Solving equations and optimization problems with uncertainty

Abstract

We study the problem of detecting zeros of continuous functions that are known only up to an error bound, extending the theoretical work of Franek and Krčál (J ACM 62(4):26:1–26:19, 2015) with explicit algorithms and experiments with an implementation (https://bitbucket.org/robsatteam/rob-sat). Further, we show how to use the algorithm for approximating worst-case optima in optimization problems in which the feasible domain is defined by the zero set of a function $f\colon X\rightarrow {\mathbbm{R}}^n$ which is only known approximately. The algorithm first identifies a subdomain A where the function f is provably non-zero, a simplicial approximation $f'\colon A\rightarrow S^{n-1}$ of f/|f|, and then verifies non-extendability of $f'$ to X to certify a zero. Deciding extendability is based on computing the cohomological obstructions and their persistence. We describe an explicit algorithm for the primary and secondary obstruction, two stages of a sequence of algorithms with increasing complexity. Using elements and techniques of persistent homology, we quantify the persistence of these obstructions and hence of the robustness of zero. We provide experimental evidence that for random Gaussian fields, the primary obstruction—a much less computationally demanding test than the secondary obstruction—is typically sufficient for approximating robustness of zero.

Appendices

Appendix 1: Secondary obstruction

1.1 Persistence of the secondary obstruction: the algorithm for $n>3$

Assume that a filtration $X\supseteq A_{r_0}\supseteq A_{r_1}\supseteq \cdots$ and a simplicial map $f'\colon A_{r_0}\rightarrow \varSigma ^{n-1}$ are given, $n>3$, and vertices on X and $\varSigma ^{n-1}$ are ordered so that $f'$ is order-preserving: this order is used in the implementation of the $\smile _{n-3}$ operation on the level of cochains. Further, we assume that the persistence of primary obstruction $r_j$ has already been computed by the algorithm described in Sect. 4.2. That is, the restriction of f to $A_{r_j}$ is not extendable to some continuous map $A_{r_j}\cup X^{(n)}\rightarrow \varSigma ^{n-1}$, but the restriction to $A_{r_{j+1}}$ is extendable. We continue to use the notation of Sect. 4: in particular, z is the characteristic cocycle of a fixed $(n-1)$-simplex in $\varSigma ^{n-1}$, $\bar{y}\in C^{n-1}(X,A_{r_0})$ is a cochain extending the pullback $y=(f')^\sharp (z)\in Z^{n-1}(A_{r_0})$ of z and $\emptyset \ne \varOmega (A_{r_{j+1}})$ is the set of all $(n-1)$-cocycles on X that extend y on $A_{r_{j+1}}$.

By Proposition 2, the persistence of secondary obstruction is the largest number $r_k$ such that

$$\begin{aligned} \delta c=v(x), \quad c\in C^n(X,A_{r_k};{\mathbbm{Z}}_2),\,\, x\in \varOmega (A_{r_k}) \end{aligned}$$

(5)

has no solution (where v is defined by (3)).

Let $x\in \varOmega (A_{r_{j+1}})$ be a fixed extension of $\bar{y}$, computed in the algorithm for primary persistence. Then also $x\in \varOmega (A_{r_k})$ for each $k > j$. For any such k, $\varOmega (A_{r_k})$ is a coset in $Z^{n-1}(X)$

$$\begin{aligned} \varOmega (A_{r_k})=x+Z^{n-1}(X,A_{r_k};{\mathbbm{Z}}) \end{aligned}$$

and hence Eq. (5) reduces to

$$\begin{aligned} \delta c=v(x-w), \quad \text{for some } w\in Z^{n-1}(X,A_{r_k};{\mathbbm{Z}}), \,\, c\in C^n(X,A_{r_k};{\mathbbm{Z}}_2). \end{aligned}$$

(6)

The crucial property we will use is that $v(x-w)$ is a relative coboundary iff $v(x)-v(w)$ is a relative coboundary: this follows directly from the linearity of the Steenrod square operation $H^{n-1}(X,A;{\mathbbm{Z}}_2)\rightarrow H^{n+1}(X,A;{\mathbbm{Z}}_2)$ for $n>3$. Thus we can reformulate (6) to the problem of finding the maximal $r_k$ such that

$$\begin{aligned} \delta c+v(w) = v(x), \quad w\in Z^{n-1}(X,A_{r_k};{\mathbbm{Z}}), \,\, c\in C^n(X,A_{r_k};{\mathbbm{Z}}_2). \end{aligned}$$

(7)

has no solution. To simplify the computations, we don’t need to consider all cocycles $w\in Z^{n-1}(X,A_r)$ but only generators of the cohomology $H^{n-1}(X,A_r;{\mathbbm{Z}})$: the Steenrod square of any relative coboundary is again a relative coboundary $\delta c'$ for some $c'\in C^{n}(X,A_r;{\mathbbm{Z}}_2)$, so adding it has no impact on the solvability of (7).

The right-hand side of (7), v(x), is a cocycle that does not depend on k (assuming $k>j$). The left-hand side is a combination of coboundaries of characteristic cocycles of n-simplices $\varDelta ^n$ and cochains of the form v(w) for $(n-1)$-cocycles w. To each $\varDelta$ and w is assigned a filtration value and we want to minimize the value $r_k$ such that v(x) can be expressed as a combination of $\delta c$’s and v(w)’s such that c and w have filtration values $\le r_k$, but cannot be expressed as a combination of such cochains with filtrations of c and w strictly smaller than $r_k$.

Summarizing the above steps, we obtain the following algorithm:

• Order the vertices of $\varSigma ^{n-1}$ and the vertices of X so that f is order-preserving

• For a precomputed $x\in \varOmega (A_{r_{j+1}})$, compute the relative cocycle $v(x)\in Z^{n+1}(X,A_{r_{j+1}};{\mathbbm{Z}}_2)$ by the definition of the Steenrod operation $\smile _{n-3}$

• Compute a subset $W\subseteq Z^{n-1}(X)$ that contains all cohomology generators w of all $H^{n-1}(X, A_r;{\mathbbm{Z}})$ for all $r > r_j$. (It may be a set of generators of $Z^{n-1}(X)$.) To each $w\in W$, assign its filtration value r(w) to be the minimal r such that w is zero on $A_r$.

• Compute the filtration value of all n-simplices, using (1).

• Order all n-simplices and elements of W by their filtration.

• Choose a basis of $(n+1)$-simplices and express the right-hand side v(x) as a vector ${\varvec{a}}\in ({\mathbbm{Z}}_2)^q$.

• Create a matrix M whose column set consists of

  • all coboundaries of characteristic cochains of n-simplices expressed in the basis of $(n+1)$-simplices over ${\mathbbm{Z}}_2$

  • all elements v(w) expressed in the basis of $(n+1)$-simplices.

• Order the columns of M by filtration.

Computing the persistence of the secondary obstruction then reduces to solving the EARLIEST SOLUTION problem for $M{\varvec{x}}= {\varvec{a}}$, this time over the ${\mathbbm{Z}}_2$-field.

The hardest part is to compute the cohomology generators: this algorithm is summarized as follows.

We give implementation details of this part on a lower-level in appendix subsection “Algorithm for the Persistent Generators problem”. Our algorithm will give the output with the number of generators $\nu$ minimal in a certain sense. Notably, when working over $\mathbbm {Q}$ or ${\mathbbm{Z}}_p$ instead of integers, the output would correspond to a persistence bar code with all the death information erased but including representative (co)cycles for each bar.

1.2 The special case $f\colon [0,1]^4\rightarrow {\mathbbm{R}}^3$

This section justifies the claim that for $X=[0,1]^4$ and $n=3$, the general algorithm for computing persistence of secondary obstruction works, if we completely ignore the persistence generators w and perform all computations over ${\mathbbm{Z}}$-coefficients.

Assume first that $n=3$ and X is arbitrary. The two main differences compared to the algorithm above are:

• The Steenrod operation and final coboundary matrix have to be computed over ${\mathbbm{Z}}$, not ${\mathbbm{Z}}_2$. This goes back to the fact that the homotopy group $\pi _3(S^2)\simeq {\mathbbm{Z}}$, unlike $\pi _n(S^{n-1})\simeq {\mathbbm{Z}}_2$ for $n>3$. The homotopy group serves as cohomology coefficients in the theory of obstructions.

• The operation $\smile _{n-3}$ reduces to the cup product $\smile$ and the operation $w\mapsto w\smile w$ is not linear on the level of cohomology but quadratic.

This means that while for any particular extension $x\in \varOmega (A)$ of $\bar{y}$, we may test satisfiability of $\delta c=v(x)$, we cannot test the existence of such x using a linear system of equations. However, if $X=[0,1]^4$, we claim that for any two extensions $x,y\in \varOmega (A)$ of $\bar{y}$, $v(x)-v(y)$ is a relative coboundary and thus we need to check the equation $\delta c=v(x)$ only for one x.

Lemma 2

Let X be a triangulation of $[0,1]^4$, $A\subseteq X$, $x\in Z^{2}(X)$ and $w\in Z^2(X,A)$. Then

$$\begin{aligned} (x-w)\smile (x-w) -(x\smile x) \in B^4(X,A). \end{aligned}$$

Using the parametrization $\varOmega (A)=\{x-w:\,\,w\in Z^{n-1}(X,A;{\mathbbm{Z}})\}$ we immediately obtain that our general algorithm for the persistence of secondary obstruction works once we replace the ${\mathbbm{Z}}_2$-coefficients by ${\mathbbm{Z}}$-coefficients in its final step. Moreover, we may completely ignore the persistent generators w and don’t need to compute them at all.¹⁰

Proof

By bilinearity of the cup product, $(x-w)\smile (x-w) \, - \, x\smile x=- \, x\smile w \, -\, w\smile x\, +\, w\smile w$. The mixed-term $x\smile w$ is a relative coboundary, because $\smile$ induces a bilinear product on the level of cohomology $H^2(X)\times H^2(X,A)\rightarrow H^4(X,A)$ and $H^2(X)=0$, as X is contractible. It remains to show that $w\smile w$ is a relative coboundary. Let CA be the cone over A and $A\hookrightarrow CA$ be an inclusion. The inclusion $A\hookrightarrow X$ can be extended to a map $CA\rightarrow X$, because X is contractible. The map of pairs $(CA,A)\rightarrow (X,A)$ induces the commutative diagram in which the rows are the long exact sequences of cohomology groups.

$$\begin{aligned} \begin{array}{ccccccccc} H^{*-1}(X) &{} \rightarrow &{} H^{*-1}(A) &{} \rightarrow &{} H^*(X,A) &{}\rightarrow &{} H^*(X) &{}\rightarrow &{} H^*(A) \\ \downarrow &{} &{}\downarrow &{} &{}\downarrow &{} &{}\downarrow &{} &{} \downarrow \\ H^{*-1}(CA) &{} \rightarrow &{} H^{*-1}(A) &{} \rightarrow &{} H^*(CA,A) &{}\rightarrow &{} H^*(CA) &{}\rightarrow &{} H^*(A) \end{array} \end{aligned}$$

The vertical arrows $H^{*(-1)}(X)\rightarrow H^{*(-1)}(CA)$ are trivial as both spaces are contractible and $H^{*(-1)}(A)\rightarrow H^{*(-1)}(A)$ are identities. By the five-lemma (Hatcher 2001, p. 129), the middle homomorphism $H^*(X,A)\rightarrow H^*(CA,A)$ is an isomorphism. Further, $H^j(CA,A)\simeq H^j(CA/A)$ for $j>0$. The space $CA/A=:\varSigma A$ is the suspension of A. The cup product $H^2(\varSigma A)\times H^2(\varSigma A)\rightarrow H^{4}(\varSigma A)$ of a suspension is trivial (Bredon 1993, Corollary 4.11) and the naturality of cup product implies that the cup product $H^2(X,A)\times H^2(X,A)\rightarrow H^{4}(X,A)$ is trivial for $j>0$ as well. This shows that $w\smile w\in B^4(X,A)$.

Appendix 2: Persistent integral homology computations

1.1 Algorithm for the Earliest Solution problem

The earliest_solution algorithm is used to find the persistence of a (co)cycle. This is closely related to computing persistent homology, which is a well-studied problem, at least for coefficient in a finite field. We adapt the boundary matrix reduction algorithm by Edelsbrunner et al. (2002), originally developed for persistent homology. Note that this algorithm, unlike classical Gaussian elimination or Smith normal form algorithms, is incremental, which is required by our application. Moreover, it only uses column operations, making efficient implementation for sparse matrices relatively easy.

1.1.1 Efficiency over finite fields

Recent work on computing persistent homology over finite fields resulted in significant performance improvements, see Bauer et al. (2014) and Maria et al. (2014). Despite the cubic worst-case bound, linear scaling is achieved on practical datasets, involving sparse matrices of size $10^9 \times 10^9$ and more. This encouraged us to adapt the modern version of the classical persistence algorithm to solve our problem over the integers, rather than adapting classical Smith or Hermite normal form algorithms to the persistent setting.

1.1.2 Reduced form and reduction

We adapt the notation common in computational topology literature: the lowest nonzero of a nonzero column is defined as the lowest position (largest index) with nonzero coefficient. A sub-matrix is called reduced if all the lowest nonzeros are unique. In other words, given a lowest nonzero, there may be other nonzero entries in the same row, but they must not be lowest nonzeros. When this invariant is not satisfied, we say there is a collision. By lowest value, we refer to the value of the lowest nonzero entry.

The algorithm starts from an empty matrix and adds one column at a time, maintaining the reduced prefix of the matrix, R. The rightmost column of each prefix is called the current column.

Procedure reduce_column reduces the column curr with respect to the reduced prefix R.

Procedure earliest_solution solves the stated problem.

1.1.3 Correctness

The basic operation is the addition of two different columns, without affecting the column span of the relevant matrix prefix. Therefore the solution is unaffected.

To retrieve the solution, at each step we attempt to reduce the input column vector a with respect to the currently reduced prefix R. The solution exists iff a becomes zero, and is encoded by (the negation of) the change of basis column of a. Since we solve the equation $Ax = a$ (and not $Ax = ka$) we perform additional divisibility check: see force_divisibility in the reduce_column procedure.

1.1.4 From finite fields to integers

In the integral case, we may modify both the current column and the colliding column. This is in contrast with the finite field case, in which only the current column is modified. To determine the required linear combination of columns, we use the extended Euclid algorithm. As a result, the lowest value of a certain column might change (decrease) many times during the reduction of other columns, but the position is fixed once the column is reduced. Because of this, while reducing column a, we need to take into account previously reduced columns, and not only the current one. Moreover, after a colliding column is affected, it may not be in the column span of the prefix of the original matrix ending at this column. At this stage, however, this column may only affect columns that succeed the currently reduced column. Therefore, correctness of the algorithm is unaffected.

1.1.5 Efficiency

For efficiency reasons we use one technique suggested in Bauer et al. (2014): The current column is stored in a data-structure handling fast column additions and maximum element queries. One natural choice is a balanced binary search tree; more efficient alternatives are available. This way we avoid the following common bad case: Let n be the total number of columns in M. When an current column becomes dense (the number nonzero entries is $\varTheta (n)$), adding $\varTheta (n)$ sparse columns takes time quadratic in n. Avoiding this situation does not imply that we can efficiently handle matrices that become dense due to fill-in. However in practice, often a small number of columns display this behavior.

We also perform the computations in an on-line fashion—we read columns one by one and stop once a solution is found. This gives significant practical improvements, because often the necessary matrix prefix is very small compared to the matrix of the entire complex.

Overall the algorithm performs well, exhibiting roughly linear scaling in the number of nonzero entries of the original matrix. In particular, we didn’t observe coefficient blowup. Note that the lowest nonzero position in the current column decreases after resolving each collision, so the number of collision resolutions is quadratic in n. Each such resolution requires combining two columns, potentially of size O(n), due to fill-in. This leads to cubic worst case running time bound, assuming that the magnitude of coefficients can be bounded by a constant. Of course, there exist cases when the blowup does occur, and the above analysis does not apply. More advanced algorithms can be used to alleviate the effect of the blowup, possibly at the cost of simplicity and efficiency in the situations that we encountered thus far.

1.2 Algorithm for the Persistent Generators problem

After possibly refining the sequence $(A_i)_i$ we may assume that for each filtration value i there is exactly one $(n-1)$-simplex $\varDelta _i^{n-1}$ of that filtration value, that is $\varDelta _i^{n-1} \in A_{i-1}{\setminus } A_{i}$ (and possibly several $(n-2)$-simplices in $A_{i-1}{\setminus } A_{i}$). This makes the algorithm and its analysis simpler.

The algorithm for Persistent Generators problem follows. By the statement “reduce the column” we refer to the procedure reduce_column from the appendix subsection “Algorithm for the Earliest Solution problem”

• Let M be the coboundary matrix for $\delta ^{n-1}\colon C^{n-1}(X;{\mathbbm{Z}})\rightarrow C^n(X;{\mathbbm{Z}})$, that is, it consists of columns ${\varvec{m}}$ for integral combination for $\delta \varDelta ^{n-1}$ for each $\varDelta ^{n-1}\in X$ of growing filtration values. In exactly the same way we create the coboundary matrix N for $\delta ^{n-2}$. During the reduction of the matrix M we keep track of the change of basis. (On low level, we augment the matrix M and then the change of basis vector for a given column is encoded in the augmented part of that column.)

• We initialize $G:=()$ to be an empty sequence of $(n-1)$-cocycles on X and $\mu (-1):=0$.

• For each filtration value $i=0,\ldots ,h$ do the following:

  • Reduce all columns of N of the filtration value i.

  • Reduce the column ${\varvec{m}}$ of M of filtration value i (i.e., corresponding to the cochain $\delta \varDelta ^{n-1}_i$). Let ${\varvec{g}}$ be the change of basis vector after the reduction.

  • If the reduced column ${\varvec{m}}$ equals to 0 and there is no reduced column ${\varvec{n}}$ in N such that ${{\mathrm{lowest}}}({\varvec{n}})=i$ and ${\varvec{n}}_{{{\mathrm{lowest}}}({\varvec{n}})}|{\varvec{g}}_{{{\mathrm{lowest}}}({\varvec{g}})}$ then add ${\varvec{g}}$ to G and set $\mu (i):=\mu (i-1)+1$.

  • Otherwise set $\mu (i):=\mu (i-1).$

• Output G and $\mu (1),\ldots ,\mu (h)$.

Theorem 2

The above algorithm solves the Persistent Cycles problem. Namely, after its i th iteration, the cohomology group $H^{n-1}(X,A_i)$ is generated by $[g_1],\ldots ,[g_{\mu (i)}]$ where cycles $g_1,\ldots ,g_{\mu (i)}\in Z^{n-1}(X,A_i;{\mathbbm{Z}})$ correspond to the columns ${\varvec{g}}$ added to G during the iterations $0,\ldots ,i.$

Proof

We proceed by induction on the value i. For $i=-1$ the claim trivially holds.

Let us assume that $i\ge 0$ and that the claim holds for $i-1$. First observe that once the new column ${\varvec{m}}$ is not reduced to a zero column, then $Z^{n-1}(X,A_i;{\mathbbm{Z}})=Z^{n-1}(X,A_{i-1};{\mathbbm{Z}})$. Otherwise $Z^{n-1}(X,A_i;{\mathbbm{Z}})=Z^{n-1}(X,A_{i-1};{\mathbbm{Z}})\oplus \langle g\rangle$ where the cocycle $g\in Z^{n-1}(X,A_i;{\mathbbm{Z}})$ corresponds to the change of basis vector ${\varvec{g}}$ in the ith iteration. We only have to check whether [g] is a linear combination of $[g_1], \ldots , [g_{\mu (i-1)}]$. By the induction hypothesis, it happens if and only if g is cohomologous to a cocycle in $Z^{n-1}(X,A_{i-1};{\mathbbm{Z}})$. And this in turn is true if and only if we can reduce to zero the lowest nonzero component (that is, the ith component) of the vector ${\varvec{g}}$ by adding a combination of columns from N of filtration value at most i (these columns generate the group of coboundaries $B^{n-1}(X,A_{i};{\mathbbm{Z}})$). Since this is exactly the reduced part of the matrix N, it is enough to find if there is a column with the lowest nonzero index equal to i and check the divisibility condition.

Appendix 3: Some details of our implementation: exploiting the cubical structure

The domain X we consider in the implementation is a cube $[0,1]^m$ triangulated as follows. We define $I_1,\ldots , I_m$ to be unit intervals subdivided into $n_i$ equidistant intervals of length $1/n_i$, $i=1,\ldots , m$. This yields a cubical set structure $\prod _j I_j$ on the unit cube. Further, we subdivide each m-cube into m! simplices via the Freudenthal triangulation (Allgower and Georg 2003, p. 154). The resulting triangulation naturally corresponds to the product $\prod I_j$, understood as a product $\prod _j I_j$ of simplicial sets (Friedman 2012). We call the set of vertices a grid. A function $f\colon X\rightarrow {\mathbbm{R}}^n$ is then given by a set of ${\mathbbm{R}}^n$-vectors in each vertex (a multidimensional rasterized image), together with a simplexwise Lipschitz constant $\alpha$.

Many operations in the algorithm outlined in the paper body—such as the computation of the pullback $y=f^\sharp (z)$, computing codifferentials of cochains and the $\smile _{n-3}$ operation—can be done locally, without having to work with the full lists of simplices of a given dimension. The only step where global structure is needed is the Earliest Solution algorithm, where the matrix M has rows indexed by the set of all k-simplices and columns indexed by $(k-1)$-simplices. Indeed, the construction of this matrix and computation with it are the most memory- and time-consuming operations, as the number of k-simplices increases rapidly with dimension.

Therefore, we implemented a modification of the algorithm described above based on the more feasible cubical structure. Instead of working with the simplicial filtration $A_r$, we use the filtration $A_r^\square \subseteq A_s^\square$ where $A_r^\square$ is defined to be the triangulation of the set of all cubes c such that $|f(v)|\ge v$ for all vertices v of c. These are still simplicial complexes and $f'$, y, the extension $\tilde{y}$ of y and $\delta \tilde{y}$ are defined with no changes. For the computation of persistence, however, we switch to the cubical setting via the Eilenberg–Zilber reduction (Eilenberg and Zilber 1953). We denote by $C^*(\varPi _j I_j)$ the simplicial cochains and by $\otimes _j C^*(I_j)$ the cubical cochains: there exist chain homomorphisms of degree 0

$$\begin{aligned}&AW:\otimes _j C^*(I_j)\rightarrow C^*(\varPi _j I_j)\\&EML:C^*(\varPi _j I_j)\rightarrow \otimes _j C^*(I_j) \end{aligned}$$

such that $EML\circ AW$ is the identity and $AW\circ EML$ is chain homotopic to the identity. Both maps induce cohomology isomorphisms and can be relatively easily implemented using common formulas (Gonzalez-Diaz and Real 2005). This allows us to switch between the simplicial and cubical cochains anytime we need. Within the computation of persistence of the primary obstruction, we compute the smallest j so that there exists a cubical cochain $c_\square \in Z^{n-1}(X,A_{r_j}^\square )$ such that $\delta c_\square =(\delta \tilde{y})_\square :=EML (\delta \tilde{y})$. The matrix computation part Earliest Solution deals with the cubical coboundary matrix, which is significantly smaller than the simplicial one. For an illustration, the number of k simplices in the triangulation of one m-cube is 50 already for $(m,k)=(4,2)$ and more than 4 millions for $(m,k)=(8,6)$.

For the secondary obstruction, we need to convert $c_\square$ back into a simplicial cochain and construct a simplicial $(n-1)$-cochain $x\in \varOmega (A_{r_j})$ that extends y on $A_{r_j}$ and is a global cocycle. We denote by SHI the cochain homotopy map $SHI\colon C^*(\varPi _j I_j)\rightarrow C^{*-1} (\varPi _j I_j)$, satisfying $AW\circ EML - {\mathrm{id}}=\delta \circ SHI + SHI\circ \delta$. Then, using $\delta ^2=0$, we obtain

$$\begin{aligned} \delta AW c_\square= & {} AW \delta c_\square = AW (\delta \tilde{y})_\square = AW\, EML\,\delta \tilde{y}\\= & {} \delta AW \,\,EML\, \tilde{y}=\delta \tilde{y} + \delta \,SHI \,\delta \tilde{y}. \end{aligned}$$

We compute $\tilde{c}:=AW (c_\square ) - SHI(\delta \tilde{y})$ which is a simplicial cochain zero on $A_{r_j}^\square$ and the last equation asserts that $\delta \tilde{c}=\delta \tilde{y}$ which allows us to compute the simplicial cocycle

$$\begin{aligned} x:=\tilde{y} - \tilde{c} \in \varOmega (r_j) \end{aligned}$$

having avoided to work with large lists of all simplices. The computation of $v(x):=(x\mod 2)\smile _{n-3} (x\mod 2)$ is done on the simplicial level and the property $x\in Z^{n+1}(X,A_{r_j}^\square ;{\mathbbm{Z}}_2)$ depends on the fact that the chosen ordering of vertices of X and $\varSigma ^{n-1}$ is compatible with $f'$. Then we again apply the EML operator to v(x) and convert it to a cubical cochain $v(x)_\square$. Persistent generators $w\in Z^{n-1}(X,A_{r}^\square ;{\mathbbm{Z}})$ and their Steenrod images can be computed on the cubical level (Krčál and Pilarczyk 2016) and the cubical persistence of the secondary obstruction is done via a cubical coboundary matrix, this time in dimension $n,(n+1)$.

There is a price to pay for the more convenient cubical filtration: it is courser than the simplicial one and we can no more use the estimate on the robustness of zero set derived in Sect. 3. On one hand, we have the relation $A_r^\square \subseteq A_r$. This implies that whenever $f'|_{A_r}$ is homotopic to f/|f|, then so is $f'|_{A_r^\square }$, and non-extendability of $f'|_{A_r^\square }$ to higher skeleta of X implies the non-extendability of $f'|_{A_r}$. Thus whenever the algorithm certifies non-extendability of $f|_{A_r^\square }$ and $r>\alpha n^{1/p}$, then $r-\alpha$ is a lower bound on the robustness of zero. If we can additionally prove that $A_s\subseteq A_r^\square$, then the extendability of $f'|_{A_r^\square }$ implies the extendability of $f'|_{A_s}$. The simplexwise $\alpha$-Lipschitz condition implies that any two vertices u, v of a cube c satisfy $|f(u)-f(v)|\le 2\alpha$, as u, v can be connected by path of at most two simplicial edges. This implies $A_{r+2\alpha }\subseteq A_r^\square$ and extendability of $f'|_{A_r^{\square }}$ to all of X implies the upper bound $r+3\alpha$ on the robustness.

Based on the experience with various functions in low dimensions, the practical performance has not one significant bottleneck. The most resources-consuming steps in the primary obstruction computation include the computation of the cubical filtration and the EARLIER SOLUTION subroutine, especially if the matrix has millions of columns.

Appendix 4: Experiments with formulas

Our prototype is implementation in Python using numpy. While the efficiency is severely limited by this choice, we were able to run examples for $\dim X\le 8$.

Although the strength of our methods is primarily in cases where we have uncertainty on f, the following examples provide some observations about the performance in cases where the function values at the vertices are computed using exact formulas.

In what follows, we assume a subdivision of the interval $[-1,1]$ into a set $I_g$ of g equidistant points and consider a grid $I_g^m\subseteq [-1,1]^m$ of size $g^m$. We chose the max-norm $\ell _\infty$ on ${\mathbbm{R}}^n$ which yields the smallest value of $\alpha n^{1/p}$ and allows smaller “initial $r_0$”. This is desired, as for small grids it often happens that $\alpha$ is large and $\alpha n^{1/p}$ may be larger then the robustness of zero in which case we fail to detect anything.

1.1 Testing primary obstruction on a quadratic function

First we used the function $f\colon [-1,1]^n\rightarrow {\mathbbm{R}}^n$ given by

$$\begin{aligned}&f_1(x)=x_1^2-x_2^2-\cdots -x_n^2 \\&f_2(x)=2x_1 x_2\\&\ldots \\&f_n(x)=2x_1 x_n. \end{aligned}$$

This function has a single zero in the origin and has been used as a benchmark example in Franek and Krčál (2015b). If n is even, then the zero is robust and its robustness equals $\min _{x\in \partial [-1,1]^m} |f(x)|$: the zero has index two in this case. If n is odd, then the zero has index zero and can be removed by arbitrary small perturbations. Using the max-norm, we use calculus to derive the estimate $|f(x)-f(y)|_{\infty }\le 2n\,|x-y|_\infty$. Whenever x, y are contained in one simplex of the triangulation, then the max-norm satisfies $|x-y|_\infty \le 2/(g-1)$ and we can use the piecewise Lipschitz constant $\alpha :=4n/(g-1)$.

For our experiments, we now assume that f is only given via a list of function values in the grid $g^n$ and the simplexwise Lipschitz constant $\alpha =4n/(g-1)$. In this case, we have $\dim X=n$ so only primary obstruction can be nontrivial. Using Lemma 1 for $p=\infty$ yields that whenever $r>\alpha$, then $f'$ is simplicial on $A_r$ and homotopic to f/|f|. The relation $A_r^\square \subseteq A_r$ then implies that $f'$ is simplicial and homotopic to f/|f| on $A_r^\square$ as well. Thus we can run our algorithm with an initial $r_0=\alpha =4n/(g-1)$, compute the persistence of primary obstruction $r_1\ge r_0$ and—using the estimates at the end of Sect. 3—conclude that whenever $r_1>r_0$ then the robustness of zero is between $r_1-\alpha$ and $r_1+3\alpha$.

The following table illustrates some properties of the computations: n refers to the dimension, g is the number of points subdividing the intervals in each dimension, the initial $r_0=\alpha$ is as above, $r_1$ is the persistence of the primary obstruction, $\delta ^{n-1}$ the number of columns of the cubical coboundary matrix $C^{n-1}_\square \rightarrow C^n_\square$, the “computed robustness” displays lower and upper bounds $r_1-\alpha$, $r_1+3\alpha$ on the robustness of zero and the “true robustness” column the approximation of the real robustness of zero of the function f defined exactly by the above formula,¹¹ all wrt. the $|\cdot |_\infty$ norm.

n	g	$r_0:=\alpha$	$r_1$	# columns of $\delta ^{n-1}$	Computed robustness	True robustness
2	20	0.421	0.8643	760	[0.443, 2.127]	0.828
2	100	0.08	0.8285	19, 800	[0.748, 1.07]	0.828
2	500	0.016	0.83	499, 000	[0.814, 0.878]	0.828
3	20	0.632	$=r_0$	21, 660	$\le 2.379$	0
3	50	0.245	$=r_0$	360, 150	$\le 0.971$	0
3	100	0.121	$=r_0$	2, 940, 300	$\le 0.483$	0
4	20	0.842	$=r_0$	548, 720	$\le 3.37$	0.667
4	30	0.552	0.711	2, 926, 680	[0.159, 2.367]	0.667
4	40	0.41	0.667	9, 491, 040	[0.257, 1.897]	0.667

The total running times of these 9 computations is displayed in Fig. 3. The data suggest that the at least in these simple cases, the running time is approximately linear in the number of $(n-1)$-simplices. The $n=3$ case is below the other two, because the primary obstruction is trivial there and the EARLIEST SOLUTION subroutine terminates almost immediately, using only very few columns in the matrix reduction.

Fig. 3

Total running time of the computations, performed on Intel Xeon CPU X5680, 3.3 GHz, as a function of the number of $(n-1)$-simplices (which equals the number of columns in the coboundary matrix)

In higher dimensions, the condition $r_0\ge \alpha :=4n(g-1)$ is too strict and we can only hardly start with $r_0$ that is smaller than the robustness. However, we could take $r_0$ to be the minimal value for which $f'|_{A_{r_0}^\square }$ is simplicial: in all cases that we have tried, we verified that f was nowhere zero on $A_{r_0}^\square$ with $f'$ homotopic to f/|f| on $A_{r_0}^\square$.

Surprisingly, starting with minimal $r_0$ for which $f'|_{A_{r_0}^\square }$ is simplicial and computing the minimal $r_1$ for which $f'$ is extendable, yields in many cases a much better estimate of the robustness of zero set than the estimates based on Lemma 1: this can already be seen in the above table where the $r_1$ column is a good approximation of the true robustness. We give the table with smaller grids that continues to higher dimensions.

n	g	Min simplicial $r_0$	$r_1$	True robustness
2	10	0.025	0.889	0.828
3	10	0.025	$=r_0$	0
4	10	0.025	0.667	0.667
5	10	0.074	$=r_0$	0
6	10	0.074	0.667	0.58
7	6	0.241	$=r_0$	0
8	5	0.251	1.0	0.522

It is an interesting question to find natural conditions on functions, other than the relatively weak simplexwise Lipschitz property, that justify the usage of smaller grids and guarantee that the robustness of zero set is close to the computed minimal r for which $f'|_{A_{r}^\square }$ becomes extendable.

1.2 A function with nontrivial secondary obstruction

A second function we experimented with is $h\colon [-1,1]^{n+1}\rightarrow {\mathbbm{R}}^n$ given by

$$\begin{aligned}&h_1(x)= 2x_0 x_2 + 2x_1 x_3\\&h_2(x)= 2x_1 x_2 - 2x_0 x_3\\&h_3(x)= x_0^2+x_1^2-x_2^2-x_3^2\\&h_4(x)=x_4 \\&\ldots \\&h_n(x)=x_n. \end{aligned}$$

The restriction of h to $\partial [-1,1]^{n+1}$ is the generator of the nontrivial homotopy group $\pi _n(S^{n-1})$: in case $n=3$ it is the Hopf map and for $n>3$ its iterated suspension. The robustness of zero equals $\min _{x\in \partial [-1,1]^{n+1}} |h(x)|$ and¹² it is the simplest example where a nontrivial secondary obstruction occurs. Common tests for zero verification such as the degree test would fail here.

Again, we work with the max-norm $\ell _\infty$ and derive, via elementary calculus, the estimate $2(n+1)$ on the global Lipschitz constant. This yields $|h(x)-h(y)|\le \frac{4(n+1)}{g-1}=:\alpha$. Assume that only $\alpha$ and a list of function values in a grid $g^{n+1}$ is given. If g is large enough so that $r_0:=\alpha$ is smaller than $\min _{\partial } |h(x)|_{\infty }=\sqrt{3}-1$, then the algorithm computes the minimal $r_1>r_0$ for which $h|_{A_r^\square }$ is extendable to a nowhere zero function $[-1,1]^{n+1}\rightarrow {\mathbbm{R}}^{n}{\setminus }\{0\}$. For this to succeed, we need to take g at least 22 for $n=3$.

Again, the program gives surprisingly good results for much smaller grids, although Lemma 1 gives no guarantees. In all cases that we tried, we verified that whenever $r_0$ is large enough so that $h'|_{A_{r}^\square }$ is simplicial, then it is homotopic to h/|h| and the algorithm computes that the secondary obstruction dies close to the real robustness of zero.

n	g	Min simplicial $r_0$	$r_1$	True robustness
3	10	0.1	0.79	0.732
4	10	0.111	0.79	0.732
5	8	0.163	0.816	0.732