Matrix language conjecture for combinatorics: Combinatorial set generation via nested cartesian products

Rubik's cube (Wikipedia)

Preamble 

Counting is one of the most important concepts in probability and statistical mechanics as well. Two primary characteristics of choosing $n$ items from $N$ items are (no)-order and (no)-repeat. This leads to four  possible cases that leads to combinations and permutations of different kinds to found the resulting set size. Specially the case for repeated no-ordering would be quite hard to memorise or derive on the spot. There is a catch, we can compute the set sizes in combinatorics but usually there is no explicit explanation how could we get the entire combinatorial set member via a computation.  In this short exposition, we have demonstrated a conjecture  that obtaining resulting combinatorial set members is possible via nested cartesian products of the set of the $N$ items: Concepts from matrices help to identify resulting set members in the given combinatorics case.  

Compact Combinatorics

Combinatorics over choosing $n$ items over $N$ has four cases as mentioned. Here is the formulations of choosing sets over $N$ items of size $n$ with different constraints, resulting set sizes will be: 

• Ordering matters with replacement : $N^n$
• Ordering matters without replacement: $\Large \frac{N!}{(N-n)!} $
• Ordering doesn't matter with replacement: $\Large \frac{(n+N-1)!}{n!(N-1)!}$
• Ordering doesn't matter without replacement: $\Large \frac{N!}{n!(N-n)!} $

However, it is often we forget the formulations of the each case, even to identify the case given a problem.

Obtaining resulting sets: Combinatorial Matrix

We know the size of the resulting sets if we do the choose over a set given $n$. However, the members of the resulting set is not usually known via a computational procedure in standard texts. However, this is possible via cartesian product of the initial set by itself if $n=2$, i.e., pairwise matrices over $N$ item, a combinatorial matrix.     

Matrix language conjecture for combinatorics: A combinatorial matrix has different portions. An entire matrix (E), diagonals (D), upper diagonal (UD) and lower diagonal (LD).  These portions or combinations of them corresponds to the cases above, somehow obvious after a close look: 

• Ordering matters with replacement : E
• Ordering matters without replacement: E-D
• Ordering doesn't matter with replacement:   UD+D or LD+D   
• Ordering doesn't matter without replacement: UD or LD

By choosing certain portions of the matrix we full-fill constraints of generating new sets. 

Nested Combinatorial Matrices ($n \ge 3$  cases) 

It is obvious that the above conjecture is valid for $n=2$. In the case of where $n \ge 3$ we would need to do more cartesian products over a combinatorial matrix selectively. The idea is as follows: we generate new cartesian products with $N$ items row-wise with removal of the base row at each level due to portions selected. Here is the sketch of the nested cartesian products: 

1. We start with the case of $n=2$ and generate the matrix.  Retain the corresponding matrix portion.
2. Then repeat portion for the next levels till exhaustion that items are size $n$: We do cartesian product of the given row to all items again to generate new combinatorial matrix of each element, except the elements deleted from the row due to portion selection, if deleted.
3. At the end we obtain multiple matrices of possibly different sizes, then choose the portions again to form the resulting set of combinatorial items. 

Conclusion

We have discussed shortly how to leverage a matrix language in finding the resulting sets in core combinatorics counting operations. This could be used as a pedagogical tool, algorithmic development and bringing new relationships between matrices and combinatorial formula. This is still a conjecture, but taught provoking one that connects matrix generation to combinatorial structures.  

Cite this as follows: 

 @misc{suezen24mcg, 

     title = { Matrix language conjecture for combinatorics: Combinatorial set generation via nested cartesian products}, 

     howpublished = {\url{https://memosisland.blogspot.com/2024/09/combinatorics-touch-of-linear-algebra.html}}, 

     author = {Mehmet Süzen},

     year = {2024}

}  

Book review: A tutorial introduction to the mathematics of deep learning

Preamble

Artificial Intelligence Engines:
An introduction to the Mathematics
of Deep Learning by Dr James V. Stone
the book and Github repository.
(c) 2019 Sebtel Press

Deep learning and associated connectionist approaches are now applied routinely in industry and academic research from image analysis to natural language processing and areas as cool as reinforcement learning. As practitioners, we use these techniques and utilise them from well designed and tested reliable libraries like Tensorflow or Pytorch as shipped black-boxed algorithms. However, most practitioners lack mathematical foundational knowledge and core algorithmic understanding. Unfortunately, many academic books and papers try to make an impression of superiority show subliminally and avoid a simple pedagogical approach. In this post we review, a unique book trying to fill this gap with a pedagogical approach to the mathematics of deep learning avoiding showing of mathematical complexity but aiming at conveying the understanding of how things work from the ground up. Moreover, the book provides pseudo-codes that one can be used to implement things from scratch along with a supporting implementation in Github repo. Author Dr James V. Stone, a trained cognitive scientist and researcher in mathematical neuroscience provides such approaches with other books many years now, writing for students, not for his peers to show off. One important note that this is not a cookbook or practice tutorial but an upper-intermediate level academic book.

Building associations and classify with a network

The logical and conceptual separation of associations and classification tasks are introduced in the initial chapters. It is ideal to start with from learning one association with one connection to many via gentle introduction to Gradient descent in learning the weights before going to 2 associations and 2 connections. This reminds me of George Gamow's term 1, 2 and infinity as a pedagogical principle. Perceptron is introduced later on how classification rules can be generated via a network and the problems it encounters with XOR problem.

Backpropagation, Hopfield networks and Boltzmann machines

Detail implementation of backpropagation is provided from scratch without too many cluttering index notation in such clarity. Probably this is the best explanation I have ever encountered. Following chapters introduced Hopfield networks and Boltzmann machines from the ground up to applied level. Unfortunately, many modern deep learning books skip these two great models but Dr Stone makes these two models implementable for a practitioner by reading his chapters.  It is very impressive. Even though I am a bit biased in Hopfield networks as I see them as an extension to Ising models and its stochastic counterparts, but I have not seen anywhere else such explanations on how to use Hopfield networks in learning and in a pseudo-code algorithm to use in a real task.

Advanced topics

Personally, I see the remaining chapters as advanced topics: Deep Boltzmann machines, variational encoders, GANs and introduction to reinforcement learning. Probably exception of deep backpropagation in Chapter 9. I would say what is now known as deep learning now was the inception of the architectures mentioned in sections 9.1 till 9.7.

Glossary, basic linear algebra and statistics

Appendices provide a fantastic conceptual introduction to jargon and basics to main mathematical techniques. Of course, this isn't a replacement to fully-fledged linear algebra and statistics book but it provides immediate concise explanations.

Not a cookbook: Not import tensorflow as tf book

One other crucial advantage of this book is that it is definitely not a cookbook. Unfortunately, almost all books related to deep learning are written in a cookbook style. This book is not. However, it is supplemented by full implementation in a repository supporting each chapter, URL here.

Conclusion

This little book archives so much with down to earth approach with introducing basic concepts with a respectful attitude, assuming the reader is very smart but inexperience in the field. If you are a beginner or even experienced research scientist this is a must-have book.  I still see this book as an academic book and can be used in upper-undergraduate class as the main book in an elective such as
"Mathematics of Deep Learning".

Enjoy reading and learning from this book. Thank you, Dr Stone, for your efforts on making academic books more accessible.

Disclosure: I received a review copy of the book but I have bought another copy for a family member. 

Core principles of sustainable data science, machine learning and AI product development: Research as a core driver

Kindly reposto to KDnuggets  by Gregory Piatetsky-Shapiro

Preamble 

Almost all businesses and industry embraced Machine learning (ML) technologies. Apart from ROI concerns, as it is an expensive endeavour to develop and deploy a service driven by ML techniques, sustainability as in going beyond proof-of-concept core development appears to be one of the roadblocks in data science. In this post, we will outline basic logical core principles that can help organisations for sustainable AI product development cycle, apart from reproducibility issues. The aim is giving a coarser view, rather than listing fine-grain good practice advice.

Research as a core driver: Research Environment

Regardless of the size of your organisation, if you are developing machine learning or AI products, the core asset you have is a research professional, data scientist or AI scientist, regardless of their academic background. Developing a model using software libraries blindly won't resolve issues you might encounter after deployment of the product. For example, even if you need to do a simple hyperparameter search, this can easily yield to research. Why? Because most probably no one ever tried building a model or try a modelling task using your dataset and you might need a different approach than ML libraries provide. A simplest different angle or deviation from ML libraries will yield to a research problem. 

• No full 'black-box' approaches.
• No blind usage of software libraries.
• Awareness and skills in the mathematical and algorithmic aspect in detail.

Figure: A schematic of core principles for AI product development.

Separate out research code and production code

Software development is an integral part of ML product development. However, during research, a code development can go very wild and a scientist, even if they are very good software developers, would end up creating hard to follow and poor code. Once there is a confidence in reproducibility and robustness of results, the production code should be re-written with high-quality software engineering principles.

Data Standardisation: Release data-sets for research 

A cold start problem for ML products is to release and design data-sets before even doing any research like work. This, of course, has to be aligned with industrial requirements. Imagine datasets like MINST or imagenet for benchmarking. Released sets will be the first step for any model building or product development, and would constitute a data product themselves. Data versioning is also a must.

Do not obsess with workflows: All workflows are ad-hoc

There is no such thing as a universal or generic workflow. A workflow depends on a human understanding of processes and steps. Human understanding is based on language and linguistically there is no such thing as universal language, at least it isn't practical yet c.f., universal grammar. Loosely defined steps are sufficient for research steps. However, once it entered into production, then much more strict workflow design might be needed, but be aware all workflows are ad-hoc.

Do not run sprints for core data science 

Agile principles are suitable for software development innovations. Sprints or Agile is not suitable for AI research and research environment, it is a different kind of innovation than software engineering. Thinking that Agile is a cure to do scientific innovation is naive wishful thinking. Structuring a research group,  periodic reviews and releases of the results via presentations and detailed technical reports are much more suitable for data science on top of mini-workshops. A simple proposal runs can also be made to decide which direction to invest, akin to research proposals.

Feedback loop: Service to Business decision making back to research

A service using ML technologies should produce more data. The very first service monitoring is A/Null testing, meaning that what would happen in the absence of the AI product. Detailed analysis of the service data would bring more insights both for business and to research. 

• produce impact assessment: A/null testing
• quality of service: Quality of service can be measured basically on what is the success of the ML model, this has to be technical.

Conclusion and outlook

There is no such thing as free-lunch and developing AI products won't be fully automated soon. Tools may improve the productivity immensely but AI replacing a data scientist or AI scientist is far from reality, at least for now. If you are investing in AI products, basically you are investing in research at the core, missing that important point may cost organisations a lot. The basic core principles or variation of them may help in sustaining AI products longer and form your teams accordingly.

MATLAB/OCTAVE crash tutorial for programmers: Useful bits

In this post, I would like to give you a crash tutorial on the basics of  MATLAB ™ - GNU Octave (MO). They are not fully compatible but quite close in grammar. For differences see here. We will concentrate on the basic data structures and object manipulation that would help you to write useful code quickly.

• Vectors/matrices and Indexing
  Vectors or vectorial operations are the core of MO
  >> A = [ 3 6 7 8 2];>> A(1:3); % range indexing will print values from index 1 to index 3
  >> A(4:end); % Last two values of this vector
  >> A > 4 % logical indexing; this will return logical vector, components satisfying the condition
  0 1 1 1 0
  >> A(A > 4)  % one can use this logical indexing directly to extract elements that satisfy the condition
  6 7 8
  A([2 3 4]) % positional indexing
  logInx = A>4
  A(logInx)
  6 7 8
  Let's define 3 by 3 matrix
  >> M = [ 1 3 4 ; 5 6 9 ; 2 3 4]
  >> M(:,1:2) % first two columns
  
• Structures and Cells
  A structure definition can be performs as follows
  persons=struct();
  Fields can be defined after .
  persons.age=12; Further index can be assigned dynamically
  persons(2).age=32;
  persons(3).age=40; 
  
• Functions
  Return vector with  function definition must be in a seperate file. For example here pov.m (in the working directory)
  
  function [ pov2, pov3] = pov(x)
    ^{pov2= x^2; pov3 = x^3;}
  end
  
  (pwd will tell you which directory you are working in. A return vector here is [pov2, pov3].
  
  
• Function handles and apply functions
  To define new function ff with function handle
  it takes a structure xx and returns its field age value
  
  ff = @(xx) xx.age
  
  We can apply this function to each element of the structure,
  returning an array (vector)
  
  arrayfun(ff, persons)
  
  
• Example Task:
  Lets align two vectors, meaning that we found matching indices and delete non matching ones. Here is the solution:
  
  p1 = [2    11    15    19    28    41    45    47    48    51];
  p2 = [2    11    31    36    41    45    51    60    68    71];
  [~,indexp1] = intersect(p1,p2);
  % if you use ~, it means don't assign anything
  [~,indexp2] = intersect(p2,p1);
  p1 = p1(indexp1);
  p2 = p2(indexp2);

Matlab/Octave: imagesc with variable colorbar

Plotting a discrete matrix with a colour map is a common task that one would need in many different fields of sciences. To plot a discrete matrix with MATLAB, a possibility is to use imagesc, while it handles discrete points better then other density utilities, where it can handle (0,0) placement better. However, if there are limited number of values in the matrix entries, you may want to keep only existing ones on the color labels. One way to achieve this is to re-write your 'colormap' via 'jet', that maps existing colours on the current matrix. However, this is not sufficient, you need to re-write your matrix as well. See example code below. So called a variable 'colormap' situation is handled with this approach. (This is tested with MATLAB, on Octave, I think one needs to be careful on 'colormap', values behaviour of 'jet' is little different.)

% bare minimum - example -
clear all
close all
allLevelNames   = {'one', 'two', 'three', 'four'};
levelValues     = [1 2 3 4];
myMatrix        =  [3 3 3 3; 3 3 3 3; 4 4 1 1; 4 4 1 1];
% Now re-write colormap
aV              = length(levelValues);
jj              = jet(aV);
availableValues = unique(myMatrix);
getIndexes      = arrayfun(@(xx) find(levelValues == xx), availableValues); 
colormap(jj(getIndexes,:)); 
% Re-write myMatrix in the availableValues range 
myMatrix = arrayfun(@(x) find(availableValues == x), myMatrix);
% Now  Plot
h    = imagesc(myMatrix);
% Handle if there is more then one color is in use
maxV = max(availableValues); 
minV = min(availableValues); 
if(length(availableValues) > 1) caxis([minV maxV]); end; 
hcb=colorbar; 
set(hcb, 'YTick', availableValues); 
set(hcb, 'YTickLabel', allLevelNames(getIndexes));
set(gca,'YDir','normal') % Not to invert x-y but putting (0,0) on the bottom left 
xlabel('x values');
ylabel('y values (care on (0,0) location from imagesc)');
title({['Variable colormap handling'];['by msuzen at gmail']});