Matrix Methods In Pattern Recognition: Fill & Download for Free

Graphs are a very flexible form of data representation, and therefore have been applied to machine learning in many different ways in the past. You can take a look to the papers that are submitted to specialized conferences like S+SSPR (The joint IAPR International Workshops on Structural and Syntactic Pattern Recognition and Statistical Techniques in Pattern Recognition) and GBR (Workshop on Graph-based Representations in Pattern Recognition) to start getting a good idea of potential applications. Some examples:Within the Computer Vision field, graphs have been used to extract structure information that can later on be used on several applications, like for instance object recognition and detection, image segmentation and so on.Spectral clustering is an example of clustering method based on graph theory. It makes use of the eigenvalues of the similarity matrix to combine clustering and dimensionality reduction.Random walks may be used to predict and recommend links in social networks or to rank webpages by relevance…

Let's first identify components Combinatoric to know how to be employed in ML and ANNs .Combinatoric arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, and combinatorics also has many applications in mathematical optimization, computer science, ergodic theory and statistical physics.In the later twentieth century, however, powerful and general Theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is graph theory, which also has numerous natural connections to other areas. Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms.-Sub Fields of combinatorics :1- Graph theory > related to information theory.2- Combinatoric optimization > related to operations research, algorithm theory Machine Learning , Image Analysis and ANNs3- Dynamical systems > related to graph dynamical system.Dynamical_systems + ML = RNNs4- Design theory > connections to coding theory and geometric systems5- Probabilistic combinatorics > related to statistical physics and analysis of algorithms in computer science, as well as classical probability, additive and probabilistic number theory, the area recently grew to become an independent field of combinatorics.6- Algebraic combinatorics > related to group theory and representation theoryMachine Learning is closely related to (and often overlaps with) computational statistics, which also focuses on prediction-making through the use of computers. It has strong ties to mathematical optimization,ٍٍSo Machine Learning needs >> advanced study in the field of mathematical optimization on discrete and combinatoric objects. and Optimization is started as a part of combinatorics and graph theory, but is now viewed as a branch of applied mathematics and computer science, related to operations research, algorithm theory and computational complexity theory and using tools from complex analysis and probability theoryand also Combinatoric OptimizationBasically, combinatorics studies countable sets. Probability uses combinatorics to assign probability (value between 0 & 1) to events. Statistics takes sample and compare them to probability models.Those fields of study have massive influence in many other fields. They are key in Machine Learning and Data Science in general.In Fact, Real-world machine learning tasks frequently involve combinatorial structure. How model, infer or predict with graphs, matchings, hierarchies, informative subsets or other discrete structure underlying the data>> ML + Combinatorics = Advance MLArtificial neural networks is The most obvious two I can think of are feature selection and parameter optimization in feed-forward artificial neural networks.In feature selection you’re trying to find an optimal combination of features to use in your dataset from a finite possible selection. Greedy algorithms, meta-heuristics and information gain filtering are all common approaches.Back-propagation is an algorithm used in artificial neural networks to find a near-optimal set of weights/parameters. It’s incredibly effective.There is a use App in this field >>> Integrated automated system for combinatorial data analysis and topological data analysisCombinatoric optimization + neural networks + reinforcement learning (ML + control_systems) = Neural Combinatorial Optimization with Reinforcement LearningReference :Vincent-Philippe Lauzon's blog6.883 Advanced Machine Learning - Learning with Combinatorial Structure[1611.09940] Neural Combinatorial Optimization with Reinforcement LearningCombinatorial (Logical) Data Analysis in Pattern Recognition ...Combinatorial Analysis of Generic Matrix PencilsData Complexity in Pattern Recognitionhttp://mat.uab.es/~alseda/MasterOpt/PotvinSmith_NeuralNetworks-Corrected.pdfhttp://www-2.dc.uba.ar/materias/rn/Aplicaciones/Hopfield/neural-networks-for-combinatorial.pdf

It took me quite some time to understand the difference between discriminative and generative models. There is a nice discussion about the three approaches to modeling in section 1.5.4 in 'Pattern Recognition & Machine Learning' textbook by Christopher Bishop.Given input data point x, the aim is to predict continuous (regression) or discrete (classification) output. That is given x, we are interested in modeling p(y|x). There are three approaches to this:1. Generative Models:One way is to model p(x, y) directly. Once we do that, we can obtain p(y|x) by simply conditioning on x. And we can then use decision theory to determine class membership i.e. we can use loss matrix, etc. to determine which class the point belongs to (such an assignment would minimize the expected loss). For e.g. in Naive Bayes model, you can learn p(y), the prior class probabilities from the data. You can also learn p(x|y) from the data using say maximum likelihood estimation (or you can Bayes estimator, if you will). Once you have p(y) and p(x|y), p(x, y) is not difficult to find out.2. Discriminative Models:Instead of modeling p(x, y), we can directly model p(y|x), for e.g. in logistic regression p(y|x) is assumed to be of the form 1 / (1 + exp(-sigma(wi. xi))). All we have to do in such a case is to learn weights that would minimize the squared loss.3. Encoding a Function:We find a function f(.) that directly maps x to a class. Decision trees do that.There are pros and cons of such methods. First notice that generative models encompass discriminative models (i.e. you can always obtain p(y|x) from p(x, y)) and that means generative modeling must be more expensive - you would need to assume the form of p(x|y) or p(x, y) - for e.g. in case of mixture of two Gaussians you would need to model parameters corresponding to mean and variances of two Gaussians. However if your aim is to simply classify which class i.e. Gaussian the point belongs to, then you need only find the class separating decision boundary (which is what logistic regression does and thus has fewer parameters). And what if the data did not come from mixture of Gaussians but from mixture of something else for which logistic regression can still find reasonable separating boundary? You can read famous paper by Michael Jordan on trade off between Generative and Discriminative models here: http://www.ics.uci.edu/~smyth/courses/cs274/readings/jordan_logistic.pdfAlso, once you model p(x, y) then you can generate synthetic data points by sampling from p(x, y) - not possible in latter two methods. You can also generate points with low probability p(x) which would be otherwise difficult to observe.There are several other trade offs, such as what if prior probabilities change or if the change is made to the loss matrix. In that case discriminative model would need to be learnt again, not so much of a problem with generative modeling.I would suggest you read the suggested section from PRML book. Most of the above is borrowed from the book.