Geometric Linearization of Ordinary Differential Equations

The linearizability of differential equations was first considered by Lie for scalar second order semi-linear ordinary differential equations. Since then there has been considerable work done on the algebraic classification of linearizable equations and even on systems of equations. However, little has been done in the way of providing explicit criteria to determine their linearizability. Using the connection between isometries and symmetries of the system of geodesic equations criteria were established for second order quadratically and cubically semi-linear equations and for systems of equations. The connection was proved for maximally symmetric spaces and a conjecture was put forward for other cases. Here the criteria are briefly reviewed and the conjecture is proved.Comment: This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Tensor Networks for Medical Image Classification

With the increasing adoption of machine learning tools like neural networks across several domains, interesting connections and comparisons to concepts from other domains are coming to light. In this work, we focus on the class of Tensor Networks, which has been a work horse for physicists in the last two decades to analyse quantum many-body systems. Building on the recent interest in tensor networks for machine learning, we extend the Matrix Product State tensor networks (which can be interpreted as linear classifiers operating in exponentially high dimensional spaces) to be useful in medical image analysis tasks. We focus on classification problems as a first step where we motivate the use of tensor networks and propose adaptions for 2D images using classical image domain concepts such as local orderlessness of images. With the proposed locally orderless tensor network model (LoTeNet), we show that tensor networks are capable of attaining performance that is comparable to state-of-the-art deep learning methods. We evaluate the model on two publicly available medical imaging datasets and show performance improvements with fewer model hyperparameters and lesser computational resources compared to relevant baseline methods.Comment: Accepted for publication at International Conference on Medical Imaging with Deep Learning (MIDL), 2020. Reviews on Openreview here: https://openreview.net/forum?id=jjk6bxk07

Active classification with comparison queries

We study an extension of active learning in which the learning algorithm may ask the annotator to compare the distances of two examples from the boundary of their label-class. For example, in a recommendation system application (say for restaurants), the annotator may be asked whether she liked or disliked a specific restaurant (a label query); or which one of two restaurants did she like more (a comparison query). We focus on the class of half spaces, and show that under natural assumptions, such as large margin or bounded bit-description of the input examples, it is possible to reveal all the labels of a sample of size $n$ using approximately $O(\log n)$ queries. This implies an exponential improvement over classical active learning, where only label queries are allowed. We complement these results by showing that if any of these assumptions is removed then, in the worst case, $\Omega(n)$ queries are required. Our results follow from a new general framework of active learning with additional queries. We identify a combinatorial dimension, called the \emph{inference dimension}, that captures the query complexity when each additional query is determined by $O(1)$ examples (such as comparison queries, each of which is determined by the two compared examples). Our results for half spaces follow by bounding the inference dimension in the cases discussed above.Comment: 23 pages (not including references), 1 figure. The new version contains a minor fix in the proof of Lemma 4.