Approximation of symmetric total variation on point clouds

APPR O XIMA TION OF SYMMETRIC TOT AL V ARIA TION ON POINT CLOUDS S. ALMI, A. KUBIN, AND E. T ASSO Abstract. The pap er in vestigates the approximation of the symmetric T otal V ariation func- tional on graphs. Such an approximation is giv en in terms of a discrete and symmetric finite difference mo del defined on point clouds obtained by randomly sampling a reference probabilit y measure. W e identify suitable scalings of the p oint distribution that guarantee an almost surely Γ-con vergence to an anisotropic w eigh ted symmetric T otal V ariation. 2020 Mathematics Subje ct Classification: 49Q20, 49J45, 26B30 Keywor ds and phr ases: F ree-discon tin uit y problems, graph approximations, symmetric gradien t, Γ-con vergence, transp ort maps. 1. Introduction The analysis of v ariational problems defined on random data has b ecome increasingly signifi- can t in a wide range of applications, including mac hine learning, imaging, and materials science [ 8 , 13 , 19 , 20 , 22 ]. T o a p oint cloud, one can naturally asso ciate a weigh ted graph: the sampled p oin ts form the vertices of the graph, and edges are introduced b etw een pairs of p oin ts that are sufficien tly close to eac h other. These interactions are enco ded by weigh ts that dep end on the distance b et ween points through a k ernel function with a prescribed interaction length scale. The choice of this length scale plays a crucial role. On the one hand, reducing the num ber of edges is desirable in order to low er the computational complexit y . On the other hand, if the distance b etw een the no des falls b elo w a certain threshold, the graph may fail to capture the relev ant geometric features of the underlying p oint cloud. In the general context outlined ab ov e, man y machine learning tasks are then formulated as a minimization problem of a functional defined on the graph representing the data set. The consistency of such problems as the num b er of samples tends to infinity b ecomes a fundamen- tal question. In a mathematical p ersp ective, one aims at sho wing the conv ergence of discrete v ariational mo dels p osed on random p oint clouds tow ards their con tinuum counterparts. This issue has b een addressed in the scalar setting in several different settings: for the total v ariation and p erimeter functionals in [ 15 , 14 , 12 ], for the Ginzburg-Landau functional in [ 23 ], for the Diric hlet energy in [ 21 ], and in [ 10 ] for the Mumford–Shah functional. In such works, a p oint cloud is mo deled by a set of random p oints { X 1 , . . . , X n } obtained by sampling a giv en proba- bilit y distribution ν = ρ d x in a b ounded domain D ⊂ R d . The p oints are assumed to interact at a giv en scale ε n > 0, which con verges to 0 with a suitable rate dep ending on the am bien t dimension d . Such a rate is dictated by the use of a transp ort-lik e distance T L p for p ∈ [1 , + ∞ ) (see Definition 2.1 and, e.g., [ 15 , 16 ]), whic h simultaneously describ es the W asserstein conv er- gence of the empirical measure ν n := 1 n P i δ X i to w ards ν and the conv ergence of L p -functions on graphs tow ards targets in L p ( D ; ν ). W e further refer to [ 6 , 7 , 9 ] for results of compactness, Γ-con v ergence, and homogenization on Poisson p oin t clouds, where the use of the T L p -distance is not p ermitted. Recen t works (see [ 13 , 22 ]) suggest the relev ance of studying v ariational mo dels from fracture mec hanics on p oin t cloud data. In particular, these contributions indicate that geometric infor- mation extracted from large-scale vectorial p oin t clouds can b e used to identify and c haracterize fracture systems, therefore motiv ating the form ulation of v ariational fracture mo dels, such as the Griffith functional [ 5 ], in the discrete setting. As a first step tow ard extending [ 15 , 10 ] to the v ectorial setting, we consider a discrete approximation of the ve ctorial symmetric total variation 1 2 S. ALMI, A. KUBIN, AND E. T ASSO defined ov er a set of random p oints, which can b e viewed as a simplified v ariational mo del of linear elasticit y . F or a reference probabilit y measure ν = ρ d x absolutely contin uous with resp ect to the Leb esgue measure in R d with density ρ having supp ort in an op en b ounded subset D of R d , we assume ρ to b e contin uous, b ounded, and b ounded a w ay from 0 (see also ( ρ 1)–( ρ 2) b elow). W e consider { X 1 , . . . , X n } random points i.i.d. as ν and fix an interaction length-scale ε n > 0, which determines the neigh bourho o d within whic h pairs of p oints are allo w ed to in teract. Given a non- negativ e, radially symmetric k ernel η : R d → [0 , ∞ ) (see Section 2 for the precise assumptions on η ), w e define its rescaling at scale ε n b y η ε n := 1 ε d n η  · ε  . F or a vector-v alued function u : { X 1 , . . . , X n } → R d , the gr aph ve ctorial symmetric total varia- tion is defined as GT V n,ε n ( u ) := 1 ε 2 n 1 n 2 n X i,j =1 η ε n ( X i − X j ) | ( u ( X i ) − u ( X j )) · ( X i − X j ) | . (1.1) The normalization factor 1 /n 2 a v erages the contributions o v er all interacting pairs of p oints. The factor 1 /ε 2 n pro vides the correct scaling with resp ect to the interaction length scale. Indeed, due to the presence of the rescaled k ernel η ε n , only pairs of p oints at a distance of order ε n con tribute significan tly to the sum. F or such pairs, the increment u ( X i ) − u ( X j ) is t ypically of order ε n if u is smo oth. Hence, its pro jection on to the edge X i − X j is of order ε 2 n . The prefactor 1 /ε 2 n comp ensates for this scaling, th us ensuring consistency with the corresp onding contin uum symmetric total v ariation in the limit ε n → 0. Compared to [ 15 , 10 ], the extra pro jection on X i − X j represen ts the key c hange in the graph-functional, as GT V n,ε n only inv olv es symmetric finite differences, and therefore leads to a B D -total v ariation functional. Our main result consists in showing that, under suitable assumptions on the kernel and the distribution of the p oint clouds, the functionals in ( 1.1 ) almost surely Γ-conv erge in the T L 1 - top ology to a functional which only dep ends on a weigh ted and anisotropic symmetric total v ariation. More precisely w e hav e the following theorem. W e refer to Section 2 for the set of assumptions. Theorem 1.1. Assume (K1) – (K3) , ( ρ 1) – ( ρ 2) , and ( 2.2 ) – ( 2.5 ) . Then, the se quenc e of function- als GT V n almost sur ely Γ -c onver ges with r esp e ct to the T L 1 -c onver genc e to T V η ( u ; ρ 2 ) := ˆ D ρ 2 ( x ) ϕ η  E u ( x ) | E u ( x ) |  d | E u ( x ) | , wher e the function ϕ η : M d sy m → [0 , + ∞ ) is define d as ϕ η ( A ) := ´ R d η ( ξ ) | Aξ · ξ | d ξ . As in [ 15 ], our analysis relies the existence of a transp ort map T n b et ween ν and ν n suc h that lim n →∞ n 1 /d ∥ I d − T n ∥ L ∞ log 1 /d ( n ) = 0 , where I d : D → D denotes the identit y function. Such a map was prov en to exist in [ 16 , 10 ] and allo ws to rewrite the functional ( 1.1 ) in a contin uum setting as GT V n,ε n ( u ) = 1 ε 2 n ¨ D × D η ε n ( T n ( x ) − T n ( y ))   ( u ◦ T n ( x ) − u ◦ T n ( y )) (1.2) · ( T n ( x ) − T n ( y ))   ρ ( x ) ρ ( y ) d x d y . Notice that, ev en defining v := u ◦ T n as a function ov er D and exploiting the monotonicity prop erties of η to reduce to work with η ε n ( x − y ), w e are not in a p osition to treat the right- hand side of ( 1.2 ) as an auxiliary nonlo cal energy defined ov er L 1 ( D ; R d ), as it was done in [ 15 ], SYMMETRIC TOT AL V ARIA TION ON POINT CLOUDS 3 as the transp ort map T n still app ears in the pro jection part of the integrand. Under the scaling assumption lim sup n →∞ log 1 /d ( n ) n 1 /d 1 ε 2 n < + ∞ , (1.3) w e are able to p erform a slicing argument similar to that of [ 17 , 18 , 1 ]. W e p oint out that condition ( 1.3 ) is stronger than the scaling considered in the B V setting of [ 15 , 10 ], as it implies a uniform con trol on a discrete second order deriv ative of T n (see ( 2.5 ) below). F rom a geometric p oin t of view, ( 1.3 ) means that the cut-off function η ε in ( 1.2 ) has to weigh t a larger n um b er of p oin ts in order to reconstruct the B D -total v ariation in the limit. A similar phenomenon is en- coun tered in other discrete finite-difference mo dels approximating free-discontin uit y functionals in v olving B D -type functions. F or instance, we refer to the Ambrosio-T ortorelli approximation on square lattices in [ 11 ], where the authors hav e to consider next-to-next-nearest neighbours in teractions, which are instead not necessary in the B V -setting [ 4 ]. Hyp othesis ( 1.3 ) is used in the Γ-liminf inequalit y (see Theorem 3.1 ) as well as to ensure that for a sequence ( u n , ν n ) con v erging to ( u, ν ) in the T L 1 -metric the limit map u b elongs to B D ( D ). In this resp ect, we notice that, by construction, the ε n -discrete second order deriv ativ e of T n con v erges to 0 in the sense of distributions. Condition ( 1.3 ) impro ves it to a weak ∗ con v ergence in L ∞ ( D ; R d ), which is in duality with the L 1 -con v ergence of u n ◦ T n , implied by the T L 1 -con v ergence. An interesting research line would b e to understand the Γ-conv ergence of GT V n,ε n in the scaling (log n ) 1 /d n 1 /d ≪ ε n ≪ (log n ) 1 / 2 d n 1 / 2 d , th us reco v ering the scalings of [ 15 , 10 ]. T o do this, one ma y ha v e to inv estigate how the geometrical and top ological prop erties of the graph influence the regularit y of the transp ort maps T n constructed in [ 16 ]. 2. Energy and assumptions Let η : R d → [0 , + ∞ ) b e a radially symmetric kernel, η ( x ) := η ( | x | ) where η : [0 , + ∞ ) → [0 , + ∞ ) is such that (K1) η (0) > 0 and η is con tin uous at 0; (K2) η is non-increasing; (K3) the integral ´ R d η ( x ) | x | 2 d x is finite. F or x ∈ R d and ε > 0 we define η ε ( x ) := 1 ε d η  x ε  . Let (Ω , P , F ) be a probabilit y space, let D ⊂ R d b e a b ounded op en set with Lipsc hitz b oundary , and ρ : D → R b e such that ( ρ 1) ρ is conti nuous; ( ρ 2) there exist 0 < α ≤ β < + ∞ such that α ≤ ρ ( x ) ≤ β for every x ∈ D . W e define ν := ρ L d and let X 1 , . . . , X n : Ω → D b e n random p oin ts i.i.d. according to ν . Let ν n b e the empirical measure asso ciated with the n data p oints, i.e., ν n := 1 n n X i =1 δ X i . Notice that ν n ∈ P ( D ) is itself a random v ariable. If not explicitly needed, we will not indicate the dep endence on the realization ω ∈ Ω, as our analysis holds almost surely (i.e., for P -a.e. ω ∈ Ω). F or every u : { X 1 , . . . , X n } → R d w e define the gr aph ve ctorial symmetric total variation by GT V n,ε ( u ) := 1 ε 2 1 n 2 X i,j η ε ( X i − X j ) | ( u ( X i ) − u ( X j )) · ( X i − X j ) | . (2.1) W e fix a sequence ε n > 0 such that lim sup n →∞ (log n ) 1 /d n 1 /d 1 ε 2 n < + ∞ . (2.2) 4 S. ALMI, A. KUBIN, AND E. T ASSO In [ 16 , 10 ] it w as sho wn that, for P -a.e. ω ∈ Ω, there exist C > 0 and a sequence of transp ortation maps { T n } n ∈ N suc h that ( T n ) # ν = ν n and lim n →∞ n 1 /d ∥ I d − T n ∥ L ∞ (log n ) 1 /d ≤ C. (2.3) W e notice that if ε n and T n satisfy ( 2.2 )–( 2.3 ), then it holds lim n →∞ ∥ I d − T n ∥ L ∞ ε n = 0 , (2.4) lim sup n →∞ ∥ T n ( · + ε n ) − 2 T n ( · ) + T n ( · − ε n ) ∥ L ∞ ε 2 n < + ∞ . (2.5) Giv en T n as abov e, we write GT V n ( u ) := GT V n,ε n ( u ) (2.6) = 1 ε 2 n ¨ D × D η ε n ( T n ( x ) − T n ( y ))   ( u ◦ T n ( x ) − u ◦ T n ( y )) · ( T n ( x ) − T n ( y ))   ρ ( x ) ρ ( y ) d x d y . F or u ∈ B D ( D ) w e define T V η ( u ; ρ 2 ) := ˆ D ρ ( x ) 2 ϕ η  E u ( x ) | E u ( x ) |  d | E u ( x ) | , where w e hav e in tro duces the norm on symmetric matrices ϕ η : M d sy m → [0 , + ∞ ) as ϕ η ( A ) := ˆ R d η ( ξ ) | Aξ · ξ | d ξ . In the following definition we recall the T L 1 -con v ergence in tro duced in [ 15 ]. Definition 2.1. Let µ 1 , µ 2 ∈ P ( D ), w 1 ∈ L 1 ( D ; R d ; µ 1 ) and w 2 ∈ L 1 ( D ; R d ; µ 2 ). W e define the T L 1 -distance as d T L 1  ( w 1 , µ 1 ) , ( w 2 , µ 2 )  := inf γ ∈ Γ( µ 1 ,µ 2 ) ¨ D × D | x − y | + | w 1 ( x ) − w 2 ( y ) | d γ ( x, y ) , where Γ( µ 1 , µ 2 ) denotes the set of transp ort plans b et w een µ 1 and µ 2 . F or µ n , µ ∈ P ( R d ), w n ∈ L 1 (Ω; R d ; µ n ), and w ∈ L 1 ( D ; R d ; µ ), w e say that ( w n , µ n ) → ( w, µ ) in the T L 1 -metric if lim n →∞ d T L 1  ( w n , µ n ) , ( w , µ )  = 0 . The follo wing characterization can b e found in [ 15 , Prop osition 3.12]. Prop osition 2.2. L et µ n , µ ∈ P ( R d ) , w n ∈ L 1 ( D ; R d ; µ n ) , and w ∈ L 1 ( D ; R d ; µ ) , and assume that µ ≪ L d . Then, the fol lowing ar e e quivalent: (1) ( w n , µ n ) → ( w, µ ) in the T L 1 -metric; (2) for every tr ansp ort map S n : D → D (i.e., such that ( S n ) # µ = µ n ) such that lim n →∞ ˆ D | x − S n ( x ) | d µ ( x ) = 0 we have that lim n →∞ ˆ D | w ( x ) − w n ( S n ( x )) | d µ ( x ) = 0 . The pro of of Theorem 1.1 is carried out in the next tw o sections, where we prov e the Γ-liminf and the Γ-limsup inequalities, resp ectively . SYMMETRIC TOT AL V ARIA TION ON POINT CLOUDS 5 3. Gamma liminf inequality In this section we establish the liminf inequality for the functionals GT V n . Theorem 3.1. Assume (K1) – (K3) , ( ρ 1) – ( ρ 2) , and ( 2.2 ) . F or P -a.e. ω ∈ Ω , for every u ∈ L 1 ( D ; R d ; ν ) and { u n } n ∈ N ⊂ L 1 ( { X 1 , . . . , X n } ; R d ; ν n ) such that ( ν n , u n ) → ( ν, u ) in T L 1 , ther e exists a subse quenc e ε n → 0 such that lim inf n →∞ GT V n ( u n ) ≥ T V η ( u ; ρ 2 ) . In p articular, u ∈ B D ( D ) . R emark 3.2 . As in [ 15 , Section 5], we ma y assume that the kernel η is of the form η ( t ) = c for t < b and η ( t ) = 0 for t ≥ b . Indeed, if w e can establish the liminf inequality under this assumption, then, by the sup eradditivit y of the liminf, the same inequality also holds for functions of the form η = P l k =1 η k , for some l ∈ N , where eac h η k is of the ab ov e form. Finally , the case of general η follows by considering an increasing sequence of piecewise constan t functions η n : [0 , + ∞ ) → [0 , + ∞ ) suc h that η n ↗ η almost everywhere, and by the contin uit y of the map η 7→ T V η . By [ 16 , 10 ], we consider ω ∈ Ω and T n : D → D such that ( 2.3 ) (and th us ( 2.4 )–( 2.5 )) holds. In view of Remark 3.2 , we assume for the remaining part of this section that η is of the form η ( t ) = c for t < b and η ( t ) = 0 for t ≥ b . F or almost ev ery ( x, y ) ∈ D × D we ha ve | T n ( x ) − T n ( y ) | > bε n ⇒ | x − y | > bε n − 2 ∥ I d − T n ∥ L ∞ . (3.1) Thanks to ( 2.4 ), for n large enough it holds ˜ ε n := ε n − 2 b ∥ I d − T n ∥ L ∞ > 0 . By ( 3.1 ) and (K2), for n large enough and for almost ev ery ( x, y ) ∈ D × D we get η  | x − y | ˜ ε n  ≤ η  | T n ( x ) − T n ( y ) | ε n  . Hence, w e hav e the low er b ound GT V n ( u n ) = GT V n,ε n ( u n ) ≥  ˜ ε n ε n  d +2 1 ˜ ε 2 n ¨ D × D 1 ˜ ε d n η  | x − y | ˜ ε n  | ( u n ◦ T n ( x ) − u n ◦ T n ( y )) · ( T n ( x ) − T n ( y )) | ρ ( x ) ρ ( y )d x d y . Since ˜ ε n ε n → 1, it is enough to pro v e the liminf estimate for 1 ˜ ε 2 n ¨ D × D η ˜ ε n ( x − y )   ( u n ◦ T n ( x ) − u n ◦ T n ( y )) · ( T n ( x ) − T n ( y ))   ρ ( x ) ρ ( y )d x d y . F or notational conv enience, we set v n := u n ◦ T n and we drop the tilde in the ab ov e expression. Hence, after a change of v ariables, we consider the follo wing sequence 1 ε 2 n ˆ D − D ε n ˆ D ∩ ( D − ε n ξ ) η ( ξ )   ( v n ( x + ε n ξ ) − v n ( x )) · ( T n ( x + ε n ξ ) − T n ( x ))   ρ ( x + ε n ξ ) ρ ( x )d x d ξ . In particular, for any function w : D → R d w e define the following functional, whic h only tak es in to accoun t the integral in the x -v ariable: F ξ n ( w , D ) := 1 ε 2 n ˆ D ∩ ( D − ε n ξ )   ( w ( x + ε n ξ ) − w ( x )) · ( T n ( x + ε n ξ ) − T n ( x ))   ρ ( x + ε n ξ ) ρ ( x )d x. As w e are going to w ork with 1-dimensional slices of the functions v n , for a function w : R d → R d w e in tro duce the notation w ξ ,y ( t ) := w ( y + tξ ) for ξ ∈ R d \ { 0 } , y ∈ Π ξ , and t ∈ R . 6 S. ALMI, A. KUBIN, AND E. T ASSO Let y ∈ Π ξ and I ⊂ ( D ∩ ( D − εξ )) ξ y , w e also define for an y v : I → R d the functional F ξ ,y n ( v , ρ, I ) := 1 ε 2 n ˆ I   ( v ( t + ε n ) − v ( t )) · ( T ξ ,y n ( t + ε n ) − T ξ ,y n ( t ))   ρ ξ ,y ( t + ε n ) ρ ξ ,y ( t ) d t. (3.2) W e no w pro v e a one-dimensional lemma similar to [ 17 , Lemma 3.2], where w e take the weigh t ρ to be identically equal to 1. Lemma 3.3. Assume that ( 2.2 ) holds, and let ξ ∈ R d and y ∈ Π ξ . L et I = [ a, b ] ⊂ R b e a finite interval and v n , v ∈ L 1 ( I ; R d ) b e such that: (i) v n → v in L 1 ( I ; R d ) ; (ii) a and b ar e L eb esgue p oints of v . Then lim inf n →∞ F ξ ,y n ( v n , 1 , I ) ≥ | ( v ( b ) − v ( a )) · ξ | . (3.3) Pr o of. T o simplify the notation, let us assume that a = 0. Let T n b e the transp ort map as in ( 2.3 ). W e notice that in view of ( 2.2 ) and ( 2.5 ) we ha ve that there exists L ∈ (0 , + ∞ ) (indep enden t of ξ and y ) such that lim n →∞ ∥ T ξ ,y n − ( y + · ξ ) ∥ L ∞ ε n = 0 , (3.4) sup n ∥ T ξ ,y n ( · + ε n ) − 2 T ξ ,y n ( · ) + T ξ ,y n ( · − ε n ) ∥ L ∞ ε 2 n ≤ L . (3.5) T o simplify the notation we drop the dep endence on ξ and y . Let us set J := | ( v ( b ) − v (0)) · ξ | . If J = 0 there is nothing to prov e. Hence, we can assume that J > 0. Moreov er, up to a subsequence, we can also assume that lim inf n →∞ F n ( v n , 1 , I ) = lim n →∞ F n ( v n , 1 , I ) and v n ( t ) → v ( t ) for a.e. t ∈ I . Set C := 4(1 + | ξ | ) + | v ( b ) | + | v (0) | . Fix σ ∈ (0 , J /C ], and let N ε n = [ | I | /ε n ]. W e define the set C n :=  t ∈ [0 , ε n ] : N ε n X k =1 1 ε n | v n ( t + kε n ) − v n ( t + ( k − 1) ε n ) · ( T n ( t + kε n ) − T n ( t + ( k − 1) ε n )) | ≥ J − 2 C σ  . W e sub divide the pro of in tw o steps. Step 1: W e show that lim n →∞ | C n | ε n = 1 . (3.6) Notice that, by a change of v ariables, ( 3.6 ) is equiv alent to lim n →∞ | C ε n n | = 1 , (3.7) where w e hav e set C ε n n :=  τ ∈ [0 , 1] : N ε n X k =1 1 ε n | v n ( ε n τ + k ε n ) − v n ( ε n τ + ( k − 1) ε n ) · ( T n ( ε n τ + k ε n ) − T n ( ε n τ + ( k − 1) ε n )) | ≥ J − 2 C σ  . T o this end, for all δ > 0 we set A δ := { t ∈ [0 , δ ] : | v ( t ) − v (0) | < σ } , B δ := { t ∈ [ b − δ, b ] : | v ( t ) − v ( b ) | < σ } . SYMMETRIC TOT AL V ARIA TION ON POINT CLOUDS 7 By h yp othesis (ii) we hav e that lim δ → 0 | A δ | δ = lim δ → 0 | B δ | δ = 1 . (3.8) By Sev erini–Egorov Theorem, there exis ts I δ ⊂ I such that | I \ I δ | < δ 2 , v n ( t ) → v ( t ) and T n ( t ) → y + tξ uniformly in t ∈ I δ . W e set c n ( t ) := T n ( t ) − T n ( t − ε n ) and observ e that c n /ε n → ξ uniformly in I δ . Then, there exists ¯ n = ¯ n ( σ, δ, I δ ) suc h that for n ≥ ¯ n it holds t ′ ∈ B δ ∩ I δ , t ∈ A δ ∩ I δ = ⇒    v n ( t ′ ) · c n ( t ′ ) ε n − v n ( t ) · c n ( t + ε n ) ε n    ≥ J − C σ . (3.9) Moreo v er, w e ha v e | A δ ∩ I δ | ≥ | A δ | − δ 2 , | B δ ∩ I δ | ≥ | B δ | − δ 2 . (3.10) Let τ ∈ [0 , 1] be such that there exists M n,τ ≤ N ε n suc h that ε n τ ∈ A δ ∩ I δ and ε n τ + M n,τ ε n ∈ B δ ∩ I δ . By the triangle inequality and by ( 3.9 ) we hav e lim inf n →∞ 1 ε n N ε n X k =1 | ( v n ( ε n τ + k ε n ) − v n ( ε n τ + ( k − 1) ε n )) · ( T n ( ε n τ + k ε n ) − T n ( ε n τ + ( k − 1) ε n )) | ≥ lim inf n →∞ 1 ε n M n,τ X k =1 | ( v n ( ε n τ + k ε n ) − v n ( ε n τ + ( k − 1) ε n )) · ( T n ( ε n τ + k ε n ) − T n ( ε n τ + ( k − 1) ε n )) | ≥ lim inf n →∞ 1 ε n     M n,τ X k =1 ( v n ( ε n τ + k ε n ) − v n ( ε n τ + ( k − 1) ε n )) · c n ( ε n τ + k ε n )     ≥ lim inf n →∞ 1 ε n     v n ( ε n τ + M n,τ ε n ) · c n ( ε n τ + M n,τ ε n ) − v n ( ε n τ ) · c n ( ε n τ + ε n )    −     M n,τ − 1 X k =1 ( c n ( ε n τ + ( k + 1) ε n ) − c n ( ε n τ + k ε n )) · v n ( ε n τ + k ε n )      ≥ J − C σ − lim sup n →∞     M n,τ − 1 X k =1 c n ( ε n τ + ( k + 1) ε n ) − c n ( ε n τ + k ε n ) ε n · v n ( ε n τ + k ε n )     . (3.11) W e now sho w that lim sup n →∞     M n,τ − 1 X k =1 c n ( ε n τ + ( k + 1) ε n ) − c n ( ε n τ + k ε n ) ε 2 n · v n ( ε n τ + k ε n ) ε n     = 0 . (3.12) Let us consider the functions f n,τ : I → R d and v n,τ : I → R d defined b y f n,τ ( s ) := N ε n − 1 X k =1 c n ( ε n τ + ( k + 1) ε n ) − c n ( ε n τ + k ε n ) ε 2 n 1 [ ε n τ + k ε n ,ε n τ +( k +1) ε n ) ( s ) , v n,τ ( s ) := N ε n − 1 X k =1 v n ( ε n τ + k ε n ) 1 [ ε n τ + k ε n ,ε n τ +( k +1) ε n ) ( s ) for s ∈ I . W e remark that b y ( 3.5 ) the functions f n,τ satisfy a uniform L ∞ -b ound. Consider 0 < c < d < b and let n c ε n := [ c/ε n ], N d ε n := [ d/ε n ]. W e first show that lim n →∞    ˆ d c f n,τ ( s ) d s    = 0 . (3.13) Indeed, for w ∈ R d it holds 0 = lim sup n →∞    1 ε n N d ε n X k = n c ε n +1 ( w − w ) · ( T n ( ε n τ + k ε n ) − T n ( ε n τ + ( k − 1) ε n ))    8 S. ALMI, A. KUBIN, AND E. T ASSO ≥ lim sup n →∞ 1 ε n  −    w · c n ( ε n τ + N d ε n ε n ) − w · c n ( ε n τ + n c ε n ε n )    +    N d ε n − 1 X k = n c ε n +1 ( c n ( ε n τ + ( k + 1) ε n ) − c n ( ε n τ + k ε n )) · w     = lim sup n →∞    N d ε n − 1 X k = n c ε n +1 ( c n ( ε n τ + ( k + 1) ε n ) − c n ( ε n τ + k ε n )) ε 2 n · w ε n    = lim sup n →∞    ˆ ( N d ε n + τ ) ε n ( n c ε n +1+ τ ) ε n w · f n,τ ( s ) d s    . This pro ves ( 3.13 ). Com bining ( 3.13 ) together with ( 3.4 ) we deduce f n,τ ⇀ 0 weakly ∗ in L ∞ ( I ; R d ). This in turn implies ( 3.12 ) after showing v n,τ → v strongly in L 1 ( I ; R d ) for a.e. τ ∈ [0 , 1]. T o this end, w e consider ˆ 1 0  ˆ b 0 | v n,τ ( s ) − v ( s ) | d s  d τ = ˆ 1 0  ˆ b 0    N ε n − 1 X k =1 1 [ ε n τ + k ε n ,ε n τ +( k +1) ε n ) ( s )( v n ( ε n τ + k ε n ) − v ( s ))    d s  d τ = N ε n − 1 X k =1 ˆ 1 0  ˆ ε n τ +( k +1) ε n ε n τ + k ε n | v n ( ε n τ + k ε n ) − v ( s ) | d s  d τ = N ε n − 1 X k =1 1 ε n ˆ ( k +1) ε n kε n  ˆ ε n 0 | v n ( t ) − v ( t + h ) | d h  d t ≤ ˆ b 0  ε n 0 | v n ( t ) − v ( t + h ) | d h  d t ≤ ˆ b 0 | v n ( t ) − v ( t ) | d t + ˆ b 0  ε n 0 | v ( t ) − v ( t + h ) | d h  d t . The first term on the righ t-hand side tends to zero since v n → v in L 1 ( I ; R d ). The second term tends to zero by standard con volution estimates, since v ∈ L 1 ( I ; R d ). Hence, lim n →∞ ˆ 1 0  ˆ b 0 | v n,τ ( s ) − v ( s ) | d s  d τ = 0 , whic h implies the claim. Therefore, ( 3.12 ) is prov ed. Combining ( 3.11 ) and ( 3.12 ) we infer that for τ ∈ [0 , 1] such that there exists M n,τ ≤ N ε n with ε n τ + M n,τ ε n ∈ B δ ∩ I δ and ε n τ ∈ A δ ∩ I δ lim inf n →∞ 1 ε n M n,τ X k =1 | ( v n ( ε n τ + k ε n ) − v n ( ε n τ + ( k − 1) ε n )) · ( T n ( ε n τ + k ε n ) − T n ( ε n τ + ( k − 1) ε n )) | ≥ J − C σ . (3.14) W e now pro ceed as in [ 17 , Lemma 3.2]. Let us set K 0 n :=  k ∈ N : [ k ε n , ( k + 1) ε n ] ⊂ [0 , δ ]  , K b n :=  k ∈ N : [ k ε n , ( k + 1) ε n ] ⊂ [ b − δ, b ]  . It is easy to see that # K 0 n = [ δ /ε n ] , # K b n ≥ ([ δ /ε n ] − 1) . (3.15) Let us further set A n,k δ :=  τ ∈ [0 , 1] : ε n τ + k ε n / ∈ A δ ∩ I δ  for k ∈ K 0 n , (3.16) SYMMETRIC TOT AL V ARIA TION ON POINT CLOUDS 9 B n,k δ :=  τ ∈ [0 , 1] : ε n τ + k ε n / ∈ B δ ∩ I δ  for k ∈ K b n . (3.17) By ( 3.16 ), ( 3.17 ) and ( 3.14 ) it follo ws that τ ∈ [0 , 1] \  [ k ∈ K 0 n A n,k δ ∪ [ k ∈ K b n B n,k δ  (3.18) = ⇒ τ ∈ C ε n n for n sufficiently large (dep ending on τ and δ ). By ( 3.10 ) w e ha v e that | A n,k δ | ≤ 1 ε n ( δ − | A δ | + δ 2 ) for k ∈ K 0 n , | B n,k δ | ≤ 1 ε n ( δ − | B δ | + δ 2 ) for k ∈ K b n . Hence, b y ( 3.18 ) lim inf n →∞ | C ε n n | ≥ lim inf n →∞     [0 , 1] \  [ k ∈ K 0 n A n δ ∪ [ k ∈ K b n B n δ      −      [0 , 1] \  [ k ∈ K 0 n A n δ ∪ [ k ∈ K b n B n δ  \ C ε n n     ≥ 1 − X k ∈ K 0 n | A n,k δ | − X k ∈ K b n | B n,k δ | ≥ 1 − 1 ε n ( δ − | A δ | + δ 2 ) h δ ε n i − 1 − 1 ε n ( δ − | B δ | + δ 2 )  h δ ε n i − 1  − 1 = 1 −  1 − | A δ | δ + δ  −  1 − | B δ | δ + δ  = | A δ | δ + | B δ | δ − 1 − 2 δ . Since δ > 0 is arbitrary , w e can send δ → 0 + and b y ( 3.8 ) w e obtain lim inf n →∞ | C ε n n | ≥ 1 . Since | C ε n n | ≤ 1, it must b e lim n →∞ | C ε n n | = 1 . Th us, equalities ( 3.7 ) and ( 3.6 ) are pro v ed. Step 2: Let us prov e ( 3.3 ). By definition of C n w e ha v e F n ( v n , 1 , I ) ≥ F n ( v n , 1 , [0 , ε n N ε n ]) = 1 ε 2 n ˆ ε n N ε n 0   ( v n ( t + ε n ) − v n ( t ) · ( T n ( t + ε n ) − T n ( t ))   d t = 1 ε 2 n ˆ ε n 0 N ε n X k =1   ( v n ( t + kε n ) − v n ( t + ( k − 1) ε n ) · ( T n ( t + kε n ) − T n ( t + ( k − 1) ε n ))   d t ≥ | C n | ε n ( J − 2 C σ ) . By sending n → ∞ and then σ → 0, we deduce lim inf n →∞ F n ( v n , 1 , I ) ≥ J. This concludes the pro of of ( 3.3 ) and of the lemma. □ F or the pro of of the liminf w e need the follo wing tec hnical lemma whic h is a generalization of [ 17 , Lemma 3.3]. Lemma 3.4. L et u ∈ L 1 loc ( R ) . Then ther e exists a ∈ R such that 10 S. ALMI, A. KUBIN, AND E. T ASSO (i) a + q is a L eb esgue p oint of u for every q ∈ Q ; (ii) every se quenc e { u n } n ∈ N ⊂ L 1 loc ( R ) that satisfies the c onditions: – u n  a + z n  = u  a + z n  for al l z ∈ Z , – if x ∈  a + z n , a + z +1 n  , then u n ( x ) b elongs to the interval with endp oints u  a + z n  and u  a + z +1 n  , has a subse quenc e c onver ging to u in L 1 loc ( R ) . Pr o of. W e only need to show the v alidity of (ii) , since (i) is trivially true. W e use the same notation of [ 17 , Lemma 3.3]. F or n ≥ 1, z ∈ Z , and a ∈ [0 , 1], w e define for ev ery x ∈ R v a n ( x ) = u  a + [ n ( x − a )] n  . F or every fixed in terv al I ⊂ R , up to a subsequence, w e ha v e v a n → u in L 1 ( I ) for a.e. a ∈ [0 , 1] . Indeed, it holds lim n →∞ ˆ I ˆ 1 0 | v a n ( x ) − u ( x ) | d a d x = lim n →∞ ˆ I 0 − 1 n | u ( x + y ) − u ( x ) | d y d x = 0 , b y the dominated conv ergence theorem since u ∈ L 1 ( R ). The proof then follows in the same wa y as in [ 17 , Lemma 3.3]. □ W e are no w in a p osition to prov e Theorem 3.1 . Pr o of of The or em 3.1 . Without loss of generality we can assume lim inf n →∞ GT V n ( u n ) < ∞ . (3.19) W e define v n := u n ◦ T n . Since ( u n , ν n ) → ( u, ν ) in T L 1 , it holds v n → u in L 1 ( D ; R d ). Let us recall that, in view of Remark 3.2 , we can reduce to the case of η of the form η ( t ) = c for t < b and η ( t ) = 0 for t ≥ b . F urthermore, we only hav e to prov e the lo w er semicontin uity of the one-dimensional functional ( 3.2 ). Indeed, we hav e lim inf n →∞ 1 ε 2 n ¨ D × D η ε n ( x − y ) | ( v n ( x ) − v n ( y )) · ( T n ( x ) − T n ( y )) | ρ ( x ) ρ ( y ) d x d y (3.20) = lim inf n →∞ ˆ D − D ε n F ξ n ( v n , D ) η ( ξ ) d ξ ≥ ˆ R d lim inf n →∞  ˆ Π ξ F ξ ,y n ( v ξ ,y n , ρ, ( D ∩ ( D − εξ )) ξ y ) d H d − 1 ( y )  | ξ | η ( ξ ) d ξ ≥ ˆ R d ˆ Π ξ lim inf n →∞ F ξ ,y n ( v ξ ,y n , ρ, ( D ∩ ( D − εξ )) ξ y ) | ξ | η ( ξ ) d H d − 1 ( y )d ξ . Recall that, for an y function v : D → R d , we hav e set, for ξ ∈ R d and y ∈ Π ξ , D ξ := { t ∈ R : y + tξ ∈ D } and v ξ ,y ( t ) := v ( y + tξ ) for t ∈ D ξ y . Since v n → v in L 1 ( D ; R d ), we hav e that v ξ ,y n → v ξ ,y in L 1 ( D ξ y ; R d ) for a.e. ξ ∈ R d and H d − 1 -a.e. y ∈ Π ξ . By arguing as in [ 17 , 1 ], w e consider j ∈ N , a ∈ R giv en by Lemma 3.4 applied to the limit function ˆ u ξ y := u ξ ,y · ξ , and I z j = h a + z j , a + z +1 j i for ev ery z ∈ Z suc h that I z j ⊂ D ξ y . W e also define ρ ξ ,y [ I z j ] := min t ∈ I z j ρ ξ ,y ( t ) . F or ev ery K ⋐ D ξ y let and w ξ ,y j : K → R d to be the piecewise affine function interpolating b et ween u ξ ,y ( a + z j ) and u ξ ,y ( a + z j ) in I z j . By Lemma 3.4 we hav e, up to a subsequence, w ξ ,y j · ξ → ˆ u ξ y in L 1 ( K ) . Let us further denote by Z j,K := { z ∈ Z : I z j ∩ K  = ∅} . Notice that for n sufficien tly large, S z ∈ Z j,K I z j ⊆ ( D ∩ ( D − ε n ξ )) ξ y . SYMMETRIC TOT AL V ARIA TION ON POINT CLOUDS 11 W e hav e the following estimates lim inf n →∞ F ξ ,y n ( v ξ ,y n , ρ, ( D ∩ ( D − ε n ξ )) ξ y ) ≥ lim inf n →∞ X z ∈ Z j,K F ξ ,y n ( v n , ρ, I z j ) (3.21) ≥ X z ∈ Z j,K lim inf n →∞ F ξ ,y n ( v ξ ,y n , ρ, I z j ) ≥ X z ∈ Z j,K  ρ ξ ,y [ I z j ]  2 lim inf n →∞ F ξ ,y n ( v ξ ,y n , 1 , I z j ) Lemma 3.3 ≥ X z ∈ Z j,K  ρ ξ ,y [ I z j ]  2    h u ξ ,y  a + z + 1 j  − u ξ ,y  a + z j i · ξ    ≥ ˆ K | D w ξ ,y j ( t ) · ξ | X z  ρ ξ ,y [ I z j ]  2 1 I z j ( t ) d t. Since, b y assumption ( ρ 1), ρ is a contin uous function we infer X z ∈ Z j,K  ρ ξ ,y [ I z j ]  2 1 I z j → ( ρ ξ ,y ) 2 uniformly in K as j → ∞ . Therefore, using also ( ρ 2), b y the dominated conv ergence theorem we get for j → ∞ lim inf j →∞ ˆ K | D w ξ ,y j ( t ) · ξ | X z ∈ Z j,K  ρ ξ ,y [ I z j ]  2 1 I z j ( t ) d t (3.22) ≥ lim inf j →∞ ˆ K | D w ξ ,y j ( t ) · ξ | ( ρ ξ ,y ( t )) 2 d t − lim sup j →∞ ˆ K | D w ξ ,y j ( t ) · ξ |     X z ∈ Z j,K  ρ ξ ,y [ I z j ]  2 1 I z j ( t ) − ( ρ ξ ,y ( t )) 2     d t = lim inf j →∞ ˆ K | D w ξ ,y j ( t ) · ξ | ( ρ ξ ,y ( t )) 2 d t = lim inf j →∞ ˆ K | ξ |     D w ξ / | ξ | ,y j ( t ) · ξ | ξ |     ( ρ ξ / | ξ | ,y ( t )) 2 | ξ | d t , where w e hav e set K | ξ | := { t ∈ R : t/ | ξ | ∈ K } ⋐ D ξ / | ξ | y . F urthermore, ( ρ 2) also implies that lim inf j →∞ ˆ K | D w ξ ,y j ( t ) · ξ | X z ∈ Z j,K  ρ ξ ,y [ I z j ]  2 1 I z j ( t ) d t (3.23) ≥ lim inf j →∞ ˆ K | ξ | α 2     D w ξ / | ξ | ,y j ( t ) · ξ | ξ |     | ξ | d t . By standard low er semicontin uit y of the total v ariation and by the conv ergence w ξ / | ξ | ,y j · ξ | ξ | → ˆ u ξ / | ξ | y in L 1 ( K | ξ | ), from ( 3.23 ), w e deduce that for a.e. ξ ∈ R d \ { 0 } and H d − 1 -a.e. y ∈ Π ξ w e ha v e lim inf j →∞ ˆ K | D w ξ ,y j ( t ) · ξ | X z ∈ Z j,K  ρ ξ ,y [ I z j ]  2 1 I z j ( t ) d t ≥ α 2 | ξ |   D ˆ u ξ / | ξ | ,y   ( K | ξ | ) . (3.24) The com bination of ( 3.24 ) with ( 3.19 ) and ( 3.21 ) yields lim inf n →∞ F ξ ,y n ( v ξ ,y n , ρ, ( D ∩ ( D − ε n ξ )) ξ y ) ≥ α 2 | ξ |   D ˆ u ξ / | ξ | ,y   ( K | ξ | ) for ev ery K ⋐ D ξ y . T aking the limit as K ↗ D ξ y w e ha v e that K | ξ | ↗ D ξ / | ξ | y and lim inf n →∞ F ξ ,y n ( v ξ ,y n , ρ, ( D ∩ ( D − ε n ξ )) ξ y ) ≥ α 2 | ξ |   D ˆ u ξ / | ξ | ,y   ( D ξ / | ξ | y ) . Th us, w e infer that u ∈ B D ( D ) (cf. [ 2 ]). In a similar wa y , ( 3.21 )–( 3.22 ) imply that lim inf n →∞ F ξ ,y n ( v ξ ,y n , ρ, ( D ∩ ( D − ε n ξ )) ξ y ) ≥ ˆ D ξ/ | ξ | y ρ ξ / | ξ | ,y ( t ) 2 | ξ | d | ˆ u ξ / | ξ | ,y | ( t ) . (3.25) 12 S. ALMI, A. KUBIN, AND E. T ASSO F rom ( 3.20 ) and ( 3.25 ) we get that lim inf n →∞ 1 ε 2 n ¨ D × D η ε n ( x − y ) | ( v n ( x ) − v n ( y )) · ( T n ( x ) − T n ( y )) | ρ ( x ) ρ ( y ) d x d y (3.26) ≥ ˆ R d ˆ Π ξ  ˆ D ξ/ | ξ | y ρ ξ / | ξ | ,y ( t ) 2 d | ˆ u ξ / | ξ | ,y | ( t )  | ξ | 2 η ( ξ ) d H d − 1 ( y ) d ξ = ˆ R d  ˆ D ρ ( x ) 2 d    E u ( x ) ξ | ξ | · ξ | ξ |     | ξ | 2 η ( ξ ) d ξ = ˆ D  ˆ R d η ( ξ )    E u ( x ) | E u ( x ) | ξ · ξ    d ξ  ρ ( x ) 2 d | E u ( x ) | . Recalling the norm on symmetric matrices ϕ η : M d sy m → [0 , ∞ ) defined as ϕ η ( A ) = ˆ R d η ( ξ ) | Aξ · ξ | d ξ , w e rewrite ( 3.26 ) as lim inf n →∞ 1 ε 2 n ¨ D × D η ε n ( x − y ) | ( v n ( x ) − v n ( y )) · ( T n ( x ) − T n ( y )) | ρ ( x ) ρ ( y )d x d y = ˆ D ρ ( x ) 2 ϕ η  E u ( x ) | E u ( x ) |  d | E u ( x ) | = T V η ( u ; ρ 2 ) . This concludes the pro of of the liminf inequality . □ 4. Construction of a recover y sequence In this section we establish the limsup inequality . Theorem 4.1. Assume (K1) – (K3) , ( ρ 1) – ( ρ 2) , and ( 2.2 ) . F or every u ∈ L 1 ( D ; R d ; ν ) ther e exists { u n } n ∈ N ⊂ L 1 ( { X 1 , . . . , X m } ; R d ; ν n ) such that ( ν n , u n ) → ( ν, u ) in T L 1 and lim sup n →∞ GT V n ( u ε n ) ≤ T V η ( u ; ρ 2 ) . (4.1) Pr o of of The or em 4.1 . Without loss of generality we can assume that T V η ( u ; ρ 2 ) < ∞ . Hence, b y ( ρ 2) we hav e that u ∈ B D ( D ). Th us, we can approximate u with a sequence u k ∈ C ∞ ( D ; R d ) ∩ Lip( D ; R d ) such that u k → u in L 1 ( D ; R d ), E u k ∗ ⇀ E u , and | E u k | ( D ) → | E u | ( D ). By Reshetn y ak con tin uit y theorem (see, e.g., [ 3 , Theorem 2.39]) w e hav e T V η ( u k ; ρ 2 ) → T V η ( u ; ρ 2 ). Therefore it is enough to pro v e the limsup inequalit y for u k . T o simplify the notation w e drop the dep endence on k . Arguing as in [ 15 , Section 5], we may assume that the k ernel η is of the form η ( t ) = c for t < b and η ( t ) = 0 for t ≥ b . Indeed, if we can establish the limsup inequality under this assumption, then, b y the subadditivit y of the limsup, the same inequalit y also holds for functions of the form η = P l k =1 η k , for some l ∈ N , where each η k is of the ab ov e form. Then, we can extend to the case η compactly supp orted by appro ximation b y a sequence of piecewise constant functions η n : [0 , + ∞ ) → [0 , + ∞ ) such that η n ↘ η almost ev erywhere. Finally , to pro v e the limsup inequality in the general case of η with p ossibly unbounded supp ort, w e consider η α : [0 , + ∞ ) → [0 , + ∞ ) defined b y η α ( t ) := η ( t ) for t ≤ α and η α ( t ) := 0 for t > α . Then the energy can b e rewritten as GT V n ( u ) = GT V α n ( u ) + 1 ε 2 n ˆ {| T n ( x ) − T n ( y ) | >αε n } η ε n ( T n ( x ) − T n ( y )) | ( u ◦ T n ( x ) − u ◦ T n ( y )) · ( T n ( x ) − T n ( y )) | ρ ( x ) ρ ( y ) d x d y , where GT V α n denotes the energy with η α in place of η . Since it is enough to prov e the limsup estimate for Lipschitz functions, pro ceeding as in [ 15 , Pro of of Theorem 1.1, Step 4] one can SYMMETRIC TOT AL V ARIA TION ON POINT CLOUDS 13 sho w that the error GT V n ( u ) − GT V α n ( u ) tends to zero. Hence, it suffices to reduce to the case in whic h η is compactly supp orted. Set ˜ ε n := ε n − 2 b ∥ I d − T n ∥ L ∞ . By the assumption (K2) we infer η  | T n ( x ) − T n ( y ) | ε n  ≤ η  | x − y | ˜ ε n  , for almost every ( x, y ) ∈ D × D . Th us w e ha v e the upp er b ound GT V n ( u ) ≤  ˜ ε n ε n  d +2 1 ˜ ε 2 n ¨ D × D 1 ˜ ε d n η  | x − y | ˜ ε n  | ( u ◦ T n ( x ) − u ◦ T n ( y )) · ( T n ( x ) − T n ( y )) | ρ ( x ) ρ ( y )d x d y =  ˜ ε n ε n  d +2 1 ˜ ε 2 n ¨ D × D η ˜ ε n ( x − y ) | ( u ◦ T n ( x ) − u ◦ T n ( y )) · ( T n ( x ) − T n ( y )) | ρ ( x ) ρ ( y )d x d y . Since ˜ ε n ε n → 1, it suffices to pro v e the limsup inequality for 1 ε 2 n ¨ D × D η ε n ( x − y ) | ( u ◦ T n ( x ) − u ◦ T n ( y )) · ( T n ( x ) − T n ( y )) | ρ ( x ) ρ ( y ) d x d y for ev ery u ∈ C ∞ ( D ; R d ) ∩ Lip( D ; R d ), where, for simplicit y of notation, w e hav e dropp ed the tilde. By slicing, w e can rewrite the energy as ˆ D − D ε n  ˆ Π ξ F ξ ,y n (( u ◦ T n ) ξ ,y , ρ, ( D ∩ ( D − ε n ξ )) ξ y ) d H d − 1 ( y )  | ξ | η ( ξ )d ξ , where w e recall that F ξ ,y n ( v , ρ, ( D ∩ ( D − ε n ξ )) ξ y ) = 1 ε 2 n ˆ ( D ∩ ( D − ε n ξ )) ξ y | v ( t + ε n ) − v ( t )) · ( T ξ ,y n ( t + ε n ) − T ξ ,t n ( t )) | ρ ξ ,y ( t + ε n ) ρ ξ ,y ( t ) d t. Set I n := ( D ∩ ( D − ε n ξ )) ξ y . W e claim that for a.e. ξ ∈ R d \ { 0 } and H d − 1 -a.e. y ∈ Π ξ it holds lim n →∞ F ξ ,y n (( u ◦ T n ) ξ ,y , ρ, I n ) (4.2) = lim n →∞ 1 ε n ˆ I n | ( u ξ ,y ( t + ε n ) − u ξ ,y ( t )) · ξ | ρ ξ ,y ( t + ε n ) ρ ξ ,y ( t ) d t . T o prov e ( 4.2 ), we prov e that the follo wing error term tends to zero: 1 ε 2 n     ˆ I n  | ( u ξ ,y ( t + ε n ) − u ξ ,y ( t )) · ε n ξ | (4.3) − | (( u ◦ T n ) ξ ,y ( t + ε n ) − ( u ◦ T n ) ξ ,y ( t )) · ( T ξ ,y n ( t + ε n ) − T ξ ,y n ( t )) |  d t     ≤ 1 ε 2 n     ˆ I n  | ( u ξ ,y ( t + ε n ) − u ξ ,y ( t )) · ( ε n ξ − ( T ξ ,y n ( t + ε n ) − T ξ ,y n ( t )) | + | ( u ξ ,y ( t + ε n ) − u ξ ,y ( t )) · ( T ξ ,y n ( t + ε n ) − T ξ ,y n ( t )) | − | (( u ◦ T n ) ξ ,y ( t + ε n ) − ( u ◦ T n ) ξ ,y ( t )) · ( T ξ ,y n ( t + ε n ) − T ξ ,y n ( t )) |  d t     ≤ 1 ε 2 n     ˆ I n  | ( u ξ ,y ( t + ε n ) − u ξ ,y ( t )) · ( ε n ξ − ( T ξ ,y n ( t + ε n ) − T ξ ,y n ( t )) | + | ( u ξ ,y ( ε n + t ) − u ξ ,y ( t ) − (( u ◦ T n ) ξ ,y ( t + ε n ) − ( u ◦ T n ) ξ ,y ( t ))) · ( T ξ ,y n ( t + ε n ) − T ξ ,y n ( t )) |  d t     . Since u ∈ Lip( D ; R d ), w e can b ound the first term on the right-hand side of ( 4.3 ) by | I n | ε 2 n Lip( u ) | ε n ξ |∥ I d − T n ∥ L ∞ , 14 S. ALMI, A. KUBIN, AND E. T ASSO where Lip( u ) denotes the Lipsc hitz constant of u . After integration ov er D − D ε n and Π ξ , this quan tit y conv erges to zero as n → ∞ by ( 2.4 ). T o b ound the second term on the right-hand side of ( 4.3 ), we observe that | ( u ξ ,y ( ε n + t ) − u ξ ,y ( t ) − (( u ◦ T n ) ξ ,y ( t + ε n ) − ( u ◦ T n ) ξ ,y ( t ))) · ( T ξ ,y n ( t + ε n ) − T ξ ,y n ( t )) | ≤ | ( u ξ ,y ( ε n + t ) − u ξ ,y ( t ) − (( u ◦ T n ) ξ ,y ( t + ε n ) − ( u ◦ T n ) ξ ,y ( t ))) · ( T ξ ,y n ( t + ε n ) − y − ( t + ε n ) ξ ) | + | ( u ξ ,y ( ε n + t ) − u ξ ,y ( t ) − (( u ◦ T n ) ξ ,y ( t + ε n ) − ( u ◦ T n ) ξ ,y ( t ))) · ( y + ( t + ε n ) ξ − ( y + tξ )) | + | ( u ξ ,y ( ε n + t ) − u ξ ,y ( t ) − (( u ◦ T n ) ξ ,y ( t + ε n ) − ( u ◦ T n ) ξ ,y ( t ))) · ( T ξ ,y n ( t ) − ( y + tξ )) | . Arguing as ab o v e, each of these con tributions tends to zero after integration ov er D − D ε n , Π ξ , and I n in view of ( 2.4 ) and of the Lipschitz contin uit y of u . Thus, we hav e shown that lim n →∞ 1 ε 2 n ˆ D − D ε n ˆ Π ξ     ˆ I n  | ( u ξ ,y ( t + ε n ) − u ξ ,y ( t )) · ε n ξ | (4.4) − | (( u ◦ T n ) ξ ,y ( t + ε n ) − ( u ◦ T n ) ξ ,y ( t )) · ( T ξ ,y n ( t + ε n ) − T ξ ,t n ( t )) |  d t     d H d − 1 ( y ) η ( ξ )d ξ = 0 . Since ρ is uniformly b ounded from ab ov e and b elow (cf. ( ρ 2)), ( 4.4 ) implies ( 4.2 ) for a.e. ξ ∈ R d \ { 0 } and H d − 1 -a.e. y ∈ Π ξ . By the fundamental theorem of calculus, w e ha v e 1 ε n ˆ I n | ( u ξ ,y ( t + ε n ) − u ξ ,y ( t )) · ξ | ρ ξ ,y ( t + ε n ) ρ ξ ,y ( t ) d t = 1 ε n ˆ I n    ˆ t + ε n t ∇ u ξ ,y ( τ )d τ · ξ    ρ ξ ,y ( t + ε n ) ρ ξ ,y ( t ) d t. Recalling that u ∈ C ∞ ( D ; R d ) and ρ ∈ C ( D ), w e ha v e, for ev ery t ∈ J ⋐ D ξ y , lim n →∞ t + ε n t |∇ u ξ ,y ( τ )d τ · ξ | ρ ξ ,y ( t + ε n ) ρ ξ ,y ( t ) = |∇ u ξ ,y ( t ) · ξ | ( ρ ξ ,y ( t )) 2 . Since ( ρ 2) holds and 1 ε n    ˆ t + ε n t ∇ u ξ ,y ( τ )d τ · ξ    ≤ Lip( u ) | ξ | 2 , w e can apply the dominated con vergence theorem and infer lim n →∞ 1 ε n ˆ D − D ε n  ˆ Π ξ ˆ I n | ( u ξ ,y ( t + ε n ) − u ξ ,y ( t )) · ξ | ρ ξ ,y ( t + ε n ) ρ ξ ,y ( t ) d t d y  | ξ | η ( ξ )d ξ ≤ lim n →∞ ˆ D − D ε n  ˆ Π ξ ˆ I n t + ε n t |∇ u ξ ,y ( τ )d τ · ξ | ρ ξ ,y ( t + ε n ) ρ ξ ,y ( t ) d t d y  | ξ | η ( ξ )d ξ = ˆ R d  ˆ Π ξ ˆ Ω ξ y |∇ u ξ ,y ( t ) · ξ | ρ ξ ,y ( t ) 2 d t d y  | ξ | η ( ξ )d ξ = T V η ( u ; ρ 2 ) . This concludes the pro of of the limsup inequality ( 4.1 ). □ A cknowledgements This research has been supported b y the Austrian Science F und (FWF) through grants 10.55776/F65, 10.55776/Y1292, 10.55776/P35359, b y the OeAD-WTZ pro ject CZ04/2019 (M ˇ SM T ˇ CR 8J19A T013), by the Univ ersit y of Naples F ederico I I through the FRA Pro ject “ReSinA- pas”, and by the INdAM-GNAMP A pro ject “Sistemi multi-agen te e replicatore: deriv azione particellare e ottimizzazione” CUP E53C25002010001. S.A. is member of Grupp o Nazionale p er l’Analisi Matematica, la Probabilit` a e le loro Applicazioni (GNAMP A) of the Istituto Nazionale di Alta Matematica (INdAM). SYMMETRIC TOT AL V ARIA TION ON POINT CLOUDS 15 References [1] S. Almi, E. Da voli, A. Kubin, and E. T asso , On De Gior gi’s c onje cture of nonlo c al appr oximations for fr e e-disc ontinuity pr oblems: The symmetric gr adient c ase , Preprin t, (2024). [2] L. Ambrosio, A. Coscia, and G. 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(Stefano Almi) Dep ar tment of Ma thema tics and Applica tions “R. Caccioppoli”, University of Naples Federico I I, Via Cintia, Monte S. Angelo, 80126 Napoli, It al y. Email addr ess : stefano.almi@unina.it (Anna Kubin) Institute of Anal ysis and Scientific Computing, TU Wien, Wiedner Hauptstr. 8-10, 1040 Vienna, Austria. Email addr ess : anna.kubin@tuwien.ac.at (Eman uele T asso) Institute of Anal ysis and Scientific Computing, TU Wien, Wiedner Hauptstr. 8- 10, 1040 Vienna, Austria. Email addr ess : emanuele.tasso@tuwien.ac.at