## Hi! * https://hackmd.io/@story645/rkpjEbxN3 * grad student at GC studying viz * matplotlib community manager (contributor/maintainer since $\approx$ 2016) * on most of the socials: @story645 * email: haizenman@gmail.com --- # What do visualization libraries do? <!--provide components for converting data to graphics-->  ---- <!-- the artist object-->  --- # What problems are we trying to solve? $data \rightarrow graphic$ ---- # What problems are we trying to solve? $data \rightarrow graphic$ $data \leftarrow graphic$ --- How do we verify structure is preserved? $data \leftrightarrow graphic$ ---- # What is structure? ---- ## Topology + Fields [Bertin]  ---- ## Relations between elements [Mackinlay]  --- # Abstract math to build better tools? :confused: --- ## Define and test structure preservation: * **Homemoporhism** bijective $g(f(x))=f(g(y)) = y, \;\forall x \in X, y \in Y$, continuous $\lim_{x\rightarrow c}f(x) = f(c)$, and has a continous inverse function $f^{-1}$. * **Homomorphism**: $f(x\circ y) = f(x) \circ f(y)$ * **Equivariance**: $f(act(g,x)) = act(g,f(x))$ --- ### Homeomorphism  --- ### Homomorphism: $f(x\circ y) = f(x) \circ f(y)$  --- ### Equivariance: $f(act(g,x)) = act(g,f(x))$  # What does dimensionality mean? --- # Why fiber bundles? [Butler] ----  ----  ---- ### What's the advantage?  ----  ---- ### We can also represent graphics the same way  --- ## Topology as categories Partitioning of a space into its subspaces such that those subspaces can be composed back into the total space; the choice of partitions is how we encode the connectivity of the data and visualizations that we are interested in. * objects: subspaces of the data $U_i$ * morphisms: inclusion $i: U_i \rightarrow U_j$ ---- ### Joining subsets: $\oplus: K \sqcup K \rightarrow K$  --- ## Fields [Spivak] as categories * objects - set of all possible values for a given type * $\pi^{1}(\texttt{int}) = \mathbb{Z}$ * morphisms: * closed operations: eg +, -, *, % * arbitrary closed functions $\tilde\phi \in Hom(F, F)$ ---- ## Joining records: $\otimes: F \times F \rightarrow F$  ---- # Data? $\tau: k \mapsto record$  ---- ## Data containers: sheaves  --- # What does preserving topology really mean?  --- # What is visualization?  ---- | | topology | fields | dataset| |---|------|-----|-----| |data| $k \in U \subseteq K$ | $r\in F$ | $\tau: K \rightarrow E$| |visual| $k \in U \subseteq K$ | $v \in P$ | $\mu: K \rightarrow V$ | | graphic| $s \in W \subseteq S$ | $d \in D$ | $\rho: S \rightarrow H$| ----  --- ## Homeomomorphism: $\xi$  --- ## Homomorphism, equivariance, etc..  ----  --- ## How do we test structure: change(input)==change(output)  ## Multivariate complex dimensionality ---  ---  ---  --- # So how do we build it?  --- ## Indexing: $\xi: W \mapsto U$ \begin{equation} \underbrace{\xi: K\times[0,1]^{n}}_{S}\mapsto K \end{equation} \begin{equation*} n = \begin{cases} dim(S) - dim(K) & dim(K)<dim(S)\\ 0 & otherwise \end{cases} \end{equation*} --- Everything in the graphic maps back to data: for $s \in W$ exists $k \in U$ s.t. $\xi(s) = k$  --- ## Encoding: $\nu: E \mapsto V$  --- Moving values does not change encoding: $\nu(\tau_{E}(\hat\phi_{k}(k^{\prime})))= = \mu(\hat\phi_{k})(k^{\prime}))$ Changing values changes the encoding in a consistent way: and $\widetilde\phi_{V}\nu(\tau(k)) = \nu(\widetilde\phi_{E}(\tau(k)))$ ---- ### Infinite composability  ----  --- ## Pre-rendering: $Q: V \rightarrow G$ Moving values retains relative position while not changing visual encoding: $\mu(\hat\phi_{K}(k^{\prime})) = \lambda(\rho|_\xi^{1}(k))$ Changing values changes the prerendering in a consistent way: $\widetilde\phi_{V}(\mu(k)) = \widetilde\phi(\lambda(Q(\mu(k))))$ ----  ----  ----  ---