Mathematics Commons^™

Fuzzy Techniques Provide A Theoretical Explanation For The Heuristic L^P-Regularization Of Signals And Images, Fernando Cervantes, Bryan E. Usevitch, Leobardo Valera, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

One of the main techniques used to de-noise and de-blur signals and images is regularization, which is based on the fact that signals and images are usually smoother than noise. Traditional Tikhonov regularization assumes that signals and images are differentiable, but, as Mandelbrot has shown in his fractal theory, many signals and images are not differentiable. To de-noise and de-blur such images, researchers have designed a heuristic method of l^p-regularization.

l^p-regularization leads to good results, but it is not used as widely as should be, because it lacks a convincing theoretical explanation -- and thus, practitioners are often reluctant to …

Membership Functions Representing A Number Vs. Representing A Set: Proof Of Unique Reconstruction, Hung T. Nguyen, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

In some cases, a membership function m(x) represents an unknown number, but in many other cases, it represents an unknown crisp set. In this case, for each crisp set S, we can estimate the degree m(S) to which this set S is the desired one. A natural question is: once we know the values m(S) corresponding to all possible crisp sets S, can we reconstruct the original membership function? In this paper, we show that the original membership function m(x) can indeed be uniquely reconstructed from the values m(S).

Why Lp-Methods In Signal And Image Processing: A Fuzzy-Based Explanation, Fernando Cervantes, Bryan E. Usevitch, Vladik Kreinovich

Departmental Technical Reports (CS)

In signal and image processing, it is often beneficial to use semi-heuristic Lp-methods, i.e., methods that minimize the sum of the p-th powers of the discrepancies. In this paper, we show that a fuzzy-based analysis of the corresponding intuitive idea leads exactly to the Lp-methods.

Limitations Of Realistic Monte-Carlo Techniques, Andrzej Pownuk, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Because of the measurement errors, the result Y = f(X1, ..., Xn) of processing the measurement results X1, ..., Xn is, in general, different from the value y = f(x1, ..., xn) that we would obtain if we knew the exact values x1, ..., xn of all the inputs. In the linearized case, we can use numerical differentiation to estimate the resulting difference Y -- y; however, this requires >n calls to an algorithm computing f, and for complex algorithms and large $n$ this can take too long. In situations when for each input xi, we know the probability distribution …

Why Min-Based Conditioning, Salem Benferhat, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we do not have full information about which alternatives are possible and which are not. In such situations, an expert can estimate, for each alternative, the degree to which this alternative is possible. Sometimes, experts can produce numerical estimates of their degrees, but often, they can only provide us with qualitative estimates: they inform us which degrees are higher, but do not provide us with numerical values for these degrees. After we get these degrees from the experts, we often gain additional information, because of which some alternatives which were previously considered possible are now excluded. …

Bell-Shaped Curve For Productivity Growth: An Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A recent analysis of the productivity growth data shows, somewhat surprisingly, that the dependence of the 20-century productivity growth on time can be reasonably well described by a Gaussian formula. In this paper, we provide a possible theoretical explanation for this observation.

Adjoint Fuzzy Partition And Generalized Sampling Theorem, Irina Perfilieva, Michal Holčapek, Vladik Kreinovich

Departmental Technical Reports (CS)

A new notion of adjoint fuzzy partition is introduced and the reconstruction of a function from its F-transform components is analyzed. An analogy with the Nyquist-Shannon-Kotelnikov sampling theorem is discussed.

Robustness As A Criterion For Selecting A Probability Distribution Under Uncertainty, Songsak Sriboonchitta, Hung T. Nguyen, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

Often, we only have partial knowledge about a probability distribution, and we would like to select a single probability distribution $\rho(x)$ out of all probability distributions which are consistent with the available knowledge. One way to make this selection is to take into account that usually, the values $x$ of the corresponding quantity are also known only with some accuracy. It is therefore desirable to select a distribution which is the most robust -- in the sense the x-inaccuracy leads to the smallest possible inaccuracy in the resulting probabilities. In this paper, we describe the corresponding most robust probability distributions, …

Why Dependence Of Productivity On Group Size Is Log-Normal, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Empirical analysis shows that, on average, the productivity of a group log-normally depends on its size. The current explanations for this empirical fact are based on reasonably complex assumptions about the human behavior. In this paper, we show that the same conclusion can be made in effect, from first principles, without making these complex assumptions.

Why Locating Local Optima Is Sometimes More Complicated Than Locating Global Ones, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In most applications, practitioners are interested in locating global optima. In such applications, local optima that result from some optimization algorithms are an unnecessary side effect. In other words, in such applications, locating global optima is a much more computationally complex problem than locating local optima. In several practical applications, however, local optima themselves are of interest. Somewhat surprisingly, it turned out that in many such applications, locating all local optima is a much more computationally complex problem than locating all global optima. In this paper, we provide a theoretical explanation for this surprising empirical phenomenon.

Voting Aggregation Leads To (Interval) Median, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

When we have several results of measuring or estimating the same quantities, it is desirable to aggregate them into a single estimate for the desired quantities. A natural requirement is that if the majority of estimates has some property, then the aggregate estimate should have the same property. It turns out that it is not possible to require this forall possible properties -- but we can require it for bounds, i.e., for properties that the value of the quantity is in between given bounds a and b. In this paper, we prove that if we restrict the …

Why Superellipsoids: A Probability-Based Explanation, Pedro Barragan Olague, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, it turns out that the set of possible values of the deviation vector is (approximately) a super-ellipsoid. In this paper, we provide a theoretical explanation for this empirical fact -- an explanation based on the natural notion of scale-invariance.

On Geometry Of Finsler Causality: For Convex Cones, There Is No Affine-Invariant Linear Order (Similar To Comparing Volumes), Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Some physicists suggest that to more adequately describe the causal structure of space-time, it is necessary to go beyond the usual pseudo-Riemannian causality, to a more general Finsler causality. In this general case, the set of all the events which can be influenced by a given event is, locally, a generic convex cone, and not necessarily a pseudo-Reimannian-style quadratic cone. Since all current observations support pseudo-Riemannian causality, Finsler causality cones should be close to quadratic ones. It is therefore desirable to approximate a general convex cone by a quadratic one. This cane be done if we select a hyperplane, and …

Solving Equations (And Systems Of Equations) Under Uncertainty: How Different Practical Problems Lead To Different Mathematical And Computational Formulations, Vladik Kreinovich

Departmental Technical Reports (CS)

Many practical problems are naturally reduced to solving systems of equations. There are many efficient techniques for solving well-defined systems of equations, i.e., systems in which we know the exact values of all the parameters and coefficients. In practice, we usually know these parameters and coefficients with some uncertainty -- uncertainty usually described by an appropriate granule: interval, fuzzy set, rough set, etc. Many techniques have been developed for solving systems of equations under such granular uncertainty. Sometimes, however, practitioners use previously successful techniques and get inadequate results. In this -- mostly pedagogical -- paper, we explain that to obtain …

From Tertullian's Credo Quia Absurdum To Bohr's Crazy Theories: A Rational Explanation Of A Seemingly Irrational Idea, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

At first glance, Tertullian's idea -- that absurdity of a statement makes it more believable -- sounds irrational, maybe appropriate for theology but definitely not for science. However, somewhat surprisingly, a similar idea was successfully used by the Nobelist Niels Bohr in theoretical physics -- an epitome of rationality in science. In this paper, we show that this Tertullian-Bohr idea actually has a simple rational explanation. Specifically, if previous attempts to construct a theory which is consistent with what is perceived as common sense were unsuccessful, this implies that a true theory much contradict common sense -- and thus, the …

Towards Selecting The Best Abstraction For A Patrolling Game, Anjon Basak, Chris Kiekintveld, Vladik Kreinovich

Departmental Technical Reports (CS)

When the number of possible strategies is large, it is not computationally feasible to compute the optimal strategy for the original game. Instead, we select our strategy based on an approximate approximate description of the original game. The quality of the resulting strategy depends on which approximation we select. In this paper, on an example of a simple game, we show how to find the optimal approximation, the approximation whose use results in the best strategy.

How To Predict Nesting Sites?, Stephen Escarzaga, Craig Tweedie, Vladik Kreinovich

Departmental Technical Reports (CS)

How to predict nesting sites? Usually, all we know is the past nesting sites, and the fact that the birds select a site which is optimal for them (in some reasonable sense), but we do not know the exact objective function describing this optimality. In this paper, we propose a way to make predictions in such a situation.

Why It Is Healthy To Regularly Challenge Authority: An Algorithmic Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One way to make group decisions is to select the best decision maker(s) in the group as the authority, and to follow his or her decisions. At first glance, it seems that if the selected authority is indeed the best decision maker, it is beneficial for everyone to obey his or her authority. However, history shows that in many cases, challenges to the authority (even to the authority of the best decision maker) were beneficial to the group. In this paper, we provide an algorithmic explanation for this phenomenon. The main idea behind this explanation is that most practical general …

Interval Methods For Data Fitting Under Uncertainty: A Probabilistic Treatment, Vladik Kreinovich, Sergey P. Shary

Departmental Technical Reports (CS)

How to estimate parameters from observations subject to errors and uncertainty? Very often, the measurement errors are random quantities that can be adequately described by the probability theory. When we know that the measurement errors are normally distributed with zero mean, then the (asymptotically optimal) Maximum Likelihood Method leads to the popular least squares estimates. In many situations, however, we do not know the shape of the error distribution, we only know that the measurement errors are located on a certain interval. Then the maximum entropy approach leads to a uniform distribution on this interval, and the Maximum Likelihood Method …

Simplest Innovations Are, Empirically, The Most Promising: An Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Many examples show that the simplest innovation are the most promising. In this paper, we provide a theoretical explanation for this empirical observation.

Science Is Helpful For Engineering Applications: A Theoretical Explanation Of An Empirical Observation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Empirical evidence shows that when engineering design uses scientific analysis, we usually get a much better performance that for the system designed by using a trial-and-error engineering approach. In this paper, we provide a quantitative explanation for this empirical observation.

Why The Range Of A Robust Statistic Under Interval Uncertainty Is Often Easier To Compute, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In statistical analysis, we usually use the observed sample values x₁, ..., x_n to compute the values of several statistics v(x₁, ..., x_n) -- such as sample mean, sample variance, etc. The usual formulas for these statistics implicitly assume that we know the exact values x₁, ..., x_n. In practice, the sample values X₁, ..., X_n come from measurements and are, thus, only approximations to the actual (unknown) values x₁, ..., x_n of the corresponding quantity. Often, the only information that we have …

Positive Consequences Of Negative Attitude: Game-Theoretic Analysis, Mahdokhat Afravi, Vladik Kreinovich

Departmental Technical Reports (CS)

At first glance, the world would be a better place if all the people had positive attitude towards each other. It is known that this is not always the case: sometimes, the excess of positive attitude can lead to negative consequences. In this paper, we show that, vice versa, a reasonable amount of negative attitude can make life better for everyone. What is therefore needed is not the exclusive appearance of positive attitude, but rather a balance -- titled towards moderately positive attitude.

Maximum Entropy Approach Is Not As Arbitrary As It May Seem At First Glance, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

When we only have partial information about the probability distribution, i.e., when several different probability distributions are consistent with our knowledge, then it makes sense to select a distribution with the largest entropy. In particular, when we only know that the quantity is located within a certain interval -- and we have no information about the probability of different values within this intervals -- then it is reasonable to assume that all these values are equally probable, i.e., that we have a uniform distribution on this interval. The problem with this idea is that if we apply it to the …

Decision Making Under Interval (And More General) Uncertainty: Monetary Vs. Utility Approaches, Vladik Kreinovich

Departmental Technical Reports (CS)

In many situations, e.g., in financial and economic decision making, the decision results either in a money gain (or loss) and/or in the gain of goods that can be exchanged for money or for other goods. In such situations, interval uncertainty means that we do not know the exact amount of money that we will get for each possible decision, we only know lower and upper bounds on this amount. In this case, a natural idea is to assign a fair price to different alternatives, and then to use these fair prices to select the best alternative. In the talk, …

On The Importance Of Duality And Multi-Ality In Mathematics Education, Mourat Tchoshanov, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

For each mathematical object, there are usually several different equivalent representations: for example, a spatial object can be represented either in geometric terms, or by a function that describes its shape. The need for several representations comes from the fact that each of these representations is useful in solving some problems for which the use of other representations is less helpful. Thus, the more representations a student knows, the more capable this student is of solving mathematical problems. In this paper, we propose a general formal description of the corresponding notion of duality (and, more generally, "multi-ality"), and we explain …

From Equations To Tri-Quations And Multi-Quations, Mourat Tchoshanov, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In general, an equation A(x₁, ..., x_n)= B(x₁, ..., x_n) corresponds to the situation when we have two quantities A(x₁, ..., x_n) and B(x₁, ..., x_n) which are known to be equal, we know how each of these quantities depends on the unknown parameters x₁, ..., x_n, and we want to find the values of the unknowns x_i from this equality -- and from other similar equalities. In some practical situations, instead of two equal values, we have three …

Waning Influence Of History: Why?, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the past, history played an important role in education: students learned history of science, history of mathematics, etc. In the last decades, the influence of history has waned. In this paper, we provide a natural explanation for this waning.

Explaining Boris Pasternak's Observation That Complex Ideas Are Sometimes Easier To Understand, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Probably the most cited lines from the poetry of the Nobel-prize winning Russian writer Boris Pasternak contain the observation that complex ideas are sometimes easier to understand than simpler ones. This is not just a paradoxical poetic statement: many teachers have observed the same seemingly counter-intuitive phenomenon. In this paper, we provide a possible explanation for this phenomenon, by showing that indeed, many easier-to-describe mathematical models lead to more-difficult-to-solve mathematical problems.

Why The Graph Isomorphism Problem Is Easier Than Propositional Satisfiability: A Possible Qualitative Explanation, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

A recent result has shown that the graph isomorphism problem can be solved in quasi-polynomial time, while the general belief is that only exponential time algorithms are possible for propositional satisfiability. This is somewhat counter-intuitive, since for propositional satisfiability, we need to look for one of 2ⁿ options, while in graph isomorphism, we need to look for one of n! options, and n! is much larger than 2ⁿ. Our qualitative explanation for this counter-intuitive fact comes from the fact that, in general, a graph isomorphism problem has a unique solution -- in contrast to propositional satisfiability which, …