Loschimidt's Paradox and Causality: Can we establish Pearlian expression for Boltzmann's H-theorem?

Boltzmann (Wikipedia)

• Post covers the papers: H-theorem do-conjecture, M. Süzen,  arxiv:2310.01458 (2023) 

Preamble

Probably the most important achievement for humans is the ability to produce scientific discoveries, that  helps us objectively understand how nature works and build artificial tools where no other species can.  Entropy is an elusive concept and one of the crown achievements of human race. We question here if causal inference and Loschmidt's paradox can be reconciled. 

Mimicking analogies are not physical

Before even try to understand what is a physical entropy, we should make sure that there is only one kind of physical entropy from thermodynamics, formulated by Gibbs-Boltzmann ($S_{G}$ and $S_{B}$).  Other entropies such as Shannon's information entropy are all analogies to physics, and mimicking concepts.

Why counting microstates are associated with time?

The following definition of entropy is due to Boltzmann but Gibbs' formulation tend to provide equivalence, technically different formulations aside, they are actually equivalent.

Definition 1: An entropy of a macroscopic material is associated with larger number of states its constituted elements take different states, $\Omega$. This is associated with $S_{B}$, Boltzmann's entropy.  

Now, as we know from basic thermodynamics classes that entropy change of a system can not decrease, so the time's arrow. 

Definition 2: Time's arrow is identified with change in entropy of material systems, that $\delta S \ge 0$.

We put aside the distinction between open and close systems and equilibrium and non-equilibrium dynamics, but concentrate on how come counting system's state's are associated with time's arrow? 

Loschimidt's Paradox: Irreversible occupancy on discrete states and causal inference

The core idea probably can be explained via discrete lattice and occupancy on them over chain of dynamics. 

Conjecture 1: Occupancy of $N$ items on $M$ discrete states, $M>N$, evolving with dynamical rules $\mathscr{D}$ necessarily increases $\Omega$, compare to the number of sampling if it were $M=N$. 

This conjecture might explain the entropy increase, but irreversibility of the dynamical rule $\mathscr{D}$ is required addressing Loschimidt's Paradox, i.e., how to generate irreversible evolution given time-reversal dynamics. Actually, do-calculus may provide a language to resolve this, by inducing interventional notation on Boltzmann's H-theorem with Pearlian view. The full definition of H-function is a bit more involved, but here we summarise it in condensed form with a do operator version of it.

Conjecture 2 (H-Theorem do-conjecture): Boltzmann's H-function provides a basis for entropy increase, it is associated with conditional probability of a system $\mathscr{S}$ being in state $X$ on ensemble $\mathscr{E}$. Hence, $P(X|\mathscr{E})$. Then, an irreversible evolution from time-reversal dynamics should use interventional notation, $P(X|do(\mathscr{E}))$. Then information on how time reversal dynamics leads to time's arrow encoded on, how dynamics provides an interventional ensembles, $do(\mathscr{E})$.

Conclusion

We provided some hints on why would counting states lead to time's arrow, an irreversible dynamics.  In the light of the development of mathematical language for causal inference in statistics, the concepts are converging. Along with understanding Loschmidt's Paradox via do-calculus, it can establish an asymmetric notation. Loschmidt's question is long standing problem in physics and philosophy with great practical implications in different physical sciences.

Further reading

• Loschimid's Paradox
• H-Theorem
• do-Calculus revisited, J. Pearl (2012) pdf
• Causal Inference : Looper Repository for collection of resources.
• H-theorem do-conjecture, M. Süzen,  arxiv:2310.01458 (2023) 

Please cite as follows:

 @misc{suezen23lpc, 

     title = {Loschimidt's Paradox and Causality: Can we establish Pearlian expression for Bolztmann's H-theorem?}, 

     howpublished = {\url{https://science-memo.blogspot.com/2023/02/loschimidts-do-calculus.html}}, 

     author = {Mehmet Süzen},

     year = {2023}

}  

@article{suzen23htd,

    title={H-theorem do-conjecture},

    author={Mehmet Süzen},

    preprint={arXiv:2310.01458},

    url = {https://arxiv.org/abs/2310.01458}

    year={2023}

}

Differentiating ensembles and sample spaces: Alignment between statistical mechanics and probability theory

Preamble 

Sample space is the primary concept introduced in any probability and statistics books and in papers. However, there needs to be more clarity about what constitutes a sample space in general: there is no explicit distinction between the unique event set and the replica sets. The resolution of this ambiguity lies in the concept of an ensemble.  The concept is first introduced by American theoretical physicist and engineer Gibbs in his book Elementary principle of statistical mechanics.  The primary utility of an ensemble is a mathematical construction that differentiates between samples and how they would form extended objects. 

In this direction, we provide the basics of constructing ensembles in a pedagogically accessible way from sample spaces that clears up a possible misconception. This usage of ensemble prevents the overuse of the term sample space for different things. We introduce some basic formal definitions.

    Figure: Gibbs's book
 introduced the concept of
ensemble (Wikipedia).

What Gibbs's had in mind by constructing statistical ensembles?

A statistical ensemble is a mathematical tool that connects statistical mechanics to thermodynamics. The concept lies in defining microscopic states for molecular dynamics; in statistics and probability, this corresponds to a set of events. Though these events are different at a microscopic level, they are sampled from a single thermodynamics ensemble, a representative of varying material properties or, in general, a set of independent random variables. In dynamics, micro-states samples an ensemble. This simple idea has helped Gibbs to build a mathematical formalism of statistical mechanics companion to Boltzmann's theories.

Differentiating sample space and ensemble in general

The primary confusion in probability theory on what constitutes a samples space is that there is no distinction between primitive events or events composed of primitive events. We call both sets sample space. This terminology easily overlooked in general as we concentrate on events set but not the primitive events set in solving practical problems.   

Definition: A primitive event $\mathscr{e}$ implies a logically distinct unit of experimental realisation that has not composed of any other events.

Definition: A sample space $\mathscr{S}$ is a set formed by all $N$ distinct primitive events $\mathscr{e}_{i}$.  

By this definition, regardless of how many fair coins are used or if a coin toss in a sequence for the experiment, the sample space is always ${H,T}$, because these are the most primitive distinct events a system can have, i.e., a single coin outcomes. However, the statistical ensemble can be different.  For example for two fair coins or coin toss in sequence of length two, corresponding ensemble of system size two reads ${HH, TT, HT, TH}$. Then, the definition of ensemble follows. 

Definition: An ensemble  $\mathscr{E}$ is a set of ordered set of primitive events $\mathscr{e}_{i}$. These event sets can be sampled with replacement but order matters, i.e., $ \{e_{i}, e_{j} \} \ne  \{e_{j}, e_{i} \}$, $i \ne j$.

Our two coin example's ensemble should be formally written as $\mathscr{E}=\{\{H,H\}, \{T,T\}, \{H,T\}, \{T,H\}\}$, as order matters members $HT$ and $TH$ are distinct. Obviously for a single toss ensemble and a sample space will be the same. 

Ergodicity makes the need for differentiation much more clear : Time and ensemble averaging 

The above distinction makes building time and ensemble averaging much easier. The term ensemble averaging is obvious as we know what would be the ensemble set and averaging over this set for a given observable.  Time averaging then could be achieved by curating a much larger set by resampling with replacement from the ensemble. Note that the resulting time-average value would not be unique, as one can generate many different sample sets from the ensemble. However, bear in mind that the definition of how to measure convergence to ergodic regime is not unique.

Conclusion

Even though the distinction we made sounds very obscure,  this alignment between statistical mechanics and probability theory may clarify the conception of ergodic regimes for general practitioners.

Further reading

• Practical understanding of ergodicity.
• A misconception in ergodicity: Identify ergodic regime not ergodic process

Please Cite:

 @misc{suezen22dess, 

     title = {Differentiating ensembles and sample spaces: Alignment between statistical mechanics and probability theory}, 

     howpublished = {\url{https://science-memo.blogspot.com/2022/11/ensembles-probability-theory.html}, 

     author = {Mehmet Süzen},

     year = {2022}

}  

Postscript

• If there are multiple events coming from set of primitive events, compositional outcomes considered to be ensemble not sample space. Sample space is a set that we sample from, either one or multiple times to build an ensemble. Ensemble notion within pure ML context was also noticed by late David J. C. MacKay, in his book Information Theory, Inference and Learning, Cambridge University Press (2003).

Bayesian rabbit holes: Decoding conditional probability with non-commutative algebra

Preamble

    The White Rabbit
(Wikipedia)

A novice analyst or even experienced (data) scientist would have thought that the bar notation $|$ in representing conditional probability carries some different operational mathematics. Primarily when written in explicit distribution functions $p(x|y)$. Similar approach applies to joint probabilities such as $p(x, y)$ too. One could see a mixture of these, such as $p(x, y | z)$. In this short exposition, we clarify that none of these identifications within arguments of probability do have any different resulting operational meaning. 

Arguments in probabilities: Boolean statement and filtering 

Arguments in any probability are mathematical statements of discrete mathematics that correspond to events in the experimental setting. These are statements declaring some facts with a boolean outcome. These statements are queries to a data set. Such as, if the temperature is above $30$ degrees, $T > 30$. Temperature $T$ is a random variable. Unfortunately, the term random variable is often used differently in many textbooks. It is defined as a mapping rather than as a single variable. The bar $|$ in conditional probability $p(x|y)$, implies statement $x$ given that statement $y$ has already occurred, i.e., if. This interpretation implies that $y$ first occurred before $x$, but it doesn't imply that they are causally linked. The condition plays a role in filtering, a where clause in query languages. $p(x|y)$ boils down to $p_{y}(x)$, where the first statement $y$ is applied to the dataset before computing the probability on the remaining statement $x$.

In the case of joint probabilities $p(x, y)$, events co-occur, i.e., AND statement. In summary, anything in the argument of $p$ is written as a mathematical statement. In the case of assigning a distribution or a functional form to $p$, there is no particular role for conditionals or joints; the modelling approach sets an appropriate structure.

Conditioning does not imply casual direction: do-Calculus do

A filtering interpretation of conditional $p(x|y)$ does not imply causal direction, but $do$ operator does, $p(x|do(y))$. 

Non-commutative algebra: When frequentist are equivalent to Bayesian

Most of the simple filtering operations would result in identical results if reversed. $p(x|y) = p(y|x)$, prior being equal to posterior. This remark implies we can't apply Bayesian learning with commutative statements. We need non-commutative statements; as a result, one can do Bayesian learning with the newly arriving data, i.e., the arrival of new subjective evidence. The reason seems to be due to the frequentist nature of filtering.

Outlook 

Even though we provided some revelations on decoding the operational meaning of conditional probabilities, we suggested that any conditional, joint or any combination of these within the argument of probabilities has no operational purpose other than pre-processing steps. However, the philosophical and practical implications of probabilistic reasoning are always counterintuitive. Probabilistic reasoning is a complex problem computationally. From a causal inference perspective, we are better equipped to tackle these issues with do-Bayesian analysis.  

Further reading

• Discrete Mathematics, Oscar Levin
• Causal Inference in Statistics, A Primer, Judea Pearl, Madelyn Glymour, Nicholas P. Jewell
• Indicative Conditionals Stanford Encyclopedia of Philosophy 
• Bayesian Epistemology Stanford Encyclopedia of Philosophy 

Please Cite as:

 @misc{suezen22brh, 

     title = {Bayesian rabbit holes: Decoding conditional probability with non-commutative algebra}, 

     howpublished = {\url{https://science-memo.blogspot.com/2022/07/bayesian-conditional-noncommutative.html}}, 

     author = {Mehmet Süzen},

     year = {2022}

}  

A misconception in ergodicity: Identify ergodic regime not ergodic process

Preamble 

    Figure 1: Two observable's approach to
 ergodicity for Bernoulli Trials. 

Ergodicity appears in many fields, in physics, chemistry and natural sciences but in economics to machine learning as well. Recall that, ergodicity in physics and mathematical definition diverges significantly due to Birkhoff's statistical definition against Boltzmann's physical approach. Here we will follow Birkhoff's definition of ergodicity which is a statistical one. The basic notion of ergodicity is confusing even among experienced  academic circles. The primary misconception is that ergodicity is attributed to a process, a given process being ergodic. We address this by pointing out that ergodicity appears as a regime or a window so to speak for a given process's time-evolution and it can't be attributed to an entire generating process.    

No such thing as ergodic process but ergodic regime given observable

A process being ergodic is not entirely true identification. Ergodicity is a regime over a given time window for a given observable derived from the process. This is the basis of ensemble theory from statistical physics.  Most of the processes generates initially a non-ergodic regime given an observable.  In order to identify an ergodic regime,  we need to define for a discrete setting : 

1. the ensemble (sample space) : In discrete dynamics we also have an alphabet that ensemble is composed of.
2. an observable defined over the sample space.
3. a process (usually dynamics on the sample space evolving over-time).
4. a measure and threshold to discriminate ergodic to non-ergodic regimes. 

Interesting thing is that different observables on the  same ensemble and the process may generate different ergodic regimes.  

 What are the processes and regime mathematically?

A process is essentially a dynamical system mathematically.  this includes stochastic models and as well as deterministic systems sensitive to initial conditions. Prominently these both combined in Statistical Physics. A regime mathematically implies a range of parameters or a time-window that a system behaves very differently. 

 Identification of ergodic regime

Figure 2: Evolution of
time-averaged OR observable.

The main objective of finding out if dynamics produced by the process on our observable enters or in an ergodic regime is to measure if ensemble-averaged observable is equivalent to time-averaged observable value. Here equivalence is a difficult concept to address quantitatively. The simplest measure would be to check if $\Omega = \langle A \rangle_{ensemble} - \langle A \rangle_{time}$ is close to zero, i.e., vanishing. $ \Omega$ being the ergodicity measure and $A$ is the observable with different averaging procedure. This is the definition we will use here. However, beware that in the physics literature there are more advanced measures to detect ergodicity, such as considering diffusion-like behaviour, meaning that the transition from non-ergodic to ergodic regime is not abrupt but have a diffusing approach to ergodicity.  

Figure 3: Evolution of
time-averaged mean.

In some other academic fields approach to ergodic regime has different names not strictly but closely related, such as in chemical physics or molecular dynamics, equilibration time, relaxation time, equilibrium, steady-state for a given observable, in statistics Monte Carlo simulations, it is usually called burn out period. Not always, but in ergodic regime, observable is stationary and time-independent. In Physics, this is much easier to distinguish because time-dependence, equilibrium and stationarity are tied to energy transfer to the system. 

Ergodic regime not ergodic process : An example of Bernoulli Trials

Apart from real physical processes such as Ising Model, a basic process we can use to understand how ergodic regime could be detected using Bernoulli Trials. 

Here for a Bernoulli trials/process, we will use random number generators for a binary outcome, i.e., RNG Marsenne-Twister  to generate time evolution of an observables on two sites:  Let's say we have two sites  $x, y \in \{1, 0\}$.  The ensemble of this two site system $xy$ is simply the sample space of all possible outcomes $S=\{10, 11, 01, 00\}$. Time evolution of such two site system is formulated here as choosing $\{0,1\}$  for a given site at a given time, see Appendix Python notebook. 

Now, the most important part of checking ergodic regime is that we need to define an observable over two side trials. We denote two observable as $O_{1}$, which is an OR operation between sites, and $O_{2}$ is averaged over two sites. Since our sample space is small, we can compute the ensemble average observables analytically:

• $O_{1}  =  (x+y)/2$  then  $10, 11, 01, 00  ;  (1/2 + 2/2 + 1/2 + 0 ) /4 = 0.5$
• $O_{2}  =  x OR y$ then   $10, 11, 01, 00  ;  ( 1  + 1 + 1 + 0 )/4 = 0.75$   

We can compute the time-averaged observables over time via simulations, but their formulation are know as follows: 

•  Time average for $O_{1}$ at time $t$  (current step)  is   $ \frac{1}{t} \sum_{i=0}^{t} (x_{i}+y_{i})/2.0$
• Time average for $O_{2}$ at time $t$  (current step)  is   $ \frac{1}{t} \sum_{i=0}^{t} (x_{i} OR y_{i})$.

One of the possible trajectories are shown in Figure 2 and 3. For approach to ergodicity measure, we shown this at Figure 1. Even though, we should run multiple trajectories to have error estimates, we can clearly see that ergodicity regime starts after 10K steps, at least. Moreover, different observables have different decay rates to ergodic regime.  From preliminary simulation, it appears to be OR observable converges slower, though this is a single trajectory.

Conclusion

We have shown that manifestation of the ergodic regime depends on the time-evolution of the observable given a measure of ergodicity, i.e., a condition how ergodicity is detected. This exposition should clarify that a generating process does not get an attribute of "ergodic process" rather we talk about "ergodic regime" depending on observable and the process over temporal evolution. Interestingly, from Physics point of view, it is perfectly possible that an observable attains ergodic regime and then falls back to non-ergodic regime.

Further reading

• Practical Understanding of Ergodicity :  Elementary ergodicity and some basic references.
• Is ergodicity a reasonable hypothesis?  : Boltzmann's definition of ergodicity.
• Scaling of ergodicity in binary systems : An idea on extending Bernoulli trials ergodicity to N-dimensions (sites).
• Effective ergodicity in single-spin-flip dynamics PRE Approach to ergodicity in magnetic systems, extents to neural networks. 
• IsingLenzMC R package : Effective ergodicity convergence R utilities and Ising-Lenz 1-D Monte Carlo.
• Diffusive behaviour of ergodicity convergence in Ising-Model.

Appendix: Code

Bernoulli Trial example we discussed is available as a Python notebook on github here. 

Please cite as follows:

 @misc{suezen22ergoreg, 

     title = {A misconception in ergodicity: Identify ergodic regime not ergodic process}, 

     howpublished = {\url{http://science-memo.blogspot.com/2022/05/ergodic-regime-not-process.html}, 

     author = {Mehmet Süzen},

     year = {2022}

}  

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.

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 

Statistical Physics Origins of Connectionist Learning: Cooperative Phenomenon to Ising-Lenz Architectures

This is an informal essay in aiming at raising awareness that Statistical Physics played a foundational role in deep learning and neural networks in general beyond being a mare analogy but its origin. 

Article version of this post is available here: doi. and on HAL Open Science

Preamble

A short account of origins of mathematical formalism of neural networks is presented for physicists and computer scientist in basic discrete mathematical setting informally. The discourse of the development of mathematical formalism on the dynamics of lattice models in statistical physics and learning internal representations of neural networks as discrete architectures as quantitative tools evolve in two almost distinct fields more than half a century with limited overlap. We aim at bridging the gap by claiming that the analogy between two approaches are not artificial but naturally occuring due to how modelling cooperative phenomenon is constructed. We define the Lenz-Ising architectures (ILAs) for this purpose.

Introduction

Figure: Tartan Ising Model
(Linas Viptas-Wikipedia)

Understanding natural or artificial phenomenon in the language of discrete mathematics is probably one of the most powerful toolbox scientist use [1]. Large portion of computer science and statistical physics deals with such finite structures. One of the most prominent successful usage of such approach was Lenz and Ising’s work on modelling ferromagnetic materials [2–5] and neural networks as a model to biological neuronal structures [6–8].

The analogy between two areas of distinct research have been pointed out by many researchers [9–13]. However, the discourse and evolution of these approaches were kept as two distinct research fields and many innovative approaches rediscovered under different names.

Cooperative Phenomenon

Statistical definition of cooperative phenomenon pioneered by Wannier and Kremer [14–16]. Even though their technical work focused on extension of Ising model to 2D with cyclic boundary condition and introduction of exact solutions with matrix algebra, they were the first to document the potential of how Lenz-Ising model actually represent a more generic system than merely model to ferromagnets, namely anything falls under cooperative phenomenon can be addressed with Lenz-Ising type model, summarised in Definition 1.

Definition 1: Cooperative phenomenon of Wannier type  [14]: Set of $N$ discrete units, $\mathscr{U}$, identified with a function $s_{i}$, i=1,..,N forms a collection or assembly. The function that identifies the units is a mapping $s_{i}: \mathbb{R} \rightarrow \mathbb{R}$. A statistic $\mathscr{S}$ applied on $\mathscr{U}$ is called cooperative phenomenon of Wannier type $\mathscr{W}$.

A statistic $\mathscr{S}$ can be any mapping or set of operations on the assembly of units $\mathscr{U}$ . For example inducing ordering on the assembly of units and summation over  $s_{i}$ values, would corresponds to non-interacting magnetic system with unit external field or non-connected set of neurons capacity of inhibition or exhibition. However, amazingly, Definition 1 is so generic that Rosenblatt’s perceptron [17], current deep learning systems [18] and complex networks [19] falls into this category as well. 

The originality of Cooperative phenomenon of Wannier type comes on a secondary concept, so called event propagation as given in Definition 2.

Definition 2. Event propagation [14] An event is defined as a snapshot of cooperative phenomenon of Wannier type $\mathscr{W}$. If an event takes place of one unit of assembly $\mathscr{U}$, the same event will be favored by other units, this is expressed as event propagation between two disjoint set of units $\mathscr{E}(u_{1}, u_{2})$, and $u_{1} \cap u_{2} = \varnothing$ and $u_{1}, u_{2} \in \mathscr{U}$ and with an additional statistic $\mathscr{S}$ is defined.

The parallels between Wannier’s event propagations are remarkably the same as of neural network formalism defined by McCulloch-Pitts-Kleene [6,7], not only conceptually but matematical treatment is identical and originates from Lenz-Ising model’s treatment of discrete units. As we mentioned, this goes beyond doubt not a simple analogy but forms a generic framework as envisioned by Wannier. The similarity between ferromagnetic systems and neural networks is probably first documented directly by Little [8]: Spin states of magnetic spins corresponds to firing state of a neuron. Unfortunately, Little only see it as simple analogy, and missed the opportunity provided by Wannier as a generic natural phenomenon of cooperation.

The conceptual similarity and inference on Wannier’s event propagation appears to be quite close to Hebb’s learning [20] and gives natural justification for backpropagation for multilayered networks. History of backpropagation is exhaustively studied elsewhere [18].

Lenz-Ising Architectures (ILAs): Ferromagnets to Nerve Nets

Ernst Ising
 Image owner APS - Physics Today :
Obituary

As we established two basic definitions of cooperative phenomenon, we can now define a generic setting of Lenz-Ising model that captures both physics literature that extensively used this in so called spin-glasses research and for neural networks. A guiding principle will be based on Wannier’s definition of cooperative phenomenon.

Definition: Lenz-Ising Architectures (ILAs) 
Given Wannier type cooperative phenomenon $\mathscr{W}$, imposing constrains on the discrete units, $\mathscr{U}^{c}$ that they should be spatially correlated on the edges $E$ of an arbitrary graph $\mathscr{G}(E, V)$ with ordering and with vertices $V$ of the arbitrary graph carring coupling weight between connected two units with biases. Set of event propagations $\mathscr{E}^{c}$ defined on the cooperative phenomeon can induce dynamics on defining vertice weights, or vice versa. ILAs are defined as statistic $\mathscr{S}$ applied to $\mathscr{U}^{c}$ with propagations $\mathscr{E}^{c}$. 

Lenz-Ising Architectures (ILAs) should not be confused with graph neural networks as it does not model data structures. It could be seen as subset of graph dynamical systems in some sense but formal connections should be established elsewhere. However, primary characteristic of ILAs are that it is conceptual and mathematical representation of spin-glass systems (including Lenz-Ising, Anderson, Sherrington-Kirkpatrick, Potts systems) and neural networks (including recurrent and convolutional networks) under the same umbrella.

 Learning representations inherent in Metropolis-Glauber dynamics

The primary originality in any neural network research papers lies in so called learning representation from data and generalisation. However, it isn’t obvious to the that community that actually spin-glasses are capable of learning representations inherently by induced dynamics such as Metropolis or Glauber dynamics by construction, as an inverse problem.

In physics literature this appears as finding a solution to the problem of how to express free energy and minimisation of this with respect to weights or coupling coefficients, This is noting but a learning represenations. Usually a simulation approach is taken as a route, for example Monte Carlo techniques [5, 21, 22] via Metropolis or Glauber dynamics. The intimate connection between concepts of ergodicity and learning in deep learning is recently shown [13,23,24] in this context.

Roy J. Glauber (Wikipedia)  
Glauber dynamics

As we argued earlier the generic definition provided by Wannier on cooperative phenomenon and ILAs; there is an intimate connection with learning and so called solving spin-glasses that usually boils down to computing free energies as mentioned. And a link between two distinct fields, computing backpropagation and free energies are natural candidates to establish equivalence relations.

Conclusions and Outlook

Apart from honouring physicists Lenz and Ising, based on understanding of cooperative phenomenon’s origins, naming the research outpus from of spin-glasses and neural networks under an umbrella term Lenz-Ising architectures (ILAs) is historically accurate and technically a resonable naming scheme under the overwhelming evidence given in the literature. This is akin to naming current computers with von Neumann architectures. This forms the origins of connectionist learning from statistical physics, where this approach currently enjoying vast engineering success today.

The rich connection between two areas in computer science and statistical physics should be celebrated. For more fruitful collaborations, both literatures, embracing large statistics literature as well, should converge much more closely. This would help communities to avoid awkward situations of reinventing the wheel again and hindering recognition of the work done by physicists decades earlies, i.e., Ising and Lenz.

 Notes

No competing or other kind of conflict of interest exists. This work is produced solely with the aim of scholarly work and does not have any personal nature at all. This essay is dedicated in memory of Ernst Ising for his contribution to physics of ferromagnetic materials, now seems to have far more implications.

References

[1] Kenneth H Rosen. Handbook of Discrete and Combinatorial Mathematics. CRC Press, 1999. 

[2] W.Lenz. Beitrag zum Verstl ̈andnis der Magnetischen Erscheinungen in Festen Korpern. Phys.Z, 21:613, 1920.

[3] Ernst Ising. Beitrag zur Theorie des Ferromagnetismus. Zeitschrift furr Physik, 31(1):253–258, 1925.

[4] Thomas Ising, Reinhard Folk, Ralph Kenna, Bertrand Berche, and Yurij Holovatch. The fate of Ernst Ising and the fate of his model. arXiv preprint arXiv:1706.01764, 2017.

[5] David P Landau and Kurt Binder. A guide to Monte Carlo Simulations in Statistical Physics. Cambridge University Press, 2014.

[6] W.S. McCulloch and W.H. Pitts. A Logical Calculus of the Ideas Imminent in Nervous Activity.Bull. Math. Biophys.,(5), pages 115–133.

[7] Stephen Cole Kleene. Representation of Events in Nerve Nets and Finite Automata. Technical report, RAND Project, Santa Monica, 1951.

[8] W. A. Little. The Existence of Persistent States in the Brain. Mathematical Biosciences, 19(1-2):101–120, 1974.

[9] P Peretto. Collective Properties of Neural Networks: a Statistical Physics Approach. Biological Cybernetics, 50(1):51–62, 1984.

[10] Jan L van Hemmen. Spin-glass Models of a Neural Network. Physical Review A, 34(4):3435, 1986.

[11] Haim Sompolinsky. Statistical Mechanics of Neural Networks. Physics Today, 41(21):70–80,1988.

[12] David Sherrington. Neural Networks: the Spin Glass Approach. In North-Holland MathematicalLibrary, volume 51, pages 261–291. Elsevier, 1993.

[13] Yasaman Bahri, Jonathan Kadmon, Jeffrey Pennington, Sam S Schoenholz, Jascha Sohl-Dickstein, and Surya Ganguli. Statistical Mechanics of Deep Learning. Annual Review of Condensed Matter Physics, 2020.

[14] Gregory H Wannier. The Statistical Problem in Cooperative Phenomena. Reviews of Modern Physics, 17(1):50, 1945.

[15] Hendrik A Kramers and Gregory H Wannier. Statistics of the two-dimensional ferromagnet.Part I. Physical Review, 60(3):252, 1941.

[16] Hendrik A Kramers and Gregory H Wannier. Statistics of the two-dimensional ferromagnet.Part II. Physical Review, 60(3):263, 1941.

[17] C van der Malsburg. Frank Rosenblatt: principles of neurodynamics: perceptrons and the theory of brain mechanisms. In Brain theory, pages 245–248. Springer, 1986.

[18] J. Schmidhuber. Deep learning in Neural Networks: An overview. Neural networks, 61:85–117, 2015. & Yoshua Bengio, Yann Lecun, Geoffrey Hinton, Communications of the ACM, July 2021, Vol. 64 No. 7, Pages 58-65 (2021) link

[19] Duncan J Watts and Steven H Strogatz. Collective dynamics of ‘small-world’networks. Nature,393(6684):440, 1998.

[20] Donald Olding Hebb. The Organization of Behavior: a Neuropsychological Theory. J. Wiley;Chapman & Hall, 1949.

[21] Mehmet Suezen. Effective ergodicity in single-spin-flip dynamics. Physical Review E, 90(3):032141, 2014.

[22] Mehmet Suezen. Anomalous diffusion in convergence to effective ergodicity. arXiv preprint arXiv:1606.08693, 2016.

[23] Mehmet Suezen, Cornelius Weber, and Joan J Cerda. Spectral ergodicity in deep learning architectures via surrogate random matrices. arXiv preprint arXiv:1704.08303, 2017.

[24] Mehmet Suezen, JJ Cerda, and Cornelius Weber. Periodic Spectral Ergodicity: A Complexity Measure for Deep Neural Networks and Neural Architecture Search. arXiv preprint arXiv:1911.07831, 2019.

Postscript 1:

(Deep) Machine learning as a subfield of statistical physics

Often researchers considers some machine learning methods
under different umbrella terms compare to established
statistical physics. However, beyond being mare analogy,  
application of these methods are quite striking. Consequently,
there is a great tradition in machine learning practice 
of being sub-field of statistical physics with explicit
classification within PACS. 

Hopfield Networks <- Ising-Lenz model
Boltzmann Machines <- Sherrington-Kirkpatrick model
Diffusion Models <- Langevin Dynamics, Fokker-Planck Dynamics
Softmax <- Boltzmann-Gibbs connection to partition function 
Energy Based Models <- Spin-glasses, Hamiltonian dynamics

For this reason, we provide semi-formal mathematical definitions
in the recent article, establishing that deep learning architectures 
should be called Ising-Lenz Architectures (ILAs), akin to calling 
current computers having von Neumann architectures.