On the Matrix Form of the Quaternion Fourier Transform and Quaternion Convolution

We study matrix forms of quaternionic versions of the Fourier Transform and Convolution operations. Quaternions offer a powerful representation unit, however they are related to difficulties in their use that stem foremost from non-commutativity of quaternion multiplication, and due to that $\mu^2 = -1$ posseses infinite solutions in the quaternion domain. Handling of quaternionic matrices is consequently complicated in several aspects (definition of eigenstructure, determinant, etc.). Our research findings clarify the relation of the Quaternion Fourier Transform matrix to the standard (complex) Discrete Fourier Transform matrix, and the extend on which well-known complex-domain theorems extend to quaternions. We focus especially on the relation of Quaternion Fourier Transform matrices to Quaternion Circulant matrices (representing quaternionic convolution), and the eigenstructure of the latter. A proof-of-concept application that makes direct use of our theoretical results is presented, where we produce a method to bound the spectral norm of a Quaternionic Convolution.Comment: 22 pages, 2 figure

The Economic Resource Receipt of New Mothers

U.S. federal policies do not provide a universal social safety net of economic support for women during pregnancy or the immediate postpartum period but assume that employment and/or marriage will protect families from poverty. Yet even mothers with considerable human and marital capital may experience disruptions in employment, earnings, and family socioeconomic status postbirth. We use the National Survey of Families and Households to examine the economic resources that mothers with children ages 2 and younger receive postbirth, including employment, spouses, extended family and social network support, and public assistance. Results show that many new mothers receive resources postbirth. Marriage or postbirth employment does not protect new mothers and their families from poverty, but education, race, and the receipt of economic supports from social networks do

Matrix Factorization in Tropical and Mixed Tropical-Linear Algebras

Matrix Factorization (MF) has found numerous applications in Machine Learning and Data Mining, including collaborative filtering recommendation systems, dimensionality reduction, data visualization, and community detection. Motivated by the recent successes of tropical algebra and geometry in machine learning, we investigate two problems involving matrix factorization over the tropical algebra. For the first problem, Tropical Matrix Factorization (TMF), which has been studied already in the literature, we propose an improved algorithm that avoids many of the local optima. The second formulation considers the approximate decomposition of a given matrix into the product of three matrices where a usual matrix product is followed by a tropical product. This formulation has a very interesting interpretation in terms of the learning of the utility functions of multiple users. We also present numerical results illustrating the effectiveness of the proposed algorithms, as well as an application to recommendation systems with promising results

Best Practices for a Handwritten Text Recognition System

Handwritten text recognition has been developed rapidly in the recent years, following the rise of deep learning and its applications. Though deep learning methods provide notable boost in performance concerning text recognition, non-trivial deviation in performance can be detected even when small pre-processing or architectural/optimization elements are changed. This work follows a ``best practice'' rationale; highlight simple yet effective empirical practices that can further help training and provide well-performing handwritten text recognition systems. Specifically, we considered three basic aspects of a deep HTR system and we proposed simple yet effective solutions: 1) retain the aspect ratio of the images in the preprocessing step, 2) use max-pooling for converting the 3D feature map of CNN output into a sequence of features and 3) assist the training procedure via an additional CTC loss which acts as a shortcut on the max-pooled sequential features. Using these proposed simple modifications, one can attain close to state-of-the-art results, while considering a basic convolutional-recurrent (CNN+LSTM) architecture, for both IAM and RIMES datasets. Code is available at https://github.com/georgeretsi/HTR-best-practices/