Beliefs in Markov Trees - From Local Computations to Local Valuation

This paper is devoted to expressiveness of hypergraphs for which uncertainty propagation by local computations via Shenoy/Shafer method applies. It is demonstrated that for this propagation method for a given joint belief distribution no valuation of hyperedges of a hypergraph may provide with simpler hypergraph structure than valuation of hyperedges by conditional distributions. This has vital implication that methods recovering belief networks from data have no better alternative for finding the simplest hypergraph structure for belief propagation. A method for recovery tree-structured belief networks has been developed and specialized for Dempster-Shafer belief functionsComment: Preliminary versioin of conference paper: M.A. K{\l}opotek: Beliefs in Markov Trees - From Local Computations to Local Valuation. [in:] R. Trappl, Ed.: Cybernetics and Systems Research , Proc. 12th European Meeting on Cybernetics and System Research, Vienna 5-8 April 1994, World Scientific Publishers, Vol.1. pp. 351-35

An Aposteriorical Clusterability Criterion for kk-Means++ and Simplicity of Clustering

We define the notion of a well-clusterable data set combining the point of view of the objective of $k$-means clustering algorithm (minimising the centric spread of data elements) and common sense (clusters shall be separated by gaps). We identify conditions under which the optimum of $k$-means objective coincides with a clustering under which the data is separated by predefined gaps. We investigate two cases: when the whole clusters are separated by some gap and when only the cores of the clusters meet some separation condition. We overcome a major obstacle in using clusterability criteria due to the fact that known approaches to clusterability checking had the disadvantage that they are related to the optimal clustering which is NP hard to identify. Compared to other approaches to clusterability, the novelty consists in the possibility of an a posteriori (after running $k$-means) check if the data set is well-clusterable or not. As the $k$-means algorithm applied for this purpose has polynomial complexity so does therefore the appropriate check. Additionally, if $k$-means++ fails to identify a clustering that meets clusterability criteria, with high probability the data is not well-clusterable.Comment: 58 page

Independence, Conditionality and Structure of Dempster-Shafer Belief Functions

Several approaches of structuring (factorization, decomposition) of Dempster-Shafer joint belief functions from literature are reviewed with special emphasis on their capability to capture independence from the point of view of the claim that belief functions generalize bayes notion of probability. It is demonstrated that Zhu and Lee's {Zhu:93} logical networks and Smets' {Smets:93} directed acyclic graphs are unable to capture statistical dependence/independence of bayesian networks {Pearl:88}. On the other hand, though Shenoy and Shafer's hypergraphs can explicitly represent bayesian network factorization of bayesian belief functions, they disclaim any need for representation of independence of variables in belief functions. Cano et al. {Cano:93} reject the hypergraph representation of Shenoy and Shafer just on grounds of missing representation of variable independence, but in their frameworks some belief functions factorizable in Shenoy/Shafer framework cannot be factored. The approach in {Klopotek:93f} on the other hand combines the merits of both Cano et al. and of Shenoy/Shafer approach in that for Shenoy/Shafer approach no simpler factorization than that in {Klopotek:93f} approach exists and on the other hand all independences among variables captured in Cano et al. framework and many more are captured in {Klopotek:93f} approach.%Comment: 1994 internal repor

Identification and Interpretation of Belief Structure in Dempster-Shafer Theory

Mathematical Theory of Evidence called also Dempster-Shafer Theory (DST) is known as a foundation for reasoning when knowledge is expressed at various levels of detail. Though much research effort has been committed to this theory since its foundation, many questions remain open. One of the most important open questions seems to be the relationship between frequencies and the Mathematical Theory of Evidence. The theory is blamed to leave frequencies outside (or aside of) its framework. The seriousness of this accusation is obvious: (1) no experiment may be run to compare the performance of DST-based models of real world processes against real world data, (2) data may not serve as foundation for construction of an appropriate belief model. In this paper we develop a frequentist interpretation of the DST bringing to fall the above argument against DST. An immediate consequence of it is the possibility to develop algorithms acquiring automatically DST belief models from data. We propose three such algorithms for various classes of belief model structures: for tree structured belief networks, for poly-tree belief networks and for general type belief networks.Comment: An internal report 199

Spectral Analysis of Laplacians of an Unweighted and Weighted Multidimensional Grid Graph -- Combinatorial versus Normalized and Random Walk Laplacians

In this paper we generalise the results on eigenvalues and eigenvectors of unnormalized (combinatorial) Laplacian of two-dimensional grid presented by Edwards:2013 first to a grid graph of any dimension, and second also to other types of Laplacians, that is unoriented Laplacians, normalized Laplacians, and random walk Laplacians. While the closed-form or nearly closed form solutions to the eigenproblem of multidimensional grid graphs constitute a good test suit for spectral clustering algorithms for the case of no structure in the data, the multidimensional weighted grid graphs, presented also in this paper can serve as testbeds for these algorithms as graphs with some predefined cluster structure. The weights permit to simulate node clusters not perfectly separated from each other. This fact opens new possibilities for exploitation of closed-form or nearly closed form solutions eigenvectors and eigenvalues of graphs while testing and/or developing such algorithms and exploring their theoretical properties. Besides, the differences between the weighted and unweighted case allow for new insights into the nature of normalized and unnormalized Laplacians.Comment: 73 pages, 18 figure

Rigid Body Structure and Motion From Two-Frame Point-Correspondences Under Perspective Projection

This paper is concerned with possibility of recovery of motion and structure parameters from multiframes under perspective projection when only points on a rigid body are traced. Free (unrestricted and uncontrolled) pattern of motion between frames is assumed. The major question is how many points and/or how many frames are necessary for the task. It has been shown in an earlier paper {Klopotek:95b} that for orthogonal projection two frames are insufficient for the task. The paper demonstrates that, under perspective projection, that total uncertainty about relative position of focal point versus projection plane makes the recovery of structure and motion from two frames impossible.Comment: arXiv admin note: text overlap with arXiv:1705.0398

Reconstruction of~3-D Rigid Smooth Curves Moving Free when Two Traceable Points Only are Available

This paper extends previous research in that sense that for orthogonal projections of rigid smooth (true-3D) curves moving totally free it reduces the number of required traceable points to two only (the best results known so far to the author are 3 points from free motion and 2 for motion restricted to rotation around a fixed direction and and 2 for motion restricted to influence of a homogeneous force field). The method used is exploitation of information on tangential projections. It discusses also possibility of simplification of reconstruction of flat curves moving free for prospective projections

Machine Learning Friendly Set Version of Johnson-Lindenstrauss Lemma

In this paper we make a novel use of the Johnson-Lindenstrauss Lemma. The Lemma has an existential form saying that there exists a JL transformation $f$ of the data points into lower dimensional space such that all of them fall into predefined error range $\delta$. We formulate in this paper a theorem stating that we can choose the target dimensionality in a random projection type JL linear transformation in such a way that with probability $1-\epsilon$ all of them fall into predefined error range $\delta$ for any user-predefined failure probability $\epsilon$. This result is important for applications such a data clustering where we want to have a priori dimensionality reducing transformation instead of trying out a (large) number of them, as with traditional Johnson-Lindenstrauss Lemma. In particular, we take a closer look at the $k$-means algorithm and prove that a good solution in the projected space is also a good solution in the original space. Furthermore, under proper assumptions local optima in the original space are also ones in the projected space. We define also conditions for which clusterability property of the original space is transmitted to the projected space, so that special case algorithms for the original space are also applicable in the projected space.Comment: 38 pages, 6 Figure

Restricted Causal Inference Algorithm

This paper proposes a new algorithm for recovery of belief network structure from data handling hidden variables. It consists essentially in an extension of the CI algorithm of Spirtes et al. by restricting the number of conditional dependencies checked up to k variables and in an extension of the original CI by additional steps transforming so called partial including path graph into a belief network. Its correctness is demonstrated.Comment: M.A. K{\l}opotek: Restricted Causal Inference Algorithm. [in:] B. Pehrson, I. Simon Eds.: Proc. World Computer Congress of IFIP . Hamburg 28 August - 2 September 1994, Vol.1, Elsevier Scientific Publishers (North-Holland), Amsterdam, pp. 342-34

Beliefs and Probability in Bacchus' l.p. Logic: A~3-Valued Logic Solution to Apparent Counter-intuition

Fundamental discrepancy between first order logic and statistical inference (global versus local properties of universe) is shown to be the obstacle for integration of logic and probability in L.p. logic of Bacchus. To overcome the counterintuitiveness of L.p. behaviour, a 3-valued logic is proposed.Comment: Draft for the conference M.A. K{\l}opotek: Beliefs and Probability in Bacchus' l.p. Logic: A 3-Valued Logic Solution to Apparent Counter-intuition. [in:] R. Trappl Ed,: Cybernetics and Systems Research. Proc. 11 European Meeting on Cybernetics and System Research EMCSR'92, Wien, Osterreich, 20. April 1992. World Scientific Singapore, New Jersey, London, HongKong Vol. 1, pp. 519-52