A simple but interesting issue with analyzing high-dimensional data

Peter Landweber, Emanuel Lazar, and Neel Patel are mathematicians. I have never worked with Peter Landweber, but have written papers with Larry and Laura Landweber. Perhaps I can add Peter one day.

Today I want to report on a recent result on the fiber structure of continuous maps.

The paper by Landweber, Lazar, and Patel (LLP) is titled, “On The Fiber Diameter Of Continuous Maps.” Pardon me, but I assume that some of you may not be familiar with the fiber of a map. Fiber has nothing to do with the content of food or diets, for example. Fibers are a basic property of a map.

Their title does not give away any suggestion that their result is relevant to those studying data sets. Indeed even their full abstract only says at the end:

${\dots}$ Applications to data analysis are considered.

I just became aware of their result from reading a recent Math Monthly issue. The paper has a number of interesting results—all with some connection to data analytics. I must add that I had not seen it earlier because of a recent move, and the subsequent lack of getting US mail. Moves are disruptive—Bob Floyd used to tell me that “two moves equal a fire”—and I’ve just moved twice. Oh well.

## The Result

The fiber of a map ${f}$ at ${y}$ is the set of points ${x}$ so that ${f(x)=y}$. The diameter of a fiber is just what you would expect: the maximum distance of the points in the fiber. LLP prove this—they say they have a “surprisingly short proof” and give earlier sources for it at the end of their paper:

Theorem: Let ${f \colon \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}}$ be a continuous function where ${n > m}$. Then for any ${M > 0}$, there exists ${y \in \mathbb{R}^m}$ whose fiber has diameter greater than ${M}$.

## The Proof

The following figure from their paper conveys the essence of the proof in the case ${m = 1}$:

For ${m > 1}$ one might expect a difficult dimension-based agument. However, they leverage whatever difficult reasoning went into the following theorem by Karol Borsuk and Stanislaw Ulam. We have mentioned both of them multiple times on this blog but never this theorem:

Theorem: Let ${f}$ be any continuous function from the ${m}$-sphere to ${\mathbb{R}^m}$. Then there are antipodal points that give the same value, i.e., some ${x}$ on the sphere such that ${f(x) = f(-x)}$.

The proof then simply observes that ${m}$-spheres of radius ${M}$ live inside ${\mathbb{R}^n}$ for any ${n > m}$, and arbitrarily large ${M}$. The antipodal points belong to the same fiber of ${f}$ but are ${2M}$ apart.

## What It Means For Data Scientists

One of the main ideas in analytics is to reduce the dimension of a set of data. If we let the data lie in a Euclidean space, say ${\mathbb{R}^{n}}$, then we may wish to map the data down to a space of lower dimension. This yields lots of obvious advantages—the crux is that we can do many computational things on lower-dimensional data that would be too expensive on the original ${n}$-dimensional space.

The LLP result shows that no matter what the mapping is, as long as it is continuous, there must be points that are far apart in the original space and yet get map to the exactly same point in the lower space. This is somewhat annoying: clearly it means there will always be points that the map does not classify correctly.

## Open Problems

One of the issues I think raised by this work on LLP is that within areas like big-data people can work on it from many angles. I think that we do not always see results from another area as related to our work. I believe that many people in analytics are probably surprised by this result, and I would guess that they may have not known about the result previously. This phenomenon seems to be getting worse as more researchers work on similar areas, but come at the problems with different viewpoints.

Can we do a better job at linking different areas of research? Finally, with respect, this seems like a result that could have been proved decades ago? Perhaps one of the great consequences of new areas like big data is to raise questions that were not thought about previously.

[fixed typo R^m, corrected picture of Landweber, added note on sources for main theorem]

Teaching automata theory

Noam Chomsky is famous for many many things. He has had a lot to say over his long career, and he wrote over 100 books on topics from linguistics to war and politics.

Today I focus on work that he pioneered sixty years ago.

Yes sixty years ago. The work is usually called the Chomsky hierarchy(CH) and is a hierarchy of classes of formal grammars. It was described by Noam Chomsky in 1956 driven by his interest in linguistics, not war and politics. Some add Marcel-Paul Schützenberger’s name to the hierachry. He played a crucial role in the early development of the theory of formal languages—see his joint
paper with Chomsky from 1962. Read more…

A possible error with mathematical ramifications

 Non-technical fact-check source

Dan Brown is the bestselling author of the novel The Da Vinci Code. His most recent bestseller, published in 2013, is Inferno. Like two of his earlier blockbusters it has been made into a movie. It stars Tom Hanks and Felicity Jones and is slated for release on October 28.

Today I want to talk about a curious aspect of the book Inferno, since it raises an interesting mathematical question. Read more…

Some CS reflections for our 700th post

 MacArthur Fellowship source

Lin-Manuel Miranda is both the composer and lyricist of the phenomenal Broadway musical Hamilton. A segment of Act I covers the friendship between Alexander Hamilton and Gilbert du Motier, the Marquis de Lafayette. This presages the French co-operation in the 1781 Battle of Yorktown, after which the British forces played the ballad “The World Turned Upside Down” as they surrendered. The musical’s track by the same name has different words and melodies.

Today we discuss some aspects of computing that seem turned upside down from when we first learned and taught them.

We revisit a paper from 1994

Richard Lipton is, among so many other things, a newlywed. He and Kathryn Farley were married on June 4th in Atlanta. The wedding was attended by family and friends including many faculty from Georgia Tech, some from around the country, and even one of Dick’s former students coming from Greece. Their engagement was noted here last St. Patrick’s Day, and Kathryn was previously mentioned in a relevantly-titled post on cryptography.

Today we congratulate him and Kathryn, and as part of our tribute, revisit a paper of his on factoring from 1994.

What is the role of theory today?

Anna Gilbert and Atri Rudra are top theorists who are well known for their work in unraveling secrets of computation. They are experts on anything to do with coding theory—see this for a book draft by Atri with Venkatesan Guruswami and Madhu Sudan called Essential Coding Theory. They also do great theory research involving not only linear algebra but also much non-linear algebra of continuous functions and approximative numerical methods.

Today we want to focus on a recent piece of research they have done that is different from their usual work: It contains no proofs, no conjectures, nor even any mathematical symbols.
Ernie Croot, Vsevolod Lev, and Péter Pach (CLP) found a new application of polynomials last month. They proved that every set ${A \subseteq U = \mathbb{Z}_4^n}$ of size at least ${|U|^{1-\epsilon}}$ has three distinct elements ${a,b,c}$ such that ${2b = a + c}$. Jordan Ellenberg and Dion Gijswijt extended this to ${\mathbb{F}_q^n}$ for prime powers ${q}$. Previous bounds had the form ${|U| n^{-c}}$ at best. Our friend Gil Kalai and others observed impacts on other mathematical problems including conjectures about sizes of sunflowers.