Principal Components Along Quiver Representations

Abstract

Quiver representations arise naturally in many areas across mathematics. Here we describe an algorithm for calculating the vector space of sections, or compatible assignments of vectors to vertices, of any finite-dimensional representation of a finite quiver. Consequently, we are able to define and compute principal components with respect to quiver representations. These principal components are solutions to constrained optimisation problems defined over the space of sections and are eigenvectors of an associated matrix pencil.

Sequential Measurements, Topological Quantum Field Theories, and Topological Quantum Neural Networks

“…First of all, we recall the basic definitions of a quiver and its representation [ 76 ] as applied in Ref. [77]. A quiver Q is a directed graph consisting of a pair $$(V,E)$$, where V denotes a finite set of vertices and E a finite set of edges (arrows), and two maps $$s,t: E \longrightarrow V$$, the source (tail) and target (head), respectively.…”

“…As in Ref. [77], on taking a path $$p = (e_1, e_2, \ldots , e_k)$$ in Q of distinct edges and sources, we associate to each such path p the map $$\mathbf {A}_p: \mathbf {A}_{s(p)} \longrightarrow \mathbf {A}_{t(p)}$$, defined via $$\begin{equation} \mathbf {A}_p := \mathbf {A}_{e_{k}} \,\circ\, \mathbf {A}_{e_{k-1}} \circ \cdots \circ\, \mathbf {A}_{e_{2}} \circ \mathbf {A}_{e_{1}} \,. \end{equation}$$The total space of $$\mathbf {A}_{\bullet }$$ is the direct product $$\text{Tot}(\mathbf {A}_{\bullet }) : = \prod _{v \in V} \mathbf {A}_v$$.…”

“…We next turn to the definition of a section as given in [77]: Definition Let $$\mathbf {A}_{\bullet }$$ be a representation of $$Q = (s,t: E \longrightarrow V)$$. A section of $$\mathbf {A}_{\bullet }$$ is an element $$\gamma = \lbrace \gamma _v \in \mathbf {A}_v: v \in V\rbrace$$ in $$\text{Tot}(\mathbf {A}_{\bullet })$$ satisfying the compatibility requirement $$\gamma _{t(e)} = \mathbf {A}_e(\gamma _{s(e)})$$ across each edge e in E .…”

“…The set of all sections of $$\mathbf {A}_{\bullet }$$ is a vector subspace of $$\text{Tot}(\mathbf {A}_{\bullet })$$ denoted $$\Gamma (Q, \mathbf {A}_{\bullet })$$. As noted in [77], Definition 12 commences an analogy between quiver representations and the theory of sheaves and vector bundles as they arise in algebraic geometry.…”

“…As in Ref. [77], on taking a path p = (e 1 , e 2 , … , e k ) in Q of distinct edges and sources, we associate to each such path p the map A p : A s(p) ←→ A t(p) , defined via…”