A simplicial complex is a collection K of non-empty subsets of a finite set V. The elements of V are called vertices of K, and the elements of K are called simplices. A simplicial complex has the following properties: 1) { υ } in K for all υ   ∈   V , and 2) τ   ⊂   σ and σ   ∈   K guarantees that τ   ∈   K .

Additionally, we say that a simplex has dimension n or is an n-simplex if it has cardinality n + 1. We use S n ( K ) to denote the collection of n-simplices contained in K.

While there are a variety of approaches to create a simplicial complex from data, our examples use a standard construction for approximation of point clouds. Given a metric space X with metric d and real number ε   ≥   0 , the Vietoris-Rips complex for X, denoted by VR ε   ( X ) , is defined as VR ε   ( X )   =   { σ ∈ S n ( K ) | d ( x , y )   ≤   ε   for all   x , y ∈ σ } .

That is, given a set of discrete points X and a metric d, we build a VR complex at scale ε by forming an n-simplex if and only if n + 1 points in X are pairwise within ε distance of each other.

Given a simplicial complex K and an abelian group G, the group of n-chains in K with coefficients in G is defined as C n   ( K ; G )   : =   G S n ( K ) .

Formally, we regard G S n ( K ) as a group of functions S n ( K )   →   G under element-wise addition. Alternatively, we may view C n   ( K ; G ) as a group of formal G-linear combinations of n-simplices, i.e., { ∑ σ x σ σ | x σ   ∈   G  and   σ   ∈   S n   ( K ) } .

Remark 2.1.

We will focus on the cases where G is ℚ (the field of rationals), ℤ (the group of integers), or F 2 (the 2-element field). Since we are most interested in the case G   =   ℚ , we adopt the shorthand C n   ( K )   =   C n ( K ; ℚ ) .

 An element x   =   ( x σ ) σ ∈ S n ( K )   ∈   G S n ( K ) is called an n-chain of K. As in this example, see we will generally use a bold-face symbol for the tuple x and corresponding light-face symbols for entries x σ . The support of an n-chain is the set of simplices on which x σ is nonzero: supp   ( x )   : =   { σ   ∈   S n   ( K )   |   x σ   ≠   0 } .

The ℓ 0 norm ¹ and ℓ 1 norm ² of x are defined as ‖ x ‖ 0   : =   | supp   ( x ) | ‖ x ‖ 1   : =   ∑ σ ∈ S n ( K ) | x σ | .

Remark 2.2

 (Indexing conventions for chains and simplices). As chains play a central role in our discussion, it will be useful to establish some special conventions to describe them. These conventions depend on the availability of certain linear orders, either on the set of vertices or the set of simplices.

Case 1: Vertex set V has a linear order ≤ . Every vertex set V discussed in this text will be assigned a (possibly arbitrary) linear order. Without risk of ambiguity, we may therefore write ( υ 0 , … , υ n ) for the n-chain that places a coefficient of 1 on σ   =   { υ 0   ≤ … ≤   υ n } and 0 on all other simplices.

Case 2: Simplex set S n ( K ) has a linear order ≤ . We will sometimes define a linear order on S n ( K ) . This determines a unique bijection σ ( n )   :   { 1 , … , | S n ( K ) | } → S n ( K ) such that σ i ( n )   ≤   σ j ( n ) iff i   ≤   j . This bijection determines an isomorphism. ϕ   :   C n ( K ; G )   =   G S n ( K )   →   G | S n ( K ) | . such that ϕ ( x ) i   =   x σ i for all i. Provided a linear order ≤ , we will use x to denote both x and ϕ ( x ) and rely on context to clarify the intended meaning.

 For each n   ≥   1 , the boundary map ∂ n   :   C n ( K )   →   C n ‐ 1 ( K ) is the linear transformation defined on a basis vector ( υ 0 ,   υ 1   , … ,   υ n ) by ∂ n   ( υ 0 , υ 1 , … , υ n )   =   ∑ i = 0 n   ( ‐ 1 ) i   ( υ 0 , … , υ i ^ , … , υ n ) , where υ i ^ omits υ i from the vector. This map extends linearly from the basis of n-simplices to any n-chain in C n ( K ) . By an abuse of notation, we also denote the matrix representation of this boundary map, known as the boundary matrix, as ∂ n . The boundary matrix is parametrized by the n-simplices S n   ( K ) along the columns and n   ‐   1 simplices S n ‐ 1   ( K ) along the rows.

 The collection ( C n ( K ) ) n ≥ 0 along with the boundary maps ( ∂ n ) n ≥ 0 form a chain complex … C n + 1 ( K ) → ∂ n + 1 C n ( K ) → ∂ n C n ‐ 1 ( K ) → ∂ n − 1 … → ∂ 3 C 2 ( K ) → ∂ 2 C 1 ( K ) → ∂ 1 C 0 ( K ) → ∂ 0 0 .

Remark 2.3

 (Indexing conventions for boundary matrices). In general, boundary matrix ∂ n is regarded as an element of G S n ‐ 1 ( K )   ×   S n ( K ) , that is, as an array with columns labeled by n-simplices and rows labeled by n   ‐   1 simplices. However, given linear orders on S n ‐ 1 ( K ) and S n ( K ) , we may naturally regard ∂ n as an element of G | S n ‐ 1 ( K ) |   ×   | S n ( K ) | , see Remark 2.2.

The boundary of an n-chain x is ∂ n ( x ) . An n-cycle is an n-chain with zero boundary. The set of all n-cycles forms a subspace Z n ( K )   : =   ker ( ∂ n ) of C n ( K ) . An n-boundary is an n-chain that is the boundary of ( n   +   1 ) chains. The set of all n-boundaries forms a subspace B n ( K )   : =   im   ( ∂ n + 1 ) of C n ( K ) . We refer to Z n and B n as the space of cycles and space of boundaries, respectively.

It can be shown that ∂ n ∘ ∂ n + 1 ( x )   =   0 for all x   ∈   C n + 1 ( K ) ; colloquially, “a boundary has no boundary.” Equivalently, ∂ n   ∘   ∂ n + 1 is the zero map. Since the boundary map takes a boundary to 0, an n-boundary must also be an n-cycle. Therefore, B n ( K )   ⊆   Z n ( K ) .

The nth homology group of K is defined as the quotient. H n ( K )   : =   Z n ( K ) / B n ( K ) .

Concretely, elements of H n ( K ) are cosets of the form [ z ]   =   { z ′   ∈   Z n ( K ) |   z ′   ‐   z   ∈   B n ( K ) } . ³ An element h   ∈   H n ( K ) is called an n-dimensional homology class. We say that a cycle z   ∈   Z n ( K ) represents h, or that z is a cycle representative of h if h   =   [ z ] . We say that z and z ′ are homologous if [ z ]   =   [ z ′ ] .

Example: Consider the example in Figure 2A, which illustrates two homologous 1-cycles and the example in Figure 2B, which illustrates two non-homologous cycles.

Remark 2.4. The term homological generator has been used differently by various authors: to refer to an arbitrary nontrivial homology class, an element in a (finite) representation of H n ( K ) , as a set of cycles which generate the homology group, or (particularly in literature surrounding optimal cycle representatives) interchangeably with cycle representative. We favor the term cycle representative, to avoid ambiguity.

A (dimension-n) homological cycle basis for H n ( K ) is a set of cycles ℬ   =   { z 1 , … , z m } such that [ z i ]   ≠   [ z j ] when i   ≠   j , and { [ z 1 ] , … , [ z m ] } is a basis for H n ( K ) . Modulo boundaries, every n-cycle can be expressed as a unique linear combination in ℬ .

Homological cycle bases have several useful interpretations. It is common, for example, to think of a 1-cycle as a type of “loop,” generalizing the intuitive notion of a loop as a simple closed curve to include more intricate structures, and to regard the operation of adding boundaries as a generalized form of “loop-deformation.” Framed in this light, a homological cycle basis ℬ for H 1   ( K ) can be regarded as a basis for the space of loops-up-to-deformation in K. Higher dimensional analogs of loops involve closed “shells” made up of n-simplices.

Another interpretation construes each nontrivial homology class [ z ]   ≠   0 as a hole in K. Such holes are “witnessed” by loops or shells that are not homologous to the zero cycle. Viewed in this light, H n ( K ) can naturally be regarded as the space of ( n + 1 ) dimensional holes in K. The rank of the nth homology group β n ( K )   : =   dim ( H n ( K ) )   =   dim ( Z n ( K ) )   ‐   dim ( B n ( K ) ) , therefore quantifies the “number of gray independent holes” in K. We call β n the nth Betti number of K.

Example: Consider the gray disks in Figure 1 [similar to Carlsson (2009)] with different numbers of holes and cycle representatives.

A filtration on a simplicial complex K is a nested sequence of simplicial complexes K • = ( K ε i ) i ∈ { 1 , … , T } such that K ε 1   ⊆   K ε 2   ⊆ … ⊆   K ε T   =   K , where ε 1   < ⋯ <   ε T are real numbers. A filtered simplicial complex is a simplicial complex equipped with a filtration K • .

Example Let X be a metric space with metric d, and let ε 1   < ⋯ <   ε T be an increasing sequence of non-negative real numbers. Then the sequence K • = ( K ε i ) i ∈ { 1 , … , T } defined by K ε i = V R ε i ( X ) is a filtration on K.

The data of a filtered complex is naturally captured by the birth function on simplices, defined Birth   :   K   →   ℝ ,   σ   ↦   min   { ε i   :   σ ∈ K ε i } .

We regard the pair ( K , Birth ) as a simpilicial complex whose simplices are weighted by the birth function. For convenience, we will implicitly identify the sequence K • with this weighted complex. Thus, for example, when we say that σ   ∈   K has birth parameter t, we mean that σ   ∈   K and Birth ( σ )   =   t .

Definition 2.5. A filtration K • is simplex-wise if one can arrange the simplices of K into a sequence ( σ 1 , … , σ | K | ) such that K ε i   =   { σ 1 , … , σ i } for all i. A simplex-wise refinement of K • is a simplex-wise filtration K • ′ such that each space in K • can be expressed in form { σ 1 , … , σ j } for some j.

 As an immediate corollary, given a simplex-wise refinement of K • , we may naturally interpret each boundary matrix ∂ n as an element of G | S n - 1 ( K ) |   ×   | S n ( K ) | , see Remark 2.3 Under this interpretation, columns (respectively, rows) with larger indices correspond to simplices with later birth times; that is, birth time increases as one moves left-to-right and top-to-bottom.

If we regard C n ( K ε i ; G ) as a family of formal linear combinations in S n ( K ε i ) , then it is natural to consider C n ( K ε i ; G ) as a subgroup of C n ( K ε j ; G ) for all i   <   j . In particular, we have an inclusion map ι : C n ( K ε i ; G ) → C n ( K ε j ; G ) ,   ∑ σ ∈ S n ( K ε i ) x σ σ   ↦   ∑ σ ∈ S n ( K ε i ) x σ σ   +   ∑ τ ∉ S n ( K ε i ) 0 ⋅ τ .

Given a simplex-wise refinement K • ′ , one can naturally regard c as an element ( c 1 , c 2 , … ) of G | S n ( K ε i ) | . From this perspective, ι has a particularly simple interpretation, namely “padding” by zeros: ι ( c )   =   ( c 1 , c 2 , … ︸ c , 0 , … , 0 ) .

Similar observations hold when one replaces C n with either Z n , the space of cycles, or B n , the space of boundaries.

The notion of birth for simplices has a natural extension to chains, as well as a variant called death. Formally, the birth and death parameters of c   ∈   C n ( K ) are Birth ( c ) = min { ε i   :   c   ∈   C n ( K ε i ) }   Death   ( c )   =   { min { ε i   :   c   ∈   B ( K ε i ) } c   ∈   B ( K ) ∞ else .

In the special case where c is a cycle, Birth   ( c ) is the first parameter value where [ c ] represents a homology class, and Death   ( c ) is the first parameter value where [ c ] represents the zero homology class. Thus, the half-open lifespan interval ℒ ( c )   =   [ Birth ( c ) , Death ( c ) ) , is the range of parameters over which c represents a well-defined, nonzero homology class.

A (dimension-n) persistent homology cycle basis is a subset ℬ   ⊆   Z n ( K ) with the following two properties:

1. Each z   ∈   ℬ has a nonempty lifespan interval.

2. For each i   ∈   { 1 , … , T } , the set ℬ ε i   : =   { z   ∈   ℬ   :   ε i   ∈   ℒ ( z ) } , is a homological cycle basis for H n ( K ϵ i ) .

Every filtration of simplicial complexes ( K ε i ) i ∈ { 1 , … , T } admits a persistent homological cycle basis ℬ (Zomorodian and Carlsson, 2005). Moreover, it can be shown that the multiset of lifespan intervals (one for each basis vector), called the dimension-n barcode of K • , Barcode n   =   { ℒ ( z )   :   z   ∈   ℬ } , is invariant over all possible choices of persistent homological cycle bases ℬ (Zomorodian and Carlsson, 2005).

Example: Consider the sequence of simplicial complexes ( K 1 , K 2 , K 3 ) shown in Figure 3E. The set ℬ   =   { x 4 , x 5 , x 6 } is a (dimension-1) persistent homological cycle basis of the filtration. The associated dimension-1 barcode is Barcode 1 = { [ 1,2 ) , [ 2 , ∞ ) , [ 3 , ∞ ) } where [ 2 , ∞ ) and [ 3 , ∞ ) are the lifespans of x 5 and x 6 , respectively.

Barcodes are among the foremost tools in topological data analysis (Ghrist, 2008; Edelsbrunner and Harer, 2008), and they contain a great deal of information about a filtration. For example, it follows immediately from the definition of persistent homological cycle bases that β n ( K ε i )   =   | ℬ ε i | for all n and i. Consequently, β n ( K ε i ) = | { J   ∈   Barcode n   :   ε i ∈ J } | .

Barcodes and persistent homology bases may be computed via the so-called R   =   D V decomposition (Cohen-Steiner et al., 2006) of the boundary matrices ∂ n . Details are discussed in the Supplementary Material.

Linear programming seeks to find a set of decision variables x   = ( x 1 , … , x η ) T which optimize a linear cost (or objective) function c T x subject to a set of linear (in)equality constraints a 1 T x   =   b 1 , … , a μ T x   =   b μ . Any linear optimization problem can be written as a Linear Program (LP) in standard form minimize  c T x  subject to  A x   =   b x   ≥   0 (1) where A is the μ   ×   η matrix with coefficients of the constraints as rows and b   =   ( b 1 , … , b μ ) T . Linear programming is well-studied and discussed in many texts (Bertsimas and Tsitsiklis, 1997; Vanderbei, 2014; Boyd and Vandenberghe, 2004).

The optimal solution x * satisfies the constraints while optimizing the objective function, yielding the optimal cost c T x * . The feasible set of solutions in a linear optimization problem is a polyhedron defined by the linear constraints. In general, the optimal solution of a (non-degenerate) LP will occur at a vertex of the polyhedron and can be solved with the standard simplex algorithm, which traverses through the edges of the polytope to vertices in a cost reducing manner, or interior point methods, which traverse along the inside of the polytope to reach an optimal vertex. In the worst-case, the complexity of the simplex method is exponential, yet it often runs remarkably fast, while interior point methods are polynomial time algorithms.

Standard LPs search for real-valued optimal solutions, but in some instances, a restriction of the decision variables, such as requiring integral solutions, may be necessitated. The mixed integer programming (MIP) problem is written minimize  c T x + d T y  subject to  A x + B y = b x , y ≥ 0 x  integer, (2) for matrices A , B and vectors b , c , d . A standard LP has fewer constraints, and thus, will have optimal cost less than or equal to that of the analogous MIP. MIPs are much more challenging to solve than LPs, as they are discrete as opposed to convex optimization problems, and no efficient general algorithm is known (Bertsimas and Tsitsiklis, 1997). However, LP relaxations, (exponential-time) exact, (polynomial-time) approximation, and heuristic algorithms can be used to obtain solutions to MIPs.

In this paper, we determine optimal cycle representatives with both LP and MIP formulations.

Given a homology class h   =   [ x O r i g ]   ∈   H n ( K ; G ) and a function loss :   Z n ( K ; G )   →   ℝ , how does one find a cycle representative of h on which loss attains minimum? This problem is equivalent to solving the following program defined in Dey et al. (2011): minimize  loss ( x )  subject to  x = x O r i g + ∂ n + 1 w w ∈ C n + 1 ( K ; G ) . (3)

This formulation considers all cycle representatives homologous to x Orig , i.e. that differ by a boundary, and selects the optimal representative x which minimizes loss . The program in Eq. 3 is correct because the coset h can be expressed in the form h   =   x Orig   +   B n ( K ; G )   =   { x O r i g   +   ∂ n + 1 w | w   ∈   C n + 1 ( K ; G ) } .

In practice, a cycle representative x O r i g is almost always provided together with the initial problem data (which consists of K, G, loss, and h), so the central challenge lies with solving the program in Eq. 3.

Several variants of the program in Eq. 3 have been studied, especially where loss ( x )   =   ‖ x ‖ 0 or loss ( x )   =   ‖ x ‖ 1 . For a survey of results when G   =   F 2 , see Chen and Freedman (2010b). For a discussion of results when G   =   ℤ , see Dey et al. (2011). Broadly speaking, minimizing against ℓ 0 tends to be hard, even when K has attractive properties such as embeddability in a low-dimensional Euclidean space (Borradaile et al., 2020). Minimizing against ℓ 1 is hard when G   =   F 2 (since, in this case, ℓ 1   =   ℓ 0 ), but tractable via linear programming when G   ∈   { ℚ , ℝ } .

An interesting variant of the minimal cycle representative problem is the minimal persistent cycle representative problem. This problem was described in Chen et al. (2008) and may be formulated as follows: given an interval [ a , b ) ∈ Barcode n ( K • ) , solve minimize  loss ( x )  subject to  Birth ( x ) = a Death ( x ) = b x ∈ Z n ( K a ; G ) , (4) for x . An advanced treatment of this problem can be found in (Chen et al., 2008) for special case where 1) G   =   F 2 , 2) loss is a weighted sum of incident edges, and 3) the birth function assigns distinct values to any two simplices of the same dimension, and 4) n = 1 .

The program in Eq. 3 has a natural extension when G is a field. This extension focuses not on the smallest representative of a single homology class, but the smallest homological cycle basis. It may be formally expressed as follows: minimize  ∑ x ∈ ℬ loss ( x )  subject to  ℬ ∈ HCB n ( K ; G ), (5) where HCB n ( K , G ) is the family of dimension-n homological cycle bases of H n ( K ; G ) . Thus, the program is finding a complete generating set ℬ for all of the homological cycles of dimension n where each element has been minimized in some sense.

It is natural to wonder whether a solution to the program in Eq. 5 could be obtained by first calculating an arbitrary (possibly non-minimal) homological cycle basis ℬ = { x 1 , … , x m } and then selecting an optimal cycle representative z i from each homology class [ x i ] . Unfortunately, the resulting basis need not be optimal. To see why, consider the simplicial complex K 3 shown in Figure 3E, taking G to be ℚ and loss to be the ℓ 0 norm. Complex K • has several different homological cycle bases in degree 1, including ℬ 0 : = { x ^ 6 , x 6 } , ℬ 1 : = { x 5 , x 6 } , and ℬ 2 : = { x 5 , x ^ 6 + x 4 } . However, only ℬ 0 is ℓ 0 minimal. Moreover, each of the cycle representatives x 5 , x 6 , x ^ 6 is already minimal within its homology class, so element-wise minimization will not transform ℬ 1 or ℬ 2 into optimal bases, as might have been hoped.

As with the minimal cycle representative problem, the minimal homological cycle basis problem has been well-studied in the special case where loss is the ℓ 0 norm and G = F 2 . In this case, the program in Eq. 5 is NP-hard to approximate for n   >   1 , but O ( n 3 ) when n = 1 (Dey et al., 2018). Several interesting variants and special cases have been developed in the n = 1 case, as well Erickson and Whittlesey (2009), Dey et al. (2010), and Chen and Freedman (2010). We are not currently aware of a systematic treatment for the case G   ∈   { ℚ , ℝ } .

A natural variant of the minimal homological cycle basis program in Eq. 5 is the minimal persistent homological cycle basis problem minimize   ∑ x ∈ ℬ loss ( x )   subject to   ℬ ∈ PrsHCB n ( K • ; G ), (6) where PrsHCB n ( K • ; G ) is the set of persistent homological cycle bases. This is a stricter condition than the program in Eq. 5 in that not only does it require that the elements of ℬ form a generating set of all cycles of dimension n, but the barcode associated to ℬ must match Barcode n ( K • ) . That is, the multisets of birth/death pairs must be identical.

The program in Eq. 6 is much more recent than the program in Eq. 5, and consequently appears less in the literature. In the special case where every bar in the multiset Barcode n ( K • ) has multiplicity 1 (i.e. there are no duplicate bars), the program in Eq. 6 can be solved by making one call to the minimal persistent cycle representative program in Eq. 4 for each bar. In particular, the method of Chen et al. (2008) may be applied to obtain a minimal persistent basis when the correct hypotheses are satisfied: G = F 2 , loss is a weighted sum of incident simplices, there are distinct birth times for all simplices of the same dimension, and n = 1 . In general, however, bars of multiplicity two are possible, and in this case repeated application of the program in Eq. 4 will be insufficient.

A close cousin of the minimal homological cycle basis the program in Eq. 5 is the minimal filtered cycle basis problem, which may be formulated as follows minimize   ∑ x ∈ C loss ( x )   subject to   C ∈ FCB ( K • ; G ), (7) where FCB ( K • ) is the family of all bases C of Z n ( K ε T ) such that C contains a basis for each subspace Z n ( K ε i ) , for i ∈ { 1 , … , T } .

Escolar and Hiraoka (2016) provide a polynomial time solution via linear programming when.

1. loss is the ℓ 1 norm,

Their key observation is that C is an optimal solution to the program in Eq. 6 if and only if C can be expressed as a collection { z j   :   j ∈ J } where

The authors formulate this problem as minimize  ‖ x ‖ 1  subject to  x = x O r i g + ∑ r ∈ R w r g r + ∑ s ∈ S υ s f s w ∈ ℚ R v ∈ ℚ S , (8) where x O r i g ∈ Z N ( K j ) \ Z N ( K j ‐ 1 ) is a novel cycle representative at filtration j; { g r : r ∈ R } is a basis for B n ( K j ‐ 1 ) ⁴ ; and { g r : r ∈ R } ∪ { f s : s ∈ S } is an extension of the given basis for B n ( K j ‐ 1 ) to a basis for Z n ( K j ‐ 1 ) . That is, x O r i g is a cycle that has just appeared in the filtration. To optimize it, we are allowed to consider linear combinations of both boundaries, { g r } , and cycles, { f s } , born before x O r i g . The cycle x obtained in this way cannot have a birth time before that of x O r i g , but may have a different death time if [ ∑ s ∈ S υ s f s ] dies later than [ x O r i g ] .

The algorithm developed in Escolar and Hiraoka (2016) is cleverly constructed to extract x O r i g , { g r : r ∈ R } , and { f s : s ∈ S } from matrices which are generated in the normal course of a barcode calculation.

Remark 3.1.

 It is important to distinguish between PrsHCB and FCB , hence between the optimization the programs in Eqs 6, 7. As Escolar and Hiraoka (2016) point out, given ℬ ∈ PrsHCB and C ∈ FCB , one can always find an injective function ϕ : ℬ → C such that Birth ( z ) = Birth ( ϕ ( z ) ) for all z . However, this does not imply that ϕ ( ℬ ) ∈ PrsHCB , as the deaths of each cycle may not coincide. Indeed, the question of whether a persistent homological cycle basis can be extracted from C by any means is an open question, so far as we are aware. We provide an example in Figure 4 where the cycle basis obtained by optimizing each cycle using the program in Eq. 7 is not a persistent homology cycle basis ℬ .

 Though Remark 3.1 is a bit disappointing for those interested in persistent homology, the machinery developed to study the program in Eq. 7 is nevertheless interesting, and we will discuss an adaptation.

Schweinhart (2015) and Obayashi (2018) consider a different notion of minimization: volume ⁵ optimality. This approach focuses on the “size” of a bounding chain; it is specifically designed for cycle representatives in a persistent homological cycle basis.

Obayashi (2018) formalizes the approach as follows. First, assume a simplex-wise filtration K • ; without loss of generality, K • = ( K 1 , … , K T ) , and we may enumerate the simplices of K T such that K i = { σ 1 , … , σ i } for all i. Since each simplex has a unique birth time, each interval in Barcode n ( K • ) = { [ b 1 , d 1 ) , … , [ b N , d N ) } has a unique left endpoint. Fix [ b i , d i ) ∈ Barcode n ( K • ) such that d i < ∞ (in the case d i = ∞ , volume is undefined). It can be shown that σ b i is an n-simplex and σ d i is an ( n + 1 ) simplex. We use τ _k = σ _k below when the dimension of σ _k is equal to n + 1.

A persistent volume v for [ b i , d i ) is an ( n + 1 ) chain v ∈ C n + 1 ( K d i ) such that ⁶ v = τ d i + ∑ τ k ∈ ℱ n + 1 α k τ k (9) ( ∂ n + 1 v ) σ = 0   ∀ σ ∈ ℱ n (10) ( ∂ n + 1 v ) σ b i ≠ 0 , (11) where ℱ n = { σ k ∈ S n ( K ) : b i < k < d i } denotes the n-simplices alive in the window between the birth and death time of the interval under consideration.

We interpret these equations as follows: Given a persistence interval [ b i , d i ) , condition Eq. 9 implies that v only contains n + 1 simplices born between b i and d i and must contain the n + 1 simplex born at d i . Condition (Eq. 10) ensures that the boundary of v contains no n-simplex born after b i , and condition (Eq. 11) ensures that the boundary of v contains the n-simplex born at b i . This guarantees that ∂ n + 1 v exists at step b i , does not exist before step b i , and dies at step d i .

Theorem 3.2.

(Obayashi, 2018). Suppose that [ b i , d i ) ∈ Barcode n ( K • ) and d i < ∞ .

 1. Interval [ b i , d i ) has a persistent volume.

By Theorem 3.2 , for any barcode composed of finite intervals, one can construct a persistent homological cycle basis from nothing but (boundaries of) persistent volumes! Were we to build such a basis, it would be natural to ask for volumes that are optimal with respect to some loss function; that is, we might like to solve minimize  loss ( v ) subject to  ( 9 ) , ( 10 ) , ( 11 ) v ∈ C n + 1 ( K d i ) , (12) for each barcode interval [ b i , d i ) . A solution v to the program in Eq. 12 is called an optimal volume; its boundary, x = ∂ n + 1 v is called a volume-optimal cycle.

It is interesting to contrast ℓ 0 minimal cycle representatives for an interval ⁷ [ b i , d i ) with ℓ 0 volume-optimal cycle for the same interval. Consider, for example, Figure 5. For the persistence interval [ b i , d i ) , the cycle with minimal number of edges is ( a , b ) + ( b , c ) + ( c , d ) + ( d , a ) . However, the volume-optimal cycle would be found as follows: considering K d i , we must find the fewest 2-simplices whose boundary captures the persistence interval. In this case, we would have an optimal volume ( a , b , e ) + ( b , c , e ) + ( a , d , e ) and volume-optimal cycle ( a , b ) + ( b , c ) + ( c , e ) + ( e , d ) + ( d , a ) .

As mentioned above, it is common to choose loss ( x ) = ‖ x ‖ 0 or loss ( x ) = ‖ x ‖ 1 . ⁸ A linear program (LP) with ℓ 1 objective function is polynomial time solvable. However, an objective function with the ℓ 0 norm restricted to { 0,1 , ‐ 1 } coefficients is often preferred as the output of such a problem is highly interpretable: a cycle representative with minimal number of edges or enclosing the minimal number of triangles. Yet, ℓ 0 optimization is known to be NP-hard (Tahbaz-Salehi and Jadbabaie, 2010).

The ℓ 1 norm promotes sparsity and often gives a good approximation of ℓ 0 optimization (Tahbaz-Salehi and Jadbabaie, 2008; Tahbaz-Salehi and Jadbabaie, 2010), but the solution may not be exact. Yet, if all of the coefficients of the solution x are restricted to 0 or ± 1 in the optimization problem, then the ℓ 0 and ℓ 1 norms are identical. A looser restriction, as proposed in Escolar and Hiraoka (2016), would be to solve an optimization with ℓ 1 objective function with integer constraints on the solution.

Requiring the solution to be integral also allows us to understand the optimal solution more intuitively than having fractional coefficients. Such an optimization problem is called a mixed integer program (MIP), which is known to be slower than linear programming and is NP-hard (Obayashi, 2018). Many variants of integer programming special to optimal homologous cycles, in particular, have been shown to be hard as well (Borradaile et al., 2020). In Section 4, we discuss the optimization problems we implement, where each is solved both as an LP with an ℓ 1 norm in the objective function and an MIP by adding the constraint that x is integral.

Dey et al. (2011) gives the totally unimodularity sufficient condition which guarantees that an LP and MIP give the same optimal solution. A matrix is totally unimodular if the determinant of each square submatrix is ‐ 1,0 , or 1. Dey et al. (2011) give conditions for when the ∂ n + 1 matrix is totally unimodular. If the totally unimodularity condition is not satisfied, then an LP may not give the desired result. As totally unimodularity is not guaranteed for all boundary matrices (Henselman and Dłotko, 2014), we cannot rely on this condition.

Edge-minimal cycles: Software implementing the edge-loss method introduced in Escolar and Hiraoka (2016) can be found at Escolar and Hiraoka (2021). This is a C++ library specialized for 3-dimensional point clouds.

Triangle-loss optimal cycles: The volume optimization technique introduced in Obayashi (2018) is available through the software platform HomCloud, available at Obayashi et al. (2021). The code can be accessed by unarchiving the HomCloud package (for example, https://homcloud.dev/download/homcloud-3.1.0.tar.gz) and picking the file homcloud-x.y.z/homcloud/optvol.py.

Each program addressed in our empirical study may be expressed in the following form minimize   ‖ W x ‖ 1 = ∑ σ W [ σ , σ ] ( x σ + + x σ ‐ ) subject to   x = x + ‐ x ‐ x + , x ‐ ≥ 0 x ∈ X , (13) where X is a space of feasible solutions and W is a diagonal matrix with nonnegative entries. These programs vary along 3 parameters:

 1. Chain dimension of x. If X is a family of 1-chains, then we say that the program in Eq. 13 is an edge-loss program. If X is a family of 2-chains, we say that the program in Eq. 13 is a triangle-loss program.

Edge-loss and triangle-loss programs will be denoted Edge and Tri , respectively. Integrality will be indicated by a superscript I (integer) or N I (non-integer). Uniform weighting will be denoted by a subscript U n i f (uniform); non-uniform weighting will be indicated by subscript L e n (for edge-loss programs) or A r e a (for triangle-loss programs). Thus, for example, Edge L e n I denotes a length-weighted edge-loss program with integer constraints.

Our approach to edge-loss minimization, based on work by Escolar and Hiraoka (2016), is summarized in Algorithm 1. As in Escolar and Hiraoka (2016), we obtain x by taking a linear combination of x O r i g with not only boundaries but cycles as well; consequently x need not be homologous to x O r i g .

Our pipeline differs from Escolar and Hiraoka (2016) in three respects. First, we perform all optimizations after the persistence calculation has run. On the one hand, this means that our persistence calculations fail to benefit from the memory advantages offered by optimized cycles; on the other hand, separating the calculations allows one to “mix and match” one’s favorite persistence solver with one’s favorite linear solver, and we anticipate that this will be increasingly important as new, more efficient solvers of each kind are developed. Second, we introduce additional constraints which guarantee that ℬ * ∈ PrsHCB [and, moreover, ℒ ( x ) = ℒ ( x O r i g ) for each x O r i g ∈ ℬ ]. Third, we remove the hypothesis of a simplex-wise filtration; this requires some technical modifications, whose motivation is explained in the Supplementary Material. The crux of this modification lies with the for loop, which replaces cycles that have been optimized in the cycle basis for later cycle optimization.

The program in Eq. 14 optimizes the jth element of an ordered sequence of cycle representatives Z = ( z 1 , … , z m ) . In particular, it seeks to minimize x O r i g : = z j . To define this program, we first construct a matrix A such that A [ : , i ] = z i for i = 1 , … , m . We then define three index sets, Q ,   ℛ ,   ℛ such that Q = { i : Birth ( z i ) ≤ Birth ( x O r i g ) ,   Death ( z i ) ≤ Death ( x O r i g ) ,   i ≠ j } ℛ = { τ ∈ S n + 1 ( K ) : Birth ( τ ) ≤ Birth ( x O r i g ) } ℛ = { σ ∈ S n ( K ) : Birth ( σ ) ≤ Birth ( x O r i g ) } , That is, ℙ indexes the set of cycles z i such that z i is born (respectively, dies) by the time that z j is born (respectively, dies), excluding the original cycle z j itself. Set ℛ is the family of triangles born by Birth ( x O r i g ) , and set ℛ is the family of edges born by Birth ( x O r i g ) .

With these definitions in place, we now formalize the general edge-loss problem as the program in Eq. 14, where ∂ n + 1 [ ℛ , ℛ ] denotes the submatrix of ∂ n + 1 indexed by triangles born by Birth ( x O r i g ) (along columns) and edges indexed by edges born by Birth ( x O r i g ) . Likewise A [ ℛ   ,   Q ] is the column submatrix of A corresponding to cycles that are born before the birth time of x O r i g (and which die before the death time of x O r i g ), excluding x O r i g itself. minimize   ‖ W x ‖ 1 = ∑ σ ∈ R W [ σ , σ ] ( x σ + + x σ ‐ ) subject to ( x + ‐ x ‐ ) = x O r i g [ ℛ ] + ∂ n + 1 [ ℛ , ℛ ] q + A [ ℛ , Q ] p p ∈ ℚ Q q ∈ ℚ ℛ x ∈ G ℛ x + , x ‐ ≥ 0. (14)

Recall from Section 4.1 that this program varies along two parameters (integrality and weighting). In integral programs G = ℤ , whereas in nonintegral programs G = ℚ . The weight matrix W is always diagonal, but in uniform-weighted programs W [ σ , σ ] = 1 for all σ = R, whereas in length-weighted programs W [ σ , σ ] is the length of edge σ. The program in Eq. 14 thus results in four variants:

Edge U n i f N I : Nonintegral edge-loss with uniform weights.

The program in Eq. 14 may have many more variables than needed, because ∂ n + 1 is often highly singular. Indeed, in applications, ∂ n + 1 can have hundreds or thousands of times as many columns as rows!

A simple means to reduce the size of the program in Eq. 14, therefore, is to replace ℛ with a subset ℛ ^ ⊆ ℛ such that ∂ n + 1 [ ℛ , ℛ ^ ] is a column basis for ∂ n + 1 [ ℛ , ℛ ] . Replacing ℛ with ℛ ^ will not change the space of feasible values for x in the program in Eq. 14, but it can cut the number of decision variables significantly. In particular, one may take ℛ ^ : = { σ : R [ : , σ ] ≠ 0 } in the R = ∂ n + 1 V decomposition of ∂ n + 1 described in the Supplementary Material. We also show correctness of this choice of ℛ ^ there.

Our approach to triangle-loss optimization is essentially that of Obayashi (2018), plus a preprocessing step that converts more general problem data into the simplex-wise filtration format assumed in Obayashi (2018). There are several noteworthy methods for time and memory performance enhancement developed in Obayashi (2018), which we do not implement (e.g., using restricted neighborhoods ℱ q ( r ) to reduce problem size), but which may substantially improve runtime and memory performance.

The original method makes the critical assumption that K • is a simplex-wise filtration, more precisely, that there exists a linear order σ 1 ≤ ⋯ ≤ σ | K | such that K i = { σ 1 , … , σ i } . This hypothesis allows one to map each finite-length interval [ i , j ) ∈ Barcode n ( K • ) to a unique pair of simplices ( σ i , σ j ) , called a birth/death pair, where σ i ∈ S n ( K ) and σ j ∈ S n + 1 ( K ) . This mapping makes it possible to formulate the program in Eq. 12. Unlike the general edge-loss the program in Eq. 13, one can formulate the program in Eq. 12 without ever needing to choose an initial (non-optimal) cycle. Thus, for simplex-wise filtrations, the method of Obayashi (2018) has the substantial advantage of being “parameter free.”