Perron-based algorithms for the multilinear pagerank

We consider the multilinear pagerank problem studied in [Gleich, Lim and Yu, Multilinear Pagerank, 2015], which is a system of quadratic equations with stochasticity and nonnegativity constraints. We use the theory of quadratic vector equations to prove several properties of its solutions and suggest new numerical algorithms. In particular, we prove the existence of a certain minimal solution, which does not always coincide with the stochastic one that is required by the problem. We use an interpretation of the solution as a Perron eigenvector to devise new fixed-point algorithms for its computation, and pair them with a homotopy continuation strategy. The resulting numerical method is more reliable than the existing alternatives, being able to solve a larger number of problems

Interpretable Image Recognition with Hierarchical Prototypes

Vision models are interpretable when they classify objects on the basis of features that a person can directly understand. Recently, methods relying on visual feature prototypes have been developed for this purpose. However, in contrast to how humans categorize objects, these approaches have not yet made use of any taxonomical organization of class labels. With such an approach, for instance, we may see why a chimpanzee is classified as a chimpanzee, but not why it was considered to be a primate or even an animal. In this work we introduce a model that uses hierarchically organized prototypes to classify objects at every level in a predefined taxonomy. Hence, we may find distinct explanations for the prediction an image receives at each level of the taxonomy. The hierarchical prototypes enable the model to perform another important task: interpretably classifying images from previously unseen classes at the level of the taxonomy to which they correctly relate, e.g. classifying a hand gun as a weapon, when the only weapons in the training data are rifles. With a subset of ImageNet, we test our model against its counterpart black-box model on two tasks: 1) classification of data from familiar classes, and 2) classification of data from previously unseen classes at the appropriate level in the taxonomy. We find that our model performs approximately as well as its counterpart black-box model while allowing for each classification to be interpreted.Comment: Published as a full paper at HCOMP 201

Generalized Induced Norms

Let ||.|| be a norm on the algebra M_n of all n-by-n matrices over the complex field C. An interesting problem in matrix theory is that "are there two norms ||.||_1 and ||.||_2 on C^n such that ||A||=max{||Ax||_2: ||x||_1=1} for all A in M_n. We will investigate this problem and its various aspects and will discuss under which conditions ||.||_1=||.||_2.Comment: 8 page

GSplit LBI: Taming the Procedural Bias in Neuroimaging for Disease Prediction

In voxel-based neuroimage analysis, lesion features have been the main focus in disease prediction due to their interpretability with respect to the related diseases. However, we observe that there exists another type of features introduced during the preprocessing steps and we call them "\textbf{Procedural Bias}". Besides, such bias can be leveraged to improve classification accuracy. Nevertheless, most existing models suffer from either under-fit without considering procedural bias or poor interpretability without differentiating such bias from lesion ones. In this paper, a novel dual-task algorithm namely \emph{GSplit LBI} is proposed to resolve this problem. By introducing an augmented variable enforced to be structural sparsity with a variable splitting term, the estimators for prediction and selecting lesion features can be optimized separately and mutually monitored by each other following an iterative scheme. Empirical experiments have been evaluated on the Alzheimer's Disease Neuroimaging Initiative\thinspace(ADNI) database. The advantage of proposed model is verified by improved stability of selected lesion features and better classification results.Comment: Conditional Accepted by Miccai,201

A Compact Linear Programming Relaxation for Binary Sub-modular MRF

We propose a novel compact linear programming (LP) relaxation for binary sub-modular MRF in the context of object segmentation. Our model is obtained by linearizing an $l_1^+$-norm derived from the quadratic programming (QP) form of the MRF energy. The resultant LP model contains significantly fewer variables and constraints compared to the conventional LP relaxation of the MRF energy. In addition, unlike QP which can produce ambiguous labels, our model can be viewed as a quasi-total-variation minimization problem, and it can therefore preserve the discontinuities in the labels. We further establish a relaxation bound between our LP model and the conventional LP model. In the experiments, we demonstrate our method for the task of interactive object segmentation. Our LP model outperforms QP when converting the continuous labels to binary labels using different threshold values on the entire Oxford interactive segmentation dataset. The computational complexity of our LP is of the same order as that of the QP, and it is significantly lower than the conventional LP relaxation

Estimation of System Reliability Using a Semiparametric Model

An important problem in reliability engineering is to predict the failure rate, that is, the frequency with which an engineered system or component fails. This paper presents a new method of estimating failure rate using a semiparametric model with Gaussian process smoothing. The method is able to provide accurate estimation based on historical data and it does not make strong a priori assumptions of failure rate pattern (e.g., constant or monotonic). Our experiments of applying this method in power system failure data compared with other models show its efficacy and accuracy. This method can be used in estimating reliability for many other systems, such as software systems or components

Functional Multi-Layer Perceptron: a Nonlinear Tool for Functional Data Analysis

In this paper, we study a natural extension of Multi-Layer Perceptrons (MLP) to functional inputs. We show that fundamental results for classical MLP can be extended to functional MLP. We obtain universal approximation results that show the expressive power of functional MLP is comparable to that of numerical MLP. We obtain consistency results which imply that the estimation of optimal parameters for functional MLP is statistically well defined. We finally show on simulated and real world data that the proposed model performs in a very satisfactory way.Comment: http://www.sciencedirect.com/science/journal/0893608

Some extremal functions in Fourier analysis, III

We obtain the best approximation in $L^1(\R)$, by entire functions of exponential type, for a class of even functions that includes $e^{-\lambda|x|}$, where $\lambda >0$, $\log |x|$ and $|x|^{\alpha}$, where $-1 < \alpha < 1$. We also give periodic versions of these results where the approximating functions are trigonometric polynomials of bounded degree.Comment: 26 pages. Submitte