A geodesic learning method based on riemannian geometry prior
By adjusting the control point set on the Bézier curve and using an autoencoder mapping, combined with Riemannian geometric priors, the robustness and accuracy issues of geodesic path calculation in high-dimensional data are solved, achieving efficient geodesic path calculation under noisy and non-uniform data.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- UNIV OF ELECTRONICS SCI & TECH OF CHINA
- Filing Date
- 2023-12-29
- Publication Date
- 2026-07-24
AI Technical Summary
Existing technologies exhibit poor robustness and insufficient accuracy in calculating geodesic paths when dealing with noisy or unevenly distributed high-dimensional data. They also struggle to accurately find geodesic distances and paths that match the geometric characteristics of the original samples in invisible low-dimensional Riemannian manifolds.
Using the geodesic equation as a geometric prior, the control point set of the Bézier curve for generating the geodesic path is adjusted. The Bézier curve is generated by the Bézier generator, and unsupervised homeomorphism is performed using an autoencoder. Combined with Riemannian metric and Riemannian connection calculation, the control point set is adjusted by gradient descent so that the sampling points satisfy the geodesic equation on the Riemannian manifold.
It exhibits better robustness under noisy or unevenly distributed high-dimensional data conditions, can calculate geodesic paths more accurately, avoids path distortion and elongation, and improves the accuracy and iteration efficiency of the model.
Smart Images
Figure CN117909623B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of machine learning technology, and in particular to a geodesic learning method based on Riemannian geometric priors. Background Technology
[0002] With the explosive growth of information in the era of big data, the analysis and classification of high-dimensional data has become crucial. High-dimensional data often exhibits dimensional redundancy, requiring us to identify and preserve the hidden data structures within the original data while reducing its dimensionality.
[0003] Existing research has shown that many high-dimensional datasets are actually attached to lower-dimensional Riemannian manifolds. Therefore, how to reduce the dimensionality of data to a lower spatial dimension while ensuring that the hidden information in the data is preserved as little as possible during the dimensionality reduction process is a key problem facing manifold learning. Using autoencoders to encode the "latent variables" of the manifold space after dimensionality reduction to describe the original high-dimensional data allows machine learning to perform faster and more efficient optimization calculations.
[0004] Distance metrics are one of the most important and core concepts in geometry, forming the foundation for all computational methods derived from geometry. In machine learning, because the statistical distribution of target sample data is often unclear, it is crucial to consider the distance between samples. Distance metrics are used to determine the similarity between sample points, thereby facilitating classification and representation learning. However, in Riemannian space, the shortest distance between points is no longer the "shortest line between two points" as in Euclidean space, but rather requires the use of geodesics to describe the local shortest distance between two points. Therefore, finding geodesic distances and paths that conform to the geometric characteristics of the original samples in an invisible low-dimensional Riemannian manifold is currently one of the key problems in machine learning.
[0005] Chinese patent document CN106599909A, published on April 26, 2017, discloses a stochastic gradient geodesic Markov chain Monte Carlo method and apparatus. The method includes determining the posterior distribution function and stochastic gradient function of the latent variables of a target Bayesian model; determining a Riemannian manifold to be sampled based on the latent variables and determining an isometric embedding map of the manifold; and determining the transformation relationship between the coordinate space and the embedding space based on the mapping. The Riemannian manifold to be sampled is the Riemannian manifold in which the latent variables reside. Multiple simulations are performed on a specific dynamical system determined by the stochastic gradient function in the embedding space corresponding to the isometric embedding map to determine N samples of the latent variables. N is the number of latent variable samples used to estimate the posterior distribution of the latent variables.
[0006] The stochastic gradient geodesic Markov chain Monte Carlo method disclosed in this patent document can remove the constraints of the global coordinate system and internal iterations, improving the efficiency of processing large-scale data. However, it has poor robustness and poor accuracy in geodesic path calculation when dealing with noisy or unevenly distributed high-dimensional data. Summary of the Invention
[0007] To overcome the shortcomings of the prior art, this invention provides a geodesic learning method based on Riemannian geometric priors. This invention uses the geodesic equation as a geometric prior and adjusts the set of control points of the Bézier curve that generates the geodesic path, so that any sampling point on the Bézier curve can maximize the satisfaction of the geodesic equation on the Riemannian manifold. This method has better robustness when facing noisy or unevenly distributed high-dimensional data and can calculate the geodesic path more accurately.
[0008] This invention is achieved through the following technical solution:
[0009] A geodesic learning method based on Riemannian geometric priors, characterized by the following steps:
[0010] a. Initialize the control point set P, the starting point b, and the ending point e. The control point set P consists of coordinate points on n manifolds.
[0011] b. Input the control point set P into the Bézier generator, which generates the Bézier curve B controlled by parameter t. P (t), where t∈[0,1], B P (0) = b, B P (1) = e;
[0012] c. Perform uniform sampling on the Bézier curve according to the sampling interval gap to obtain a set of 1 / gap sampling parameters s. Substitute these parameters into the Bézier curve to obtain the set of sampling points B on the Bézier curve. P，gp (s);
[0013] d. Set the sampling points B P，gp (s) Input the decoder of the trained autoencoder In the middle, the Riemann metric of the locations of all sampling points is calculated. Contact with Riemann
[0014] e. Obtain the geodesic equation values of all sampling point locations through the geodesic equation and use them as the network output. Then, substitute all the geodesic equation values into the mean square error loss function. Finally, adjust the position P+ΔP of all points in the control point set P through gradient descent to minimize the difference between the geodesic equation values and 0.
[0015] In step b, the Bézier generator generates a Bézier curve B controlled by parameter t.P (t) specifically refers to completion through Equation 1;
[0016]
[0017] In the formula, B(t) is an nth-order Bézier curve, and P i For the i-th control point, B i，n (t) is the i-th Benstein basis function.
[0018] The i-th Bensench basis function is obtained by Equation 2;
[0019]
[0020] In the formula, Let be the combination number, representing the number of ways to choose i from n, and t be the parameter.
[0021] The number of combinations Calculated using Equation 3;
[0022]
[0023] In step c, the sampling interval is 0 < gap < 1.
[0024] In step d, the Riemann metric Calculated using Equation 4;
[0025]
[0026] in, Let be the differential component of the tangent vector, du be the differential component along the u-axis, and dv be the differential component along the v-axis.
[0027] In step d, Riemann communication Calculated using Equation 5;
[0028]
[0029] In the formula, g is the inverse matrix of the Riemannian metric. , To find the sign of the partial derivative, Let u be the tangent vector. j Let u be the j-axis, u k For the uk axis, u i Let ui be the axis and im be the element in the i-th row and m-th column of the matrix.
[0030] In step e, the geodesic equation is Equation 6;
[0031]
[0032] Where d represents the derivative symbol, u mThe um axis.
[0033] In this invention, ΔP refers to the adjustment step size.
[0034] The basic principle of this invention is as follows:
[0035] In Riemannian space, only distance operations within local coordinates can use the inner product of Euclidean space as the unit of distance description. On a macroscopic level, any two points outside their neighborhood require geodesics to describe their shortest distance. In the development of deep learning, research has shown that high-dimensional data may not reside in Euclidean space, but rather on a low-dimensional Riemannian manifold embedded within a higher dimension. However, because high-dimensional data is difficult to visualize, and the distribution of unknown datasets cannot be uniformly described by a single probability distribution formula, accurately describing the distance relationships between these high-dimensional data points remains a challenge in deep learning and data mining research.
[0036] This invention presents a universally applicable trainable parameterized model. First, under unsupervised conditions, high-dimensional data homeomorphisms are mapped to a low-dimensional manifold space, and the Riemannian metric properties in the low-dimensional manifold space can be obtained by calculating the Jacobian matrix from the decoder. Second, based on this, a new trainable parameterized model is designed, incorporating the Riemann geodesic equation as prior knowledge, enabling the model to more accurately calculate the Riemann geodesic path.
[0037] The beneficial effects of this invention are mainly reflected in the following aspects:
[0038] 1. This invention uses the geodesic equation as a geometric prior and adjusts the set of control points of the Bézier curve that generates the geodesic path so that any sampling point on the Bézier curve can maximize the satisfaction of the geodesic equation on the Riemann manifold. This invention has better robustness when facing noisy or unevenly distributed high-dimensional data and can calculate the geodesic path more accurately.
[0039] 2. Compared to traditional nonlinear geodesic learning algorithms, which tend to bypass sparse regions with uneven data distribution and favor denser regions when calculating geodesics, causing the geodesic path to be distorted and elongated, this invention uses an autoencoder to perform unsupervised homeomorphic mapping on high-dimensional data. This ensures that the high-dimensional Euclidean space is equivalent to the mapped low-dimensional manifold in terms of macroscopic topology. Furthermore, the autoencoder can train a model and has a certain generalization learning effect. Even when faced with uneven data distribution or noise, it can better reconstruct the main feature distribution of the original data, avoiding the distortion and elongation of the geodesic path.
[0040] 3. In this invention, the discretized approximate geodesic sampling results are used as the initialization parameters of the unsupervised geodesic learning model, thereby shortening the iteration cycle of the geodesic learning model. Furthermore, the geodesic equation is used as the geometric prior knowledge of the model, further improving the accuracy of the model in unsupervised geodesic learning.
[0041] 4. This invention maps spatial encoded information in Euclidean space to a low-dimensional Riemannian manifold space through an autoencoder. The Riemannian geometric feature attributes of any spatial location on the manifold can be obtained through a decoder. Based on the geodesic learning model of Riemannian geometric prior, it is possible to find approximate geodesic path samples that conform to the Riemannian geodesic equation more accurately from the low-dimensional manifold. Attached Figure Description
[0042] The present invention will now be further described in detail with reference to the accompanying drawings and specific embodiments:
[0043] Figure 1 This is a flowchart of the present invention. Detailed Implementation
[0044] Example 1
[0045] See Figure 1 A geodesic learning method based on Riemannian geometric priors includes the following steps:
[0046] a. Initialize the control point set P, the starting point b, and the ending point e. The control point set P consists of coordinate points on n manifolds.
[0047] b. Input the control point set P into the Bézier generator, which generates the Bézier curve B controlled by parameter t. P (t), where t∈[0,1], B P (0) = b, B P (1) = e;
[0048] c. Perform uniform sampling on the Bézier curve according to the sampling interval gap to obtain a set of 1 / gap sampling parameters s. Substitute these parameters into the Bézier curve to obtain the set of sampling points B on the Bézier curve. P，gp (s);
[0049] d. Set the sampling points B P，gp (s) Input the decoder of the trained autoencoder In the middle, the Riemann metric of the locations of all sampling points is calculated. Contact with Riemann
[0050] e. Obtain the geodesic equation values of all sampling point locations through the geodesic equation and use them as the network output. Then, substitute all the geodesic equation values into the mean square error loss function. Finally, adjust the position P+ΔP of all points in the control point set P through gradient descent to minimize the difference between the geodesic equation values and 0.
[0051] This embodiment is the most basic implementation method. It uses the geodesic equation as a geometric prior and adjusts the set of control points of the Bézier curve that generates the geodesic path so that any sampling point on the Bézier curve can maximize the satisfaction of the geodesic equation on the Riemann manifold. It has better robustness when facing noisy or unevenly distributed high-dimensional data and can calculate the geodesic path more accurately.
[0052] Example 2
[0053] See Figure 1 A geodesic learning method based on Riemannian geometric priors includes the following steps:
[0054] a. Initialize the control point set P, the starting point b, and the ending point e. The control point set P consists of coordinate points on n manifolds.
[0055] b. Input the control point set P into the Bézier generator, which generates the Bézier curve B controlled by parameter t. P (t), where t∈[0,1], B P (0) = b, B P (1) = e;
[0056] c. Perform uniform sampling on the Bézier curve according to the sampling interval gap to obtain a set of 1 / gap sampling parameters s. Substitute these parameters into the Bézier curve to obtain the set of sampling points B on the Bézier curve. P，gp (s);
[0057] d. Set the sampling points B P，gp (s) Input the decoder of the trained autoencoder In the middle, the Riemann metric of the locations of all sampling points is calculated. Contact with Riemann
[0058] e. Obtain the geodesic equation values of all sampling point locations through the geodesic equation and use them as the network output. Then, substitute all the geodesic equation values into the mean square error loss function. Finally, adjust the position P+ΔP of all points in the control point set P through gradient descent to minimize the difference between the geodesic equation values and 0.
[0059] In step b, the Bézier generator generates a Bézier curve B controlled by parameter t. P (t) specifically refers to completion through Equation 1;
[0060]
[0061] In the formula, B(t) is an nth-order Bézier curve, and P i For the i-th control point, B i，n (t) is the i-th Benstein basis function.
[0062] This embodiment is a preferred implementation. Compared to traditional nonlinear geodesic learning algorithms, which tend to bypass sparse regions with uneven data distribution and favor denser regions when calculating geodesics, thus distorting and elongating the geodesic path, this embodiment uses an autoencoder to perform unsupervised homeomorphic mapping on high-dimensional data. This ensures that the high-dimensional Euclidean space is equivalent to the mapped low-dimensional manifold in terms of macroscopic topology. Furthermore, the autoencoder can train a model and has a certain generalization learning effect. Even when faced with uneven data distribution or noise, it can better reconstruct the main feature distribution of the original data, avoiding the distortion and elongation of the geodesic path.
[0063] Example 3
[0064] See Figure 1 A geodesic learning method based on Riemannian geometric priors includes the following steps:
[0065] a. Initialize the control point set P, the starting point b, and the ending point e. The control point set P consists of coordinate points on n manifolds.
[0066] b. Input the control point set P into the Bézier generator, which generates the Bézier curve B controlled by parameter t. P (t), where t∈[0,1], B P (0) = b, B P (1) = e;
[0067] c. Perform uniform sampling on the Bézier curve according to the sampling interval gap to obtain a set of 1 / gap sampling parameters s. Substitute these parameters into the Bézier curve to obtain the set of sampling points B on the Bézier curve. P，gp (s);
[0068] d. Set the sampling points B P，gp (s) Input the decoder of the trained autoencoder In the middle, the Riemann metric of the locations of all sampling points is calculated. Contact with Riemann
[0069] e. Obtain the geodesic equation values of all sampling point locations through the geodesic equation and use them as the network output. Then, substitute all the geodesic equation values into the mean square error loss function. Finally, adjust the position P+ΔP of all points in the control point set P through gradient descent to minimize the difference between the geodesic equation values and 0.
[0070] In step b, the Bézier generator generates a Bézier curve B controlled by parameter t. P (t) specifically refers to completion through Equation 1;
[0071]
[0072] In the formula, B(t) is an nth-order Bézier curve, and P i For the i-th control point, B i，n (t) is the i-th Benstein basis function.
[0073] The i-th Bensench basis function is obtained by Equation 2;
[0074]
[0075] In the formula, Let be the combination number, representing the number of ways to choose i from n, and t be the parameter.
[0076] The number of combinations Calculated using Equation 3;
[0077]
[0078] In step c, the sampling interval is 0 < gap < 1.
[0079] This embodiment is another preferred implementation method. It uses the discretized approximate geodesic sampling results as the initialization parameters of the unsupervised geodesic learning model, thereby shortening the iteration cycle of the geodesic learning model. Furthermore, it uses the geodesic equation as the geometric prior knowledge of the model, thereby further improving the accuracy of the model in unsupervised geodesic learning.
[0080] Example 4
[0081] See Figure 1 A geodesic learning method based on Riemannian geometric priors includes the following steps:
[0082] a. Initialize the control point set P, the starting point b, and the ending point e. The control point set P consists of coordinate points on n manifolds.
[0083] b. Input the control point set P into the Bézier generator, which generates the Bézier curve B controlled by parameter t. P (t), where t∈[0,1], B P (0) = b, BP (1) = e;
[0084] c. Perform uniform sampling on the Bézier curve according to the sampling interval gap to obtain a set of 1 / gap sampling parameters s. Substitute these parameters into the Bézier curve to obtain the set of sampling points B on the Bézier curve. P，gp (s);
[0085] d. Set the sampling points B P，gp (s) Input the decoder of the trained autoencoder In the middle, the Riemann metric of the locations of all sampling points is calculated. Contact with Riemann
[0086] e. Obtain the geodesic equation values of all sampling point locations through the geodesic equation and use them as the network output. Then, substitute all the geodesic equation values into the mean square error loss function. Finally, adjust the position P+ΔP of all points in the control point set P through gradient descent to minimize the difference between the geodesic equation values and 0.
[0087] In step b, the Bézier generator generates a Bézier curve B controlled by parameter t. P (t) specifically refers to completion through Equation 1;
[0088]
[0089] In the formula, B(t) is an nth-order Bézier curve, and P i For the i-th control point, B i，n (t) is the i-th Benstein basis function.
[0090] The i-th Bensench basis function is obtained by Equation 2;
[0091]
[0092] In the formula, Let be the combination number, representing the number of ways to choose i from n, and t be the parameter.
[0093] The number of combinations Calculated using Equation 3;
[0094]
[0095] In step c, the sampling interval is 0 < gap < 1.
[0096] In step d, the Riemann metric Calculated using Equation 4;
[0097]
[0098] in, Let be the differential component of the tangent vector, du be the differential component along the u-axis, and dv be the differential component along the v-axis.
[0099] In step d, Riemann communication Calculated using Equation 5;
[0100]
[0101] In the formula, g is the inverse matrix of the Riemannian metric. , To find the sign of the partial derivative, Let u be the tangent vector. j Let u be the j-axis, u k For the uk axis, u i Let ui be the axis and im be the element in the i-th row and m-th column of the matrix.
[0102] In step e, the geodesic equation is Equation 6;
[0103]
[0104] Where d represents the derivative symbol, u m The um axis.
[0105] This embodiment is the best implementation method. It uses an autoencoder to map spatial encoded information in Euclidean space to a low-dimensional Riemannian manifold space. The decoder can obtain the Riemannian geometric feature attributes of any spatial location on the manifold. Then, based on the geodesic learning model of Riemannian geometric prior, it is possible to find approximate geodesic path samples that conform to the Riemannian geodesic equation more accurately from the low-dimensional manifold.
[0106] The trainable model architecture based on Riemannian geometric priors in this invention is as follows:
[0107] First, an autoencoder is used to unsupervisedly map samples from high-dimensional space to a low-dimensional Riemannian manifold. A trained decoder can then obtain the Riemannian geometric properties at any position on the manifold. Next, a geodesic path is constructed using a polynomial Bézier curve. By setting a trainable model loss function, the control points of the Bézier curve are generated through back-optimization, ensuring that the sampling points on the curve satisfy the Riemann geodesic equation as much as possible.
[0108] Since the geodesic equation is always equal to zero, any point on the geodesic line will also equal zero when substituted into the geodesic equation. Therefore, the geodesic equation can be regarded as a geometric prior. The starting point and the ending point can be set as B(0) and B(1) of the Bézier curve B(t). A smooth curve can be determined through the Bézier curve, ensuring that any sampling point B(n) on this curve, n∈[0,1], equals zero when substituted into the geodesic equation. The Bézier curve then determines a geodesic path between the starting point and the ending point.
[0109] Riemannian metric The computation is performed through the decoder in the upstream task. The Jacobian matrix is calculated as follows:
[0110] decoder Jacob's matrix:
[0111] The following Riemannian metric can be derived from the Jacobian matrix. :
[0112]
[0113]
[0114]
[0115] The self-encoder described in this invention is a differential homeomorphic self-encoder, consisting of an encoder and decoder constitute;
[0116] encoder Receive an R m Spatial data, mapping it to E n In space, the decoder Receive E n Data in space and mapping it back to R m space;
[0117] Assumption This demonstrates that an autoencoder can losslessly encode and decode to reconstruct the original data x. i ,but Furthermore, it is known that backpropagation is required for neural network training. and They are all continuous functions. Therefore, based on the mapping function... European Space E n With latent space R m Belongs to topological homeomorphism prove It is a homeomorphism, and if in E n A set of open covering {O} can be found on the table. a}, it can be known It will definitely make It's also the opening of the market.
[0118] In E n Find any pair of open neighborhoods O a O b satisfy
[0119] according to for Then we have:
[0120]
[0121] Therefore E is certified n It is an m-dimensional differential manifold. Therefore, under the assumption that the autoencoder can encode and decode the original data in near lossless manner, the autoencoder is a differential homeomorphism.
Claims
1. A geodesic learning method based on Riemannian geometric priors, characterized in that, Includes the following steps: a. Initialize the control point set P, the starting point b, and the ending point e. The control point set P consists of coordinate points on n manifolds. b. Input the control point set P into the Bézier generator, which generates the Bézier curve B controlled by parameter t. P (t), where t∈[0,1], B P (0) = b, B P (1) = e; c. Perform uniform sampling on the Bézier curve according to the sampling interval gap to obtain a set of 1 / gap sampling parameters s. Substitute these parameters into the Bézier curve to obtain the set of sampling points B on the Bézier curve. P，gp (s); d. Set the sampling points B P，gp (s) Input the decoder of the trained autoencoder In the middle, the Riemann metric of the locations of all sampling points is calculated. Contact with Riemann e. Obtain the geodesic equation values of all sampling point locations through the geodesic equation and use them as the network output. Then, substitute all the geodesic equation values into the mean square error loss function. Finally, adjust the positions of all points in the control point set P in reverse through gradient descent to minimize the difference between the geodesic equation values and 0.
2. The geodesic learning method based on Riemannian geometric priors according to claim 1, characterized in that: In step b, the Bézier generator generates a Bézier curve B controlled by parameter t. P (t) specifically refers to completion through Equation 1; In the formula, B(t) is an nth-order Bézier curve, and P i For the i-th control point, B i，n (t) is the i-th Benstein basis function.
3. The geodesic learning method based on Riemannian geometric priors according to claim 2, characterized in that: The i-th Bensench basis function is obtained by Equation 2; In the formula, Let be the combination number, representing the number of ways to choose i from n, and t be the parameter.
4. The geodesic learning method based on Riemannian geometric priors according to claim 3, characterized in that: The number of combinations Calculated using Equation 3; 5. The geodesic learning method based on Riemannian geometric priors according to claim 1, characterized in that: In step c, the sampling interval is 0 < gap < 1.
6. The geodesic learning method based on Riemannian geometric priors according to claim 1, characterized in that: In step d, the Riemann metric Calculated using Equation 4; in, Let be the differential component of the tangent vector, du be the differential component along the u-axis, and dv be the differential component along the v-axis.
7. The geodesic learning method based on Riemannian geometric priors according to claim 1, characterized in that: In step d, Riemann communication Calculated using Equation 5; In the formula, g is the inverse matrix of the Riemannian metric. , To find the sign of the partial derivative, Let u be the tangent vector. j Let u be the j-axis, u k For the uk axis, u i Let ui be the axis and im be the element in the i-th row and m-th column of the matrix.
8. The geodesic learning method based on Riemannian geometric priors according to claim 1, characterized in that: In step e, the geodesic equation is Equation 6; Where d represents the derivative symbol, u m The um axis.