Concepts from Riemannian geometry are fundamental in various fields such as Robotics, Computer Vision, Machine Learning, Quantitative Finance, and Medical Imaging.
This article utilizes the Geomstats library to explore and experiment with metrics, connections, and parallel transport in these contexts.
Table of contents
What you will learn: How to implement Riemann metric and connection in Python using Geomstats library.
- Environments: Python 3.10.10, Geomstats 2.7.0
- This article assumes that the reader is somewhat familiar with differential and tensor calculus [ref 1, 2]. Please refer to the previous articles related to geometric learning [ref 5, 6, 7].
- Source code is available at Github.com/patnicolas/Data_Exploration/manifolds
- To enhance the readability of the algorithm implementations, we have omitted non-essential code elements like error checking, comments, exceptions, validation of class and method arguments, scoping qualifiers, and import statements.
Introduction
Geometric learning addresses the difficulties of limited data, high-dimensional spaces, and the need for independent representations in the development of sophisticated machine learning models, including graph-based and physics-informed neural networks.
The primary goal of learning Riemannian geometry is to understand and analyze the properties of curved spaces that cannot be described adequately using Euclidean geometry alone. Riemannian geometry enables to describe the geometric structures of manifolds equipped with a metric, which defines the concept of distance and angle on these spaces.
This article is the seventh part of our ongoing series focused on geometric learning. In this installment, we utilize the Geomstats Python library [ref. 3] and explore the hypersphere manifold, which was introduced in a previous piece, Geometric Learning in Python: Manifolds - Hypersphere and is detailed in the Geomstats API [ref. 4].
I highly recommend watching the comprehensive series of 22 YouTube videos Tensor Calculus - Eigenchris to familiarize yourself with fundamental concepts of differential geometry. Summaries of my earlier articles on this topic can be found in the Appendix
Riemann geometry
First, let's review some basic concepts in Riemannian geometry that we described in previous articles [ref 5, 6, 7].
Vector fields
In tensor calculus a vector field is an assignment of a vector X to each point p on a Euclidean space. For instance, a vector field on a 3D sphere can be visualized as a collection of arrows with a given direction and length.
Fig. 1 Visualization of vector fields on a sphere in 3D Euclidean space
Formally, a vector field X(p) at point p is defined as: \[X: S \rightarrow \mathbb{R}^{n} \ \ \ \ \ X(p) = \sum_{i=1}^{n} X_{i}(p)\frac{\partial }{\partial x_{i}}\]The gradient of a vector field on a differential manifold is defined as: \[\triangledown _{\vec{v}}f= \triangledown f. \vec{v}=\sum_{i=1}^{n}v^{i}\frac{\partial f}{\partial x^{i}} \ \ \ (1) \]
Covariant derivative
The gradient of a vector field is defined by its components along the basis vectors, which are orthogonal in Euclidean space. The covariant derivative, on the other hand, calculates a gradient along a specific vector, particularly the tangent vector of a manifold.
The covariant derivative measures the rate of change of vector fields, considering the variation of basis vectors. For instance, in flat space, the covariant derivative of a vector field corresponds to the ordinary derivative, calculating both the derivative of the vector components and the derivative of the basis vectors.
The covariant derivative can be expressed as a directional derivative in extrinsic coordinates as \[\triangledown_{\frac{\partial }{\partial u^{i}}}\vec{v}=\frac{\partial \vec{v}}{\partial u^{i}}-\vec{n} \ \ \ \ (2)\]
Fig. 2 Visualization of covariant derivative on a sphere from formula (2)
Parallel transport
The covariant derivative is often conceptualized as an affine connection because it links the tangent spaces (or planes) at two points on the manifold. Parallel transport involves moving a geometric vector or tensor along smooth curves within the manifold. This process is facilitated by the covariant derivative, which enables the transportation of manifold vectors along these curves, ensuring they remain parallel according to the connection.
Fig. 3 Illustration of parallel transport along smooth curves on a sphere
In a parallel transport the rate of change of the tangent vector as illustrated in the previous figure is 0. Therefore the covariant derivative along the tangent vector is 0.
\[\triangledown _{\frac{\partial }{\partial u^{i}}}\vec{v} = 0 \ \ \ (3)\]
Fig. 4 Illustration of Christoffel symbols for the covariant derivative on a sphere
The covariant derivative can be expressed using the Christoffel symbols as\[\triangledown _{\frac{\partial }{\partial u^{i}}}\vec{v} =\left \lfloor \frac{\partial v^{k}}{\partial u^{i}} +v^{j}.\Gamma^{k}_{ij}\right \rfloor \vec{e_{k}} \ \ \ (4)\]
Metric & connection
A metric tensor is a structure (such as a tensor or matrix) on a manifold which facilitates the definition of distances, angles, and norms, in a manner comparable to the inner product in Euclidean space. Given a set of basis vector ei the metric tensor gij and the inverse metric tensor are defined as \[g_{ij}=\vec{e_{i}}.\vec{e_{j}}=\left \langle \frac{\partial }{\partial x^{i}} \frac{\partial }{\partial x^{j}} \right \rangle \ \ (5) \ \ \ Inverse \ metric \ \mathfrak{g^{ij}} \ \ g_{ij}.\mathfrak{g^{ij}} =\delta _{ij}\]
An affine connection that incorporates a metric is described as a bilinear map. Given a manifold M, a tangent spaceTM and vector fields space (TM). \[\begin{matrix} \Gamma(T_{M})\ X \ \Gamma(T_{M})\rightarrow \Gamma(T_{M}) \\ \triangledown : \ \ \ (X, Y) \rightarrow \triangledown_{X}Y \ \ \ \ \ \ \ \ \ \end{matrix}\]For any infinitely differentiable and continuous function f, \[\triangledown_{fX}Y=f\triangledown_{X}Y \ \ \ \ \triangledown_{X}(fY)=\frac{\partial f}{\partial X}Y+\triangledown_{fX}Y\]An affine connection that preserves the Riemann metric under parallel transport and is symmetric (commutative covariant derivative) is known as the Levi-Civita connection.
Levi-Civita coefficients
The Levi-Civita connection is a type of metric connection that is compatible with the metric tensor of a Riemannian manifold. Its covariant derivative of the metric tensor with respect to the Levi-Civita connection is zero. The property of parallel transport ensures that the connection preserves the inner product structure defined by the metric tensor.
It is related to the fundamental Theorem of riemannian geometry which states "There is a unique connection (or covariant derivative) that is Torsion free and has metric compatibility"
The Levi-Civita connection is essential is the computation of geodesics, curvature tensor and parallel transport on Riemannian manifolds.
The coefficients of the Levi-Civita used to compute the covariant derivative is defined by the Christoffel symbols. Using the intrinsic coordinates, the Christoffel symbols are computed from the metric tensors g as \[\Gamma _{jk}^{m}=\frac{1}{2}\mathfrak{g}^{im}\left ( \frac{\partial }{\partial u^{k}} .g_{ij} +\frac{\partial }{\partial u^{j}} .g_{ki} - \frac{\partial }{\partial u^{i}} .g_{jk} \right ) \ \ \ (6) \]
Use case: hypersphere
Setup
As a reminder, we reuse the structure of data points on a manifold defined by the class ManifoldPoint described in a previous post ManifoldPoint definition.
@dataclass
class ManifoldPoint:
id: AnyStr
location: np.array
tgt_vector: Optional[List[float]] = None
geodesic: Optional[bool] = False
As in previous articles, we utilize the Geomstats Python library to explore, implement, and assess the Riemannian manifold and its connection. For ease of use, we encapsulated the Riemann metric within a class named RiemannianConnection as follows:
import geomstats.backend as gs
from geomstats.geometry.riemannian_metric import RiemannianMetric
from geomstats.geometry.base import LevelSet
from typing import AnyStr, Optional
from manifoldpoint import ManifoldPoint
from geometricexception import GeometricException
class RiemannianConnection(object):
def __init__(self, space: LevelSet, manifold_type: AnyStr):
self.riemannian_metric = RiemannianConnection.__get_metric(space)
self.manifold_descriptor = f'{manifold_type}\nDimension: {space.dim}\nShape: {space.shape}' \
f'\nCoordinates type: {space.default_coords_type}'
# Add 1 dimension for extrinsic coordinates
self.ndim = space.dim+1 if space.default_coords_type == "extrinsic" else space.dim
The Riemann metric is inferred from the type of manifold (or level set), space. The variable ndim captures the dimension of tangent vectors in either intrinsic or extrinsic coordinates.
Riemann metric
Let's implement the inner product of two tangent vectors, tgt_vec1 and tgt_vec2 at a base point base_pt defined in formula (4) for the RiemannianConnection class.
def inner_product(self, tgt_vec1: np.array, tgt_vec2: np.array, base_pt: np.array) -> np.array:
if len(tgt_vec1) != len(tgt_vec2):
raise GeometricException(f'Inner product of vector size {len(tgt_vec1)} and vector size {len(tgt_vec1)}')
return 0 if len(tgt_vec1) == 0 or len(tgt_vec2) == 0 \
else self.riemannian_metric.inner_product(tgt_vec1, tgt_vec2, base_pt)
The inner product on the hypersphere is computed for a vector and its negative value.
dim = 2
coordinates_type = 'extrinsic'
hypersphere = Hypersphere(dim=dim, equip=True, default_coords_type=coordinates_type)
riemann_connection = RiemannianConnection(hypersphere, 'hypersphere')
vector = np.array([0.4, 0.1, 0.8])
base_point = np.array([0.4, 0.1, 0.0])
inner_product = riemann_connection.inner_product(vector, -vector, base_point #. -0.81
Output:
Inner product: -0.81
Riemann connection
Parallel transport
Let's implement the formula (5) by adding the method parallel_transport to the class RiemannianConnection. Its implementation invoked the Geomstats method, RiemannianMetric.parallel_transport. The parallel transport is initialized from the manifold base point, manifold_base_pt, to an optional end point, manifold_end_pt. The direction for the parallel transport can be optionally specified with the tangent vector at the base point as default value.
def parallel_transport(
self,
manifold_base_pt: ManifoldPoint,
manifold_end_pt: Optional[ManifoldPoint] = None,
direction: Optional[np.array] = None) -> np.array:
if manifold_base_pt.ndim() != self.ndim:
raise GeometricException(f'Base pt dimension {manifold_base_pt.ndim()} should be {self.ndim}')
return self.riemannian_metric.parallel_transport(
manifold_base_pt.tgt_vector,
manifold_base_pt.location,
direction,
manifold_end_pt.location)
The evaluation consists of one manifold base point with a tangent vector and another manifold end point with a translation +0.4
vector = np.array([0.5, 0.1, 2.1])
base_pt = np.array([1.5, 2.0, 1.6])
manifold_base_pt = ManifoldPoint(id='base', location=base_pt, tgt_vector=vector
manifold_end_pt = ManifoldPoint(id='end',location=base_pt + 0.4)
parallel_trans = riemannian_connection.parallel_transport(manifold_base_pt, manifold_end_pt)
print(f'Parallel transport: {parallel_trans}')
Output:
Parallel transport: [[-1.20097215 -2.09879718 0.29946284]]
Christoffel symbols
Lastly, let's compute the Christoffel symbols for the hypersphere as a given point of intrinsic coordinates (u, v).\[\begin{bmatrix} \Gamma _{11}^{1} & \Gamma _{12}^{1}\\ \Gamma _{11}^{1} & \Gamma _{22}^{1} \end{bmatrix} = \begin{bmatrix} 0 & 0\\ 0 & - \frac{1}{2}sin\left ( 2u \right ) \end{bmatrix} \begin{bmatrix} \Gamma _{11}^{2} & \Gamma _{12}^{2}\\ \Gamma _{11}^{2} & \Gamma _{22}^{2} \end{bmatrix} = \begin{bmatrix} cot(u) & 0\\ 0 & cot(u) \end{bmatrix} \ (7) \]We add the method levi_civita_coefficients to the class RiemannianConnection. Its implementation of this formula leverages the Geomstats, RiemannianMetric.christoffels method.
def levi_civita_coefficients(self, base_pt: np.array) -> np.array:
if len(base_pt) != self.ndim:
raise GeometricException(f'Base pt dimension {len(base_pt)} should be {self.ndim}')
return self.riemannian_metric.christoffels(base_pt)
Let's validate the formula (6) for the Christoffel symbols on the Hypersphere with u = 1.5 and v = 2.0
base_pt = np.array([1.5, 2.0, 1.6])
levi_civita_coefs = riemann_connection.levi_civita_coefficients(base_pt)
print(f'Levi-Civita coefficients:\n{str(levi_civita_coefs)}')
Output:
Levi-Civita coefficients:
[[[ 0. 0. ]
[ 0. -0.07056 ]]
[[ 0. 0.07091484]
[ 0.07091484 0. ]]]
We validate that 0.5*sin(2u) = 0.0756 and cot(u) = 0.0709148
References
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Patrick Nicolas has over 25 years of experience in software and data engineering, architecture design and end-to-end deployment and support with extensive knowledge in machine learning.
He has been director of data engineering at Aideo Technologies since 2017 and he is the author of "Scala for Machine Learning", Packt Publishing ISBN 978-1-78712-238-3 and Geometric Learning in Python Newsletter on LinkedIn.
Appendix
Keywords: geometriclearning, riemanngeometry, manifold, differential geometry, ai, python, geomstats
