Examples

Filtering also by density

Filtering also by density

Let us add a density filtration to $\mathrm{VR}(\mathcal{P},r)$. Fix $k\in {\mathbb{Z}}_{\ge 0}$ and define $\rho :\mathcal{P}\to \mathbb{R}$ by:

$$\rho (x_i)=\frac{1}{d_k(x_i)},d_k(x_i)=\text{distance to kth nearest neighbor}$$

Now the points in $\mathcal{P}$, and the $\mathrm{VR}$-complex itself, can be filtered by a density minimum:

$$\mathrm{VR}(\mathcal{P},r,s)=\mathrm{VR}(\{x\in \mathcal{P}\mid \rho (x)\ge s\},r)$$

This bi-filtered complex determines a 2-parameter persistence module:

$$H_k(\mathrm{VR}(\mathcal{P},r,s))$$

For large $s$, only points very close to $k$ distinct neighbors are included. As $s$ decreases, more points are brought in.

With data like this, the $s$ filtration (for middling $s$ values) removes points inside the disk because they are too sparse.

Filtering also by distance to a subspace

Filtering also by distance to a subspace

Suppose $\iota :{\mathbb{R}}^k\hookrightarrow {\mathbb{R}}^n$ is an embedded $k$-dimensional subspace of the background space ${\mathbb{R}}^n$ that contains the data:$\mathcal{P}\subset {\mathbb{R}}^n$.

Let $f_{\iota }:\mathcal{P}\to \mathbb{R}$ measure the distance from data points to the image of $\iota$.

Define ${\mathcal{P}}_R=f^{-1}((-\infty ,R))$. This gives a filtration in terms of the parameter $R$. This filtration could be useful if the complete data set $\mathcal{P}$ is too big to process all at once. The filtration allows one to process the data in layers.

These filtrations determine inclusions of $\mathrm{VR}$ complexes. For fixed $R$ and $r<s$, we have:

$$\mathrm{VR}({\mathcal{P}}_R,r)\subset \mathrm{VR}({\mathcal{P}}_R,s)$$

Upon applying homology, for $k\in {\mathbb{Z}}_{\ge 0}$, we have:

$$H_k(\mathrm{VR}({\mathcal{P}}_R,r))\stackrel{L_{r,s}^R}{⟶}{H}_K(\mathrm{VR}({\mathcal{P}}_R,s)))$$

And for fixed $r$ and $R<S$, we have:

$$\mathrm{VR}({\mathcal{P}}_R,r)\subset \mathrm{VR}({\mathcal{P}}_S,r)$$

Upon applying homology, we have:

$$\mathrm{VR}({\mathcal{P}}_R,r)\stackrel{L_r^{R,S}}{⟶}\mathrm{VR}({\mathcal{P}}_R,s)$$

Finally, these induced maps should commute with each other:

$$L_s^{R,S}\circ L_{r,s}^R=L_{r,s}^S\circ L_r^{R,S}$$

Morse layers

Morse layers

An application of zig-zag filtration arises by dividing the entire space into small layers for which the homology is easier to compute. For example, with a Morse function $f:\mathcal{K}\to {\mathbb{R}}_{\ge 0}$, we have this zig-zag filtration:

$$f^{-1}(0)\hookrightarrow f^{-1}([0,1])\hookleftarrow f^{-1}(1)\hookrightarrow f^{-1}([1,2])\hookleftarrow f^{-1}(2)\hookrightarrow \cdots$$

Repeated samples

Repeated samples

Suppose some data is sampled repeatedly and data sets are generated in a sequence:

$$X_1,X_2,X_3,X_4,\dots$$

Then we can form a zig-zag filtration by:

$$X_1\hookrightarrow X_1\cup X_2\hookleftarrow X_2\hookrightarrow X_2\cup X_3\hookleftarrow X_3\hookrightarrow X_3\cup X_4\hookleftarrow \cdots$$