Improved Ingham-type result on Rd\mathbb R^d and on connected, simply connected nilpotent Lie Groups

In \cite{BRS} we have characterized the existance of a non zero function vanishing on an open set in terms of the decay of it's Fourier transform on the $d$-dimensional Euclidean space, the $d$-dimensional torus and on connected, simply connected two step nilpotent Lie groups. In this paper we improved these results on $\mathbb R^d$ and prove analogus results on connected, simply connected nilpotent Lie groups

On cluster systems of tensor product systems of Hilbert spaces

It is known that the spatial product of two product systems is intrinsic. Here we extend this result by analyzing subsystems of the tensor product of product systems. A relation with cluster systems is established. In a special case, we show that the amalgamated product of product systems through strictly contractive units is independent of the choices of the units. The amalgamated product in this case is isomorphic to the tensor product of the spatial product of the two and the type I product system of index one.Comment: 9 page

Doc2Im: document to image conversion through self-attentive embedding

Text classification is a fundamental task in NLP applications. Latest research in this field has largely been divided into two major sub-fields. Learning representations is one sub-field and learning deeper models, both sequential and convolutional, which again connects back to the representation is the other side. We posit the idea that the stronger the representation is, the simpler classifier models are needed to achieve higher performance. In this paper we propose a completely novel direction to text classification research, wherein we convert text to a representation very similar to images, such that any deep network able to handle images is equally able to handle text. We take a deeper look at the representation of documents as an image and subsequently utilize very simple convolution based models taken as is from computer vision domain. This image can be cropped, re-scaled, re-sampled and augmented just like any other image to work with most of the state-of-the-art large convolution based models which have been designed to handle large image datasets. We show impressive results with some of the latest benchmarks in the related fields. We perform transfer learning experiments, both from text to text domain and also from image to text domain. We believe this is a paradigm shift from the way document understanding and text classification has been traditionally done, and will drive numerous novel research ideas in the community

Solve-Select-Scale: A Three Step Process For Sparse Signal Estimation

In the theory of compressed sensing (CS), the sparsity $\|x\|_0$ of the unknown signal $\mathbf{x} \in \mathcal{R}^n$ is of prime importance and the focus of reconstruction algorithms has mainly been either $\|x\|_0$ or its convex relaxation (via $\|x\|_1$). However, it is typically unknown in practice and has remained a challenge when nothing about the size of the support is known. As pointed recently, $\|x\|_0$ might not be the best metric to minimize directly, both due to its inherent complexity as well as its noise performance. Recently a novel stable measure of sparsity $s(\mathbf{x}) := \|\mathbf{x}\|_1^2/\|\mathbf{x}\|_2^2$ has been investigated by Lopes \cite{Lopes2012}, which is a sharp lower bound on $\|\mathbf{x}\|_0$. The estimation procedure for this measure uses only a small number of linear measurements, does not rely on any sparsity assumptions, and requires very little computation. The usage of the quantity $s(\mathbf{x})$ in sparse signal estimation problems has not received much importance yet. We develop the idea of incorporating $s(\mathbf{x})$ into the signal estimation framework. We also provide a three step algorithm to solve problems of the form $\mathbf{Ax=b}$ with no additional assumptions on the original signal $\mathbf{x}$

Hashing Image Patches for Zooming

In this paper we present a Bayesian image zooming/super-resolution algorithm based on a patch based representation. We work on a patch based model with overlap and employ a Locally Linear Embedding (LLE) based approach as our data fidelity term in the Bayesian inference. The image prior imposes continuity constraints across the overlapping patches. We apply an error back-projection technique, with an approximate cross bilateral filter. The problem of nearest neighbor search is handled by a variant of the locality sensitive hashing (LSH) scheme. The novelty of our work lies in the speed up achieved by the hashing scheme and the robustness and inherent modularity and parallel structure achieved by the LLE setup. The ill-posedness of the image reconstruction problem is handled by the introduction of regularization priors which encode the knowledge present in vast collections of natural images. We present comparative results for both run-time as well as visual image quality based measurements.Comment: 7 pages, 6 figure

Additive Non-negative Matrix Factorization for Missing Data

Non-negative matrix factorization (NMF) has previously been shown to be a useful decomposition for multivariate data. We interpret the factorization in a new way and use it to generate missing attributes from test data. We provide a joint optimization scheme for the missing attributes as well as the NMF factors. We prove the monotonic convergence of our algorithms. We present classification results for cases with missing attributes.Comment: General extension of the NMF framewor

Uncertainty Principles of Ingham and Paley-Wiener on Semisimple Lie Groups

Classical results due to Ingham and Paley-Wiener characterize the existence of nonzero functions supported on certain subsets of the real line in terms of the pointwise decay of the Fourier transforms. Viewing these results as uncertainty principles for Fourier transforms, we prove certain analogues of these results on connected, noncompact, semisimple Lie groups with finite center. We also use these results to show unique continuation property of solutions to the initial value problem for time-dependent Schr\"odinger equations on Riemmanian symmetric spaces of noncompact type

Measuring the Effect of Discourse Relations on Blog Summarization

The work presented in this paper attempts to evaluate and quantify the use of discourse relations in the context of blog summarization and compare their use to more traditional and factual texts. Specifically, we measured the usefulness of 6 discourse relations - namely comparison, contingency, illustration, attribution, topic-opinion, and attributive for the task of text summarization from blogs. We have evaluated the effect of each relation using the TAC 2008 opinion summarization dataset and compared them with the results with the DUC 2007 dataset. The results show that in both textual genres, contingency, comparison, and illustration relations provide a significant improvement on summarization content; while attribution, topic-opinion, and attributive relations do not provide a consistent and significant improvement. These results indicate that, at least for summarization, discourse relations are just as useful for informal and affective texts as for more traditional news articles.Comment: In Proceedings of the 6th International Joint Conference on Natural Language Processing (IJCNLP 2013), pages 1401-1409, October 2013, Nagoya, Japa

An Uncertainty Principle of Paley and Wiener on Euclidean Motion Group

A classical result due to Paley and Wiener characterizes the existence of a non-zero function in $L^2(\mathbb{R})$, supported on a half line, in terms of the decay of its Fourier transform. In this paper we prove an analogue of this result for compactly supported continuous functions on the Euclidean motion group $M(n)$. We also relate this result to a uniqueness property of solutions to the initial value problem for time-dependent Schr\"odinger equation on $M(n)$

On Manipulation in Prediction Markets When Participants Influence Outcomes Directly

Prediction markets are often used as mechanisms to aggregate information about a future event, for example, whether a candidate will win an election. The event is typically assumed to be exogenous. In reality, participants may influence the outcome, and therefore (1) running the prediction market could change the incentives of participants in the process that creates the outcome (for example, agents may want to change their vote in an election), and (2) simple results such as the myopic incentive compatibility of proper scoring rules no longer hold in the prediction market itself. We introduce a model of games of this kind, where agents first trade in a prediction market and then take an action that influences the market outcome. Our two-stage two-player model, despite its simplicity, captures two aspects of real-world prediction markets: (1) agents may directly influence the outcome, (2) some of the agents instrumental in deciding the outcome may not take part in the prediction market. We show that this game has two different types of perfect Bayesian equilibria, which we term LPP and HPP, depending on the values of the belief parameters: in the LPP domain, equilibrium prices reveal expected market outcomes conditional on the participants' private information, whereas HPP equilibria are collusive -- participants effectively coordinate in an uninformative and untruthful way