Exploiting the ever-growing corpus of scientific content calls for new ways and means to effectively organize, search, and retrieve scientific formulae. We propose a new data-driven framework for retrieving similar scientific formulae via learned formula representations based on tree embeddings. FORTE (for FOrmula Representation learning via Tree Embeddings) leverages operator tree representations of symbolic scientific formulae (such as math equations) to explicitly capture their inherent structural and semantic properties. FORTE employs i) a tree encoder that encodes the formula’s operator tree into an embedding vector and ii) a tree decoder that directly generates a formula’s operator tree from the embedding vector. We also develop a novel tree beam search algorithm that improves the quality of the decoded operator trees. We demonstrate that FORTE (sometimes significantly) outperforms various baseline methods on formula reconstruction and retrieval using a real-world dataset comprising 770k scientific formulae collected on-line.

The 2016 United States presidential election has been characterized as a period of extreme divisiveness that was exacerbated on social media by the influence of fake news, trolls, and social bots. However, the extent to which the public became more polarized in response to these influences over the course of the election is not well understood. In this paper we propose IdeoTrace, a framework for (i) jointly estimating the ideology of social media users and news websites and (ii) tracing changes in user ideology over time. We apply this framework to the last two months of the election period for a group of 47508 Twitter users and demonstrate that both liberal and conservative users became more polarized over time.

Kernel matrices that encode the distance (or similarity) between data points are widely used throughout the computational sciences for classification, clustering, and dimensionality reduction. For large datasets, the cost of computing and factorizing such matrices becomes intractable. Thus instead of operating on the entire matrix, approximate methods such as the Nyström method and the incomplete Cholesky decomposition (ICD) generate a low rank matrix factorization using only a subset of the matrix columns (or rows). Here, we present an adaptive column sampling strategy for the Nyström method that we dub Accelerated Sequential Incoherence Selection (oASIS). This sampling strategy reveals a missing link between Nyström methods and ICD: we demonstrate that ICD is actually a special case of the Nyström method with the oASIS adaptive sampling rule. Numerical experiments and theoretical results suggest that oASIS achieves performance comparable to state-of-the-art greedy Nyström methods but with shorter runtimes and less memory consumption.

We propose SPARFA-Trace, a new machine learning-based framework for time-varying learning and content analytics for educational applications. We develop a novel message passing-based, blind, approximate Kalman filter for sparse factor analysis (SPARFA) that jointly traces learner concept knowledge over time, analyzes learner concept knowledge state transitions (induced by interacting with learning resources, such as textbook sections, lecture videos, etc., or the forgetting effect), and estimates the content organization and difficulty of the questions in assessments. These quantities are estimated solely from binary-valued (correct/incorrect) graded learner response data and the specific actions each learner performs (e.g., answering a question or studying a learning resource) at each time instant. Experimental results on two online course datasets demonstrate that SPARFA-Trace is capable of tracing each learner's concept knowledge evolution over time, analyzing the quality and content organization of learning resources, and estimating the question--concept associations and the question difficulties. Moreover, we show that SPARFA-Trace achieves comparable or better performance in predicting unobserved learner responses compared to existing collaborative filtering and knowledge tracing methods.

Compressive sensing (CS) is a new signal acquisition technique for sparse and compressible signals. Rather than uniformly sampling the signal, CS computes inner products with randomized basis functions; the signal is then recovered by a convex optimization. Random CS measurements are universal in the sense that the same acquisition system is sufficient for signals sparse in any representation. This paper examines the quantization of strictly sparse, power-limited signals and concludes that CS with scalar quantization uses its allocated rate inefficiently. The results complement related work on the quantization of CS measurements of compressible signals.

A new algorithm for compressing three-dimensional triangular mesh data was used for representing surfaces. The estimation-quantization (EQ) algorithm was applied. EQ was originally designed for still image compression to the normal mesh wavelet coefficient. The EQ algorithm models the wavelet coefficients as a Gaussian random field with slowly varying standard deviation. By designing the quantizers in a rate-distortion optimal fashion, the previously proposed zerotree normal mesh compression algorithm has improved distortion by 0.5 to 1 dB.

Wavelet-based image coders optimally represent smooth regions and isolated point singularities. However, wavelet coders are less adept at representing perceptually important edge singularities, and coding performance suffers significantly as a result. We propose a novel two-stage image coder framework based on modeling images as edge cartoons + textures. In stage 1, we infer and efficiently code the edge information from the image using a multiscale wedgelet decomposition. In stage 2, we code the residual, "edgeless" texture image using a standard wavelet coder. Our preliminary coder improves significantly over standard wavelet coding techniques in terms of visual quality.

Summary form only given. Image compression relies on efficient representations of images, and within smooth image regions, the wavelet transform provides such a representation. However, near edges, wavelet coefficients decay slowly and are expensive to code. We focus on improving the transform by incorporating adaptivity. Construction of nonlinear filter banks has been discussed, but the question of how to utilize the nonlinearities remained. We answer this question by describing our transform via lifting. Lifting provides a spatial domain framework for the wavelet transform. In the lifting formalism, wavelet coefficients are seen as prediction residuals from a linear prediction operation. Wavelet coefficients are large near edges because the linear predictors are built to interpolate low order polynomials. Our goal is to avoid this problem by adapting the predictor based on local image properties. In smooth regions of the image, we use high order polynomial predictors. We adaptively reduce the prediction order to avoid attempting to predict values across discontinuities.