Even though data is abundant, it is often subjected to some form of censoring or truncation which inherently creates biases. Removing such biases and performing parameter estimation is a classical challenge in Statistics. In this paper, we focus on the problem of estimating the means of a mixture of two balanced d-dimensional Gaussians when the samples are prone to truncation. A recent theoretical study on the performance of the Expectation-Maximization (EM) algorithm for the aforementioned problem showed EM almost surely converges for d=1 and exhibits local convergence for d>1 to the true means. Nevertheless, the EM algorithm for the case of truncated mixture of two Gaussians is not easy to implement as it requires solving a set of nonlinear equations at every iteration which makes the algorithm impractical. In this work, we propose a gradient based variant of the EM algorithm that has global convergence guarantees when d=1 and local convergence for d>1 to the true means. Moreover, the update rule at every iteration is easy to compute which makes the proposed method practical. We also provide numerous experiments to obtain more insights into the effect of truncation on the convergence to the true parameters in high dimensions.

We introduce algebraic machine reasoning, a new reasoning framework that is well-suited for abstract reasoning. Effectively, algebraic machine reasoning reduces the difficult process of novel problem-solving to routine algebraic computation. The fundamental algebraic objects of interest are the ideals of some suitably initialized polynomial ring. We shall explain how solving Raven's Progressive Matrices (RPMs) can be realized as computational problems in algebra, which combine various well-known algebraic subroutines that include: Computing the Gröbner basis of an ideal, checking for ideal containment, etc. Crucially, the additional algebraic structure satisfied by ideals allows for more operations on ideals beyond set-theoretic operations. Our algebraic machine reasoning framework is not only able to select the correct answer from a given answer set, but also able to generate the correct answer with only the question matrix given. Experiments on the I-RAVEN dataset yield an overall 93.2% accuracy, which significantly out-performs the current state-of-the-art accuracy of77.0% and exceeds human performance at 84.4% accuracy.

We propose a new approach, Knowledge Distillation using Optimal Transport (KNOT), to distill the natural language semantic knowledge from multiple teacher networks to a student network. KNOT aims to train a (global) student model by learning to minimize the optimal transport cost of its assigned probability distribution over the labels to the weighted sum of probabilities predicted by the (local) teacher models, under the constraints that the student model does not have access to teacher models’ parameters or training data. To evaluate the quality of knowledge transfer, we introduce a new metric, Semantic Distance (SD), that measures semantic closeness between the predicted and ground truth label distributions. The proposed method shows improvements in the global model’s SD performance over the baseline across three NLP tasks while performing on par with Entropy-based distillation on standard accuracy and F1 metrics. The implementation pertaining to this work is publicly available at https://github.com/declare-lab/KNOT.

We showcase a topological mapping framework for a challenging indoor warehouse setting. At the most abstract level, the warehouse is represented as a Topological Graph where the nodes of the graph represent a particular warehouse topological construct (e.g. rackspace, corridor) and the edges denote the existence of a path between two neighbouring nodes or topologies. At the intermediate level, the map is represented as a Manhattan Graph where the nodes and edges are characterized by Manhattan properties and as a Pose Graph at the lower-most level of detail. The topological constructs are learned via a Deep Convolutional Network while the relational properties between topological instances are learnt via a Siamese-style Neural Network. In the paper, we show that maintaining abstractions such as Topological Graph and Manhattan Graph help in recovering an accurate Pose Graph starting from a highly erroneous and unoptimized Pose Graph. We show how this is achieved by embedding topological and Manhattan relations as well as Manhattan Graph aided loop closure relations as constraints in the backend Pose Graph optimization framework. The recovery of near ground-truth Pose Graph on real-world indoor warehouse scenes vindicate the efficacy of the proposed framework.