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‎M2/Macaulay2/packages/PathSignatures.m2‎

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newPackage("PathSignatures",
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Version => "1.0",
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Authors => {{Name => "Felix Lotter", HomePage => "https://felixlotter.gitlab.io"}, {Name => "Oriol Reig"}, {Name => "Angelo El Saliby"}, {Name => "Carlos Amendola"}},
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Headline => "A package for working with signatures of algebraic paths",
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AuxiliaryFiles => true,
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Keywords => {"Applied Algebraic Geometry"},
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PackageExports => {"NCAlgebra", "Permutations"}
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);
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export {
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--types
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"Path",
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--methods
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"sig",
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"polyPath",
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"linPath",
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"pwLinPath",
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"concatPath",
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"matrixAction",
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"CAxisTensor",
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"CMonTensor",
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"tensorParametrization",
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"wordAlgebra",
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"sgnVolTensor",
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"shuffle",
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"halfshuffle",
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"wordFormat",
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"wordString",
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"getDimension",
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"getPieces",
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"getCoefficientRing",
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"getNumberOfPieces",
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-- symbols
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"adjointWord",
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"tensorArray",
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"inner",
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"lyndonWords",
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"lie",
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"lieBasis",
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"tensorExp",
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"tensorLog",
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"lyndonShuffle",
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"VarWordTable"
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-- "type",
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-- "pieces",
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-- "dimension",
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-- "numberOfPieces",
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-- "bR"
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};
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exportFrom("NCAlgebra",{"NCRingElement", "NCPolynomialRing"})
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--------------------------------------------
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--Include interface for NCAlgebra
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load "./PathSignatures/interfaceNCAlgebra.m2"
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--Include tensor algebra
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load "./PathSignatures/algebra.m2"
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--Include types
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load "./PathSignatures/types.m2"
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--Include signatures
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load "./PathSignatures/signatures.m2"
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--Include documentation
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load "./PathSignatures/documentation.m2"
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--------------------------------------------
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----------------------------------------------------
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--Landing page and case use documentation
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----------------------------------------------------
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beginDocumentation()
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doc ///
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Node
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Key
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PathSignatures
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Headline
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a package for working with signatures of algebraic paths
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Description
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Text
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{\em PathSignatures} is a package for studying the signature of piecewise polynomial paths.
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Text
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The package heavily simplifies the process of obtaining data related to signature varieties, e.g. as in @HREF("#ref1","[1]")@. See @TO "Computing Path Varieties"@.
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Text
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A polynomial path is a path $X: [0,1] \to \mathbb R^d$ whose coordinate functions are given by polynomials. A piecewise polynomial path is a path $X: [0,1] \to \mathbb R^d$ which is polynomial on each interval in a partition of $[0,1]$.
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Text
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Given such a path $X$, its signature is the linear form $\sigma: T((\mathbb{R}^d)^*)\rightarrow \mathbb{R}$ on the tensor algebra of the dual of
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$\mathbb{R}^d$, whose image on a decomposable tensor $\alpha_1\otimes \dots\otimes \alpha_k$ is the iterated integral $$
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\alpha_1\otimes \dots\otimes \alpha_k\overset{\sigma}{\mapsto} \int_0^1\int_0^{t_k}\dots\int_0^{t_2}\partial(\alpha_1 X)\dots \partial (\alpha_k X) d t_1\dots dt_k.
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$$
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This form is invariant under translation, reparametrization and tree-like equivalence of $X$ and characterizes $X$ uniquely up to these relations.
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Text
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In this package, we identify $T((\mathbb{R}^d)^*)$ with the free associative algebra over the alphabet $\{\texttt{1},\dots,\texttt{d}\}$ via $\texttt{i} \mapsto e_i^*$ where $e_1^*, \dots, e_d^*$ is the dual of the canonical basis of $\mathbb{R}^d$. For example, the word $\texttt{12}$ corresponds to $e_1^* \otimes e_2^*$.
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Text
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It is easy to create a polynomial path:
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Example
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R = QQ[t];
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X = polyPath({t + t^2, t^3})
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Text
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A piecewise polynomial path is obtained by concatenating polynomial paths:
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Example
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Y = X ** X
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Text
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Any @TO2 {"NCAlgebra::NCPolynomialRing", "NCPolynomialRing"}@ can serve as a tensor algebra. Use @TO wordAlgebra@ to quickly create one in variables $\texttt{Lt}_i$. The package introduces a convenient notation for words in this algebra.
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Example
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A2 = wordAlgebra(2)
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[1,2]_A2 -- the word 12.
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Text
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To evaluate the signature of $\mathtt{X}$ at a tensor $\mathtt{w}$, use @TO sig@. The following computes the @ITALIC "signed volume"@ of the path; also see @TO sgnVolTensor@.
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Example
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sig(X,[1,2]_A2-[2,1]_A2)
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Text
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@TO sig@ can also be used to obtain the @ITALIC "$k$-th level signature tensor"@. Use @TO wordFormat@ or @TO tensorArray@ to display the tensor in a nicer way.
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Example
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T = sig(X,2)
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T // wordFormat
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T // tensorArray
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Text
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The package allows for the computation of signatures for parametrized families of paths.
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Example
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S = QQ[a,b,c]
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R = S[t]
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X = polyPath({a*t+b*t^2,c*t^3})
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Y = X ** X
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sig(X, sgnVolTensor(A2))
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References
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@LABEL("[1]","id" => "ref1")@ Améndola, C., Friz, P., & Sturmfels, B. (2019, January). Varieties of signature tensors. In Forum of Mathematics, Sigma (Vol. 7, p. e10). Cambridge University Press.
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Node
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Key
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"Computing Path Varieties"
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Description
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Text
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@TO PathSignatures@ simplifies the computation of varieties coming from signature tensors. We showcase this in a number of examples:
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Subnodes
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"Polynomial paths of degree m"
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"A family of paths on a cone"
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"The universal variety and toric coordinates"
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Node
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Key
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"Polynomial paths of degree m"
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Description
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Text
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We reproduce a computation from @HREF("#ref1","[1]")@.
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We briefly recall the setting for this computation. We are interested in paths $X:[0,1]\rightarrow \mathbb{R}^\mathtt{d}$ whose coordinates are polynomials of degree $\mathtt{m}$. These can be
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represented by a $\mathtt{d}\times \mathtt{m}$ matrix with real entries whose coordinates are determined by the expressions
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$$ X_i(t) = x_{i,1}t+x_{i,2}t^2+\dots+ x_{i,m}t^m$$
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When we restrict to the $\mathtt{k}$-th level signature of $X$, $\sigma:= \sigma^{(\mathtt{k})}(X)$, each of its coordinates $\sigma_{i_1, \dots, i_{\mathtt{k}}}$ is a homogeneous
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polynomial of degree $\mathtt{k}$ in the $\mathtt{d}\cdot \mathtt{m}$ unknowns $x_{i,j}$ corresponding to the matrix representation of the path $X$. Let $\mathtt{CMon}$ be the canonical monomial path
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in $\mathbb{R}^{\mathtt{m}}$ introduced in @TO CMonTensor@. Then $\sigma$ can be computed through the @TO matrixAction@ of $X$ on $\sigma^{(k)}(\mathtt{CMon})$.
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Moreover, we can view the entries of $X$ as coordinates in the projective space $\mathbb{P}^{\mathtt{d}\cdot \mathtt{m}-1}$ over some algebraically closed field $\mathbb{K}$ containing
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$\mathbb{R}$. Then the matrix action just described gives rise to a rational map $$
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\sigma^{(\mathtt{k})}:\mathbb{P}^{\mathtt{d}\cdot \mathtt{m}-1}\rightarrow \mathbb{P}^{\mathtt{d}^{\mathtt{k}}-1}$$
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determined by $X\mapsto \sigma^{(\mathtt{k})}(X)$, of degree $\mathtt{k}$. The {\em polynomial signature variety}, denoted $\mathcal{P}_{\mathtt{d},\mathtt{k},\mathtt{m}}$, is then defined to be
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the Zariski closure of the image of this map (informally, the closure of the space of all tensors of order $\mathtt{k}$ that arise as signatures of paths of the specified type), while its homogeneous prime ideal is called
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{\em polynomial signature ideal} and denoted $P_{{\mathtt{d},\mathtt{k},\mathtt{m}}}$.
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Text
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We set up the procedure to compute the ideal $P_{\mathtt{d},\mathtt{k},\mathtt{m}}$.
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Example
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d = 2; k = 3; m = 2;
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Text
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We create a ring $\mathtt{R}$ with $\mathtt{d}\cdot\mathtt{m}$ variables, corresponding to the entries of a path. Then we create the free algebra $\mathtt{A}$ over $\mathtt{R}$ with $\mathtt{m}$ generators, where we can compute the $\mathtt{k}$-th level signature of the canonical monomial path in $\mathbb{R}^{\mathtt{m}}$.
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Example
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R = CC[x_1..x_(d*m)]; --x_1, ..., x_(d*m) are the entries of the degree m paths in d-dimensional space, seen as matrices
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A = wordAlgebra(m, CoefficientRing => R); --Signatures in m-dimensional space, where the signature of CMon lives.
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sigmaCMon = CMonTensor(k, A); sigmaCMon // wordFormat -- The 2nd level signature of CMon
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Text
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Next we create the @TO genericMatrix@ with $\mathtt{d}\times \mathtt{m}$ variables.
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Example
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M = genericMatrix (R, d, m)
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Text
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Finally we compute the matrix action on $\sigma^{(k)}(\mathtt{CMon})$ with @TO (symbol *, Matrix, NCRingElement)@, and then compute the corresponding map on rings using @TO tensorParametrization@.
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Example
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f = M * sigmaCMon;
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sigVarietyParam = tensorParametrization(f, CoefficientRing => CC);
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Text
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Now that we have the map, any tool for implicitization can be used. We compute its dimension and degree with @TO2 {"NumericalImplicitization::NumericalImplicitization", "NumericalImplicitization"}@.
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Example
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needsPackage "NumericalImplicitization";
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numericalImageDim(sigVarietyParam,ideal 0_R)
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numericalImageDegree(sigVarietyParam,ideal 0_R, Verbose => false)
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Text
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This agrees with the result in Table 3 of @HREF("#ref1","[1]")@, where dimension and degree of the corresponding projective variety is computed.
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References
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@LABEL("[1]","id" => "ref1")@ Améndola, C., Friz, P., & Sturmfels, B. (2019, January). Varieties of signature tensors. In Forum of Mathematics, Sigma (Vol. 7, p. e10). Cambridge University Press.
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Node
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Key
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"A family of paths on a cone"
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Description
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Text
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We consider the following family of polynomial paths of degree 6:
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Example
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S = QQ[a_1..a_6]
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R = S[t]
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u = a_1*t + a_2*t^2 + a_3*t^3;
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v = a_4*t + a_5*t^2 + a_6*t^3;
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X = polyPath({u^2 - v^2, 2*u*v, u^2 + v^2});
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Text
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Let us take a look at its signature matrix variety. We obtain its parametrization as follows:
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Example
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sigMatrix = sig(X,2);
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S = QQ[s_(1,1)..s_(3,3)]
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vwtable = hashTable apply(gens S, i-> (i, new Array from last baseName i))
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m = tensorParametrization(sigMatrix, VarWordTable => vwtable);
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Text
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Let us use numericalImplicitization to obtain information about the dimension of the image.
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Example
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needsPackage "NumericalImplicitization";
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Snum = CC[a_1..a_6];
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Rnum = Snum[t];
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unum = sub(u, Rnum);
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vnum = sub(v, Rnum);
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Xnum = polyPath({unum^2 - vnum^2, 2*unum*vnum, unum^2 + vnum^2});
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sigMatrixnum = sig(Xnum,2);
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mnum = tensorParametrization(sigMatrixnum,CoefficientRing => CC);
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numericalImageDim(mnum,ideal 0_Snum)
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Text
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The universal variety has dimension 6, so we expect at least one additional relation. We use MultigradedImplicitization:
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Example
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needsPackage "MultigradedImplicitization";
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I = sub(ideal flatten values componentsOfKernel(2, m, Grading => matrix {toList(9:1)}), S);
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dim I
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isPrime I
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betti mingens I
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degree I
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Text
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We conclude that our variety is cut out by one linear relation and 6 quadrics. Let us take a look at the linear relation:
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Example
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lin = select(flatten entries gens I, i-> (degree i == {1}))
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Text
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We recognize it as a shuffle polynomial in the letters $\mathtt{1},\mathtt{2},\mathtt{3}$. It corresponds to the constraint $X(1) - X(0) \in V(x^2 + y^2 - z^2)$ for paths in our family.
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Text
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Recall that the universal variety is cut out by the 2-minors of the symmetric part of the matrix. We check if the linear relation is the only additional one on our path family:
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Example
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A = genericMatrix(S,3,3)
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univI = minors(2, A + transpose(A));
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I == univI + ideal lin
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Node
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Key
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"The universal variety and toric coordinates"
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Description
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Text
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Recall that $K \langle \mathtt{1}, \ldots, \mathtt{d} \rangle$ is isomorphic to the free commutative algebra over the Lyndon words and if we grade each word by its length, the algebra homomorphism
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$$\phi: K[\mathtt{w} \ | \ \mathtt{w} \ \mathrm{Lyndon}] \cong K \langle \mathtt{1}, \ldots, \mathtt{d} \rangle, \ \mathtt{w} \mapsto \mathtt{w}$$
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is an isomorphism of graded vector spaces. The inverse $\psi$ of this isomorphism is computed by @TO lyndonShuffle@. Dually, we have the isomorphism of graded vector spaces
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$$\psi^*: K[\mathtt{w} \ | \ \mathtt{w} \ \mathrm{Lyndon}]^* \cong K \langle \mathtt{1}, \ldots, \mathtt{d} \rangle^*, \ \alpha \mapsto \alpha \circ \psi.$$
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Text
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We view the two vector spaces $K\langle \mathtt{1}, \ldots, \mathtt{d}\rangle^*$ and $K[\mathtt{w} \ | \ \mathtt{w} \ \mathrm{Lyndon}]^*$ as infinite-dimensional affine spaces. We are interested in the subset $\mathcal U_d$ of those points of $K\langle \mathtt{1}, \ldots, \mathtt{d}\rangle^*$ that define shuffle algebra homomorphisms. This is a Zariski closed set. As $\psi$ is an algebra homomorphism, they correspond to the points of $K[\mathtt{w} \ | \ \mathtt{w} \ \mathrm{Lyndon}]^*$ under $\psi^*$ that define algebra homomorphisms; these are parametrized by the points of the vector space $K^{\mathcal L}$, where $\mathcal L$ is the set of Lyndon words, via the map
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$$\eta: K^{\mathcal L} \to K[\mathtt{w} \ | \ \mathtt{w} \ \mathrm{Lyndon}]^*, \ x \mapsto ev_x.$$
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In particular, $\mathcal U_d$ is parametrized by $\psi^* \circ \eta, \ x \mapsto (\mathtt{w} \mapsto \psi(\mathtt{w})(x))$.
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Projecting $\mathcal U_d$ to the degree $k$ component, we obtain a subvariety of $((K^d)^{\otimes k})^* \cong (K^d)^{\otimes k}$, called the universal variety, $\mathcal U_{d,k}$.
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As the map $\psi$ is compatible with the projection, $\psi^* \circ \eta$ restricts to a parametrization of $\mathcal U_{d,k}$: it is the image of the induced morphism
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$$K^{\mathcal L_k} \to (K\langle \mathtt{1}, \ldots, \mathtt{d}\rangle_k)^* \cong (K^d)^{\otimes k}, \ x \mapsto (\psi(\mathtt{w})(x) \cdot \mathtt{w})$$
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where $\mathcal L_k$ is the set of Lyndon words of length at most $k$.
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As usual, we can compute the ideal that cuts out the image variety as the kernel of the corresponding ring map $$K[x_{\mathtt w} \ | \ \mathtt{w} \text{ of length } k] \to K[y_{\mathtt w} \ | \ \mathtt{w} \text{ Lyndon of length } \leq k].$$ This map can be computed via @TO lyndonShuffle@. Let us do this in the example $d=3, k=3$.
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Example
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words = toList apply((3:1)..(3:3), i -> new Array from i);
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lwords = lyndonWords(3,3)
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R = QQ new Array from apply(words, i->x_i);
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Q = QQ new Array from (apply(lwords,i->y_i) | {Degrees => apply(lwords, i->length(i))});
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A3 = wordAlgebra(3);
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lpolyHT = apply(words, i -> lyndonShuffle(i_A3));
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lpols = apply(lpolyHT, f -> sum(pairs f, (term,coef) -> coef * product(pairs term, (word,ex)-> y_word^ex)));
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lpols_{0..4}
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m = map(Q,R,lpols);
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Text
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Let us compute the kernel.
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Example
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I = ker m;
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dim I
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degree I
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betti mingens I
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(mingens I)_(0,0)
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Text
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This agrees with the result in Table 2 of @HREF("#ref1","[1]")@.
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Text
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As $\phi$ is an isomorphism of graded vector spaces, we see that the variety $\mathcal U_{d,k}$ is parametrized by the vector of degree $k$ monomials in Lyndon words after a linear coordinate change on $(K^d)^{\otimes k}$. We can use this to simplify the computation of the universal variety.
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Example
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mons = flatten entries basis(3,Q);
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S = QQ[z_1..z_(length mons)];
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m = map(Q,S,mons);
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I = ker m;
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dim I
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degree I
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betti mingens I
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Text
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Let us compute the matrix of the coordinate change for $d=2, k=4$.
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Example
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words = toList apply((4:1)..(4:2), i-> new Array from i);
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lwords = lyndonWords(2,4);
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R = QQ new Array from apply(words,i->x_i);
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Q = QQ new Array from (apply(lwords,i->y_i) | {Degrees => apply(lwords, i->length(i))});
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A2 = wordAlgebra(2);
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lpolyHT = apply(words, i -> lyndonShuffle(i_A2));
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lpols = apply(lpolyHT, f -> sum(pairs f, (term,coef) -> coef * product(pairs term, (word,ex)-> y_word^ex)));
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mons = basis(4,Q)
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M = sub(matrix apply(lpols, i -> (flatten entries (coefficients(i, Monomials => mons))#1) ),QQ)
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M^(-1)
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Text
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Note that the coordinate change differs from the one described in Example 21 of @HREF("#ref2","[2]")@ as our toric parametrization does not arise from the exponential map on the Lie algebra. We can easily construct this coordinate change as well, by using @TO lieBasis@ and @TO tensorExp@:
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Example
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A2 = wordAlgebra(2, CoefficientRing => Q)
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lbasis = apply(lwords, i -> lieBasis(i,A2));
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lT = sum(0..length(lbasis)-1, i-> Q_i * lbasis_i);
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gT = tensorExp(lT,4);
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lpols = apply(words, i-> gT @ i_A2)
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M = sub(matrix apply(lpols, i -> (flatten entries (coefficients(i, Monomials => mons))#1) ),QQ);
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M^(-1)
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Text
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This is the matrix from Example 21 in @HREF("#ref2","[2]")@. Note that while we obtained the coordinate change by inverting the map that sends a word to its coefficient in the exponential (which is a linear combination of Lyndon word monomials), in @HREF("#ref2","[2]")@ the coordinate change is obtained directly without computing the exponential. Both strategies yield the same result by Lemma 18 in loc. cit..
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-- Text
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-- Let us compute a path variety after toric coordinate change.
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-- Example
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-- S = QQ[a_(1,1)..a_(3,2)]
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-- A2 = wordAlgebra(3, CoefficientRing => S)
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-- T = CAxisTensor(4,A2)
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-- A = genericMatrix(S,2,3)
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-- m = tensorParametrization(A * T, VarWordTable => hashTable apply(gens R, i-> (i, last baseName i)))
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-- cc = map(R,R,flatten entries (M^(-1) * (transpose basis(1,R))))
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-- mcc = m * cc;
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-- needsPackage "MultigradedImplicitization"
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-- I = ideal componentOfKernel({2},source mcc,mcc)
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-- J = ideal componentOfKernel({3},source mcc,mcc)
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-- dim (I+J)
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-- isPrime (I+J)
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References
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@LABEL("[1]","id" => "ref1")@ Améndola, C., Friz, P., & Sturmfels, B. (2019, January). Varieties of signature tensors. In Forum of Mathematics, Sigma (Vol. 7, p. e10). Cambridge University Press.
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@LABEL("[2]","id" => "ref2")@ Galuppi, F. (2019). The rough Veronese variety. Linear algebra and its applications, 583, 282-299.
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///
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endPackage;
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