Improved method for finding optimal formulae for bilinear maps in a finite field

In 2012, Barbulescu, Detrey, Estibals and Zimmermann proposed a new framework to exhaustively search for optimal formulae for evaluating bilinear maps, such as Strassen or Karatsuba formulae. The main contribution of this work is a new criterion to aggressively prune useless branches in the exhaustive search, thus leading to the computation of new optimal formulae, in particular for the short product modulo X 5 and the circulant product modulo (X 5 -- 1). Moreover , we are able to prove that there is essentially only one optimal decomposition of the product of 3 x 2 by 2 x 3 matrices up to the action of some group of automorphisms

Topological Optimization of the Evaluation of Finite Element Matrices

We present a topological framework for finding low-flop algorithms for evaluating element stiffness matrices associated with multilinear forms for finite element methods posed over straight-sided affine domains. This framework relies on phrasing the computation on each element as the contraction of each collection of reference element tensors with an element-specific geometric tensor. We then present a new concept of complexity-reducing relations that serve as distance relations between these reference element tensors. This notion sets up a graph-theoretic context in which we may find an optimized algorithm by computing a minimum spanning tree. We present experimental results for some common multilinear forms showing significant reductions in operation count and also discuss some efficient algorithms for building the graph we use for the optimization

Learning to Rank Question Answer Pairs with Holographic Dual LSTM Architecture

We describe a new deep learning architecture for learning to rank question answer pairs. Our approach extends the long short-term memory (LSTM) network with holographic composition to model the relationship between question and answer representations. As opposed to the neural tensor layer that has been adopted recently, the holographic composition provides the benefits of scalable and rich representational learning approach without incurring huge parameter costs. Overall, we present Holographic Dual LSTM (HD-LSTM), a unified architecture for both deep sentence modeling and semantic matching. Essentially, our model is trained end-to-end whereby the parameters of the LSTM are optimized in a way that best explains the correlation between question and answer representations. In addition, our proposed deep learning architecture requires no extensive feature engineering. Via extensive experiments, we show that HD-LSTM outperforms many other neural architectures on two popular benchmark QA datasets. Empirical studies confirm the effectiveness of holographic composition over the neural tensor layer.Comment: SIGIR 2017 Full Pape

Sum-factorization techniques in Isogeometric Analysis

The fast assembling of stiffness and mass matrices is a key issue in isogeometric analysis, particularly if the spline degree is increased. We present two algorithms based on the idea of sum factorization, one for matrix assembling and one for matrix-free methods, and study the behavior of their computational complexity in terms of the spline order $p$. Opposed to the standard approach, these algorithms do not apply the idea element-wise, but globally or on macro-elements. If this approach is applied to Gauss quadrature, the computational complexity grows as $p^{d+2}$ instead of $p^{2d+1}$ as previously achieved.Comment: 34 pages, 8 figure

OPML: A One-Pass Closed-Form Solution for Online Metric Learning

To achieve a low computational cost when performing online metric learning for large-scale data, we present a one-pass closed-form solution namely OPML in this paper. Typically, the proposed OPML first adopts a one-pass triplet construction strategy, which aims to use only a very small number of triplets to approximate the representation ability of whole original triplets obtained by batch-manner methods. Then, OPML employs a closed-form solution to update the metric for new coming samples, which leads to a low space (i.e., $O(d)$) and time (i.e., $O(d^2)$) complexity, where $d$ is the feature dimensionality. In addition, an extension of OPML (namely COPML) is further proposed to enhance the robustness when in real case the first several samples come from the same class (i.e., cold start problem). In the experiments, we have systematically evaluated our methods (OPML and COPML) on three typical tasks, including UCI data classification, face verification, and abnormal event detection in videos, which aims to fully evaluate the proposed methods on different sample number, different feature dimensionalities and different feature extraction ways (i.e., hand-crafted and deeply-learned). The results show that OPML and COPML can obtain the promising performance with a very low computational cost. Also, the effectiveness of COPML under the cold start setting is experimentally verified.Comment: 12 page

Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains

We explore the connection between fractional order partial differential equations in two or more spatial dimensions with boundary integral operators to develop techniques that enable one to efficiently tackle the integral fractional Laplacian. In particular, we develop techniques for the treatment of the dense stiffness matrix including the computation of the entries, the efficient assembly and storage of a sparse approximation and the efficient solution of the resulting equations. The main idea consists of generalising proven techniques for the treatment of boundary integral equations to general fractional orders. Importantly, the approximation does not make any strong assumptions on the shape of the underlying domain and does not rely on any special structure of the matrix that could be exploited by fast transforms. We demonstrate the flexibility and performance of this approach in a couple of two-dimensional numerical examples