albanese maps and off diagonal long time asymptotics for the heat kernel we discuss long time asymptotic behaviors of the heat kernel on a non compact riemannian manifold which admits a discontinuous free action of an abelian isometry group with a compact quotient a local central limit theorem and the asymptotic power series expansion for the heat kernel as the time parameter goes to infinity are established by employing perturbation arguments on eigenvalues and eigenfunctions of twisted laplacians our ideas and techniques are motivated partly by analogy with floque bloch theory on periodic schrodinger operators for the asymptotic expansion we make careful use of the classical laplace method in the course of a discussion we observe that the notion of albanese maps associated with the abelian group action is closely related to the asymptotics a similar idea is available for asymptotics of the transition probability of a random walk on a lattice graph the results obtained in the present paper refine our previous ones 4 in the asymptotics the euclidean distance associated with the standard realization of the lattice graph which we call the albanese distance plays a crucial role

effects of domain characteristics on instance based learning algorithms this paper presents average case analyses of instance based learning algorithms the algorithms analyzed employ a variant of k nearest neighbor classifier k nn our analysis deals with a monotone m of n target concept with irrelevant attributes and handles three types of noise relevant attribute noise irrelevant attribute noise and class noise we formally represent the expected classification accuracy of k nn as a function of domain characteristics including the number of training instances the number of relevant and irrelevant attributes the threshold number in the target concept the probability of each attribute the noise rate for each type of noise and k we also explore the behavioral implications of the analyses by presenting the effects of domain characteristics on the expected accuracy of k nn and on the optimal value of k for artificial domains

characterizations of monotonic and dual monotonic language learning the present paper deals with monotonic and dual monotonic language learning from positive as well as from positive and negative examples the three notions of monotonicity reflect different formalizations of the requirement that the learner has to produce better and better generalizations when fed more and more data on the concept to be learned the three versions of dual monotonicity describe the concept that the inference device has to produce specializations that fit better and better to the target language we characterize strong monotonic monotonic weak monotonic dual strong monotonic dual monotonic and monotonic dual monotonic as well as finite language learning from positive data in terms of recursively generable finite sets these characterizations provide a unifying framework for learning from positive data under the various monotonicity constraints moreover they yield additional insight into the problem of what a natural learning algorithm should look like

large deviation and the tangent cone at infinity of a crystal lattice we discuss a large deviation property of a periodic random walk on a crystal lattice in view of geometry and relate it to a rational convex polyhedron in the first homology group of a finite graph which as we shall observe has remarkable combinatorial features and shows up also in the gromov hausdorff limit of a crystal lattice

identification criteria and lower bounds for perceptron like learning rules perceptron like learning rules are known to require exponentially many correction steps in order to identify boolean threshold functions exactly we introduce criteria that are weaker than exact identification and investigate whether learning becomes significantly faster if exact identification is replaced by one of these criteria probably approximately correct pac identification order identification and sign identification pac identification is based on the learning paradigm introduced by valiant 1984 and is known to be easier than exact identification order identification uses the fact that each threshold function induces an ordering relation on the input variables that can be represented by weights of linear size sign identification is based on a property of threshold functions known as unateness and requires only weights of constant size we show that perceptron like learning rules cannot satisfy these criteria when the number of correction steps is to be bounded by a polynomial we also pre

how to construct random functions a constructive theory of randomness for functions based on computational complexity is developed and a pseudorandom function generator is presented this generator is a deterministic polynomial time algorithm that transforms pairs g r where g is any one way function and r is a random k bit string to polynomial time computable functions r 1 2 k 1 2 k these r s cannot be distinguished from random functions by any probabilistic polynomial time algorithm that asks and receives the value of a function at arguments of its choice the result has applications in cryptography random constructions and complexity theory

parameterized learnability of juntas we study the parameterized complexity of learning k juntas and some variations of juntas we show the hardness of learning k juntas and subclasses of k juntas in the pac model by reductions from a w 2 complete problem on the other hand as a consequence of a more general result we show that k juntas are exactly learnable with improper equivalence queries and access to a w p oracle

interpolation and approximation of sparse multivariate polynomials over gf 2 a function f 0 1 n to 0 1 is called t sparse if the n variable polynomial representation of f over gf 2 contains at most t monomials such functions are uniquely determined by their values at the so called critical set of all binary n tuples of hamming weight geqq n lfloor log 2 t rfloor 1 an algorithm is presented for interpolating any t sparse function f given the values of f at the critical set the time complexity of the proposed algorithm is proportional to n t and the size of the critical set then the more general problem of approximating 1 sparse functions is considered in which case the approximating function may differ from f at a fraction varepsilon of the space 0 1 n it is shown that o t varepsilon cdot n evaluation points are sufficient for the deterministic varepsilon approximation of any t sparse function and that an order t varepsilon alpha t varepsilon cdot log n points are necessary for this purpose wh

inductive inference of approximations in this paper we investigate inductive inference identification criteria which permit infinitely many errors in explanations but which require that the density of these errors be no more than a certain prespectified amount we introduce three hierarchies of such criteria each of which has the same order type as the real unit interval these three hierarchies are progressively more strict in the way they measure density of errors of explanations the strictest of the three turns out to have all of its members save one incomparable to the identification criterion which permits finitely many errors in explanations

on the power of probabilistic strategies in inductive inference abstract inductive inference of programs of recursive functions from input output examples by probabilistic strategies with an a priori bound n n of changes of hypotheses is investigated advantages of probabilistic strategies over deterministic ones are shown concerning their power in principle for every n 2 with probability arbitrarily close to 1 inference of function classes which cannot be inferred by any deterministic strategy with n changes of hypotheses as well as their computational complexity linear speed up of the number of changes of hypotheses necessary for inference by deterministic strategies

comparison of identification criteria for machine inductive inference abstract a natural pl 1 hierarchy of successively more general criteria of success for inductive inference machines is described based on the size of sets of anomalies in programs synthesized by such machines these criteria are compared to others in the literature some of our results are interpreted as tradeoff results or as showing the inherent relative computational complexity of certain processes and others are interpreted from a positivistic mechanistic philosophical stance as theorems in philosophy of science the techniques of recursive function theory are employed including ordinary and infinitary recursion theorems

standard realizations of crystal lattices via harmonic maps an eells sampson type theorem for harmonic maps from a finite weighted graph is employed to characterize the equilibrium configurations of crystals it is thus observed that the mimimum principle frames symmetry of crystals

random walks on periodic graphs this paper concerns random walks on periodic graphs embedded in the d dimensional euclidian space r d and obtains asymptotic expansions of the green functions of them up to the second order term which expressed fairly explicitly are easily computable for many examples the result is used to derive an asymptotic form of the hitting distribution of a hyperplane of codimension one which involves not only the first but also second order terms of the expansion of the green function we also give similar expansions of the transition probabilities of the walks

graph distance measures based on topological indices revisited graph distance measures based on topological indices have been already explored by dehmer et al also inequalities for those graph distance measures have been proved in this paper we continue studying such comparative graph measures based on the well known wiener index graph energy and randic index respectively we prove extremal properties of the graph distance measures for some special classes of graphs to demonstrate useful properties of the measures we also discuss numerical results to conclude the paper we state some open problems

fast identification of geometric objects with membership queries we investigate the number of membership queries that are needed to identify polygons i e intersections of halfplanes over a two dimensional grid 0 n 1 2 we exhibit a learning algorithm that learns 100 correctly while requiring no random examples and not more membership queries than previous algorithms needed in addition to their random examples for probably almost correctly learning even for moderate values of furthermore the learning algorithm in this paper uses only grid points for its membership queries this appears to be appropriate in situations where the probing device has only a limited resolution and for applications to the set of pixels on a two dimensional screen the learning algorithm that is described in this paper overcomes a fundamental obstacle related to the zigzag borderline that is generated by a halfplane over a discrete grid this obstacle had thwarted previous attempts to show that in angluin s model for on line learning with equivalence and membership queries one can learn in an efficient manner even those classes of geometric objects which cannot be learnt fast by equivalence queries alone such as rectangles in general position

equivalence of models for polynomial learnability abstract in this paper we consider several variants of valiant s learnability model that have appeared in the literature we give conditions under which these models are equivalent in terms of the polynomially learnable concept classes they define these equivalences allow comparisons of most of the existing theorems in valiant style learnability and show that several simplifying assumptions on polynomial learning algorithms can be made without loss of generality we also give a useful reduction of learning problems to the problem of finding consistent hypotheses and give comparisons and equivalences between valiant s model and the prediction learning models of haussler littlestone and warmuth in 29th annual ieee symposium on foundations of computer science 1988

long time asymptotics of non symmetric random walks on crystal lattices abstract in the present paper we study long time asymptotics of non symmetric random walks on crystal lattices from a view point of discrete geometric analysis due to kotani and sunada 11 25 we observe that the euclidean metric associated with the standard realization of the crystal lattice called the albanese metric naturally appears in the asymptotics in the former half of the present paper we establish two kinds of functional central limit theorems for random walks we first show that the brownian motion on the euclidean space with the albanese metric appears as the scaling limit of the usual central limit theorem for the random walk next we introduce a family of random walks which interpolates between the original non symmetric random walk and the symmetrized one we then capture the brownian motion with a constant drift of the asymptotic direction on the euclidean space with the albanese metric associated with the symmetrized random walk through another kind of central limit theorem for the family of random walks in the latter half of the present paper we give a spectral geometric proof of the asymptotic expansion of the n step transition probability for the non symmetric random walk this asymptotic expansion is a refinement of the local central limit theorem obtained by sunada 22 23 and is a generalization of the result in 11 for symmetric random walks on crystal lattices to non symmetric cases

asymptotic behavior of the transition probability of a random walk on an infinite graph ideas cultivated in spectral geometry are applied to obtain an asymptotic property of a reversible random walk on an infinite graph satisfying a certain periodic condition in the course of our argument we employ perturbation theory for the maximal eigenvalues of twisted transition operator as a result an asymptotic of the probabilityp n x y that a particle starting atxreachesyat timenasngoes to infinity is established

version spaces and the consistency problem a version space is a collection of concepts consistent with a given set of positive and negative examples mitchell artificial intelligence 18 1982 203 226 proposed representing a version space by its boundary sets the maximally general g and maximally specific consistent concepts s for many simple concept classes the size of g and s is known to grow exponentially in the number of positive and negative examples this paper argues that previous work on alternative representations of version spaces has disguised the real question underlying version space reasoning we instead show that tractable reasoning with version spaces turns out to depend on the consistency problem i e determining if there is any concept consistent with a set of positive and negative examples indeed we show that tractable version space reasoning is possible if and only if there is an efficient algorithm for the consistency problem our observations give rise to new concept classes for which tractable version space reasoning is now possible e g 1 decision lists monotone depth two formulas and halfspaces

computational limitations on learning from examples the computational complexity of learning boolean concepts from examples is investigated it is shown for various classes of concept representations that these cannot be learned feasibly in a distribution free sense unless r np these classes include a disjunctions of two monomials b boolean threshold functions and c boolean formulas in which each variable occurs at most once relationships between learning of heuristics and finding approximate solutions to np hard optimization problems are given