The decomposition of the higher-order homology embedding constructed from the $k$-Laplacian

1 Motivation . The k-th homology vector space Hk provides rich geometric information on manifolds/networks . For instance , the zeroth , the first , and the second homology vector spaces identify the connected components , the loops , and the cavities in the manifold , respectively . Topological Data Analysis ( TDA ) [ 24 , 58 ] ( as well as other early works in this field ) has found wide use in analyzing biological [ 31 , 47 ] , human behavior [ 1 , 64 ] , or other complex systems [ 58 ] . Even though they easily generalize to k ≥ 1 , additional efforts are needed to extract topological features ( e.g. , instances of loops ) besides ranks due to the combinatorial complexity of the structures that support them . Spectral methods based on k-Laplacians ( Lk ) , by contrast , investigate Hk in a linear algebraic manner ; abundant geometric information can be extracted from the homology embedding Y ( the null space eigenvectors of Lk ) of Hk . Analysis of the eigenfunctions ( of H0 ) [ 14 , 37 , 40 , 49 ] of the graph Laplacian L0 is pivotal in providing guarantees for spectral clustering and community detection algorithms in well-separated datasets . Recent advances in this field [ 4 , 12 , 48 ] extend the existing spectral algorithms based on L0 to k ≥ 1 ; however , the theoretical analysis of Y of Hk , unlike spectral clustering , is less developed , in spite of intriguing empirical results by [ 21 ] . Here , we put these observations on a formal footing based on the concepts of connected sum and prime decomposition of manifolds ( Section 2 and 3 ) . We examine these operations through the lens of the ( subspace ) perturbation to the homology embedding Y of the discrete k-Laplacian Lk ( an estimator of the continuous k-Hodge Laplacian operator ∆k ) on finite samples ( Section 4 ) . This framework 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . finds applications in , i.e. , identifying the shortest homologous loops ( Section 5 ) . Lastly , we support our theoretical claims with numerous empirical results from point clouds and images . 2 Background in Hodge theory and topology . Simplicial and cubical complex . An abstract complex is a natural extension of a graph designed to capture higher-order relationships between its vertices . A simplicial k-complex ( used when the data are point clouds or networks ) is a tuple SCk = ( Σ0 , · · · , Σk ) , with Σℓ being a set of ℓ dimensional simplices , such that every face of a simplex σ ∈ Σℓ is in Σℓ−1 for ℓ ≤ k. As a side note , a graph G = ( V , E ) is an SC1 ; and SC2 = ( V , E , T ) , which is commonly used in edge flow learning [ 12 , 48 ] , is obtained by adding a set of 3-cliques ( triangles ) T of G. This procedure extends to defining Σℓ as the set of all ℓ-cliques of G , with the resulting complex called a clique complex of the graph G. This complex is also known as a Vietoris-Rips ( VR ) complex if G is the ϵ-radius neighborhood graph used in the manifold learning literature [ 12 , 14 , 53 ] , The cubical k-complex CBk = ( K0 , · · · , Kk ) is a complex widely used with image data . The difference between this complex and the SCk is that a CBk is a collection of sets of ℓ-cubes , for ℓ < k. Note that we write Σ0 = K0 = V the vertex set and Σ1 = K1 = E the edge set . Σ2 = T and K2 = R are the triangle and rectangle set , respectively . Additionally , we define nℓ = |Σℓ| ( or = |Kℓ| ) to be the cardinality of the ℓ-dimensional cells and let n = n0 for simplicity . For more information about building various complexes on different datasets please refer to Otter et al . [ 43 ] . k-cochain . By choosing an orientation for every k-simplex σk , i ∈ Σk ( or Kk ) , one can define a finite-dimensional vector space Ck ( k-cochain space1 ) . An element ωk = ∑ i ωk ( σk , i ) σk , i ∈ Ck is called a k-cochain ; one can further express ωk as ωk = ( ωk,1 , · · · , ωk , nk ) ⊤ ∈ Rnk by identifying each σk , i with the standard basis vector ei ∈ Rnk . Functions on nodes and edge flows , for example , are elements of C0 and C1 , respectively . Boundary matrix . The k-th boundary matrix Bk [ 35 ] maps a k-cochain of k-cells ( simplices/cubes ) σk to the ( k− 1 ) -cochain of its faces , i.e. , Bk : Ck → Ck−1 . Bk ∈ { 0 , ±1 } nk−1×nk is a sparse binary matrix , with the sign of the non-zero entries σk−1 , σk given by the orientation of σk w.r.t . its face σk−1 . Hence , different SC or CB will induce different Bk . For k = 1 on either the SC or CB , the boundary map is the graph incidence matrix , i.e. , ( B1 ) [ x ] , [ x , y ] = 1 , ( B1 ) [ y ] , [ x , y ] = −1 , and zero otherwise ; for k = 2 , each column of B2 contains the orientation of a triangle/rectangle w.r.t . its edges . Specifically , for an SC , ( B2 ) [ x , y ] , [ x , y , z ] = ( B2 ) [ y , z ] , [ x , y , z ] = 1 , ( B2 ) [ x , z ] , [ x , y , z ] = −1 , and 0 otherwise ; for a CB , ( B2 ) [ x , y ] , [ x , y , z , w ] = ( B2 ) [ y , z ] , [ x , y , z , w ] = ( B2 ) [ z , w ] , [ x , y , z , w ] = 1 , ( B2 ) [ x , w ] , [ x , y , z , w ] = −1 , and 0 otherwise . Simplex σk+1 is a coface of σk iff σk is a face of σk+1 ; let coface ( σk ) be the set of all cofaces of σk . The ( k− 1 ) -th coboundary matrix B⊤k ( adjoint of Bk ) maps σk−1 , as a ( k − 1 ) -cochain , to the k-cochain of coface ( σk−1 ) . k-Hodge Laplacian . Let Wℓ be a diagonal non-negative weight matrix of dimension nℓ , with [ Wℓ ] σ , σ representing the weight of the ℓ-simplex/cube σ and wℓ ← diag ( Wℓ ) . The weighted k-Hodge Laplacian2 [ 26 ] is defined as Lk = A⊤k Ak +Ak+1A ⊤ k+1 , where Aℓ = W −1/2 ℓ−1 BℓW 1/2 ℓ for ℓ = k , k + 1 . ( 1 ) The weights capture combinatorial or geometric information and must satisfy the consistency relation wℓ ( σℓ ) = ∑ σℓ+1∈coface ( σℓ ) wℓ+1 ( σℓ+1 ) ( in matrix form : wℓ = |Bℓ+1|wℓ+1 ) for ℓ = k , k − 1 . Hence Ak can be seen as normalized boundary matrix . To determine the weight for the ( k + 1 ) - simplices , one can selected wk+1 to be constant [ 48 ] or based on ( a product of ) pairwise distance kernel ( for k = 0 , 1 ) so that the large sample limit exists [ 12 , 14 ] . The first and second terms of ( 1 ) are called respectively the down ( Ldownk = A ⊤ k Ak ) and up ( L up k = Ak+1A ⊤ k+1 ) Laplacians . For k = 0 , the down component disappears and the resulting L0 is the symmetric normalized graph Laplacian used in spectral clustering [ 55 ] and Laplacian Eigenmap [ 5 ] . 1We use chain and cochain interchangeably , see Lim [ 35 ] for the distinction between them . 2In this paper , these matrices are also called the k-Laplacian for simplicity . k-th homology vector space and embedding . The homology vector space Hk is a subspace of Ck ( loop space ) such that every k-cycle ( expressed as a k-cochain ) in Hk is not the boundary of any ( k+ 1 ) -cochain . In mathematical terms , Hk : = ker ( Ak ) /im ( Ak+1 ) . The rank of the subspace is called the k-th Betti number βk = dim ( Hk ) , which counts the number of “ loops ” ( homology classes ) in the SC . Hk is equivalent to the null space of Lk [ 35 , 48 ] ; therefore , a basis ofHk can be obtained by the eigenvectors Y = [ y1 , · · · , yβk ] ∈ R nk×βk of Lk with eigenvalue 0 . The homology embedding maps a k-simplex σk to Yσk , : = [ y1 ( σk ) , · · ·yβk ( σk ) ] ⊤ ∈ Rβk . Note that the basis Y is only identifiable up to a unitary transformation ; hence , individual homology embedding coordinates might change with a different basis Y . Continuous operators on manifolds . The k-cochains are the discrete analogues of k-forms [ 59 ] . For k = 1 , the following path integral [ 59 ] ( along the geodesic γ ( t ) connecting x and y ) relates a 1-cochain ω to a 1-form v ( vector field ) : ω ( [ x , y ] ) = ∫ y x v ( γ ( t ) ) γ′ ( t ) dt . To estimate a vector field from ω , one can solve a least-squares problem [ 12 ] , which is the inverse operation of the path integral ( e.g. , the vector fields in Figure 1 are estimated from Y ) . Similarly , one can define the differential d and codifferential δ operators which are analog to B⊤k+1 and Bk , respectively . The ( continuous ) k-Hodge Laplacian operators , which act on k-forms , can be defined for manifolds too , i.e. , by ∆k = dk−1δk + δk+1dk . The homology group ( the continuous version ofHk ) is defined as the null of ∆k . Its elements are harmonic k-forms ζk , computed by solving ∆kζk = 0 with proper boundary conditions ; they represent the continuous version of the discrete homology basis Y . Connected sum and manifold ( prime ) decomposition . The connected sum [ 33 ] of two d dimensional manifoldsM =M1♯M2 is built from removing two d dimensional “ disks ” from each manifoldM1 , M2 and gluing together two manifolds at the boundaries ( technical details in [ 33 ] ) . The analog of the connected sum for the abstract complexes will be defined in Section 3 . The connected sum is a core operation in topology and is related to the concept of manifold ( prime ) decomposition . Informally speaking , the prime decomposition aims to factorize a manifoldM into κ smaller building blocks ( M =M1♯ · · · ♯Mκ ) so that eachMi can not be further expressed as a connected sum of other manifolds . The well-known classification theorem of surfaces [ 3 ] states that any oriented and compact surface is the finite connected sum of manifolds homeomorphic to either a circle S1 , a sphere S2 , or a torus T2 . Classification theorems for d > 2 are currently unknown ; fortunately , the uniqueness of the prime decomposition for d = 3 ( up to homeomorphism ) was shown ( Kneser-Milnor theorem [ 39 ] ) . Recently , Bokor et al . [ 9 ] ( Corollary 2.5 ) showed the existence of factorizations of manifolds with d ≥ 5 , even though they might not be unique . In this paper , we are interested in the following : given finite samples from M , which is a κ-fold connected sum ofMi , can this decomposition be recovered from the discrete homology embedding Y ofM ? Namely , we would like to understand how Y relates to the homology embedding of each prime manifoldMi .

Summary: This paper studies the kernel of the higher-order combinatorial Laplacian, which equivalently amounts to a study of higher-order homology. In particular, given finite samples from a manifold with a ground truth decomposition into a connected sum of smaller ("prime") manifolds, they study the question of when it is possible to computationally recover a "nice" homology basis such that the basis vectors are resolved into independent components contributed from each of the prime manifolds. In this sense, the theoretical aim is quite nice and fundamental. The authors are able to provide a positive answer to their question in the setting where the prime manifolds are connected over a small, sparsely connected region. The proof technique is similar to that for obtaining bounds in spectral graph theory, but is carefully implemented because the problem has lots of block matrices floating around. Finally, the authors propose algorithmic applications of this method to the problem of detecting shortest loops in data, and exemplify their constructions on several point cloud and image datasets.