Definition 4 (Definition: Fundamental Group) Let \(X\) be a topological space and \(x_0 \in X\) a basepoint. A loop based at \(x_0\) is a continuous map \(\gamma: [0,1] \to X\) with \(\gamma(0) = \gamma(1) = x_0\). Two loops \(\gamma, \delta\) are homotopic relative to \(x_0\) (written \(\gamma \simeq \delta\)) if there exists a continuous map \(H: [0,1] \times [0,1] \to X\) such that \[H(s,0) = \gamma(s), \quad H(s,1) = \delta(s), \quad H(0,t) = H(1,t) = x_0\] for all \(s,t \in [0,1]\). This is an equivalence relation; denote the equivalence class of \(\gamma\) by \([\gamma]\).
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One challenge is having enough training data. Another is that the training data needs to be free of contamination. For a model trained up till 1900, there needs to be no information from after 1900 that leaks into the data. Some metadata might have that kind of leakage. While it’s not possible to have zero leakage - there’s a shadow of the future on past data because what we store is a function of what we care about - it’s possible to have a very low level of leakage, sufficient for this to be interesting.
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