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.
A Brown University undergraduate, Noah Golowich, disproved a long-standing conjecture in data science related to the "Kadison-Singer problem." This problem, with implications for signal processing and quantum mechanics, asked about the possibility of extending certain "frame" functions while preserving their key properties. A 2013 proof showed this was possible in specific high dimensions, leading to the conjecture it was true for all higher dimensions. Golowich, building on recent mathematical tools, demonstrated a counterexample, proving the conjecture false and surprising experts in the field. His work, conducted under the mentorship of Assaf Naor, highlights the potential of exploring seemingly settled mathematical areas.
Hacker News users discussed the implications of the undergraduate's discovery, with some focusing on the surprising nature of such a significant advancement coming from an undergraduate researcher. Others questioned the practicality of the new algorithm given its computational complexity, highlighting the trade-off between statistical accuracy and computational feasibility. Several commenters also delved into the technical details of the conjecture and its proof, expressing interest in the specific mathematical techniques employed. There was also discussion regarding the potential applications of the research within various fields and the broader implications for data science and machine learning. A few users questioned the phrasing and framing in the original Quanta Magazine article, finding it slightly sensationalized.
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.