Comparative study of the critical behavior in one-dimensional random and aperiodic environments

We consider cooperative processes (quantum spin chains and random walks) in one-dimensional fluctuating random and aperiodic environments characterized by fluctuating exponents omega>0. At the critical point the random and aperiodic systems scale essentially anisotropically in a similar fashion: length (L) and time (t) scales are related as t ~ log^{1/omega}. Also some critical exponents, characterizing the singularities of average quantities, are found to be universal functions of omega, whereas some others do depend on details of the distribution of the disorder. In the off-critical region there is an important difference between the two types of environments: in aperiodic systems there are no extra (Griffiths)-singularities.Comment: 13 pages RevTeX, 10 eps-figures include

Online hashing for fast similarity search

In this thesis, the problem of online adaptive hashing for fast similarity search is studied. Similarity search is a central problem in many computer vision applications. The ever-growing size of available data collections and the increasing usage of high-dimensional representations in describing data have increased the computational cost of performing similarity search, requiring search strategies that can explore such collections in an efficient and effective manner. One promising family of approaches is based on hashing, in which the goal is to map the data into the Hamming space where fast search mechanisms exist, while preserving the original neighborhood structure of the data. We first present a novel online hashing algorithm in which the hash mapping is updated in an iterative manner with streaming data. Being online, our method is amenable to variations of the data. Moreover, our formulation is orders of magnitude faster to train than state-of-the-art hashing solutions. Secondly, we propose an online supervised hashing framework in which the goal is to map data associated with similar labels to nearby binary representations. For this purpose, we utilize Error Correcting Output Codes (ECOCs) and consider an online boosting formulation in learning the hash mapping. Our formulation does not require any prior assumptions on the label space and is well-suited for expanding datasets that have new label inclusions. We also introduce a flexible framework that allows us to reduce hash table entry updates. This is critical, especially when frequent updates may occur as the hash table grows larger and larger. Thirdly, we propose a novel mutual information measure to efficiently infer the quality of a hash mapping and retrieval performance. This measure has lower complexity than standard retrieval metrics. With this measure, we first address a key challenge in online hashing that has often been ignored: the binary representations of the data must be recomputed to keep pace with updates to the hash mapping. Based on our novel mutual information measure, we propose an efficient quality measure for hash functions, and use it to determine when to update the hash table. Next, we show that this mutual information criterion can be used as an objective in learning hash functions, using gradient-based optimization. Experiments on image retrieval benchmarks confirm the effectiveness of our formulation, both in reducing hash table recomputations and in learning high-quality hash functions

Analytic solution for tachyon condensation in open string field theory

We propose a new basis in Witten's open string field theory, in which the star product simplifies considerably. For a convenient choice of gauge the classical string field equation of motion yields straightforwardly an exact analytic solution that represents the nonperturbative tachyon vacuum. The solution is given in terms of Bernoulli numbers and the equation of motion can be viewed as novel Euler--Ramanujan-type identity. It turns out that the solution is the Euler--Maclaurin asymptotic expansion of a sum over wedge states with certain insertions. This new form is fully regular from the point of view of level truncation. By computing the energy difference between the perturbative and nonperturbative vacua, we prove analytically Sen's first conjecture.Comment: 60 pages, 4 figures, v2: typos corrected, references adde

Universality classes of interaction structures for NK fitness landscapes

Kauffman's NK-model is a paradigmatic example of a class of stochastic models of genotypic fitness landscapes that aim to capture generic features of epistatic interactions in multilocus systems. Genotypes are represented as sequences of $L$ binary loci. The fitness assigned to a genotype is a sum of contributions, each of which is a random function defined on a subset of $k \le L$ loci. These subsets or neighborhoods determine the genetic interactions of the model. Whereas earlier work on the NK model suggested that most of its properties are robust with regard to the choice of neighborhoods, recent work has revealed an important and sometimes counter-intuitive influence of the interaction structure on the properties of NK fitness landscapes. Here we review these developments and present new results concerning the number of local fitness maxima and the statistics of selectively accessible (that is, fitness-monotonic) mutational pathways. In particular, we develop a unified framework for computing the exponential growth rate of the expected number of local fitness maxima as a function of $L$, and identify two different universality classes of interaction structures that display different asymptotics of this quantity for large $k$. Moreover, we show that the probability that the fitness landscape can be traversed along an accessible path decreases exponentially in $L$ for a large class of interaction structures that we characterize as locally bounded. Finally, we discuss the impact of the NK interaction structures on the dynamics of evolution using adaptive walk models.Comment: 61 pages, 9 figure