Sparse representation based classification (SRC) has gained great success in image recognition. Motivated by the fact that kernel trick can capture the nonlinear similarity of features, which may help improve the separability and margin between nearby data points, we propose Euler SRC for image classification, which is essentially the SRC with Euler sparse representation. To be specific, it first maps the images into the complex space by Euler representation, which has a negligible effect for outliers and illumination, and then performs complex SRC with Euler representation. The major advantage of our method is that Euler representation is explicit with no increase of the image space dimensionality, thereby enabling this technique to be easily deployed in real applications. To solve Euler SRC, we present an efficient algorithm, which is fast and has good convergence. Extensive experimental results illustrate that Euler SRC outperforms traditional SRC and achieves better performance for image classification.
We study first-order optimization methods obtained by discretizing ordinary differential equations (ODEs) corresponding to Nesterov's accelerated gradient methods (NAGs) and Polyak's heavy-ball method. We consider three discretization schemes: an explicit Euler scheme, an implicit Euler scheme, and a symplectic scheme. We show that the optimization algorithm generated by applying the symplectic scheme to a high-resolution ODE proposed by Shi et al.  achieves an accelerated rate for minimizing smooth strongly convex functions. On the other hand, the resulting algorithm either fails to achieve acceleration or is impractical when the scheme is implicit, the ODE is low-resolution, or the scheme is explicit.
Project Euler is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve. Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems. The motivation for starting Project Euler, and its continuation, is to provide a platform for the inquiring mind to delve into unfamiliar areas and learn new concepts in a fun and recreational context. The intended audience include students for whom the basic curriculum is not feeding their hunger to learn, adults whose background was not primarily mathematics but had an interest in things mathematical, and professionals who want to keep their problem solving and mathematics on the cutting edge. The problems range in difficulty and for many the experience is inductive chain learning.
In the 20 20 grid below, four numbers along a diagonal line have been marked in red. The product of these numbers is 26 63 78 14 1,788,696. What is the greatest product of four adjacent numbers in the same direction (up, down, left, right, or diagonally) in the 20 by 20 grid? The solution applies straightforward vector arithmetic. The product of all verticals is an array of the product of rows 1 to 4, rows 2 to 5 and so on.