Configuration Space

Configuration space is a construction closely related to state spaces and phase spaces. In physics, these are used to describe the state of a whole system as a single point in a high-dimensional space, here configuration refers to the position of all the constituent point particles of the system. In mathematics, they are used to ascribe assignments of a collection of points to positions in a topological space.

In classical mechanics, the parameters that define configuration of a system are referred to as generalized coordinates, and space defined by these coordinates is called the configuration space of the system. In mathematics, in particular in topology, a notion of "restricted" configuration space is mostly used, in which the diagonals, representing "colliding" particles, are removed.

# Configuration Spaces in Mathematics

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## Braid group

Main article: Braid

The n-strand braid group on a connected topological space $X$ is the fundamental group of the n-point unordered configuration space on $X$.

$$B_n(X) := \pi_1(\mathrm{UC}_n(X))$$

The n-strand pure braid group on $X$ is, similarly,

$$P_n(X) := \pi_1(\mathrm{C}_n(X))$$

First studied braid groups were the Artin braid groups $B_n \cong \pi_1(\mathrm{UC}_n(\mathbb{R}^2)$. It follows from this definition and the fact that $\mathrm{C}(\mathbb{R}^2$ and $\mathrm{UC}(\mathbb{R}^2$ are Eilenberg-MacLane space of the type $K(\pi, 1)$ with both being classifying spaces for respective Artin braid groups when considered as discrete groups.

# Configuration Spaces in Physics

## Phase Space

Main article: Phase space, cotangent bundle

The configuration space is insufficient to completely describe a mechanical system: it fails to take into account velocities. The set of velocities available to a system defines a plane tangent to the configuration manifold of the system. At a point $q \in Q$, that tangent plane is denoted by $T_q Q$. Momentum vectors are linear functionals of the tangent plane, known as cotangent vectors; defined at every point as a cotangent plane $T_q^\ast Q$. The set of positions and momenta of a mechanical system forms the cotangent bundle $T^\ast Q$ of the configuration manifold $Q$. This larger manifold $T^\ast Q$ is the phase space of the system.

## State Space

In quantum mechanics, particularly quantum computing, an analog concept of a state space is used. The analog of a "point particle" is a single point in $\mathbb{CP}^1$, also known as the Bloch sphere. It is complex because quantum mechanical wavefunction has a complex phase, and it is projective because the wavefunction is normalized to unit probability.

## Feature Space

In machine learning and pattern recognition, an object is often represented by an n-dimensional feature vector, a collection of measurable characteristics known as features. This abstract mathematical construct, known as the feature space, allows algorithms to discern patterns and make predictions. A feature vector is an n-dimensional vector of numerical features that represent some object. The vector space associated with these vectors is often called the feature space.

Topological data analysis and machine learning share deep connections, particularly when viewed through the lens of their respective abstraction spaces - the configuration spaces in TDA, and feature spaces in machine learning. Both these spaces serve as abstract mathematical constructs to encapsulate complex data structures. The connections between these two spaces become apparent when leveraging TDA to analyze and understand feature spaces in machine learning. Essentially, TDA can be used to extract shape information from the feature spaces, providing insight into their topological properties and complexities, such as connectivity and clustering, which could have been otherwise obscured in high dimensions.

Dimensionality reduction, applied to both feature and configuration spaces, is a critical method to tackle high-dimensionality issues, simplifying complex data into lower-dimensional constructs while preserving their intrinsic characteristics. In this process, a form of coupling takes place, causing a loss of degrees of freedom, as multiple original dimensions or features are often merged or transformed into fewer representative ones. This coupling can lead to a constrained exploration of the space, potentially limiting the range of patterns and structures that can be identified, but simultaneously reducing computational complexity and overfitting risks.

# Notable Concepts

A remarkable result[4] provides great visual intuition for the relationship between (generalized) rigidity and topological properties of configuration spaces by the way of kinematics of mechanical linkages:

Theorem. Any smooth compact manifold is diffeomorphic to the configuration space of some planar linkage. [4]

Consider a planar mechanical linkage consisting of several flat, rigid rods joined at their ends by pins that permit free rotation in the plane. One can use out-of-plane height (or mathematical license) to assert that interior intersections of rods are ignorable. The configuration space of the linkage is a topological space that assigns a point to each configuration of the linkage — a relative positioning of the rods up to equivalence generated by rotations and translations in the plane — and which assigns neighborhoods in the obvious manner. A neighborhood of a configuration is all configurations obtainable via a small perturbation of the mechanical linkage. The configuration space of a planar linkage is almost always a manifold, the dimension of which conveys the number of mechanical degrees of freedom of the device.

A canonical example of a simple useful linkage is the crank-rocker, where four rods of lengths $\{L_i\}^4_1$ are linked in a cyclic chain. When one rod is anchored, the system is seen to have one mechanical degree of freedom. The configuration space is thus one-dimensional and almost always a manifold. If one has a single short rod, then this rod can be rotated completely about an anchor, causing the opposing rod to rock back and forth. This linkage is used to transform spinning motion into rocking motion. The configuration space of such a linkage is $\mathbb{S}^1 \sqcup \mathbb{S}^1$, the corproduct of two circles. The second circle comes from taking the mirror image of the linkage along the axis of its fixed rod in the plane and repeating the circular motion there: this forms an entirely separate circle's worth of configuration states.

## Motion Planning

• Braids can be used to find non-colliding routes as paths in spacetime.
• Navigation fields use flow dynamics to plan motion of individual points in terms of fields rather than paths.

## Information Fusion

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