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.
The blog post showcases efficient implementations of hash tables and dynamic arrays in C, prioritizing speed and simplicity over features. The hash table uses open addressing with linear probing and a power-of-two size, offering fast lookups and insertions. Resizing is handled by allocating a larger table and rehashing all elements, a process triggered when the table reaches a certain load factor. The dynamic array, built atop realloc, doubles in capacity when full, ensuring amortized constant-time appends while minimizing wasted space. Both examples emphasize practical performance over complex optimizations, providing clear and concise code suitable for embedding in performance-sensitive applications.
Hacker News users discuss the practicality and efficiency of Chris Wellons' C implementations of hash tables and dynamic arrays. Several commenters praise the clear and concise code, finding it a valuable learning resource. Some debate the choice of open addressing over separate chaining for the hash table, with proponents of open addressing citing better cache locality and less memory overhead. Others highlight the importance of proper hash functions and the potential performance degradation with high load factors in open addressing. A few users suggest alternative approaches, such as using C++ containers or optimizing for specific use cases, while acknowledging the educational value of Wellons' straightforward C examples. The discussion also touches on the trade-offs of manual memory management and the challenges of achieving both simplicity and performance.
Summary of Comments ( 129 )
https://news.ycombinator.com/item?id=43002511
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.
The Hacker News post titled "Undergraduate Upends a 40-Year-Old Data Science Conjecture" has generated a moderate number of comments discussing various aspects of the linked Quanta Magazine article. Several commenters focus on the nature of the problem itself, Kadison-Singer, explaining its significance in different fields like quantum mechanics and operator theory. There's a discussion on how seemingly abstract mathematical concepts can have unexpected real-world applications, and how this particular problem relates to things like signal processing.
Some comments express admiration for the undergraduate student, Noah Stephens-Davidowitz, and his advisor, John Tao, for their achievement in solving this long-standing conjecture. They highlight the collaborative aspect of research and the importance of mentorship. One commenter mentions being personally acquainted with Tao and praises his mentorship abilities.
A few commenters delve into the technical details of the proof, discussing concepts like paving and discrepancy theory. They try to explain the core ideas of the proof in a more accessible way, acknowledging the complexity of the original work. One comment thread explores the difference between the original Kadison-Singer problem and the Weaver conjecture, which was the focus of Stephens-Davidowitz's work.
There's also some discussion about the nature of mathematical breakthroughs and the process of peer review. One commenter questions whether the proof has been fully vetted by the mathematical community, highlighting the importance of rigorous scrutiny in such cases.
Finally, a couple of commenters offer links to related resources, like Terence Tao's blog post discussing the problem, which provides further context and insights for those interested in learning more. Overall, the comments demonstrate a mix of appreciation for the mathematical achievement, attempts to understand the complex concepts involved, and reflections on the broader implications of such discoveries.