Thank you very much for the renewed invitation to speak in this beautiful series of seminars. So I will essentially read my notes and start by defining, so I will speak on these bounded generation linear groups and exponential parametrizations since the object of essentially two papers with the authors that have written down, Pietro Corvaya, Julian de Mayo, Andrea Pinchuk and Gene Borelli. So I shall start by defining bounded generation, which is a notion that can be given in any abstract group and it may be seen as a strong form of finite generation and the precise definition is that the group gamma is said to be boundedly generated if there exists gamma 1, gamma r in gamma such that gamma is, so each element of gamma may be written as a product of powers in order powers of gamma 1 times the power of gamma 2 times the power of gamma r. Of course this coincides with the finite generation if the group is commutative but non-commutativity is very essential in this context. So the bounded generator, so let me remark they need not be distinct and the terminology is there is also an alternative one called finite cyclic width and this number r, the minimum such number is called the cyclic width sometimes. Let me give some examples. So every virtually a billion or virtually nilpotent group has bounded generation even only if it has finite generation. By virtually a billion for instance I mean that the group has an a billion subgroup of finite index, so virtually we'll denote this property of having a subgroup with the property and being of finite index. So here is a criterion to construct new examples of bounded generation from known ones. If we have an extension of the groups g, an extension of h by n, so an exact sequence like displayed there, if n and h have bounded generation then the same holds for g. But generally not the converse. And in particular a solvable group has bounded generation provided however all pieces in solvability filtration are finite regenerated. And however warning here there are solvable groups already inside sl2 which are finitely generated but not bounded regenerated. We give an example in one of the papers that I mentioned. Another example result by MULCHER it follows that every solvable subgroup of glmz has bounded generation so in the context of glmz the solvability suffices. Our context is indeed near this context of glmz because we shall be concerned with linear groups so called for me a linear group it means a subgroup of g of k so the k rational points of an algebraic group g for some field k. And that we shall always assume that the field is of a characteristic zero and usually it will be a number field. And typically the entries of the matrices so we have a group of matrices and that the entries shall be restricted to lying some subring of the field usually. For instance the ring of integers of the number field k. And this context is a context of the so called arithmetic groups. Let me give now some examples which show why we can be interested in bounded generation. This is because the bounded generation property has been found to have several relevant consequences toward many other independent questions on linear groups. For instance here are some theorems some results for instance are a pinchuk in 1990 found that the bounded generation has strong implications toward the so called SS rigidity which is an important property of linear groups which means the finalness of completely reducible complex representation in any given dimension. This also has implication in the dimension of the character so called character varieties I omit precise statements for time reasons for instance. And then we may mention a work of Lubotsky and the Platon of a pinchuk in 92 there are two series of independent works who proved that the congruence subgroup property which is a famous series of problems the congruence subgroup property regards for instance subgroup solve gl and z whether the every finite index subgroup is a congruent subgroup. And there is a generalization for arithmetic groups and this property this important congruent subgroup property was deduced from bounded generation in some cases by the works that I mentioned. Shalom Willis used the crucially this bounded generation for proving the so called Margolis Zimmer conjecture this is again technical it has to do with commensurability of subgroups of arithmetic groups of Chevrolet groups so but it was a completely independent problem and it was found that bounded generation had influence on this and the further implications appear work of Avni, Lubotsky and Mayer. Let me now give some simple examples but where it holds and not for instance bounded generation does not hold for the perhaps the simplest arithmetic group non commutative SL2Z in fact this group contains a noncyclic free group of finite index for instance the group gamma two which is a subgroup of index two of gamma of two and it is not difficult to see that this fact that it contains a noncyclic free group excludes the bounded generation. So, however, it is somewhat surprising that it does not hold for SL2Z but already for and at least three Carter Keller proved in 1983 for the bounded generation for all SLNO so the points with coordinates in a given ring of integers of number field and actually they prove the bounded generation using only elementary matrices in particular unipotent matrices this distinction will be relevant in the following in the sequel and let me remark that for n equal to anyway the property holds not for SL2Z but for other rings of integers and it holds as soon as the ring of integer has an infinity unit group so we may say almost always so as shown by Morgana Pinchuk and Suri in 2018. Now let me pause a bit on parametrizations which is another aspect of my talk and its connection with bounded generation so let us observe that bounded generation by unipotent matrices allows in particular to parametrize polynomially SLNO for n at least three. What do I mean with this in the sense that there is a polynomial map which is subjective from some power of the ring into two SLONTO SLNO namely we have a matrix in SLNO whose entries are polynomials in some number of variables so we may substitute integers in all in place of the variables and we obtain elements of SLNO and we may obtain we may find the polynomials so as to obtain every element this is easy after the theorem of Carter Keller because we observe that if gamma is a unipotent matrix the entries of the powers of this matrix are polynomials in the variable in the in the exponent and hence if we have bounded generation we need only a finite bounded number of variables to parameterize. This parametrization property is a kind of the euphantine property which can be of independent motivation and interest. Indeed for instance there was a question of Scholem going back to long ago who asked already for n equal to whether SL2z was parametrizable by polynomials and there were some negative answers in few small number of variables but surprisingly Wasserstein in 2010 approved that indeed gave a counter example to the expectation by proving that in sufficiently many variables we can parametrize SL2z by polynomials and then Lars and Gouyen recently they extended the thing to rings of integers other than z and this is remarkable because this shows in particular that whereas bounded generation implies the polynomial parametrization if the bounded generation occurs with unipotence the converse does not fold because we have remarked above that SL2z is not boundedly generated so there is there are connections between these parametrizations and the connections but not double implications so let us note now that what I say the work for unipotent matrices but if we also have some simple matrices in a in a bounded generation we still obtain parametrization but not anymore by polynomials but we have to introduce also exponential polynomials and the other extreme we have so-called purely exponential parametrizations if we have only semi-simple elements in a bounded generation we shall meet this restriction also below and these observations are easy in fact on diagonalizing a semi-simple matrix we see that the entries of the power are linear forms in the powers of the eigenvalues and if the matter is semi-simple so we find exponentials and not polynomials so let us go back to bounded generation further examples of bounded generation were found for linear groups the result of Carter-Teller on SLN was extended by Tavgen in 1991 to all Chevrolet groups of rank greater than one and to most so-called quasi-split groups which has to do with Borrel sub-groups I will skip the definitions so anyway these results and some other ones raised the expectation that bounded generation would hold in even greater generality let me comment again on the dichotomy unipotent and semi-simple bounded generation in many of the quoted results for instance for SLN bounded generation occurs with the unipotent but we would like also to allow semi-simple elements for other potential applications of the bounded generation property and with this in mind I shall call an isotropic any subgroup gamma of GLN k that contains only semi-simple elements and I shall for the moment think of the bounded generation for such a group this restriction is relevant because these groups there are very interesting groups which are an isotropic here are some examples that is first the most easy we can call it trivial case is constituted by virtually a billion groups generated by semi-simple matrices so I said this is trivial case one quaternions of norm one over a certain ring of s integers and more generally the same with quaternions replaced with other division algebras and the other series of interesting examples come from orthogonal groups of quadratic forms not represented zero these these forms are called an isotropic and this is the origin most probably of the terminology an isotropic for the groups that I that I am considering for me an isotropic group is a group I repeat containing only semi-simple matrices one gets many examples of orthogonal groups working in rings such as z allowing some denominators some ring ring of s integers we have remarked before that a group with bounded generation by semi-simple elements may be parameterized by purely exponential polynomials and let me give formally the definition of a purely exponential polynomial so you see it is a linear combination of with constant coefficients of exponentials of free where the bases are fixed and the exponents are linear forms in our variables supposed to take integer values so we have the dichotomy that I have already mentioned so polynomials on one side and exponential polynomials on the other side this dichotomy appears here in discussing this property but it appears in the euphanta in theory for instance it may remind of the Hilbert-Tent problem and Matiasiewicz made the last step for proving the undecidability raised by Hilbert by reducing a certain exponential day of hunting equation to usual day of hunting equation and the same dichotomy appears for instance already with the integral points on curves of genus 0 if we have an affine curve so where we seek for integral points if it has one point at infinity we may find infinitely many integral points parameterized by polynomials whereas if there are two points at infinity there are when there are infinitely many integral points they may be parameterized by exponentials and with three points we have only finitely many integral points with three points at infinity so this dichotomy appeared already in the euphanta in theory and it appears here again and despite the behavior so polynomials being quite different from the exponential polynomial there was expectations somewhat that some non-trivial finitely generated that anisotropic groups could be found over rings of S integers and satisfying the bounded generation however no such example was found and starting this problem with Corvaya-Rapinchuk and Reng in 2021 we found strong limitations to bounded generation in linear finitely generated anisotropic groups and so let me fix some notation so k any field of characteristic 0 and gamma in glnk be a linear group we have this theorem that gamma is boundedly generated by semi-simple elements so it need not be an isotropic but it may be boundedly generated by semi-simple elements then it is virtually solvable so it has a solvable subgroup of finite index let me make some remarks on this one may a little bit generalized the theorem by allowing not only semi-simple elements in a bounded generation but one may allow a single non-semi-simple generator and one keeps the same conclusion and this supplementary result costs some effort and indeed already allowing work there is some effort in the arguments and an open question which is the maximum number of such elements that we can allow in order to keep the conclusion the question is also motivated by examples showing that we can't allow more than four for instance a cell two over certain rings of s integers inside z can be generated by by by five non-semi-simple elements so and non-semi-simple elements lead to ordinary day of fronting equations so to usual polynomials so this explains somewhat the difficulty and they might even lead to undecidable issues so another remark is that there exists a virtually solvable finitely generated linear groups without bounded generation so a pure converse of the theorem does not hold so and a consequence regards another question which was open for some time regarding a profiled notion of bounded generation there is indeed a profiled version of the concept I skip the precise definition but you may imagine them it uses the topology of a profiled group and it works the same way it may be proved that if gamma hat denotes the profiled completion of a group gamma then if gamma is boundedly generated then gamma hat is boundedly generated in the profiled sense and it was an open question where whether the converse held and the above theorem leads to a negative answer providing examples of finitely generated groups gamma without bounded generation but such that the profiled bounded generation instead holds assuming moreover that gamma hat is so-called a residual finite which amounts to the topology of the completion being housed so to obtain this example this negative answer to the question I will not be able to do this because it uses a number of results by Platonov and Pinchuk and it uses the following corollary of the above theorem the above theorem I recall it works for every bounded generation by semi-simple elements without requiring that the group is an isotropic but if we also insist that the group is an isotropic then we have a stronger conclusion it has bounded generation if and only if it is finitely generated and it is virtually a billion this corollary is a quick consequence of the previous theorem it may be proved because a quick consequence given taken for granted some general theory because some general theory proved that a virtually solvable for a 9-9 isotropic group implies virtually a billion without disregarding bounded generation it is a general property so this virtually a billion is the most trivial case of bounded generation and I profit of this corollary for recalling we were calling zero characteristic but in positive characteristic indeed it was shown by Abert Lubbocki-Piber in 2003 that bounded generation implies virtually a billion without further conditions and they use completely different methods and conversely our methods would not apply without as they stand to their context let me add with further results in work also with Julian de Mayo and by means of a partially different method so related to the previous one but in fact different we obtained more precise conclusions concerning not only bounded generation but concerning also purely exponential parametrizations so which we have remarked that bounded generation implies either polynomial or exponential parametrization depending on the nature of the bounded generation either by important or semi-simple matrices but we have also remarked that the converse would not hold so there are groups which are like SL2Z which admit a polynomial parametrization but are not boundedly generated whereas for purely exponential parametrization the the situation is different is different we have the following theorem we Corvaya de Mayo, Rapinchuk and Ren and we have three equivalent condition that gamma has a purely exponential parametrization that gamma is an isotropic and has bounded generation so in this case purely exponential parametrization implies bounded generation and the third equivalence is that gamma is finitely generated and the identity component of its Zariski closure is a torus and this in particular essentially implies that gamma is virtually a P and I call a usual polynomial parametrization does not generate a bounded generation so this result this theorem which is more precise than the previous one is a consequence of sparseness of sets obtained from a purely exponential parametrization this feature of sparseness did not directly appear in the arguments for the former result let me mention a bit what I mean for sparseness you'll meet detail for time reasons and this sparseness is expressed by estimates related to heights so I use the capital according to paulaco and the redhoff notation the capital height means the exponential and the small lower case height means the logarithmic veil affine heights so the estimates have this shape we bounded the number of elements in gamma of height exponential height at most t and coming from a purely exponential parametrization we estimate them by a power of the logarithm of t and as is natural to expect somewhat but it's not clear a priori because we could have exponential polynomials in many variables producing small values so such estimates may be maybe even turned into an asymptotic formula so that the number of elements of height at most t are asymptotic to a constant time a power of the logarithm for some exponent less or equal than the number of variables these more precise information may be not entirely free of independent interest but it is not strictly needed for the present applications so the estimate looked natural but a priori we could have large in a large variables producing small values for the exponential polynomials polynomials so we could have many many values of the exponential polynomials even if the variables are large similar estimates were produced in the past by other authors let me mention just Everest and Linsky in 1999 but they worked with restrictions on the exponential polynomials and these restrictions would prevent the applications that we need the estimates are essentially derived from the following lower bound we let e be a purely exponential polynomial and I call a vector of integers minimal for the polynomial if its absolute value if it's normally clear the norm is minimal among all the vectors such that the polynomial assumes at those vectors the same value as at the vector x so the point is that the polynomial could assume the same value for a whole bunch of integral points and so we cannot expect to have a lower bound for the value the height of the value in terms of the height of the vector of variables but this lower bound exists if we restrict to minimal vectors there is a constant positive constant such that the logarithmic height of the value of the polynomial is greater than the constant times the norm of the vector for all minimal vectors except a finite set of values so not except the finite set of vectors but a finite set of values of the values of the polynomials so this condition of minimality is easily seen to be necessary for the conclusions and this estimate is the essential estimate to derive the previously mentioned estimates and let us compare these upper estimates with some lower bounds so by looking at the total elements number of elements in the group with the given bound on the height so the ones coming from a polynomial purely exponential polynomial parameterization there are at most the power of the logarithm of the height let me mention some results which say that on the contrary if we look at all the elements of the group we have many more so let us restrict to s arithmetic groups so of the shape s integral points of the of g where g is an algebraic subgroup of some glm and always is the ring of s integers in a number field in general with mild assumptions that I will skip but they really mild we have estimates that show that the number of elements of height at most t in our arithmetic group have polynomial growth so more than t to the delta for some delta greater than zero these estimates go back to seagull bail and others in some important cases related for instance to quadratic forms orthogonal groups of quadratic forms seagull wrote a number of paper and there was much research on this in our paper we have not really new results on this but we put together a number of separate the results obtained until recently and these lower bounds are not strictly needed for the above theorem but they illustrated the sparseness so they illustrated that the number of elements in the group of interest for us the number of elements which we may obtain from an exponential polynomial parameterization and so from a bounded generator from a bounded generating set are really really very few compared to the whole and so the bounded generated subsets are very small and unfortunately the group is not boundedly generated so let me employ the last few minutes of the talk by so serving to some aspects of some of the proofs let me think of the first theorem one so that i restated bounded generation by semi simple elements implies that the group is virtually solvable and we have to prove this for all linear groups in over any field of characteristic zero and we assume by contradiction that gamma is not virtually solvable but it has bounded generation with semi simple elements and the first point is to reduce to linear groups of a number field so from a field of characteristic zero this is a common practice here there are some features that do not appear in other in other contexts so by specialization it is easy to find the specialization which maintain the second property so which maintain the semi simplicity of the elements this is easy it is a little bit more intricate to maintain the fact that gamma is not virtually solvable because the the derived series could be a priority very long but one instead uses the uniform boundedness of the length of the derived series which produces solvability and and it is known this is known inside the glm and the ones that we have this uniform boundedness one can prove the existence of good specializations which preserve this effect so this kind of reduction for theorem one has alternatives indeed because then we use other theorems of the euphantine type that are known to hold in for arbitrary fields of zero characteristic and that they are even easier to prove over fields which are truly not a number fields but this reduction becomes more more essential for the second theorem because the second theorem depends for number fields it depends on estimates involving heights and we haven't very easy notions of heights in fields which are not number fields there are notions of course even in function fields but these notions do not have the good properties that we have on number fields and so to work with heights in general fields would be difficult probably and so the reduction to number fields is to be preferred especially for the second theorem so another feature is existence of multiplicatively independent eigenvalues so to state this we let gamma one gamma are bounded generators we assume that they exist and we assume that gamma is not virtually solvable and then to use this hypothesis we take the quotient g prime g over the radical so called and the radical of g of the component identity component of g and the working in g prime it is not difficult to see that we may further assume that g is g zero is the non-trivial semi-simple group so with trivial radical and at this point we use the theory of Prasad and Rappinchuk of generic elements and one that uses the existence of an element of the group such that the eigenvalues of the element are multiplicatively independent from those of the given hypothetical bounded generators and we mean that the respective eigenvalues generate subgroups of q star with trivial intersection or q bar star with trivial intersection and here we use for this the theory of Prasad and Rappinchuk and I would like to remark that this multiplicative independence of values of rational functions because after all these eigenvalues may be seen as a rational functions perhaps not on the group itself but on a finite cover of the group on a variety which is a finite cover and in a completely independent context especially of unlikely intersections this multiplicative independence of values has been studied and in this view the result of Prasad and Rappinchuk is not unrelated to theorems obtained jointly with Bombardian master and later refined by Moran and by other authors and so I wanted to mention this because presumably that could be a useful and interesting interplay between the theory of Prasad and Rappinchuk and the unlikely intersection theory of multiplicative independence I hope that someone perhaps undertakes this analysis so to sum up we have obtained the matrix having an eigenvalue not a root of unity and such that denoting by mu 1 mu s the eigenvalues of the hypothetical bounded semi-simple generators we have that the corresponding groups have trivial intersections and we may assume that all of these eigenvalues lie in a given number field and now the conclusion at least for the first theorem is a simple application of the theory of integral points on sub-varieties of Torai gm to the air we have just to so this theory is the so-called mortal land context in the somewhat easiest case of Toric which was known before the general mortal land and we exploit this theory and we want to prove that not all the powers of the matrix gamma that we have constructed lie may be expressed as products of powers of the given matrices gamma 1 gamma r products of powers in this order however and assuming the contrary for each m in z there would exist exponents depending on m such that the displayed equation holds and by diagonalization we derive from this equation certain exponential equations like the one which is displayed where q is a fixed polynomials and the news are fixed monomials and the bias are integers and I can now conclude and skip the rest this equation these kind of equations give give a variety a variety defined in the in the torus gm to the t plus one defined by the displayed equation y equal q of x1 x t and the point belongs to a finitely generated subgroup of the torus generated by these points and now we use the theorem of Laurent which is essentially the mortal land statement for the toric case if we have a finitely generated subgroup of some power of gm and if we have sigma be any subset then that's the risk closure of this subset has a very special shape it is a finite union of translates of algebraic subgroups of gm to dm so this result goes back to Laurent but it goes back actually also to produce a result by Everts of underportage liquify and everything depends on the subspace theorem of Schmidt which in turn is a higher dimensional version of the theorem of Laurent and the algebraic subgroups of gm to dm are defined by finitely many binomial equation as of the type x1 to the a1 xn to the am equal to 1 in integers and this yields a very explicit application of the theorem and to conclude with this result in practice we apply the result with the variety that we have obtained and the points in it and we deduce a contradiction with the multiplicative independence of the eigenvalue which we had found with the theory of Prasadra Finchuk so this concludes the argument and I would stop here.


Today I will be talking about fuzzy C means algorithm, I will be talking about fuzzy C means algorithm, before that let me just say little bit about soft computing techniques where fuzzy sets is one of the components of soft computing techniques. Soft computing techniques were first introduced by someone named Lot Fijade in 1965, actually he introduced fuzzy sets in his famous paper which was published in information sciences. Fuzzy sets is a variation from the ordinary sets, note that in ordinary sets if you have a set A and if you have a point x we know that either x belongs to A or x does not belong to A, only one of them is true, another way of saying it is you have an indicator function of the set A where this Iax is equal to 0 if x does not belong to A and 1 if x belongs to A, I is said to be indicator function of the set indicator function, Jade defined fuzzy sets as you have a set A with the membership function value membership function which is mu for a fuzzy set A, it takes values not only 0 and 1 but it can also take values in between 0 and 1 that is this is known as membership function membership function and mu A x takes values in the interval 0 to 1 not only 0 or 1 it may take a value in between, there are many examples one example is suppose you want to look at the set of all let us say tall persons and then if someone height is let us just say 5 feet 5 inches and would you just call him tall probably all the persons whose height is more than 5 feet 7 inches you may call them tall and probably all the persons whose height is less than let us just say 5 feet 2 inches you may call them not all but from 5 feet 2 inches to 5 feet 7 inches you have you may not have a clear cut idea of what exactly is tall probably let us say this is 155 centimeters and then say it is 64 let us just say 166 centimeters just I have taken some values so and here I am writing membership function say this is 0 then say this is 1 if the height is 5155 centimeters probably you may not want to call the person as tall if it is 166 centimeters surely you would like to call the person as tall and anything greater than 166 you would like to call the person as tall anything less than or equal to 155 it is surely not all that is this one and between 155 to 166 probably you have like this and this is called membership function the membership for the attribute tall is will be increasing and it will go to one this many times in reality we use adjectives without clear cut mathematical formulations without clear cut mathematical formulation exact and precise mathematical formulation is not there and by using those adjectives when I speak to you or when you speak to me when you use those adjectives or when I use those adjectives we understand each other when we are able to understand each other and we would like to have we would like to have our computer also to understand these words we would like to have the computer also to understand these words that is the basis for soft computing I mean that is the basis for fuzzy set theoretic that is the basis for fuzzy sets it is also the basis for what is known as fuzzy logic what is known as fuzzy logic in the usual binary valued logic that is 0 or 1 a statement is either true or it is not true but in fuzzy logic that is a statement is true with some membership value let us just say 0.5 0.6 0.7 and it is not true with some membership value point again with some membership like that you can have I mean the membership functions in many many problems which is actually the case in reality in reality we do not always make mathematically precise statements in fact many times most of the times we do not make mathematically precise statements even then we understand each other then we would like to have our computer also to understand the language which we are speaking then the come then the analysis should be done in such a way that this understanding is possible so one of the ways that the suggested is to enter to use these membership values in the analysis so there are very many developments after fuzzy sets neural networks is considered to be a part of soft computing multi-layer perceptrons and radial basis function networks you have principle component analysis networks these are considered to be part of soft computing genetic algorithms are considered to be part of soft computing rough sets are considered to be part of soft computing soft computing actually means that we can have input imprecise we can have ambiguous input and sometimes even incorrect see when we discuss many things sometimes we make incorrect statements but even then we understand each other always whenever I mean the basic significance of all these things is we would like to make our computer to understand and to behave in such a way as a human being since we sometimes make incorrect statements still we understand each other then how do you make the computer also understand these sort of things basically with this brief introduction to soft computing like fuzzy sets neural network genetic algorithms and rough sets okay I am not going to discuss all these things in this class I will I have only talked about fuzzy sets because I will be using them in the fuzzy C means algorithm fuzzy sets means you basically have some membership values you basically have some membership values and the membership values are used in several several applications and that is the reason why I introduced fuzzy sets a set a is he said to be a fuzzy set if the membership values they lie between 0 and 1 membership values they lie between 0 and 1 and there is at least one point for which the membership value is strictly in between 0 and 1 if all the membership values are either 0s or 1s then it is going to be called as a crisp set if all the membership values are either 0s or 1s and nothing in between 0 and 1 then the set is called as a crisp set fuzzy set means there exist at least one membership value which lies strictly in between 0 and 1 it is neither 0 nor 1 at least one x for which Mu x is strictly in between 0 and 1 so that is fuzzy sets now using this fuzzy sets we can also discuss what is known as fuzzy partition fuzzy C partition fuzzy C partition first let me tell you the meaning of ordinary partition that we all know but I will repeat it again then we will go for fuzzy C partition now the usual C partition C is greater than or equal to 2 and C is an integer usual C partition of a set S is let us just say x1 to xn let us say say say S is x1 to xn is what it is like this usual C partition of a set S represented as let me just call it P is for partition a1 a2 a C you need to give partition of S into C subsets partition of S into C subsets C is the number of subsets okay is what represented as this and the definition is what is the definition definition is first each AI is not is equal to 5 right and AI intersection AJ equal to 5 right and union partition of S into C subsets that is a1 a2 a C if I represent the C subsets as a1 a2 a C then the definition is that each AI should be not is equal to 5 and AI intersection AJ is 5 and union AI 1 to C is equal to S so this is a usual C partition now what is fuzzy C partition now fuzzy C partition partition of S basically you need to give membership values basically you need to give membership values fuzzy C partition of S represented by represented by U is U S S is the set okay where U is n by C matrix where U is an n by C matrix U is equal to so I throw this column element is written as Uij okay where Uij denotes membership value membership value of ith point the jth set jth fuzzy set jth fuzzy set naturally the number of points is n so I lies between 1 to n and C1 less than or equal to j less than or equal to C fuzzy C partition of S represented by U S where U is an n by C matrix Uij denotes membership value of ith point to the jth okay satisfying the following properties satisfying the properties stated I do not I do not want to write below because I need to write there so satisfying the properties stated let me just try it stated here just one minute I will erase this portion shall I start the first property is naturally 0 less than or equal to Uij less than or equal to 1 for all ij membership values to summation Uij j is equal to 1 to C this means you have taken the ith point you have fixed the ith point it is membership for the first set is UI1 membership for the second set is UI2 membership for the jth set is UIC membership for the cth set is UIC so this submission it is equal to 1 okay its summation is equal to 1 in the usual partition if a point x belongs to one of the sets it is not going to belong to any other set so for that point if you sum up all the membership values one place you will get 1 and at all other places you are going to get 0 so summation is 1 here the summation is 1 but it is not necessarily true that exactly at one place you here you have 1 and at other places you have 0 you may get some fractional values here 3 3 is summation i is equal to 1 to n Uij that is you are fixing the set now and you are looking at the sum of all the membership values for a particular set it should be strictly greater than 0 it should be strictly less than n the corresponding thing here in this properties is AI is not is equal to 5 for all I that means every set has at least one point if a set has no point then all the membership values will be 0 then the summation also will be 0 right if a set has no points all the membership values will be 0 then the summation is also 0 so this is actually an extension of this property this is actually an extension of this property so if Uij satisfies all these three properties then you say that that is a fuzzy C partition of the data set that is a fuzzy C partition of the data set now when we have usual partition what did we do we have defined an objective function if you remember for the C means or K means algorithm when we did it we defined an objective function what was the objective function the objective function is you have this partition and we defined Vi mean of AI I is equal to 1 to C we have got a partition for this partition for every I mean of AI okay is defined and then what did we do we looked at this double summation of X- suppose we have this set S and this set S is a subset of m dimensional space S is a subset of m dimensional space Euclidean space then we can talk about mean of AI some of all the points in AI divided by the number of points in AI you are going to get mean of AI then this is objective function what is this objective function for each point X we just look at which cluster it belongs to it belongs to the cluster say AI then for that point X you look at the mean of that cluster the mean is Vi find out the distance between X and this Vi and then take the square of the distance that you do it for every X and AI and I is equal to 1 to C so this is basically the within cluster distance that partition that which provides the minimum within cluster distance is taken as the best partition that partition that provides the minimum within cluster distance that is taken as the best partition so basically this is to be minimized over all these partitions this is to be minimized over all the partitions this is the minimum within cluster distance criterion and the usual C means or K means algorithm I am using both these words C means and K means the reason is that when this algorithm was proposed sometime in 60s it was proposed as K means algorithm and that is what people have been using it and fuzzy set theory people they devised this algorithm that was in late 70s and early 80s late 70s and early 80s by best debt and he used C for the number of clusters he used to see for the number of clusters so number of clusters C K fine I mean you give some name for the number of clusters and then you devise the algorithm so CRK there is a number of clusters and here C is the number of clusters so it is C means algorithm now it is the minimum within cluster distance criterion is this and since we are not in a position to get the minimum within cluster distance since we need to do the search over the number of such partitions is too many so they are the algorithm K means the C means algorithm was devised the C means algorithm was devised here also in the fuzzy set theory setup also we have defined a partition and there is an objective function what is the objective function the objective function here is the following we take a parameter let R be greater than or equal to get strictly greater than 1 let R be strictly greater than 1 let V I that is mean for the ith cluster is summation X I into sorry let me write it as J J is for the clusters so V J is summation X I X I multiplied by U I J whole to the power of R we are multiplying X I by U I J whole to the power of R if U I Js are either 0s or 1s the place when it has 1 the corresponding excise will be summed divided by the number of such points the place where U I Js are 1 if you just sum up all those things that will give you the number of points in that cluster that is U I J to the power of R summation that will give you the number of points and this is summation of excise divided by the number of point so that is mean summation of excise divided by the number of points so that is mean this is the mean now objective function the objective function is dependent upon this value R objective function if you start with if you have a fuzzy C partition U of U of a set S and R is with respect to the parameter R what is this one this is actually going to be U I J to the power R norm of actually X I – V J prime a and let me just write it like this U S a where a is a positive definite matrix a is a positive definite matrix note that it should have m rows and m columns note that it should have m rows and m columns if a is identity matrix and U I Js are either zeros or ones then actually what you are going to get is this let me repeat it if a is identity matrix and U I Js are either zeros or ones and nothing in between then this summation is same as what I wrote here so this is we are looking at it in a much generalized setup first this U it is not either zeros or ones something in between and we introduced a positive definite matrix and there is an R so this is to be minimized for a given a and S given a S and R you need to get a U which minimizes this that is the problem formulation I will write it here given S the data set a the positive definite matrix and R one needs to find the optimal U which minimizes J R U given S is a data set a the positive definite matrix and R the exponent term R is the exponent term one needs to find the optimal U which minimizes J R U S U is the membership value matrix S is the data set and a is the positive definite matrix so till now we have been discussing the problem formulation in the fuzzy seems algorithm we have defined an objective function J we have defined an objective function J this function depends for every fuzzy see partition U of the set S and for a given R greater than one and for a given positive definite matrix a the objective function value is given by this expression we are supposed to find the best U given S a and R one needs to find the optimal U or the best U best from the point of view of that U which minimizes J R U S a now how does one do it before I go into the algorithm part I would like to mention a few things this algorithm was first proposed by James Bestek a very famous figure and a proof was also given to the algorithm that it converges and then the proof was modified and then a modified proof was given but then ultimately after a I should say a few iterations ultimately the correct proof was found the algorithm was same throughout in all these things the proof needs knowledge quite a bit of knowledge of mathematics including topology and other such fields I will not go into the proof of the algorithm I will basically tell you the steps of the algorithm so this is FCM algorithm F for fuzzy C means fuzzy C means algorithm what are the steps first one is we are given the set S which has smaller number of points and it is a subset of RMM dimensional space we are given the positive definite matrix A and then exponent R I am saying that we are given okay usually people choose the value of R some R greater than 1 usually people choose the value of R something that is greater than 1 and people choose it as something which is very close to 1 from 1.01 some people may choose to also but this value is generally chosen by users and of course the number of clusters C the C is same as the C we will start with a fuzzy C partition U of S so once you is there you can always calculate the means VJs what is the expression for mean the expression for mean is this this is the expression for mean so you can always calculate this you can always calculate this expression so you can always calculate VJs And once you calculate vj's you can calculate uij's how the expression for uij is this summation k is equal to 1 to c of dij square by dik square k is equal to 1 this k is 1 to c this thing to the power 1 by r minus 1 and hold to the power of minus 1 here r is greater than 1 that is used then you can write r minus 1 what are this dij square and dik square actual k is equal to 1 to c and dij square is this portion dij square is this portion that is you understand why it is written as d d is the distance distance of the ith point to the jth mean that is dij square this is dij square in the numerator and the denominator dik square k is equal to 1 to c this thing hold to the power 1 by r minus 1 this hold to the power of minus 1 okay so you compute given you you are computing v's from their u and from you again v then again you and again v the fifth step is go to 3 the convergence criterion is not satisfied go to 3 if the convergence criterion is not satisfied what is the convergence criterion you have now a previous fuzzy C partition and present fuzzy C partition to fuzzy C partitions you take the difference that is again a matrix okay and then you take the difference take the modulus values if each of them is say less than say some epsilon then you stop it otherwise you do this you can so you can just go on and on and on doing till that is satisfied you can have a difference stopping criterion also here I use the stopping criterion on use you can do the same thing on v's also if two consecutive set of vectors means the previous mean and the present mean of the same cluster if the distance between them is less than epsilon and if that happens for all the C clusters then also you can stop it you do that or you do this does not matter you just go on and on doing this so convergence for the convergence criterion you need to have a small value epsilon and you need to calculate the difference between either the two fuzzy C partitions or the two C mean vectors right either you have to calculate the difference between the two fuzzy C partitions consecutive two fuzzy C partitions are consecutive two fuzzy C means if the each difference is a less than epsilon then you stop it otherwise you go to that calculate the next just go on doing this and the convergence theorem says that this process really converges that is if you go on and on doing this thing then you will surely I mean ultimately the UAJs the previous UAJ the present UAJ they will become same or the means also the previous mean in the present mean they will become same ultimately the differences they will go towards 0 that is basically the convergence theorem now the next point is what sort of clusters are we going to get right so we have got an algorithm and we can use it and it is converges so you get a you right what you may do is that since it is fuzzy clustering and you are ultimately interested in getting the ordinary clustering you can proceed in a few ways one way is that all the membership values which are greater than 0.5 okay or for each point what you can do is that find the maximal membership value to whichever cluster it goes give their one okay give that thing one and the rest as 0 this is a way which is followed by many okay in that way you will get a crisp clustering that is one thing that you can do it you can also do one thing you can keep the fuzziness as it is many times you need overlapping clusters what this fuzzy seems algorithm does is ultimately it provides you overlapping clusters okay because the same point has membership to more than one class so what you are ultimately going to get is overlapping clusters now when you get overlapping clusters in many situations overlapping clusters are important why they are important you do not want to say that it is always this cluster or always that cluster maybe you do really have I mean that point may be actually put in more than one cluster why do you want to force it to go to only one cluster maybe keeping the information that it can go to both the clusters may help you these sort of situations they arise many times in I will tell you one example where this sort of situation arises many times these things they occur in the satellite image processing when you have taken when you take the image of a place from the satellite say the resolution is say let us just say even if it is one meter resolution suppose you take the bridge on the Cauvery river okay the photograph is taken from above so you see the bridge and under that there is water now that those pixels they can go to either water or they can go to bridge they can go to concrete structures or they can go to water you may want to keep these two different two clusters information intact you may not force it to go to always to either water or to I mean either the water or to the other one other cluster the reason is that if someone is interested in crossing the river by boat then he would want the information that this is water if you are interested in crossing the river by railway line and by using railway line then you need the bridge so why do you force it to go to either always this or to that you may want to keep this overlapping information intact so that wherever you need that part then you may want to keep it there are you understanding me you need not force it always to go to one of the clusters you need not force it to go to always one of the clusters keep it as it is so that only when ultimately you are forced to do a classification are forced to do clustering depending on the surrounding depending on some information then okay in this depending under these circumstances you would like to use this water so we use the water these sort of things are extremely necessary in very many real life problems in very many real life problems for the same thing you may have more than one alternative you may want to keep both the alternatives intact you may not force yourself to go to only one of them so that exactly at the right at the right time you will decide which alternative you need to take okay in fact this is the reason for I mean fuzzy set theory based classification or fuzzy set theory based clustering is has become popular because of this particular reason you may you may not always want to force it to go to one of the clusters or one of the classes you may do it at the right point of time depending on the situation there okay so you can keep the overlapping information intact so that you will use it for future purposes depending on the problem and thirdly there is one more thing what sort of clusters are we going to get they are essentially again convex sort of clusters if if this a is the usual identity matrix okay then you will get basically convex sort of clusters again the problems regarding non-convexity and other things they need to be properly addressed in this context with this I stop okay do you have questions how to choose the value of our exponent I do not want to always I do not want to make a strong statement that R should be chosen always in this way or in that way I do not want to make some such statement what I can say is that people choose are sometimes many persons choose are in some some particular ways for example someone may choose R as something very close to one someone 0.0001 because you may want to get something very close to the usual came in algorithm as you can see if R is actually equal to 1 then this is like came in algorithm right the usual came in algorithm so that is why one of the reasons that people choose R to be very close to one is this they want to make this thing to be as close to came in as possible the usual came in algorithm so they may choose R to be a value that is very close to one if your question is what is the performance of this algorithm for different values of R how do you compare the performance of this algorithm for different values of R R has some impact naturally R has some impact on the final output R has some impact on the final output output changes if R value changes that is also true I think except that I do not want to make any statement on the I mean if R is very very large then what is going to happen so these things I do not want to talk about it now because it depends on the data set there are very many other things involved here so I do not want to make a statement about what will happen if R is very very large and very very small okay other questions this is one of the most popular algorithms in I mean by algorithms of fuzzy sets fuzzy seems algorithm many people apply this one in fact probably the word fuzzy is a misnomer see summation uij is equal to 1 for each point some of the memberships for different clusters is 1 then basically you are looking at a probabilistic setup right probability of the ith point going to the cluster 1 is some value cluster 2 is some value so some of this probabilities is equal to 1 so probably the word fuzzy is a misnomer because you are making it the summation as 1 so maybe that is also one of the reasons why many probabilists like this method many probabilists like this method okay because of that and it has nice mathematical proof though the proof is complicated I went through each and every step of the proof proof is good but it is a complicated one other questions


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