The Binary Space Partitioning-Tree Process Supplementary Material

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1 The inary Space Partitioning-Tree Process Suppementary Materia Xuhui Fan in Li Scott. Sisson Schoo of omputer Science Fudan University Schoo of Mathematics and Statistics University of New South Waes Schoo of Mathematics and Statistics University of New South Waes Justification on the accumuated cut cost Let = sup x,x { x x } denotes the diameter of the poygon. θ, it is obviousy that the ength of the newy generated cut ine L(θ, u) is smaer or equa to, i.e., L(θ, u). Thus, we have the resut for the sum of perimeters in the -th partitioning resut as: k= P ( τ ) P ( ) + ( ) () h $ h θ # $ θ # w$ w # ccording to the Fatou s emma, we get im inf im inf k= P ( τ ) k= P ( τ ) [P ( )] + ( ) im inf < () k= which eads to im inf P ( τ ) surey. Since surey, we get = surey. k= P ( τ < amost ) is increasing for amost [ k= P ( τ )] = amost Mathematica formuation of the three-restrictions on the measure invariance. transation t: ( θ ) = λ t v t v ( θ ), where t v (x) = x + v, v R ;. rotation r: = λ rθ r θ, where r θ (x) = [ ] cos θ sin θ sin θ cos θ x refers to rotate the point x in an ange of θ ; Figure : The design of a set of divisions in Lemma. Red dashed ines denote the rough divisions, whie green dashed ines denote the grained divisions based on each consecutive rough divisions. 3. restriction ψ: ( θ ) = λ ψ ψ ( θ ), where refers to a sub-domain of ; ψ = {x x }, and θ refers to the set of cut ines for a the potentia cuts crossing through. Proof of Proposition Lemma. ssume two convex poygons and have the same ength on the ine segment (θ). There exists a set of countabe divisions passing through (θ) in the direction of θ + π. ach ine sub-segment of (θ) cut by consecutive divisions is covered by the intersection of the,, whie, can move in the direction of θ + π/. Proof. The set of divisions (a in the direction of θ + π ) can be designed into two stages. Stage, a set of rough divisions that pass through each vertices of the two poygons. Stage, sets of grained divisions based on the each consecutive rough divisions. Let denotes the distance between two seected consecutive divisions, w (w ) denotes

2 Running heading tite breaks the ine the maximum width of poygon ( ) between these two rough divisions and θ (θ ) is the smaest ange between the edge of poygon ( ) and division. We proceed the grained division in the foowing way. In the case of min{w tan θ, w tan θ }, there is no need to do further grained division; otherwise, et (θ, d ) = arg θ,d min{d tan θ, d tan θ }, the first grained division is paced in the position of d tan θ (see h in Figure ). The second grained division woud design based on the first one and proceed in a simiar way. Given the condition of d tan θ <, we can do the partition at the positions of: { d tan θ ( d tan θ ) } = where = d tan θ ( ) d tan θ =. Proposition. The famiy of partition probabiity measure ( θ ) keeps invariant under the operations of transation, rotation and restriction if and ony if we have a constant such that ( θ ) = (θ), R+. Proof. The reverse case is fuy discussed as in the main part of the paper. On the other hand, assume we have two sets of cut ines θ, θ with (θ ) = (θ). Given that the measure ( θ ) is invariant under the operations of transation, rotation and restriction, we need to prove the foowing equity: (3) ( θ ) = ( θ ) (4) To compete this, we first do rotation and transation operations on, which is := r θ t v, in a way that and project into the same image (θ). ased on Lemma, we divide and into countabe parts, where the intersection of these pair parts projects to the same images. That is: θ, θ = k,, = k (5) Y =,, (θ) Y, k N (6) = {L(θ, u) crossing, θ is fixed, u ies on (θ)} = {L(θ, u) crossing θ is fixed,, u ies on (θ)} (7) The additivity of measures indicates that: ( θ ) = k ( θ ) = k, ( θ,) ( θ ) q. (4) is correct if we can prove the foowing, ( θ, ) = ( θ ), k N (8) From the measure invariance under rotation and transation, we get ( θ ) = λ tv(r θ ( ))(t v (r θ ( θ ))). We aso get Thus, we get λ tvr θ, Π Y π = {Y } = Π Y t (ρπ) (9) Π Y θ, ( θ,) restriction = Π Y θ = λ ΠY t vr θ, q.(9) = λ ΠY, q.() restriction = λ ΠY = Proof of Proposition ( θ ) () (Π Y θ,) (Π Y θ,) (Π Y θ ) () Our partition woud resut in convex poygon. We have the integration resuts for the convex poygon. Lemma. The integration of the intersection ine in a triange over [, π] equas to the triange s perimeter. Proof. We first consider the acute triange (Top row of Figure ) case. Let {,, 3 } being the engths of the triange s edges and {,, } being the corresponding anges. ccording to the aw of sines, we have = sin = sin = 3 sin () where we use to denote the ratio between the ength and its corresponding ange. W..o.g., we are cutting the bock in the direction within. The projection scaar of is cacuated as =

3 Xuhui Fan, in Li, Scott. Sisson Here the nd equation hods due to the aw of Sines. The case of right triange (Midde row of Figure ) is straight forward. We can get 3 I = cos + cos + cos + 3 cos = = P ( ) (5) On the case of obtuse triange (ottom row of Figure ) 3 I = cos 3 cos + cos cos + 3 cos + cos = = P ( ) (6) Lemma 3. The integration of the ength of the bock s projected image in the direction of θ over (, π] equas to the perimeter of the bock, which is π (θ) dθ = P ( ). 3 F Figure : Top: cute Triange; Midde: Right Triange; ottom: Obtuse Triange. cos θ(θ = ). Whie θ is ranging from to, the integration of is dθ = cos θdθ = sin θ = cos( ) (3) y using the simiar routines, we can get the integration of a the projection ines I as: I = cos + cos + cos + cos + 3 cos + 3 cos = sin cos + sin cos + sin cos + sin cos + sin cos + sin cos = sin( + ) + sin( + ) + sin( + ) = sin + sin + sin = = P ( ) (4) Figure 3: From convex poygon with n vertices to convex poygon with n vertices. Proof. onvex poygon with n vertices can be divided into n trianges. Since we have the resut for the case of trianges, mathematica induction is used to get the concusion for any convex poygons. ssume we have the resut for convex poygon with n vertices, the additiona part for its transformation to convex poygon with n vertices is the triange. orrespondingy, the increase in the scaar projection is composed of two parts: L increase = sin θdθ = cos = (7)

4 Running heading tite breaks the ine where θ = F and L increase refers to the integration of F in the ange of. L increase = sin θdθ = cos = (8) Thus, the tota add amount is L increase = + ( + ) = +. This is exacty the increase of perimeter from the convex poygon with n vertices to convex poygon with n vertices. Thus, we can get the resut for a the convex poygons. Proposition is a direct resut of Lemma 3. onsistency Some notations are firsty defined for convenient reference. We use and to denote a domain and its subdomain, which is. M τ and N τ are individuay defined as the SP-Tree processes on and respectivey. The restriction is denoted as Π, in which we have Π M τ = N τ. so, we et c( ) denote the measure over the bock, which is c( ) = θ ω(θ) (θ) dθ and et O n denote the partition after n-th cut on the convex poygon. xtending partition from to For the SP-Tree process N τ, we et Z and {σ } N denotes the reated Markov chain and the corresponding time stops. For t, define m t to be the index such that t [σ mt, σ mt+ ], N t = Z mt. To extend N τ from to, et τ = and Y =. For n N, we define τ n+ and Y n+ inductivey as: τ n+ := min{σ mτn +, τ n + ξ n c(y n ) c(z mτn ) } (9) where ξ n is generated from the exponentia distribution with mean. { iftyn, (Z Y n+ = ), τ mτn n+ = σ mτn + ; gencut (Y n ), otherwise. () where ift Yn, (Z mτn ) denotes extending the existing cut to the arger domain and gencut (Y n ) refers to the case that there wi be a new cut generated in that does not cross into. ccording to the resuts of Proposition V.6 of chapter VI in [], the defined process {Y n, τ n } are we-defined. Prove the correctness t >, the waiting time for the next cut in X is: ζ t = τ n+ t () ccording to τ n+ s definition (q. (9)), ζ t foows the exponentia distribution with the rate being c(y nt ). What is more, the probabiity of the event τ n+ = σ mτn + occurs with probabiity c(o n )/c(o n ). For the newy extended case {Y n } n, whie Y n crosses through, the probabiity measure on θ is in proportion to ω(θ) (θ) and u ocates ony on (θ). Thus, we get P = c(o n ) c(o n ) = ω(θ) c(o n ) ω(θ) (θ) θ ω(θ) (θ) dθ (θ) () whie Y n does not cross through,, the probabiity measure on θ \ is in proportion to ω(θ)( (θ) (θ) ) and u ocates ony on \ (θ) (with the ength (θ) (θ) ). Thus, we get P = c(o n ) c(on ) ω(θ)( (θ) (θ) ) c(on ) θ ω(θ)( (θ) (θ) )dθ (θ) (θ) = ω(θ) (3) c(on ) q. () and q. (3) show that the probabiity measure of Y n equas to the one that directy generated in the domain of. Therefore, the partition constructed by q. (9) and q. () is a reaization of SP-Tree process in. ccording the transfer theorem (Theorem V.3 of chapter VI in []), the partition distribution is consistent from to. F MM for the SP-RM gorithm dispays an MM soution for the SP-RM. gorithm MM for SP-RM Input: Training data X, udget τ, Number of partices Output: reaization of the SP-Tree process; coordinates {(ξ i, η i )} n i= of X : Initiaize the partition and nodes coordinates : for t = : T do 3: Use -SM agorithm to update the partition structure, according to gorithm ; 4: Update nodes coordinates {ξ i, η i } n i= according to q. (4). 5: end for F. Updating nodes coordinates (ξ i, η i ) n i= (ξ i, η i ) s updating is impemented through the Metropois- Hastings agorithm. We propose the new vaues of ξ i, η i

5 Xuhui Fan, in Li, Scott. Sisson Figure 4: Toy ata Partition Visuaization (ase ). with the uniform distribution in [, ] and the acceptance ratio min(, α) is as foows: α(ξ i, ξ i ) = α(η j, η j ) = j = P (e ij ξ i, ξ \i, η j, θ) j = P (e ij ξ i, ξ \i, η j, θ) ; i = P (e i j η j, η \j, ξ i, θ) i = P (e i j η j, η \j, ξ i, θ) (4) G Visuaization of ase Figure 4 shows the visuaization of ase. References [] anie M. Roy. omputabiity, Inference and Modeing in Probabiistic Programming. Ph thesis, MIT,.