Polynomial-Time Decomposition Algorithms for Support Vector Machines

This paper studies the convergence properties of a general class of decomposition algorithms for support vector machines (SVMs). We provide a model algorithm for decomposition, and prove necessary and sufficient conditions for stepwise improvement of this algorithm. We introduce a simple rate certifying condition and prove a polynomial-time bound on the rate of convergence of the model algorithm when it satisfies this condition. Although it is not clear that existing SVM algorithms satisfy this condition, we provide a version of the model algorithm that does. For this algorithm we show that when the slack multiplier C satisfies 1/2 C mL, where m is the number of samples and L is a matrix norm, then it takes no more than 4LC ² m ⁴/ iterations to drive the criterion to within of its optimum.     

From Conditional Events to Conditional Measures: A New Axiomatic Approach

Our starting point is a definition of conditional event EH which differs from many seemingly similar ones adopted in the relevant literature since 1935, starting with de Finetti. In fact, if we do not assign the same third value u (undetermined) to all conditional events, but make it depend on EH, it turns out that this function t(EH) can be taken as a general conditional uncertainty measure, and we get (through a suitable – in a sense, compulsory – choice of the relevant operations among conditional events) the natural axioms for many different (besides probability) conditional measures.     

Unification in partially commutative semigroups

Unification algorithms have been constructed for semigroups and commutative semigroups. This paper considers the intermediate case of partially commutative semigroups. We introduce classesN and of such semigroups and justify their use. We present an equation-solving algorithm for any member of the classN. This algorithm is relative to having an algorithm to determine all non-negative solutions of a certain class of diophantine equations of degree 2 which we call -equations. The difficulties arising when attempting to solve equations in members of the class are discussed, and we present arguments that strongly suggest that unification in these semigroups is undecidable.     

Obvious inferences

The notion of obvious inference in predicate logic is discussed from the viewpoint of proof-checker applications in logic and mathematics education. A class of inferences in predicate logic is defined and it is proposed to identify it with the class of obvious logical inferences. The definition is compared with other approaches. The algorithm for implementing the obviousness decision procedure follows directly from the definition.     

Surface reconstruction from stereoscopy and “shape from shading” in SEM images

The computational reconstruction of surface topographies from scanning electron microscope (SEM) images has been extensively investigated in the past, but fundamental image processing problems still exist. Since conventional approaches adapted from general-purpose image processing have not sufficiently met the requirements in terms of resolution and reliability, we have explored combining different methods to obtain better results.This paper presents a least-squares combination of conventional stereoscopy with shape from shading and a way of obtaining self-consistent surface profiles from stereoscopy and stereo-intrinsic shape from shading using dynamic programming techniques. Results are presented showing how this combined analysis of multi-sensorial data yields improvements of the reconstructed surface topography that cannot be obtained from individual sensor signals alone.     

Almost surely continuous solutions of a nonlinear stochastic integral equation

A nonlinear stochastic integral equation of the Hammerstein type in the formx(t; ) = h(t, x(t; )) + _s k(t, s; )f(s, x(s; ); )d(s) is studied wheret S, a measure space with certain properties, , the supporting set of a probability measure space (,A, P), and the integral is a Bochner integral. A random solution of the equation is defined to be an almost surely continuousm-dimensional vector-valued stochastic process onS which is bounded with probability one for eacht S and which satisfies the equation almost surely. Several theorems are proved which give conditions such that a unique random solution exists. AMS (MOS) subject classifications (1970): Primary; 60H20, 45G99. Secondary: 60G99.     

Fuzzy measurement in the Mishnah and the Talmud

I discuss the attitude of Jewish law sources from the 2nd–:5th centuries to the imprecision of measurement. I review a problem that the Talmud refers to, somewhat obscurely, as impossible reduction. This problem arises when a legal rule specifies an object by referring to a maximized (or minimized) measurement function, e.g., when a rule applies to the largest part of a divided whole, or to the first incidence that occurs, etc. A problem that is often mentioned is whether there might be hypothetical situations involving more than one maximal (or minimal) value of the relevant measurement and, given such situations, what is the pertinent legal rule. Presumption of simultaneous occurrences or equally measured values are also a source of embarrassment to modern legal systems, in situations exemplified in the paper, where law determines a preference based on measured values. I contend that the Talmudic sources discussing the problem of impossible reduction were guided by primitive insights compatible with fuzzy logic presentation of the inevitable uncertainty involved in measurement. I maintain that fuzzy models of data are compatible with a positivistic epistemology, which refuses to assume any precision in the extra-conscious world that may not be captured by observation and measurement. I therefore propose this view as the preferred interpretation of the Talmudic notion of impossible reduction. Attributing a fuzzy world view to the Talmudic authorities is meant not only to increase our understanding of the Talmud but, in so doing, also to demonstrate that fuzzy notions are entrenched in our practical reasoning. If Talmudic sages did indeed conceive the results of measurements in terms of fuzzy numbers, then equality between the results of measurements had to be more complicated than crisp equations. The problem of impossible reduction could lie in fuzzy sets with an empty core or whose membership functions were only partly congruent. Reduction is impossible may thus be reconstructed as there is no core to the intersection of two measures. I describe Dirichlet maps for fuzzy measurements of distance as a rough partition of the universe, where for any region A there may be a non-empty set of - _A (upper approximation minus lower approximation), where the problem of impossible reduction applies. This model may easily be combined with probabilistic extention. The possibility of adopting practical decision standards based on -cuts (and therefore applying interval analysis to fuzzy equations) is discussed in this context. I propose to characterize the uncertainty that was presumably capped by the old sages as U-uncertainty, defined, for a non-empty fuzzy set A on the set of real numbers, whose -cuts are intervals of real numbers, as U(A) = 1/h(A) ₀ ^h(A) log [1+(A)]d, where h(A) is the largest membership value obtained by any element of A and (A) is the measure of the -cut of A defined by the Lebesge integral of its characteristic function.     

The dynamic clusters method in nonhierarchical clustering

Given a finite setE R ⁿ, the problem is to find clusters (or subsets of similar points inE) and at the same time to find the most typical elements of this set. An original mathematical formulation is given to the problem. The proposed algorithm operates on groups of points, called samplings (samplings may be called multiple centers or cores); these samplings adapt and evolve into interesting clusters. Compared with other clustering algorithms, this algorithm requires less machine time and storage. We provide some propositions about nonprobabilistic convergence and a sufficient condition which ensures the decrease of the criterion. Some computational experiments are presented.     

Thiele Rational Interpolation for the Numerical Computation of the Reversible Randles–Sevcik Function in Electrochemistry

This paper uses Thiele rational interpolation to derive a simple method for computing the Randles–Sevcik function ^1/2(x), with relative error at most 1.9 × 10^–5 for – < x < . We develop a piecewise approximation method for the numerical computation of ^1/2(x) on the union (–, –10) [–10, 10] (10, ). This approximation is particularly convenient to employ in electrochemical applications where four significant digits of accuracy are usually sufficient. Although this paper is primarily concerned with the approximation of the Randles–Sevcik function, some examples are included that illustrate how Thiele rational interpolation can be employed to generate useful approximations to other functions of interest in scientific work.     

The informed neuron: Issues in the use of information theory in the behavioral sciences

The concept of information is virtually ubiquitous in contemporary cognitive science. It is claimed to be processed (in cognitivist theories of perception and comprehension), stored (in cognitivist theories of memory and recognition), and otherwise manipulated and transformed by the human central nervous system. Fred Dretske's extensive philosophical defense of a theory of informational content (semantic information) based upon the Shannon-Weaver formal theory of information is subjected to critical scrutiny. A major difficulty is identified in Dretske's equivocations in the use of the concept of a signal bearing informational content. Gibson's alternative conception of information (construed as analog by Dretske), while avoiding many of the problems located in the conventional use of signal, raises different but equally serious questions. It is proposed that, taken literally, the human CNS does not extract or process information at all; rather, whatever information is construed as locatable in the CNS is information only for an observer-theorist and only for certain purposes.Blood courses through our veins, andinformation through our central nervous system.— A Neuropsychology Textbook.     

How to find Steiner minimal trees in euclideand-space

This paper has two purposes. The first is to present a new way to find a Steiner minimum tree (SMT) connectingN sites ind-space,d >- 2. We present (in Appendix 1) a computer code for this purpose. This is the only procedure known to the author for finding Steiner minimal trees ind-space ford > 2, and also the first one which fits naturally into the framework of backtracking and branch-and-bound. Finding SMTs of up toN = 12 general sites ind-space (for anyd) now appears feasible.We tabulate Steiner minimal trees for many point sets, including the vertices of most of the regular and Archimedeand-polytopes with <- 16 vertices. As a consequence of these tables, the Gilbert-Pollak conjecture is shown to be false in dimensions 3–9. (The conjecture remains open in other dimensions; it is probably false in all dimensionsd withd 3, but it is probably true whend = 2.)The second purpose is to present some new theoretical results regarding the asymptotic computational complexity of finding SMTs to precision .We show that in two-dimensions, Steiner minimum trees may be found exactly in exponential time O(C ^N ) on a real RAM. (All previous provable time bounds were superexponential.) If the tree is only wanted to precision , then there is an (N/)^O(N)-time algorithm, which is subexponential if 1/ grows only polynomially withN. Also, therectilinear Steiner minimal tree ofN points in the plane may be found inN ^O(N) time.J. S. Provan devised an O(N ⁶/⁴)-time algorithm for finding the SMT of a convexN-point set in the plane. (Also the rectilinear SMT of such a set may be found in O(N ⁶) time.) One therefore suspects that this problem may be solved exactly in polynomial time. We show that this suspicion is in fact true—if a certain conjecture about the size of Steiner sensitivity diagrams is correct.All of these algorithms are for a real RAM model of computation allowing infinite precision arithmetic. They make no probabilistic or other assumptions about the input; the time bounds are valid in the worst case; and all our algorithms may be implemented with a polynomial amount of space. Only algorithms yielding theexact optimum SMT, or trees with lengths (1 + ) × optimum, where is arbitrarily small, are considered here.     

One Connection between Standard Invariance Conditions on Modal Formulas and Generalized Quantifiers

The language of standard propositional modal logic has one operator ( or ), that can be thought of as being determined by the quantifiers or , respectively: for example, a formula of the form is true at a point s just in case all the immediate successors of s verify .This paper uses a propositional modal language with one operator determined by a generalized quantifier to discuss a simple connection between standard invariance conditions on modal formulas and generalized quantifiers: the combined generalized quantifier conditions of conservativity and extension correspond to the modal condition of invariance under generated submodels, and the modal condition of invariance under bisimulations corresponds to the generalized quantifier being a Boolean combination of and .     

A Full Bayesian Approach to Curve and Surface Reconstruction

When interpolating incomplete data, one can choose a parametric model, or opt for a more general approach and use a non-parametric model which allows a very large class of interpolants. A popular non-parametric model for interpolating various types of data is based on regularization, which looks for an interpolant that is both close to the data and also smooth in some sense. Formally, this interpolant is obtained by minimizing an error functional which is the weighted sum of a fidelity term and a smoothness term.The classical approach to regularization is: select optimal weights (also called hyperparameters) that should be assigned to these two terms, and minimize the resulting error functional.However, using only the optimal weights does not guarantee that the chosen function will be optimal in some sense, such as the maximum likelihood criterion, or the minimal square error criterion. For that, we have to consider all possible weights.The approach suggested here is to use the full probability distribution on the space of admissible functions, as opposed to the probability induced by using a single combination of weights. The reason is as follows: the weight actually determines the probability space in which we are working. For a given weight , the probability of a function f is proportional to exp(– f² _uu du) (for the case of a function with one variable). For each different , there is a different solution to the restoration problem; denote it by f. Now, if we had known , it would not be necessary to use all the weights; however, all we are given are some noisy measurements of f, and we do not know the correct . Therefore, the mathematically correct solution is to calculate, for every , the probability that f was sampled from a space whose probability is determined by , and average the different f's weighted by these probabilities. The same argument holds for the noise variance, which is also unknown.Three basic problems are addressed is this work: Computing the MAP estimate, that is, the function f maximizing Pr(f/D) when the data D is given. This problem is reduced to a one-dimensional optimization problem. Computing the MSE estimate. This function is defined at each point x as f(x)Pr(f/D) f. This problem is reduced to computing a one-dimensional integral.In the general setting, the MAP estimate is not equal to the MSE estimate. Computing the pointwise uncertainty associated with the MSE solution. This problem is reduced to computing three one-dimensional integrals.     

The network complexity and the Turing machine complexity of finite functions

Summary Let L(f) be the network complexity of a Boolean function L(f). For any n-ary Boolean function L(f) let . Hereby p ranges over all relative Turing programs and ranges over all oracles such that given the oracle , the restriction of p to inputs of length n is a program for L(f). p is the number of instructions of p. T _p (n) is the time bound and S _p of the program p relative to the oracle on inputs of length n. Our main results are (1) L(f) O(TC(L(f))), (2) TC(f) O(L(f) ^{2 2+}) for every O.The results of this paper have been reported in a main lecture at the 1975 annual meeting of GAMM, April 2–5, Göttingen     

Property preserving abstractions for the verification of concurrent systems

We study property preserving transformations for reactive systems. The main idea is the use of simulations parameterized by Galois connections (, ), relating the lattices of properties of two systems. We propose and study a notion of preservation of properties expressed by formulas of a logic, by a function mapping sets of states of a systemS into sets of states of a systemS'. We give results on the preservation of properties expressed in sublanguages of the branching time -calculus when two systemsS andS' are related via (, )-simulations. They can be used to verify a property for a system by verifying the same property on a simpler system which is an abstraction of it. We show also under which conditions abstraction of concurrent systems can be computed from the abstraction of their components. This allows a compositional application of the proposed verification method.This is a revised version of the papers [2] and [16]; the results are fully developed in [28].This work was partially supported by ESPRIT Basic Research Action REACT.Verimag is a joint laboratory of CNRS, Institut National Polytechnique de Grenoble, Université J. Fourier and Verilog SA associated with IMAG.     

Personalized and Adaptive Services on Board a Car: An Application for Tourist Information

Personalization and adaptation techniques are an interesting opportunity to design new services on-board vehicles. In this context, in fact, the need of an individual user to receive the right service at the right time and in the right way is more critical than in other cases, where personalization and adaptation already showed interesting advantages. At the same time, this context of application can provide new interesting insights for user modeling and adaptation. In the paper we present an architecture for providing personalized services on-board vehicles and we discuss an application to the case of tourist information. We focus on the choices we made to design an on-board system which was as less intrusive and distracting as possible and that could adapt its recommendations, the way it presents them and its own behavior to the user's preferences/interests and to the context of interaction (especially the driving conditions).     

The connection between an event structure semantics and an operational semantics forTCSP

The relation between an operational interleaving semantics forTSCP based on a transition system and a compositional true concurrency semantics based on event structures is studied. In particular we extend the consistency result of Goltz and Loogan [15] forTCSP processes without recursion to the general case. Thus we obtain for everyTCSP processP that its operational meaningO(P) and the interleaving behaviourO( M3P3) which is derived from the event structureM3P3 associated withP are bisimilar.     

Approximate syllogisms – on the logic of everyday life

Since Aristotle it is recognised that a valid syllogism cannot have two particular premises. However, that is not how a lay person sees it; at least as long as the premises read many, most etc, instead of a plain some. The lay people are right if one considers that these syllogisms do not have strict but approximate (Zadeh) validity. Typically there are only particular premises available in everyday life and one is dependent on such syllogisms. – Some rules on the usage of particular premises are given below.     

Petri net based verification of distributed algorithms: An example

A technique to model and to verify distributed algorithms is suggested. This technique (based on Petri nets) reduces the modelling and analysis effort to a reasonable level. The paper outlines the technique using the example of a typical network algorithm, theecho algorithm.Supported by the DFG-projects Verteilte Algorithmen and Konsensalgorithmen     

The input/output complexity of transitive closure

Suppose a directed graph has its arcs stored in secondary memory, and we wish to compute its transitive closure, also storing the result in secondary memory. We assume that an amount of main memory capable of holdings values is available, and thats lies betweenn, the number of nodes of the graph, ande, the number of arcs. The cost measure we use for algorithms is theI/O complexity of Kung and Hong, where we count 1 every time a value is moved into main memory from secondary memory, or vice versa.In the dense case, wheree is close ton ², we show that I/O equal toO(n ³/s) is sufficient to compute the transitive closure of ann-node graph, using main memory of sizes. Moreover, it is necessary for any algorithm that is standard, in a sense to be defined precisely in the paper. Roughly, standard means that paths are constructed only by concatenating arcs and previously discovered paths. For the sparse case, we show that I/O equal toO(n ²e/s) is sufficient, although the algorithm we propose meets our definition of standard only if the underlying graph is acyclic. We also show that(n ²e/s) is necessary for any standard algorithm in the sparse case. That settles the I/O complexity of the sparse/acyclic case, for standard algorithms. It is unknown whether this complexity can be achieved in the sparse, cyclic case, by a standard algorithm, and it is unknown whether the bound can be beaten by nonstandard algorithms.We then consider a special kind of standard algorithm, in which paths are constructed only by concatenating arcs and old paths, never by concatenating two old paths. This restriction seems essential if we are to take advantage of sparseness. Unfortunately, we show that almost another factor ofn I/O is necessary. That is, there is an algorithm in this class using I/OO(n ³e/s) for arbitrary sparse graphs, including cyclic ones. Moreover, every algorithm in the restricted class must use(n ³e/s/log³ n) I/O, on some cyclic graphs.The work of this author was partially supported by NSF grant IRI-87-22886, IBM contract 476816, Air Force grant AFOSR-88-0266 and a Guggenheim fellowship.