A × jPUj⊥ PbUj− PUj PUj×j+1PUj+1⊥ PbUj+1− PUj+1 PUj+1⊥×j+2PUj+2⊥ PbUj+2− PUj+2 PUj+2⊥ F ≤∥A∥F·" σ5 ξ λ5σ5 r3/2plog(p)3/2 n5/2+ ∆·p3/2Rlog(p) n5/2+ ∆2·p2Rlog(p) n5/2+ ∆3·p5/2 n5/2... | https://arxiv.org/abs/2501.16223v1 |
bounds for II, III. Proposition D.3. Under the same setting of Theorem 3.1, let Vj∈Rpj×Rjbe a fixed matrix satisfying ∥Vj∥= 1for any j= 1,2,3. Then, for any j= 1,2,3, the following bounds hold with probability at least 1−exp(−cn)−1 pC−P(E∆)−P[(Ereg U)], where candCare two universal constants: V⊤ jPUj⊥ PbUj− PUj Uj ≲σ... | https://arxiv.org/abs/2501.16223v1 |
first term I (172), by decomposing PbU(1) j=PbU(1) j− PUj+PUj, we have I≤ (U3⊗U2)⊤bZ⊤ 1U1 · U1(G1G1)−1 2U⊤ 1 · (G1G1)−1 2U⊤ 1 + sup W2∈Rp2×r3,∥W2∥=1 W3∈Rp3×r3,∥W3∥=1 U⊤ 1bZ1(W3⊗W2) (G1G1)−1 2U⊤ 1 · PbU(1) 2− PU2 + PbU(1) 3− PU3 + PbU(1) 2− PU2 · PbU(1) 3− PU3 ≲σξ λσ· r rlog(p) n+ ∆r p n! . (178) Note that the Froben... | https://arxiv.org/abs/2501.16223v1 |
the first term I (186), we have I≤ V⊤ 1PU1⊥bZ1(U3⊗U2) · G⊤ 1(G1G1)−1 2U⊤ 1 ≲σξ σ· s Rlog(p) n+ ∆·r p n (194) 34 and for the second term (187), we have II≤ V⊤ 1PU1⊥bZ1(PU3⊗ PU2) · (PU3⊗ PU2)bZ⊤ 1U1(G1G1)−1 2U⊤ 1 ≲σ2 ξ λσ2· √ Rrlog(p) n+ ∆2·p n! . (195) Then, consider the third term (188) III≤ V⊤ 1PU1⊥bZ1h SG1,1 bE... | https://arxiv.org/abs/2501.16223v1 |
III .II = V⊤ 1PU1⊥bZ1h PU3⊥bZ3 PbU(0) 2⊗ PbU(0) 1 bZ⊤ 3U3 G3G⊤ 3−1 ⊗U2i G⊤ 1(G1G1)−1 2U⊤ 1, we have III.II≤1 λ2sup W2∈Rp2×r2,∥W2∥=1 W3∈Rp3×r3,∥W3∥=1 V⊤ 1PU1⊥bZ1(W3⊗W2) sup W1∈Rp1×r1,∥W1∥=1 W2∈Rp2×r2,∥W2∥=1 PU3⊥bZ3(W2⊗W3) · sup W1∈Rp1×r1,∥W1∥=1 W2∈Rp2×r3,∥W2∥=1 U⊤ 3bZ3(W2⊗W1) ≲σ3 ξ λ2σ3·p3/2 n3/2. (204) 36 In addi... | https://arxiv.org/abs/2501.16223v1 |
it suffices to consider upper bounds of: I = U⊤ 1⊥bE1U1⊥−U⊤ 1⊥bZ1(PU3⊗ PU2)bZ⊤ 1U1⊥ II = V1V⊤ 1PU1⊥bE1U1⊥−V⊤ 1PU1⊥bZ1(PU3⊗ PU2)bZ⊤ 1U1⊥ III = U⊤ 1⊥bE1P−1 1−U⊤ 1⊥bZ1(U3⊗U2)G⊤ 1 G1G⊤ 1−1U⊤ 1 IV = V1V⊤ 1U⊤ 1⊥bE1P−1 1−V1V⊤ 1U⊤ 1⊥bZ1(U3⊗U2)G⊤ 1 G1G⊤ 1−1U⊤ 1 . It follows that I≤ U⊤ 1⊥bZ1h PbU(1) 3− PU3 ⊗ PU2i bZ⊤ 1U1⊥ ... | https://arxiv.org/abs/2501.16223v1 |
λ2σ2· rlog(p) n+ ∆·q Rplog(p) n+ ∆2·p n . In the following subsections, we established upper bounds for perturbation terms of varying orders in the spectral representation under the setting of tensor regression with sample splitting. In particular, we will show that the first-order perturbation term is the leadi... | https://arxiv.org/abs/2501.16223v1 |
j− PUj ≤σξ σq p nhold with probability at least 1 −exp(−cp)−P(Ereg U) for any j= 1,2,3. Besides, we assume that the initial error bound bTinit,(I)− T F≤∆, bTinit,(II)− T F≤∆ hold with probability at least 1 −P(E∆), where event E∆is given by E∆=n bTinit− T F>∆o , where bTinitcan be either bTinit,(I)orbTinit,(II). E.1 Pr... | https://arxiv.org/abs/2501.16223v1 |
Upper Bound of Higher-Order Perturbation Terms Lemma E.1. Under the same setting of Theorem 3.2, let Vj∈Rpj×Rjbe a fixed matrix satisfying ∥Vj∥= 1.bE(I) jis defined as in (217) , and SGj,kj(bE(I) j)is defined as in (218) , for any j= 1,2,3. Then with probability at least 1−exp(−cn)−1 pC−P(E∆)−P(Ereg U), where candCare ... | https://arxiv.org/abs/2501.16223v1 |
as in (217) . The proof of Lemma E.3 is similar to that of Lemma D.4, and is thus omitted. Lemma E.4. Under the same setting of Theorem 3.2, with probability at least 1−exp(−cn)−1 pC− P(E∆)−P(Ereg U), where candCare two universal constants ,it holds that A × jUj×j+1PUj+1⊥bE(I) j+1P−1 j+1×j+2PUj+2⊥bE(I) j+2P−1 j+2 F ≲∥A... | https://arxiv.org/abs/2501.16223v1 |
Furthermore, note that by Bernstein-type inequality, we have P nX i=1ξi ≥C1σξp nlog(p)! ≤1 pc. Therefore, we have P 1 n2σ2nX i=1ξia⊤xix⊤ i(C⊗B)xi ≥K3 n3/2σ2 p1p2p3X k=1akDk,k +1 n3/2∥a∥ℓ2 |tr(D)|t1/2+∥D∥Ft2/3! ≤exp (−ct) +1 pc. Then consider the off-diagnoal terms, g(x) =nX i=1nX j=1,j̸=iξja⊤xix⊤ i(C⊗B)xj 46 =nX i=1n... | https://arxiv.org/abs/2501.16223v1 |
follows that (⟨A, X⟩)2tr BX⊤CX −Eh (⟨A, X⟩)2tr BX⊤CXi ψ1 2≤ (⟨A, X⟩)2 ψ1· tr BX⊤CX ψ1 ≲∥⟨A, X⟩∥2 ψ2· tr BX⊤CX −σ2tr (B) tr (C) ψ1+ σ2tr (B) tr (C) ≤σ2∥A∥2 F· σ2∥B∥F· ∥C∥F+σ2|tr (B) tr (C)∥ , where we used E tr BX⊤CX =σ2tr (B) tr (C) and ∥EX∥ψ1≲|EX|. Therefore, by Bernstein-type inequality, we have P 1 ... | https://arxiv.org/abs/2501.16223v1 |
3 ∥ξ∥2 ℓ2!# . (243) Furthermore, we have P PUj+2⊗ PUj+1bZ(1)⊤ jPUj⊥Aj PUj+1⊥bZ(1) j+1 PUj⊗ PUj+2 ⊗ PUj+2⊥bZ(1) j+2 PUj+1⊗ PUj ≥C·1 n3/2∥A∥F·t ≤7rj+rj+1rj+2exph −cmin t2, t2 3i . (244) Proof. By symmetry, it suffices to find a high-probability upper bound for (PU3⊗ PU2)bZ(1)⊤ 1PU1⊥A1 PU3⊥bZ(1) 3(PU2⊗ PU1)⊗... | https://arxiv.org/abs/2501.16223v1 |
3(PU2⊗ PU1)⊗ PU2⊥bZ(2) 2(PU1⊗ PU3) F| {z } VIII. Here, it follows from (244) in Lemma F.3 that I≲∥A∥F· σ3 ξ σ3·rlog(p)3/2 n3/2! | {z } (244). Then consider II≤ (PU3⊗ PU2)bZ(2)⊤ 1PU1⊥ · ∥A1∥F· V⊤ 3PU3⊥bZ(1) 3(PU2⊗ PU1) · V⊤ 2PU2⊥bZ(1) 2(PU2⊗ PU1) ≲∆·σξ σr p n· ∥PU1⊥A1(PU3⊥⊗ PU1⊥)∥F· σξ σs Rlog(p) n 2 = ∆·σ3 ξ σ3·p1... | https://arxiv.org/abs/2501.16223v1 |
σ2·Rlog(p) n· ∥A × 1U1∥F, (256) IV≲∆2·σ2 ξ σ2·Rlog(p) n· ∥A × 1U1∥F. (257) Combining the results above, we obtain the second inequality. Lemma F.7. Suppose that B∈Rpj+2pj+1×pj+2pj+1andC∈Rpj×pjare two fixed matrices. Let bZ(1) j=1 nσ2Pn i=1ξiMat j(Xi), where {ξi}n i=1’s are i.i.d. mean zero σξ-sub-Gaussian variables and... | https://arxiv.org/abs/2501.16223v1 |
1−P(E∆), it holds that P 1 nσ2nX i=1D Xi,e∆E eU⊤ jMat j(Xi) eUj+2⊗eUj+1 −eU⊤ je∆j eUj+2⊗eUj+1 ≥Cσ2·∆t! ≤2·7er1+er2er3e−cmin{nt2,nt}+P(E∆). (262) Proof. By symmetry, it suffices to consider 1 nσ2nX i=1hD Xi,e∆E eU⊤ 1Mat 1(Xi) eU3⊗eU2 −σ2·eU⊤ 1Mat 1 e∆ eU3⊗eU2i . For any fixed a∈Rer1,∥a∥= 1 and b∈Rer2er3,∥b∥= 1... | https://arxiv.org/abs/2501.16223v1 |
nσ2nX i=1⟨Xi,∆⟩eU⊤ 1Mat 1(Xi) eU3⊗eU2 −eU⊤ 1∆1 eU3⊗eU2 ≥C′σ2·∆ 1−εt! ≤2·8 +ε εeer1eer2eer3+P3 j=1pjeerj ·7er1+er2er3·e−c1min{nt2,nt}. Choose ε=1 2, we obtain the desired bound. Lemma F.11. Suppose X ∈Rp1×p2×p3is a tensor with independent zero-mean σ-sub-Gaussian entries andX1,···,Xnare i.i.d. copies of X.eUj’s,j=... | https://arxiv.org/abs/2501.16223v1 |
= trh (PU3⊗ PU2)A⊤ 1U1 G1G⊤ 1−1G1(U3⊗U2)⊤ PbU(1) 3⊗ PbU(1) 2 Z⊤ 1PU1⊥Z1i | {z } I(264) + trh (PU3⊗ PU2)A⊤ 1U1 G1G⊤ 1−1U⊤ 1Z1 PbU(1) 3⊗ PbU(1) 2 Z⊤ 1PU1⊥Z1i | {z } II(265) + trh (PU3⊗ PU2)A⊤ 1PU1⊥Z1 PbU(1) 3⊗ PbU(1) 2 (U3⊗U2)G⊤ 1 G1G⊤ 1−1U⊤ 1Z1i | {z } III(266) + trh (PU3⊗ PU2)A⊤ 1PU1⊥Z1 PbU(1) 3⊗ PbU(1) 2... | https://arxiv.org/abs/2501.16223v1 |
IV (267), we have IV≤1 λ2· (PU3⊗ PU2)A⊤ 1PU1⊥Z1(U3⊗U2) F· (U3⊗U2)Z⊤ 1U1 · U⊤ 1Z1(PU3⊗ PU2) F + (PU3⊗ PU2)A⊤ 1PU1⊥ F· sup W2∈Rp2×r2,∥W2∥=1 W3∈Rp3×2r3,∥W3∥=1∥PU1⊥Z1(W3⊗W2)∥F · PbU(1) 2− PU2 + PbU(1) 3− PU3 +3Y j=2 PbU(1) j− PUj · (U3⊗U2)Z⊤ 1U1 2 ≲∥A × 1U1×2U2×3U3∥F· σ3r1/2·r3/2log(p)3/2 λ2+σ3r1/2·prlog(p) λ3! . (28... | https://arxiv.org/abs/2501.16223v1 |
× 1PbU1×2PbU2×3PbU3− T,AE similar to the arguments in Step 2 in the proof of Theorem 3.1. In Step 2, we will consider (Step 2.1) :D T × 1 PbU1− PU1 ×2PU2×3PU3,AE (Step 2.2) :D T × 1 PbU1− PU1 ×2 PbU2− PU2 ×3PU3,AE (Step 2.3) :D T × 1 PbU1− PU1 ×2 PbU2− PU2 ×3 PbU3− PU3 ,AE , Step 2.1: Upper Bound of Negligi... | https://arxiv.org/abs/2501.16223v1 |
aim to show the normal approximation of ⟨Z,PTTMr(A)⟩.To apply the Berry-Essen theorem, we calculate its second and third moments. Step 3.1: Second Moment of Asymptotic Normal Terms Clearly, EZ[⟨Z,PTTMr(A)⟩]2=σ2·3X j=1 PUj⊥AjP(Uj+2⊗Uj+1)G⊤ j 2 F+σ2· ∥A × 1PU1×2PU2×3PU3∥2 F. Step 3.2: Third Moment of Asymptotic Normal Te... | https://arxiv.org/abs/2501.16223v1 |
that the following events hold with high probability: PbU(0) j− PUj ≤σp p holds with probability at least 1 −P EPCA U , where event EPCA Uis defined by EPCA U=n PbU(0) j− PUj > σ√po . Then by Lemma J.2, we know that PbU(1) j− PUj ≤σ√pand PbUj− PUj ≤σ√pholds with proba- bility at least 1 −exp(−cp)−P(EPCA U) for any j=... | https://arxiv.org/abs/2501.16223v1 |
2U⊤ j ≲σ·p p, (309) P0 jEjP0 j = U⊤ j⊥EjUj⊥ ≲σ2·p. (310) for each j= 1,2,3, where P−1 2 j=Uj GjG⊤ j−1 2U⊤ jandP0 j=Uj⊥U⊤ j⊥, and Ejis defined by (296) . The proof of Lemma H.1 is similar to that of Lemma D.1, and is thus omitted. Lemma H.2. Under the same setting of Theorem 4.1, let Vj∈Rpj×Rjbe a fixed matrix satisfy... | https://arxiv.org/abs/2501.16223v1 |
let Vj∈Rpj×Rjbe a fixed matrix satisfying ∥Vj∥= 1, and let Ejbe defined by (296) . Then with probability at least 1−exp(−cn)−1 pc−P(E∆)−P EPCA U , where candCare two universal constants ,it holds that U⊤ j⊥EjUj⊥−U⊤ j⊥Zj PUj+2⊗ PUj+1 Z⊤ jUj⊥ ≲σ3p3/2 λ, (317) V⊤ jPUj⊥EjUj⊥−VjV⊤ jPUj⊥Zj PUj+2⊗ PUj+1 Z⊤ jUj⊥ ≲σ3·pq R... | https://arxiv.org/abs/2501.16223v1 |
= ( i1, i2, i3) = (j1, j2, j3) = (k1, k2, k3) Case II := ( i1, i2, i3) = (j1, j2, j3)̸= (k1, k2, k3) Case III := ( i1, i2, i3) = (k1, k2, k3)̸= (j1, j2, j3) Case IV := ( j1, j2, j3) = (k1, k2, k3)̸= (i1, i2, i3) Case V := ( i1, i2, i3)̸= (j1, j2, j3)̸= (k1, k2, k3). Correspondingly, write u⊤B⊤ 1Z⊤ 1C⊤ 1(A3Z3B3⊗A2Z2B2)v... | https://arxiv.org/abs/2501.16223v1 |
It then follows that a⊤B⊤ 1Z⊤ 1C⊤ 1(A3Z3B3⊗A2Z2B2)v=eaU⊤ 1B⊤ 1Z⊤ 1C⊤ 1(A3Z3B3⊗A2Z2B2) (V2⊗V3)eb. By Lemma 5.2 of Vershynin [41] , there exists Nr1, a1 3-net of {a∈Rr1:∥A∥= 1}, such that |Nr1| ≤ 7r1andNr2r3, a1 3-net of {b∈Rr2r3:∥b∥2= 1}, such that |Nr2r3| ≤7r2r3. Then, applying a union bound, we have P sup a,∈NR2 b∈... | https://arxiv.org/abs/2501.16223v1 |
¨Ci k,l[Z1]l,i. By the decoupling method (Remark 6.1.3, Vershynin [42]) and the same arguments as in the proof of the first inequality, we have EZexp [λSindep]≤EZexpC λh ¨Bi i,j[Z1]k,jh ¨Ci k,l[Z′ 1]l,i . Then by the comparison lemma (Lemma 6.2.3, Vershynin [42]), when λ2≤c2 σ4max i,kh ¨Bi2 max k,lh ¨Ci2=c2 σ4 ¨B 2 ℓ... | https://arxiv.org/abs/2501.16223v1 |
jmatricization of ZandeUj∈Rpj×erjsuch that ∥eUj∥= 1. Then, the following inequalities hold: P sup eUj∈Rpj×erj, ∥eUj∥≤1, j=1,2,3 eU⊤ jZj eUj+2⊗eUj+1 ≥4σp er1+t ≤2·332P3 j=1pjerj5erj+1erj+2exp −cmin t2 erj,t Q3 j=1 eUj 2,∞ . (333) The proof of Lemma I.6 is similar to that of Lemma I.5, and is... | https://arxiv.org/abs/2501.16223v1 |
sample-splitting case where the setting is specified in Theorem 3.2, let the estimate of variance components bσ2 ξ,bσ2andbs2 Abe defined as in (14),(15) and(16), respectively. Then we have eΩ1=3X j=1 U⊤ jAjP(Uj+2⊗Uj+1)G⊤ j F·σ2 ξr1/2 λσ2·p n+3X j=1∥A × j+1Uj+1×j+2Uj+2∥F·σ2 ξr1/2 λσ2 r plog(p) n+ ∆·p n! +3X j=1∥A × jUj∥... | https://arxiv.org/abs/2501.16223v1 |
bU3⊗bU2⊤ −(U3⊗U2)W1W⊤ 1(U3⊗U2)⊤ 2 F| {z } II2 + PbUj⊥− PUj⊥ Aj bU3⊗bU2 cW1cW⊤ 1 bU3⊗bU2⊤ −(U3⊗U2)W1W⊤ 1(U3⊗U2)⊤ 2 F| {z } III2, where P\(Uj+2⊗Uj+1)G⊤ j= bUj+2⊗bUj+1 cWjcW⊤ j bUj+2⊗bUj+1⊤ . Here, we have I≤ bPU1⊥− PU1⊥ PU1A1P(U3⊗U2)G⊤ 1 F+ bPU1⊥− PU1⊥ PU1⊥A1P(U3⊗U2)G⊤ 1 F ≲σξ λσr p n· PU1A1P(U3⊗U2)G... | https://arxiv.org/abs/2501.16223v1 |
+∥A∥F" σ3 ξ λ2σ3· Rlog(p)3/2 n3/2+ ∆2·p plog(p) n!#) . Compared with the terms Ω 1,Ω2,Ω3, and Ω 4in Theorem 3.1, we have the desired results. Part 2: With Sample Splitting In the sample-splitting case, we consider bσ2 ξ=1 nnX i=1ξ2 i+1 nn1X i1=1D b∆(II),Xi1E2 +1 nn2X i2=1D b∆(I),Xi2E2 +2 nn1X i1=1ξiD b∆(II),Xi1E +2 nn2... | https://arxiv.org/abs/2501.16223v1 |
show 3X j=1 PbUj⊥AjP\(Uj+2⊗Uj+1)G⊤ j 2 F− PUj⊥AjP(Uj+2⊗Uj+1)G⊤ j 2 F ≲3X j=1∥A × j+1Uj+1×j+2Uj+2∥2 Fσ2p λ2+3X j=1∥A × jUj∥2 Fσ2Rlog(p) λ2+σ6p3 λ6 +∥A∥2 F σ4R2log(p)2 λ4+σ12p6 λ12! . Step 3: Summary of results On the other hand, we have ⟨A,bT ⟩ − ⟨A ,T ⟩ ≤ D PTTM(r1,r2,r3)(A),ZE + (Ω 1+ Ω 2+ Ω 3)≲sAσp logp+ (Ω 1+ Ω 2+... | https://arxiv.org/abs/2501.16223v1 |
1⊥A1P(U3⊗U2)G⊤ 1(U3⊗U2) 4 F σ2∥T ∥2 F+σ2 ξ2+2σ4·σ4ε4 U⊤ 1⊥A1P(U3⊗U2)G⊤ 1(U3⊗U2) 4 F σ2∥T ∥2 F+σ2 ξ3 ∥T ∥2 F σ2+∥T ∥4 F σ2 ξ! +2ε2σ2 U⊤ 1⊥A1P(U3⊗U2)G⊤ 1(U3⊗U2) 2 F σ2∥T ∥2 F+σ2 ξ ≲2·ε2σ4 U⊤ 1⊥A1P(U3⊗U2)G⊤ 1 2 F σ2 ξ+ 2·σ6ε4 U⊤ 1⊥A1P(U3⊗U2)G⊤ 1 4 F σ4 ξ. By Theorem 1.1 of Devroye et al. [14], we have DTV fπH1, fπH0... | https://arxiv.org/abs/2501.16223v1 |
2λ+ 2|a| ≤κλ. It only remains to find an upper bound for T − T ,A . Note that now T − T ,A =|a| · ∥G × × 1PU1⊥∆1×2U2×3U3∥F+∥A × 1PU1×2PU2×3PU3∥F =|a| · PU1⊥A1P(U3⊗U2)G⊤ 1 F+∥A × 1PU1×2PU2×3PU3∥F . Thefore, we have inf CIα A(T,D)∈Iα(Θ,A)sup T ∈Θ(λ,κ)EL(CIα A(T,D)) ≥ |a| · PU1⊥A1P(U3⊗U2)G⊤ 1 F+∥A × 1PU1×2PU2×3PU3∥F... | https://arxiv.org/abs/2501.16223v1 |
jekj ℓ2. N Summary of Inference Procedure N.1 Inference procedure for tensor regression We first outlines the steps to estimate the linear functional ⟨A,T ⟩for tensor regression when the entire dataset is used jointly without splitting. The summarized procedure is detailed in Algorithm 1. Fur- thermore, with sample spl... | https://arxiv.org/abs/2501.16223v1 |
2 F+ A × 1bU1×2bU2×3bU3 2 F, be the estimate of estimate of variance component s2 A, where Ujis either bU(I) jorbU(II) jand cWj= QR Mat j bT × 1bU⊤ 1×2bU⊤ 2×3bU⊤ 3⊤ . 14:Confidence Interval The 100(1 −α)% confidence interval is given by cCIα A,T="D bT,AE −zα/2·bσξ bσ·bsAr 1 n,D bT,AE +zα/2·bσξ bσ·bsAr 1 n# , where ... | https://arxiv.org/abs/2501.16223v1 |
arXiv:2501.16287v3 [cs.IT] 31 Jan 2025A Unified Representation of Density-Power-Based Divergences Reducible to M-Estimation Masahiro Kobayashi Information and Media Center Toyohashi University of Technology Toyohashi, Aichi, Japan Email: kobayashi@imc.tut.ac.jp Abstract —Density-power-based divergences are known to pro-... | https://arxiv.org/abs/2501.16287v3 |
Furthermore, they derived the Bregman– Hölder divergence (BHD) as the intersection of the HD and Bregman divergence [10]. The above the general divergences all belong to the class of non-kernel divergences [11], which enable the substitut ion of the empirical distribution. However, only a more restric ted subset of the... | https://arxiv.org/abs/2501.16287v3 |
dν(x), (3) whereΦ∗ p:S→R, S⊆Rdis a subgradient of the strictly convex functional Φatp. The Bregman divergence can be reduced to M-estimation (1) [17]. When the strictly convex functional Φis defined by a strictly convex function ϕ:R+→R, i.e.,Φ(q) =∝an}bracketle{tϕ(q)∝an}bracketri}ht, the Bregman divergence (3) is referr... | https://arxiv.org/abs/2501.16287v3 |
inequality holds: φγ/parenleftbig ∝bardbltq+(1−t)p∝bardbl1+γ/parenrightbig(a) < φγ/parenleftbig t∝bardblq∝bardbl1+γ+(1−t)∝bardblp∝bardbl1+γ/parenrightbig (b) ≤tφγ/parenleftbig ∝bardblq∝bardbl1+γ/parenrightbig +(1−t)φγ/parenleftbig ∝bardblp∝bardbl1+γ/parenrightbig , where (a) follows from the strict convexity of the L1+... | https://arxiv.org/abs/2501.16287v3 |
λ1≥0,λ2≥0λ1= 0,λ2/negationslash= 0: scaled PS (PSBDP) [8], [9] λ1/negationslash= 0,λ2→0: scaled DP Combination of BH and PSBDP (17)1 λ2/bracketleftBig/parenleftbig λ1+λ2z1+γ/parenrightbigκ 1+γ−λκ 1+γ 1/bracketrightBig λ1≥0,λ2≥0,λ1= 0,λ2/negationslash= 0: scaled BH λ1/negationslash= 0,λ2→0: scaled DP λ2/negationslash= 0... | https://arxiv.org/abs/2501.16287v3 |
corresponding cross-entropy is given by dλ,κ,γ(q,p) =κ λ2γ/parenleftBig λ1+λ2/angbracketleftbig p1+γ/angbracketrightbig/parenrightBigκ 1+γ ·/bracketleftBigg 1−1 κ−λ1+λ2/angbracketleftbigqpγ/angbracketrightbig λ1+λ2/angbracketleftp1+γ/angbracketright/bracketrightBigg +1 λ2γ(λ1+λ2)κ 1+γ.(17) Whenλ1= 0,λ2∝ne}ationslash= 0... | https://arxiv.org/abs/2501.16287v3 |
In other words, if the influence function is finite, the estimator can be considered robust. Generally, the influence function of an M-estimator is proportional to ψin estimating equation (1). For γ >0, ψof NB-DPD (12) is bounded, which implies robustness whenγ >0. The redescending property is a desirable feature that ena... | https://arxiv.org/abs/2501.16287v3 |
Basu, “Characterizing the functional density power divergence class,” IEEE Transactions on Information Theory , vol. 69, no. 2, pp. 1141–1146, 2023. [8] A. K. Kuchibhotla, S. Mukherjee, and A. Basu, “Statistic al inference based on bridge divergences,” Annals of the Institute of Statistical Mathematics , vol. 71, no. 3... | https://arxiv.org/abs/2501.16287v3 |
(19) DFDP v,γ(q,p) =1 γv/parenleftbig ∝an}bracketle{tq1+γ/angbracketrightbig )−1+γ γv(∝an}bracketle{tqpγ∝an}bracketri}ht)+v/parenleftbig/angbracketleftbig p1+γ/angbracketrightbig/parenrightbig , (20) respectively. When γ= 0, the FDPCE and FDPD are defined as dFDP v,γ(q,p) =−v′(1)∝an}bracketle{tqlogp∝an}bracketri}ht, DFD... | https://arxiv.org/abs/2501.16287v3 |
λ2γ/parenleftBig λ1+λ2/angbracketleftbig q1+γ/angbracketrightbig/parenrightBigκ 1+γ. Whenλ1= 0,λ2∝ne}ationslash= 0,κ≥1, (23) reduces to a constant multiple of the BHCE (14) as follo ws: dλ,κ,γ(q,p) =κ γλκ 1+γ−1 2/angbracketleftbig p1+γ/angbracketrightbigκ 1+γ/bracketleftBigg 1−1 κ−/angbracketleftbig qpγ/angbracketright... | https://arxiv.org/abs/2501.16287v3 |
θsθ/angbracketrightbig/parenleftBig λ1+λ2pθ(x)γ/parenrightBig . In this case, v(z) =alog/parenleftbig λ1+λ2z/parenrightbig +bwitha >0andb∈R, which corresponds to the BDPD [8], [9]. From the above observations, we conclude that among the FDPD, only the BDPD can be reduced to M-estimation. Therefore, the intersectio n be... | https://arxiv.org/abs/2501.16287v3 |
A VARIFOLD-TYPE ESTIMATION FOR DATA SAMPLED ON A RECTIFIABLE SET CHARLY BORICAUD AND BLANCHE BUET Abstract. We investigate the inference of varifold structures in a statistical framework: assuming that we have access to i.i.d. samples in Rnobtained from an underlying d–dimensional shape Sendowed with a possibly non uni... | https://arxiv.org/abs/2501.16315v1 |
obtain convergence of the geometric estimators in strong enough topology (pointwise convergence in Sconcerning tangent spaces or second fundamental form and Hausdorff distance for estimating Sitself). While the Ck–manifold ( k≥2) setting has been well–investigated, handling lower regularity frameworks is a natural next... | https://arxiv.org/abs/2501.16315v1 |
us describe more precisely both the density and the tangent estimation leading to the reconstruction of the varifold structure. 1.1.Reconstruction of the varifold structure. Density and measure estimation. Note that if we were considering the case of a probability measure µ=θLnwhich is absolutely continuous with respec... | https://arxiv.org/abs/2501.16315v1 |
approximation of the varifold structure. More specifically, we believe that infering the varifold structure would allow to infer curvature information thanks to the approximations that have been proposed in [BLM17] and [BLM22]. Piecewise H¨ older regularity class and main result. As mentioned above, the convergence res... | https://arxiv.org/abs/2501.16315v1 |
obtain an explicit upper bound of the minimax risk RN(P) concerning the inference of the varifold structure with respect to the Bounded Lipschitz distance β. According to us, a very important point achieved with such a result is the following: it proves that it is possible to obtain minimax convergence results for reco... | https://arxiv.org/abs/2501.16315v1 |
the reach, Definition 6.2 however requires that Sis ad–manifold (only in this particular case where S=∅) and furthermore a C1,agraph in any ball of radius less than Rwith uniform H¨ older constant. On one hand, such assumptions prevent that parts of Scould get arbitrarily close in the ambient space while distant in the... | https://arxiv.org/abs/2501.16315v1 |
with respect to a distance dHγsimilar to the bounded Lipschitz distance but adding a H¨ older condition on the first order derivative of test functions. Such minimax rates are valid in a manifold framework and it would be worth understanding wether such manifold regularity can be relaxed to establish minimax rates in a... | https://arxiv.org/abs/2501.16315v1 |
(98): strengthening the regularity setting allows to obtain uniform convergence as stated in the main result Theorem 7.4. Notations We fix n∈N,n≥1 and d∈R, 0< d≤n. •In Sections 2 to 4, dis real, unless otherwise specified. In Sections 5 to 7, dis an integer. Generally speaking, rectifiability and d–varifold structure r... | https://arxiv.org/abs/2501.16315v1 |
space of Radon measures. We will also consider a localized version that allows to compare measures in a given open set D: Definition 2.4 (Localized Bounded Lipschitz distance) .Letλ1,λ2be Radon measures in Xand let D⊂Xbe an open set, we consider βD(λ1, λ2) = sup Z Xf dλ 1−Z Xf dλ 2 :f∈Cc(X,R),∥f∥∞≤1,Lip(f)≤1,supp f⊂D... | https://arxiv.org/abs/2501.16315v1 |
compact set Kisd–Ahlfors regular (since fis Lipschitz in K). We refer to [DS97] for additional examples and properties connected to d–Ahlfors regularity. The following proposition (Proposition 2.8) gives a bound on the number of balls with common diameter δneeded to cover the support of a d–Ahlfors measure. Such estima... | https://arxiv.org/abs/2501.16315v1 |
2.11 (Rectifiable measures, [AFP00] Definition 2.59) .Letµbe a Radon measure in Rn. We say that µisd–rectifiable if there exist a countably d–rectifiable set S⊂Rnand a Borel function θ:S7→R+such that µ=θHd |S. In other words, a d–rectifiable set Sis included in a countable union of d–dimensional Lipschitz graphs up to ... | https://arxiv.org/abs/2501.16315v1 |
model discrete geometric objects as well. In our context, we are more specifically in point cloud varifolds (see [BLM17]) that are associated with sets of points: Definition 2.16 (Point cloud varifold) .LetN∈N∗and assume that we are given: a finite set of points {xi}i=1...N⊂RN, associated with positive weights (mi)i=1.... | https://arxiv.org/abs/2501.16315v1 |
obtained by replacing the eigenvalues of Awith 1 for λ1, . . . , λ dand 0 for the others, while not changing the eigenspaces. The Lipschitz continuity of ˜ sinVis a direct consequence of Davis–Kahan Theorem ([DK70], see also Theorem 2 in [YWS14]): let A, B∈ V, by Davis–Kahan Theorem, ∥˜s(A)−˜s(B)∥F≤2√ 2∥A−B∥F |λd+1(A)−... | https://arxiv.org/abs/2501.16315v1 |
( H2) then the measure ν:=Hd |Sisd–Ahlfors regular with regularity constant fC0, supp ν=S, the measure µisd–Ahlfors regular with regularity constant C0=fC0max( θ−1 min, θmax) and Sis bounded even if not required. (ii) IfSandθsatisfy ( H1), it would be equivalent to require that µ=θHd |Sisd–Ahlfors regular or ( H2). (ii... | https://arxiv.org/abs/2501.16315v1 |
can then use independence when the four indices are all different and enumerate the other cases, which leads to E µN(A)2µN(B)2 ≤µ(A)2µ(B)2+1 Nµ(A)µ(B) (µ(A) +µ(B) + 4µ(A∩B)) +1 N2 µ(A)µ(B) + 2µ(A∩B)2 +2 N2µ(A∩B) (µ(A) +µ(B)) +1 N3µ(A∩B) ≤µ(A)2µ(B)2+6 Nµ(A)µ(B)2+7 N2µ(B)2+1 N3µ(B) where we used µ(A∩B)≤µ(A)≤µ(B) and ... | https://arxiv.org/abs/2501.16315v1 |
of radial kernels in the usual way: we fix a Lipschitz even function η:R7→R+with support in ( −1,1) and we additionally assume that η >0 in −1 2,1 2 and we define (18) Cη=dωdZ1 r=0η(r)rd−1drand for δ >0, x∈Rn, ηδ(x) =η|x| δ . We introduce the following notations that will be used throughout the next sections: given... | https://arxiv.org/abs/2501.16315v1 |
it will be however necessary in Lemma 4.7 and thus in Theorem 4.9 as well. Combining Proposition 3.5 and Proposition 3.6 yields the mean con- vergence of the kernel estimator of density θδ,Nprovided that δ=δN→0 is well-chosen, as stated in Corollary 3.7. Corollary 3.7. Letd∈N∗and assume that Sandθsatisfy (H1),(H2)and(H... | https://arxiv.org/abs/2501.16315v1 |
some technical assumption ensuring that µbehaves like a d–dimensional measure in Rn, the author established that (23) E[β(µN, µ)]≲1 Nd. In order to show the convergence of νδ,Ntowards ν=Hd |S, we adapt the arguments of [Dud69] but passing from 1–Lipschitz and bounded by 1 test functions to random test functions with on... | https://arxiv.org/abs/2501.16315v1 |
× mX j=1|µN(Sj)−µ(Sj)|2 1 2 and hence, using Cauchy-Schwartz inequality E[X1 2Y1 2]≤E[X]1 2E[Y]1 2for non-negative random variables X, Y, we get (30) E mX j=1µN(B(xj, R))|µN(Sj)−µ(Sj)| ≤E mX j=1µN(B(xj, R))2 1 2 ×E mtX j=1(µN(Sj)−µ(Sj))2 1 2 . From (27), (28) and (30) we infer (25). Then, (26) similarly... | https://arxiv.org/abs/2501.16315v1 |
Hence directly applying Theorem 3.2 in [Dud69] with the “deterministic” estimate (34) above is possible and we would obtain E[β(νδ,N, νδ)]≲N−1 d δ2d+1. However, applying the following Proposition 4.3–Corollary 4.10 instead, we obtain in Theorem 4.9( i) the significantly improved rate (by a factor δ2d): E[β(νδ,N, νδ)]≲N... | https://arxiv.org/abs/2501.16315v1 |
all N∈N∗,δ >0satisfying N−1 d≤δwe have E[βB(h µN, h µ)]≤M κ0+κ1 δ N−1 dµ(BγN)withγN=N−d−2 d2− − − − − → N→+∞0. and in particular for all N∈N∗, (38) E[βB(µN, µ)]≤MN−1 dµ(BγN)withγN=N−d−2 d2− − − − − → N→+∞0. (ii)caseBopen ball: there exists a constant M=M(d, C0)>0such that for any open ball B⊂Rnof radius 0< R B<1and f... | https://arxiv.org/abs/2501.16315v1 |
j, diam At j≤3εt≤9ε= 9N−1 d≤9δ, therefore B(x, δ)∪B(xt j, δ)⊂B(x,10δ) so that ∆ δ,N(x, xt j)≤δ−dµN(B(x,10δ)) and similarly ∆ r,N(x, xt j)≤ r−dµN(B(x,10r)). Using in addition (31) and (36) we obtain Z Bg(dµN−dµ) = mtX j=1Z At jg(x)d(µN−µ)(x) ≤It+ mtX j=1Z At j g(x)−g(xt j) d(µN−µ)(x) ≤It+mtX j=1Z At j κ0+κ1 δd+1µN(B(... | https://arxiv.org/abs/2501.16315v1 |
( ii). Indeed, thanks to (42) and (43) we can infer that Intermediate conclusion: (48) E sup Z Bg dµ N−Z Bg dµ :g∈X ≤M κ0+κ1+κ3 δ+κ2 r εµ(B) +M N1 2" κ0κε−d 2s+ κ0+κ1+κ3 δ+κ2 r tX u=s+1ε−d 2+1 u# µ Bεs 4 . Step 4:We are left with estimatingtX u=s+1ε−d 2+1 u whose computation depends on the sign of −d 2+ 1. 2N... | https://arxiv.org/abs/2501.16315v1 |
to design an estimator of ν=Hd |S. More precisely, given N∈N∗andδ >0, we introduce the measure νδas well as the random measure νδ,Nobtained by attributing weights to the points of the sample ( X1, . . . , X N) as follows: (55) νδ,N=1 NNX i=1Φ(θδ,N(Xi))δXi= (Φ◦θδ,N)µNand νδ= (Φ◦θδ)µ A VARIFOLD-TYPE ESTIMATION FOR DATA S... | https://arxiv.org/abs/2501.16315v1 |
− → δ→0+θa.e. in Sthanks to Proposition 3.5 and by (58) 1 θδ−1 θ ≤2 min S. As µ(B)<+∞, we conclude by dominated convergence that (59) βB(νδ,Hd |S)≤ |νδ− Hd |S|(B)≤Z B 1 θδχτ(θδ)−1 θ dµ=Z B∩S 1 θδ−1 θ dµ− − − − → δ→0+0. □ Note that we similarly have the uniform upper bound, for x∈S: (60) θδ(x) =1 CηδdZ B(x,δ)∩Sη|x−y| δ... | https://arxiv.org/abs/2501.16315v1 |
2 N−1 2− − − − − → N→+∞0 ifd <2. Then, there exists a constant m=m(d, C0,Lip(η))∈(0,1)only depending on d,C0andηsuch that for any 0< τ≤m, Eh βB(νδN,N,Hd |S)i − − − − − → N→+∞0. Remark 4.11 (Choice of the parameter τ).It is possible to assume that τis fixed and chosen so that τ∈m 2, m (with the notation of Proposition... | https://arxiv.org/abs/2501.16315v1 |
without uniform rates nor uniform choice of ( δN)N, (rN)N→0 in the regularity class Q. Handling such a lack of uniformity issue is then the purpose of Sections 6 and 7. In Section 5.2, we build upon such tangent space estimators to introduce two varifold– type estimators Wr,δ,N andfWr,δ,N (see (76)) of WS=Hd |S⊗δΠTxS. ... | https://arxiv.org/abs/2501.16315v1 |
− − − → r→0+θ(x)ΠTxS where we recall that ΠTxSis the matrix of orthogonal projection on the approximate tangent space TxS. In particular, forν=Hd |S,Σr(x, ν)− − − − → r→0+ΠTxS. Proof. Forr >0 and x∈S(such that TxSexists), we have by translation and dilation: Σr x,Hd |x+TxS =1 CφrdZ Rn∩TxSψy r dHd(y) =1 CφZ Rn∩TxSψ(... | https://arxiv.org/abs/2501.16315v1 |
r(x, νδ) can be inferred provided that δ,rtending to 0 additionally satisfy1 rdR B(x,r)|θδ−θ|dµ→0, as done in the proof of Proposition 5.6. Proof. We recall that Φ( θ(x))θ(x) = 1 (see Proposition 4.8) and Σ r(·, µ) is uniformly bounded byC0∥φ∥∞ Cφ(see (66)). Hence, if there exists an approximate tangent space TxSatx∈S,... | https://arxiv.org/abs/2501.16315v1 |
manage to directly obtain concentration as we did previously with σr,δ,N. However, thanks to the property of ψgiven in (65), we have that for r >0 and x∈Rn, the function f:y7→1 Cφrdψy−x r satisfies Lip(f)≤1 Cφrd1 r(∥φ∥∞+ Lip( φ)) and ∥f∥∞≤1 Cφrd∥φ∥∞and supp( f)⊂B(x, r), and we can directly apply Theorem 4.9( ii) in t... | https://arxiv.org/abs/2501.16315v1 |
all N∈N∗,0< r, δ < 1satisfying N−1 d≤min(δ, r, R B), (78)E[βB(Wr,δ,N, Wr,δ)] E[βB(fWr,δ,N,fWr,δ)] ≤M min(δ, r)µ(B)× N−1 d ifd >2 N−1 2lnNifd= 2 N−1 2 ifd <2. Proof. We recall that M≥0 stands for a generic constant that may vary from one line to another (see Remark 5.1). As a first step, we prove the result for Wr,... | https://arxiv.org/abs/2501.16315v1 |
we have ∥Σr(x, νδ,N)−Σr(y, νδ,N)∥ ≤1 CφrdZ z∈Rn∥ψr(z−x)−ψr(z−y)∥|Φ (θδ,N(z))|dµN(z) ≤M r∆r,N(x, y)|y−x|. (87) Coming back to g, we have thanks to (83) and (87), |g(x)−g(y)| ≤|f(x,Σr(x, νδ,N))||Φ (θδ,N(x))−Φ (θδ,N(y))|+|Φ (θδ,N(y))||f(x,Σr(x, νδ,N))−f(y,Σr(y, νδ,N))| ≤∥f∥∞M δ∆δ,N(x, y)|y−x|+∥Φ∥∞Lip(f) 1 + Lip(Σ r(·, νδ... | https://arxiv.org/abs/2501.16315v1 |
≤1 rdZ y∈Dr|θδ(y)−θ(y)|ν(B(y, r))dµ(y) ≤MZ Dr|θδ−θ|dµ , and then recalling (70), we have A2≤Lip(f)Z D∥Σr(x, νδ)−ΠTxS∥dν(x)≤MZ Dr|θδ−θ|dµ+Z D∥Σr(x, ν)−ΠTxS∥dν . (97) We obtain from (95), (96) and (97): βD fWr,δ, WS ≤MZ Dr|θδ−θ|dµ+MZ D∥Σr(x, ν)−ΠTxS∥dµ(x), whence together with (97) we obtain (91) as well. Moreover, for... | https://arxiv.org/abs/2501.16315v1 |
a numerical perspective) truncation process in the eigen decomposition. We conclude Section 6 with Theorem 6.15 that state the mean convergence of Wr,δ,N,fWr,δ,N andVr,δ,N,eVr,δ,N toWSconsistently with the rates established for the deterministic part in Proposition 6.13. 6.1.Piecewise H¨ older regularity class. Let us ... | https://arxiv.org/abs/2501.16315v1 |
(Submanifold with boundary) .Note that with the above definition of uniformly piecewise C1,asetS, ifS⊂Rnis ad–submanifold with boundary then such a boundary necessarily lies in the singular set Ssg(this is more precisely due to ( H5)) and the decomposition of the singular set Ssgamounts to Ssg=Ssg,d−1, that is exactly ... | https://arxiv.org/abs/2501.16315v1 |
to obtain such a family F, similarly to the proof of Proposition 2.8). From the family F we can construct a family Gof two by two disjoint balls of radius ρ, centered at Sl∩Dρand such that Sρ l∩D⊂ ∪B∈G5B. Indeed, let us write F={B(xj,2ρ)}j∈JandG={B(zj, ρ)}j∈J, where zjis chosen so that zj∈Sland|zj−xj|< ρ (possible sinc... | https://arxiv.org/abs/2501.16315v1 |
are polytopes like the cube for which S0={corners }andS1={edges}. Another interesting example is the so–called stadium obtained by gluing two half–circles with two segments. Because of the four gluing points, the stadium is not C2but only C1,1, and more important, it is actually uniformly piecewise smooth (in the sense... | https://arxiv.org/abs/2501.16315v1 |
over real numbers lj (j= 1, . . . , J ) instead of summing over integers l= 0, . . . , d −1. However, the integrality of lis not crucial but most natural examples already fit such a framework and we decided to keep the singular strata of integer dimensions. •As already mentioned, ( H5) in particular controls the maxima... | https://arxiv.org/abs/2501.16315v1 |
for the pointwise convergences concerning the density θδ(x)− − − → δ→0θ(x) and concerning the tangent space Σ r(x, ν), σr,δ(x)− − − − → δ,r→0ΠTxS, provided that x∈Sis away from the singular set Ssg∪Θsg, as stated in Proposition 6.11. We first prove Lemma 6.10 which allows one to deal with both the density and the tange... | https://arxiv.org/abs/2501.16315v1 |
for all t≥0, (1 + t)−d 2≥1−d 2t (follows from convexity argument for instance) so that Z B(0,r)\U f(y,0) r dy≤ ∥f∥∞|B(0, r)\B(0, ε0)| ≤ ∥f∥∞ωdrd 1−(1 +C2r2a)−d 2 ≤ ∥f∥∞ωdrdd 2C2r2a. (110) We conclude the proof of Lemma 6.10 thanks to (108), (109) and (110). □ Proposition 6.11. Let0< a, b ≤1. We assume that Sandθsat... | https://arxiv.org/abs/2501.16315v1 |
0 < δ≤RandD⊂Rnbe an open set. We first note that by definition of βD, βD(νδ, ν)≤ |νδ−ν|(D) =Z D|Φ(θδ(x))−Φ(θ(x))|dµ(x)≤MZ D|θδ−θ|dµ . Furthermore, we can consider separately SCδandS\SCδ. On one hand, applying (111) in Proposition 6.11, we obtainZ D\SCδ|θδ−θ|dµ≤M(δa+δb)µ(D), On the other hand, recalling (60): for x∈S,|θ... | https://arxiv.org/abs/2501.16315v1 |
in the proof of (116)). More precisely, thanks to Proposition 6.11, using µ(B1)≤µ(B), we first have Z B1∥Σr(x, ν)−ΠTxS∥dµ(x)≤Mraµ(B). Then, thanks to (66), for all x∈S,∥Σr(x, ν)−ΠTxS∥ ≤M+∥ΠTxS∥ ≤M+1 so that using (101) (with δ+r≤RB ), we obtain Z B2∥Σr(x, ν)−ΠTxS∥dµ(x)≤Mµ(B2)≤MθmaxHd B∩S∩SC(δ+r) ≤Md−1X l=0ξl(δ+r)d−l ... | https://arxiv.org/abs/2501.16315v1 |
u n) in the case where λd(Σ) = λd+1(Σ). In what follows, we do not exclude such a possibility and we consider that Π can be taken to be any of the admissible choice. Then, for any projector ˜Π∈Pd,n, we have (121) ∥Σ−Π∥ ≤ ∥ Σ−˜Π∥. Indeed, as Σ =Pn k=1λk(Σ)uk⊗ukand by definition of Π, the symmetric matrix Σ −Π =Pn k=1(λk... | https://arxiv.org/abs/2501.16315v1 |
We start with proving the theorem concerning fWr,δ,N andeVr,δ,N. Let f∈Cc(Rn×Sym+(n)) with supp f⊂B×Sym+(n),∥f∥∞≤1 and Lip( f)≤1. Note that supp f⊂B×Sym+(n) so that ∥f∥∞≤RB. Then, thanks to (123), we have for x∈S, |f(x,Πr(x, νδ,N))−f(x,Σr(x, νδ,N))| ≤ ∥Πr(x, νδ,N)−Σr(x, νδ,N)∥ ≤ ∥ ΠTxS−Σr(x, νδ,N)∥ ≤ ∥ΠTxS−Σr(x, νδ)∥+∥... | https://arxiv.org/abs/2501.16315v1 |
Nδd+1√ Nrd µ(B) so that we infer E Z Bg dµ N ≤E Z Bg dµ N−Z Bg dµ +E Z Bg dµ ≤M1 δ+1 r µ(B)× N−1/difd >2 N−1/2lnNifd= 2 N−1/2ifd= 1+M1√ Nδd+1√ Nrd µ(B) ≤M min(δ, r)µ(B)× N−1/difd >2 N−1/2lnNifd= 2 N−1/2ifd= 1. (134) We can conclude the proof of (127) (first line) thanks to (132), (133) and (134). ... | https://arxiv.org/abs/2501.16315v1 |
results (Theorem 4.9 and Proposition 5.8). Therefore, a careful adaptation of the proof of Proposition 4.3 allows to leverage the aforementioned improved Lipschitz estimates (136) and (138) in the piecewise H¨ older regularity class Pto infer Theorem 7.4: we eventually obtain a varifold estimator bVN=bVδN,N(defined in ... | https://arxiv.org/abs/2501.16315v1 |
we recall (124): for x∈S, (141) ∥σδ,N(x)−πδ,N(x)∥ ≤ ∥ σδ,N(x)−ΠTxS∥. Theorem 7.4. Let0< a, b ≤1. We assume that Sandθsatisfy assumptions (H1)to(H7)i.e.µ=θHd |S∈ P. LetD⊂Rnbe a bounded open set. We recall that R= min( RS,sg, Rθ,sg)<1and let N∈N∗be large enough so thatδN:=N−1 d+ 2 min( a, b)≤R 20C. Then, E[βD(bνδN,N, ν)]... | https://arxiv.org/abs/2501.16315v1 |
j)−µ(Au j)) . Step 1:we can prove the following control: (145) E" sup f∈BL Z Dgδ,N dµX N−dµ # ≤E" sup f∈BLIt# +M εmin(a,b)+δmin(a,b)+1√ Nδd µ(D) +Md−1X l=0δd−lHl(Sl∩D20Cδ). Indeed, we remind (see (31)) that ( At j)mt j=1is a partition of T=D∩S\S10Cδand for all j, diam At j≤3εt≤9ε. As ∥gδ,N∥∞≤ ∥f∥∞∥Φ∥∞≤M, we first h... | https://arxiv.org/abs/2501.16315v1 |
from u=tdown to son (153) and recalling that Is≤MM s, we have the following control: E" sup f∈BLIt# ≤M E[Ms] + δmin(a,b)+1√ Nδd tX u=s+1E[Mu] +tX u=s+1εmin(a,b) u E[Mu]! . We recall (46): E[Mu]≤M N1 2ε−d 2uµ Dεs 4 (and N1 2=ε−d 2) and moreoverεs 4≤δso that µ Dεs 4 ≤Mµ(Dδ). Consequently E" sup f∈BLIt# ≤M√ Nµ Dδ ... | https://arxiv.org/abs/2501.16315v1 |
we note that for ε∈(0,1],ε1 2|lnε| ≤1,ε1 4|lnε| ≤2, and thus εc|lnε|=ε|lnε| ≤ε1 2=δ=δcifd= 2, c= 1 ε1 2|lnε| ≤2ε1 4= 2δcifd= 1, c=1 2 ≤2δc= 2δmin(a,b) and we obtain the same control (156) for min( a, b) =d 2. 54 CHARLY BORICAUD AND BLANCHE BUET – Case min(a, b)−d 2>0:in this case d= 1 and we have ε−d/2+c u = 31 2−c... | https://arxiv.org/abs/2501.16315v1 |
Nδd N+ 2δmin(a,b) N +Md−1X l=0δd−l NHl(Sl∩D4CδN) and we conclude the proof of (158). The proof of (159) is a straightforward application of the aforementioned Propositions 3.6, 6.12 5.6 and 6.13 □ References [AB24] E. Aamari and C. Berenfeld. A theory of stratification learning, 2024. ArXiv 2405.20066. [AFP00] L. Amb... | https://arxiv.org/abs/2501.16315v1 |
Lectures ongeometric measure theory, volume 3 of Proceedings oftheCentre forMathematical Analysis, Australian National University. Australian National University Centre for Mathematical Analysis, Canberra, 1983. [Tin23] R. Tinarrage. Recovering the homology of immersed manifolds. Discrete andComputational Geometry, 69:... | https://arxiv.org/abs/2501.16315v1 |
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