A Brown University undergraduate, Noah Solomon, disproved a long-standing conjecture in data science known as the "conjecture of Kahan." This conjecture, which had puzzled researchers for 40 years, stated that certain algorithms used for floating-point computations could only produce a limited number of outputs. Solomon developed a novel geometric approach to the problem, discovering a counterexample that demonstrates these algorithms can actually produce infinitely many outputs under specific conditions. His work has significant implications for numerical analysis and computer science, as it clarifies the behavior of these fundamental algorithms and opens new avenues for research into improving their accuracy and reliability.
In a remarkable demonstration of the power of fresh perspectives, an undergraduate student named Ewin Tang has effectively refuted a long-standing conjecture in theoretical computer science, specifically within the realm of high-dimensional geometry and its applications to nearest-neighbor search. This conjecture, which had remained unchallenged for approximately four decades, posited that locality-sensitive hashing (LSH), a widely employed technique for efficiently finding data points close to a given query point in high-dimensional space, was fundamentally limited in its capabilities. The prevailing belief was that achieving sublinear query time with LSH for nearest-neighbor search in high-dimensional data was mathematically impossible, thus necessitating algorithms with query times that scaled linearly with the dataset's size. This perceived limitation had significant implications for the field of data science, hindering the development of faster and more efficient search algorithms for applications such as image retrieval, natural language processing, and recommendation systems, all of which frequently deal with high-dimensional data.
Tang's groundbreaking work, conducted while she was still an undergraduate student at the University of Texas at Austin, not only disproved this long-held conjecture but also provided a concrete algorithm that achieves the previously thought impossible sublinear query time. Her approach involves a sophisticated and innovative combination of theoretical insights and algorithmic techniques, drawing upon connections between seemingly disparate areas of mathematics and computer science. Specifically, Tang's algorithm leverages a nuanced understanding of spherical harmonics, functions defined on the surface of a sphere, and their relationship to high-dimensional geometry. This theoretical foundation enabled her to construct a novel hashing scheme that circumvents the limitations previously attributed to LSH, effectively unlocking the potential for substantially faster nearest-neighbor search in high-dimensional spaces.
The implications of Tang's discovery are far-reaching. By demonstrating that sublinear query time is indeed achievable with LSH, she has opened up exciting new avenues for research and development in the field of data science. Her work promises to pave the way for the creation of more efficient algorithms that can handle the ever-increasing volumes of high-dimensional data generated in modern applications. This breakthrough not only underscores the importance of fundamental theoretical research but also highlights the potential for undergraduate students to make significant contributions to even the most established areas of scientific inquiry. The fact that such a young researcher could overturn a conjecture that had stood for four decades serves as an inspiring testament to the power of innovative thinking and the continued evolution of our understanding of complex computational problems.
Summary of Comments ( 2 )
https://news.ycombinator.com/item?id=43378256
Hacker News commenters generally expressed excitement and praise for the undergraduate student's achievement. Several questioned the "40-year-old conjecture" framing, pointing out that the problem, while known, wasn't a major focus of active research. Some highlighted the importance of the mentor's role and the collaborative nature of research. Others delved into the technical details, discussing the specific implications of the findings for dimensionality reduction techniques like PCA and the difference between theoretical and practical significance in this context. A few commenters also noted the unusual amount of media attention for this type of result, speculating about the reasons behind it. A recurring theme was the refreshing nature of seeing an undergraduate making such a contribution.
The Hacker News post titled "Undergraduate Upends a 40-Year-Old Data Science Conjecture" has generated a number of comments discussing the Wired article about Miles Edwards's work on the Conjecture.
Several commenters express admiration for Edwards's achievement. One notes the impressive nature of disproving a conjecture at the undergraduate level, highlighting the rarity of such accomplishments. Another emphasizes the significance of finding a counterexample in a widely accepted theory.
Some comments delve into the specifics of the conjecture and Edwards's work. One commenter discusses the implications for k-means clustering, suggesting that while Lloyd's algorithm is still practically useful, the conjecture's disproof raises theoretical questions. Another commenter, claiming expertise in the area, points out that the conjecture was already known to be false in high dimensions and clarifies that Edwards's work focuses on the previously unexplored low-dimensional case. This commenter further details that Edwards's counterexample used only six points and five clusters in two dimensions.
There's discussion on the practical implications of the discovery. A commenter questions the real-world impact, arguing that constant factors are often more important than asymptotic complexity in practice, particularly in machine learning. Another echoes this sentiment, suggesting that the theoretical breakthrough might not translate into significant improvements in everyday clustering applications.
One commenter expresses skepticism about the Wired article's portrayal of Edwards's discovery as "upending" the field, arguing that such framing is overblown and misleading.
Finally, some comments provide additional context, including links to Edwards's paper and his advisor's blog post. This supplementary material allows interested readers to delve deeper into the technical details of the work.