This post explains the connection between convolutions and polynomial multiplication. It demonstrates how discrete convolution can be interpreted as multiplying two polynomials where one polynomial's coefficients represent the input signal and the other represents the convolution kernel (filter). The seemingly strange "flipping" of the kernel in the typical convolution operation arises naturally from the process of aligning terms with the same exponent during polynomial multiplication. By viewing convolution through this polynomial lens, the author illuminates the underlying mathematical structure and provides a clearer intuition for why the kernel is flipped. This perspective also bridges the gap between the discrete and continuous forms of convolution, highlighting their fundamental similarity.
Eli Bendersky's blog post, "Convolutions, Polynomials, and Flipped Kernels," meticulously explores the deep connection between the mathematical concept of convolution and its application in signal processing, particularly within the realm of convolutional neural networks (CNNs). The post begins by establishing a clear definition of discrete convolution, emphasizing the characteristic "flipping" of the kernel (a small matrix or vector of weights) before sliding it across the input signal (another vector or matrix representing data like an image) and performing element-wise multiplication and summation at each position. This operation produces the convolved output signal.
Bendersky then delves into the fascinating relationship between convolution and polynomial multiplication. He explains that if the input signal and the kernel are viewed as coefficients of polynomials, then the convolution operation is equivalent to multiplying those polynomials. This equivalence is demonstrated through a detailed example, carefully illustrating how the coefficients of the resulting product polynomial correspond to the output of the convolution. This perspective offers a powerful and intuitive way to understand why the kernel is flipped during convolution. When multiplying polynomials, terms are combined based on matching exponents, which effectively mirrors the sliding and aligning process of the flipped kernel in the convolution operation.
The blog post proceeds to connect this polynomial interpretation to the concept of the z-transform. The z-transform represents a discrete-time signal as a polynomial in the complex variable 'z'. By taking the z-transforms of the input signal and the kernel, their convolution in the time domain translates to a simple multiplication of their respective z-transforms in the z-domain. This transformation simplifies the analysis and manipulation of signals and systems.
Further elaborating on the practical implications, Bendersky discusses the significance of flipped kernels in CNNs. He explains how this flipping operation, inherent in the mathematical definition of convolution, contributes to the network's ability to detect patterns and features within the input data. The flipping ensures that the kernel effectively scans the input for specific arrangements of values, enabling the network to learn meaningful representations.
Finally, the post acknowledges a common practice in some CNN implementations where the kernel is not explicitly flipped. This is explained by noting that since the kernel weights are learned during the training process, the network can effectively learn a "pre-flipped" version of the kernel, thus implicitly incorporating the flip operation. This optimization doesn't change the fundamental mathematical principle but rather represents a computational shortcut. The post concludes by reiterating the important link between convolution, polynomial multiplication, and the concept of flipped kernels, providing a comprehensive mathematical understanding of this fundamental operation in signal processing and machine learning.
Summary of Comments ( 23 )
https://news.ycombinator.com/item?id=44048306
Commenters on Hacker News largely praised the article for its clear explanation of the relationship between convolutions and polynomial multiplication. Several highlighted the insightful connection made between flipping the kernel in convolution and the order of coefficients in polynomial multiplication. One commenter appreciated the focus on discrete convolution, noting its importance in computer science applications. Another pointed out the practical implications for understanding signal processing, while others discussed extensions of these concepts to areas like generating functions. A few commenters also shared resources for further exploration of related topics like fast convolution algorithms and the Fourier transform.
The Hacker News post "Convolutions, Polynomials and Flipped Kernels" has generated a modest discussion with several insightful comments.
One commenter highlights the significance of the article's explanation of the "flip" in convolutions, stating that it finally helped them understand the concept after years of confusion. They appreciate the clear connection made between polynomial multiplication and convolutions. This comment resonates with other users who express similar past struggles and praise the article for its clarity.
Another commenter delves into the practical application of flipped kernels, particularly in computer graphics. They discuss how this concept relates to blurring and sharpening filters. This comment adds a practical dimension to the theoretical discussion.
One user mentions the importance of convolutions in various fields, including image processing, signal processing, and probability theory, further emphasizing the wide applicability of the concepts discussed in the article.
A further comment expands on the mathematical background, pointing out that the commutativity of convolution allows for flipping the kernel, which can be computationally advantageous in certain situations. This adds a deeper mathematical understanding of the "flip" operation.
The discussion also touches on the use of FFT (Fast Fourier Transform) for efficient computation of convolutions, highlighting its practical significance in signal processing applications.
Finally, a comment mentions the article's connection to generating functions, expanding the mathematical scope of the discussion and linking it to other areas of mathematics.
Overall, the comments on the Hacker News post provide valuable perspectives on the article, ranging from personal experiences of understanding the "flip" to practical applications in various fields and deeper mathematical insights. They collectively enrich the understanding of convolutions, polynomials, and flipped kernels.