It is said that , feng · Neumann once attended a meeting , A physics researcher is reporting on a research progress , Using a very complex model , try
graph theory Verify that the experimental data points fall on the same curve , In line with model expectations . So Feng · Neumann just said , It's better to say that these points are on the same plane . Last , feng · Neumann left a famous saying ：「With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.」
This is Feng · Neumann classic 「 four
Parameters Draw elephants , Five
Parameters Nose shaking 」 The story of .
2010 year , The papers published by three researchers from Max Planck Institute of molecular cell biology and genetics in Germany and European Molecular Biology Laboratory have realized four
Parameters Draw elephants , As follows ：
Picture source ：https://publications.mpi-cbg.de/Mayer_2010_4314.pdf
The same idea , In recent days, , An article was published in 2019 year 4 Month's old paper 《 Real numbers, data science and chaos: How to fit any dataset with a single parameter 》, There was another wave of discussion on twitter . Author of the paper Laurent Boué He is now a Microsoft senior manager
machine learning scientists , He talked about 「 How to use a single
Parameters Fit any dataset 」.
Address of thesis ：https://arxiv.org/pdf/1904.12320.pdf
The poster is a Princeton doctoral student 、
DeepMind Research scientist intern Miles Granmer, He said ,「 This paper provides a model with a single
Parameters Scalar function of , And this function is differentiable and continuous ！」
For this study , Some people think that ：「 Technically speaking , This article has some 『 cheat 』, Because this paper uses floating-point numbers with arbitrary precision . Because the number of bits required for floating-point numbers is very small , Therefore, this article may be a good candidate for compressed representation . But it's definitely not 『 A single 』
Parameters . I agree that this paper is a way to encode data sets into numbers , Then decode it back to a clever way to reconstruct a single point .」
There is also a fit to the study
Parameters The standard error is of interest , If it's a single
Parameters , How big the error will be ？
Others say ：「1 individual
Parameters The continuous differentiable function of can generate infinite VC the Uygurs . This paper seems to be a version of the technique .」
The content of the paper is introduced
This paper introduces how to pass with a single real value
Parameters Scalar function of （ continuity 、 It's very small ...） To approximate any different modes （ The time series 、 Images 、 voice ...） Data set of . Based on the basic concept of chaos theory , The researchers used teaching methods （pedagogical） Method to demonstrate how to adjust this real value
Parameters , To achieve arbitrary precision fitting of all data samples .
Real world data come in a variety of shapes and sizes , Its mode includes from traditional structured database mode to unstructured media source , Such as video source and recording . However , Any data set can eventually be considered a list of values X = [x_0, · · · , x_n] , This list describes the data content and ignores the underlying mode of the data . And this paper aims to prove that any data set X All samples can be reproduced by a simple differential equation ：
among α ϵ R Is the real value to learn from the data
Parameters ,x ϵ [0, · · · , n] Take the whole number .（τ ϵ N It's a constant , Can effectively control the required
Accuracy rate ）. according to 「 Fit the elephant 」 Tradition , The study first shows how to select the appropriate α Values generate different animal shapes , Pictured 1 Shown .
After the demonstration f_α You can generate any type of graffiti drawing above , The paper continues to use words 「Hello world」 There was a demonstration , To further illustrate the function of the method . The figure below 2 Shows how to use carefully selected α Value to generate complex high-dimensional acoustic signals , Coding actually expresses 「Hello world」.
In the data mode of image , With the development of special hardware and new technology
neural network The continuous emergence of Architecture , It is generally believed that the available large-scale labeled training data has become a hot topic
Computer vision 「 mature 」 One of the most important factors .
under these circumstances ,CIFAR-10 Data sets are considered to be a powerful standard to measure the performance of new learning algorithms . The study shows that ： Here's the picture 3 Shown , You can always find one α value , bring f_α Be able to build a reflection CIFAR-10 Artificial images of categories .
Based on the examples of the above modes , The paper concludes that ： A model with simple and differentiable formulas f_α Can produce any type of semantically related scatter diagram 、 Audio or visual data （ The text is similar ）, You only need a single real value
Parameters . This has aroused the doubts of researchers .
Besides , This paper expounds the fact that this method can not be generalized . This is because all the information in this method is directly encoded , Without any compression or 「 Study 」. From a mathematical point of view , There are an infinite number of real numbers , Therefore, it should not be confused with the limited precision data types implemented by the programming language . Based on this ,f_α It is impossible to achieve real generalization , The figure below 9 That's one example .