# Approximate Triangulations of Grassmann Manifolds

## Abstract

**:**

## 1. Introduction

**Theorem**

**1**

**.**Every triangulation of the Grassmann manifold ${G}_{k}\left({\mathbb{R}}^{n+k}\right)$ must have at least

## 2. Materials and Methods

#### 2.1. Grassmann Manifolds

**Definition**

**1.**

**Lemma**

**1.**

#### 2.2. Schubert Cells

**Definition**

**2.**

**Lemma**

**2.**

$\sigma $ | $d\left(\sigma \right)$ |

$(1,2)$ | 0 |

$(1,3)$ | 1 |

$(1,4)$ | 2 |

$(2,3)$ | 2 |

$(2,4)$ | 3 |

$(3,4)$ | 4 |

**Theorem**

**2.**

**Proposition**

**1.**

#### 2.3. Persistent Homology

#### 2.4. Vietoris–Rips and Witness Complexes

- •
- The vertex set of $W(X,L,R)$ is L;
- •
- $\ell ,{\ell}^{\prime}\in L$ span an edge if there exists an $x\in X$, called a witness, such that$$d(x,\ell ),d(x,{\ell}^{\prime})\le R+min\{d(x,{\ell}^{\prime \prime}):{\ell}^{\prime \prime}\in L-\{\ell ,{\ell}^{\prime}\}\};$$
- •
- A collection ${\ell}_{0},\cdots ,{\ell}_{p}\in L$ spans a p-simplex if $\{{\ell}_{i},{\ell}_{j}\}$ span an edge for all $i\ne j$.

- If $R=0$, then $\ell ,{\ell}^{\prime}\in L$ form an edge if there is an ${x}_{i}\in X$ such that $d({x}_{i},\ell )$ and $d({x}_{i},{\ell}^{\prime})$ are the two smallest entries in the i-th column of D. This is analogous to the existence of an edge in the Delaunay triangulation $\mathrm{Del}\left(L\right)$.
- For $R>0$, one may think of relaxing the boundaries of the Voronoi diagram of L and taking the nerve of the resulting covering of X.
- If $0\le R<{R}^{\prime}$, then there is an inclusion of simplicial complexes $W(X,L,R)\subseteq W(X,L,{R}^{\prime})$.

- How should the landmark set L be chosen?
- What is the correct value of R?

- Select landmarks at random.
- Use the maxmin procedure: Choose a seed ${\ell}_{1}$ at random. Then, if ${\ell}_{1},\cdots ,{\ell}_{n}$ have been chosen, let ${\ell}_{n+1}\in X-\{{\ell}_{1},\cdots ,{\ell}_{n}\}$ be the point which maximizes the function$$z\mapsto min\{d(z,{\ell}_{1}),d(z,{\ell}_{2}),\cdots ,d(z,{\ell}_{n})\}.$$
- Use a density-based strategy.

#### 2.5. Sampling Procedures

**Proposition**

**2.**

**Proof.**

- Select k random vectors in ${\mathbb{R}}^{n}$.
- Perform the Gram–Schmidt orthogonalization algorithm to yield an orthornomal set ${x}_{1},\cdots ,{x}_{k}$. Let A be the matrix with ${x}_{i}$ as columns.
- Compute $A{A}^{T}$.

- Determine the percentage of sample points desired from each Schubert cell. For example, one might choose 5% from a 1-cell, 10% from a 2-cell, and so on.
- Elements of a given Schubert cell correspond to the column space of a particular matrix form. Generate such a matrix B using random vectors of the required form.
- Generate a random $n\times n$ orthogonal matrix X.
- Add the matrix $A=X\left(B{B}^{T}\right){X}^{T}$ to the point cloud.

#### 2.6. Approximate Triangulations

- Construct a sample of points on ${G}_{k}\left({\mathbb{R}}^{n}\right)$.
- Construct a collection of Vietoris–Rips or witness complexes on the point cloud.
- Compute the persistent homology of this filtration.
- Determine a range of parameters where the homology of the complexes agrees with that of ${G}_{k}\left({\mathbb{R}}^{n}\right)$.

**Definition**

**3.**

## 3. Results

#### 3.1. $\mathbb{R}{P}^{2}$, Part I

#### 3.2. $\mathbb{R}{P}^{2}$, Part II

#### 3.3. $\mathbb{R}{P}^{3}$

#### 3.4. ${G}_{2}\left({\mathbb{R}}^{4}\right)$, Part I

#### 3.5. ${G}_{2}\left({\mathbb{R}}^{4}\right)$, Part II

## 4. Conclusions

- How small of a sample can we use to generate an approximate triangulation? For example, a result in [3] asserts that any triangulation of ${G}_{2}\left({\mathbb{R}}^{4}\right)$ must have at least 14 vertices. We built an approximate triangulation using a witness complex on 100 landmarks. Surely, our algorithm will not work with only 14 points, but we plan to investigate how few we can get away with. A theorem of Niyogi–Smale–Weinberger [16] provides lower bounds on the number of points required to compute homology correctly with high probability, but these are certainly too high and can be improved in practice.
- Can we push the computations further? The next Grassmannian to study is ${G}_{2}\left({\mathbb{R}}^{5}\right)$. This is a nonorientable 6-manifold, and, using our procedure, we would embed it in ${\mathbb{R}}^{25}$. The machine used to compute the persistent homology of the witness complexes on ${G}_{2}\left({\mathbb{R}}^{4}\right)$ in MATLAB ran out of memory on 100 landmarks in ${G}_{2}\left({\mathbb{R}}^{5}\right)$. We therefore need either a bigger machine running MATLAB, or software that can handle witness complexes. The GUDHI package [17] is one option, but we have not attempted it yet.
- The author expects to gain access to a new GPU based supercomputer at his institution in the next year. This may allow for similar computations on higher-dimensional ${G}_{k}\left({\mathbb{R}}^{n}\right)$.

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) a Delaunay triangulation of a collection of points in the plane with the corresponding Voronoi diagram, and (

**b**) two associated witness complexes.

**Figure 3.**Vietoris–Rips persistence diagrams for 100 points on $\mathbb{R}{P}^{2}$ (

**a**) ${H}_{1}$ persistence and (

**b**) ${H}_{2}$ persistence.

**Figure 4.**Vietoris–Rips persistence diagrams for 200 points on $\mathbb{R}{P}^{2}$ (

**a**) ${H}_{1}$ persistence and (

**b**) ${H}_{2}$ persistence.

**Figure 5.**Vietoris–Rips persistence diagrams for 100 points on $\mathbb{R}{P}^{2}$, using the isometric embedding into ${\mathbb{R}}^{5}$ (

**a**) ${H}_{1}$ persistence and (

**b**) ${H}_{2}$ persistence.

**Figure 6.**Vietoris–Rips persistence diagrams for 200 points on $\mathbb{R}{P}^{2}$, using the isometric embedding into ${\mathbb{R}}^{5}$ (

**a**) ${H}_{1}$ persistence and (

**b**) ${H}_{2}$ persistence.

**Figure 7.**Vietoris–Rips persistence diagrams for 100 points on $\mathbb{R}{P}^{3}$, realizing it as the Lie group $SO\left(3\right)\subset {\mathbb{R}}^{9}$ (

**a**) ${H}_{1}$ persistence and (

**b**) ${H}_{2}$ persistence.

**Figure 9.**Vietoris–Rips persistence diagrams for 200 points on $\mathbb{R}{P}^{3}$ (

**a**) ${H}_{1}$ persistence and (

**b**) ${H}_{2}$ persistence.

**Figure 11.**Vietoris–Rips persistence diagrams for 150 points on ${G}_{2}\left({\mathbb{R}}^{4}\right)$ (

**a**) ${H}_{1}$ persistence and (

**b**) ${H}_{2}$ persistence.

**Figure 12.**Vietoris–Rips persistence diagrams for 150 points on ${G}_{2}\left({\mathbb{R}}^{4}\right)$ (

**a**) ${H}_{3}$ persistence and (

**b**) ${H}_{4}$ persistence.

**Figure 13.**Barcodes for a witness complex on 100 points in a 5000-point sample on ${G}_{2}\left({\mathbb{R}}^{4}\right)$ (

**a**) $\mathbb{Z}/2$ coefficients and (

**B**) $\mathbb{Z}/3$ coefficients.

**Table 1.**Computation times on a MacBook Pro, 16 GB RAM, of Vietoris–Rips persistence for various spaces. An X indicates that the software could not complete the calculation; a indicates the software was not used for the run. The ? indicates the number of simplices is unknown. Note the rapid explosion in the number of simplices.

Space | # Points | Top Dim | # Simplices | Eirene | Ripser |
---|---|---|---|---|---|

$\mathbb{R}{P}^{2}\subset {\mathbb{R}}^{4}$ | 100 | 2 | 206K | 0:00.53 | – |

200 | 2 | 2.1M | 0:01 | – | |

$\mathbb{R}{P}^{2}\subset {\mathbb{R}}^{5}$ | 100 | 2 | 436K | 0:02 | – |

200 | 2 | 4.2M | 0:07 | – | |

$\mathbb{R}{P}^{3}\subset {\mathbb{R}}^{9}$ | 100 | 3 | 7.6M | 0:09 | – |

200 | 3 | 146M | 6:54 | – | |

${G}_{2}\left({\mathbb{R}}^{4}\right)\subset {\mathbb{R}}^{16}$ | 100 | 4 | 107M | 1:51 | 1:15 |

150 | 4 | 792M | 1:04:45 | X | |

200 | 3 | 112M | 3:01 | 3:07 | |

200 | 4 | ? | X | X |

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**MDPI and ACS Style**

Knudson, K.P.
Approximate Triangulations of Grassmann Manifolds. *Algorithms* **2020**, *13*, 172.
https://doi.org/10.3390/a13070172

**AMA Style**

Knudson KP.
Approximate Triangulations of Grassmann Manifolds. *Algorithms*. 2020; 13(7):172.
https://doi.org/10.3390/a13070172

**Chicago/Turabian Style**

Knudson, Kevin P.
2020. "Approximate Triangulations of Grassmann Manifolds" *Algorithms* 13, no. 7: 172.
https://doi.org/10.3390/a13070172