Optimal transmission algorithm implemented using Julia: Earth-Mover (Wasserstein) distance, Sinkhorn entropy regularization and its variants exploration

Part One:

1. Introduction

Measuring the difference between two probability distributions is very important in fields such as number processing, machine learning, and image recognition. Among them, the optimal transmission algorithm provides a powerful framework for this requirement. The core concept of the algorithm is to find the “minimum” transfer cost between two distributions, that is, the cost of transferring one distribution to another. In recent years, this field has undergone many interesting and important developments, especially as it relates to applications of deep learning.

This article introduces several optimal transmission algorithms implemented in the Julia language, focusing on the Earth-Mover (Wasserstein) distance, the Sinkhorn algorithm for entropy regularized optimal transmission, and some of their variants or extensions.

2. Earth-Mover (Wasserstein) distance

The Earth-Mover (Wasserstein) distance provides an accurate and intuitive way to measure the difference between two probability distributions. It can be understood as moving “soil” (or mass) from one distribution to another such that the total “moving work” is minimized.

The specific Julia code implementation is as follows:

using LinearAlgebra

function wasserstein_distance(p1, p2, C)
    """
    Computes the Wasserstein distance between two distributions p1 and p2, where C is the cost matrix.
    """
    n, m = length(p1), length(p2)
    # Make sure the total mass of both distributions is the same
    if sum(p1) != sum(p2)
        error("The total mass of the two distributions is not the same!")
    end

    # Use linear programming to solve Wasserstein distance
    # Note: This is just a simplified version of the implementation. In actual applications, a special optimization library may be needed.
    P = zeros(n, m)
    for i in 1:n
        for j in 1:m
            P[i, j] = p1[i] * p2[j]
        end
    end
    return sum(P .* C)
end

This function first ensures that the total mass of the two input distributions is the same and then calculates the Wasserstein distance. A simplified linear programming method is used here, but in practical applications, we may need to use a specialized optimization library to improve efficiency.

3. Sinkhorn algorithm and entropy regularization

The Sinkhorn algorithm overcomes the computational complexity of traditional methods by introducing entropy regularization to approximately solve the optimal transmission problem. Entropy regularization not only makes the problem easier to compute, but also introduces a regularization term that prevents overfitting and improves the stability of the algorithm.

The basic idea is to transform the original optimal transmission problem into a constrained optimization problem, and then use iterative methods to solve it.

The following is a simplified version of the Sinkhorn algorithm implementation based on Julia:

using LinearAlgebra

function sinkhorn_algorithm(p1, p2, C, lambda; max_iter=1000, tol=1e-6)
    """
    Compute the entropy-regularized Wasserstein distance between two distributions p1 and p2 using the Sinkhorn algorithm.
    C is the cost matrix and lambda is the regularization parameter.
    """
    n, m = length(p1), length(p2)
    K = exp.(-C / lambda)
    u, v = ones(n), ones(m)

    for _ in 1:max_iter
        u = p1 ./ (K * v)
        v = p2 ./ (K' * u)

        if norm(p1 - K * v .* u) < tol & amp; & amp; norm(p2 - K' * u .* v) < tol
            break
        end
    end

    P = diagm(u) * K * diagm(v)
    return sum(P .* C)
end

The function first computes the matrix KKK and then uses Sinkhorn iteration to update the weights uuu and vvv. When the iterations converge, the algorithm returns the approximate Wasserstein distance.

Part Two:

4. Variations and extensions of the Sinkhorn algorithm

Although the basic Sinkhorn algorithm provides an efficient way to approximate the Wasserstein distance, in some applications this may not be sufficient. Fortunately, there are many variations and extensions that can further improve the performance and stability of the algorithm.

4.1. Stabilized Sinkhorn algorithm

To deal with numerical instabilities, a scaling parameter can be introduced to stabilize the Sinkhorn iteration. The idea is to incorporate a small scaling factor into each iteration to ensure the numerical stability of the algorithm.

The Julia code is implemented as follows:

function stabilized_sinkhorn(p1, p2, C, lambda; max_iter=1000, tol=1e-6, theta=0.5)
    """
    The Wasserstein distance is calculated using the stabilized Sinkhorn algorithm.
    theta is the scaling factor.
    """
    n, m = length(p1), length(p2)
    K = exp.(-C / lambda)
    u, v = ones(n), ones(m)

    for _ in 1:max_iter
        u_prev, v_prev = copy(u), copy(v)

        u = p1 ./ (K * (v .* theta . + (1-theta)))
        v = p2 ./ (K' * (u .* theta . + (1-theta)))

        if norm(u - u_prev) < tol & amp; & amp; norm(v - v_prev) < tol
            break
        end
    end

    P = diagm(u) * K * diagm(v)
    return sum(P .* C)
end

4.2. Entanglement regularized Sinkhorn algorithm

Entanglement regularization is another way to improve the Sinkhorn algorithm, which introduces an additional regularization term designed to prevent the algorithm from producing too sparse solutions.

The following is the Julia code implementation of entanglement regularization:

function entangled_sinkhorn(p1, p2, C, lambda, mu; max_iter=1000, tol=1e-6)
    """
    The Wasserstein distance is calculated using the entanglement regularized Sinkhorn algorithm.
    mu is the entanglement regularization parameter.
    """
    n, m = length(p1), length(p2)
    K = exp.(-C / lambda - mu)
    u, v = ones(n), ones(m)

    for _ in 1:max_iter
        u = p1 ./ (K * v)
        v = p2 ./ (K' * u)

        if norm(p1 - K * v .* u) < tol & amp; & amp; norm(p2 - K' * u .* v) < tol
            break
        end
    end

    P = diagm(u) * K * diagm(v)
    return sum(P .* C)
end

5. Conclusion and application

Wasserstein distance and its related algorithms provide us with a powerful tool to quantify the similarity between two distributions. Compared with other traditional distance measures, Wasserstein distance has better intuitive and mathematical properties.

Furthermore, the Sinkhorn algorithm and its variants provide us with an efficient method to approximate the Wasserstein distance. By introducing entropy regularization and other improvement techniques, we can get an algorithm that is both fast and stable.

Please download the complete project for the specific process.

Part Three:

6. Discussion

Although optimal transmission and Wasserstein distance provide a powerful theoretical framework for a variety of applications, they still face some challenges and limitations.

  • Computational complexity: Despite algorithms such as Sinkhorn, calculations of large-scale problems can still become very complex. To handle this complexity, we may need to further research and develop more optimization techniques.

  • Parameter selection: The Sinkhorn algorithm and its variants require the selection of appropriate regularization parameters. Different parameter selections may lead to different results, so choosing the optimal parameters becomes an important issue.

7. Practical application cases

Optimal transmission and Wasserstein distance have been widely used in many fields. Here are some examples of practical applications:

  • Image processing: In image color transfer and style transfer, Wasserstein distance provides us with a way to quantify the differences between images, which can produce more natural and high-quality transfer results.

  • Machine learning: Wasserstein distance is used as a loss function, especially in generative adversarial networks (GANs), which provides a robust way to train models.

8. Future Outlook

As deep learning and other machine learning methods continue to develop, the potential of optimal transmission and Wasserstein distance for practical applications is rapidly expanding. In order to further improve the efficiency and robustness of the algorithm, future research may focus on the following aspects:

  1. Develop faster algorithms: For large-scale datasets, current methods may still not be fast enough. Therefore, developing more efficient algorithms is a key direction for future research.

  2. Integrated deep learning: Combining optimal transmission methods with deep learning may open up new application areas and technologies.

  3. Multimodal data: With the emergence of more multimodal data applications, how to effectively combine different data sources has become an important research topic. Optimal transport might provide an interesting framework for this.

9. Conclusion

The Wasserstein distance and optimal transport have far-reaching implications both in theory and in practice. Through continuous research and development, we expect to further improve the effectiveness of these tools and open up new areas of application. Optimal transport provides us with a promising research area, both from the perspective of fundamental research and practical applications.

Thank you for your patience in reading, I hope this article can provide you with some useful guidance and enlightenment in the research and application of optimal transmission.