107 research outputs found
The output distribution of important LULU-operators
Two procedures to compute the output distribution phi_S of certain stack
filters S (so called erosion-dilation cascades) are given. One rests on the
disjunctive normal form of S and also yields the rank selection probabilities.
The other is based on inclusion-exclusion and e.g. yields phi_S for some
important LULU-operators S. Properties of phi_S can be used to characterize
smoothing properties of S. One of the methods discussed also allows for the
calculation of the reliability polynomial of any positive Boolean function
(e.g. one derived from a connected graph).Comment: 20 pages, up to trivial differences this is the final version to be
published in Quaestiones Mathematicae 201
Calculating the output distribution of stack filters that are erosion-dilation cascades, in particular LULU-filters
Original article available at http://arxiv.org/ENGLISH ABSTRACT: Two procedures to compute the output distribution 0S of certain stack filters S (so
called erosion-dilation cascades) are given. One rests on the disjunctive normal form of S
and also yields the rank selection probabilities. The other is based on inclusion-exclusion
and e.g. yields 0S for some important LULU-operators S. Properties of 0S can be used to
characterize smoothing properties.Preprin
LearnFCA: A Fuzzy FCA and Probability Based Approach for Learning and Classification
Formal concept analysis(FCA) is a mathematical theory based on lattice and order theory used for data analysis and knowledge representation. Over the past several years, many of its extensions have been proposed and applied in several domains including data mining, machine learning, knowledge management, semantic web, software development, chemistry ,biology, medicine, data analytics, biology and ontology engineering.
This thesis reviews the state-of-the-art of theory of Formal Concept Analysis(FCA) and its various extensions that have been developed and well-studied in the past several years. We discuss their historical roots, reproduce the original definitions and derivations with illustrative examples. Further, we provide a literature review of it’s applications and various approaches adopted by researchers in the areas of dataanalysis, knowledge management with emphasis to data-learning and classification problems.
We propose LearnFCA, a novel approach based on FuzzyFCA and probability theory for learning and classification problems. LearnFCA uses an enhanced version of FuzzyLattice which has been developed to store class labels and probability vectors and has the capability to be used for classifying instances with encoded and unlabelled features. We evaluate LearnFCA on encodings from three datasets - mnist, omniglot and cancer images with interesting results and varying degrees of success.
Adviser: Dr Jitender Deogu
Trading inference effort versus size in CNF Knowledge Compilation
Knowledge Compilation (KC) studies compilation of boolean functions f into
some formalism F, which allows to answer all queries of a certain kind in
polynomial time. Due to its relevance for SAT solving, we concentrate on the
query type "clausal entailment" (CE), i.e., whether a clause C follows from f
or not, and we consider subclasses of CNF, i.e., clause-sets F with special
properties. In this report we do not allow auxiliary variables (except of the
Outlook), and thus F needs to be equivalent to f.
We consider the hierarchies UC_k <= WC_k, which were introduced by the
authors in 2012. Each level allows CE queries. The first two levels are
well-known classes for KC. Namely UC_0 = WC_0 is the same as PI as studied in
KC, that is, f is represented by the set of all prime implicates, while UC_1 =
WC_1 is the same as UC, the class of unit-refutation complete clause-sets
introduced by del Val 1994. We show that for each k there are (sequences of)
boolean functions with polysize representations in UC_{k+1}, but with an
exponential lower bound on representations in WC_k. Such a separation was
previously only know for k=0. We also consider PC < UC, the class of
propagation-complete clause-sets. We show that there are (sequences of) boolean
functions with polysize representations in UC, while there is an exponential
lower bound for representations in PC. These separations are steps towards a
general conjecture determining the representation power of the hierarchies PC_k
< UC_k <= WC_k. The strong form of this conjecture also allows auxiliary
variables, as discussed in depth in the Outlook.Comment: 43 pages, second version with literature updates. Proceeds with the
separation results from the discontinued arXiv:1302.442
List, Sample, and Count
Counting plays a fundamental role in many scientific fields including chemistry, physics, mathematics, and computer science. There are two approaches for counting, the first relies on analytical tools to drive closed form expression, while the second takes advantage of the combinatorial nature of the problem to construct an algorithm whose output is the number of structures. There are many algorithmic techniques for counting, they cover the explicit approach of counting by listing to the approximate approach of counting by sampling.
This thesis looks at counting three sets of objects. First, we consider a subclass of boolean functions that are monotone. They appear naturally in great variety of contexts including combinatorics, cryptography, voting theory, and game theory. Next, we consider permutations of n pairs of numbers, called Skolem sequences. These sequences are employed in several areas including construction of Steiner triple systems, binary sequences with controllable complexity, interference resistant codes, and graph labeling. Finally, we consider a variation of the n-queens problem, called the queens of the night. This constraint satisfaction problem is not just a recreational puzzle, but rather it is useful in designing conflict free access in parallel systems. In each case we verify previously known values and provide the next unknown exact value(s) in the counting sequence. Furthermore, we approximate the count for the next unknown values in the sequence by employing a sampling procedure
Calculating the output distribution of stack filters that are erosion-dilation cascades, in particular Lulu-filters
Two procedures to compute the output distribution Ï•S of certain stack
lters S (so called erosion-dilation cascades) are given. One rests on the disjunctive
normal form of S and also yields the rank selection probabilities. The other is based
on inclusion-exclusion and e.g. yields Ï•S for some important LULU-operators S.
Properties of Ï•S can be used to characterize smoothing properties of S. Also, in
the same way as our polynomials Ï•S are computed one could compute the reliability
polynomial of a connected graph, or more generally the reliability polynomial w.r.t.
any positive Boolean function.http://www.tandfonline.com/loi/tqma202016-12-31hb201
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