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Full-Text Articles in Mathematics
How To Simulate If We Only Have Partial Information But We Want Reliable Results?, Vladik Kreinovich, Olga Kosheleva
How To Simulate If We Only Have Partial Information But We Want Reliable Results?, Vladik Kreinovich, Olga Kosheleva
Departmental Technical Reports (CS)
The main objective of a smart energy system is to make control decisions that would make energy systems more efficient and more reliable. To select such decisions, the system must know the consequences of different possible decisions. Energy systems are very complex, they cannot be described by a simple formula, the only way to reasonably accurately find such consequences is to test each decision on a simulated system. The problem is that the parameters describing the system and its environment are usually known with uncertainty, and we need to produce reliable results -- i.e., results that will be true for …
Why Moments (And Generalized Moments) Are Used In Statistics And Why Expected Utility Is Used In Decision Making: A Possible Explanation, R. Noah Padilla, Vladik Kreinovich
Why Moments (And Generalized Moments) Are Used In Statistics And Why Expected Utility Is Used In Decision Making: A Possible Explanation, R. Noah Padilla, Vladik Kreinovich
Departmental Technical Reports (CS)
Among the most efficient characteristics of a probability distribution are its moments and, more generally, generalized moments. One of the most adequate numerical characteristics describing human behavior is expected utility. In both cases, the corresponding characteristic is the sum of results of applying appropriate nonlinear functions applied to individual inputs. In this paper, we provide a possible theoretical explanation of why such functions are efficient.
How Multi-View Techniques Can Help In Processing Uncertainty, Olga Kosheleva, Vladik Kreinovich
How Multi-View Techniques Can Help In Processing Uncertainty, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Multi-view techniques help us reconstruct a 3-D object and its properties from its 2-D (or even 1-D) projections. It turns out that similar techniques can be used in processing uncertainty -- where many problems can reduced to a similar task of reconstructing properties of a multi-D object from its 1-D projections. In this chapter, we provide an overview of these techniques.
Why Do People Become Addicted: Towards A Theoretical Explanation For Eyal's Experiment-Based Hook Model, Christopher Reyes, Vladik Kreinovich
Why Do People Become Addicted: Towards A Theoretical Explanation For Eyal's Experiment-Based Hook Model, Christopher Reyes, Vladik Kreinovich
Departmental Technical Reports (CS)
Why do people become addicted, e.g., to gambling? Experiments have shown that simple lotteries, in which we can win a small prize with a certain probability, and not addictive. However, if we add a second possibility -- of having a large prize with a small probability -- the lottery becomes highly addictive to many participants. In this paper, we provide a possible theoretical explanation for this empirical phenomenon.
Why Ovals In Eliciting Intervals?, Joshua Zamora, Vladik Kreinovich
Why Ovals In Eliciting Intervals?, Joshua Zamora, Vladik Kreinovich
Departmental Technical Reports (CS)
To elicit people's opinions, we usually ask them to mark their degree of satisfaction on a scale -- e.g., from 0 to 5 or from 0 to 10. Often, people are unsure about the exact degree: 7 or 8? To cover such situations, it is desirable to elicit not a single value but an interval of possible values. However, it turns out that most people are not comfortable with marking an interval. Empirically, it turned out that the best way to elicit an interval is to ask them to draw an oval whose intersection with the 0-to-10 line is the …
Decision Making Under Uncertainty: Cases When We Only Know An Upper Bound Or A Lower Bound, Toshiki Kamio, Gavin Baechle, Vladik Kreinovich
Decision Making Under Uncertainty: Cases When We Only Know An Upper Bound Or A Lower Bound, Toshiki Kamio, Gavin Baechle, Vladik Kreinovich
Departmental Technical Reports (CS)
In situations when we have a perfect knowledge about the outcomes of several situations, a natural idea is to select the best of these situations. For example, among different investments, we should select the one with the largest gain. In practice, however, we rarely know the exact consequences of each action. In some cases, we know the lower and upper bounds on the corresponding gain. It has been proven that in such cases, an appropriate decision is to use Hurwicz optimism-pessimism criterion. In this paper, we extend the corresponding results to the cases when we only know an upper bound …
Commonsense "And"-Operations, Javier Tellez, Wenbo Xie, Vladik Kreinovich
Commonsense "And"-Operations, Javier Tellez, Wenbo Xie, Vladik Kreinovich
Departmental Technical Reports (CS)
In many practical situations, we need to estimate our degree of belief in a statement "A and B" when the only thing we know are the degrees of belief a and b in combined statements A and B. An algorithm for this estimation is known as an "and"-operation, or, for historical reasons, a t-norm. Usually, "and"-operations are selected in such a way that if one of the statements A or B is false, our degree of belief in "A and B" is 0. However, in practice, this is sometimes not the case: for example, an ideal faculty candidate must satisfy …
Fourier Transform And Other Quadratic Problems Under Interval Uncertainty, Oscar Galindo, Christopher Ibarra, Vladik Kreinovich
Fourier Transform And Other Quadratic Problems Under Interval Uncertainty, Oscar Galindo, Christopher Ibarra, Vladik Kreinovich
Departmental Technical Reports (CS)
In general, computing the range of a quadratic function on given intervals is NP-hard. Recently, a feasible algorithm was proposed for computing the range of a specific quadratic function -- square of the modulus of a Fourier coefficient. For this function, the rank of the quadratic form -- i.e., the number of nonzero eigenvalues -- is 2. In this paper, we show that this algorithm can be extended to all the cases when the rank of the quadratic form is bounded by a constant.
Why Model Order Reduction, Salvador Robles, Martine Ceberio, Vladik Kreinovich
Why Model Order Reduction, Salvador Robles, Martine Ceberio, Vladik Kreinovich
Departmental Technical Reports (CS)
Reasonably recently, a new efficient method appeared for solving complex non-linear differential equations (and systems of differential equations). In this method -- known as Model Order Reduction (MOR) -- we select several solutions, and approximate a general solution by a linear combination of the selected solutions. In this paper, we use the known explanation for efficiency of neural networks to explain the efficiency of MOR techniques.
Why Residual Neural Networks, Sofia Holguin, Vladik Kreinovich
Why Residual Neural Networks, Sofia Holguin, Vladik Kreinovich
Departmental Technical Reports (CS)
In the traditional neural networks, the outputs of each layer serve as inputs to the next layer. It is known that in many cases, it is beneficial to also allow outputs from pre-previous etc. layers as inputs. Such networks are known as residual. In this paper, we provide a possible theoretical explanation for the empirical success of residual neural networks.
How To Gauge The Quality Of A Multi-Class Classification When Ground Truth Is Known With Uncertainty, Ricardo Mendez, Osagumwenro Osaretin, Vladik Kreinovich
How To Gauge The Quality Of A Multi-Class Classification When Ground Truth Is Known With Uncertainty, Ricardo Mendez, Osagumwenro Osaretin, Vladik Kreinovich
Departmental Technical Reports (CS)
The usual formulas for gauging the quality of a classification method assume that we know the ground truth, i.e., that for several objects, we know for sure to which class they belong. In practice, we often only know this with some degree of certainty. In this paper, we explain how to take this uncertainty into account when gauging the quality of a classification method.
Kinematic Metric Spaces Under Interval Uncertainty: Towards An Adequate Definition, Vladik Kreinovich, Olga Kosheleva, Victor Selivanov
Kinematic Metric Spaces Under Interval Uncertainty: Towards An Adequate Definition, Vladik Kreinovich, Olga Kosheleva, Victor Selivanov
Departmental Technical Reports (CS)
In the physical space, we define distance between the two points as the length of the shortest path connecting these points. Similarly, in space-time, for every pair of events for which the event a can causally effect the event b, we can define the longest proper time t(a,b) over all causal trajectories leading from a to b. The resulting function is known as kinematic metric. In practice, our information about all physical quantities -- including time -- comes from measurement, and measurements are never absolutely precise: the measurement result V is, in general, different from the actual (unknown) value v …
Fuzzy Logic Beyond Traditional "And"- And "Or"-Operations, Vladik Kreinovich, Olga Kosheleva
Fuzzy Logic Beyond Traditional "And"- And "Or"-Operations, Vladik Kreinovich, Olga Kosheleva
Departmental Technical Reports (CS)
In the traditional fuzzy logic, we can use "and"-operations (also known as t-norms) to estimate the expert's degree of confidence in a composite statement A&B based on his/her degrees of confidence d(A) and d(B) in the corresponding basic statements A and B. But what if we want to estimate the degree of confidence in A&B&C in situations when, in addition to the degrees of estimate d(A), d(B), and d(C) of the basic statements, we also know the expert's degrees of confidence in the pairs d(A&B), d(A&C), and d(B&C)? Traditional "and"-operations can provide such an estimate -- but only by ignoring …
Different Concepts, Similar Computational Complexity: Nguyen's Results About Fuzzy And Interval Computations 35 Years Later, Hung T. Nguyen, Vladik Kreinovich
Different Concepts, Similar Computational Complexity: Nguyen's Results About Fuzzy And Interval Computations 35 Years Later, Hung T. Nguyen, Vladik Kreinovich
Departmental Technical Reports (CS)
When we know for sure which values are possible and which are not, we have crisp uncertainty -- of which interval uncertainty is a usual case. In practice, we are often not 100% sure about our knowledge, i.e., we have fuzzy uncertainty -- i.e., we have fuzzy knowledge, of which crisp is a particular case. Usually, general problems are more difficult to solve that most of their particular cases. It was therefore expected that processing fuzzy data is, in general, more computationally difficult than processing interval data -- and indeed, Zadeh's extension principle -- a natural formula for fuzzy computations …
Fault Detection In A Smart Electric Grid: Geometric Analysis, Hector Reyes, Dillon Trinh, Vladik Kreinovich
Fault Detection In A Smart Electric Grid: Geometric Analysis, Hector Reyes, Dillon Trinh, Vladik Kreinovich
Departmental Technical Reports (CS)
The main idea behind a smart grid is to equip the grid with a dense lattice of sensors monitoring the state of the grid. If there is a fault, the sensors closer to the fault will detect larger deviations from the normal readings that sensors that are farther away. In this paper, we show that this fact can be used to locate the fault with high accuracy.
Why Geological Regions?, Daniela Flores, Olga Kosheleva, Vladik Kreinovich
Why Geological Regions?, Daniela Flores, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In most practical applications, we approximate the spatial dependence by smooth functions. The main exception is geosciences, where, to describe, e.g., how the density depends on depth and/or on spatial location, geophysicists divide the area into regions on each of which the corresponding quantity is approximately constant. In this paper, we provide a possible explanation for this difference.
Why People Overestimate Small Probabilities?, David Amparan, Vladik Kreinovich
Why People Overestimate Small Probabilities?, David Amparan, Vladik Kreinovich
Departmental Technical Reports (CS)
It is a known empirical fact that people overestimate small probabilities. This fact seems to be inconsistent with the fact that we humans are the product of billions years of improving evolution -- and that we therefore perceive the world as accurately as possible. In this paper, we provide a possible explanation for this seeming contradiction.
Why Rectified Linear Neurons: A Possible Interval-Based Explanation, Jonathan Contreras, Martine Ceberio, Vladik Kreinovich
Why Rectified Linear Neurons: A Possible Interval-Based Explanation, Jonathan Contreras, Martine Ceberio, Vladik Kreinovich
Departmental Technical Reports (CS)
At present, the most efficient machine learning techniques are deep neural networks. In these networks, a signal repeatedly undergoes two types of transformations: linear combination of inputs, and a non-linear transformation of each value v -> s(v). Empirically, the function s(v) = max(v,0) -- known as the rectified linear function -- works the best. There are some partial explanations for this empirical success; however, none of these explanations is fully convincing. In this paper, we analyze this why-question from the viewpoint of uncertainty propagation. We show that reasonable uncertainty-related arguments lead to another possible explanation of why rectified linear functions …
How Probabilistic Methods For Data Fitting Deal With Interval Uncertainty: A More Realistic Analysis, Vladik Kreinovich, Sergey P. Shary
How Probabilistic Methods For Data Fitting Deal With Interval Uncertainty: A More Realistic Analysis, Vladik Kreinovich, Sergey P. Shary
Departmental Technical Reports (CS)
In our previous paper, we showed that a simplified probabilistic approach to interval uncertainty leads to the known notion of a united solution set. In this paper, we show that a more realistic probabilistic analysis of data fitting under interval uncertainty leads to another known notion -- the notion of a tolerable solution set. Thus, the notion of a tolerance solution set also has a clear probabilistic interpretation. Good news is that, in contrast to the united solution set whose computation is, in general, NP-hard, the tolerable solution set can be computed by a feasible algorithm.
Why Neural Networks In The First Place: A Theoretical Explanation, Jonatan Contreras, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich
Why Neural Networks In The First Place: A Theoretical Explanation, Jonatan Contreras, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Neural networks -- specifically, deep neural networks -- are, at present, the most effective machine learning techniques. There are reasonable explanations of why deep neural networks work better than traditional "shallow" ones, but the question remains: why neural networks in the first place? why not networks consisting of non-linear functions from some other family of functions? In this paper, we provide a possible theoretical answer to this question: namely, we show that of all families with the smallest possible number of parameters, families corresponding to neurons are indeed optimal -- for all optimality criteria that satisfy some reasonable requirements: : …
Why Daubechies Wavelets Are So Successful, Solymar Ayala Cortez, Laxman Bokati, Aaron Velasco, Vladik Kreinovich
Why Daubechies Wavelets Are So Successful, Solymar Ayala Cortez, Laxman Bokati, Aaron Velasco, Vladik Kreinovich
Departmental Technical Reports (CS)
In many applications, including analysis of seismic signals, Daubechies wavelets perform much better than other families of wavelets. In this paper, we provide a possible theoretical explanation for the empirical success of Daubechies wavelets. Specifically, we show that these wavelets are optimal with respect to any optimality criterion that satisfies the natural properties of scale- and shift-invariance.
Uncertainty: Ideas Behind Neural Networks Lead Us Beyond Kl-Decomposition And Interval Fields, Michael Beer, Olga Kosheleva, Vladik Kreinovich
Uncertainty: Ideas Behind Neural Networks Lead Us Beyond Kl-Decomposition And Interval Fields, Michael Beer, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In many practical situations, we know that there is a functional dependence between a quantity q and quantities a1, ..., an, but the exact form of this dependence is only known with uncertainty. In some cases, we only know the class of possible functions describing this dependence. In other cases, we also know the probabilities of different functions from this class -- i.e., we know the corresponding random field or random process. To solve problems related to such a dependence, it is desirable to be able to simulate the corresponding functions, i.e., to have algorithms that transform simple intervals or …
While, In General, Uncertainty Quantification (Uq) Is Np-Hard, Many Practical Uq Problems Can Be Made Feasible, Anderson Gray, Scott Ferson, Olga Kosheleva, Vladik Kreinovich
While, In General, Uncertainty Quantification (Uq) Is Np-Hard, Many Practical Uq Problems Can Be Made Feasible, Anderson Gray, Scott Ferson, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In general, many general mathematical formulations of uncertainty quantification problems are NP-hard, meaning that (unless it turned out that P = NP) no feasible algorithm is possible that would always solve these problems. In this paper, we argue that if we restrict ourselves to practical problems, then the correspondingly restricted problems become feasible -- namely, they can be solved by using linear programming techniques.
Ethical Dilemma Of Self-Driving Cars: Conservative Solution, Christian Servin, Vladik Kreinovich, Shahnaz Shahbazova
Ethical Dilemma Of Self-Driving Cars: Conservative Solution, Christian Servin, Vladik Kreinovich, Shahnaz Shahbazova
Departmental Technical Reports (CS)
When designing software for self-driving cars, we need to make an important decision: When a self-driving car encounters an emergency situation in which either the car's passenger or an innocent pedestrian have a good change of being injured or even die, which option should it choose? This has been a subject of many years of ethical discussions -- and these discussions have not yet led to a convincing solution. In this paper, we propose a "conservative" (status quo) solution that does not require making new ethical decisions -- namely, we propose to limit both the risks to passengers and risks …
Localized Learning: A Possible Alternative To Current Deep Learning Techniques, Javier Viana, Kelly Cohen, Anca Ralescu, Stephan Ralescu, Vladik Kreinovich
Localized Learning: A Possible Alternative To Current Deep Learning Techniques, Javier Viana, Kelly Cohen, Anca Ralescu, Stephan Ralescu, Vladik Kreinovich
Departmental Technical Reports (CS)
At present, the most efficient deep learning technique is the use of deep neural networks. However, recent empirical results show that in some situations, it is even more efficient to use "localized" learning -- i.e., to divide the domain of inputs into sub-domains, learn the desired dependence separately on each sub-domain, and then "smooth" the resulting dependencies into a single algorithm. In this paper, we provide theoretical explanation for these empirical successes.
Freedom Of Will, Physics, And Human Intelligence: An Idea, Miroslav Svitek, Vladik Kreinovich, Nguyen Hoang Phuong
Freedom Of Will, Physics, And Human Intelligence: An Idea, Miroslav Svitek, Vladik Kreinovich, Nguyen Hoang Phuong
Departmental Technical Reports (CS)
Among the main fundamental challenges related to physics and human intelligence are: How can we reconcile the free will with the deterministic character of physical equations? What is the physical meaning of extra spatial dimensions needed to make quantum physics consistent? and Why are we often smarter than brain-simulating neural networks? In this paper, we show that while each of these challenges is difficult to resolve on its own, it may be possible to resolve all three of them if we consider them together. The proposed possible solution is that human reasoning uses the extra spatial dimensions. This may sound …
What Is A Reasonable Way To Make Predictions?, Leonardo Orea Amador, Vladik Kreinovich
What Is A Reasonable Way To Make Predictions?, Leonardo Orea Amador, Vladik Kreinovich
Departmental Technical Reports (CS)
Predictions are usually based on what is called laws of nature: many times, we observe the same relation between the states at different moments of time, and we conclude that the same relation will occur in the future. The more times the relation repeats, the more confident we are that the same phenomenon will be re-peated again. This is how Newton's laws and other laws came into being. This is what is called inductive reasoning. However, there are other reasonable approaches. For example, assume that a person speeds and is not caught. This may be repeated two times, three times …
How The Pavement's Lifetime Depends On The Stress Level: An Explanation Of The Empirical Formula, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich, Olga Kosheleva, Hoang Phuong Nguyen
How The Pavement's Lifetime Depends On The Stress Level: An Explanation Of The Empirical Formula, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich, Olga Kosheleva, Hoang Phuong Nguyen
Departmental Technical Reports (CS)
We show that natural invariance ideas explain the empirical dependence on the pavement's lifetime on the stress level.
Why Rectified Linear Activation Functions? Why Max-Pooling? A Possible Explanation, Julio C. Urenda, Vladik Kreinovich
Why Rectified Linear Activation Functions? Why Max-Pooling? A Possible Explanation, Julio C. Urenda, Vladik Kreinovich
Departmental Technical Reports (CS)
At present, the most successful machine learning technique is deep learning, that uses rectified linear activation function (ReLU) s(x) = max(x,0) as a non-linear data processing unit. While this selection was guided by general ideas (which were often imprecise), the selection itself was still largely empirical. This leads to a natural question: are these selections indeed the best or are there even better selections? A possible way to answer this question would be to provide a theoretical explanation of why these selections are -- in some reasonable sense -- the best. This paper provides a possible theoretical explanation for this …
Why Normalized Difference Vegetation Index (Ndvi)?, Francisco Zapata, Eric Smith, Vladik Kreinovich, Nguyen Hoang Phuong
Why Normalized Difference Vegetation Index (Ndvi)?, Francisco Zapata, Eric Smith, Vladik Kreinovich, Nguyen Hoang Phuong
Departmental Technical Reports (CS)
Plants play a very important role in ecological systems -- they transform CO2 into oxygen. It is therefore very important to be able to estimate the overall amount of live green vegetation in a given area. The most efficient way to provide such a global analysis is to use remote sensing, i.e., multi-spectral photos taken from satellites, drones, planes, etc. At present, one of the most efficient ways to detect, based on remote sensing data, how much live green vegetation an area contains is to compute the value of the normalized difference vegetation index (NDVI). In this paper, we provide …