Rerepresenting and Restructuring Domain Theories: A Constructive Induction Approach

Theory revision integrates inductive learning and background knowledge by combining training examples with a coarse domain theory to produce a more accurate theory. There are two challenges that theory revision and other theory-guided systems face. First, a representation language appropriate for the initial theory may be inappropriate for an improved theory. While the original representation may concisely express the initial theory, a more accurate theory forced to use that same representation may be bulky, cumbersome, and difficult to reach. Second, a theory structure suitable for a coarse domain theory may be insufficient for a fine-tuned theory. Systems that produce only small, local changes to a theory have limited value for accomplishing complex structural alterations that may be required. Consequently, advanced theory-guided learning systems require flexible representation and flexible structure. An analysis of various theory revision systems and theory-guided learning systems reveals specific strengths and weaknesses in terms of these two desired properties. Designed to capture the underlying qualities of each system, a new system uses theory-guided constructive induction. Experiments in three domains show improvement over previous theory-guided systems. This leads to a study of the behavior, limitations, and potential of theory-guided constructive induction.Comment: See http://www.jair.org/ for an online appendix and other files accompanying this articl

Shearlets and Optimally Sparse Approximations

Multivariate functions are typically governed by anisotropic features such as edges in images or shock fronts in solutions of transport-dominated equations. One major goal both for the purpose of compression as well as for an efficient analysis is the provision of optimally sparse approximations of such functions. Recently, cartoon-like images were introduced in 2D and 3D as a suitable model class, and approximation properties were measured by considering the decay rate of the $L^2$ error of the best $N$-term approximation. Shearlet systems are to date the only representation system, which provide optimally sparse approximations of this model class in 2D as well as 3D. Even more, in contrast to all other directional representation systems, a theory for compactly supported shearlet frames was derived which moreover also satisfy this optimality benchmark. This chapter shall serve as an introduction to and a survey about sparse approximations of cartoon-like images by band-limited and also compactly supported shearlet frames as well as a reference for the state-of-the-art of this research field.Comment: in "Shearlets: Multiscale Analysis for Multivariate Data", Birkh\"auser-Springe

Probabilistic Reconstruction in Compressed Sensing: Algorithms, Phase Diagrams, and Threshold Achieving Matrices

Compressed sensing is a signal processing method that acquires data directly in a compressed form. This allows one to make less measurements than what was considered necessary to record a signal, enabling faster or more precise measurement protocols in a wide range of applications. Using an interdisciplinary approach, we have recently proposed in [arXiv:1109.4424] a strategy that allows compressed sensing to be performed at acquisition rates approaching to the theoretical optimal limits. In this paper, we give a more thorough presentation of our approach, and introduce many new results. We present the probabilistic approach to reconstruction and discuss its optimality and robustness. We detail the derivation of the message passing algorithm for reconstruction and expectation max- imization learning of signal-model parameters. We further develop the asymptotic analysis of the corresponding phase diagrams with and without measurement noise, for different distribution of signals, and discuss the best possible reconstruction performances regardless of the algorithm. We also present new efficient seeding matrices, test them on synthetic data and analyze their performance asymptotically.Comment: 42 pages, 37 figures, 3 appendixe

On the performance of algorithms for the minimization of ℓ1\ell_1-penalized functionals

The problem of assessing the performance of algorithms used for the minimization of an $\ell_1$-penalized least-squares functional, for a range of penalty parameters, is investigated. A criterion that uses the idea of `approximation isochrones' is introduced. Five different iterative minimization algorithms are tested and compared, as well as two warm-start strategies. Both well-conditioned and ill-conditioned problems are used in the comparison, and the contrast between these two categories is highlighted.Comment: 18 pages, 10 figures; v3: expanded version with an additional synthetic test problem

Fluorescence kinetics of flavin adenine dinucleotide in different microenvironments

Fluorescence kinetics of flavin adenine dinucleotide was measured in a wide time and spectral range in different media, affecting its intra- end extramolecular interactions, and analyzed by a new method based on compressed sensing

Efficient Resolution of Anisotropic Structures

We highlight some recent new delevelopments concerning the sparse representation of possibly high-dimensional functions exhibiting strong anisotropic features and low regularity in isotropic Sobolev or Besov scales. Specifically, we focus on the solution of transport equations which exhibit propagation of singularities where, additionally, high-dimensionality enters when the convection field, and hence the solutions, depend on parameters varying over some compact set. Important constituents of our approach are directionally adaptive discretization concepts motivated by compactly supported shearlet systems, and well-conditioned stable variational formulations that support trial spaces with anisotropic refinements with arbitrary directionalities. We prove that they provide tight error-residual relations which are used to contrive rigorously founded adaptive refinement schemes which converge in $L_2$. Moreover, in the context of parameter dependent problems we discuss two approaches serving different purposes and working under different regularity assumptions. For frequent query problems, making essential use of the novel well-conditioned variational formulations, a new Reduced Basis Method is outlined which exhibits a certain rate-optimal performance for indefinite, unsymmetric or singularly perturbed problems. For the radiative transfer problem with scattering a sparse tensor method is presented which mitigates or even overcomes the curse of dimensionality under suitable (so far still isotropic) regularity assumptions. Numerical examples for both methods illustrate the theoretical findings

Restricted Isometries for Partial Random Circulant Matrices

In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to succeed when the restricted isometry constants of the sampling matrix are small. Many potential applications of compressed sensing involve a data-acquisition process that proceeds by convolution with a random pulse followed by (nonrandom) subsampling. At present, the theoretical analysis of this measurement technique is lacking. This paper demonstrates that the $s$th order restricted isometry constant is small when the number $m$ of samples satisfies $m \gtrsim (s \log n)^{3/2}$, where $n$ is the length of the pulse. This bound improves on previous estimates, which exhibit quadratic scaling

Guaranteed clustering and biclustering via semidefinite programming

Identifying clusters of similar objects in data plays a significant role in a wide range of applications. As a model problem for clustering, we consider the densest k-disjoint-clique problem, whose goal is to identify the collection of k disjoint cliques of a given weighted complete graph maximizing the sum of the densities of the complete subgraphs induced by these cliques. In this paper, we establish conditions ensuring exact recovery of the densest k cliques of a given graph from the optimal solution of a particular semidefinite program. In particular, the semidefinite relaxation is exact for input graphs corresponding to data consisting of k large, distinct clusters and a smaller number of outliers. This approach also yields a semidefinite relaxation for the biclustering problem with similar recovery guarantees. Given a set of objects and a set of features exhibited by these objects, biclustering seeks to simultaneously group the objects and features according to their expression levels. This problem may be posed as partitioning the nodes of a weighted bipartite complete graph such that the sum of the densities of the resulting bipartite complete subgraphs is maximized. As in our analysis of the densest k-disjoint-clique problem, we show that the correct partition of the objects and features can be recovered from the optimal solution of a semidefinite program in the case that the given data consists of several disjoint sets of objects exhibiting similar features. Empirical evidence from numerical experiments supporting these theoretical guarantees is also provided

Learning Partially Shared Dictionaries for Domain Adaptation

Abstract. Real world applicability of many computer vision solutions is constrained by the mismatch between the training and test domains. This mismatch might arise because of factors such as change in pose, lighting conditions, quality of imaging devices, intra-class variations in-herent in object categories etc. In this work, we present a dictionary learning based approach to tackle the problem of domain mismatch. In our approach, we jointly learn dictionaries for the source and the target domains. The dictionaries are partially shared, i.e. some elements are common across both the dictionaries. These shared elements can rep-resent the information which is common across both the domains. The dictionaries also have some elements to represent the domain specific information. Using these dictionaries, we separate the domain specific information and the information which is common across the domains. We use the latter for training cross-domain classifiers i.e., we build classi-fiers that work well on a new target domain while using labeled examples only in the source domain. We conduct cross-domain object recognition experiments on popular benchmark datasets and show improvement in results over the existing state of art domain adaptation approaches.

The merit of high-frequency data in portfolio allocation

This paper addresses the open debate about the usefulness of high-frequency (HF) data in large-scale portfolio allocation. Daily covariances are estimated based on HF data of the S&P 500 universe employing a blocked realized kernel estimator. We propose forecasting covariance matrices using a multi-scale spectral decomposition where volatilities, correlation eigenvalues and eigenvectors evolve on different frequencies. In an extensive out-of-sample forecasting study, we show that the proposed approach yields less risky and more diversified portfolio allocations as prevailing methods employing daily data. These performance gains hold over longer horizons than previous studies have shown