Signature, Chen's Identity, and the ML / Feature Map¶
Job: develop the signature / Chen's identity / Hambly–Lyons, the truncated / nilpotent group / Lyons lift, the signature in ML (feature map / expected signature), and the signature kernel — with the BM signature / expected signature as the worked example.
Core¶
The signature and Chen's identity¶
| Signature / Chen / Hambly–Lyons | Description |
|---|---|
| Signature \(\mathscr{S}(X)\) | \(\sum_{k\ge0}\int dX^{\otimes k}\in T((V))\) (the tensor series of iterated integrals) |
| Chen's identity | \(\mathscr{S}_{s,t}\otimes\mathscr{S}_{t,u}=\mathscr{S}_{s,u}\) (the multiplicative functional) |
| Hambly–Lyons | the signature determines the path modulo tree-like equivalence |
The truncated signature and the nilpotent group¶
| Truncated / \(G^n\) / Lyons lift | Description |
|---|---|
| Truncated signature | \(\mathscr{S}^{(n)}(X)\in T^{(n)}(V)\) (the step-\(n\) truncation) |
| \(G^n(V)\) | the step-\(n\) free nilpotent group (the group-like elements, \(G^n\subset T^{(n)}\)) |
| Lyons lift | smooth paths embed; rough paths are limits in rough metric (the signature the lift) |
The signature in ML / statistics¶
| Signature ML / feature / expected | Description |
|---|---|
| Feature map | the signature as the universal nonlinearity (Stone–Weierstrass on path space, the ML mathematics) |
| Expected signature | \(\mathbb{E}[\mathscr{S}(X)^{\otimes k}]\) the factorial decay (the BM, the quantitative) |
| Universal | the signature features the universal approximation (the ML mathematics) |
The signature and the rough path / the kernel¶
| Kernel / ODE / controlled | Description |
|---|---|
| Signature kernel | \(K(X,Y)=\langle\mathscr{S}(X),\mathscr{S}(Y)\rangle\) (the ML mathematics) |
| Signature / ODE | the signature the ODE (the controlled, the SPDE) |
See it¶
Signature of Brownian motion and the expected signature / factorial decay. The \(\mathscr{S}(B)_{0,t}=\exp(\sum\otimes B - \frac12 t\cdot I)\) (the Stratonovich, the geometric), the \(\mathbb{E}[\mathscr{S}(B)^{\otimes k}]=\frac{t^{k/2}}{(k/2)!}\mathrm{Sym}^k\) (the factorial decay, the stochastic processes). The key point: the signature the BM the explicit, the expected the factorial decay — the signature the ML the feature.
Check¶
Show that the signature satisfies Chen's identity, and explain the Hambly–Lyons uniqueness.
The Chen: \(\mathscr{S}_{s,t}\otimes\mathscr{S}_{t,u}=\mathscr{S}_{s,u}\) (the iterated integrals the concatenation, the \(\int_s^u=\int_s^t+\int_t^u\) the Chen). The Hambly–Lyons: the signature determines the path modulo tree-like (the tree-like the excursions the signature the invariant, the Hambly–Lyons the uniqueness modulo the tree-like). So the Chen the multiplicative, the Hambly–Lyons the tree-like.
Explain the signature as a feature map for ML, and the expected signature.
The feature map: the \(\mathscr{S}(X)\) the path the feature (the universal nonlinearity, the Stone–Weierstrass on the path space, the ML mathematics). The expected: the \(\mathbb{E}[\mathscr{S}(X)^{\otimes k}]\) the moments (the factorial decay, the BM the \(\frac{t^{k/2}}{(k/2)!}\), the quantitative the approximation). The kernel: the \(K(X,Y)=\langle\mathscr{S}(X),\mathscr{S}(Y)\rangle\) the signature kernel (the ML mathematics). So the signature the feature, the expected the decay.
Pitfalls¶
- The Hambly–Lyons the tree-like (the signature determines the path modulo tree-like, not the path itself) — the tree-like the equivalence, the signature the invariant.
- The truncated / \(G^n\) (the step-\(n\) the nilpotent, the truncated the finite) — the \(G^n\) the nilpotent, the truncated the finite-dim.
- The expected signature the factorial decay (the BM the \(\frac{t^{k/2}}{(k/2)!}\), the quantitative) — the expected the decay, the approximation the rate.
- The signature kernel (the \(K(X,Y)\) the inner product, the ML mathematics) — the kernel the ML, the signature the feature.
Refs¶
- Lyons–Caruana–Lévy, Differential Equations Driven by Rough Paths (LNM 2007); the signature
- Hambly–Lyons, "Uniqueness for the signature of a rough path" (Ann. Math. 2010); the tree-like
- Chevyrev–Kormilitzin, "A primer on the signature method in machine learning" (2016); the ML mathematics
Next¶
RP index · Foundations · Controlled & Sewing · FBM & Non-Semimartingale · Branched & Hopf · ML mathematics · Stochastic processes