Physics origins of the most important statistical ideas of recent times

Figure: Maxwell's handwritings, 
state diagram (Wikipedia)

Preamble

The modern statistics now move into an emerging field called data science that amalgamate many different fields from high performance computing to control engineering. However, the emergent behaviour from researchers in machine learning and statistics that, sometimes they omit naïvely and probably unknowingly the fact that some of the most important ideas in data sciences are actually originated from Physics discoveries and specifically developed by physicist. In this short exposition we try to review these physics origins on the areas defined by Gelman and Vehtari (doi). Additional section is also added in other possible areas that are currently the focus of active research in data sciences. 

Bootstrapping and simulation based inference : Gibbs's Ensemble theory and Metropolis's simulations

Bootstrapping is a novel idea of estimations with uncertainty with given set of samples. It is mostly popularised by Efron and his contribution is immense, making this tool available to all researchers doing quantitative analysis.  However, the origins of bootstrapping can be traced back to the idea of ensembles in statistical physics, which is introduced by J. Gibbs. The ensembles in physics allow us to do just what bootstrapping helps, estimating a quantity of interest with sub-sampling, in the case of statistical physics this appears as sampling a set of different microstates. Using this idea Metropolis devised a inference in 1953, to compute ensemble averages for liquids using computers.  Note that, usage of Monte Carlo approach for pure mathematical nature, i.e., solving integrals, appear much earlier with von Neumann's efforts.

Causality : Hamiltonian systems to Thermodynamic potentials

Figure: Maxwell 
Relations as causal
diagrams.

Even though the historical roots of causal analysis in early 20th century attributed to Wright 1923 for his definition of path analysis, causality was the core tanents of Newtonian mechanics in distinguishing left and right of the equations of motions in the form of differential equations, and the set of differential equations following that with Hamiltonian Mechanics is actually forms a graph, i.e., relationships between generalised coordinates, momentum and positions. This connection is never acknowledge in early statistical literature, and probably causal constructions from classical physics were not well known in that community or did not find its way to data-driven mechanics. Similarly, causal construction of thermodynamic potentials appear as a directed graph as in, Born wheel. It appears as a mnemonic but it is actually  causally constructed via Legendre Transformations.  Of course, causality, philosophically speaking, is discussed since Ancient Greece but here we restrict the discussion on solely quantitative theories after Newton.

Overparametrised models and regularisation : Poincaré classifications and astrophysical dynamics

The current deep learning systems classified as massively overparametrized systems. However, the lower dimensional understanding of this phenomenon were well studied by Poincare's classification of classical dynamics, namely the measurement problem of having overdetermined system of differential equations, i.e., whereby inverse problems are well known in astrophysics and theoretical mechanics.     

High-performance computing: Big-data to GPUs

Similarly, using supercomputers or as now we call it high-performance computation with big data generating processes were actually can be traced back to Manhattan project and ENIAC that aims solving scattering equations and almost 50 years of development on this direction before 2000s. 

Conclusion

The impressive development of new emergent field of data science as a larger perspective of statistics into computer science have strong origins from core Physics literature and research. These connections are not sufficiently cited or acknowledged. Our aim in this short exposition is to bring these aspects into the attention of data science practitioners and researchers alike.

Further reading

Some of the mentioned works and related reading list, papers or books.

• What are the Most Important Statistical Ideas of the Past 50 Years? Gelman & Vehtari (2021)
• Leisurely Look at the Bootstrap, the Jackknife, and Cross-Validation, Bradley Efron and Gail Gong (1983)
• Elementary Principles in Statistical Mechanics, Gibbs (1902)
• Equation of State Calculations by Fast Computing Machines, Metropolis et. al. (1953)
• Generalized statistical mechanics: connection with thermodynamics, Curado-Tsallis (1992)
• Poincaré sections of Hamiltonian systems (1996)

• Statistical mechanics of ensemble learning, Anders Krogh and Peter Sollich (1997)

Please cite as follows:

 @misc{suezen22pom, 

     title = { Physics origins of the most important statistical ideas of recent times }, 

     howpublished = {\url{http://science-memo.blogspot.com/2022/02/physics-origins-of-most-important.html}, 

     author = {Mehmet Süzen},

     year = {2022}

  }

Appendix: Pearson correlation and Lattices

Auguste Bravais is famous for his contribution in foundational work on the mathematical theory for crystallography, now seems to be going far beyond periodic solids. Unknown to many, he actually first driven the expression for what we know today as correlation coefficient or  Pearson’s correlation or less commonly Pearson-Galton coefficient. Interestingly, one of the grandfathers of causal analysis Wright is mentioned this in his seminal work of 1921 titled “Correlation and causation” acknowledged Bravais for his 1849 work as the first derivation of correlation.

Appendix: Partition function and set theoretic probability

Long before Kolmogorov set forward his formal foundations of probabilities, Boltzmann, Maxwell and Gibbs build theories of statistical mechanics using probabilistic language and even define settings for set theoretic foundations by introducing ensembles for thermodynamics. For example, partition function (Z) appeared as defining a normalisation factor that summation of densities should yield to 1. Apparently Kolmogorov and contemporaries inspired a lot from physics and mechanics literature.

Appendix: Generative AI

Of course now generative AI took over the hype. Indeed physics of diffusion from Fokker-Planck equation to basic Langevin dynamics is leveraged.  

 

Appendix: Physics is fundamental for the advancement of AI research and practice 

AI as a phenomena appears to be in the domain of core physics. For this reason, studying physics as a (post)-degree or as a self-study modules will give students and practitioners alike a definitive cutting-edge insights.  

• Statistical models based on correlations originates from physics of periodic solids and astrophysical n-body dynamics.
• Neural networks originates from the modelling magnetic materials in discrete states and later named as cooperative phenomenon. Their training dynamics closely follows free-energy minimisation.
• Causality roots in ensemble theory of physical entropy.
• Almost all sampling based techniques are based on the idea of sampling  physics of energy surfaces, i.e. Potential Energy Surfaces. (PES).
• Generative AI  originates from physics of diffusion of fluids: classical  Liouville description of the classical mechanics, i.e, phase-space flows and generalised Fokker-Planck dynamics. 
• Language models based on attention are actually coarse-grained entropy-dynamics
  introduced by Gibbs: ‘Attention Layers’ behaves as coarse-graining procedure, i.e, compressed
  causal graphs mapping.

This is not about building analogies to physics but as foundational topics to AI.

Periodic Spectral Ergodicity Accurately Predicts Deep Learning Generalisation

 Preamble 

    Dali (1931),
The Persistence of Memory (Wikipedia)

One of the new mathematical concepts arise due to understanding of deep learning is called periodic spectral ergodicity (PSE). The cascading PSE (cPSE) propagates over deep learning layers which can also be used as a complexity measure. cPSE actually can also predict the generalisation ability. In this post, we review this interesting  finding in an easy and short manner.

How periodic spectral ergodicity cascades over layers

We have reviewed spectral ergodicity in a gentle fashion earlier, here.  Only difference is that in real deep learning architectures, length of the eigenvalue spectrum, i.e., the number  of bins in the histogram, generated by weight matrices are not equal in size. To align them, we use something called periodic boundary conditions or turn the eigenvalues in a cyclic fashion, up to the maximum length spectra we have seen up to that layer. Here are the steps that give, the intuition of how to compute cascading periodic spectral ergodicity (cPSE).

1. We compute eigenvalue spectrum up to a layer $i$ and align the smaller spectrum with periodic boundary conditions, i.e., cyclic.

2. Compute spectral ergodicity at layers $i$ and $i-1$.

3. Compute the cascading PSE at layer $i$ simply with a distance metric $\Omega^{i}$  and $\Omega^{i-1}$. i.e.,  KL divergence in two directions, recall earlier tutorials.  

If we repeat this up to the last layer, cPSE measures the complexity of the deep learning architecture, both capturing structural and learning algorithm-wise, in a depth of a layer fashion. 

 Generalisation Gap and cPSE

Apart from being a complexity measure, cPSE predicts the generalisation gap given reference architecture i.e., it correlates with the performance almost perfectly. These findings are presented in the paper suzen2019 .

Conclusions and Outlook

The complexity of deep learning architectures are still an open research problem.  One of the most promising direction is to use cPSE in terms of capturing structural complexity as well. While other measures in the literature did not consider depth dependency, whereby cPSE appears to be the first one.

Reference

@article{suzen2019,
  title={Periodic Spectral Ergodicity: A Complexity Measure for Deep Neural Networks and Neural Architecture Search},
  author={S{\"u}zen, Mehmet and Cerd{\`a}, Joan J and Weber, Cornelius},
  journal={arXiv preprint arXiv:1911.07831},
  year={2019}
}

Cite this post as  Periodic Spectral Ergodicity Accurately Predicts Deep Learning Generalisation, Mehmet Süzen,  https://science-memo.blogspot.com/2021/11/periodic-spectral-ergodicity-predicts-generalisation-deep-learning.html 2021

Appendix 

Bristol v0.12.2 is now supporting in computing cPSE from list of matrices

from bristol import cPSE

import numpy as np

np.random.seed(42)

matrices = [np.random.normal(size=(64,64)) for _ in range(10)]

(d_layers, cpse) = cPSE.cpse_measure_vanilla(matrices) 

Deep Learning in Mind a Gentle Introduction to Spectral Ergodicity

Preamble

    Figure: Monalisa on
Eigenvector grids (Wikipedia)

In the post, A New Matrix Mathematics for Deep Learning : Random Matrix Theory of Deep Learning, we have outlined a new mathematical concepts that are aimed at deep learning but in general belonging to applied mathematics. Here, we dive into one of the concepts,  spectral ergodicity. We aimed at conveying what does it mean and how to compute spectral ergodicity for a set of matrices, i.e., ensemble. We will use a visual aid and verbal descriptions of steps to produce a quantitative measure of spectral ergodicity. 

The idea of spectral ergodicity comes from quantum statistical physics but it is recently revived for deep learning as a new concept in order to accommodate mathematical needs of explaining and understanding the complexity of deep learning architectures.

Understanding Spectral Ergodicity

The concept of ergodicity can get quiet mathematical even for a professional mathematician.  A practical understanding of ergodicity  could lead to the law of large numbers statistically speaking. However, observed ergodicity for ensemble of matrices, i.e. over their eigenvalue spectrum, are not formally defined before in the literature, and only appeared in statistical quantum mechanics in a specialised case.  Here we do a formal definition gently.

The spectral ergodicity of snapshot of values from $M$ matrices, where they are $N \times N$ sizes,  denoted by $\Omega$, can be produce with the following steps:

1. Compute eigenvalues of $M$ matrices separately.  
2. Produce equidistance spectra of matrices out of eigenvalues, i.e., histograms with $b_{k}$ bins. Each cell in the Figure corresponds to bin in the spectra of the matrices. 
3. Compute average values over each bin across  $M$ matrices.
4. Computing root mean square deviation that went to each bin from $M$ matrices from corresponding ensemble averaged value and average over $M$ and $N$. This will give a distribution, $\Omega=\Omega(b_{k})$, which represents spectral ergodicity value, think as a snapshot value of a dynamical process.

Attentive reader would notice that normally, measures of ergodicity leads to a single value, such as in spin-glasses, but here we obtain ergodicity as a measure distribution. This stems from the fact that our observable is not univariate but it is a multivariate measure over spectra of the matrix, i.e., bins in the histogram of eigenvalues.  

Why spectral ergodicity important for deep learning? 

The reason why this measure is so important lies in dynamics and consistency in measuring observables (no nothing to do with quantum mechanics but time and ensemble averages classically). Normally we can't measure ensemble averages. In experimental conditions the measurement we do is usually a time averaged value. This is exactly what happens when we train deep neural network, i.e, ergodicity of weight matrices. Essentially, spectral ergodicity would capture deep neural network's characteristics.

Outlook

The way we express spectral ergodicity here would only consider all layer having the same size.  One would need a more advanced computation of spectral ergodicity for more realistic architectures, which is called cascading Periodic Spectral Ergodicity measure suitable as a complexity measure for deep learning.  The computation of such measure is more involved and spectral ergodicity we cover here is the first step.

Cite this post with  Deep Learning in Mind Very Gentle Introduction to Spectral Ergodicity, Mehmet Süzen, (2021) https://science-memo.blogspot.com/2021/07/deep-learning-random-matrix-theory-spectral-ergodicity.html 

A New Matrix Mathematics for Deep Learning : Random Matrix Theory of Deep Learning

 Preamble 

    Figure: Definition of Randomness
 (Compagner 1991, Delft University)

Development of deep learning systems (DLs)  increased our hopes to develop more autonomous systems. Based on the hierarchal learning of representations, deep learning defies the basic learning theory that beg the question of still rethinking generalisation. Even though DLs lacks severely the ability to reason without causal inference, they can't do that in vanilla form. However despite this limitation, they provide very rich new mathematical concepts as introduced recently. Here, we review couple of these new concepts briefly and draw attention to Random Matrix Theory's relevance in DLs and its applications in Brain networks.  These concepts in isolation are subject of applied mathematics but their interpretation and usage in deep learning architectures are demonstrated recently. In this post we provide a glossary of new concepts, that are not only theoretically interesting, they are directly practical from measuring architecture complexity to equivalance.  

Random matrices can simulate deep learning architectures with spectral ergodicity

Random Matrix Theory (RMT) has origins in foundation of mathematical statistics and mathematical physics pioneered by Wishart Distribution and Dyson Circular Ensembles.  As primary ingredient of a deep learning model as a result are set of weights, or learned parameter set, manifests as matrices and they come from a learning dynamics that are used in so called in inference time. Natural consequence of this, learning these matrices can be simulated via Random matrices of spectral radius close to unity. This provides us the following, ability to make a generic statement about deep learning systems independent of 

1. Network architecture (topology).
2. Learning algorithm. 
3. Data sizes and type.
4. Training procedure.

Why not Hessian or loss-landscape but Weight matrices? 

There are studies taking Hessian matrix as a major object, i.e., second derivative of parameters as a function of loss of the network and associate this to random matrices. However, this approach would only covers learning algorithm properties rather than architectures inference or learning capacity. For this reason, weight matrices should be taken as a primary object in any studies of random matrix theory in deep learning as they encode depth in deep learning. Similarly, loss-landscape can not capture the capacity of deep learning. 

Conclusion and outlook

In this short exposition, we tried to stimulate readers interest in exciting set of tools from RMTs for deep learning theory and practice. That is still subject of recent research with direct practical relevance. We provided glossary and reading list as well.  

Further Reading

Papers introducing new mathematical concepts in deep learning are listed here, they come with associated Python codes for reproducing the concepts.

• Spectral Ergodicity in Deep Learning Architectures via Surrogate Random Matrices
• Periodic Spectral Ergodicity: A Complexity Measure for Deep Neural Networks and Neural Architecture Search
• Equivalence in Deep Neural Networks via Conjugate Matrix Ensembles

Earlier relevant blog posts 

• Freeman Dyson's contribution to deep learning: Circular ensembles mimics trained deep neural networks
• Applying Occam's razor's to Deep Learning
• A practical understanding of ergodicity
• Statistical Physics Origins of Connectionist Learning: 
  
  Cooperative Phenomenon to Ising-Lenz Architectures

Citing this post

A New Matrix Mathematics of Deep Learning: Random Matrix Theory of Deep Learning : https://science-memo.blogspot.com/2021/07/random-matrix-theory-deep-learning.html Mehmet Süzen, 2021

Glossary of New Mathematical Concepts of Deep Learning

Summary of the definition of new mathematical concepts for new matrix mathematics.

Spectral Ergodicity Measure of ergodicity in spectra of a given random matrix ensemble sizes. Given set of matrices of equal size that are coming from the same ensemble, average deviation of spectral densities of individual eigenvalues over ensemble averaged eigenvalue. This mimic standard ergodicity, instead of over states of the observable, it measures ergodicity over eigenvalue densities.  $\Omega_{k}^{N}$, $k$-th eigenvalue and matrix size of $N$.

Spectral Ergodicity Distance A symmetric distance constructed with two Kullback-Leibler distances over two different size matrix ensembles, in two different direction. $D = KL(N_{a}|N_{b})+ KL(N_{b}|N_{a})$

Mixed Random Matrix Ensemble (MME) Set of matrices constructed from a random ensemble but with difference matrix sizes from N to 2, sizes determined randomly with a coefficient of mixture. 

Periodic Spectral Ergodicity (PSE) A measure of Spectral ergodicity for MMEs whereby smaller matrix spectrum placed in periodic boundary conditions, i.e., cyclic list of eigenvalues, simply repeating them up to N eigenvalues. 

Layer Matrices Set of learned weight matrices up to a layer in deep learning architecture. Convolutional layers mapped into a matrix, i.e. stacked up. 

Cascading Periodic Spectral Ergodicity (cPSE) Measuring PSE over feedforward manner in a deep neural network.  Ensemble size is taken up-to that layer matrices. 

Circular Spectral Deviation (CSD) This is a measure of fluctuations in spectral density between two ensembles.

Matrix Ensemble Equivalence If CSDs are vanishing for conjugate MMEs, they are said to be equivalent.

Appendix: Practical Python Example

Complexity measure for deep architectures and random matrix ensembles: cPSE.cpse_measure_vanilla Python package Bristol  (>= v0.2.12) has now a support for computing cPSE from a list of matrices, no need to put things in torch model format by default.

!pip install bristol==0.2.12

An example case:

from bristol import cPSE

import numpy as np

np.random.seed(42)

matrices = [np.random.normal(size=(64,64)) for _ in range(10)]

(d_layers, cpse) = cPSE.cpse_measure_vanilla(matrices) 

d_layers is decreasing vector, it will saturate at some point, that point is where adding more

layers won’t improve the performance. This is data, learning or architecture independent measure.

Only a French word can explain the excitement here: Voilà!