Estimating the difficulty level of math word problems is an important task for many educational applications. Identification of relevant and irrelevant sentences in math word problems is an important step for calculating the difficulty levels of such problems. This paper addresses a novel application of text categorization to identify two types of sentences in mathematical word problems, namely relevant and irrelevant sentences. A novel joint probabilistic classification model is proposed to estimate the joint probability of classification decisions for all sentences of a math word problem by utilizing the correlation among all sentences along with the correlation between the question sentence and other sentences, and sentence text. The proposed model is compared with i) a SVM classifier which makes independent classification decisions for individual sentences by only using the sentence text and ii) a novel SVM classifier that considers the correlation between the question sentence and other sentences along with the sentence text. An extensive set of experiments demonstrates the effectiveness of the joint probabilistic classification model for identifying relevant and irrelevant sentences as well as the novel SVM classifier that utilizes the correlation between the question sentence and other sentences. Furthermore, empirical results and analysis show that i) it is highly beneficial not to remove stopwords and ii) utilizing part of speech tagging does not make a significant improvement although it has been shown to be effective for the related task of math word problem type classification.

Images have become one of the most popular types of media through which users convey their emotions within online social networks. Although vast amount of research is devoted to sentiment analysis of textual data, there has been very limited work that focuses on analyzing sentiment of image data. In this work, we propose a novel visual sentiment prediction framework that performs image understanding with Deep Convolutional Neural Networks (CNN). Specifically, the proposed sentiment prediction framework performs transfer learning from a CNN with millions of parameters, which is pre-trained on large-scale data for object recognition. Experiments conducted on two real-world datasets from Twitter and Tumblr demonstrate the effectiveness of the proposed visual sentiment analysis framework.

We analyze directed, unweighted graphs obtained from $x_i\in \mathbb{R}^d$ by connecting vertex $i$ to $j$ iff $|x_i - x_j| < \epsilon(x_i)$. Examples of such graphs include $k$-nearest neighbor graphs, where $\epsilon(x_i)$ varies from point to point, and, arguably, many real world graphs such as co-purchasing graphs. We ask whether we can recover the underlying Euclidean metric $\epsilon(x_i)$ and the associated density $p(x_i)$ given only the directed graph and $d$. We show that consistent recovery is possible up to isometric scaling when the vertex degree is at least $\omega(n^{2/(2+d)}\log(n)^{d/(d+2)})$. Our estimator is based on a careful characterization of a random walk over the directed graph and the associated continuum limit. As an algorithm, it resembles the PageRank centrality metric. We demonstrate empirically that the estimator performs well on simulated examples as well as on real-world co-purchasing graphs even with a small number of points and degree scaling as low as $\log(n)$.

This paper proposed a new regression model called $l_1$-regularized outlier isolation and regression (LOIRE) and a fast algorithm based on block coordinate descent to solve this model. Besides, assuming outliers are gross errors following a Bernoulli process, this paper also presented a Bernoulli estimate model which, in theory, should be very accurate and robust due to its complete elimination of affections caused by outliers. Though this Bernoulli estimate is hard to solve, it could be approximately achieved through a process which takes LOIRE as an important intermediate step. As a result, the approximate Bernoulli estimate is a good combination of Bernoulli estimate's accuracy and LOIRE regression's efficiency with several simulations conducted to strongly verify this point. Moreover, LOIRE can be further extended to realize robust rank factorization which is powerful in recovering low-rank component from massive corruptions. Extensive experimental results showed that the proposed method outperforms state-of-the-art methods like RPCA and GoDec in the aspect of computation speed with a competitive performance.

Data association, the problem of reasoning over correspondence between targets and measurements, is a fundamental problem in tracking. This paper presents a graphical model formulation of data association and applies an approximate inference method, belief propagation (BP), to obtain estimates of marginal association probabilities. We prove that BP is guaranteed to converge, and bound the number of iterations necessary. Experiments reveal a favourable comparison to prior methods in terms of accuracy and computational complexity.

Model-based methods and deep neural networks have both been tremendously successful paradigms in machine learning. In model-based methods, problem domain knowledge can be built into the constraints of the model, typically at the expense of difficulties during inference. In contrast, deterministic deep neural networks are constructed in such a way that inference is straightforward, but their architectures are generic and it is unclear how to incorporate knowledge. This work aims to obtain the advantages of both approaches. To do so, we start with a model-based approach and an associated inference algorithm, and \emph{unfold} the inference iterations as layers in a deep network. Rather than optimizing the original model, we \emph{untie} the model parameters across layers, in order to create a more powerful network. The resulting architecture can be trained discriminatively to perform accurate inference within a fixed network size. We show how this framework allows us to interpret conventional networks as mean-field inference in Markov random fields, and to obtain new architectures by instead using belief propagation as the inference algorithm. We then show its application to a non-negative matrix factorization model that incorporates the problem-domain knowledge that sound sources are additive. Deep unfolding of this model yields a new kind of non-negative deep neural network, that can be trained using a multiplicative backpropagation-style update algorithm. We present speech enhancement experiments showing that our approach is competitive with conventional neural networks despite using far fewer parameters.

Binary classification is a common statistical learning problem in which a model is estimated on a set of covariates for some outcome indicating the membership of one of two classes. In the literature, there exists a distinction between hard and soft classification. In soft classification, the conditional class probability is modeled as a function of the covariates. In contrast, hard classification methods only target the optimal prediction boundary. While hard and soft classification methods have been studied extensively, not much work has been done to compare the actual tasks of hard and soft classification. In this paper we propose a spectrum of statistical learning problems which span the hard and soft classification tasks based on fitting multiple decision rules to the data. By doing so, we reveal a novel collection of learning tasks of increasing complexity. We study the problems using the framework of large-margin classifiers and a class of piecewise linear convex surrogates, for which we derive statistical properties and a corresponding sub-gradient descent algorithm. We conclude by applying our approach to simulation settings and a magnetic resonance imaging (MRI) dataset from the Alzheimer's Disease Neuroimaging Initiative (ADNI) study.

Correlation filters (CFs) are a class of classifiers that are attractive for object localization and tracking applications. Traditionally, CFs have been designed in the frequency domain using the discrete Fourier transform (DFT), where correlation is efficiently implemented. However, existing CF designs do not account for the fact that the multiplication of two DFTs in the frequency domain corresponds to a circular correlation in the time/spatial domain. Because this was previously unaccounted for, prior CF designs are not truly optimal, as their optimization criteria do not accurately quantify their optimization intention. In this paper, we introduce new zero-aliasing constraints that completely eliminate this aliasing problem by ensuring that the optimization criterion for a given CF corresponds to a linear correlation rather than a circular correlation. This means that previous CF designs can be significantly improved by this reformulation. We demonstrate the benefits of this new CF design approach with several important CFs. We present experimental results on diverse data sets and present solutions to the computational challenges associated with computing these CFs. Code for the CFs described in this paper and their respective zero-aliasing versions is available at http://vishnu.boddeti.net/projects/correlation-filters.html

Modeling dynamical systems with ordinary differential equations implies a mechanistic view of the process underlying the dynamics. However in many cases, this knowledge is not available. To overcome this issue, we introduce a general framework for nonparametric ODE models using penalized regression in Reproducing Kernel Hilbert Spaces (RKHS) based on operator-valued kernels. Moreover, we extend the scope of gradient matching approaches to nonparametric ODE. A smooth estimate of the solution ODE is built to provide an approximation of the derivative of the ODE solution which is in turn used to learn the nonparametric ODE model. This approach benefits from the flexibility of penalized regression in RKHS allowing for ridge or (structured) sparse regression as well. Very good results are shown on 3 different ODE systems.

Computational intensity and sequential nature of estimation techniques for Bayesian methods in statistics and machine learning, combined with their increasing applications for big data analytics, necessitate both the identification of potential opportunities to parallelize techniques such as MCMC sampling, and the development of general strategies for mapping such parallel algorithms to modern CPUs in order to elicit the performance up the compute-based and/or memory-based hardware limits. Two opportunities for Single-Instruction Multiple-Data (SIMD) parallelization of MCMC sampling for probabilistic graphical models are presented. In exchangeable models with many observations such as Bayesian Generalized Linear Models, child-node contributions to the conditional posterior of each node can be calculated concurrently. In undirected graphs with discrete nodes, concurrent sampling of conditionally-independent nodes can be transformed into a SIMD form. High-performance libraries with multi-threading and vectorization capabilities can be readily applied to such SIMD opportunities to gain decent speedup, while a series of high-level source-code and runtime modifications provide further performance boost by reducing parallelization overhead and increasing data locality for NUMA architectures. For big-data Bayesian GLM graphs, the end-result is a routine for evaluating the conditional posterior and its gradient vector that is 5 times faster than a naive implementation using (built-in) multi-threaded Intel MKL BLAS, and reaches within the striking distance of the memory-bandwidth-induced hardware limit. The proposed optimization strategies improve the scaling of performance with number of cores and width of vector units (applicable to many-core SIMD processors such as Intel Xeon Phi and GPUs), resulting in cost-effectiveness, energy efficiency, and higher speed on multi-core x86 processors.