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Mieczysław A. Kłopotek

Publications and source records attributed to Mieczysław A. Kłopotek.

At least 37 records · Page 2Linked to original sources
arXiv2018

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.

cs.CV↗
arXiv2018

An Aposteriorical Clusterability Criterion for $k$-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.

cs.LG↗
arXiv2018

Dempsterian-Shaferian Belief Network From Data

Shenoy and Shafer {Shenoy:90} demonstrated that both for Dempster-Shafer Theory and probability theory there exists a possibility to calculate efficiently marginals of joint belief distributions (by so-called local computations) provided that the joint distribution can be decomposed (factorized) into a belief network. A number of algorithms exists for decomposition of probabilistic joint belief distribution into a bayesian (belief) network from data. For example Spirtes, Glymour and Schein{Spirtes:90b} formulated a Conjecture that a direct dependence test and a head-to-head meeting test would suffice to construe bayesian network from data in such a way that Pearl's concept of d-separation {Geiger:90} applies. This paper is intended to transfer Spirtes, Glymour and Scheines {Spirtes:90b} approach onto the ground of the Dempster-Shafer Theory (DST). For this purpose, a frequentionistic interpretation of the DST developed in {Klopotek:93b} is exploited. A special notion of conditionality for DST is introduced and demonstrated to behave with respect to Pearl's d-separation {Geiger:90} much the same way as conditional probability (though some differences like non-uniqueness are evident). Based on this, an algorithm analogous to that from {Spirtes:90b} is developed. The notion of a partially oriented graph (pog) is introduced and within this graph the notion of p-d-separation is defined. If direct dependence test and head-to-head meeting test are used to orient the pog then its p-d-separation is shown to be equivalent to the Pearl's d-separation for any compatible dag.

cs.AI↗
arXiv2018

Too Fast Causal Inference under Causal Insufficiency

Causally insufficient structures (models with latent or hidden variables, or with confounding etc.) of joint probability distributions have been subject of intense study not only in statistics, but also in various AI systems. In AI, belief networks, being representations of joint probability distribution with an underlying directed acyclic graph structure, are paid special attention due to the fact that efficient reasoning (uncertainty propagation) methods have been developed for belief network structures. Algorithms have been therefore developed to acquire the belief network structure from data. As artifacts due to variable hiding negatively influence the performance of derived belief networks, models with latent variables have been studied and several algorithms for learning belief network structure under causal insufficiency have also been developed. Regrettably, some of them are known already to be erroneous (e.g. IC algorithm of [Pearl:Verma:91]. This paper is devoted to another algorithm, the Fast Causal Inference (FCI) Algorithm of [Spirtes:93]. It is proven by a specially constructed example that this algorithm, as it stands in [Spirtes:93], is also erroneous. Fundamental reason for failure of this algorithm is the temporary introduction of non-real links between nodes of the network with the intention of later removal. While for trivial dependency structures these non-real links may be actually removed, this may not be the case for complex ones, e.g. for the case described in this paper. A remedy of this failure is proposed.

cs.AI↗
arXiv2017

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 $δ$. 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 $1-ε$ all of them fall into predefined error range $δ$ for any user-predefined failure probability $ε$. 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.

cs.DS↗
arXiv2017

On (Anti)Conditional Independence in Dempster-Shafer Theory

This paper verifies a result of {Shenoy:94} concerning graphoidal structure of Shenoy's notion of independence for Dempster-Shafer theory of belief functions. Shenoy proved that his notion of independence has graphoidal properties for positive normal valuations. The requirement of strict positive normal valuations as prerequisite for application of graphoidal properties excludes a wide class of DS belief functions. It excludes especially so-called probabilistic belief functions. It is demonstrated that the requirement of positiveness of valuation may be weakened in that it may be required that commonality function is non-zero for singleton sets instead, and the graphoidal properties for independence of belief function variables are then preserved. This means especially that probabilistic belief functions with all singleton sets as focal points possess graphoidal properties for independence.

cs.AI↗
arXiv2017

Fast Restricted Causal Inference

Hidden variables are well known sources of disturbance when recovering belief networks from data based only on measurable variables. Hence models assuming existence of hidden variables are under development. This paper presents a new algorithm "accelerating" the known CI algorithm of Spirtes, Glymour and Scheines {Spirtes:93}. We prove that this algorithm does not produces (conditional) independencies not present in the data if statistical independence test is reliable. This result is to be considered as non-trivial since e.g. the same claim fails to be true for FCI algorithm, another "accelerator" of CI, developed in {Spirtes:93}.

cs.AI↗
arXiv2017

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.%

cs.AI↗
arXiv2017

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.

cs.AI↗
arXiv2017

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.

cs.AI↗
arXiv2017

What Does a Belief Function Believe In ?

The conditioning in the Dempster-Shafer Theory of Evidence has been defined (by Shafer \cite{Shafer:90} as combination of a belief function and of an "event" via Dempster rule. On the other hand Shafer \cite{Shafer:90} gives a "probabilistic" interpretation of a belief function (hence indirectly its derivation from a sample). Given the fact that conditional probability distribution of a sample-derived probability distribution is a probability distribution derived from a subsample (selected on the grounds of a conditioning event), the paper investigates the empirical nature of the Dempster- rule of combination. It is demonstrated that the so-called "conditional" belief function is not a belief function given an event but rather a belief function given manipulation of original empirical data.\\ Given this, an interpretation of belief function different from that of Shafer is proposed. Algorithms for construction of belief networks from data are derived for this interpretation.

cs.AI↗
arXiv2017

Evidence Against Evidence Theory (?!)

This paper is concerned with the apparent greatest weakness of the Mathematical Theory of Evidence (MTE) of Shafer \cite{Shafer:76}, which has been strongly criticized by Wasserman \cite{Wasserman:92ijar} - the relationship to frequencies. Weaknesses of various proposals of probabilistic interpretation of MTE belief functions are demonstrated. A new frequency-based interpretation is presented overcoming various drawbacks of earlier interpretations.

cs.AI↗
arXiv2017

Distribution of degrees of freedom over structure and motion of rigid bodies

This paper is concerned with recovery of motion and structure parameters from multiframes under orthogonal projection when only points are traced. The main question is how many points and/or how many frames are necessary for the task. It is demonstrated that 3 frames and 3 points are the absolute minimum. Closed-form solution is presented. Furthermore, it is shown that the task may be linearized if either four points or four frames are available. It is demonstrated that no increase in the number of points may lead to recovery of structure and motion parameters from two frames only. It is shown that instead the increase in the number of points may support the task of tracing the points from frame to frame.

cs.CV↗
arXiv2017

A Comment on "Analysis of Video Image Sequences Using Point and Line Correspondences"

In this paper we would like to deny the results of Wang et al. raising two fundamental claims: * A line does not contribute anything to recognition of motion parameters from two images * Four traceable points are not sufficient to recover motion parameters from two perspective To be constructive, however, we show that four traceable points are sufficient to recover motion parameters from two frames under orthogonal projection and that five points are sufficient to simplify the solution of the two-frame problem under orthogonal projection to solving a linear equation system.

cs.CV↗
arXiv2017

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 functions

cs.AI↗
arXiv2017

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.

cs.CV↗