Diffusion Representation for Asymmetric Kernels

Abstract: We extend the diffusion-map formalism to data sets that are induced by asymmetric kernels. Analytical convergence results of the resulting expansion are proved, and an algorithm is proposed to perform the dimensional reduction. In this work we study data sets in which its geometry structure is induced by an asymmetric kernel. We use a priori coordinate system to represent this geometry and, thus, be able to improve the computational complexity of reducing the dimensionality of data sets. A coordinate system connected to the tensor product of Fourier basis is used to represent the underlying geometric structure obtained by the diffusion-map, thus reducing the dimensionality of the data set and making use of the speedup provided by the two-dimensional Fast Fourier Transform algorithm (2-D FFT). We compare our results with those obtained by other eigenvalue expansions, and verify the efficiency of the algorithms with synthetic data, as well as with real data from applications including climate change studies.

• The paper introduces a novel diffusion representation method that employs a 2-D FFT to efficiently process asymmetric kernel matrices.
• It leverages the tensor product of Fourier bases to reduce dimensionality and improve the speed of computing diffusion geometry.
• The methodology is validated on synthetic, image, and climate datasets, revealing key geometric structures and temperature variations.

Diffusion Representation for Asymmetric Kernels

Introduction to Diffusion Maps and Asymmetric Kernels

The paper "Diffusion Representation for Asymmetric Kernels" presents an adaptation of diffusion maps for datasets characterized by asymmetric kernels. Traditionally, diffusion maps rely on spectral decompositions of symmetric operators, like the Laplacian matrix, to reduce dimensionality while preserving local geometry in data (2401.12251). However, asymmetric kernels pose challenges, as spectral theorems don't apply directly. This work introduces a new methodology leveraging a framework based on the tensor product of Fourier bases, exploiting the computational efficiencies of the two-dimensional Fast Fourier Transform (2-D FFT).

Methodology and Implementation

The proposed methodology offers a dimensionality reduction technique designed to efficiently handle asymmetric kernels. The key innovation is using the 2-D FFT in a coordinate system aligned with a Fourier basis, which allows the representation and efficient computation of diffusion geometry.

Algorithm Overview:

1. Initial Setup: Start with data in the form of a possibly asymmetric $M \times M$ kernel matrix.
2. Kernel Normalization: Apply a Markov process normalization if applicable.
3. 2-D FFT Application: Utilize the 2-D FFT to efficiently compute the expansion of the kernel in a Fourier basis.
4. Dimensional Reduction: Generate a lower-dimensional embedding based on the computed coefficients, effectively compressing the data while approximating diffusion distances.

This method is computationally advantageous, reducing complexity from $O(n^3)$ with traditional spectral methods to $O(n^2 \log n)$ using FFT techniques.

Empirical Evaluation

The proposed diffusion representation was tested across various synthetic and real-world datasets:

1. Sphere and M\"obius Strip Synthetic Data: Demonstrated the method's ability to capture geometric structures, revealing fine details like rotational characteristics not easily discernible with SVD.

   Figure 1: Data set X with 512 random points in the sphere $S^2$.

2. Visualization on Images: A synthetic asymmetric kernel, derived from image data, showed the method's capacity to approximate complex kernel structures with fewer coefficients effectively.

   Figure 2: Synthetic asymmetric kernel structure chosen from a picture of Tom Jobim in Ipanema Beach (Rio de Janeiro, Brazil).

3. Climate Data Application: Applied to Brazilian temperature data over several decades, the method identified areas with significant temperature variations, providing insights potentially valuable for climate change analysis.

Figure 3: Plot of the average Brazilian temperature distribution in years 2000, 2010, 2018.

Performance Metrics

Evaluations highlighted the Fourier basis's ability to closely approximate diffusion distances from large, asymmetric datasets efficiently:

• Error Metrics: The metric $M_B = \frac{1}{N} E \times t_{cpu}$ (where $E$ is absolute error and $t_{cpu}$ is computational time) demonstrated that low-order approximations achieved balance between precision and efficiency.
• Computational Load: The 2-D FFT based computations proved significantly faster than SVD, making high-dimensional problems tractable on standard computing hardware.

Conclusion and Future Directions

This study provides a viable extension of diffusion map techniques to asymmetric settings, holding promise for applications requiring efficient processing of large, complex datasets. The ability to run efficiently on low-power systems broadens the accessibility of advanced data analysis tools.

Future avenues include exploring orthonormal bases beyond Fourier, such as spherical harmonics on curved geometries, and extending the methodology to more intricate dynamic systems, particularly in evolving network analysis and signal processing applications. Additionally, potential advancements in adaptive bases could further improve efficiency and accuracy in representation.

Overall, this research opens new possibilities for handling asymmetry in data-driven fields, fostering advancements in areas from network analysis to climate science (2401.12251).